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Introduction to Economic Analysis

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Introduction to Economic Analysis
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McAfee, R. Preston
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economics, supply and demand, microeconomics, economic models, economic equations, markets, resource allocation, OGT+ isbn: 9781616100414
Economics, Microeconomics, Supply and Demand
Social Studies / Economics


This book presents introductory economics material using standard mathematical tools, including calculus. It is designed for a relatively sophisticated undergraduate who has not taken a basic university course in economics. The book can easily serve as an intermediate microeconomics text. The focus of this book is on the conceptual tools. Contents: 1) What is Economics? 2) Supply and Demand. 3) The US Economy. 4) Producer Theory. 5) Consumer Theory. 6) Market Imperfections. 7) Strategic Behavior.
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McAfee: Introduction to Economic Analysis, July 24, 2006 i Introduction to Economic Analysis by R. Preston McAfee J. Stanley Johnson Professor of Business, Economics & Management California Institute of Technology x y Initial Choice pY Compensated Choice qA* Tax p q DBefore Tax SBefore Tax qB* Tax Revenue Dead Weight Loss 0.1 0.2 0.3 0.4 -0.04 -0.02 0.02 S S Stable Equilibrium Unstable Equilibrium 1 2


McAfee: Introduction to Economic Analysis, July 24, 2006 ii Dedication to this edition : For Sophie. Perhaps by the time she goes to university, we’ll have won the war against the publishers. Disclaimer : This is the third draft. Please point out typo s, errors or poor exposition, preferably by email to Your assistance matters. In preparing this manuscript, I have receiv ed assistance from many people, including Michael Bernstein, Steve Bisset, Grant Chang-Chien, Lauren Feiler, Alex Fogel, Ben Golub, George Hines, Richard Jones, Jorg e Martnez, Joshua Moses, Dr. John Ryan, and Wei Eileen Xie. I am especially indebted to Anthony B. Williams for a careful, detailed reading of the manuscript yielding hundreds of improvements.


McAfee: Introduction to Economic Analysis, July 24, 2006 iii Introduction to Economic Analysis Version 2.0 by R. Preston McAfee J. Stanley Johnson Professor of Business, Economics & Management California Institute of Technology Begun: June 24, 2004 This Draft: July 24, 2006 This book presents introductory economic s (“principles”) material using standard mathematical tools, including calculus. It is designed for a relatively sophisticated undergraduate who has not taken a basic univ ersity course in economics. It also contains the standard intermediate microeco nomics material and some material that ought to be standard but is not. The book can easily serve as an intermediate microeconomics text. The focus of this book is on the conceptual tools and not on fluff. Most microeconomics texts are mostly fluff and the fluff market is exceedingly overserved by $100+ texts. In contrast, this b ook reflects the approach actually adopted by the majority of economists for understandin g economic activity. There are lots of models and equations and no pictures of economists. This work is licensed under the Creative Commons AttributionNonCommercial-ShareAlike License. To view a copy of this license, visit or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. Please email changes to


McAfee: Introduction to Economic Analysis,, July 24, 2006 ivTable of Contents 1 WHAT IS ECONOMICS? ..............................1-1 1.1.1 Normative and Positive Theories ..................1-2 1.1.2 Opportunity Cost ..........................................1-3 1.1.3 Economic Reasoning and Analysis ................1-5 2 SUPPLY AND DEMAND ...............................2-8 2.1 Supply and Demand .................................2-8 2.1.1 Demand and Consumer Surplus ...................2-8 2.1.2 Supply .........................................................2-13 2.2 The Market ...............................................2-18 2.2.1 Market Demand and Supply .......................2-18 2.2.2 Equilibrium .................................................2-20 2.2.3 Efficiency of Equilibrium ............................2-22 2.3 Changes in Supply and Demand ..........2-22 2.3.1 Changes in Demand ....................................2-22 2.3.2 Changes in Supply .......................................2-23 2.4 Elasticities ................................................2-27 2.4.1 Elasticity of Demand ...................................2-27 2.4.2 Elasticity of Supply .....................................2-30 2.5 Comparative Statics ...............................2-30 2.5.1 Supply and Demand Changes .....................2-30 2.6 Trade .........................................................2-32 2.6.1 Production Possibilities Frontier ................2-32 2.6.2 Comparative and Absolute Advantage ........2-36 2.6.3 Factors and Production ...............................2-38 2.6.4 International Trade .....................................2-39 3 THE US ECONOMY .....................................3-41 3.1.1 Basic Demographics ....................................3-41 3.1.2 Education ....................................................3-47 3.1.3 Households and Consumption ....................3-49 3.1.4 Production ..................................................3-56 3.1.5 Government ................................................3-65 3.1.6 Trade ...........................................................3-73 3.1.7 Fluctuations ................................................3-76 4 PRODUCER THEORY .................................4-79 4.1 The Competitive Firm ............................4-79 4.1.1 Types of Firms ............................................4-79 4.1.2 Production Functions ..................................4-81 4.1.3 Profit Maximization ....................................4-85 4.1.4 The Shadow Value .......................................4-91 4.1.5 Input Demand .............................................4-92 4.1.6 Myriad Costs ...............................................4-95 4.1.7 Dynamic Firm Behavior ..............................4-97 4.1.8 Economies of Scale and Scope ..................4-100 4.2 Perfect Competition Dynamics ..........4-104 4.2.1 Long-run Equilibrium ...............................4-104 4.2.2 Dynamics with Constant Costs ..................4-105 4.2.3 General Long-run Dynamics .....................4-109 4.3 Investment .............................................4-114 4.3.1 Present value .............................................4-114 4.3.2 Investment ................................................4-118 4.3.3 Investment Under Uncertainty .................4-120 4.3.4 Resource Extraction ..................................4-125 4.3.5 A Time to Harvest .....................................4-127 4.3.6 Collectibles ................................................4-130 4.3.7 Summer Wheat .........................................4-135 5 CONSUMER THEORY .............................5-139 5.1 Utility Maximization ...........................5-139 5.1.1 Budget or Feasible Set ..............................5-140 5.1.2 Isoquants .................................................5-143 5.1.3 Examples ..................................................5-148 5.1.4 Substitution Effects ..................................5-151 5.1.5 Income Effects ..........................................5-155 5.2 Additional Considerations .................5-158 5.2.1 Corner Solutions ......................................5-158 5.2.2 Labor Supply ............................................5-160 5.2.3 Compensating Differentials ......................5-164 5.2.4 Urban Real Estate Prices ..........................5-165 5.2.5 Dynamic Choice .......................................5-169 5.2.6 Risk ..........................................................5-174 5.2.7 Search ......................................................5-178 5.2.8 Edgeworth Box .........................................5-181 5.2.9 General Equilibrium .................................5-188 6 MARKET IMPERFECTIONS ...................6-195 6.1 Taxes .......................................................6-195 6.1.1 Effects of Taxes ........................................6-195 6.1.2 Incidence of Taxes ....................................6-199 6.1.3 Excess Burden of Taxation .......................6-200 6.2 Price Floors and Ceilings ...................6-202 6.2.1 Basic Theory .............................................6-203 6.2.2 Longand Short-run Effects .....................6-207 6.2.3 Political Motivations ................................6-209 6.2.4 Price Supports ..........................................6-210 6.2.5 Quantity Restrictions and Quotas ............6-211 6.3 Externalities .........................................6-213 6.3.1 Private and Social Value, Cost ..................6-214 6.3.2 Pigouvian Taxes .......................................6-217 6.3.3 Quotas ......................................................6-218 6.3.4 Tradable Permits and Auctions ................6-219 6.3.5 Coasian Bargaining ..................................6-220 6.3.6 Fishing and Extinction .............................6-221 6.4 Public Goods .........................................6-226 6.4.1 Examples ..................................................6-226 6.4.2 Free-Riders ..............................................6-227 6.4.3 Provision with Taxation ...........................6-229 6.4.4 Local Public Goods ...................................6-230 6.5 Monopoly ...............................................6-232 6.5.1 Sources of Monopoly ................................6-232 6.5.2 Basic Analysis ...........................................6-233 6.5.3 Effect of Taxes ..........................................6-236 6.5.4 Price Discrimination ................................6-237 6.5.5 Welfare Effects .........................................6-240 6.5.6 Two-Part Pricing ......................................6-240 6.5.7 Natural Monopoly ....................................6-241 6.5.8 Peak Load Pricing .....................................6-242 6.6 Information ..........................................6-245 6.6.1 Market for Lemons ...................................6-245 6.6.2 Myerson-Satterthwaite Theorem ..............6-246 6.6.3 Signaling ..................................................6-248 7 STRATEGIC BEHAVIOR .........................7-251 7.1 Games .....................................................7-251 7.1.1 Matrix Games ...........................................7-251 7.1.2 Nash Equilibrium .....................................7-255 7.1.3 Mixed Strategies .......................................7-257 7.1.4 Examples ..................................................7-262 7.1.5 Two Period Games ...................................7-265


McAfee: Introduction to Economic Analysis,, July 24, 2006 v7.1.6 Subgame Perfection ..................................7-266 7.1.7 Supergames ..............................................7-268 7.1.8 The Folk Theorem ....................................7-269 7.2 Cournot Oligopoly ................................7-270 7.2.1 Equilibrium ..............................................7-271 7.2.2 Industry Performance ...............................7-272 7.3 Search and Price Dispersion .............7-274 7.3.1 Simplest Theory ........................................7-275 7.3.2 Industry Performance ...............................7-277 7.4 Hotelling Model ....................................7-279 7.4.1 Types of Differentiation ............................7-279 7.4.2 The Standard Model .................................7-280 7.4.3 The Circle Model .......................................7-280 7.5 Agency Theory ......................................7-283 7.5.1 Simple Model ............................................7-284 7.5.2 Cost of Providing Incentives .....................7-286 7.5.3 Selection of Agent .....................................7-287 7.5.4 Multi-tasking ............................................7-288 7.5.5 Multi-tasking without Homogeneity .........7-292 7.6 Auctions ..................................................7-295 7.6.1 English Auction .........................................7-295 7.6.2 Sealed-bid Auction ....................................7-296 7.6.3 Dutch Auction ...........................................7-298 7.6.4 Vickrey Auction .........................................7-299 7.6.5 Winner’s Curse ..........................................7-301 7.6.6 Linkage .....................................................7-303 7.6.7 Auction Design ..........................................7-304 7.7 Antitrust .................................................7-306 7.7.1 Sherman Act .............................................7-306 7.7.2 Clayton Act ................................................7-308 7.7.3 Price-Fixing ..............................................7-309 7.7.4 Mergers .....................................................7-311 8 INDEX ..........................................................8-315 8.1 List of Figures .......................................8-315 8.2 Index .......................................................8-317


McAfee: Introduction to Economic Analysis,, July 24, 2006 1-11 What is Economics? Economics studies the allocation of scarce resources among people – examining what goods and services wind up in the hands of which people. Why scarce resources? Absent scarcity, there is no significant a llocation issue. All practical, and many impractical, means of allocating scarce re sources are studied by economists. Markets are an important means of allocating resources, so economists study markets. Markets include stock markets like the New York St ock Exchange, commodities markets like the Chicago Mercantile, but also farmer’s mark ets, auction markets like Christie’s or Sotheby’s (made famous in movies by people scratching their noses and inadvertently purchasing a Ming vase) or eBay, or more ephemeral markets, such as the market for music CDs in your neighborhood. In additi on, goods and services (which are scarce resources) are allocated by governments, using taxation as a means of acquiring the items. Governments may be controlled by a po litical process, and the study of allocation by the politics, which is kn own as political economy, is a significant branch of economics. Goods are allocated by certain means, like theft, deemed illegal by the government, and such allocation methods nevertheless fall within the domain of economic analysis; the market for marijuana remains vibrant despite interdiction by the governments of most nations. Other all ocation methods include gifts and charity, lotteries and gambling, and cooperative societie s and clubs, all of which are studied by economists. Some markets involve a physical marketplace. Traders on the New York Stock Exchange get together in a trading pit. Traders on eBay come together in an electronic marketplace. Other markets, which are more familiar to most of us, involve physical stores that may or may not be next door to each other, and customers who search among the stores, purchasing when the customer fi nds an appropriate item at an acceptable price. When we buy bananas, we don’t typi cally go to a banana market and purchase from one of a dozen or more banana selle rs, but instead go to a grocery store. Nevertheless, in buying bananas, the grocery stores compete in a market for our banana patronage, attempting to attract customer s to their stores and inducing them to purchase bananas. Price – exchange of goods and services for money – is an important allocation means, but price is hardly the only factor even in market exchanges. Other terms, such as convenience, credit terms, reliability, and trustworthiness are also valuable to the participants in a transaction. In some mark ets such as 36 inch Sony WEGA televisions, one ounce bags of Cheetos, or Ford Autolite spark plugs, the products offered by distinct sellers are identical, and for such produc ts, price is usually the primary factor considered by buyers, although delivery and other aspects of the transaction may still matter. For other products, like restau rant meals, camcorders by different manufacturers, or air travel on distinct airlines, the products differ to some degree, and thus the qualities of the product are factors in the decision to purchase. Nevertheless, different products may be considered to be in a single market if the products are reasonable substitutes, and we can consider a “quality-adjusted” price for these different goods.


McAfee: Introduction to Economic Analysis,, July 24, 2006 1-2 Economic analysis is used in many situations. When British Petroleum sets the price for its Alaskan crude oil, it uses an estimated demand model, both fo r gasoline consumers and also for the refineries to which BP sell s. The demand for oil by refineries is governed by a complex economic model used by the refineries and BP estimates the demand by refineries by estimating the econ omic model used by refineries. Economic analysis was used by experts in the antitrus t suit brought by the U.S. Department of Justice both to understand Microsoft’s in centive to foreclose (eliminate from the market) rival Netscape and consumer behavior in the face of alleged foreclosure. Stock market analysts use economic models to fore cast the profits of companies in order to predict the price of their stocks. When the government forecasts the budget deficit or considers a change in environmental regulation s, it uses a variety of economic models. This book presents the building blocks of the models in common use by an army of economists thousands of times per day. 1.1.1 Normative and Positive Theories Economic analysis is used for two main purposes. The first is a scientific understanding of how allocations of goods and services – scarce resources – are actually determined. This is a positive analysis, analogous to the study of electromagnetism or molecular biology, and involves only the attempt to understand the world around us. The development of this positive theory, howe ver, suggests other uses for economics. Economic analysis suggests how distinct ch anges in laws, rules and other government interventions in markets will affect peop le, and in some cases, one can draw a conclusion that a rule change is, on balance, socially beneficial. Such analyses combine positive analysis – predicting the effects of changes in rules – with value judgments, and are known as normative analyses. For example, a gasoline tax used to build highways harms gasoline buyers (who pay higher pr ices), but helps drivers (who face fewer potholes and less congestion). Since drivers and gasoline buyers are generally the same people, a normative analysis may suggest that everyone will benefit. This type of outcome, where everyone is made better off by a change, is relatively uncontroversial. In contrast, cost-benefit analysis weighs the gains and losse s to different individuals and suggests carrying out changes that prov ide greater benefits than harm. For example, a property tax used to build a loca l park creates a benefit to those who use the park, but harms those who own property (altho ugh, by increasing property values, even non-users obtain some benefits ). Since some of the taxp ayers won’t use the park, it won’t be the case that everyone benefits on balance. Cost-benefit analysis weighs the costs against the benefits. In the case of the park, the costs are readily monetized (turned into dollars), because the costs to the tax-payers are just the amount of the tax. In contrast, the benefits are much more challenging to estimate. Conceptually, the benefits are the amount the park users would be willing to pay to use the park if the park charged admission. However, if the park do esn’t charge admission, we would have to estimate willingness-to-pay. In principle, the park provides greater benefits than costs if the benefits to the users exceed the losses to the taxpayers. However, the park also involves transfers from one group to another. Welfare analysis provides another approach to evalua ting government intervention into markets. Welfare analysis posits social pr eferences and goals, like helping the poor. Generally a welfare analysis involves perfor ming a cost-benefit analysis taking account


McAfee: Introduction to Economic Analysis,, July 24, 2006 1-3 not just of the overall gains and losses, but also weighting those gains and losses by their effects on other social goals. For example, a property tax used to subsidize the opera might provide more value than costs, but the bulk of property taxes are paid by lower and middle income people, while the majority of opera-goers are rich. Thus, the opera subsidy represents a transfer from relatively low income people to richer people, which is not consistent with societal goals of equa lization. In contrast, elimination of sales taxes on basic food items like milk and bread generally has a relatively greater benefit to the poor, who spend a much larger percentage of their income on food, than to the rich. Thus, such schemes may be con sidered desirable not so much for their overall effects but for their redistribution effects. Econom ics is helpful not just in providing methods for determining the overall effects of taxes and programs, but also the incidence of these taxes and programs, that is, who pays, and who benefits. What economics can’t do, however, is say who ought to benefit. That is a matter for society at large to decide. 1.1.2 Opportunity Cost Economists use the idea of cost in a slightly quirky way that make s sense once you think about it, and we use the term opportunity cost to remind you occasionally of our idiosyncratic notion of cost. For an economis t, the cost of something is not just the cash payment, but all of the value given up in the process of acquiring the thing. For example, the cost of a universi ty education involves tuition, and text book purchases, and also the wages that would have been earn ed during the time at university, but were not. Indeed, the value of the time spent in acquiring the education – how much enjoyment was lost – is part of the cost of education. However, some “costs” are not opportunity costs. Room and board would not generally be a cost because, after all, you are going to be living and eating whether you are in university or not. Room and board are part of the cost of an education only in sofar as they are more expensive than they would be otherwise. Similarly, the expend itures on things you would have otherwise done – hang-gliding lessons, a trip to Europe – represent savings. However, the value of these activities has been lost while you are busy reading this book. The concept of opportunity cost can be summarized by a definition: The opportunity cost is the value of the best foregone alternative. This definition captures the idea that the cos t of something is not just its monetary cost but also the value of what you didn’t get. The opportunity cost of spending $17 on a CD is what you would have done with the $17 instead, and perhaps the value of the time spent shopping. The opportunity cost of a puppy includes not just the purchase price of the puppy, but also the food, veterinary bills carpet cleaning, and the value of the time spent dealing with the puppy. A puppy is a good example, because often the purchase price is a negligible portion of the total cost of ownership. Yet people acquire puppies all the time, in spite of their high cost of owners hip. Why? The economic view of the world is that people acquire puppies because th e value they expect to get exceeds the opportunity cost. That is, they acquire a puppy when the value of a puppy is higher than the value of what is foregone by the acquisition of a puppy. Even though opportunity costs include lo ts of non-monetary costs, we will often monetize opportunity costs, translating th e costs into dollar terms for comparison


McAfee: Introduction to Economic Analysis,, July 24, 2006 1-4 purposes. Monetizing opportunity costs is cl early valuable, because it gives a means of comparison. What is the opportunity cost of 30 days in jail? It used to be that judges occasionally sentenced convicted defendants to “thirty days or thirty dollars,” letting the defendant choose the sentence. Conceptually, we can use the same idea to find out the value of 30 days in jail. Suppose you woul d choose to pay a fine of $750 to avoid the thirty days in jail, but wouldn’t pay $1 ,000 and instead would choose time in the slammer. Then the value of the thirty da y sentence is somewhere between $750 and $1000. In principle, there exists a price where at that price you pay the fine, and at a penny more you go to jail. That price – at which you are just indifferent to the choice – is the monetized or dollar cost of the jail sentence. The same idea as choosing the jail sentence or the fine justifies monetizing opportunity costs in other contexts. For example, a gamble has a certainty equivalent which is the amount of money that makes one indifferent to choosing the gamble versus the certain amount. Indeed, companies buy and sell risk, and much of the field of risk management involves buying or selling risky it ems to reduce overall risk. In the process, risk is valued, and riskier stocks and assets must sell for a lower price (or, equivalently, earn a higher average return). This differential is known as a risk premium and it represents a monetization of the risk portion of a risky gamble. Home buyers considering various available houses are presented with a variety of options, such as one or two story, buildi ng materials like brick or wood, roofing materials, flooring materials like wood or carpet, presence or absence of swimming pools, views, proximity to parks, and so on The approach taken to valuing these items is known as hedonic pricing and corresponds to valuing ea ch item separately – what does a pool add to value on average? – and then summing the value of the components. The same approach is used to value old cars making adjustments to a base value for the presence of options like leather interior, CD changer, and so on. Again, such a valuation approach converts a bundle of disparate attributes into a monetary value. The conversion of costs into dollars is occasion ally controversial, and nowhere is it more controversial than in valuing human life. Ho w much is your life worth? Can it be converted into dollars? A certain amount of in sight into this question can be gleaned by thinking about risks. Wearing seatbelts an d buying optional safety equipment reduce the risk of death by a small but measurab le amount. Suppose a $400 airbag option reduces the overall risk of death by 0.01%. If you are indifferent to buying the option, you have implicitly valued the probability of death at $400 per 0.01%, or $40,000 per 1%, or around $4,000,000 per life. Of course, you may feel quite differently about a 0.01% chance of death than a risk ten th ousand times greater, which would be a certainty. But such an approach provides on e means of estimating the value of the risk of death – an examination what people will, and will not, pay to reduce that risk. Opportunity cost – the value of the best foregone alternative – is a basic building block of economic analysis. The conversion of costs into dollar terms, while sometimes controversial, provides a convenient means of comparing costs.


McAfee: Introduction to Economic Analysis,, July 24, 2006 1-51.1.3 Economic Reasoning and Analysis What this country needs is some one-armed economists. -Harry S Truman Economic reasoning is rather easy to satiri ze. One might want to know, for instance, what the effect of a policy change – a government program to educate unemployed workers, an increase in military spending, or an enhanced environmental regulation – will be on people and their ability to purc hase the goods and services they desire. Unfortunately, a single change may have mult iple effects. As an absurd and tortured example, government production of helium fo r (allegedly) military purposes reduces the cost of children’s birthday balloons, causing substitution away from party hats and hired clowns. The reduction in demand for clow ns reduces clowns’ wages and thus reduces the costs of running a circus. This cost re duction increases the number of circuses, thereby forcing zoos to lower admission fees to compete with circuses. Thus, were the government to stop subsidizing the manufac ture of helium, the admission fee of zoos would likely rise, even though zoos use no helium. This example is superficially reasonable, although the effects are miniscule. To make any sense at all of the effects of a ch ange in economic conditions, it is helpful to divide up the effect into pieces. Thus, we wi ll often look at the effects of a change “other things equal,” that is, assuming nothing else changed. This isolat es the effect of the change. In some cases, howeve r, a single change can lead to multiple effects; even so, we will still focus on each effect individu ally. A gobbledygook way of saying “other things equal” is to use Latin and say “ ceteris paribus .” Part of your job as a student is to learn economic jargon, and that is an exampl e. Fortunately, there isn’t too much jargon. We will make a number of assumptions that yo u may not find very easy to believe. Not all of the assumptions are required for the analysis, and instead merely simplify the analysis. Some, however, are required but deserve an explanation. There is a frequent assumption that the people we will talk abou t seem exceedingly selfish relative to most people we know. We model the choices that people make, assuming that they make the choice that is best for them. Such people – the people in the models as opposed to real people – are known occasionally as “homo e conomicus.” Real people are indubitably more altruistic than homo economicus, becaus e they couldn’t be less: homo economicus is entirely selfish. (The technical term is acting in one’s self-interest. ) That doesn’t necessarily invalidate the conclusions drawn fr om the theory, however, for at least four reasons: People often make decisions as families or households rather than individuals, and it may be sensible to consider th e household as the “consumer.” That households are fairly selfish is more plausible perhaps than individuals being selfish. Economics is pretty much silent on why consumers want things. You may want to make a lot of money so that you can bu ild a hospital or endow a library, which would be altruistic things to do. Such motives are broadly consistent with selfinterested behavior. Corporations are often required to serv e their shareholders by maximizing the share value, inducing self-interested beha vior on the part of the corporation. Even if corporations had no legal responsibi lity to act in the financial interest of


McAfee: Introduction to Economic Analysis,, July 24, 2006 1-6 their shareholders, capital markets may forc e them to act in the self-interest of the shareholders in order to raise capita l. That is, people choosing investments that generate a high return will tend to fo rce corporations to seek a high return. There are many good, and some not-so-goo d, consequences of people acting in their own self-interest, which may be another reason to focus on self-interested behavior. Thus, while there are limits to the applicability of the theory of self-interested behavior, it is a reasonable methodology for a ttempting a science of human behavior. Self-interested behavior will often be de scribed as “maximizing behavior,” where consumers maximize the value they obtain from their purchases, and firms maximize their profits. One objection to the economic methodology is that people rarely carry out the calculations necessary to literally maxi mize anything. However, that is not a sensible objection to the methodology. People don’t carry out the physics calculations to throw a baseball or thread a needle, either and yet they accomplish these tasks. Economists often consider that people act “a s if” they maximize an objective, even though no calculations are carried out. Some corporations in fact use elaborate computer programs to minimize costs or maximi ze their profits, and the entire field of operations research is used to create an d implement such maximization programs. Thus, while individuals don’t carry out the calculations, some companies do. A good example of economic reasoning is th e sunk cost fallacy. Once one has made a significant non-recoverable investment, there is a psychological tendency to invest more even when the return on the subsequent in vestment isn’t worthwhile. France and Britain continued to invest in the Concor de (a supersonic aircraft no longer in production) long after it became clear that the project would generate little return. If you watch a movie to the end, long after yo u become convinced that it stinks, you have exhibited the sunk cost fallacy. The fallacy is the result of an attempt to make an investment that has gone bad turn out to be good, even when it probably won’t. The popular phrase associated with the sunk cost fallacy is “throwing good money after bad.” The fallacy of sunk costs arises because of a psychological tendency to try to make an investment pay off when someth ing happens to render it obsolete. It is a mistake in many circumstances. The fallacy of sunk costs is often thought to be an advantage of casinos. People who lose a bit of money gambling hope to recover th eir losses by gambling more, with the sunk “investment” in gambling inducing an attemp t to make the investment pay off. The nature of most casino gambling is that th e house wins on average, which means the average gambler (and even the most skille d slot machine or craps player) loses on average. Thus, for most, trying to win ba ck losses is to lose more on average. The way economics is performed is by a prol iferation of mathematical models, and this proliferation is reflected in this book. Econ omists reason with models. Models help by removing extraneous details from a problem or issue, letting one analyze what remains more readily. In some cases the models ar e relatively simple, like supply and demand. In other cases, the models are relatively com plex (e.g. the over-fishing model of Section 6.3.6). In all cases, the models are the si mplest model that lets us understand the question or phenomenon at hand. The purpose of the model is to illuminate


McAfee: Introduction to Economic Analysis,, July 24, 2006 1-7 connections between ideas. A typical implication of a model is “when A increases, B falls.” This “ comparative static ” prediction lets us see how A affects B and why, at least in the context of the model. The real wo rld is always much more complex than the models we use to understand the world. Th at doesn’t make the model useless, indeed, exactly the opposite. By stripping out extran eous detail, the model represents a lens to isolate and understand aspects of the real world. Finally, one last introductory warning before we get started. A parody of economists talking is to add the word marginal before every word. Marginal is just economist’s jargon for “the derivative of.” For example, marginal cost is the derivative of cost; marginal value is the derivative of value. Because introductory economics is usually taught to students who have not yet studied calculus or can’t be trusted to remember even the most basic elements of it, econom ists tend to avoid using derivatives and instead talk about the value of the next unit purchased, or the cost of the next unit, and describe that as the marginal value or cost. This book uses the term marginal frequently because one of the purposes of the book is to introduce the necessary jargon so that you can read more advanced texts or take more advanced classes. For an economics student not to know the word marginal would be akin to a physics student not knowing the word mass. The book minimizes jargon where possi ble, but part of the job of a principles student is to learn the jargon, and there is no getting around that.


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-82 Supply and Demand Supply and demand are the most fundamental tools of economic analysis. Most applications of economic reasoning involve su pply and demand in one form or another. When prices for home heating oil rise in the winter, usually the reason is that the weather is colder than normal and as a result, demand is higher than usual. Similarly, a break in an oil pipeline creates a short-li ved gasoline shortage, as occurred in the Midwest in the year 2000, which is a reduction in supply. The price of DRAM, or dynamic random access memory, used in personal computers falls when new manufacturing facilities begin producti on, increasing the supply of memory. This chapter sets out the basics of supply and demand, introduces equilibrium analysis, and considers some of the factors that infl uence supply and demand and the effects of those factors. In addition, quantification is introduced in the form of elasticities. Dynamics are not considered, however, unti l Chapter 4, which focuses on production, and Chapter 5 introduces a more fundamental analysis of demand, including a variety of topics such as risk. In essence, this is th e economics “quickstart” guide, and we will look more deeply in the subsequent chapters. 2.1 Supply and Demand 2.1.1 Demand and Consumer Surplus Eating a French fry makes most people a little bit happier, and we are willing to give up something of value – a small amount of money, a little bit of time – to eat one. What we are willing to give up measures the value – ou r personal value – of the French fry. That value, expressed in dollars, is the willingness to pay for French fries. That is, if you are willing to give up three cents for a single French fry, your willingness to pay is three cents. If you pay a penny for the French fry, you’ve obtained a net of two cents in value. Those two cents – the differenc e between your willingness to pay and the amount you do pay – is known as consumer surplus. Consumer surplus is the value to a consumer of consumption of a good, minus the price paid. The value of items – French fries, eyeglasses, violins – is not necessarily close to what one has to pay for them. For people with bad vision, eyeglasses might be worth ten thousand dollars or more, in the sense that if eyeglasses and contacts cost $10,000 at all stores, that is what one would be willing to pay for vision correction. That one doesn’t have to pay nearly that amount means th at the consumer surplus associated with eyeglasses is enormous. Similarly, an orde r of French fries might be worth $3 to a consumer, but because French fries are available for around $1, the consumer obtains a surplus of $2 in the purchase. How much is a second order of French fries wo rth? For most of us, that first order is worth more than the second one. If a second order is worth $2, we would still gain from buying it. Eating a third order of fries is wo rth less still, and at some point we’re unable or unwilling to eat any more fries even when they are free, which implies that at some point the value of additional French fries is zero. We will measure consumption generally as unit s per period of time, e.g. French fries consumed per month.


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-9 Many, but not all, goods have this feature of diminishing marginal value – the value of the last unit consumed declines as the number consumed rises. If we consume a quantity q it implies the marginal value v ( q ) falls as the number of units rise.1 An example is illustrated in Figure 2-1. Here the value is a straight line, declining in the number of units. Figure 2-1: The Demand Curve Demand need not be a straight line, and inde ed could be any downward-sloping curve. Contrary to the usual convention, demand gi ves the quantity chosen for any given price off the horizontal axis, that is, given the value p on the vertical axis, the corresponding value q0 on the horizontal axis is the quantity the consumer will purchase. It is often important to distinguish the dema nd curve itself – the entire relationship between price and quantity demanded – fr om the quantity demanded. Typically, “demand” refers to the entire curve, while “quantity demanded” is a point on the curve. Given a price p a consumer will buy those units with v ( q )> p since those units are worth more than they cost. Similarly, a consumer should not buy units for which v ( q )< p Thus, the quantity q0 that solves the equation v ( q0)= p gives the quantity of units the consumer will buy. This value is also illustrated in Figure 2-1.2 Another way of 1 When diminishing marginal value fails, which some times is said to occur with beer consumption, constructing demand takes some additional effort, which isn’t of a great deal of consequence. Buyers will still choose to buy a quantity where marginal value is decreasing. 2 We will treat units as continuous, even though in real ity they are discrete units. The reason for treating them as continuous is only to simplify the mathemat ics; with discrete units, the consumer buys those units with value exceeding the price, and doesn’t buy those with value less than the price, just as before. However, since the value function isn’t continuous, much less differentiable, it would be an accident for q value v ( q ) q0 v(q0), p


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-10 summarizing this insight is that the marginal value curve is the inverse of demand function, where the demand function gives the quantity demanded for any given price. Formally, if x ( p ) is the quantity a consumer buys given a price of p then )) ( ( p p x v But what is the marginal value curve? Su ppose the total value of consumption of the product, in dollar value, is given by u ( q ). That is, a consumer who pays u ( q ) for the quantity q is just indifferent to getting nothing and paying nothing. For each quantity, there should exist one and only one price that exactly makes the consumer indifferent between purchasing it and getting nothing at all, because if the consumer is just willing to pay u ( q ), any greater amount is more than th e consumer should be willing to pay. The consumer facing a price p gets a net value or consumer surplus of CS = u ( q ) – pq from consuming q units. In order to obtain the maximal benefit, the consumer would then choose the level of q to maximize u ( q ) – pq When the function CS is maximized, its derivative is zero, which implies that, at the quantity that maximizes the consumer’s net value ) ( ) ( 0 p q u pq q u dq d Thus we see that ), ( ) ( q u q v that is, the marginal value of the good is the derivative of the total value. Consumer surplus is the value of the consumption minus the amount paid, and represents the net value of the purchase to the consumer. Formally, it is u ( q )pq A graphical form of the consumer surplus is generated by the following identity. ) ( ) ( ) ( ) ( max0 00 0 0 0 q q qdx p x v dx p x u pq q u pq q u CS This expression shows that consumer surplus can be represented as the area below the demand curve and above the price, as is illustrated in Figure 2-2. The consumer surplus represents the consumer’s gains from trade, the value of consumption to the consumer net of the price paid. marginal value to equal price. It isn’t particularly arduous to handle discreteness of the products, but it doesn’t lead to any significant insight either, so we won’t consider it here.


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-11 Figure 2-2: Consumer Surplus The consumer surplus can also be expressed using the demand curve, by integrating from the price up. In this case, if x ( p ) is the demand, we have pdy y x CS ) ( When you buy your first car, you experience an increase in demand for gasoline because gasoline is pretty useful for cars and no t so much for other things. An imminent hurricane increases the demand for plywood (to protect windows), batteries, candles, and bottled water. An increase in demand is represented by a movement of the entire curve to the northeast (up and to the righ t), which represents an increase in the marginal value v (movement up) for any given unit, or an increase in the number of units demanded for any given price (movement to the right). Figure 2-3 illustrates a shift in demand. Similarly, the reverse movement represents a decrease in demand. The beauty of the connection between demand and marginal value is that an increase in demand could in principle have meant either more units demanded at a given price, or a higher willingness to pay for each unit, but those ar e in fact the same concept – both create a movement up and to the right. For many goods, an increase in income increases the demand for the good. Porsche automobiles, yachts, and Beverly Hills homes are mostly purchased by people with high incomes. Few billionaires ride the bus. Economists aptly named goods whose demand doesn’t increase with income inferior goods, with the idea that people substitute to better quality, more expensive goods as thei r incomes rise. When demand for a good q value q0 v(q0) ) ( ) ( q u q v Consumer Sur p lus


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-12increases with income, the good is called normal It would have been better to call such goods superior, but it is too late to ch ange such a widely accepted convention. Figure 2-3: An Increase in Demand Another factor that influences demand is the price of related goods. The dramatic fall in the price of computers over the past twenty years has significantly increased the demand for printers, monitors and internet access. Such goods are examples of complements Formally, for a given good X a complement is a good whose consumption increases the value of X Thus, the use of computers increases the value of peripheral devices like printers and monitors. The consumption of coffee increases the demand for cream for many people. Spaghetti and tomato sauce, na tional parks and hiking boots, air travel and hotel rooms, tables and chairs, movies and popcorn, bathing suits and sun tan lotion, candy and dentistry are all exampl es of complements for most people – consumption of one increases the value of the other. The complementarity relationship is symmetric – if consumption of X increases the value of Y then consumption of Y must increase the value of X .3 There are many complementary goods and changes in the prices of complementary goods have pred ictable effects on the demand of their complements. Such predictable effects re present the heart of economic analysis. The opposite case of a complement is a substitute Colas and root beer are substitutes, and a fall in the price of root beer (resulting in an increase in the consumption of root beer) will tend to decrease the demand fo r colas. Pasta and ramen, computers and typewriters, movies (in theaters) and sporti ng events, restaurants and dining at home, spring break in Florida versus spring brea k in Mexico, marijuana and beer, economics 3 The basis for this insight can be seen by denoting the total value in dollars of consuming goods x and y as u ( x y ). Then the demand for x is given by the partial x u The statement that y is a complement is the statement that the demand for x rises as y increases, that is, 02 y x u But then with a continuous second derivative, 02 x y u, which means the demand for y, y u increases with x q value v ( q )


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-13courses and psychology courses, driving and bi cycling are all examples of substitutes for most people. An increase in the price of a substitute increases the demand for a good, and conversely, a decrease in the price of a substitute decreases demand for a good. Thus, increased enforcement of the drug laws, which tends to increase the price of marijuana, leads to an increase in the demand for beer. Much of demand is merely idiosyncratic to the individual – some people like plaids, some like solid colors. People like what they like. Often people are influenced by others – tattoos are increasingly common not because the price has fallen but because of an increased acceptance of body art. Popular cl othing styles change, not because of income and prices but for other reasons. While there has been a modest attempt to link clothing style popularity to economic factors,4 by and large there is no coherent theory determining fads and fashions beyond the obse rvation that change is inevitable. As a result, this course, and economics more gene rally, will accept preferences for what they are without questioning why people like what they like. While it may be interesting to understand the increasing social acceptance of tattoos, it is beyond the scope of this text and indeed beyond most, but not all, economic analyses. We will, however, account for some of the effects of the increasing acce ptance of tattoos through changes in the number of firms offering tattooing, changes in the variety of products offered, and so on. (Exercise) A reservation price is the maximum willing ness to pay for a good that most people buy one unit of, like cars or computers. Graph the demand curve for a consumer with a reservation price of $30 for a unit of a good. (Exercise) Suppose the demand curve is given by x ( p ) = 1 – p The consumer’s expenditure is px ( p ) = p (1 – p ). Graph the expenditure. What price maximizes the consumer’s expenditure? (Exercise) For demand x ( p ) = 1 – p compute the consumer surplus function as a function of p (Exercise) For demand x( p ) = p–, for > 1, find the consumer surplus as a function of p (Hint: recall that the consumer surplus can be expressed as pdy y x CS ) ( .) 2.1.2 Supply The supply curve gives the number of units, represented on the horizontal axis, as a function of the price on the vertical axis, wh ich will be supplied for sale to the market. An example is illustrated in Figure 2-4. Generally supply is upward-sloping, because if it is a good deal for a seller to sell 50 units of a product at a price of $10, then it remains a good deal to supply those same 50 at a price of $11. The seller might choose to sell 4 Skirts are allegedly shorter during econom ic booms and lengthen during recessions.


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-14more than 50, but if the first 50 weren’t wo rth keeping at a price of $10, that remains true at $11.5 Figure 2-4: The Supply Curve The seller who has a cost c ( q ) for selling q units obtains a profit, at price p per unit, of pq – c ( q ). The quantity which maximizes profit for the seller is the quantity q satisfying *). ( ) ( 0 q c p q c pq dq d Thus, price equals marginal cost is a charac teristic of profit maximization; the seller sells all the units whose cost is less than price, and doesn’t sell the units whose cost exceeds price. In constructing the demand curve, we saw that the demand curve was the inverse of the marginal value. There is an analogous property of supply: the supply curve is the inverse func tion of marginal cost Graphed with the quantity supplied on the horizontal axis and price on the vertical axis, the supply curve is the marginal cost curve, with marginal cost on the vertical axis. Exactly in parallel to consumer surplus with demand, profit is given by the difference of the price and marginal cost 5 This is a good point to remind the reader that the economists’ familiar assumption of “other things equal” is still in effect. If the increased price is an indication that prices might rise still further, or a consequence of some other change that affects the seller s’ value of items, then of course the higher price might not justify sale of the items. We hold other things equal to focus on the effects of price alone, and then will consider other changes separately. The pure effect of an increased price should be to increase the quantity offered, while the effect of increased expectations may be to decrease the quantity offered. q p q0 p


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-15Profit. ) ( *) ( ) ( max* 0 q qdx x c p q c pq q c pq This area is shaded in Figure 2-5. Figure 2-5: Supplier Profits The relationship of demand and marginal va lue exactly parallels the relationship of supply and marginal cost, for a somewhat hi dden reason. Supply is just negative demand, that is, a supplier is just the posses sor of a good who doesn’t keep it but instead offers it to the market for sale. For exampl e, when the price of housing goes up, one of the ways people demand less is by offering to rent a room in their house, that is, by supplying some of their housing to the market. Similarly, the marginal cost of supplying a good already produced is the loss of not havi ng the good, that is, the marginal value of the good. Thus, with exchange, it is possi ble to provide the theory of supply and demand entirely as a theory of net demand, where sellers are negative demanders. There is some mathematical economy in this approach, and it fits certain circumstances better than separating supply and demand. Fo r example, when the price of electricity rose very high in the western United States in 2003, several aluminum smelters resold electricity they had purchased in long-ter m contracts, that is, demanders became suppliers. However, the “net demand” approach obscures the likely outcomes in instances where the sellers are mostly different people, or co mpanies, than the buyers. Moreover, while there is a theory of complements and substitu tes for supply that is exactly parallel to the equivalent theory for demand, the nature of these complements and substitutes tends to be different. For these reasons, and also for the purpose of being consistent with common economic usage, we will distinguish supply and demand. q p q0 p Profit


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-16An increase in supply refers to either more units available at a given price, or a lower price for the supply of the same number of units. Thus, an increase in supply is graphically represented by a curve that is lowe r or to the right, or both, that is, to the south-east. This is illustrated in Figure 2-6. A decrease in suppl y is the reverse case, a shift to the northwest. Figure 2-6: An Increase in Supply Anything that increases costs of production will tend to increase marginal cost and thus reduce the supply. For example, as wages rise, the supply of goods and services is reduced, because wages are the input price of labor. Labor accounts for about twothirds of all input costs, and thus wage increases create suppl y reductions (a higher price is necessary to provide the same quantity ) for most goods and services. Costs of materials of course increase the price of goods using those materials. For example, the most important input into the manufacture of ga soline is crude oil, and an increase of $1 in the price of a 42 gallon barrel of oil incr eases the price of gasoline about two cents – almost one-for-one by volume. Another signif icant input in many industries is capital, and as we will see, interest is cost of capita l. Thus, increases in interest rates increase the cost of production, and thus tend to decrease the supply of goods. Parallel to complements in demand, a complement in supply to a good X is a good Y such that an increase in the price of Y increases the supply of X Complements in supply are usually goods that are jointly produced. In producing lumber (sawn boards), a large quantity of wood chips and sawdust are also produced as a by-product. These wood chips and saw dust are useful in the manufacture of paper. An increase in the price of lumber tends to increase the quantity of tr ees sawn into boards, thereby increasing the supply of wood chips. Thus, lumber an d wood chips are complements in supply. It turns out that copper and gold are ofte n found in the same kinds of rock – the conditions that give rise to gold compounds also give rise to copper compounds. Thus, an increase in the price of gold tends to increase the number of people prospecting for q p


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-17gold, and in the process increases not just the quantity of gold supplied to the market, but also the quantity of copper. Thus, co pper and gold are complements in supply. The classic supply-complement is beef and leather – an increase in the price of beef increases the slaughter of cows, thereb y increasing the supply of leather. The opposite of a complement in supply is a substitute in supply Military and civilian aircraft are substitutes in supply – an increase in the price of military aircraft will tend to divert resources used in the manufacture of aircraft toward military aircraft and away from civilian aircraft, thus reducing the supp ly of civilian aircraft. Wheat and corn are also substitutes in supply. An increase in the price of wheat will lead farmers whose land is reasonably well-suited to producing ei ther wheat or corn to substitute wheat for corn, increasing the quantity of wheat and decr easing the quantity of corn. Agricultural goods grown on the same type of land usually are substitutes. Similarly, cars and trucks, tables and desks, sweaters and sweatshirts, horror movies and romantic comedies are examples of substitutes in supply. Complements and substitutes are important because they are common and have predictable effects on demand and supply. Ch anges in one market spill over to the other market, through the mechanism of complements or substitutes. (Exercise) A typist charges $30/hr and type s 15 pages per hour. Graph the supply of typed pages. (Exercise) An owner of an oil well has two te chnologies for extracting oil. With one technology, the oil can be pumped out and transported for $5,000 per day, and 1,000 barrels per day are produc ed. With the other technology, which involves injecting natural gas into th e well, the owner spends $10,000 per day and $5 per barrel produced, but 2,000 barrels per day are produced. What is the supply? Graph it. (Hint: Compute the profits, as a function of the price, for each of the technologies. At what price would the producer switch from on e technology to the other? At what price would the producer shut down and spend nothing?) (Exercise) An entrepreneur has a factory the produces L widgets, where <1, when L hours of labor is used. The cost of labor (wage and benefits) is w per hour. If the entrepreneur maximizes profit, what is the supply curve for widgets? Hint: The entrepreneur’s profit, as a function of the price, is pL – wL. The entrepreneur chooses the amount of labor to maximize profit. Find the amount of labor that maximizes, which is a function of p w and The supply is the amount of output produced, which is L.


McAfee: Introduction to Economic Analysis,, July 24, 2006 2- (Exercise) In the above exercise, suppose now that more than 40 hours entails a higher cost of labor (overtime pay). Let w be $20/hr for under 40 hours, and $30/hr for each hour over 40 hours, and = Find the supply curve. Hint: Let L ( w p ) be the labor demand when the wage is w (no overtime pay) and the price is p Now show that, if L (20, p ) < 40, the entrepreneur uses L (20, p ) hours. This is shown by verifying that profits are higher at L (20, p ) than at L (30, p ). If L (30, p ) > 40, the entrepreneur uses L (30, p ) hours. Finally, if L (20, p ) > 40 > L (30, p ), the entrepreneur uses 40 hours. La bor translates into supply via L. (Exercise) In the previous exercise, for what range of prices does employment equal 40 hours? Graph the labo r demanded by the entrepreneur. Hint: The answer involves 10. (Exercise) Suppose marginal cost, as a function of the quantity q produced, is mq Find the producer’s profit as a function of the price p 2.2 The Market Individuals with their own supply or demand trade in a market, which is where prices are determined. Markets can be specific or virtual locations – the farmer’s market, the New York Stock Exchange, eBay – or may be an informal or more amorphous market, such as the market for restaurant meals in Billings, Montana or the market for roof repair in Schenectady, New York. 2.2.1 Market Demand and Supply Individual demand gives the quantity purc hased for each price. Analogously, the market demand gives the quantity purchased by a ll the market participants – the sum of the individual demands – for each price. This is sometimes called a “horizontal sum” because the summation is over the quantities fo r each price. An example is illustrated in Figure 2-7. For a given price p the quantity q1 demanded by one consumer, and the quantity q2 demanded by a second consumer are illustrated. The sum of these quantities represents the market demand, if the market has just those two-participants. Since the consumer with subscript 2 has a po sitive quantity demanded for high prices, while the consumer with subscript 1 does not, the market demand coincides with consumer 2’s demand when the price is suffici ently high. As the price falls, consumer 1 begins purchasing, and the market quantity de manded is larger than either individual participant’s quantity, and is the sum of the two quantities. Example: If the demand of buyer 1 is given by q = max {0, 10 – p }, and the demand of buyer 2 is given by q = max {0, 20 – 4 p }, what is market demand for the twoparticipants? Solution: First, note that buyer 1 buys zero at a price 10 or higher, while buyer 2 buys zero at a price of 5 or higher. For a price above 10, market demand is zero. For a price between 5 and 10, market demand is buyer 1’s demand, or 10 – p Finally, for a price between zero and 5, the market quantity demanded is 10 – p + 20 – 4 p = 30 – 5 p


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-19 Market supply is similarly constructed – the market supply is the horizontal (quantity) sum of all the individual supply curves. Figure 2-7: Market Demand Example: If the supply of firm 1 is given by q = 2 p and the supply of firm 2 is given by q = max {0, 5 p – 10}, what is market supply for the two-participants? Solution: First, note that firm 1 is in the mark et at any price, but firm 2 is in the market only if price exceeds 2. Thus, for a price be tween zero and 2, market supply is firm 1’s supply, or 2 p For p >2, market supply is 5 p – 10 + 2 p = 7 p – 10. (Exercise) Is the consumer surplus for market demand the sum of the consumer surpluses for the individual de mands? Why or why not? Illustrate your conclusion with a figure like Figure 2-7. (Exercise) Suppose the supply of firm i is i p when the price is p where i takes on the values 1, 2, 3, … n What is the mark et supply of these n firms? (Exercise) Suppose consumers in a small town choose between two restaurants, A and B Each consumer has a value vA for A and a value vB for B each of which is a uniform random draw fr om the [0,1] interval. Consumers buy whichever product offers the higher c onsumer surplus. The price of B is 0.2. In the square associated with the possible value types, identify which consumers buy from firm A Find the demand (which is t he area of the set of consumers who buy from A in the picture below). Hint: Consumers have three choices: Buy nothing (value 0), buy from A (value vA – pA) and buy from B (value vB – pB p q1 q2 q1+ q2Market Demand


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-20 = vB – 0.2). Draw the lines illustrating which choice has the highest value for the consumer. 2.2.2 Equilibrium Economists use the term equilibrium in the same way as the word is used in physics, to represent a steady state in which opposing fo rces are balanced, so that the current state of the system tends to persist. In the context of supply and demand, equilibrium refers to a condition where the pressure for higher pr ices is exactly balanced by a pressure for lower prices, and thus that the current state of exchange between buyers and sellers can be expected to persist. When the price is such that the quantity supplied of a good or service exceeds the quantity demanded, some sellers are unable to sell because fewer units are purchased than are offered. This condition is called a surplus The sellers who fail to sell have an incentive to offer their good at a slightly lo wer price – a penny less – in order to succeed in selling. Such price cuts put downward pressu re on prices, and prices tend to fall. The fall in prices generally reduces the quan tity supplied and increases the quantity demanded, eliminating the surplus. That is a surplus encourages price cutting, which reduces the surplus, a process that ends on ly when the quantity supplied equals the quantity demanded. Similarly, when the price is low enough that the quantity demanded exceeds the quantity supplied, a shortage exists. In this case, some buyers fail to purchase, and these buyers have an incentive to accept a sl ightly higher price in order to be able to trade. Sellers are obviously happy to get th e higher price as well, which tends to put upward pressure on prices, and prices rise. The increase in price tends to reduce the quantity demanded and increase the quan tity supplied, thereby eliminating the shortage. Again, the process stops when the quantity supplied equals the quantity demanded. Value of A Value of B Price of B Price of A Buy Nothing


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-21 Figure 2-8: Equilibration This logic, which is illustrated in Figure 2-8, justifies the conclusion that the only equilibrium price is the price in which th e quantity supplied equals the quantity demanded. Any other price will tend to rise in a shortage, or fall in a surplus, until supply and demand are balanced. In Figure 2-8, a surplus arises at any price above the equilibrium price p* because the quantity supplied qs is larger than the quantity demanded qd. The effect of the surplus – leading to sellers with excess inventory – induces price cutting which is illustrated with three arrows pointing down. Similarly, when the price is below p* the quantity supplied qs is less than the quantity demanded qd. This causes some buyers to fail to find goods, leading to higher asking prices and higher bid prices by buyers. The tendency for the price to rise is illustrated with the arrows pointing up. The only pr ice which doesn’t lead to price changes is p* the equilibrium price in which the quantity supplied equals the quantity demanded. The logic of equilibrium in supply and demand is played out daily in markets all over the world, from stock, bond and commodity market s with traders yelling to buy or sell, to Barcelona fish markets where an auctioneer he lps the market find a price, to Istanbul gold markets, to Los Angeles real estate markets. (Exercise) If demand is given by qd( p) = a – bp and supply is given by qs( p) = cp solve for the equilibrium price and quan tity. Find the consumer surplus and producer profits. p q Demand Supply q* p* Surplus: q d < qs Shortage: qs < q d


McAfee: Introduction to Economic Analysis,, July 24, 2006 2- (Exercise) If demand is given by qd( p) = ap-, and supply is given by qs( p) = bp, where all parameters are positive numbers, solve for the equilibrium price and quantity. 2.2.3 Efficiency of Equilibrium The equilibrium of supply and demand balances the quantity demanded and the quantity supplied, so that there is no excess of either. Would it be desirable, from a social perspective, to force more trade, or to restrain trade below this level? There are circumstances where the equilibrium level of trade has harmful consequences, and such circumstances are considered in Chapter 6. However, provided that the only people affected by a transaction are the buyer and seller, the equilibrium of supply and demand maximizes the total gains from trade This proposition is quite easy to see. To maximize the gains from trade, clearly the highest value buyers must get the goods. Ot herwise, if there is a potential buyer that doesn’t get the good with higher value than one who does, the gains from trade rise just by diverting the good to the higher value buye r. Similarly, the lowest cost sellers must supply those goods; otherwise we can increa se the gains from trade by replacing a higher cost seller with a lowe r cost seller. Thus, the only question is how many goods should be traded to maximize the gains from trade, since it will involve the lowest cost sellers selling to the highest value buyers. Adding a trade increases the total gains from trade when that trade involves a buyer with va lue higher than the se ller’s cost. Thus, the gains from trade are maximized by the set of tr ansactions to the left of the equilibrium, with the high value buyers buying from the low cost sellers. In the economist’s language, the equilibrium is efficient in that it maximizes the gains from trade, under the assumption that the on ly people affected by any given transaction are the buyers and seller. 2.3 Changes in Supply and Demand 2.3.1 Changes in Demand What are the effects of an increase in demand? As the population of California has grown, the demand for housing has risen. Th is has pushed the price of housing up, and also spurred additional development, increa sing the quantity of housing supplied as well. We see such a demand in crease illustrated in Figure 2-9, which represents an increase in the demand. In this figure, su pply and demand have been abbreviated S and D. Demand starts at D1 and is increased to D2. Supply remains the same. The equilibrium price increases from p1* to p2*, and the quantity rises from q1* to q2*.


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-23 Figure 2-9: An Increase in Demand A decrease in demand – such as occurred for typewriters with the advent of computers, or buggy whips as cars replaced horses as the major method of transportation – has the reverse effect of an increase, and implies a fall in both the price and the quantity traded. Examples of decreases in demand include pr oducts replaced by other products – VHS tapes were replaced by DVDs, vinyl records re placed by CDs, cassette tapes replaced by CDs, floppy disks (oddly named because the 1.44 MB “floppy,” a physically hard product, replaced the 720KB, 5 inch soft floppy disk) replaced by CDs and flash memory drives, and so on. Even personal computers experienced a fall in demand as the market was saturated in the year 2001. 2.3.2 Changes in Supply An increase in supply comes about from a fa ll in the marginal cost – recall that the supply curve is just the marginal cost of production. Consequently, an increased supply is represented by a curve that is lower an d to the right on the supply/demand graph, which is an endless source of confusion for many students. The reasoning – lower costs and greater supply are the same thing – is t oo easily forgotten. The effects of an increase in supply are illustrated in Figure 2-10. The supply curve goes from S1 to S2, which represents a lower marginal cost. In this case, the quantity traded rises from q1* to q2* and price falls from p1* to p2*. Computer equipment provides dramatic exam ples of increases in supply. Consider Dynamic Random Access Memory, or DRAM. DRAMs are the chips in computers and many other devices that store information on a temporary basis.6 Their cost has fallen dramatically, which is illustrated in Figure 2-11.7 Note that the prices in this figure reflect a logarithmic scale, so that a fixed perc entage decrease is illustrated by a straight line. Prices of DRAMs fell to close to 1/1000th of their 1990 level by 2004. The means 6 Information that will be stored on a longer term basis is generally embedded in flash memory or on a hard disk. Neither of these products lose their in formation when power is turned off, unlike DRAM. 7 Used with permission of computer storage expert Dr. Edward Grochowski. p q D1S q1* p1* D2 p 2 q 2 *


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-24by which these prices have fallen are themselv es quite interesting. The main reasons are shrinking the size of the chip (a “die shrink”) so that more chips fit on each silicon disk, and increasing the size of the disk itself so that more chips fit on a disk. The combination of these two, each of which required the solutions to thousands of engineering and chemistry problems, has led to dramatic reductions in marginal costs and consequent increases in supply. The effe ct has been that prices fell dramatically and quantities traded rose dramatically. Figure 2-10: An Increase in Supply An important source of supply and demand changes are changes in the markets of complements. A decrease in the price of a demand-complement increases the demand for a product, and similarly, an increase in the price of a demand-substitute increases demand for a product. This gives two me chanisms to trace through effects from external markets to a particular market via the linkage of demand substitutes or complements. For example, when the price of gasoline falls, the demand for automobiles (a complement) overall should incr ease. As the price of automobiles rises, the demand for bicycles (a substitute in so me circumstances) should rise. When the price of computers falls, the demand for operating systems (a complement) should rise. This gives an operating system seller like Mi crosoft an incentive to encourage technical progress in the computer market, in order to make the operating system more valuable. p q D S1 q1* p1* S2p 2 q 2


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-25 Figure 2-11: Price of Storage An increase in the price of a supply-substit ute reduces the supply of a good (by making the alternative good more attractive to suppli ers), and similarly, a decrease in the price of a supply complement reduces the supply of a good. By making the by-product less valuable, the returns to investing in a good ar e reduced. Thus, an increase in the price of DVD-R discs (used for recording DVDs) di scourages investment in the manufacture of CD-Rs, which are a substitute in supply, lead ing to a decrease in the supply of CD-Rs. This tends to increase the price of CD-Rs, othe r things equal. Similarly, an increase in the price of oil increases exploration for oil, tending to increase the supply of natural gas, which is a complement in supply. Ho wever, since natural gas is also a demand substitute for oil (both are used for heating homes), an increase in the price of oil also tends to increase the demand for natural gas. Thus, an increase in the price of oil increases both the demand and the supply of natural gas. Both changes increase the quantity traded, but the increase in demand tends to increase the price, while the increase in supply tends to decrease the pric e. Without knowing more, it is impossible to determine whether the net effect is an increase or decrease in the price. (Exercise) Video games and music CDs are substi tutes in demand. What is the effect of an increase in supply of video games on the price and quantity traded of music CDs? Illustrate your answer with diagrams for both markets. (Exercise) Electricity is a major input into the production of aluminum, and aluminum is a substitute in supply for steel What is the effect of an increase in price of electricity on the steel market?


McAfee: Introduction to Economic Analysis,, July 24, 2006 2- (Exercise) Concerns about terrorism reduced demand for air travel, and induced consumers to travel by car more often. What should happen to the price of Hawaiian hotel rooms? When the price of gasoline goes up, people cu rtail their driving to some extent, but don’t immediately scrap their SUVs and rush out and buy more fuel-efficient automobiles or electric cars. Similarly, when the price of el ectricity rises, people don’t replace their air conditioners and refrigerators with the most modern, energy-saving models right away. There are three significant issues raised by th is kind of example. First, such changes may be transitory or permanent, and people reasonably react differently to temporary changes than to permanent changes. The effe ct of uncertainty is a very important topic and will be considered in section 5.2.6, but only in a rudimentary way for this introductory text. Second, energy is a mo dest portion of the cost of owning and operating an automobile or refrigerator, so it doesn’t make sense to scrap a large capital investment over a small permanent increase in cost. Thus people rationally continue operating “obsolete” devices until their useful life is over, even wh en they wouldn’t buy an exact copy of that device, an ef fect with the gobbledygook name of hysteresis Third, a permanent increase in energy prices leads people to buy more fu el efficient cars, and to replace the old gas guzzlers more quickly. That is, the effects of a change are larger over a larger time interval, which economists tend to call the long-run A striking example of such delay arose when oi l quadrupled in price in 1973-4, caused by a reduction in sales by the cartel of oil-pr oducing nations, OPEC, which stands for the Organization of Petroleum Exporting Countries. The increased price of oil (and consequent increase in gasoline prices) caus ed people to drive less and to lower their thermostats in the winter, thus reducing th e quantity of oil demanded. Over time, however, they bought more fuel effici ent cars and insulated their homes more effectively, significantly reducing the quantity demanded still further. At the same time, the increased prices for oil attracted new inve stment into oil production in Alaska, the North Sea between Britain and Norway, Mexico and other areas. Both of these effects (long-run substitution away from energy, and long-run supply expansion) caused the price to fall over the longer term, undoing the supply reduction created by OPEC. In 1981, OPEC further reduced output, sending pr ices still higher, but again, additional investment in production, combined with energy-saving investment, reduced prices until they fell back to 1973 levels (adjusted fo r inflation) in 1986. Prices continued to fall until 1990, when they were at all-time low levels and Iraq’s invasion of Kuwait and the resulting first Iraqi war sent them higher again. Short-run and long-run effects represent a theme of economics, with the major conclusion of the theme that substitution do esn’t occur instantaneously, which leads to predictable patterns of prices and quantities over time. It turns out that direct estimates of demand and supply are less useful as quantifications than notions of percentage changes, which ha ve the advantage of being unit-free. This observation gives rise to the conc ept of elasticity, the next topic.


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-272.4 Elasticities 2.4.1 Elasticity of Demand Let x ( p ) represent the quantity pu rchased when the price is p so that the function x represents demand. How responsive is de mand to price changes? One might be tempted to use the derivative x to measure the responsiveness of demand, since it measures directly how much the quantity demanded changes in response to a small change in price. However, this measure has two problems. First, it is sensitive to a change in units. If I measure the quantity of candy in kilograms rather than pounds, the derivative of demand for cand y with respect to price changes even when demand itself has remained the same. Second, if I change price units, converting from one currency to another, again the derivative of demand will ch ange. So the derivative is unsatisfactory as a measure of responsiveness because it de pends on units of measure. A common way of establishing a unit-free measure is to us e percentages, and that suggests considering the responsiveness of demand in percentage terms to a small percentage change in price. This is the notion of elasticity of demand .8 The elasticity of demand is the percentage decrease in quantity that results from a small percentage increase in price. Formally, the elasticity of demand, which is generally denoted with the Greek letter epsilon (chosen to mnemonically suggest elasticity) is ) ( ) ( p x p x p dp dx x p p dp x dx The minus sign is included to make the elas ticity a positive number, since demand is decreasing. First, let’s verify that the elasti city is in fact unit free. A change in the measurement of x cancels because the proportional ity factor appears in both the numerator and denominator. Similarly, if we change the units of measurement of price to replace the price p with r = ap, x ( p ) is replaced with x ( r / a ). Thus, the elasticity is ) ( ) ( ) / ( 1 ) / ( ) / ( ) / ( p x p x p a r x a a r x r a r x a r x dr d r which is independent of a and therefore not affected by the change in units. How does a consumer’s expenditure, also known as (individual) total revenue, react to a change in price? The consumer buys x ( p ) at a price of p and thus expenditure is TR = px ( p ). Thus 1 ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( p x p x p x p p x p x p p x p px dp d Therefore, 8 The concept of elasticity was invented by Alfred Ma rshall, 1842-1924, in 1881 while sitting on his roof.


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-28 1 1 TR p TR dp d Table 2-1: Various Demand Elasticities9 Product Salt 0.1 Matches 0.1 Toothpicks 0.1 Airline travel, short-run 0.1 Residential natural gas, short-run 0.1 Gasoline, short-run 0.2 Automobiles, long-run 0.2 Coffee 0.25 Legal services, short-run 0.4 Tobacco products, short-run 0.45 Residential natural gas, long-run 0.5 Fish (cod) consumed at home 0.5 Physician services 0.6 Taxi, short-run 0.6 Gasoline, long-run 0.7 Movies 0.9 Shellfish, consumed at home 0.9 Tires, short-run 0.9 Oysters, consumed at home 1.1 Private education 1.1 Housing, owner occupied, long-run 1.2 Tires, long-run 1.2 Radio and television receivers 1.2 Automobiles, short-run 1.2-1.5 Restaurant meals 2.3 Airline travel, long-run 2.4 Fresh green peas 2.8 Foreign travel, long-run 4.0 Chevrolet automobiles 4.0 Fresh tomatoes 4.6 9 From ; cited sources: Economics: Private and Public Choice James D. Gwartney and Richard L. Stroup, eighth edition 1997, seventh edition 1995; Hendrick S. Houthakker and Lester D. Taylor, Consumer Demand in the United States 1929-1970 (Cambridge: Harvard University Pre ss, 1966,1970); Douglas R. Bohi, Analyzing Demand Behavior (Baltimore: Johns Hopkins University Press, 1981); Hsaing-tai Cheng and Oral Capps, Jr., "Demand for Fish" American Journal of Agricultural Economics August 1988; and U.S. Department of Agriculture.


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-29In words, the percentage change of total re venue resulting from a one percent change in price is one minus the elasticity of demand. Thus, a one percent increase in price will increase total revenue when the elasticity of demand is less than one, which is defined as an inelastic demand. A price increase will decrease total revenue when the elasticity of demand is greater than one, which is defined as an elastic demand. The case of elasticity equal to one is called unitary elasticity and total revenue is unchanged by a small price change. Moreover, th at percentage increase in price will increase revenue by approximately 1percent. Because it is often possi ble to estimate the elasticity of demand, the formulae can be readily used in practice Table 2-1 provides estimates on demand elas ticities for a variety of products. Figure 2-12: Elasticities for Linear Demand When demand is linear, x ( p )= a bp the elasticity of demand has the form p b a p bp a bp This case is illustrated in Figure 2-12. If demand takes the form x ( p )= ap-, then demand has constant elasticity and the elasticity is equal to (Exercise) Suppose a consumer has a constant elasticity of demand and demand is elastic ( > 1). Show that expenditure increases as price decreases. (Exercise) Suppose a consumer has a constant elasticity of demand and demand is inelastic ( < 1). What price makes expenditure the greatest? q price a a/b =1 =0=


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-302.4.1.3 (Exercise) For a consumer with constant elasticity of demand > 1, compute the consumer surplus. 2.4.2 Elasticity of Supply The elasticity of supply is analogous to the elas ticity of demand, in that it is a unit-free measure of the responsiveness of supply to a price change, and is defined as the percentage increase in quanti ty supplied resulting from a small percentage increase in price. Formally, if s ( p ) gives the quantity supplied for each price p the elasticity of supply, denoted (the Greek letter “eta”, chosen because epsilon was already taken) is ) ( ) ( p s p s p dp ds s p p dp s ds Again similar to demand, if supply takes the form s ( p )= ap, then supply has constant elasticity and the elasticity is equal to A special case of this form is linear supply, which occurs when the elasticity equals one. (Exercise) For a producer with constant elasticity of supply, compute the producer profits. 2.5 Comparative Statics When something changes – the price of a co mplement, the demand for a good – what happens to the equilibrium? Su ch questions are answered by comparative statics which are the changes in equilibrium variable s when other things change. The use of the term “static” suggests that such change s are considered without respect to dynamic adjustment, but instead just focus on the chan ges in the equilibrium level. Elasticities will help us quantify these changes. 2.5.1 Supply and Demand Changes How much do the price and quantity traded ch ange in response to a change in demand? We begin by considering the constant elastici ty case, which will let us draw conclusions for small changes in more general demand functions. We will denote the demand function by qd( p )= apand supply function by qs( p )= bp. The equilibrium price p is given by the quantity supplied equal to the quantity demanded, or the solution to the equation: *). ( *) ( p q p qs d Substituting the constant elasticity formulae, *) ( *) ( bp p q p q aps d Thus,


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-31 p b a or *1 b a p The quantity traded, q *, can be obtained from either supply or demand and the price: *) ( b a b a b bp p q qs There is one sense in which th is gives an answer to the qu estion of what happens when demand increases. An increase in demand, holding the elasticity constant, corresponds to an increase in the parameter a Suppose we increase a by a fixed percentage, replacing a by a (1+ ). Then price goes up by the multiplicative factor 11 and the change in price, as a proportion of the price, is 1 1 *1 p p Similarly, quantity rises by 1 1 q q. These formulae are problematic for two reasons. First, they are specific to the case of constant elasticity. Second, th ey are moderately complicated. Both of these issues can be addressed by considering small ch anges, that is, a small value of We make use of a trick to simplify the formula. The trick is that, for small 1 1 rr The squiggly equals sign should be read “approximately equal to.”10 Applying this insight, we have that: 10 The more precise meaning of is that, as gets small, the size of the error of the formula is small even relative to That is, 1 1 rr means 0 1 10 rr


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-32For a small percentage increase in demand, quantity rises by approximately percent and price rises by approximately percent. The beauty of this claim is that it holds even when demand and supply do not have constant elasticities, because the effect consid ered is local, and locally, the elasticity is approximately constant if the demand is “smooth.” (Exercise) Show that for a small percentage increase in supply, quantity rises by approximately percent and price falls by approximately percent. (Exercise) If demand is perfectly inelastic, wh at is the effect of a decrease in supply? Apply the formula and then graph the solution. 2.6 Trade Supply and demand offers one approach to un derstanding trade, and it represents the most important and powerful concept in the toolbox of economists. However, for some issues, especially those of international trad e, another related tool is very useful: the production possibilities frontier. Analysis us ing the production possibilities frontier was made famous by the “guns and butter” discu ssions of World War II. From an economic perspective, there is a tradeoff between guns and butter – if a society wants more guns, it must give up something, and one thing to give up is butter. That getting more guns might entail less butter often seems mysterious because butter, after all, is made with cows, and indirectly with land and hay. But the manufacture of butter also involves steel containers, tractors to turn the soil, transp ortation equipment, and labor, all of which either can be directly used (steel, labor) or require inputs that could be used (tractors, transportation) to manufacture guns. From a production standpoint, more guns entail less butter (or other things). 2.6.1 Production Possibilities Frontier Formally, the set of production possibilities is the collection of “feasible outputs” of an individual, group or society or country. You could spend your time cleaning your apartment, or you could study. The more of your time you devote to studying, the higher your grades will be, but the dirtier your apartment will be. This is illustrated, for a hypothetical student, in Figure 2-13. The production possibilities set embodies the fe asible alternatives. If you spend all your time studying, you would obtain a 4.0 (perfect) grade point average (GPA). Spending an hour cleaning reduces the GPA, but not by much; the second hour reduces by a bit more, and so on. The boundary of the production po ssibilities set is known as the production possibilities frontier This is the most important part of th e production possibilities set, because at


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-33any point strictly inside the production possibi lities set, it is possible to have more of everything, and usually we would choose to have more.11 The slope of the production possibilities frontier reflects opportunity co st, because it describes what must be given up in order to acquire more of a good. Thus to get a cleaner apartment, more time, or capital, or both, must be spent on cleani ng, which reduces the amount of other goods and services that can be had. For the two-good case in Figure 2-13, diverting time to cleaning reduces studying, which lowers the GPA. The slope dictates how much lost GPA there is for each unit of cleaning. Figure 2-13: The Production Po ssibilities Frontier One important feature of production possibiliti es frontiers is illustrated in the Figure 2-13: they are concave toward the origin. While this feature need not be universally true, it is a common feature, and there is a reason for it that we can see in the application. If you are only going to spend an hour studying, you spend that hour doing the most important studying that can be done in an hour, and thus get a lot of grades for the hour’s work. The second hour of studying produces less value than the first, and the third hour less than the second This is the principle of diminishing marginal returns Diminishing marginal returns are like pickin g apples. If you are only going to pick apples for a few minutes, you don’t need a la dder because the fruit is low on the tree; the more time spent, the fewer a pples per hour you will pick. 11 To be clear, we are considering an example with two goods, cleanliness and GPA. Generally there are lots of activities, like sleeping, eating, teeth-br ushing, and the production possibilities frontier encompasses all of these goods. Spending all your time sleeping, studying and cleaning would still represent a point on a three-dimensional frontier. Grades Cleanliness 4.0 2.0 3.0 1.0


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-34Consider two people, Ann and Bob, getting ready for a party. One is cutting up vegetables, the other is making hors d’oe uvres. Ann can cut up two ounces of vegetables per minute, or make one hors d’ oeuvre in a minute. Bob, somewhat inept with a knife, can cut up one ounce of vegeta bles per minute, or make two hors d’oeuvres per minute. Ann’s and Bob’s production possi bilities frontiers are illustrated in the Figure 2-14, given that they have an hour to work. Since Ann can produce two ounces of chopped vegetables in a minute, if she spends her entire hour on vegetables, she can produce 120 ounces. Similarly, if she devotes all her time to hors d’oeuvres, she produces 60 of them. The constant translation between the two means that her production po ssibilities frontier is a straig ht line, which is illustrated in the left side of Figure 2-14. Bob’s is the reverse – he produces 60 ounces of vegetables, or 120 hors d’oeuvres, or something on the line in between. Figure 2-14: Two Production Possibilities Frontiers For Ann, the opportunity cost of an ounce of ve getables is half of one hors d’oeuvre – to get one extra ounce of vegetable, she must spend 30 extra seconds on vegetables. Similarly, the cost of one hors d’oeuvres for Ann is two ounces of vegetables. Bob’s costs are the inverse of Ann – an ounce of vegetables costs him two hors d’oeuvres. What can Bob and Ann accomplish together? The important insight is that they should use the low cost person in the manufacture of each good, when po ssible. This means that if fewer than 120 ounces of vegetables will be made, Ann makes th em all. Similarly, if fewer than 120 hors d’oeuvres are made, Bob makes them all. This gives a joint production possibilities fronti er illustrated in the Figure 2-15. Together, they can make 180 of one and none of the other. If Bo b makes only hors d’oeuvres, and Ann makes only chopped vegetables, they will have 120 of each. With fewer than 120 ounces of vegetables, the opportunity cost of vegetables is Ann’s, and is thus half an hors d’oeuvre, but if more than 120 are needed, then the opportunity cost jumps to two. Now change the hypothetical slightly – suppose that Bob and Ann are putting on separate dinner parties, each of which will feature chopped vegetables and hors Ounces of vegetables 120 60 120 60 hors d’oeuvres Ounces of vegetables 120 60 120 60 hors d’oeuvres Ann’s Production Possibilities Frontier Bob’s Production Possibilities Frontier


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-35d’oeuvres in equal portions. By herself, A nn can only produce 40 ounces of vegetables and 40 hors d’oeuvres if she must produce equal portions. She accomplishes this by spending 20 minutes on vegetables and 40 mi nutes on hors d’oeuvres. Similarly, Bob can produce 40 of each, but using the reverse allocation of time. Figure 2-15: Joint PPF By working together, they can collectively have more of both goods. Ann specializes in producing vegetables, and Bob specializes in pr oducing hors d’oeuvres. This yields 120 units of each, which they can split equally, to have 60 of each. By specializing in the activity in which they have lower cost, Bob and Ann can jointly produce more of each good. Moreover, Bob and Ann can accomplish this by trading. At a “one for one” price, Bob can produce 120 hors d’oeuvres, and trade 60 of them for 60 ounces of vegetables. This is better than producing the vegetables hims elf, which netted him only 40 of each. Similarly, Ann produces 120 ounces of vegeta bles, and trades 60 of them for 60 hors d’oeuvres. This trading makes them both better off. The gains from specialization are potentially enormous. The grandfather of economics, Adam Smith, writes about specializa tion in the manufacture of pins: “…One man draws out the wire; another straights it; a third cuts it; a fourth points it; a fifth grinds it at the top for receiving the head; to make the head requires two or three distinct op erations ; to put it on is a peculiar business; to whiten the pins is another ; it is even a trade by itself to put them into the paper ; and the important business of making a pin is, in this 120 60 120 60 Ounces of vegetables hors d’oeuvres 180 180 Joint Production Possibilities Frontier


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-36manner, divided into about eighteen di stinct operations, which, in some manufactories, are all performed by distinct hands, though in others the same man will someti mes perform two or three of them.”12 Smith goes on to say that skilled individual s could produce at most twenty pins per day acting alone, but that with specialization, ten people can produce 48,000 pins per day, 240 times as many pins. (Exercise) The Manning Company has two factories, one that makes roof trusses, and one that makes cabinets. With m workers, the roof factory produces m trusses per day. With n workers, the cabinet plant produces n 5 The Manning Company has 400 worker s to use in the two factories. Graph the production possibi lities frontier. (Hint: Let T be the number of trusses produced. How many work ers are used making trusses?) (Exercise) Alarm & Tint, Inc., has 10 workers working a total of 400 hours per week. Tinting takes 2 hours per car. Alarm installation is complicated, however, and performing A alarm installations requires A2 hours of labor. Graph Alarm & Tint’s production po ssibilities frontier for a week. 2.6.2 Comparative and Absolute Advantage Ann produces chopped vegetables because her opportunity cost of producing vegetables, at of one hors d’oeuvre, is lower than Bob’s. A lower opportunity cost is said to create a comparative advantage That is, Ann gives up less to produce chopped vegetables than Bob, so in comparison to hors d’oeuvre s, she has an advantage in the production of vegetables. Since the cost of one good is the amount of another good foregone, a comparative advantage in one good implies a comparative disadvantage in another. If you are better at producing butter, you are necessarily worse at something else, and in particular the thing you give up less of to get more butter. To illustrate this point, let’s consider an other party planner. Charlie can produce one hors d’oeuvre, or one ounce of chopped ve getables, per minute. His production is strictly less than Ann’s, that is, his production possibilities frontier li es inside of Ann’s. However, he has a comparative advantage over Ann in the production of hors d’oeuvres, because he gives up only one ounce of vegeta bles to produce a hors d’oeuvres, while Ann must give up two ounces of vegetables. Thus, Ann and Charlie can still benefit from trade if Bob isn’t around. (Exercise) Graph the joint production possibilities frontier for Ann and Charlie, and show that collectively they can produce 80 of each if they need the same number of each product. Hint: First show that Ann will produce some of both goods, by showing that if Ann spec ializes, there are too many ounces of vegetables. Then show, if Ann devotes x minutes to hors d’oeuvres, that 60 + x = 2(60 – x ). 12 Adam Smith, “An Inquiry into the Nature and Causes of the Wealth of Nations,” originally published 1776, released by the Gutenberg project, 2002.


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-37 When one production possibilities frontier lies outside another, the larger is said to have an absolute advantage – more total things are possible. In this case, Ann has an absolute advantage over Charlie – she can, by herself, have more – but not over Bob. Bob has an absolute advantage over Char lie, too, but again, not over Ann. Diminishing marginal returns implies that the more of a good that a person produces, the higher is the cost (in terms of the good given up). That is to say, diminishing marginal returns means that supply curv es slope upward; the marginal cost of producing more is increasing in the amount produced. Trade permits specialization in activities in which one has a comparative advantage. Moreover, whenever opportunity costs differ, po tential gains from trade exist. If person 1 has an opportunity cost of c1 of producing good X (in terms of Y that is, for each unit of X that person 1 produces, person 1 gives up c1 units of Y ), and person 2 has an opportunity cost of c2, then there are gains from trade whenever c1 is not equal to c2 and neither party has specialized.13 Suppose c1 < c2. Then by having person 1 increase the production of X by c1 less of the good Y is produced. Let person 2 reduce the production of X by so that the production of X is the same. Then there is c2 units of Y made available, for a net increase of ( c2c1) The net changes are summarized in Table 2-2. Table 2-2: Construction of the Gains From Trade 1 2 Net Change Change in X + 0 Change in Y c1 c2 ( c2c1) Whenever opportunity costs differ, there are gains from re-allocating production from one producer to another, gains which are created by having th e low cost producers produce more, in exchange for greater production of the other good by the other producer, who is the low cost producer of this other good. An important aspect of this re-allocation is that it permits production of mo re of all goods. This means there is little ambiguity about whether it is a good thing to re-allocate production – it just means we have more of everything we want.14 How can we guide the reallocation of producti on to produce more goods and services? It turns out that under some circumstances, the price system does a superb job of creating efficient production. The price system posits a price for each good or service, and anyone can sell at the common price. The insight is that such a price induces efficient production. To see this, suppose we have a price p which is the number of units 13 If a party specialized in one product, it is a usefu l convention to say that the marginal cost of that product is now infinite, since no more can be produced. 14 If you are worried that more production means mo re pollution or other bad things, rest assured. Pollution is a bad, so we enter the negative of pollu tion (or environmental cleanl iness) as one of the goods we would like to have more of. The reallocation dict ated by differences in marginal costs produces more of all goods. Now with this said, we have no reason to believe that the reallocation will benefit everyone – there may be winners and losers.


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-38of Y one has to give to get a unit of X (Usually prices are in currency, but we can think of them as denominated in goods, too.) If I have a cost c of producing X which is the number of units of Y that I lose to obtain a unit of X I will find it worthwhile to sell X if p > c because the sale of a unit of X nets me p – c units of Y which I can either consume or resell for something else I want. Similarly, if c > p I would rather buy X (producing Y to pay for it). Either way, on ly producers with costs less than p will produce X and those with costs greater than p will purchase X paying for it with Y which they can produce more cheaply than its price. (The price of Y is 1/ p – that is the amount of X one must give to get a unit of Y .) Thus, a price system, with appropriate prices will guide the allocation of production to insure the low cost producers are the ones wh o produce, in the sense that there is no way of re-allocating production to obtain more goods and services. (Exercise) Using Manning’s production possibilities frontier in (Exercise), compute the marginal cost of trusses in terms of cabinets. (Exercise) Using Alarm & Tint’s production possibilities frontier in (Exercise), compute the marginal cost of alarms in terms of window tints. 2.6.3 Factors and Production Production possibilities frontiers provid e the basis for a rudimentary theory of international trade. To understand the theory, it is first necessary to consider that there are fixed and mobile factors. Factors of production are jargon for inputs to the production process. Labor is generally considered a fixed factor, because most countries don’t have borders wide open to immigrati on, although of course some labor moves across international borders. Temperature, we ather, and land are also fixed – Canada is a high-cost citrus grower because of its we ather. There are other endowments that could be exported, but are expensive to export because of transportation costs, including water and coal. Hydropower – electricity ge nerated from the movement of water – is cheap and abundant in the Pacific Northwest, and as a result, a lot of aluminum is smelted there, because aluminum smelting requir es lots of electricity. Electricity can be transported, but only with losses (higher costs), which gives other regions a disadvantage in the smelting of aluminum. Capital is generally considered a mobile factor, because plants can be built anywhere although investment is easier in some environments than in others. For example, reliable electricity and other inputs are necessary for most factories. Moreover, comparative advantage may arise from the presence of a functioning legal system, the en forcement of contracts, and the absence of bribery, because enforcement of contracts increases the return on investment by increasing the probability the economic retu rn to investment isn’t taken by others. Fixed factors of production give particular regions a comparative advantage in the production of some kinds of goods, and not in others. Europe, the United States and Japan have a relative abundance of highly skilled labor, and have a comparative advantage in goods requiring high skills, like computers, automobiles and electronics. Taiwan, South Korea, Singapore and Hong Kong have increased the available labor skills, and now manufacture more complicated goods like VCRs, computer parts and the like. Mexico has a relative abundance of middle-level skills, and a large number of


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-39assembly plants operate there, as well as clothing and shoe manufacturers. Lower skilled Chinese workers manufacture the majority of the world’s toys. The skill levels of China are rising rapidly. The basic model of international trade was fi rst described by Davi d Ricardo (1772-1823), and suggests that nations, responding to price incentives, will specialize in the production of goods in which they have a comparative advantage, and purchase the goods in which they have a comparative di sadvantage. In Ricardo’s description, England has a comparative advantage of manufacturing cloth, and Portugal in producing wine, leading to gains from trade from specialization. The Ricardian theory suggests that the Unit ed States, Canada, Australia and Argentina should export agricultural goods, especially grains that require a large land area for the value generated (they do). It suggests that complex technical goods should be produced in developed nations (they are) and that si mpler products and natural resources should be exported by the lesser developed nations (the y are). It also suggests that there should be more trade between developed and underd eveloped nations than between developed and other developed nations. The theory falter s on this prediction – the vast majority of trade is between developed nations. There is no consensus for the reasons for this, and politics plays a role – the North American Free Trade Act vastly increased the volume of trade between the United States and Mexico, for example, suggesting that trade barriers may account for some of the lack of trade between the developed and the underdeveloped world. Trade barriers don’t account for the volume of trade between similar nations, which the th eory suggests should be unne cessary. Developed nations sell each other mustard and tire s and cell phones, exchanging distinct varieties of goods they all produce. 2.6.4 International Trade The Ricardian theory emphasizes that the relative abundance of particular factors of production determines comparative advantag e in output, but there is more to the theory. When the United States exports a co mputer to Mexico, American labor, in the form of a physical product, has been sold abroad. When the United States exports soybeans to Japan, American land (or at le ast the use of American land for a time) has been exported to Japan. Similarly, when the United States buys car parts from Mexico, Mexican labor has been sold to the United States, and similarly when the Americans buy Japanese televisions, Japanese labor has been purchased. The goods that are traded internationally embody the factors of produc tion of the producing nations, and it is useful to think of international trade as directly trading th e inputs through the incorporation of inputs into products. If the set of traded goods is broad enough, the value of factors of production should be equalized through trade. The United States ha s a lot of land, relative to Japan, but by selling agricultural goods to Japan, it is as if Japan had more land, by way of access to US land. Similarly, by buying automobiles from Japan, it is as if a portion of the Japanese factories were present in the United States. With inexpe nsive transportation, the trade equalizes the values of factories in the United States and Japan, and also equalizes the value of agricultural land. One can reasonably think that soybeans are soybeans, wherever they are produced, and that trade in soybeans at a common price


McAfee: Introduction to Economic Analysis,, July 24, 2006 2-40forces the costs of the factors involved in producing soybeans to be equalized across the producing nations. The purchase of soybeans by Japanese drives up the value of American land, and drives down the value of Japanese land by giving an alternative to its output, leading toward equalization of the value of the land across the nations. This prediction, known as factor price equalization, of modern international trade theory was first developed by Paul Samu elson (1915 – ) and generalized by Eli Heckscher (1879 – 1952) and Bertil Ohlin (18 99 – 1979). It has powerful predictions, including the equalization of wages of equally skilled people after free trade between the United States and Mexico. Thus, free trade in physical goods should equalize the price of haircuts, and land, and economic consulting, in Mexico City and New York. Equalization of wages is a direct consequenc e of factor price equalization because labor is a factor of production. If economic co nsulting is cheap in Mexico, trade in goods embodying economic consulting – boring repo rts, perhaps – will bid up the wages in the low wage area, and reduce the quantity in the high wage area. An even stronger prediction of the theory is that the price of water in New Mexico should be the same as in Minnesota. If wa ter is cheaper in Minnesota, trade in goods that heavily use water – e.g. paper – will tend to bid up the value of Minnesota water, while reducing the premium on scarce New Mexico water. It is fair to say that if factor price equaliza tion works fully in practice, it works very, very slowly. Differences in taxes, tariffs and othe r distortions make it a challenge to test the theory across nations. On the other hand, wi thin the United States, where we have full factor mobility and product mobility, we still have different factor prices – electricity is cheaper in the Pacific Northwest. Neverthe less, nations with a relative abundance of capital and skilled labor export goods that us e these intensively, na tions with a relative abundance of land export land-intensive goods like food, nations with a relative abundance of natural resources export these resources, and nations with an abundance of low-skilled labor export goods that make intensive use of this labor. The reduction of trade barriers between such nations works like Ann and Bob’s joint production of party platters: by specializing in the goods in wh ich they have a comparative advantage, there is more for all.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-413 The US Economy An important aspect of economics is econom ic statistics, and an army of economists collect and analyze these statistics. This ch apter presents an overview of the economic activity of the United States. How much do yo u need to know about these statistics? It would be ridiculous to memorize them. At th e same time, it would be undesirable to be ignorant of how we are ch anging, and how we are not.15 3.1.1 Basic Demographics There are about three hundred million people in the United States, up from 76 million in 1900. 0 50,000 100,000 150,000 200,000 250,000 300,000 350,0001900 190 6 1912 191 8 1924 1 9 3 0 1936 1 9 4 2 1948 195 4 1 960 1966 1 972 1978 1984 199 0 1996 200 2YearPopulation Figure 3-1: US Resident Population During the last century, the US population has become primarily an urban population, growing from 40% to 80% urban. The population is primarily white, with 12-13% African-American and 4% classified as other. These proportions are relatively stable over the century, with the white population falling from 89% to 83%. The census is thought to understate minority populations be cause of greater difficulties in contacting minorities. The census does not attempt to classify people but instead accepts people’s description of their own race. 15 I apologize to those using the book in foreign countries; this chapter is about the US not because it is more important but because I know it better. Encourage your professor to write a chapter on your country! All of the statistics in this chapter come from Fedstats, from FRED, and from the NBER,


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-42 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 190019101920193019401950 19601970198019902000Percent Urban White Figure 3-2: US Urban and White Population The United States population has been agin g significantly, with the proportion of seniors (over 65 years old) tripling over the past century, and the proportion of young people dropping by over a third. Indeed, the proportion of children between zero and five years old has dropped from 12.1 % of the population to under 7%. 0.0 10.0 20.0 30.0 40.0 50.0 60.0 19001910192019301940195019601970198019902000Percent 65+ 0-24 25-44 45-64 Figure 3-3: Population Proportions by Age Group The baby boom – a dramatic increase in birt hs for the years 1946-1964, is visible in the Figure 3-3 as the population in the 0-24 age group begins increasing in 1950, peaking in 1970 and then declining significantly as th e baby boom moves into the 25-44 year old


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-43bracket. There is a slight “echo” of the baby boom, most readily seen by looking at the 0-5 age bracket, as in Figure 3-4. 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.01 90 0 1910 1 92 0 1930 194 0 1 95 0 196 0 1970 1980 1 99 0 2000 200 2Percent Figure 3-4: Proportion of Population under Age Five The aging of the American population is a co nsequence of greater life expectancy. When social security was created in 1935, the aver age American male lived to be slightly less than sixty years old. The social security benefits, which didn’t start until age 65, thus were not being paid to a substantial portion of the population. Figure 3-5 shows life expectancy at birth, thus including infant mortality. The significant drop in life expect ancy in 1918 – to nearly 30 years old for non-whites – is primarily a consequence of the great influenz a, which killed about 2.5% of the people who contracted it and killed more Americans in 1918 than did World War I. The Great Depression (1932-39) also reduced life ex pectancy. The steady increase in life expectancy is also visible, with white fe males now living eighty years on average.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-44 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0 75.0 80.01900 1908 1916 1924 1932 1940 1948 1956 1964 1972 1980 1988 1996 Years White Male White Female Nonwhite Male Nonwhite Female Figure 3-5: US Life Expectancy at Birth It is said that the United States is a coun try of immigrants, and a large fraction of the population had ancestors that came from el sewhere. Immigration into this United States, however, has been increasing after a long decline, and the fraction of the population that were born in foreign countries is about 11% -one in nine. 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 19001910192019301940195019601970198019902000Population Percent Europe Asia Latin Am Total Figure 3-6: US Immigrant Population, in Percent, by Continent of Origin The majority of immigrants during this century came from Europe, but immigration from Europe has been declining for most of the century, while immigration from Asia and Latin America has grown substantially. Figure 3-7 aggregates the total country of origin data over the century, to iden tify the major sources of immigrants.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-45 Germany, 4.6% Ireland, 1.9% Italy 9.4% Norway 0.8% Sweden 1.1% Soviet Union 6.7% UK 4.8% China, 2.2% Canada, 7.3% Mexico, 13.1% Caribbean 7.6% Africa 1.5% Other, 39.0% Figure 3-7: National Origin of Immigrants, 1900-2000 One hears a lot about divorce rates in the Un ited States, with statements like “fifty percent of all marriages end in divorce.” Al though it has grown, the divorced population is actually a small fraction of the population in the United States. 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 19001910192019301940195019601970198019902000Percent Never Married Widowed Divorced Figure 3-8: Male Marital Status (Percentages)


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-46 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 19001910192019301940195019601970198019902000Percent Never Married Widowed Divorced Figure 3-9: Female Marital Status (Percent) Marriage rates have fallen, but primarily because the “never married” category has grown. Some of the “never married” probab ly represent unmarried couples, since the proportion of children from unmarried wome n has risen fairly dramatically. Even so, marriage rates are greater than they were a century ago. However, a century ago there were more unrecorded and common-law marriages than probably there are today. 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.01940 1943 1946 1949 1952 1955 1958 1961 1964 1967 1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 Figure 3-10: Percent of Births to Unwed Mothers While we are on the subject, however, the mu ch discussed crisis in teen-age pregnancy doesn’t appear like such a crisis when viewed in terms of the proportion of all births that involve a teenage mother.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-47 0% 5% 10% 15% 20% 25%1940 1943 1946 1949 1952 1955 1958 1961 1964 1967 1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 Figure 3-11: Percent of Births to Women Age 19 or less 3.1.2 Education Why are the western nations rich, and many other nations poor? What creates the wealth of the developed nations? Modern economic analysis attributes much of the growth of the United States and other developed nations to its educated workforce, and not to natural resources. Japan, with a relati ve scarcity of natural resources but a highly educated workforce, is substantially richer than Brazil, with its abundance of natural resources. 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0191 0 191 7 192 4 193 1 193 8 194 5 1952 1959 1966 197 3 198 0 1 98 7 1 99 4 200 1 <5 >12 >16 Figure 3-12: Educational Attainment in Years (Percent of Population) Just less than 85% of the US population comp letes 12 years of schooling, not counting kindergarten. Not all of these students gradu ate from high school, but they spent twelve years in school. The proportion that comple tes only five or fewer years of elementary school has dropped from about a quarter of th e population to a steady 1.6%. At least


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-48four years of university now represents a bit more than a quarter of the population, which is a dramatic increase. Slightly fe wer women (25% versus 28%) complete four years of university, although women are more likely to complete four years of high school. Graduation rates are somewhat below the numb er of years completed, so that slightly less than three-quarters of the US population actually obtain their high school degree. Of those obtaining a high school degree, nearly half obtain a university or college degree. 0 10 20 30 40 50 60 70 80 901900 1907 1914 1921 1928 1935 1942 1949 1956 1963 1970 1977 1984 1991 1998 HS U Figure 3-13: Graduation Rates There are several interesting things to see in Figure 3-13. First, high school completion dropped significantly during World War II (194 0-45) but rebounded after. Second, after World War II, college graduation spiked, as many US soldiers were sent to university by the government under a program called the “GI Bill.”16 As the numbers of high school students rose the portion of high school graduates going to university fell, meaning a larger segmen t of the population became high school educated. This increase repres ents the creation of the US middle class; previously, high school completion and university were in large part a sign of wealth. The creation of a large segment of the population who graduat ed from high school, but didn’t attend university, led to a population with substant ial skills and abilities, but no inherited wealth, who became the middle class. High school completion has been declining for thirty years. This is surprising given the high rate of financial return to education in the United States. Much of the reduction in completion can be attributed to an increase in General Education Development, or GED, certification, which is a program that grants diplomas (often erroneously thought to be a 16 The etymology of GI, as slang for US soldiers, is disputed, with candidates including “Government Issue,” “General Infantry” and “Galva nized Iron,” the latter a reference to trash cans that looked like German World War I artillery.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-49“General Equivalent Degree”) after successfu lly passing examinations in five subject areas. Unfortunately, those people who obtain GED certification are not as successful as high school graduates, even marginal graduat es, and indeed the GED certification does not seem to help students succeed, in comparison with high school graduation.17 3.1.3 Households and Consumption There are approximately one hundred million households in the United States. The number of residents per household has consis tently shrunk during this century, from over four to under three. 0 1 2 3 4 5 19001910192019301940195019601970198019902000 Figure 3-14: Household Occupancy The shrinking size of households reflects a re duction not just in birthrates but also an increase in the number of people living alon e. More women live alone than men, even though four times as many families with a single adult member are headed by women. This discrepancy – many more women both livi ng on their own and li ving with children and no partner, even though there are abou t the same number of men and women born – is accounted for by the greater female longevity already noted above. 17 In performing this kind of analysis, economists are very concerned with adjusting for the type of person. Smarter people are more likely to graduate from hi gh school, but one doesn’t automatically become smart by attending high school. Thus, care has been taken to hold constant i nnate abilities, measured by various measures like IQ scores and performa nce on tests, so that the comparis on is between similar individuals, some of whom persevere to finish school, some of wh o don’t. Indeed, some st udies use iden tical twins.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-50 0 10 20 30 40 50 60 70 80 90 1001947 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2001 2002 Family Married Single Male Single Female Figure 3-15: Proportion of Households by Type Where do we live? About 60% of households live in single family detached homes, meaning houses that stand alon e. Another 5% or so live in single family attached houses, such as “row houses.” Slightly ov er 7 % live in mobile homes or trailers, and the remainder live in multi-unit housing, including apartments and duplexes. Twothirds of American families own their own homes, up from 43% in 1940. Slightly less than half a percent of the population is incar cerated in state and federal prisons. This represents a four-fold increase over 1925-1975. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51925 1 92 9 1933 1 93 7 1941 1 94 5 1949 1 95 3 19 5 7 1 96 1 1965 1 96 9 1973 1 97 7 1981 1985 19 8 9 1 99 3 1997 Figure 3-16: Percentage of Incarcerated Residents Ten percent of households do not have an au tomobile, and 97.6% have a telephone. Socalled “land line” telephones may start to fall as apartment dwellers, especially students, begin to rely exclusively on cell phones. Ju st under 99% of households have complete plumbing facilities (running water, bath or shower, flush toilet), up from 54.7% in 1940. How much income do these households make? What is the distribution of income? One way of assessing the distribution is to use quintiles to measure dispersion. A quintile (or fifth) is a group of size 20%. Thus the to p income quintile represents the top 20% of


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-51income earners, the next represents thos e ranking 60%-80%, and so on. Figure 3-17 shows the earnings of the top, middle and bottom quintiles. 0 10 20 30 40 50 601 9 4 7 1 9 5 4 1 9 6 1 1 9 6 8 197 5 1 9 8 2 1 9 8 9 1 9 9 6 Lowest 5th Middle 5th Highest 5th Figure 3-17: Income Shares for Three Quintiles 0 20,000 40,000 60,000 80,000 100,000 120,000 140,000 160,000 180,00019 47 1951 1955 1959 19 63 19 67 1971 1975 1979 19 83 19 87 1991 1995 1999 95% 80% 60% 40% 20% Figure 3-18: Family Income The earnings of the top quintile fell slightly until the late 1960s, when it began to rise. All other quintiles lost income share to the top quintile starting in the middle 1980s. Figures like these suggest that families are getting poorer, except for an elite few. However, in fact families are getting rich er, just not as fast as the top quintile. Figure 3-18 shows the income, adjusted for inflation to be in 2001 dollars, for families at various points in the income spectrum. For example, the 60% line indicates families for whom 40% of the families have higher inco me, and 60% have lower income. Incomes of


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-52all groups has risen, although the richer famili es have seen their incomes rise faster than poorer families. That is readily seen when percentage changes are plotted in Figure 3-19. -50 0 50 100 150 200 2501950 1953 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 20% 40% 60% 80% 95% Figure 3-19: Family Income, Cumulative Percentage Change Real income gains in percentage terms have been larger for richer groups, even though the poor have also seen substantially increased incomes. If the poor have fared less well than the ri ch in percentage term s, how have AfricanAmericans fared? After World War II, Africa n-Americans families earned about 50% of white family income. This has risen gradua lly, noticeably in the 1960s after the 1964 Civil Rights Act that is credited with inte grating workplaces throughout the southern United States. African-American family income lagged white income growth through the 1980s, but has been rising again.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-53 0% 10% 20% 30% 40% 50% 60% 70%1 9 47 1950 1953 1956 1959 1962 1965 1 9 68 1971 1 9 74 1977 1980 1983 1986 1989 1992 1995 1998 Figure 3-20: Black Family Income as a Percentage of White Income These income measures attemp t to actually measure purchasi ng power, and thus adjust for inflation. How much will $1 buy? This is a complicated question, because changes in prices aren’t uniform – some goods get re latively cheaper, while others become more expensive, and the overall cost of living is a challenge to calculate. The price index typically used is the consumer price index (CPI ), which adjusts for what it costs to buy a “standard” bundle of food, clothing, housin g, electricity and other items. Figure 3-21 shows the CPI over most of past century, where 1982 is set as the reference year. 0 20 40 60 80 100 120 140 160 180 2001913 1918 1923 1928 1 9 33 1 9 38 19 43 19 4 8 1953 1958 1963 1968 1973 1978 1983 1 9 88 1993 1 9 98 Figure 3-21: Consumer Price Index (1982 = 100) (Exercise) Have prices actually risen? Economists generally agree that the meaning of “prices have risen” is that you would prefer past prices to current prices. What makes this challenging is th at the set of available products change over time. Cars have gone up significantl y in price, but are also more reliable.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-54Would you be better off with your current income in 1913 than today? You would be very rich with current averag e income in 1913, but not have access to modern medicine, television, electronic s, refrigeration, highways, and many other technologies. If you made $40,000 annually in 1913, how would you live and what would you buy? (Do some research.) (Exercise) Compare a $40,000 income in 1980 to the present. What differences are there in available products? In the quality of products? How rich does $40,000 make you in each time period? In which period would you choose to live, and why? There have been three major inflations in the past century. Both World War I and World War II, with a large portion of the good s and services diverted to military use, saw significant inflations. In addition, th ere was a substantial inflation during the 1970s, after the Vietnam War in the 1960s. The price level fell during the Great Depression (1929-39). Falling price leve ls create investment problems, because inflation adjusted interest rates, which must ad just for a deflation, are forced to be high, since unadjusted interest rates cannot be negative. -15 -10 -5 0 5 10 15 2019 14 1919 19 24 1929 1934 19 39 1944 19 49 1954 19 59 1964 1969 19 74 1979 1984 19 89 1994 19 99 Figure 3-22: CPI Percent Changes The cost of food has fallen quite dramat ically over the past century. Figure 3-23 shows that the percentage of pre-tax household in come spent on food has fallen from 25% to about 10%. This fall is a reflection of greater incomes, and of the fact that the real cost of food has fallen.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-55 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.019 29 19 34 1939 19 44 19 49 1954 1 959 1964 19 69 19 74 1979 19 84 19 89 1994 1 999 Food Eating Out Figure 3-23: Food Expenditure as Percent of Income, and Proportion Spent Out Moreover, a much greater fraction of expendit ures on food are spent away from home, a fraction that has risen from under 15% to 40%. How do we spend our income? The major categories are food, clothing, housing, medical, household operation, transportati on, and recreation. The percentage of disposable income spent on these categories are shown, fo r the years 1929, 1965 and 2001, in Figure 3-24. 0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0%Foo d Cl ot h i ng H ous i ng H ous ehol d M edi c al T r ansportati on R ec r eat i on 1929 1965 2001 Figure 3-24: After Tax Income Shares Food has shrunk substantially, but we enjoy more recreation, and spend a lot more staying healthy. (Food is larger than in Figure 3-23 because these figures use after tax


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-56disposable income, rather than pre-tax income .) This is in part a consequence of our aging population, but also of the increased technology available. 3.1.4 Production We learned something about where we live and what we buy. Where do we get our income? Primarily we earn by providing go ods and services. Nationally, we produce about eleven trillion dollars worth of good s and services. Broadly speaking, we spend that eleven trillion on personal consum ption of goods and services, savings, and government. This, by the way, is often expressed as Y = C + I + G, which states that income (Y) is spent on cons umption (C), investment (which comes from savings) and government. One can consume imports as well, so the short-term constraint looks like Y + M = C + I + G + X, where M is imports and X is exports. How much does the United States produce? Economists measur e output with the gross domestic product (GDP), which is the value of traded goods and services produced within the borders of the United States. GDP measures what is produced within the United States, and thus excludes output of Japanese factories owned by Americans, but includes the output of US factories owned by Japanese. Importantly, GDP excludes non-traded goods and services. Thus, unpaid housework is not included. If you clean your own home, and your neighbor cleans his or her home, the cleaning does not contribute to GDP. On the other hand, if you and your neighbor pay each other to clean each other’s homes, GDP goes up by the pa yments, even though the actual production of goods and services remains unchanged. Thus, GDP does not measure our total output as a nation, because it neglects unpaid services. Why does it neglect unpaid services? Primarily, because we can’t readily measure them. Data on transactions is generated by tax informat ion and reporting requirements imposed on businesses. For the same reason, GDP neglects illegal activities as well, such as illegal drug sales and pirated musi c sales. Thus, GDP is not a perfect measure of the production of our society. It is just the best measure we have. Figure 3-25 shows the growth in GDP, and its components of personal consumption, government expenditures, and investment. The figures are expressed in constant 1996 dollars, that is, adjusted for inflation. The figure for government includes the government’s purchases of goods and services – weapons, highways, rockets, pencils – but does not include transfer payments like social security and welfare programs. Transfer payments are excluded from this calculation because the actual dollars are spent by the recipient, not by the government. The cost of making the transfer payments (e.g. printing and mailing the checks), however, is included in the cost of government.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-57 0.0 1,000.0 2,000.0 3,000.0 4,000.0 5,000.0 6,000.0 7,000.0 8,000.0 9,000.0 10,000.01929 1934 1939 1944 1949 1954 1959 1964 1969 1974 1979 1984 1989 1994 1999 GDP Consumption Investment Government Figure 3-25: Output, Consumption, Investment and Government It is often visually challenging to draw useful information from graphs like Figure 3-25, because economic activity is growing at a constant percentage. Consequently, economists often use a logarithmic scale, rath er than a dollar scale. A logarithmic scale has the useful property that a straight line gives constant percentage growth. Consider a variable X that takes on values xt at time t Define %x to be the percentage change: 1 1% t t tx x x x. Then x x x x x xt t t t t % 1 log ) log( log ) log( ) log(1 1 1 Thus, if the percentage change is constant over time, log( xt) will be a straight line over time. Moreover, for small percentage changes: x x % % 1 log so that the slope is approximately the growth rate.18 Figure 3-26 shows these statistics with a logarithmic scale. 18 The meaning of throughout this book is ‘to the first order.’ Here that means 0 % % % 1 log lim0 % x x xx. Moreover, in this case the errors of the approximation are modest up to about 25% changes.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-58 1.0 10.0 100.0 1,000.0 10,000.01929 1934 1939 1944 1949 1954 1959 1964 1969 1974 1979 1984 1989 1994 1999 GDP Consumption Investment Government Figure 3-26: Major GDP Components in Log Scale Immediately noticeable is the approximately constant growth rate from 1950 to the present, because a straight line with a log scale represents a constant growth rate. In addition, government has grown much more slowly (although recall that transfer payments, another aspect of government, aren’t shown). A third feature is the volatility of investment – it shows much greater chan ges than output and consumption. Indeed, during the great depression (1929-39), in come fell somewhat, consumption fell less, government was approximately flat, and invest ment plunged to 10% of its former level. 0 5,000 10,000 15,000 20,000 25,000 30,000 35,000 19301940195019601970198019902000 Consumption GDP Figure 3-27: Per Capita Income and Consumption Some of the growth in the American economy has arisen because there are more of us. Double the number of people, and consume tw ice as many goods, and individually we aren’t better off. How much are we pr oducing per capita, and how much are we consuming?


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-59 US output of goods and services, and consum ption, have grown substantially over the past 75 years. In addition, consumption has b een a steady percentage of income. This is more clearly visible when income shares are plotted in Figure 3-28. -10.0% 0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 70.0% 80.0% 90.0%192 9 193 4 1939 1944 1 9 4 9 1954 1959 1964 1969 1974 1979 1984 1989 1994 1999 C/Y I/Y (X-M)/Y G/Y Figure 3-28: Consumption, Investment and Government (% GDP) Consumption was a very high portion of inco me during the Great Depression (1929-39), because income itself fell. Little investment took place. The wartime economy of World War II reduced consumption to below 50% of output, with government spending a similar fraction as home consumers. Othe rwise, consumption has been a relatively stable 60-70% of income, rising modestly duri ng the past twenty years, as the share of government shrank, and net imports grew. Ne t imports rose to 4% of GDP in 2001. The most basic output of our economic sy stem is food, and the US economy does a remarkable job producing food. The US has about 941 million acres under cultivation to produce food, which represents 41 % of the surface area of the United States. Land use for agriculture peaked in 1952, at 1,20 6 million acres, and has been dwindling ever since, especially in the northeast where fa rms are being returned to forest through disuse. Figure 3-29 shows the output of agricultural products in the United States, adjusting to 1982 prices.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-60 0 50 100 150 200 2501913 1918 1923 1928 1933 1938 1943 1948 1953 1958 1963 1968 1973 1978 1983 1988 1993 1998 Figure 3-29: US Agricultural Output, 1982 constant dollars The growth in output is more pronounced when viewed per worker involved in agriculture. 1 10 100 10001947 1951 1955 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 Per Worker Total Figure 3-30: Agricultural Output, Total and Per Worker (1982 $, Log Scale) Where do we work? Economists divide prod uction into goods and services. Goods are historically divided into mi ning, construction and manufacturing. Mining includes production of raw materials of all kinds, including metals, oil, bauxite and gypsum. Construction involves production of ho using and business space. Manufacturing involves the production of everything from co mputers to those little chef’s hats that are placed on turkey legs. Figure 3-31 describes the major sectors of the US economy. Because the data come from firms, agricultur e is excluded, although goods and services provided to farms would be included.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-61 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0Mi n ing C o nst r u ct i o n Manufacturin g T & P U Wholesale Ret a il F I R E Other Servi ce s G o vernment 1940 1970 2000 Figure 3-31: Major Non-Agricultural Sectors of US Economy, percent of GDP Mining has diminished as a major factor in the US economy, a consequence of the growth of other sectors, and the reduction in the prices for raw materials. Contrary to many popular predictions, the prices of ra w materials have fallen even as output and population have grown. We wi ll see later in this book that the fall in prices of raw materials – ostensibly in fi xed supply given the limited cap acity of the earth – means that people expect a relative future ab undance, either because of technological improvements in thei r use or because of large as yet undiscovered pools of the resources. An example of technological impr ovements is the substi tution of fiber optic cable for copper wires. An enormous am ount of copper has been recovered from telephone lines, and we can have more te lephone lines and use less copper than was used in the past. Manufacturing has become less important, fo r several reasons. Many manufactured goods cost less, pulling down the overall value. In addition, we import more manufactured goods than in the past. We produce more services. T& PU stands for transportation and public utilities, and incl udes electricity and telephone services and transportation including rail and air travel. This sector has shrunk as a portion of the entire economy, although the components have grown in abso lute terms. We take more airplane trips than we have historically.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-62 0 0.5 1 1.5 2 2.51928 193 2 1 9 36 1 9 40 1 9 44 194 8 1 9 52 1 9 56 1 9 60 1 9 64 1968 197 2 1 9 76 1 9 80 1 9 84 1 9 88 1992 199 6 2 0 00 Figure 3-32: Air Travel Per Capita Electricity production has risen dramatically. 0 500,000 1,000,000 1,500,000 2,000,000 2,500,000 3,000,000 3,500,000 190019201940196019802000 Figure 3-33: Electricity Production (M kwh) However, energy use more ge nerally has not grown as much just doubling over the post-war period.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-63 0 10 20 30 40 50 60 70 801949 1954 1959 1964 1969 1974 1979 1984 1989 1994 1999 Figure 3-34: Energy Use (Quadrillion BTUs) The number of automobiles per capita in the United States peaked in the early 1980s, which looks like a reduction in transportation since then. However, we still drive more than ever, suggesting the change is actually an increase in the reliability of automobiles. 0 2000 4000 6000 8000 10000 120001900 1910 1920 1930 1940 1950 1960 1970 1980 1990 20000.0 100.0 200.0 300.0 400.0 500.0 600.0 Miles Cars Figure 3-35: Cars Per Thousand Population and Miles Driven Per Capita The cost of selling goods – wholesale and retail costs – remains relatively stable, as does “FIRE,” which stands for finance, insurance, and real estate costs. Other services, ranging from restaurants to computer tutoring have grown substantially. This is the socalled “service economy” that used to be in th e news frequently, but is less so these days.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-64A bit more than 60% of the population works. The larger numbers in recent years are partially a reflection of the baby boom’s entry into working years, reducing the proportion of elderly and children in American society. However, it is partially a reflection of an increased propensity for households to have two income earners. 48.0 50.0 52.0 54.0 56.0 58.0 60.0 62.0 64.0 66.01940 1943 1946 1948 1951 1954 1957 1960 1963 1966 1969 1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 Figure 3-36: Percentage of Population Employ ed (Military & Prisoners Excluded) 0 10 20 30 40 50 60 70 190019201940196019802000 All Married Figure 3-37: Labor Force Participation Rates, All Women and Married Women Female participation in the labor force has rise n quite dramatically in the United States. Figure 3-37 shows female labor force participation. The overall participation rate has roughly tripled during the century, and signif icantly exceed the rate prevailing during World War II, when many women went to work In addition, participation of married women has now risen above the level for unma rried women. The participation rate for single women is even higher, currently at 68% it is higher than the overall average participation rate of all residents. The di fference is primarily elderly women, who are


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-65disproportionately more likely to be widowed rather than married or single, and who are less likely to be working. Another sector of the economy which has been much in the news is national defense. How much do we spend on the military? In this century, the large expenditure occurred during World War II, when about 50% of GD P was spent by the government, and 37% of GDP went to the armed forces. During the Korean War, we spent about 15% of GDP on military goods, and less than 10% of GDP duri ng the Vietnam war. The military buildup during Ronald Reagan’s presidency (19801988) increased our military expenditures from about 5% to 6% of GDP – a large perc entage change in military expenditures, but a small diversion of GDP. The fall of the Soviet Union led the United States to reduce military expenditures, in what was called the “peace dividend,” but again the effects were modest. 0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0% 19301940195019601970198019902000 Figure 3-38: Defense as a Percentage of GDP Historically, defense represents the largest expenditure by the federal government. However, as we see, defense has become a mu ch smaller part of the economy overall. However, the federal government plays many other roles in the modern US economy. 3.1.5 Government With a budget over two trillion dollars, th e federal government represents just under 20% of the US economy. It is one of the largest organization s in the world; only nations are larger organizations, and only a handful of nations are larger. The size of the federal government, as a percentage of GDP, is shown in Figure 3-39. Federal expenditures boomed during World War II (1940-45), but shrank back to nearly pre-war levels shortly afterwar d, with much of the differenc e attributable to veterans’ benefits and continuing international in volvement. Federal expenditures, as a percentage of GDP, continue to grow until Ronald Reagan’s presidency in 1980, when they begin to shrink slightly af ter an initial growth. Figure 3-39 also shows federal revenues, and the deficit – the difference between expenditures and revenues, is apparent, especially for World War II and 1970-1998.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-66 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.01930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 Revenue Expenditures Figure 3-39: Federal Expenditures and Revenues (Percent of GDP) Much has been written about the federal gove rnment’s “abdication” of various services, which are pushed onto state and local government Usually this behavior is attributed to the Reagan presidency (1980-88). There is so me evidence of this behavior in the postwar data, but the effect is very modest and long term. Most of the growth in state and local government occurred between 1947 and 1970, well before the Reagan presidency; state and local government has been stable since then. Moreover, the expenditure of the federal government, which shows ups and downs, has also been fairly stable. In any event, such effects are modest overall. 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.01947 1951 1955 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 Total Federal Regional Figure 3-40: Federal, State & Local and Total Government Receipts (% GDP) Figure 3-40 sets out the taxation at both the federal and state and local (merged to be regional) level. Figure 3-41 shows expenditures of the same entities. Both figures are


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-67stated as a percentage of GDP. State and local taxation and expenditures doubled over the postwar period. The two figures are very similar. The federal government’s expenditures have varied more significantly than its revenues. 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.01947 1951 1955 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 Total Federal Regional Figure 3-41: Federal, Regional and Total Ex penditures as a Percent of GDP A peculiarity of the US federal government is a penchant for “off-budget” expenditures. Originally, such off-budget items involv ed corporations like Intelsat (which commercialized satellite technology) and RCA (the Radio Corporation of America, which commercialized radio) and other semi-auton omous and self-sustaining operations. Over time, however, off-budget items became a way of hiding the growth of government, through a process that became kn own as “smoke and mirrors.” 0.0 5.0 10.0 15.0 20.0 25.01947 1951 1955 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 Budget Off-Budget Figure 3-42: Federal Expenditures, On and Off Budget, Percent of GDP


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-68During the 1980s, the public became aware of off-budget items. Political awareness made off-budget items cease to work as a device for evading balanced-budget requirements, and few new ones were created, although they continue to be debated. Sporadically there are attempts to push social security off-budget. 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000 20,00019 6 2 1 9 65 1968 19 7 1 1 9 7 4 1977 19 8 0 1 9 8 3 1 9 86 19 8 9 1 9 9 2 1995 1998 20 0 1 Executive Military Federal Regional Figure 3-43: Federal and Regional Government Employment (000s) Federal employees includes two major categor ies, uniformed military personnel and the executive branch. State and Local government is much larger and has tripled in size since 1962. The biggest growth areas involve public school teachers, police, corrections (prisons) and hospitals. About 850,000 of the federal employees work for the postal service. 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.01940 1944 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 Defense Transfers To Regional Other Grants Interest Other Figure 3-44: Major Expenditures of the Federal Government


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-69 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00%1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 Social Security Medical Welfare Unemployment Veterans Figure 3-45: Major Transfer Payments (% of Federal Budget) Transfers to individuals represent almost 50 % of federal expenditures. These transfers are direct payments to individu als, in the form of a check. Such transfers include social security, Medicare, Medicaid, unemployment insurance, and veteran’s benefits. Transfers to state and local governments are listed as regional. “Other grants” also involve sending checks, usually with strings attached. The growth in social security during the 1950s and 1960s is primarily a consequence of increasing benefit levels. The growth in Medicare and Medicaid payments over the period 1970-90, in contrast, is primarily a consequence of increased costs of existing programs rather than increases in benefit levels. 1 10 100 1,000 10,000 100,000 1,000,0001937 1943 1949 1955 1961 1967 1973 1979 1985 1991 1997 Revenue Expenditure Figure 3-46: Social Security Revenue and Expenditure, $ millions


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-70 A question you may ask, quite reasonably, is whether the social security program can survive to the time when you retire. A co mmon misunderstanding about social security is that it is an investment program – that the taxes individuals paid in are invested and returned at retirement. As Figure 3-46 makes clear, for most of its existence the social security program has paid out approximately what it took in. The social security administration has b een ostensibly investing money and has a current value of approximately 1.5 trillion do llars, which is a bit less than four times the current annual expenditure on social security Unfortunately, this money is “invested” in the federal government, and thus is an obligation of the federal government, as opposed to an investment in the stock mark et. Consequently, from the perspective of someone hoping to retire in, say, 2050, this investment isn’t much comfort, since the investment won’t make it easier for the fede ral government to make the social security payments. The good news is that the government can prin t money. The bad news is that when it prints a lot of it, and the obliga tions of the social security administration are in the tens of trillions of do llars, it isn’t worth very much. 0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0194 0 19 4 5 19 5 0 1 95 5 1960 19 6 5 19 7 0 197 5 19 8 0 19 8 5 19 9 0 199 5 20 0 00 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 % GDP Debt ($B) Figure 3-47: Federal Debt, Total and Percent of GDP The federal government runs deficits, spending more than it earned. In most of the past 75 years we see from Figure 3-39 that the government runs a deficit, bringing in less than it spends. Interest has been as high as 15% of the cost of the federal government (see Figure 3-44). How large is the debt, and how serious is it? Figure 3-47 gives the size of the federal debt, in absolute dollars and as a percent of GDP. The debt was increased dramatically during World War II (1 940-45), but over the following 25 years, little was added to it, so that as a po rtion of growing GDP, the debt fell. Starting in the late 1970s, the US began accu mulating debt faster than we were growing, and the debt began to rise. That trend wasn’t stabilized until the 1990s, and then only because the economy grew at an extraordinar y rate by historical standards. The


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-71expenditures following the September 11, 2001 terrorist attacks, combined with a recession in the economy, have sent the debt rising again. The national debt isn’t out of control. At current 4% interest rates on federal borrowing, we spend about 2 % of GDP on interest servicing the federal debt. The right evaluation of the debt is as a percentage of GDP; viewed as a percentage, the size of the debt is of moderate size – serious but not cr itical. The serious side of the debt is the coming retirement of the baby boom genera tion, which is likely to put additional pressure on the government. An important distinction in many economic activities is the distinction between a stock and a flow Your bank account represents a st ock of money, the expenditures and income a flow. The national debt is a stock; the deficit is the addition to the debt and is a flow. If you think about a lake with inco ming water and evaporation, the amount of water in the lake is the stock of water, the incoming stream minus evaporation the flow. Table 3-1: Expenditures on Agencies as Percent of Non-Transfer Expenditures Department or Agency 1977 1990 2002 Legislative 0.4 0.4 0.5 Judiciary 0.2 0.3 0.6 Agriculture 2.1 2.2 2.7 Commerce 3.2 0.7 0.7 Education 3.9 3.8 6.7 Energy 3.1 3.2 2.9 Health 3.7 4.6 8.3 Defense 43.8 59.2 46.9 Homeland Security 4.1 Housing & Urban Dev. 13.4 2.9 4.3 Interior 1.6 1.3 1.4 Justice 1.0 1.7 2.7 Labor 6.1 1.7 1.7 State 0.6 0.9 1.3 Transportation 2.2 2.6 2.1 Treasury 1.7 1.6 1.4 Veterans 2.3 2.6 3.3 Corps of Engineers 1.0 0.6 0.6 Environmental P.A. 1.1 1.1 1.1 Fed Emergency M.A. 0.2 0.4 0.0 GSA 0.2 0.5 0.0 Intl Assistance 2.8 2.7 1.9 NASA 1.6 2.5 2.0 NSF 0.3 0.4 0.7 Table 3-1 gives the expenditures on various agencies, as a percentage of the “discretionary” expenditures, where discretion ary is a euphemism for expenditures that


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-72aren’t transfers. Transfers, which are al so known as entitlements, include social security, Medicare, aid to families with dependent children, unemployment insurance and veteran’s benefits. In contrast, Table 3-1 gives the expenditures by the “Alphabet Soup” of federal agencies. The National Science Foundation (NSF) provid es funds for basic research. The general idea of government-funded research is that it is useful for ideas to be in the public domain, and moreover that some research isn’t commercially viable, but is valuable nevertheless. Studying astero ids and meteors produces little if any revenue, but could, perhaps, save humanity one day in the ev ent that we can deflect a large incoming asteroid. (Many scientists appear pessimistic about actually deflecting an asteroid.) Similarly, research into nuclear weapons migh t be commercially viable but as a society, we don’t want firms selling nuclear weapons to the highest bidder. In addition to the NSF, the National Institutes of Health, also a government agency, fund a great deal of research. How much does the gove rnment spend on R&D? Figure 3-48 shows the history of R&D expenditures. The 1960s “s pace race” competition between the US and the Soviet Union led to the greatest federal expenditure on research and development, and it topped 2% of GDP. There was a modest increase during the Reagan presidency (1980-88) in defense R&D, which promptly returned to earlier levels. 0.0 0.5 1.0 1.5 2.0 2.51 9 4 9 1953 1957 1961 1965 1 9 6 9 1973 1977 1981 1985 1 9 8 9 1993 1997 2001 Military Civilian Total Figure 3-48: Federal Spending on R&D, as a Percent of GDP Where does the government get the money to bu y all these things? As we see in Figure 3-49, the federal income tax currently produc es just under 50% of federal revenue. Social Security and Medicare taxes produce the next largest portion, with around 3035% of revenue. The rest comes from corpor ate profits taxes (about 10%), excise taxes like those imposed on cigarettes, liquor an d cigarettes (under 5%) and other taxes like tariffs, fees, sales of property like radio sp ectrum and oil leases, and fines. The major change since World War II is the dramatic incr ease in social security, a consequence of the federal government’s attempt to insure th e future viability of the program, in the face of severely adverse demographics in the form of the retirement of the baby boom generation.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-73 0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0%1 934 1 93 9 1 94 4 1 94 9 19 5 4 1959 1 964 1 969 1 974 1 97 9 1 98 4 1989 1994 1999 Income Corp Soc Sec Excise Other Figure 3-49: Sources of Federal Government Revenue An important aspect of tax collection is that the income taxes, like the federal income tax as well as Social Security and Medicare taxe s, are very inexpensive to collect, relative to sales taxes and excise taxes. Income taxe s are straightforward to collect even relative to corporate income taxes. Quite reasonably corporations can deduct expenses and the costs of doing business and are taxed on thei r profits, not on revenues. What is an allowable deduction, and what is not, make corporate profits complicated to administer. Moreover, from an economic perspective, corp orate taxes are paid by consumers, in the form of higher prices for goods, at least when industries are competitive. 3.1.6 Trade The United States is a major trading nation. Figure 3-50 presents total US imports and exports, including foreign investment and earnings (for example, earnings from US owned foreign assets). As is clear from th is figure, the net trade surplus ended in the 1970s, and the US now runs substantial trade deficits, around 4% of GDP. In addition, trade is increasingly important in the economy.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-74 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 16.00% 18.00% 20.00%19 60 1964 1968 19 72 1 976 198 0 198 4 1988 1992 19 96 20 00 Exports Imports Figure 3-50: Total Imports and Exports as a Proportion of GDP Figure 3-50 includes investment and earnings. When we think of trade, we tend to think of goods traded – American soybeans, movies and computers sold abroad, and automobiles, toys, shoes and wine purcha sed from foreign countries. Figure 3-51 shows the total trade in goods and services, as a percentage of US GDP. These figures are surprisingly similar, which shows that investment and earnings from investment are roughly balanced – the US invests abroad to a similar extent as foreigners invest in the US. 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 16.00%19 60 1964 1968 19 72 19 76 19 80 1984 1988 19 92 19 96 20 00 Exports Imports Figure 3-51: US Trade in Goods and Services Figure 3-52 shows the earnings on US assets abroad, and the payments from US-based assets owned by foreigners. These accounts ar e roughly in balance, while the US used to earn about 1% of GDP from its ownership of foreign assets.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-75 0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50% 4.00%196 0 19 6 4 196 8 1 97 2 1976 198 0 19 8 4 198 8 19 9 2 199 6 2 00 0 Payments Earnings Figure 3-52: Income and Payments as a Percent of GDP Who does the US trade with? Table 3-2 details the top fifteen trading partners., and the share of trade. The US and Canada remain the top trading countries of all pairs of countries. Trade with Mexico has grown su bstantially since the enactment of the 1994 North American Free Trade Act (NAFTA), which extended the earlier US – Canada agreement to include Mexico and Mexico is the US’s second largest trading partner. Together, the top fifteen account for three-quarters of US foreign trade.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-76Table 3-2: Top US Trading Partners and Trade Volumes ($B) Rank Country Exports Year-to-Date Imports Year-to-Date Total Percent All Countries 533.6 946.6 1,480.2 100.0% Top 15 Countries 400.7 715.4 1,116.2 75.4% 1 Canada 123.1 167.8 290.9 19.7% 2 Mexico 71.8 101.3 173.1 11.7% 3 China 22.7 121.5 144.2 9.7% 4 Japan 36.0 85.1 121.0 8.2% 5 Germany 20.4 50.3 70.8 4.8% 6 United Kingdom 23.9 30.3 54.2 3.7% 7 Korea, South 17.5 29.6 47.1 3.2% 8 Taiwan 14.0 22.6 36.5 2.5% 9 France 13.4 20.0 33.4 2.3% 10 Italy 6.9 18.6 25.5 1.7% 11 Malaysia 7.2 18.0 25.2 1.7% 12 Ireland 5.2 19.3 24.5 1.7% 13 Singapore 13.6 10.1 23.7 1.6% 14 Netherlands 15.7 7.9 23.6 1.6% 15 Brazil 9.3 13.2 22.5 1.5% 3.1.7 Fluctuations The US economy has recessions, a term which re fers to a drop in gross domestic output. Recessions are officially called by the Nati onal Bureau of Economic Research, which keeps statistics on the economy and engages in various kinds of economic research. Generally a recession is called whenever output drops for half of a year. Figure 3-53 shows the overall industrial produc tion of the United States since World War II. Drops in output are clearly noticeable The official recessions are also marked. There are three booms that lasted about a decade; these are the longest booms in US history and much longer than booms ordinarily lasted. Prior to World War II, a normal boom lasted 2 years and the longest was fo ur years. Recession s have historically lasted a 1 to two years, a pattern that co ntinues. Indeed, the average recession since World War II has been shorter than the average recession prior to that time.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-77 0 20 40 60 80 100 120 140 194619511956196119661971197619811986199119962001 IP Recession Figure 3-53: Postwar Industrial Production and Recessions These fluctuations in output are known as th e business cycle, which is not an exactly periodic cycle but instead a random cycle. 48.0 50.0 52.0 54.0 56.0 58.0 60.0 62.0 64.0 66.01940 1943 1946 194 8 19 51 19 54 1 957 1960 1963 1966 1969 197 2 197 5 19 78 19 81 1 984 1987 1990 1993 1996 199 9 200 2 Figure 3-54: Percentage of the Population Employed An important aspect of the business cycle is that many economic variables move together, or covary Some economic variables vary less with the business cycle than others. Investment varies very strongly with the business cycle, while overall employment varies weakly. Interest rates, inflation, stock prices, unemployment and many other variables also vary systematica lly over the business cycle. Recessions are clearly visible in the percentage of the po pulation employed, illustrated in Figure 3-54.


McAfee: Introduction to Economic Analysis,, July 24, 2006 3-78Some economic variables are much more variable than others. For example, investment, durable goods purchases, and utilization of production capacity vary more dramatically over the business cycle than consumption and employment. Figure 3-55 shows the percentage of industrial capacity utilized to produce manufactured goods. This series is more volatile than producti on itself, and responds more strongly to economic conditions. 70.0 72.0 74.0 76.0 78.0 80.0 82.0 84.0 86.0 88.0 90.0Jan-67 Jan-70 Jan-73 Jan-76 Jan-79 Jan-82 Jan-85 Jan-88 Jan-91 Jan-94 Jan-97 Jan-00 Jan-03 Figure 3-55: Industrial Factory Capacity Utilitzation (Source: FRED) Most of the field of macroeconomics is de voted to understanding the determinants of growth and of fluctuations, but further consideration of this important topic is beyond the scope of a microeconomics text.


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-794 Producer Theory The most basic theory of the firm views the firm as a means of transforming things into other, more valuable things, which is known as production Thus, smelting of copper or gold removes impurities and makes the result ing product more valuable. Silicon valley transforms silicon, which is the primary ingre dient of sand, along with a thousand other chemicals and metals, into computer chips used in everything from computers to toasters. Cooking transforms raw food, adding flavor and killing bacteria. Moving things to locations where they have higher va lue is a form of production. Moving stone to the location of a house where the stone can be installed in the exterior, or bringing the King Tut museum exhibit temporarily to Chicago, or a basketball team to the playoffs, are all examples of production. In this simplistic view, a firm is comprised of a technology or set of technologies for tr ansforming things an d then chooses the transformation to maximize the net profits. This “firm as a production function” view of the firm is adequate for some purposes, es pecially when products or services are relatively standardized and technologies wi dely available, but fares poorly when the internal organization of the firm matters a great deal. Nevertheless, the “firm as a production function” model is a natural starting point in the investigation of competition. 4.1 The Competitive Firm 4.1.1 Types of Firms There are four major types of firms created in law, although these broad types have many subtypes. At the smallest end is the proprietorship, in which a firm is owned by a single individual (the propriet or) or perhaps a family, and op erated by a relatively small number of people. The family farm, many restaurants, convenience stores, and laundromats are operated this way. Debt s accrued by the proprietorship are the personal responsibility of the proprietor. Professionals like attorneys and accountants are often organized as partnerships Partnerships share profits according to a formula (some equally by partner, some assigning shares or points to partners so that ‘rainmakers’ who generate more of the busine ss obtain a larger share of the profits) and usually all are liable for losse s incurred by the partnership. Thus, if a partner in a law firm steals a client’s money and disappears, the other partners are generally responsible for the loss. In contrast, a corporation is, by a legal fiction, a person, which means a corporation itself can incur debt and the respon sibility for repayment of that debt is with the corporation, not with the officers or ow ners of the corporation. When the energy trader company Enron collapsed, the shareholde rs in Enron lost thei r investment in the stock, but were not responsible for the rema ining debts of the corporation. Moreover, executives of the company are also not financially responsible for debts of the corporation, provided the executives act le gally and carry out th eir responsibilities appropriately. If a meteor strikes a manufacturing facility and bankrupts the corporation, the executives are not persona lly responsible for the debts the corporation fails to pay. On the other hand, breaking the law is not permitted, and executives at Archer Daniels Midland, the large agricultur e firm, who colluded in the fixing of the price of lysine went to jail and were persona lly fined. The corporation was fined as well.


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-80Corporations shield company executives and sh areholders from liability, and are said to offer “limited liability.” So why would anyone in their right mind organize a firm as a proprietorship or a partnership? Corporat ions cost money to organize, about $1,000 per year at the time of this writing, and are taxed, which is why many small businesses are organized as proprietorships: it is chea per. Moreover, it may not be possible for a corporation owned by a family to borrow mone y to open a restaurant: potential lenders fear not being repaid in the event of bankrupt cy, so insist on some personal liability on the part of the owners. So why are professional groups o rganized as partnerships and not corporations? The short answer is that a large variety of hybrid organizational forms exist. The distinctions have been blurred and organizations like “Chapter S Corporations” and “Limited Liability Partners hips” offer the advantages of partnerships (including avoidance of taxation) and corporat ions. The disadvantages to these forms is primarily larger legal fees, and limitations on the nature of ownership and rules specific to individual states. It is usually the case that proprietorsh ips are smaller than partnerships, and partnerships smaller than corporations, althou gh there are some very large partnerships (e.g. the big four accounting firms) and some tiny corporations. The fourth kind can be of any size, for its distinction is not how it is organized internally bu t what it does with the revenue. The non-profit firm is prohibited from distributing a profit to its owners. Religious operations, academic associations, environmental groups, most zoos, industry associations, lobbying groups, many hospitals, credit unions (a type of bank), labor unions, private universities and charities are all organized as non-profit corporations. The major advantage of non-profit firms is that the government doesn’t tax them. In exchange for avoiding taxes, non-profits must be engaged in government-approved activities, meaning generally that the nonprofit operates for the benefit of some segment of society. So why can’t you establ ish your own non-profit, that operates for the benefit of you, and avoid taxes? Genera lly you alone aren’t enough of a socially worthy purpose to meet the requ irements to form a non-profit.19 Moreover, you can’t establish a non-profit for a worthy goal and no t serve that goal but just pay yourself all the money the corporation raises, because non-profits are prohibited from overpaying their managers, since overpaying the manage r means not serving the worthy corporate goal as well as possible. Finally, commercial activities of non-profits are taxable. Thus, when the non-profit zoo sells stuffed animals in the gift-shop, generally the zoo collects sales tax and is potentially subject to corporate taxes. The modern corporation is a surprisingly re cent invention. Prior to World War I, companies were generally organized in a pyrami d structure, with a president at the top, and vice-presidents who reported to him, etc. In a pyramid structure, there is a welldefined chain of command, and no one is ever below two distinct managers of the same level. The problem with a pyramid structure is that two retail st ores that want to coordinate have to contact their managers and possibly their managers’ managers, and so on up the pyramid until a common manager is reached. There are circumstances where such rigid decision-making is unwi eldy, and the larger the operation of a corporation, the more unwieldy it gets. 19 Certainly some of the non-profit religious organiza tions created by televangelists suggest that the nonprofit established for the benefit of a single individual isn’t far-fetched.


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-81Four companies – Sears, DuPont, General Motors and Standard Oil of New Jersey (Exxon) – found that the pyramid structure di dn’t work well for them. Sears found that its separate businesses of retail stores and mail order required a mix of shared inputs (purchased goods) but distinct marketing and warehousing of these goods. Consequently, retail stores and mail order needed to be separate business units, but purchasing had to answer to both of them. Similarly, DuPont’s military business (e.g. explosives) and consumer chemicals were very different operations serving very different kinds of customers, yet often selli ng the same things, so again the inputs needed to be centrally produced and to coordinate with two separate corporate divisions. General Motors’ many car divi sions employ ‘friendly rivalry,’ in which technology and parts are shared across th e divisions but the di visions compete in marketing their cars to consumers. Again, te chnology can’t be under just one division, but instead is common to all. Finally, Stan dard Oil of New Jersey was attempting to create a company that managed oil products from oil exploration all the way through pumping gasoline into automobile gas tanks. With such varied operations all over the globe, Standard Oil of New Jersey required extensive coordination and found that the old business model needed to be replaced. These four companies independently invented the modern corporation, which is organized into separate business units. These business units run as semi-autonomou s companies themselves, with one business unit purchasing, at a negotiated price, inputs from another unit, and selling outputs to a third. The study of the internal organization of firms and its ramifications for competitiveness is fascinating, but beyond the scope of this book.20 4.1.2 Production Functions The firm transforms inputs into outputs. For example, a bakery takes inputs like flour, water, yeast, labor, and heat and makes lo aves of bread. An earth-moving company takes capital equipment, ranging from shovels to bulldozers, and labor and digs holes. A computer manufacturer buys parts, genera lly “off-the-shelf” like disk-drives and memory, along with cases and keyboards and other parts that may be manufactured specially for the computer manufacturer, and uses labor to produce computers. Starbucks takes coffee beans, water, some capital equipment, and labor and produces brewed coffee. Many if not all firms produce several outputs. However, we can view a firm producing multiple outputs as using several distinct prod uction processes, and thus it is useful to start by looking at a firm that produces only one output. Generally, we can describe this firm as buying an amount x1 of the first input, x2 of the second input, and so on (we’ll use xn to denote the last input) and producing an amount y of the output, that is, the production function is y = f ( x1, x2, … xn). Mostly we will focus on two inpu ts in this section, but carry ing out the analysis for more than two inputs is straightforward. 20 If you want to know more about organization theory, I happily recommend Competitive Solutions: The Strategist’s Toolkit, by R. Preston McAfee, Princeton: Princeton University Press, 2002.


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-82Example: The Cobb-Douglas production function is the product of the x ’s raised to powers, and comes in the form: na n a a nx x x a x x x f ... ) ,..., (2 12 1 0 2 1 The constants a1 through an are positive numbers, generally individually less than one. For example, with two goods, capital K and labor L Cobb-Douglas can be expressed as a0KaLb. We will use this example frequently. It is illustrated, for a0 = 1, a =1/3 and b=2/3, in Figure 4-1. 0.5 1 1.5 2 L 20 40 60 80 100 K Figure 4-1: Cobb-Douglas Isoquants Figure 4-1 shows three isoquants for the Cobb-Douglas production function. An isoquant, meaning “equal quantity,” illust rates the input mixes that produce a given output level. In this case, gi ven a=1/3 and b=2/3, we can solve y = KaLb for K to obtain K = y3 L-2. Thus, K = L-2 gives the combinations of inputs yielding an output of 1, and that is what the dark, solid line represents. The middle, grey dashed line represents an output of 2, and finally the dotted light-grey line represents an output of 3. Isoquants are familiar contour plots used, for example, to show the height of terrain or temperature on a map. Temperature isoquants are, not surprisingly, called isotherms. Figure 4-2: The Production Function


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-83 Isoquants provide a natural way of looking at production functions and are a bit more useful to examine than 3-D plots like the one provided in Figure 4-2. The fixed-proportions production function comes in the form } ..., { ) ,..., (2 2 1 1 2 1n n nx a x a x a Min x x x f Figure 4-3: Fixed Proportions The fixed proportions production function has the property that adding an input beyond a necessary level does no good. For example, the productive value of having more than one shovel per worker is pretty low, so th at shovels and diggers are reasonably modeled as producing holes using a fixed proportions production function. Moreover, without a shovel or other digging implement like a b ackhoe, a bare-handed worker produces so little digging as to be nearly useless, so extra workers beyond the number of shovels have little effect. Ultimately, the size of the holes is pretty much determined by Min {number of shovels, number of diggers}. The Figure 4-3 illustrates the isoquants for fixed proportions. As we will see, fixed proportions makes the inputs “perfect complements.” Two inputs K and L are perfect substitutes in a production function f if they enter as a sum, that is, f ( K L x3, … xn) = g ( K + cL x3, … xn), for a constant c With an appropriate scaling of the units of one of the variables, all that matters is the sum of the two variables, not the individual values. In this case, the isoquants are straight lines that are parallel to each other, as illustrated in the Figure 4-4.


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-84 Figure 4-4: Perfect Substitutes The marginal product of an input is just the derivative of the production function with respect to that input.21 An important aspect of margi nal products is that they are affected by the level of other inputs. Fo r example, in the Cobb-Douglas case with two inputs22 and for constant A : ) ( L AK L K f the marginal product of capital is ) (1 L AK L K K f If and are between zero and one (the usual case), then the marginal product of capital increases in the amount of labor, an d decreases in the amount of capital. For example, an extra computer is very productive in a situation with lots of workers and few computers, but not so productive in a situation where there are lots of computers and few people to operate them. The value of the marginal product of an input is just the marginal product times the price of the output. If the value of the marg inal product of an input exceeds the cost of that input, it is profitable to use more of the input. Some inputs are more readily changed than ot hers. It can take five years or more to order and obtain new passenger aircraft, four years to build an electricity generation 21 This is a partial derivative, since it holds the other inputs fixed. Partial derivatives are denoted with the symbol 22 The symbol is the Greek letter “alpha.” The symbol is the Greek letter “beta.” These are the first two letters of the Greek alphabet, and the word alph abet itself originates from these two letters.


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-85facility or a pulp and paper mill. Very ski lled labor – experienced engineers, animators, patent attorneys – is often hard to find and ch allenging to hire. It usually takes three to five years to hire even a small number of ac ademic economists. On the other hand, it is possible to buy shovels, telephones, and comp uters and to hire a variety of temporary workers quite rapidly, in a matter of a day or so. Moreover, additional hours of work can be obtained by an existing labor force simp ly by hiring them “overtime,” at least on a temporary basis. The amount of water or el ectricity a production facility uses can be varied second by second. If you run a re staurant, you can use more water tonight to wash dishes if you need it. If you start in the morning, you can probably get a few additional workers by that evening by paying overtime to those who aren’t scheduled to work. It will probably take a few days or mo re to hire additional waiters and waitresses, and perhaps more than a few days to find a skilled chef. You can obtain more ingredients, generally the same day, and more plates and silverware pretty quickly. You can lease more space, but it will probably take more than a month to actually occupy a larger space, what with finding the space for rent, renting it, remodeling it and obtaining the necessary permits. That some inputs or factors can be varied quickly, othe rs only slowly, leads to the notions of the long-run and short-run. In the short-run, only some inputs can be adjusted, while in the long-run, all inputs can be adjusted. Traditionally, economists viewed labor as quickly adjustable, and capita l equipment as more difficult to adjust. That is certainly right for airlines – obtainin g new aircraft is a very slow process – and for large complex factories, and for relatively low-skilled and hence substitutable labor. On the other hand, obtaining workers with unusual skills is a slower process than obtaining warehouse or office space. Genera lly speaking, the long-run inputs are those that are expensive to adjust quickly, while the short-run factors can be adjusted in a relatively short time frame. What factors be long in which category is dependent on the context or application under consideration. (Exercise) For the Cobb-Douglas production function, suppose there are two inputs K and L and the sum of the exponents is on e. Show that if each input is paid the value of the marginal product per unit of the input, the entire output is just exhausted. That is, for this production function, show ) ( L K f L f L K f K 4.1.3 Profit Maximization Consider an entrepreneur that would like to maximize profit, perhaps by running a delivery service. The entrepre neur uses two inputs, capital K (e.g. trucks) and labor L (e.g. drivers), and rents the capital at cost r per dollar of capital. The wage rate for drivers is w The production function is F ( K, L ), that is, given inputs K and L the output is F ( K, L ). Suppose p is the price of the output. This gives a profit of:23 23 Economists often use the Greek letter to stand for profit. There is little risk of confusion because economics doesn’t use the ratio of the circumference to the diameter of a circle very often. On the other hand, the other two named constants, Euler’s e and i, the square root of -1, appear fairly frequently in economic analysis.


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-86. ) ( wL rK L K pF First, consider the case of a fixed level of K The entrepreneur chooses L to maximize profit. The value L of L that maximizes the function must satisfy: *) ( 0 w L K L F p L This expression is known as a first order condition because it says the first derivative of the function is zero.24 The first order condition show s that we add workers to the production process until reaching a worker who just pays his salary, in that the value of the marginal product for that worker is equal to the cost of the worker. Figure 4-5: Profit-Maximizing Labor Input In addition, a second characteristic of a maximum is that the second derivative is negative (or non-positive). This arises be cause, at a maximum, the slope goes from positive (since the function is increasi ng up to the maximum), to zero (at the maximum), to a negative number (because the function is falling as the variable rises past the maximum). This means that the de rivative is falling, that is, the second derivative is negative. This lo gic is illustrated in the Figure 4-5. 24 It is possible that L=0 is the best that entrepreneur can do. In this case, the derivative of profit with respect to L is not necessarily zero. The first order condition instead would be: Either L 0, or L=0 and L 0. The latter pair of conditions reflects the logic that either the derivative is zero and we are at a maximum, or L = 0, in which case a small increase in L must not cause to increase. L* L Slope positive to left of maximum Slope zero at maximum Slope negative to right of maximum


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-87 The second property is known as the second order condition because it is a condition on the second derivative.25 It is expressed as: *). ( ) ( ) ( 02 2 2 2L K L F p L This is enough of a mathematical treatmen t to establish comparative statics on the demand for labor. Here, we treat the choice L as a function of another parameter – the price p the wage w or the level of capital K For example, to find the effect of the wage on the labor demanded by the entrepreneur, we can write: )) ( ( 0 w w L K L F p This expression recognizes that the choice L* that the entrepreneur makes satisfies the first order condition, and results in a value that depends on w But how does it depend on w ? We can differentiate this expression to obtain: 1 ) ( )) ( ( ) ( 02 2 w L w L K L F p or 0 )) ( ( ) ( 1 ) ( *2 2 w L K L F p w L The second order condition lets the derivati ve be signed. This form of argument assumes that the choice L* is differentiable, which is not necessarily true. Digression: In fact, there is a revealed preference form of argument that makes the point without calculus and makes it su bstantially more generally. Suppose w1 < w2 are two wage levels, and that the entrepreneur chooses L1 when the wage is w1 and L2 when the wage is w2. Then profit maximizati on requires that these choices are optimal. In particular, when the wage is w1, the entrepreneur earns higher profit with L1 than with L2: 2 1 2 1 1 1) ( ) ( L w rK L K pf L w rK L K pf When the wage is w2, the entrepreneur earns higher profit with L2 than with L1. 25 The orders refer to considering small but positive terms which are sent to zero to reach derivatives. The value 2, the “second order term” goes to zero faster than the first order term.


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-88. ) ( ) (1 2 1 2 2 2L w rK L K pf L w rK L K pf The sum of the left hand sides of these two expressions is at least as large as the sum of the right hand side of the two expressions: 2 2 2 1 1 1) ( ) ( L w rK L K pf L w rK L K pf 2 1 2 1 2 1) ( ) ( L w rK L K pf L w rK L K pf A large number of terms cancel, to yield 2 1 1 2 2 2 1 1L w L w L w L w This expression can be re-arranged to yield 0 ) )( (1 2 2 1 L L w w This shows that the higher labor choice must be associated with the lower wage. This kind of argument, sometimes know n as a “revealed preference” kind of argument because choices by consumers were the first place the type of argument was applied, can be very powerful and gene ral, because issues of differentiability are avoided. However, we will use th e more standard differentiability type argument, because such arguments are usually more readily constructed. The effect of an increase in the capital level K on the choice by the entrepreneur can be calculated by considering L* as a function of the capital level K )) ( ( 0 w K L K L F p Differentiating this expression with respect to K we obtain ), ( )) ( ( ) ( )) ( ( 02 2 2K L K L K L F p K L K L K F p or, )) ( ( ) ( )) ( ( ) ( *2 2 2K L K L F K L K L K F K L We know the denominator of this expression is not positive, thanks to the second order condition, so the unknown part is the numera tor. We then obtain the conclusion that:


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-89An increase in capital increases the la bor demanded by the entrepreneur if 0 )) ( (2 K L K L K F, and decreases the labor demanded if 0 )) ( (2 K L K L K F This conclusion looks like gobbledygook but is actually quite intuitive. Note that 0 )) ( (2 K L K L K F means that an increase in capital increases the derivative of output with respect to labor, that is, an increase in capital increases the marginal product of labor. But this is the definition of a complement! That is, 0 )) ( (2 K L K L K F means that labor and capital are complements in prod uction – an increase in capital increases the marginal productivity of labor. Thus an increase in capital will increase the demand for labor when labor and capital are comple ments, and will decrease the demand for labor when labor and capital are substitutes. This is an important conclusion because di fferent kinds of capital may be complements or substitutes for labor. Are computers co mplements or substitute s for labor? Some economists consider that computers are complements to highly skilled workers, increasing the marginal value of the most skilled, but substitute for lower skilled workers. In academia, the ratio of secretar ies to professors has fallen dramatically since the 1970s as more and more professors use ma chines to perform secretarial functions. Computers are thought to have increased the marginal product of professors and reduced the marginal product of secretaries, so the number of professors rose and the number of secretaries fell. The revealed preference version of the effect of an increase in capital is to posit two capital levels, K1 and K2, with associated profit-maximizing choices L1 and L2. The choices require, for profit maximization, that 2 1 2 1 1 1 1 1) ( ) ( wL rK L K pF wL rK L K pF and ) ( ) (1 2 1 2 2 2 2 2wL rK L K pF wL rK L K pF Again, adding the left-hand-sides together pr oduces a result at least as large as the sum of the right hand sides: 2 2 2 2 1 1 1 1) ( ) ( wL rK L K pF wL rK L K pF ) ( ) (2 1 2 1 1 2 1 2wL rK L K pF wL rK L K pF Eliminating redundant terms yields ), ( ) ( ) ( ) (2 1 1 2 2 2 1 1L K pF L K pF L K pF L K pF


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-90or, ) ( ) ( ) ( ) (1 1 1 2 2 1 2 2L K F L K F L K F L K F or, ) ( ) (2 1 2 11 2 K K K Kdx L x K F dx L x K F26 or 0 ) ( ) (2 11 2 K Kdx L x K F L x K F and finally, 0 ) (2 1 2 12 K K L Ldx dy y x L K F Thus, if K2 > K1 and 0 ) (2 L K L K F for all K and L then L2 L1, that is, with complementary inputs, an increase in one input increases the optimal choice of the second input. In contrast, with substitute s, an increase in one input decreases the other input. While we still used differentia bility of the production function to carry out the revealed preference argument, we did not need to establish that the choice L* was differentiable to perform the analysis. Example: Labor Demand with the Cobb-Do uglas production function. The CobbDouglas production function has the form ) ( L AK L K F for constants A and all positive. It is necessary for <1 for the solution to be fi nite and well-defined. The demand for labor satisfies )) ( ( 01w L AK p w K L K L F p or 26 Here we use the standard convention that ... ... a b b adx dx


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-91 *1 1 w AK p L When + =1, L is linear in capital. Cobb-D ouglas production is necessarily complementary, that is, an increase in capital increases labor demanded by the entrepreneur. (Exercise) For the fixed proportions production function Min { K L }, find labor demand (capital fixed at K ). 4.1.4 The Shadow Value When capital K can’t be adjusted in the short-run, it creates a constraint on the profit available on the entrepreneur – the desire to change K reduces the profit available to the entrepreneur. There is no direct value of capi tal, because capital is fixed. That doesn’t mean we can’t examine its value, howeve r, and the value of capital is called a shadow value because it refers to the value associated with a constraint. Shadow value is wellestablished jargon. What is the shadow-value of capital? Le t’s return to the constrained, short-run optimization problem. The pr ofit of the entrepreneur is: ) ( wL rK L K pF The entrepreneur chooses the value L to maximize profit, but is stuck in the short-run with the level of capital inherited from a past decision. The shadow value of capital is the value of capital to profit, given the optimal decision L *. Because w L K L F p L *) ( 0, the shadow value of capital is r L K K F p K L K dK L K d *) ( *) ( *) ( Note that this could be negative; the entrep reneur might like to sell some capital but can’t, perhaps because it is installed in the factory. Any constraint has a shadow value. The term re fers to the value of relaxing a constraint. The shadow value is zero when the constrai nt doesn’t bind; for example, the shadow value of capital is zero when it is set at the profit-maximizing level. Technology binds the firm; the shadow value of a superior techno logy is the increase in profit associated with it. For example, parameterize th e production technology by a parameter a so that aF ( K, L ) is produced. The shadow value of a given level of a is, in the short-run,


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-92 *). ( *) ( *) ( L K pF a L K da L K d A term is vanishing in the process of establ ishing the shadow value. The desired value L varies with the other parameters like K and a but the effect of these parameters on L doesn’t appear in the expression for th e shadow value of the parameter because L 0 at L *. 4.1.5 Input Demand Over a long period of time, an entrepreneur can adjust both the capital and the labor used at the plant. This le ts the entrepreneur maximize profit with respect to both variables K and L We’ll use a double star, **, to denote variables in their long-run solution. The approach to maximizing prof it over two variables is to maximize it separately over each variable, thereby obtaining *) *, ( 0 w L K L F p and *) *, ( 0 r L K K F p We see for both capital and labor, the value of the marginal product is equal to purchase price of the input. It is more of a challenge to carry out comp arative statics exercises with two variables, and the general method won’t be developed here.27 However, we can illustrate one example as follows. Example: The Cobb-Douglas production func tion implies choices of capital and labor satisfying two first order conditio ns, one each for labor and capital.28 *) *, ( 01w L AK p w L K L F p *) *, ( 01r L AK p w L K K F p To solve this expression, first rewrite to obtain 27 If you want to know more, the approach is to arrange the two equations as a vector with x = (K, L), z=(r/p, w/p), so that *) ( z x F 0 and then differentiate to obtain *) ( *1dz x F dx which can then be solved for each comparative static. 28 It is necessary for + <1 for the solution to be finite and well-defined.


McAfee: Introduction to Economic Analysis,, July 24, 2006 4-93 1* L AK p w and *1L AK p r, then divide the firs t by the second to yield L K r w or L r w K This can be substituted into either equation to obtain 1 1 1 1* w r Ap L and *1 1 1 1 w r Ap K While these expressions appear complicated, in fact the dependence on the output price p and the input prices r and w are quite straightforward. How do equilibrium values of capital and labor respond to a change in input prices or output price for the Cobb-Douglas production function? It is useful to cast these changes in percentage terms. It is straigh tforward to demonstrate that both capital and labor respond to a small percentage change in any of these variables with a constant percentage change. (Exercise) For the Cobb-Douglas production function ) ( L AK L K F show 1 r L L r 1 1 w L L w 1 1 p L L p 1 1 r K K r 1 w K K w and 1 1 p K K p An important insight of profit maximization is that it implies minimization of costs of yielding the chosen output, that is, profit-maximization entails efficient production The logic is straightforward. The profit of an entrepreneur is revenue minus costs, and the revenue is price times output. For the chosen output, then, the entrepreneur earns the revenue associated with the output, which is fixed since we are considering only the chosen output, minus the costs of producing that output. Thus, for the given output, maximizing profits is equivale nt to maximizing a constant (revenue) minus costs. Since maximizing – C is equivalent to minimizing C the profit-maximizing entrepreneur minimizes costs. This is important beca use profit-maximization implies not being wasteful in this regard: a profit-maximizing entrepreneur produces at least cost.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-94 Figure 4-6: Tangency and Isoquants There are circumstances where the cost-minimiz ation feature of profit maximization can be used, and this is especially true when a graphical approach is taken. The graphical approach to profit-maximization is illustrated in Figure 4-6. The curve represents an isoquant, which holds constant the output. Th e straight lines represent “isocost” lines, which hold constant the expenditure on inputs. Isocost lines solve the problem rK + wL = constant and thus have slope r w dL dK Isocost lines are necessarily parallel – they have the same slope. Moreover, the cost associated with an isocost line rises the further northeast we go in the graph, or the further away from the origin. What point on an isoquant minimizes total cost? The answer is the point associated with the lowest (most southwest) isocost that intersects the isoquant. This point will be tangent to the isoquant and is denoted by a star. At any lower cost, it isn’t possible to produce the desired quantity. At any higher cost, it is possible to lower cost and still produce the quantity. That cost minimization requires a tangency between the isoquant and the isocost has a useful interpretation. The slope of the isoc ost is minus the ratio of input prices. The slope of the isoquant measures the substitutabi lity of the inputs in producing the output. Economists call this slope the marginal rate of te chnical substitution which is the amount of one input needed to make up for a decrease in another input and hold output constant. Thus, one feature of cost minimizati on is that the input price ratio equals the marginal rate of technical substitution. L K q

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-954.1.6 Myriad Costs How much does it cost to produce a given quantity q ? We already have a detailed answer to this question, but now need to focu s less on the details and more on the “big picture.” First, let’s focus on the short-run, and suppose L is adjustable in the short-run, but K is not. Then the shortrun total cost of producing q given the capital level, is SRTC ( q |K) = wL rK L min, over all L satisfying F ( K L ) q In words, this equation says the short-run total cost of the quantity q given the existing level K is the minimum cost, where L gets to vary (which is denoted by “min over L”), where the L considered is large enough to produce q The vertical line | is used to indicate a condition or conditional requirement; here | K indicates that K is fixed. The minimum lets L vary but not K Finally, there is a constraint F ( K L ) q which indicates that one has to be able to produce q with the mix of inputs because we are considering the short-run cost of q The short-run total cost of q given K has a simple form. First, since we are minimizing cost, the constraint F ( K L ) q will be satisfied with equality, F ( K L ) = q This equation determines L since K is fixed, that is, q K q L K FS )) ( ( gives the short-run value of L LS( q,K ). Finally, the cost is then rK + wL = rK + wLS( q,K ). The short-run marginal cost given K is just the derivative of total cost with respect to q To establish the short-run marginal cost, note that the equation F ( K L ) = q gives )) ( ( dq dL K q L K L FS or )) ( ( 1 K q L K L F dq dLS q F The tall vertical line, subscripted with F = q is used to denote the constraint F ( K L ) = q that is being differentiated. Thus, the short-run marginal cost is SRMC ( q|K ) = )) ( ( ) ( ) ( K q L K L F w dq dL w wL rK dq d q C SRTS q F There are two other short-run costs that will be needed to complete the analysis. First, there is the notion of the short-run averag e cost of production, which we obtain by dividing the total cost by the quantity:

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-96 q K q SRTC K q SRAC ) | ( ) | ( Finally, we need one more short-run cost: the short-run average variable cost. The variable cost eliminates the fixed costs of operation, which in this case are rK That is, ) | ( ) | 0 ( ) | ( ) | ( q K q wL q K SRTC K q SRTC K q SRAVCS The short-run average variable cost is the aver age cost ignoring the investment in capital equipment. The short-run average cost could also be called the short-run average total cost, since it is the average of the total cost per unit of output, but “average total” is a bit of an oxymoron.29 Consequently, when total, fixed or va riable is not specified, the convention is to mean total. Note that the marginal va riable cost is the same as the marginal total costs, because the difference between variable co st and total cost is a constant – the cost of zero production, also known as the fixed cost of production. At this point, we have identified four distinct costs, all relevant to the short-run. These are the total cost, the marginal cost, the aver age cost, and the average variable cost. In addition, all of these can be considered in the long-run as well. There are three differences in the long-run. First, the long-run lets all inputs vary, so the long-run total cost is LRTC ( q ) = wL rKK L,min over all L and K combinations satisfying F ( K L ) q Second, since all inputs can vary, the long-run cost isn’t conditioned on K Finally, the long-run average variable cost is the same as the long-run average total cost. Because in the long-run a firm could use no inputs and thus incur no costs, the cost of producing zero is zero. Therefore, in the long-run, all costs are variable, and the long-run average variable cost is the long-run average total cost. (Exercise) For the Cobb-Douglas production function ) ( L AK L K F with + <1, with K fixed in the short-run but not in the long-run, and cost r of capital and w for labor, show SRTC( q | K ) = ,1 AK q w rK 29 An oxymoron is a word or phrase which is self-con tradictory, like “jumbo shrimp,” “stationary orbit,” “virtual reality,” “modern tradition,” or “pretty ug ly.” Oxymoron comes from the Greek oxy, meaning sharp, and moros, meaning dull. Thus oxymoron is itself an oxymoron, so an oxymoron is selfdescriptive. Another word which is self-descriptive is “pentasyllabic.”

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-97 SRAVC( q | K ) = ,1 1 AK q w SRMC( q | K ) = ,1 1 AK q w LRTC( q | K ) = .1 A q r w Note that the easiest way to find the long-r un total cost is to minimize the short-run total cost over K Since this is a function of one variable, it is straightforward to identify the K that minimizes cost, and then plug that K into the expression for total cost. One might want to distinguish the very shortrun, from the short-run, from the medium run, from the long-run, from the very long -run. But a better approach is to view adjustment as a continuous process, with a gradual easing of the constraints. Faster adjustment costs more. Continuous adjustment is a more advanced topic, requiring an Euler equation approach. 4.1.7 Dynamic Firm Behavior In this section, we consider a firm or entrepre neur that can’t affect the price of output or the prices of inputs, that is, a competitiv e firm. How does such a competitive firm respond to price changes? When the price of the output rises, the firm earns profits ), | ( K q c pq where c( q | K ) is the total cost of producing give n that the firm currently has capital K Assuming the firm produces at all, the firm maximizes profits by choosing the quantity qs satisfying ) | ( 0 K q c ps that is, choosing the quantity where price equals marginal cost. However, this is a good strategy only if producing a positive quantity is desirable, that is, if ), 0 ( ) | ( K c K q c pqs s which can be rewritten as s sq K c K q c p ) 0 ( ) | ( The right-hand-side of this inequality is th e average variable cost of production, and thus the inequality implies that a firm wi ll produce provided price exceeds the average variable cost. Thus, the profit-maximizing firm produces the quantity qs where price equals marginal cost, provided price is as large as minimum average variable cost. If price falls below minimum average va riable cost, the firm shuts down.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-98 The behavior of the competitive firm is illustrated in Figure 4-7. The thick line represents the choice of the firm as a function of the price, which is on the vertical axis. Thus, if the price is below the minimum aver age variable cost (AVC), the firm shuts down. When price is above the minimum aver age variable cost, the marginal cost gives the quantity supplied by the firm. Thus, the choice of th e firm is composed of two distinct segments – the marginal cost, where the firm produces the output where price equals marginal cost, and shutdown, where the firm makes a higher profit, or loses less money, by producing zero. Figure 4-7 also illustrates the average total co st, which doesn’t affect the short term behavior of the firm but does affect the long term behavior, because when price is below average total cost, the firm is not making a pr ofit, but instead would prefer to exit over the long term. That is, when the price is between the minimum average variable cost and the minimum average total cost, it is better to produce than to shut down, but the return on capital was below the cost of capital With a price in this intermediate area, a firm would produce, but would not replace th e capital, and thus woul d shut down in the long-term if the price is expected to persis t. As a consequence, minimum average total cost is the long-run “shut down” point for th e competitive firm. (Shutdown may refer to reducing capital rather that literally setting ca pital to zero.) Similarly, in the long term, the firm produces the quanti ty where the price equals the long-run marginal cost. Figure 4-7: Short-Run Supply Figure 4-7 illustrates one other fact: the minimum of average cost occurs at the point that marginal cost equals average cost. To see this, let C ( q ) be total cost, so that average cost is C ( q )/ q Then the minimum of average cost occurs at the point satisfying: MC ATC AVC q p

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-99 ) ( ) ( ) ( 02q q C q q C q q C dq d But this can be rearranged to imply q q C q C ) ( ) ( that is, marginal cost equals average cost at the minimum of average cost. The long-run marginal cost has a complicated relationship to short-run marginal cost. The problem in characterizing the relati onship between long-run and short-run marginal costs is that some co sts are marginal in the long-run that are fixed in the shortrun, tending to make long-run marginal co sts larger than short-run marginal costs. However, in the long-run, the assets can be configured optimally, while some assets are fixed in the short-run, and this optimal co nfiguration tends to make long-run costs lower. Instead, it is more useful to compare the long-run average total costs and short-run average total costs. The advantage is that cap ital costs are included in short-run average total costs. The result is a picture like Figure 4-8. Figure 4-8: Average and Marginal Costs In Figure 4-8, the short-run is unchanged – there is a short-run average cost, short-run average variable cost, and short-run marginal cost. The long-run average total cost has been added, in such a way that the minimum average total cost occurs at the same point as the minimum short-run average cost, which equals the short-run marginal cost. This is the lowest long-run average cost, and has th e nice property that long-run average cost equals short-run average total cost equals short-run marginal cost. However, for a SRMC SRAC SRAVC q p LRATC

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-100different output by the firm, there would ne cessarily be a different plant size, and the three-way equality is broken. Such a point is illustrated in Figure 4-9. In Figure 4-9, the quantity produced is larger than the quantity that minimizes long-run average total cost. Consequently, as is visi ble in the picture, the quantity where shortrun average cost equals long-run average cost does not minimize short-run average cost. What this means is that a factory design ed to minimize the cost of producing a particular quantity won’t necessarily mini mize short-run average cost. Essentially, because the long-run average total cost is increasing, larger plant sizes are getting increasingly more expensive, and it is chea per to use a somewhat “too small” plant and more labor than the plant size with th e minimum short-run average total cost. However, this situation wouldn ’t likely persist indefinitely because, as we shall see, competition tend to force price to the minimu m long-run average total cost, and at that point, we have the three-way equality betw een long-run average total cost, short-run average total cost, and short-run marginal cost. Figure 4-9: Increased Plant Size (Exercise) Suppose a company has total cost given by K q rK 22 where capital K is fixed in the short-run. What is short-run average total cost and marginal cost? Plot these curves. For a given quantity q0, what level of capital minimizes total cost? What is the minimum average total cost of q0? 4.1.8 Economies of Scale and Scope An economy of scale – that larger scale lowers cost – arises when an increase in output reduces average costs. We met economies of scale, and their opposite, diseconomies of SRMC SRAC SRAVC q p LRATC

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-101scale, in the previous section, with an exam ple where long-run average total cost initially fell, then rose, as quantity was increased. What makes for an economy of scale? La rger volumes of productions permit the manufacture of more specialized equipment. If I am producing a million identical automotive tail lights, I can spend $50,000 on an automate d plastic stamping machine and only affect my costs by five cents each. In contrast, if I am producing 50,000 units, the stamping machine increases my costs by a dollar each, and is much less economical. Indeed, it is somewhat more of a puzzle as to what produces a diseconomy of scale. An important source of diseconomies are mana gerial in nature – organizing a large, complex enterprise is a challenge, and la rger organizations tend to devote a larger percentage of their revenues to management of the operation. A bookstore can be run by a couple of individuals who rarely if ev er engage in management activities, where a giant chain of bookstores needs finance, human resource, risk management and other “overhead” type expenses just in order to function. Informal operation of small enterprises is replaced by formal procedural ru les in large organizations. This idea of managerial diseconomies of scale is reflected in the aphorism that “A platypus is a duck designed by a committee.” In his influential 1975 book The Mythical Man-Month IBM software manager Fred Books describes a particularly se vere diseconomy of scale. Adding software engineers to a project increases the number of conversation s necessary between pairs of individuals. If there are n engineers, there are n ( n – 1) pairs, so that communication costs rise at the square of the project size. This is pithily summarized in Brooks’ Law : "Adding manpower to a late software project makes it later." Another related source of diseconomies of scal e involves system slack. In essence, it is easier to hide incompetence and laziness in a large organization than in a small one. There are a lot of familiar examples of this insight, starting with the Peter Principle, which states that people rise in organizati ons to the point of their own incompetence, which means eventually people cease to do the jobs that they do well.30 That slack grows as an organization grows impl ies an diseconomy of scale. Generally, for many types of products, econom ies of scale from production technology tend to reduce average cost, up to a point where the operation becomes difficult to manage, at which point diseconomies tend to prevent the firm from economically getting larger. Under this view, improvements in information technologies over the past twenty years have permitted firms to get large r and larger. While that seems logical, in fact firms aren’t getting that much larger than they used to be, and the share of output produced by the top thousand firms has been relatively steady. That is, the growth in the largest firms just mirro rs world output growth. Related to an economy of scale is an economy of scope An economy of scope is a reduction in cost associated with producin g several distinct goods. For example, Boeing, which produces both commercial and military jets, can amortize some of its R&D costs over both types of aircraft, thereb y reducing the average costs of each. Scope 30 Laurence Johnston Peter (1919–1990).

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-102economies work like scale economies, except they account for advantages of producing multiple products, where scale economies invo lve an advantage of multiple units of the same product. Economies of scale can operate at the level of the individual firm but can also operate at an industry level. Suppose there is an econom y of scale in the production of an input. For example, there is an econom y of scale in the production of disc drives for personal computers. That means an increase in the pr oduction of PCs will tend to lower the price of disc drives, reducing the cost of PCs, which is a scale economy. In this case, it doesn’t matter to the scale economy whether one firm or many firms are responsible for the increased production, and this is known as an external economy of scale or an industry economy of scale, because the scale economy operates at the level of the industry rather than in the individual firm. Thus, the long-run average cost of individual firms may be flat, while the long-run average cost of the industry slopes downward. Even in the presence of an external econom y of scale, there may be diseconomies of scale at the level of the firm. In such a situat ion, the size of any individual firm is limited by the diseconomy of scale, bu t nonetheless the average cost of production is decreasing in the total output of the industry, through th e entry of additional firms. Generally there is an external diseconomy of scale if a large r industry drives up input prices, for example increasing land costs. Increasing the produc tion of soybeans significantly requires using land that isn’t so well suited for them, tending to increase the average cost of production. Such a diseconomy is an extern al diseconomy rather than operating at the individual farmer level. Second, there is an external economy if an increase in output permits the creation of more specialized techniques and a greater effort in R&D to lower costs. Thus, if an increase in output in creases the development of specialized machine tools and other production inputs, an external economy will be present. An economy of scale arises when total average cost falls as the number of units produced rises. How does this relate to production functions? We let y = f ( x1, x2,…, xn) be the output when the n inputs x1, x2,…, xn are used. A rescaling of the inputs involves increasing the inputs by a fixed percentage, e.g. multiplying them all by the constant (the Greek letter lambda), where >1. What does this do to output? If output goes up by more than we have an economy of scale (also known as increasing returns to scale ): scaling up production increases output proportionately more. If output goes up by less than we have a diseconomy of scale or decreasing returns to scale And finally, if output rises by exactly we have constant returns to scale. How does this relate to average cost? Formally, we have an economy of scale if ) ( ) (2 1 2 1n nx x x f x x x f if >1. This corresponds to decreasing average cost. Let w1 be the price of input 1, w2 the price of input 2, and so on. Then the average cost of producing y = f ( x1, x2,…, xn) is AVC = ) ( ...2 1 2 2 1 1n n nx x x f x w x w x w

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-103What happens to average cost as we scale up production by >1? Call this AVC( ). ) ( ... ) ( ... ) AVC(2 1 2 2 1 1 2 1 2 2 1 1n n n n n nx x x f x w x w x w x x x f x w x w x w ) 1 ( ) ( ) (2 1 2 1AVC x x x f x x x fn n Thus, average cost falls if there is an econom y of scale and rises if there is a diseconomy of scale. Another insight about the returns to scale co ncerns the value of the marginal product of inputs. Note that, if there are constant returns to scale: 1 2 1 2 2 1 1) ( ...n n nx x x f d d x f x x f x x f x ) ( 1 ) ( ) ( lim2 1 2 1 2 1 1 n n nx x x f x x x f x x x f The value 1x f is the marginal product of input x1, and similarly 2x f is the marginal product of input 2, and so on. Consequent ly, if the production function exhibits constant returns to scale, it is possible to di vide up output in such a way that each input receives the value of the marginal product. That is, we can give 1 1x f x to the suppliers of input 1, 2 2x f x to the suppliers of input 2, and so on, and this exactly uses up the all the output. This is known as “paying the marginal product,” because each supplier is paid the marginal product associated with the input. If there is a diseconomy of scale, then payi ng the marginal product is feasible, but there is generally something left over, too. If there are increasing returns to scale (an economy of scale), then it is not possible to pay all the inputs their marginal product, that is, ). (2 1 2 2 1 1 n n nx x x f x f x x f x x f x

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-1044.1.8.1 (Exercise) Given the Cobb-Douglas production function na n a a nx x x x x x f ... ) ,..., (2 12 1 2 1, show there is constant returns to scale if 12 1 na a a, increasing returns to scale if 12 1 na a a, and decreasing returns to scale if 12 1 na a a. (Exercise) Suppose a company has total cost given by K q rK 22 where capital K can be adjusted in the long-run. Does this company have an economy of scale, diseconomy of scale, or constant returns to scale in the long-run? (Exercise) A production function f is homogeneous of degree r if ) ( ) (2 1 2 1 n r nx x x f x x x f Consider a firm with a production function that is homogeneous of degree r Suppose further that the firm pays the value of marginal product for all its inputs. Show that the portion of revenue left over is 1 – r. 4.2 Perfect Competition Dynamics The previous section developed a detailed an alysis of how a competitive firm responds to price and input cost changes. In this se ction, we consider how a competitive market responds to demand or cost changes. 4.2.1 Long-run Equilibrium The basic picture of a long-run equilibrium is presented in Figure 4-10. There are three curves, all of which are already familiar. First, there is demand, considered in the first chapter. Here demand is taken to be the “per period” demand. Second, there is the short-run supply, which reflects two components – a shut down point at minimum average variable cost, and quantity such that price equals short-run marginal cost above that level. The short-run su pply, however, is the market supply level, which means it sums up the individual firm effects. Finally there is the long-run average total cost at the industry level, thus reflecting any exte rnal diseconomy or economy of scale. As drawn in Figure 4-10, there is no long-run scale effect. The long-run average total cost is also the long-run industry supply.31 As drawn, the industry is in equilibrium, with price equal to P0, which is the long-run average total cost, and also equating short-ru n supply and demand. That is, at the price of P0, and industry output of Q0, no firm wishes to shut down no firm can make positive profits from entering, there is no excess outp ut, and no consumer is rationed. Thus, no market participant has an incentive to change their behavior, so the market is in both long-run and short-run equilibrium. 31 This may seem confusing, because supply is generally the marginal cost, not the average cost. However, because a firm will quit producing in the long term if price falls below its minimum average cost, the long-term supply is just th e minimum average cost of the indivi dual firms, because this is the marginal cost of the industry.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-105 Figure 4-10: Long-Run Equilibrium 4.2.2 Dynamics with Constant Costs Now consider an increase in demand. Dema nd might increase because of population growth, or because a new use for an existing product is developed, or because of income growth, or because the product becomes more useful. For example, the widespread adoption of the Atkins diet increased demand for high protein prod ucts like beef jerky and eggs. Suppose that the change is expect ed to be permanent. This is important because the decision of a firm to enter is ba sed more on expectations of future demand than on present demand. Figure 4-11 reproduces the equilibrium figure, but with the curves “grayed out” to indicate a starting position, and a darker new demand curve, labeled D1. The initial effect of the increased demand is that the price is bid up, because there is excess demand at the old price P0. This is reflected by a change in both price and quantity to P1 and Q1, to the intersection of the short-run supply SRS and the new demand curve. This is a short-run equilibriu m, and persists temporarily because, in the short-run, the cost of additional supply is higher. At the new, short-run equilibrium, price exc eeds the long-run supply cost. This higher price attracts new investment in the industry. It takes some time for this new investment to increase the quantity supplied but over time the new investment leads to increased output, and a fall in the price, as illustrated in Figure 4-12. Q0 Q p LRATC=LRS D SRS P0

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-106 Figure 4-11: A Shift in Demand Figure 4-12: Return to Long-Run Equilibrium LRATC=LRS Q0 Q P D0 SRS0P0 D1 P1 Q1 Q2 SRS2 LRATC=LRS Q0 Q P D0 SRS P0 D1 P1 Q1

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-107 As new investment is attracted into the indust ry, the short-run supply shifts to the right, because with the new investment, more is prod uced at any given price level. This is illustrated with the darker short-run supply, SRS2. The increase in price causes the price to fall back to its initial level, and the quantity to increase still further to Q2. It is tempting to think that the effect of a decrease in demand just retraces the steps of an increase in demand, but that isn’t correct In both cases, the first effect is the intersection of the new demand with the ol d short-run supply. Only then does the short-run supply adjust to equilibrate the dema nd with the long-run supply. That is, the initial effect is a short-run equilibrium, fo llowed by adjustment of the short-run supply to bring the system into long-run equilibriu m. Moreover, a small decrease in demand can have a qualitatively different effect in the short-run than a large decrease in demand, depending on whether the decrease is large enough to induce immediate exit of firms. This is illustrated in Figure 4-13. Figure 4-13: A Decrease in Demand In Figure 4-13, we start at the long-run equilibrium where LRS and D0 and SRS0 all intersect. If demand falls to D1, the price falls to the intersection of the new demand and the old short-run supply, along SRS0. At that point, exit of firms reduces the short-run supply and the price rises, follo wing along the new demand D1. If, however, the decrease in demand is large enough to push the industry to minimum average variable cost, there is immediate exit. In Figure 4-14, the fall in demand from D0 to D1 is sufficient to push the price to minimum average variable cost, which is the shutdown point of suppliers. Enough supplier s have to shutdown to keep the price at this level, which induces a shift in of the short-run supply, to SRS1. Then there is Q P SRS0 LRS D0 D1SRS2

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-108additional shutdown, shifting the short-run supply in still further, but driving up the price (along the demand curve) until the long-term equilibrium is reached. Figure 4-14: A Big Decrease in Demand Consider an increase in the price of an inpu t into production. For example, an increase in the price of crude oil increases the cost of manufacturing gasoline. This tends to decrease (shift up) both the long-run supply and the short-run supp ly, by the amount of the cost increase. The effect is illustrated in Figure 4-15. The increased costs reduce both the short-run supply (prices have to be higher to in order to produce the same quantity) and the long-run supply. The short-run supply shifts upward to SRS1, and the long-run supply to LRS2. The short-run effect is to move to the intersection of the shortrun supply and demand, which is at the price P1 and the quantity Q1. This price is below the long-run average cost, which is the long-r un supply, so over time some firms don’t replace their capital and there is disinvestment in the industry. This disinvestment causes the short-run supply to be reduced (move left) to SRS2. The case of a change in supply is more ch allenging because both the long-run supply and the short-run supply are shifted. But th e logic – start at a long-run equilibrium, then look for the intersection of current de mand and short-run supply, then look for the intersection of current demand and long-run supply – is the same whether demand or supply has shifted. Q P SRS0 LRS D0 D1 SRS1 SRS2

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-109 Figure 4-15: A Decrease in Supply 4.2.3 General Long-run Dynamics The previous section made two simplifyin g assumptions that won’t hold in all applications of the theory. First, it assumed constant returns to scale, so that long-run supply is horizontal. A perfectly elastic long-run supply means that price always eventually returns to the same point. Seco nd, the theory didn’t distinguish long-run from short-run demand. But with many prod ucts, consumers will adjust more over the long-term than immediately. As energy prices rise, consumers buy more energyefficient cars and appliances, reducing demand. But this effect takes time to be seen, as we don’t immediately scrap our cars in response to a change in the price of gasoline. The short-run effect is to drive less in respon se to an increase in the price, while the long-run effect is to choose the appropriate car for the price of gasoline. To illustrate the general analysis, we star t with a long-run equilibrium. Figure 4-16 reflects a long-run economy of scale, because the long-run supply slopes downward, so that larger volumes imply lower cost. The syst em is in long-run equilibrium because the short-run supply and demand intersection occu rs at the same price and quantity as the long-run supply and demand intersection. Both short-run supply and short-run demand are less elastic than their long-run co unterparts, reflecting greater substitution possibilities in the long-run. Q0 Q P P1 Q1 Q2 SRS1 LRS2 SRS2P0 D0 SRS0 LRS0 P2

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-110 Figure 4-16: Equilibrium with External Scale Economy Now consider a decrease in demand, decreasi ng both short-run and long-run demand. This is illustrated in Figure 4-17. To reduce the proliferation of curves, we color the old demand curves very faintly, and mark the init ial long-run equilibrium with a zero inside a small rectangle.32 The intersection of short-run supply and short-run demand is marked with the number 1. Both long-run supply and long-run demand are more elastic than their short-run counterparts, which ha s an interesting effect. The short-run demand tends to shift down over time, becau se the price associated with the short-run equilibrium is above the long-run demand price for th e short-run equilibrium quantity. However, the price associated with the sh ort-run equilibrium is below the long-run supply price at that quantity. The effect is that buyers see the price as too high, and are reducing their demand, while sellers see the pr ice as too low, and so are reducing their supply. Both short-run supply and short-ru n demand fall, until a long-run equilibrium is achieved. In this case, the long-run equilibrium involv es higher prices, at the point labeled 2, because of the economy of scale in supply. This economy of scale means that the reduction in demand causes prices to rise over the long-run. The short-run supply and demand eventually adjust to bring the syst em into long-run equilibrium, as Figure 4-18 illustrates. The new long-run equilibriu m has short-run demand and supply curves associated with it, and the system is in long-run equilibrium because the short-run demand and supply, which determine the curre nt state of the system, intersect at the 32 The short-run demand and long-run demand have been shifted down by the same amount, that is, both reflect an equal reduction in value. This kind of sh ift might arise if, for instance, a substitute had become cheaper, but the equal reduction is not essential to th e theory. In addition, the fact of equal reductions often isn’t apparent from the diagram, because of the di fferent slopes – to most ob servers, it appears that short-run demand fell less than long-run demand. This isn’t correct, however, and one can see this because the intersection of the new short-run demand and long-run demand occurs directly below the intersection of the old curves, implying both fell by equal amounts. Q P SRS0LRS LRD0 SRD0 0

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-111same point as the long-run demand and su pply, which determine where the system is heading. Figure 4-17: Decrease in Demand Figure 4-18: Long-run After a Decrease in Demand SRS2 SRD2LRD0 SRD0Q P SRS0LRS LRD1 SRD1 1 0 2 LRD0 SRD0Q P SRS0LRS LRD1 SRD1 1 0 2

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-112There are four basic permutations of the dynamic analysis – demand increase or decrease, and a supply increase or decrease. Generally, it is possible for long-run supply to slope down – this is the case of an economy of scale – and for long-run demand to slope up.33 This gives sixteen variations of the basic analysis. In all sixteen cases, the procedure is the same. Start with a long-run equilibrium, shift both the short-run and long-run levels of either demand or supply. The first stage is th e intersection of the short-run curves. The system will then go to the intersec tion of the long-run curves. An interesting example of competitive dyna mics concepts is the computer memory market, which was discussed above. Most of the costs of manufacturing computer memory are fixed costs. The modern DRAM plant costs several billion dollars; the cost of other inputs – chemicals, energy, labor, silicon wafers – are modest in comparison. Consequently, the short-run supply is vertical until prices are very, very low; at any realistic price, it is optimal to run these plants 100% of the time.34 The nature of the technology has let manufacturers cut the cost s of memory by about 30% per year over the past forty years, demonstrating that there is a strong economy of scale in production. These two features – vertical sh ort-run supply, strong economies of scale – are illustrated in the Figure 4-19. The system is started at the point labeled with the number 0, with a relatively high price, an d technology which has made costs lower that this price. Responding to the profitability of DRAM, short-run supply shifts out (new plants are built and die-shrinks permits increa sing output from existing plants). The increased output causes prices to fall, relati vely dramatically because short-run demand is inelastic, and the system moves to the poin t labeled 1. The fall in profitability causes DRAM investment to slow, which lets demand catch up, boosting prices to the point labeled 2. (One should probably think of Figure 4-19 as being in a logarithmic scale.) The point labeled with the number 2 looks qualitatively similar to the point labeled 1. The prices have followed a “saw-tooth” pattern and the reason is du e to the relatively slow adjustment of demand compared to supply, as well as the inelasticity of short-run demand, which creates great price swings as sh ort-run supply shifts out. Supply can be increased quickly, and is increased “in lump s” because a die-shrink (making the chips smaller so that more fit on a given silicon wa fer) tends to increase industry production by a large factor. This process can be repeated starting at the point labeled 2. The system is marching inexorably toward a long-run equilibrium in which electronic memory is very, very cheap even by 2004 standards and used in applications that haven’t yet been considered, but the process of getting there is a wild ride, indeed. The saw-tooth pattern is illustrated in Figure 4-20, which shows DRAM industry revenues in billions of dollars from 1992 to 2003 and projections of 2004 and 2005.35 33 The demand situation analogous to an economy of scale in supply is a network externality in which the addition of more users of a product increases the va lue of the product. Telephones are a clear example – suppose you were the only person wi th a phone – but other products li ke computer operating systems and almost anything involving adoption of a standard represent examples of network externalities. When the slope of long-run demand is greater than the slop e of long-run supply, the system will tend to be inefficient, because an increase in production produc es higher average value and lower average cost. This usually means there is another equilibrium at a greater level of production. 34 The plants are expensive in part because they are so clean, because a single speck of dust falling on a chip ruins the chip. The Infineon DRAM plant in Vi rginia stopped operations only when a snow-storm prevented workers and materials from reaching the plant. 35 Two distinct data sources were used, which is wh y there are two entries for each of 1998 and 1999.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-113 Figure 4-19: DRAM Market $0 $10 $20 $30 $40 $501992199419961998200020022004 Figure 4-20: DRAM Revenue Cycle (Exercise) Land close to the center of a city is in fixed supply, but it can be used more intensively by us ing taller buildings. When the population of a city increases, illustrate the longand sh ort-run effects on the housing markets using a graph. (Exercise) Emus can be raised on a wide vari ety of ranch land, so that there are constant returns to scale in the produc tion of emus in the long-run. In the short-run, however, the population of emus is limited by the number of breeding pairs of emus and the supply is es sentially vertical. Illustrate the longand short-run effects of an increase in demand for emus. (In the late 1980s, LRS LRD Q P SRS0 SRD0 SRS1 0 1 SRD2 2

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-114there was a speculative bubble in emus with prices reaching $80,000 per breeding pair, in contrast to $2,000 or so today.) (Exercise) There are long-run economies of scale in the manufacture of computers and their components. Ther e was a shift in demand away from desktop computers and toward notebook computers around the year 2001. What are the shortand long-run effect s? Illustrate your answer with two diagrams, one for the notebook market and one for the desktop market. Account for the fact that the two products are substitutes, so that if the price of notebook computers rises, some consumer s shift to desktops. (To answer this question, start with a time 0 and a market in long-run equilibrium. Shift demand for notebooks out and demand for desktops in. What happens in the short-run? What happens in the long-run to the prices of each? What does that price effect do to demand for each?) 4.3 Investment The distinction between the short-run supply and the long-run supply is governed by the time that investment takes. Some of the difference between the short-run demand and the long-run demand arises because we don’t scrap capital goods – cars, fridges, and air conditioners – in response to price changes. In both cas es, investment is an important component of the responsiveness of supply and demand. In this section, we take a first look at investment. We will take a second look at investment from a somewhat different perspective later when we consider basic finance tools near the end of the book. Investment goods require expenditures today to produce future value, so we begin the analysis by examinin g the value of future payments. 4.3.1 Present value The promise of $1 in the future is not worth $1 today. There are a variety of reasons why a promise of future payments is not worth th e face value today, some of which involve risk that the money may not be paid. Let’s set aside such risk for the moment; we’ll consider risk separately later. Even when the future payment is perceived to occur with negligible risk, nevertheless mo st people prefer $1 today to $1 payable a year hence. One way of expressing this is that the present value – the value today – of a future payment of a dollar is less than a dollar. From a present value perspective, future payments are discounted. From the individual perspective, one reason that you should value a future payment less than a current payment is due to arbitrage .36 Suppose you are going to need $10,000 one year from now, to put a down-payment on a house. One way of producing $10,000 is to buy a government bond that pays $10, 000 a year from now. What will that bond cost you? At current interest rates, a secure bond37 will cost around $9700. This means 36 Arbitrage is the process of buying and selling in such a way to make a profit. For example, if wheat is selling for $3 per bushel in New York, but $2.50 per bushel in Chicago, one can buy in Chicago and sell in New York and profit by $0.50 per bushel, minus any tr ansaction and transportation cost. Such arbitrage tends to force prices to differ by no more than tran saction costs. When these transaction costs are small, as with gold, prices will be about the same worldwide. 37 Economists tend to consider US federal government securi ties secure, because the probability of such a default is very, very low.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-115that no one should willing to pay $10,000 for a future payment of $10,000, because instead one can have the future $10,000, by bu ying the bond, and have $300 left over to spend on cappuccinos or economics textbooks. In other words, if you will pay $10,000 for a secure promise to repay the $10,000 a year hence, then I can make a successful business selling you the secure promise for $10,000, and pocketing $300. This arbitrage consideration also suggests ho w to value future payments: discount them by the relevant interest rate. Example (Auto loan): You are buying a $20,000 car, and you are offered the choice to pay it all today in cash, or to pay $21,000 in one year. Should you pay cash (assuming you have that much in cash) or take the loan? The loan is at a 5% annual interest rate, because the repayment is 5% higher than the lo an amount. This is a good deal for you if your alternative is to borrow money at a higher interest rate, e.g. on (most) credit cards. It is also a good deal if you have savings th at pay more than 5% -if buying the car with cash entails cashing in a certificate of deposi t that pays more than 5%, then you would be losing the difference. If, on the other ha nd, you are currently saving money that pays less than 5% interest, paying off the car is a better deal. The formula for present value is to discount by the amount of interest. Let’s denote the interest rate for the next year as r1, the second year’s rate as r2, and so on. In this notation, a $1 invested would pay $1+ r1 next year, or $(1+ r1) (1+ r2) after 2 years, or $(1+ r1) (1+ r2) (1+ r3) after 3 years. That is, ri is the interest rate that determines the value, at the end of year i of $1 invested at the start of year i Then, if we obtain a stream of payments A0 immediately, A1 at the end of year 1, A2 at the end of year 2, and so on, the present value of that stream is PV = ... ) 1 )( 1 )( 1 ( ) 1 )( 1 ( 13 2 1 2 2 1 2 1 1 0 r r r A r r A r A A Example (Consolidated annuities or Consols ): What is the value of $1 paid at the end of each year forever, with a fixed interest rate r ? Suppose the value is v Then38 1 1 1 1 1 1 ... ) 1 ( 1 ) 1 ( 1 1 13 2r r r r r v At a 5% interest rate, $1 million per year pa id forever is worth $20 million today. Bonds that pay a fixed amount every year forever ar e known as consols; no current government issues consols. Example (Mortgages): Again, fix an interest rate r but this time let r be the monthly interest rate. A mortgage implies a fixe d payment per month for a large number of 38 This development uses the formula, for -1
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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-116months (e.g. 360 for a 30 year mortgage). What is the present va lue of these payments over n months? A simple way to compute this is to use the cons ol value, because .... ) 1 ( 1 ) 1 ( 1 ) 1 ( 1 1 ) 1 ( 1 ... ) 1 ( 1 ) 1 ( 1 1 13 2 1 3 2 n n n nr r r r r r r r M ... ) 1 ( 1 ) 1 ( 1 ) 1 ( 1 ) 1 ( 1 13 2r r r r rn ) 1 ( 1 1 1 1 ) 1 ( 1 1 n nr r r r r Thus, at a monthly interest rate of %, payi ng $1 per month for 360 months produces a present value M of 79 166 ) 005 1 ( 1 1 005 1360 Thus, to borrow $100,000, one would have to pay $599.55 79 166 000 100 $ per month. It is important to remember that a different loan amount just changes the scale; borrowing $150,000 requires a payment of $899.33 79 166 000 150 $ per month, because $1 per mont h generates $166.79 in present value. Example (Simple and Compound Interest): In the days before calculators, it was a challenge to actually solve interest rate form ulas, so certain simplifications were made. One of these was “simple” interest, whic h means that daily or monthly rates are translated into annual rates by incorrect form ulas. For example, with an annual rate of 5%, the simple interest daily rate is % 07692 365 % 5 That this is incorrect can be seen from the calculation that % 051267 1 365 05 1365 Simple interest increases the annual rate, so it benefits lenders and harm s borrowers. (Consequently, banks advertise the accurate annual rate on savings accounts – when consumers like the number to be larger – and not on mortgages, although bank s are required by law to disclose – but not to advertise widely – actual annual interest on mortgages.) Obligatory Lottery Example: You win the lott ery, and the paper reports you’ve won $20 million. You’re going to be paid $20 million but is it worth $20 million? In fact, you get $1 million per year for 20 years. Howeve r, in contrast to our formula, you get the first million right off the bat, so the value is ) 1 ( 1 1 1 1 ) 1 ( 1 ... ) 1 ( 1 ) 1 ( 1 1 1 119 19 3 2 r r r r r r PV

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-117Table 3.1 computes the present value of our $20 million dollar lottery, listing the results in thousands of dollars, at various interest rate s. At ten percent interest, the value of the lottery is less than half the “number of dolla rs” paid, and even at 5%, the value of the stream of payments is 65% of the face value. r 3% 4% 5% 6% 7% 10% PV (000s) $15,324 $14,134 $13,085 $12,158 $11,336 $9,365 The lottery example shows that interest rate s have a dramatic impact on the value of payments made in the distant future. Present value analysis is the number one tool used in MBA programs, where it is known as Ne t Present Value or NPV analysis. It is accurate to say that the majority of corpor ate investment decisions are guided by an NPV analysis. Example (Bond prices): A standard treasury bill has a fixed future value. For example it may pay $10,000 in one year. It is sold at a discount off the face value, so that a oneyear $10,000 bond might sell for $9,615.39, producing a 4% interest rate. To compute the effective interest rate r the formula relating the future value FV the number of years n and the price is Price 1 FV rn or 1 Price1 nFV r We can see from either formula that treasury bill prices move inversely to interest rates – an increase in interest rates reduces treasury prices. Bonds are a bit more complicated. Bonds pay a fixed interest rate set at the time of issue during the life of the bond, generally collected semi-annually, and th e face value is paid at the end of the term. These bonds were often sold on long te rms, as much as 30 years. Thus, a threeyear $10,000 bond at 5% with semi-annual pa yments would pay $250 at the end of each half year for three years, and pay $10,000 at the end of the three years. The net present value, with an annual interest rate r is 1 10000 $ 1 250 $ 1 250 $ 1 250 $ 1 250 $ 1 250 $ 1 250 $3 3 2 5 2 2 3 1 2 1r r r r r r r NPV The net present value will be the price of th e bond. Initially, the price of the bond should be the face value, since the interest rate is set as a market rate. The U.S. Treasury quit issuing such bonds in 2001, replacing them with bonds in which the face value is paid and then interest paid semi-annually. (Exercise) At a 7% annual interest rate, what is the present value of $100 paid at the end of one year, and $200 paid at the end of the second year? (Exercise) Compute the NPV of the 3 year, $10,000 bond, with $250 payments semi-annually, that was described abov e, at an interest rate of 4%.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-1184.3.1.3 (Exercise) You can finance your $20,000 car with a straight 5% loan paid monthly over 5 years, or get one year interest free, but then pay 7% over the following four years. Which is a better de al? (Hint: In both cases, figure out the fixed monthly payments that produce a net present value equal to $20,000.) (Exercise) You win the lottery. At what interest rate should you accept $7 million today over twenty annu al payments of $500,000? 4.3.2 Investment A simple investment project involves spending an investment, I and then reaping a return over time. If you dig a mine, drill an oil well, build an apartment building or a factory, or buy a share of stock, you spen d money now, in the hopes of earning money subsequently. We will set asid e the very important risk issu e until the next subsection, and ask how to make the decision to invest. The NPV approach involves assigning a rate of return r that is reasonable for, and specific to, the project and th en computing the present value of the expected stream of payments. Since the investment is initially expended, it is counted as negative revenue. This gives an expression that looks like: .... ) 1 ( ) 1 ( 13 3 2 2 1 r R r R r R I NPV where R1 represents first year revenues, R2 represents second year revenues, etc.39 The investment is then made when NPV is positive – since this would add to the net value of the firm. Carrying out an NPV analysis essentially requ ires two things. First, investment and revenues must be estimated. This is a ch allenge, especially fo r new products where there is no direct way of estimating demand, or with uncertain outcomes like oil wells or technological research.40 Second, an appropriate rate of return must be identified. The rate of return is a problem, mostly because of risk associated with the payoffs to the investment, but also because of the incentives of project managers to inflate the payoffs and minimize the costs to make the project look more attractive to upper management. In addition, most investment undertaken by corporations is financed not with borrowing but with retained earnings, that is, with profits from previous activities. Thus a company that undertakes one investment can’t carry out some other investment, and the interest rate has to account for the internal corporate value of funds. As a result of these factors, interest rates of 15%-20% ar e common for evaluating the NPV of projects of major corporations. 39 The most common approach is to treat revenues with in a year as if they are received at the midpoint, and then discount appropriately for that mid-year poin t. The present discussion oversimplifies in this regard. 40 The building of the famed Sydney Opera House, wh ich looks like billowing sails over Sydney harbor, was estimated to cost $7 million and actually cost $1 05 million. A portion of the cost overrun was due to the fact that the original design neglected to install air conditioning. When this oversight was discovered, it was too late to install a standard unit, which would interfere with the excellent acoustics, so instead an ice hockey floor was installed as a means of cooling the building.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-119 Example (Silver Mine): A company is consider ing whether to develop a silver mine in Mexico. The company estimates that develo ping the mine (building roads and opening a large hole in the ground) would require $4 million per year for four years and no revenues would accrue during this time. St arting in year 5 the expenses fall to $2 million per year, and $6 million in net revenue is earned off the mined silver for each of the subsequent 40 years. If the company va lues funds at 18%, should it develop the mine? The earnings from the mine are calculated in the following table. First, the NPV of the investment phase during years 0, 1, 2, and 3 is 697 12 ) 18 1 ( 4 ) 18 1 ( 4 18 1 4 43 2 NPV A dollar earned in each of years 4 through 43 have a present value of 377 13 ) 1 ( 1 1 1 ) 1 ( 1 ) 1 ( 1 ... ) 1 ( 1 ) 1 ( 1 ) 1 ( 140 3 43 6 5 4 r r r r r r r The mine is just profitable at 18%, in spite of the fact that its $4 million payments are made in four years, after which point $4 million dollar revenues are earned for forty years. The problem in the economics of mini ng is that 18% makes those future revenues have quite modest present values. Year Earnings ($M) / yr PV ($M) 0-3 -4 -12.697 4-43 4 13.377 Net 0.810 There are other approaches to deciding whether to take an investment. In particular, the Internal Rate of Return approach solves the equation NPV=0 for the interest rate, and then the project is undertaken if the ra te of return is sufficiently high. This approach is flawed because the equation may have more than one solution, or no solutions and it is not transparent what the ri ght thing to do should be in these events. Indeed, the IRR approach gets the profit-maximiz ing answer only if it agrees with NPV. A second approach is the payback period, wh ich asks how many years a project must be run before profitability is reached. The problem with the payback period is deciding between projects – if I can only do one of two projects, the one with the higher NPV makes the most money for the company. Th e one with the faster payback may make a quite small amount of money very quickly; it isn’t apparent that this is a good choice. When a company is in risk of bankruptcy, a short payback period might be valuable, although this would ordinarily be handled by employing a higher interest rate in an NPV analysis. NPV does a good job when the ques tion is whether to undertake a project or not, and it does better than other approaches to investment decisions. For this reason, NPV has become the most common approach to investment decisions. Indeed, NPV analysis is more common than all other appr oaches combined. NPV does a poor job,

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-120however, when the question is whether to unde rtake a project, or delay the project. That is, NPV answers “yes or no” to investment, bu t when the choice is “yes or wait,” NPV requires amendment. (Exercise) Suppose that, without a university education, you’ll earn $25,000 per year. A university education costs $20,000 per year, and you forgo the $25,000/year you would have earned fo r four years. However, you earn $50,000 per year for the following forty years. At 7%, what is the NPV of the university education? (Exercise) Now that you’ve decided to go to university based on the previous answer, suppose that you can attend Eas t State U, paying $3,000 per year for four years and earning $40,000 when you graduate, or North Private U, paying $22,000 per year for the four years and earning $50,000 when you graduate. Which is the better deal at 7%? 4.3.3 Investment Under Uncertainty Risk has a cost, and people, and corporatio ns, buy insurance against financial risk.41 The standard approach to investment under un certainty is to compute an NPV, with the revenues composed of expected values, an d the interest rate used adjusted to compensate for the risk. For example, consider a project like oil explor ation. The risks are enormous. Half of all underwater tracts in the Gulf Coast near Louisiana and Texas that are leased are never drilled, because later information makes them a bad bet. Half of all the tracts that are drilled are dry. So right off the bat, threequarters of the tracts that are sold produce zero or negative revenue, and positive costs. To see how the economics of such a risky investment might be developed, suppose that the relevant rate of return for such investments is 18%. Suppose further the tr act can be leased for $500,000 and the initial exploration costs $1 million. If the tract has oil (with a 25% probability), it produces $1 million per year for twenty years, and then ru ns dry. This gives an expected revenue of $250,000 per year. To compute the expected net present value, we first compute the returns: Table 4-1: Oil Tract Return Expected revenue EPV 0 -$1.5M -$1.5M 1-20 $0.25M $1.338M Net -$0.162 At 18%, the investment is a loss – the ri sk is too great given the average returns. A very important consideration for investment under uncertainty is the choice of interest rate. The most important thing to unde rstand is that the interest rate is specific 41 For example, NBC spent $6 million in buying an in surance policy against US nonparticipation in the 1980 Moscow summer Olympic games, and the US didn’t participate (because of the Soviet invasion of Afghanistan), and NBC was paid $94 million from the policy.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-121to the project, and not to the investor. Th is is perhaps the most important insight of corporate finance generally: the interest rate should adjust for the risk associated with the project and not the invest or. For example, suppose hamburger retailer McDonald’s is considering investing in a cattle ranch in Pe ru. McDonald’s is overall a very low-risk firm, but this particular project is quite risk y, because of local conditions. McDonald’s still needs to adjust for the market value of th e risk it is undertaking, and that value is a function of the project risk, not the ri sk of McDonald’s other investments. This basic insight of corporate finance – the appropriate interest rate is determined by the project, not the investor – is counter-intu itive to most of us because it doesn’t apply to our personal circumstances. For individual s, the cost of borrowing money is mostly a function of their own personal circumstances, and thus the decision of whether to pay cash for a car or borrow the money is not so much a function of the car being purchased but of the wealth of the borrower. Even so, personal investors borrow money at distinct interest rates. Mortgage rates on houses ar e lower than interest rates on automobiles, and interest rates on automobiles lower than on credit cards. This is because the “project” of buying a house has less risk a ssociated for it: the percentage loss to the lender in event of borrower default is lower on a house than on a car. Credit cards carry the highest interest rates because they are unsecured by any asset. One way of understanding why the interest ra te is project-specific but not investorspecific is to think about undertaking the proj ect by creating a separate firm to make the investment. The creation of subsidiary un its is a common strategy, in fact. This subsidiary firm created to operate a project has a value equal to the NPV of the project using the interest rate specific to the subsidia ry, which is the interest rate for the project, independent of the parent. For the parent company, owning such a firm is a good thing if the firm has positive value, and not otherwise.42 Investments in oil are subject to another kind of uncertainty: price risk. Prices of oil fluctuate and aren’t constant. Moreover, oil pumped and sold today is not available for the future. Should you develop and pump the oil you have today, or should you hold out and sell in the future? Th is question, known as the option value of investment is generally somewhat challenging and arcane, but a simple example provides a useful insight. To develop this example, let’s set aside some extraneous issues first. Consider a very simple investment, in which either C is invested or not.43 If C is invested, a value V is generated. The cost C is a constant; it could correspond to drilling or exploration costs, or in the case of a stock option, the strike price of the option, which is the amount one pays to obtain the share of stock. The value V in contrast, varies from time to time in a random fashion. To simplify the analysis, we assume that V is uniformly distributed on the interval [0,1], so that the probability of V falling in an interval [ a, b ] is b a if 0 a b 1. The option only has value if C <1, which we assume for the rest of this section. 42 It may seem that synergies between parent and subsidiary are being neglected here, but synergies should be accounted for at the time they produce valu e, i.e. as part of the stream of revenues of the subsidiary. 43 This theory is developed in striking gene rality by Avinash Dixit and Robert Pindyck, Investment Under Uncertainty, Princeton University Press, 1994

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-122 The first thing to note is that the op timal rule to make the investment is cutoff value that is, to set a level V0 and exercise the option if, and only if, V V0. This is because, if you are willing to exercise the option and generate value V you should be willing to exercise the option and obtain even mo re value. The NPV rule simply says V0 = C that is, invest whenever it is profitable. The purpose of the example developed below is to provide some insight into how far wrong the NPV rule will be when option values are potentially significant. Now consider the value of option to in vest, given that the investment rule V V0 is followed. Call this option value J ( V0). If the realized value V exceeds V0, one obtains V C Otherwise, one delays the investment, producing a discounted level of the same value. This logic says ) ( 1 1 2 1 ) 1 ( ) (0 0 0 0 0 V J r V C V V V J This expression for J ( V0) arises as follows. First, th e hypothesized distribution of V is uniform on [0,1]. Consequently, the value of V will exceed V0 with probability 1V0. In this event, the expected value of V is the midpoint of the interval [ V0, 1], which is ( V0+1). The value ( V0+1) C is the average payoff from the strategy of investing whenever V V0, which is obtained with probability 1V0. Second, with probability V0, the value falls below the cutoff level V0. in this case, no investment is made, and instead, we wait until the next period. The expected profits of the next period are J ( V0) and these profits are discounted in the standard way. The expression for J is straightforward to solve: 1 1 2 1 ) 1 ( ) (0 0 0 0r V C V V V J Rudimentary calculus shows 1 1 ) 1 ( 2 ) 1 ( 2 2 1 ) (2 0 0 2 0 0 r V r V r V rC V J First, note that 0 ) ( C J and 0 ) 1 ( J, which together imply the existence of a maximum at a value V0 between C and 1, satisfying 0 ) (0 V J Second, the solution occurs at ) 1 ( 2 ) 1 ( ) 2 1 ( ) 1 ( ) 1 (2 2 0C r r r rC r r V

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-123 The positive root of the quadratic has V0>1, which entails never investing, and hence is not a maximum. The profit-maximizing invest ment strategy is to invest whenever the value exceeds V0 given by the negative root in the formula. There are a couple of notable features about this solution. First, at r =0, V0 = 1. This is because r =0 corresponds to no discounting, so there is no loss in holding ou t for the highest possible value. Second, as r V0 C As r the future is valueless, so it is worth investing if the return is anything over costs. These are not surprising findings, quite the opposite – they should hold in any reasonable formulation of such an investment strategy. Moreover, they show that the NPV rule, which requires V0 = C is correct only if the future is valueless. How does this solution behave? The solution is plotted as a function of r for C =0, 0.25 and 0.5, in Figure 4-21. The horizontal axis represents inte rest rates, so this picture shows very high interest rates by current standards, up to 200%. Even so, V0 remains substantially above C That is, even when the future has very little value because two-thirds of the value is destroyed by discounting each period, the op timal strategy deviates significantly from the NPV strategy. Figure 4-22 shows a close-up of that picture for a more reasonable range of interest rates, for interest rates of zero to ten percent Figure 4-21: Investment Strike Price Given Interest Rate r in Percent Figure 4-22 shows the cutoff values of in vestment for three values of C the cost of the investment. These three values are 0 (lowes t curve), 0.25 (the middle dashed curve), and 0.5, the highest, dotted line. Consider the lowest curve, with C =0. The NPV of this project is always positive – there are no costs and re venues are positive. Nevertheless, because the investment can only be made once, it pays to hold out for a higher level of payoff, indeed, for 65% or more of the maxi mum payoff. The economics at an interest rate of 10% is as follows. By waiting, th ere is a 65% chance that ten percent of the potential value of the investment is lost. However, there is a 35% of an even higher value. The optimum value of V0 trades these considerations off against each other. 50 100 150 200 0.2 0.4 0.6 0.8 C=0.5 C=0.25 C=0

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-124For C = 0.25, at 10% the cutoff value for taking an investment is 0.7, nearly three times the actual cost of the investment. Indeed the cutoff value incorporates two separate costs: the actual expenditure on the investment C and the lost opportunity to invest in the future. The latter cost is much larger than the expenditure on the investment in many circumstances, and in th is example, can be quantitati vely much larger than the actual expenditure on the investment. Some investments can be replicated. There are over 13,000 McDonald’s restaurants in the United States, and building another doesn’ t foreclose building even more. For such investments, NPV analysis gets the right an swer, provided that appropriate interest rates and expectations are used. Other investments are difficult to replicate or logically impossible to replicate – having pumped and sold the oil from a tract, that tract is now dry. For such investments, NPV is consiste ntly wrong because it neglects the value of the option to delay the investment. A correct analysis adds a lost va lue for the option to delay the cost of the investment, a value whic h can be quantitatively large, as we have seen. Figure 4-22 Investment Strike Price Given Interest Rate r in Percent Example: When should you refinance a mort gage? Suppose you are paying 10% on a $100,000 mortgage, and it costs $5,000 to re finance, but refinancing permits you to lock in a lower interest rate, and hence pay less. When is it a good idea? To answer this question, we assume that the $5,000 cost of re financing is built into the loan, so that in essence you borrow $105,000 at a lower intere st rate when you refinance. This is actually the most common method of refinancing a mortgage. To simplify the calculations, we will consider a mortgage that is never paid off, that is, one pays the same amount per year forever. If the mortgage isn’t refinanced, one pays ten percent of the $100,000 face value of the mortgage each year, or $10,000 per year. If one refinances at interest rate r one pays r $105,000 per year, so the NPV of refinancing is 2 4 6 8 10 0.65 0.7 0.75 0.8 0.85 0.9 0.95 C=0.25 C=0.5 C=0

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-125NPV = $10,000 r $105,000. Thus NPV is positive whenever % 52 9 105 10 r Should you refinance when the interest rate drop s to this level? No. At that level, you would exactly break even, but would also be carrying a $105,000 mortgage rather than a $100,000 mortgage, making it harder to be nefit from any further interest rate decreases. The only circumstance in which refi nancing at 9.52% is sensible is if interest rates can’t possibly fall further. When should you refinance? That depends on the nature and magnitude of the randomness governing interest rates, preferen ces over money today versus money in the future, and attitudes to risk. The model developed in this section is not a good guide to answering this question, primar ily because the interest rates are strongly correlated over time. However, an approximate guide to impl ementing the option theory of investment is to seek an NPV of twice the investment, which would translate into a refinance point of around 8.5%. (Exercise) You are searching for a job. The net value of jobs that arise is uniformly distributed on the interval [0,1 ]. When you accept a job, you must stop looking at subsequent jobs. If you can interview with one employer per week, what jobs should you accept? Use a 7% annual interest rate. Hint: Relate the job search problem to the in vestment problem, where accepting a job is equivalent to making the investment. What is c in the job search problem? What is the appropriate interest rate? 4.3.4 Resource Extraction For the past sixty years, the wo rld has been “running out of oil.” There are news stories about the end of the reserves being only ten, fifteen or twenty years away. The tone of these stories is that, at that time, we will run out of oil completely and prices will be extraordinarily high. Industry studies counte r that more oil continues to be found and that the world is in no danger of running out of oil. If you believe that the world will run out of oil, what should you do? You should buy and hold That is, if the price of oil in twenty years is going to be $1,000 per barrel, then you can buy oil at $40 and hold it for twenty years, and sell it at $1,000. The rate of return from this behavior is the solution to 40 1000 ) 1 (20 r This equation solves for r = 17.46%, which represents a healthy rate of return on investment. This substitution is part of a general conclusion known as the Ramsey44 44 The solution to this problem is known as Ramsey pricing, after the discoverer Frank Ramsey (19031930).

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-126rule : for resources in fixed supply, prices rise at the interest rate. With a resource in fixed supply, owners of the resource will se ll at the point maximizing the present value of the resource. Even if they do not, others can buy the resource at the low present value of price point and resell at the high present value, and make money. The Ramsey rule implies that prices of resource s in fixed supply rise at the interest rate. An example of the Ramsey rule in action concerns commodities that are temporarily fixed in supply, such as grai ns, after the harvest. During the period between harvests, these products rise in price on average at the interest rate, where the interest rate includes storage and insurance costs, as well as the cost of funds. Example: Let time run t = 0, 1, … and suppose the demand for a resource in fixed supply has constant elasticity: 1) ( aQ Q p Suppose there is a total stock R of the resource, and the interest rate is fixed at r What is the price and consumption of the resource at each time? Solution: Let Qt represent the quantity consumed at time t Then the arbitrage condition requires: 1 0 1 0) ( ) 1 )( ( ) 1 (t t t taQ Q p r Q p r aQ Thus, t tr Q Q ) 1 (0. Finally, the resource constraint implies ) 1 ( 1 ... ) 1 ( ) 1 ( 1 ...) (0 2 0 2 1 0 r Q r r Q Q Q Q R This solves for the initial consumption Q0. Consumption in future periods declines geometrically, thanks to the co nstant elasticity assumption. Market arbitrage insures the availability of the resource in the future, and drives the price up to ration the good. The world runs out slowly, and the price of a resource in fixed supply rises on average at the interest rate. Resources like oil and minerals are ostensibly in fixed supply – there is only so much oil, or gold, or bauxite, or palladium in the earth. Markets, however, behave as if there is an unlimited supply, and with good reason. Peop le are inventive, and find substitutes. England’s wood shortage of 1651 didn’t resu lt in England being cold permanently, nor was England limited to the wood it could grow as a source of heat. Instead, coal was discovered. The shortage of whale oil in the mid-nineteenth century led to the development of oil resources as a replacement. If markets expect that price increases will lead to substitutes, then we rationa lly should use more today, trusting that

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-127technological developments will provide substitutes.45 Thus, while some believe we are running out of oil, most investors are betting that we are not, and that energy will not be very expensive in the future, either because of continued discovery of oil, or because of the creation of alternative energy sources. If you disagree, why not invest and take the bet? If you bet on future price increases, that will tend to increase the price today, encouraging conservation today, and increase the supply in the future. (Exercise) With an elasticity of demand of 2, compute the percentage of the resource that is used each year if the interest rate is 10%. If the interest rate falls, what happens to the proportion quantity used? 4.3.5 A Time to Harvest A tree grows slowly, but is renewabl e, so the analysis of Section 4.3.4 doesn’t help us understand when it is most profitable to cu t the tree down. Consider harvesting for pulp and paper use. In this use, the amount of w ood chips is what matters to the profitability of cutting down the tree, and the biomass of th e tree provides a direct indication of this. Suppose the biomass sells for a net price p which has the costs of harvesting and replanting deducted from it and the biomass of the tree is b ( t ) when the tree is t years old. It simplifies the analysis slight ly to use continuous time discounting t tr e 1 1 where ) 1 log( r Consider the policy of cutting down trees when they are T years old. This induces a cutting cycle of length T A brand new tree will produce a present value of profits of: 1 ) ( 1 ) ( ) ( ) ( ) (3 2 T T T T T Te T pb e T pb e T pb e T pb e T pb e. This profit arises because th e first cut occurs at time T with discounting e-T, and produces a net gain of pb ( T ). The process then starts ov er, with a second tree cut down at time 2T and so on. Profit maximization gives a first order condition on the optimal cycle length T of 21 ) ( 1 ) ( 1 ) ( 0 T T T Te e T pb e T b p e T pb dT d. This can be rearranged to yield: 45 Unlike oil and trees, whales were overfished and th ere was no mechanism for arbitraging them into the future, that is, no mechanism for capturing and savin g whales for later use. This problem, known as the tragedy of the commons, results in too much use and is taken up in Section 6.3.6. Trees have also been over-cut, most notably on Easter Island.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-128 Te T b T b 1 ) ( ) (. The left hand side of this equation is the grow th rate of the tree. The right hand side is approximately the continuous-time discount factor, at least when T is large, as it tends to be for trees, which are usually on a 20 to 80 year cycle, depending on the species. This is the basis for a conclusion: cut down th e tree slightly before it is growing at the interest rate. The higher are interest rate s, the shorter the cycl e on which the trees should be cut down. The pulp and paper use of trees is special, be cause the tree is going to be ground up into wood chips. What happens wh en the object is to get boards from the tree, and larger boards sell for more? In particular, it is more profitable to get a 4 4 than two 2 4s. Doubling the diameter of the tree, which appr oximately raises the biomass by a factor of six to eight, more than increases the value of the timber by the increase in the biomass. It turns out our theory is already capable of ha ndling this case. The only adaptation is a change in the interpretation of the function b Now, rather than representing the biomass, b ( t ) must represent the value in boards of a tree that is t years old. (The parameter p may be set to one.) The only amendment to the rule for cutting down trees is that the most profitable point in time to cut down the tree occurs slightly before the time when the value (in boards) of the tree is growing at the interest rate. For example, lobsters become more valuable as they grow; the profit-maximizing time to harvest lobsters is governed by the same equation, where b ( T ) is the value of a lobster of age T Prohibiting the harves t of lobsters under age T is a means of insuring the profit-maximizing capture of lobsters, and prev enting over-fishing, a topic considered in section 6.3.6. The implementation of the formula is illustrated in Figure 4-23. The dashed line represents the growth rate ) ( ) ( T b T b while the solid line represents the discount rate, which was set at 5%. Note that the best ti me to cut down the trees is when they are approximately 28.7 years old, and at that time, they are growing at 6 %. Figure 4-23 also illustrates another feature of the optimi zation – there may be multiple solutions to the optimization problem, and the pr ofit-maximizing solution involves ) ( ) ( T b T b cutting Te 1 from above.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-129 Figure 4-23: Optimal Solution for T The U.S. Department of the Interior is in charge of selling timber rights on federal lands. The Department uses the policy of maximum sustainable yield to determine the time that the tree is cut down. Maximum sustainable yield maximizes the long-run average value of the trees cut down, that is, it maximizes T T b ) ( (Exercise) Show maximum sustainable yield results in cutting down the tree when it is T years old, where T satisfies T T b T b 1 ) ( ) ( Maximum sustainable yield is actually a specia l case of the policies considered here, and arises for a discount factor of 0. It turn s out (thanks to a formula known variously as L’Hpital’s or L’Hospital’s rule) that 1 1 lim0T eT Thus, the rule T e T b T bT1 1 ) ( ) ( as 0, and this is precisel y the same rule that arises under maximum sustainable yield. Thus, the Department of the Interior acts as if th e interest rate is zero, when it is not. The justification given is that the Department is valuing future generations at the same level as current generations, that is, increasi ng the supply for future generations, while slightly harming the current generation of buyers. The major consequence of the 10 20 30 40 5 10 15 20 25 30 Growth Rate T e 1 T

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-130Department’s policy of maximum sustainable yield is to force cutting of timber even when prices are low during recessions. (Exercise) Suppose the growth rate of trees satisfies tte T b T b ) ( ) ( Numerically approximate the efficient time to cut the tree if =0.1. How does this compare to the solution of maximum sustainable yield? 4.3.6 Collectibles Many people purchase durable goods as in vestments, including Porsche Speedsters, Tiffany lamps, antique telephones, postage stamps and coins, baseball cards, original Barbie dolls, antique credenzas, autogr aphs, original rayon Hawaiian shirts, old postcards, political campaign buttons, old cloc ks and even Pez dispensers. How is the value of, say, a 1961 Porsche Speedster or a $500 bill from the confederacy, which currently sells for over $500, determined? Figure 4-24: The Porsche Speedster The theory of resource prices can be adapted to cover these items, which are in fixed supply. There are four major differences that are relevant. First, using the item doesn’t consume it; the goods are durable. I can own an “I Like Ike” campaign button for years, then sell the same button. Second, these it ems may depreciate. Cars wear out even when they aren’t driven, and the brilliant color of Pez disp ensers fades. Every time a standard 27 pound gold bar, like the kind in the Fort Knox depository, is moved, approximately $5 in gold wears off the bar. Third, the goods may cost something to store. Fourth, the population grows, and some of the potential buyers are not yet born. To understand the determinants of the prices of collectibles, it turns out to create a major simplification to perform the an alysis in continuous time. Let t, ranging from zero to infinity, be the continuous time va riable. If the good depreciates at rate and q0 is the amount available at time 0, the quantity available at time t is te q t q 0) (. For simplicity, assume that there is constant elasticity of demand If g is the population growth rate, the quantity demanded, for any price p, is given by

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-131 p ae t p xgt d) (, for a constant a which represents the demand at time 0. This represents demand for the good for direct use, but neglects the investme nt value of the good – that the good can be resold for a higher price later. In other words, xd captures the demand for looking at Pez dispensers or driving Porsche Speedsters, bu t does not incorporate the value of being able to resell these items. The demand equation can be used to generate the lowest use value to a person owning the good at time t That marginal use value v arises from the equality of supply and demand: v ae t v x t q e qgt d t) ( ) (0 or t ge q a v) ( 0 Thus, the use value to the margin al owner of the good at time t satisfies t ge q a v 1 0. An important aspect of this development is that the value to the owner is found without reference to the price of the good. The reas on this calculation is possible is that the individuals with high values will own the go od, and the number of goods and the values of people are assumptions of the theory. Es sentially, we already know that the price will ration the good to the individuals with hi gh values, so computing the lowest value individual who holds a good at time t is a straightforward “supply equals demand” calculation. Two factors increase the marginal value to the owner – there are fewer units available because of depreciation, and there are more high-value people demanding them, because of population gr owth. Together, these factors make the marginal use value grow at the rate g. Assume that s is the cost of storage per unit of time and per unit of the good, so that storing x units for a period of length costs sx This is about the simplest possible storage cost technology. The final assumption that we make is that all potential buyers use a common discount rate r, so that the discount of money or value received units of time in the future is e-r. It is worth a brief digression why it is sensible to assume a common discount rate, when it is evident that many people have different discount rates. Di fferent discount rates induce gains from trade in borrowing and le nding, and create an incentive to have

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-132banks. While banking is an interesting thin g to study, this section is concerned with collectibles, not banks. If we have differe nt discount factors, then we must also introduce banks, which would complicate the model substantially. Otherwise, we would intermingle the theory of bankin g and the theory of collectibles. It is probably a good idea to develop a joint theory of banking an d collectibles given the investment potential of collectibles, but it is better to start wi th the pure theory of either one before developing the joint theory. Consider a person who values the collectible at v. Is it a good thing for this person to own a unit of the good at time t? Let p be the function that gives the price across time, so that p(t) is the price at time t. Buying the good at time t and then selling what remains (recall that the good depreciates at rate ) at time t+ gives a net value of ) ( ) ( ) (0 t p e e t p du s v er ru. For the marginal person, that is, the person who is just indifferent to buying or not buying at time t, this must be zero at every moment in time, for =0. If v represents the value to a marginal buyer (indifferent to ho lding or selling) holding the good at time t, then this expression should come out to be zero. Thus, dividing by ) ( ) ( ) ( 1 lim 0) ( 0 0t p e t p du s v er ru ) ( ) ( ) ( ) ( 1 ) ( ) ( lim) ( 0t p r t p s v t p e t p t p s vr Recall that the marginal value is t ge q a v 1 0, which gives t ge q a s t p r v s t p r t p 1 0) ( ) ( ) ( ) ( ) (. The general solution to this differential equation is g r e q a s r e p e t pt g r t r t r1 ) ( 1 ) 0 ( ) (1 0 ) ( ) (.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-133It turns out that this equation only makes sense if 0 g r for otherwise the present value of the marginal value goes to in finity, so there is no possible finite initial price. Provided demand is elastic and discou nting is larger than growth rates (which is an implication of equilibrium in the cred it market), this condition will be met. What is the initial price? It must be the case that the present value of the price is finite, for otherwise the good would always be a good investment for everyone at time 0, using the “buy and hold for resale” strategy. That is, ) ( lim t p ert t This condition implies g r e q a s r e p et g r t r t t1 ) ( 1 ) 0 ( lim1 0 ) ( and thus 0 1 ) ( 1 ) 0 (1 0 g r q a s r p. This equation may take on two different form s. First, it may be solvable for a nonnegative price, which happens if 0 ) ( 1 1 ) 0 (1 0 s r g r q a p. Second, it may require destruction of some of the endowment of the good. Destruction must happen if the quantity of the good q0 at time 0 satisfies 0 ) ( 1 11 0 s r g r q a. In this case, there is too much of the good and an amount must be destroyed to make the initial price zero. Since the initial price is zero, the good is valueless at time zero, and destruction of the good makes sense – at the current quantity, the good is too costly

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-134to store for future profits. Enough is de stroyed to insure indifference between holding the good as a collectible and destroying it. Consider, for example, the $500 confederate bill pictured in Figure 4-25. Many of these bills were destroyed at the end of the US Civil War, when the currency became valueless, burned as a source of heat. Now, an uncirculated version retails for $900. The amount of the good that must be destroyed is such that the initial price is zero. As q0 is the initial (pre-destruction) quantity, th e amount at time zero after the destruction is the quantity q(0) satisfying s r g r q a p ) ( 1 1 ) 0 ( ) 0 ( 01 Figure 4-25: $500 Confederate States Bill Given this construction, we have that 0 1 ) 0 ( ) ( 1 ) 0 (1 g r q a s r p where either q (0)= q0 and p (0) 0, or q (0)< q0 and p (0)=0. Destruction of a portion of the stock of a collectible, followed by price increases, is actually a quite common phenomenon. In particular, consider the “Model 500” telephone by Western Electric illustrated in Figure 4-26. This ubiquitous classic phone was retired as the US switched to tone dial ing and push-button phones in the 1970s, and millions of phones – perhaps over 100 million – wound up in landfills. Now, the phone is a collectible and rotary phone enthus iasts work to keep them operational.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-135 Figure 4-26: Western Electric Model 500 Telephone The solution for p (0) dramatically simplifies the expression for p ( t ): g r e q a s r e p e t pt g r t r t r1 ) 0 ( ) ( 1 ) 0 ( ) (1 ) ( ) ( g r e q a s r e et g r t r t r1 ) ( ) () 0 ( ) ( r s g r e q at g 1) 0 ( This formula lets us compare different collectib les. The first insight is that storage costs enter linearly into prices, so that growth rates are approximately unaffected by storage costs. That gold is easy to store, while st amps and art require control of humidity and temperature to preserve value and are hence more expensive to store, affects the level of prices but not the growth rate. However, depreciation and the growth of population affect the growth rate, and they do so in com bination with the demand elasticity. With more elastic demand, prices grow more slowly and start at a lower level. 4.3.7 Summer Wheat Typically, wheat harvested in the fall has to la st until the following harvest. How should prices evolve over the season? If I know that I need wheat in January, should I buy it at harvest time and store it myself, or wait and buy it in January? We can use a theory analogous to the theory of collectibles developed in Section 4.3.6 to determine the evolution of prices for commodities like wh eat, corn, orange juice, and canola oil.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-136Unlike collectibles, buyers need not hold for th eir personal use, since there is no value in admiring the wheat in your home. Let p ( t ) be the price at time t and suppose that the year has length T Generally there is a substantial amount of uncertainty regarding the size of wheat harvests and most countries ma intain an excess inventory as a precaution. However, if the harvest were not uncertain, there would be no need for a precautionary holding, and instead we would consume the enti re harvest over the course of a year, at which point the new harvest comes in. It is such a model that is investigated in this section. Let represent the depreciation rate (which for wheat includes the quantity eaten by rodents) and s be the storage cost. Buying at time t and reselling at t + should be a break-even proposition. If one purchases at time t it costs p ( t ) to buy the good. Reselling at t + the storage cost is about s (This is not the precisely relevant cost, but rather it is the present value of the storage cost, and hence the restriction to small values of .) The good depreciates to only have e left to sell, and discounting reduces the value of that amount by the factor re. For this to be a breakeven proposition, for small ) ( ) ( 0t p s t p e er or s t p e t p t pr ) ( 1 ) ( ) () (. Taking the limit as 0, s t p r t p ) ( ) ( ) (. This arbitrage condition insures that it is a break-even proposition to invest in the good; the profits from the price appreciation are ex actly balanced by depreciation, interest and storage costs. We can solve the differential equation to obtain: 1 ) 0 ( 1 ) 0 ( ) () ( ) ( ) ( ) (s r e p e s r e p e t pt r t r t r t r The unknown is p(0). The constraint on p(0), however, is like the resource extraction problem – p(0) is determined by the need to use up the harvest over the course of the year. Suppose demand has constant elasticity Then the quantity used comes in the form ) ( ) (t ap t x. Let z(t) represent the stock at time t. Then the equation for the evolution of the stock is ). ( ) ( ) (t z t x t z This equation is obtained by noting that

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-137the flow out of stock is composed of two elements: depreciation z and consumption x. The stock evolution equation solves for t u tdu u x e q e t z0) ( ) 0 ( ) (. Thus, the quantity of wheat is consumed exactly if ) 0 ( ) (0q du u x eT u. But this equation determines the initial price through T u r u r u T u T udu s r e p e a e du u ap e du u x e q0 ) ( ) ( 0 01 ) 0 ( ) ( ) ( ) 0 ( This equation doesn’t lead to a closed form for p (0) but is readily estimated, which provides a practical means of computing expected prices for commodities in temporarily fixed supply. Generally, the price equation produces a “s aw-tooth” pattern, which is illustrated in Figure 4-27. The increasing portion is actually an exponential, but of such a small degree that it looks linear. When the new ha rvest comes in, prices drop abruptly as the inventory grows dramatically, and the same pattern is repeated. 1 2 3 4 0.1 0.2 0.3 0.4 Figure 4-27: Prices over a Cycle for Seasonal Commodities

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McAfee: Introduction to Economic Analysis,, July 24, 2006 4-138 Log of Price of Gold versus Time2.67 2.68 2.69 2.7 2.71 2.72 2.73 2.74 2.75 2.76 5/28/200510/10/20062/22/20087/6/2009 DateLog Gold Price Figure 4-28: Log of Price of Gold over Time How well does the theory work? Figure 4-28 shows the log of the future price of gold over time. The relevant data comes from a futures market which establishes, at one moment in time, the price of gold for f uture delivery, and thus represents today’s estimate of the future price of gold. Thes e data, then, represent the expected future price at a particular moment in time (the afternoon of October 11, 2005), and thus correspond to the prices in the theory, since perceived risks are fixed. (Usually in the real world, risk plays a salient role.) We can observe that prices are approximately an exponential, because the log of prices is appr oximately linear. However, the estimate of r + is surprisingly low, at an annual level of less than 0.03, or 3% for both discounting and depreciation. Depreciation of gold is low, but this still represen ts a very low interest rate.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-1395 Consumer Theory Consumer theory is to demand as producer th eory is to supply. The major difference is that producer theory assumes that sellers are motivated by profit, and profit is something that one can usually directly meas ure. Moreover, the costs that enter into profit arise from physical properties of the production process – how many coffee cups come from the coffee cup manufacturing plant? In contrast, consumer theory is based on what people like, so it begins with so mething that we can’t directly measure, but must infer. That is, consumer theory is ba sed on the premise that we can infer what people like from the choices they make. Now, inferring what people like from choices they make does not rule out mistakes. But our starting point is to consider the implicat ions of a theory in which consumers don’t make mistakes, but make choices that give them the most satisfaction. Economists think of this approach as anal ogous to studying gravitation in a vacuum before thinking about the effects of air fricti on. There is a practical consideration that dictates ignoring mistakes. There are many kinds of mistakes, e.g. “I meant to buy toothpaste but forgot and bought a toothb rush,” a memory problem, “I thought this toothpaste was better but it is actually wors e,” a learning issue, and “I meant to buy toothpaste but I bought crack instead,” a se lf-control issue. All of these kinds of mistakes lead to distinct theories. Moreov er, we understand these alternative theories by understanding the basic theory first, an d then seeing what ch anges these theories lead to. 5.1 Utility Maximization Economists use the term utility in a peculiar and idiosyncratic way. Utility refers not to usefulness but to the flow of pleasure or ha ppiness that a person enjoys – some measure of the satisfaction a person experiences. Usefulness might contribute to utility, but so does style, fashion, or even whimsy. The term utility is unfortunate not just because it suggests usefulness, but because it makes the economic approach to behavior appear more limite d than it actually is. We will make very few assumptions about the form of utility that a consumer might have. That is, we will attempt to avoid making value judgments about the preferences a consumer holds – whether they like smoking ci garettes or eating only carrots, watching Arnold Schwarzenegger movies or spending time with a hula hoop. Consumers like whatever it is that they like; the economic a ssumption is that they attempt to obtain the goods that they like. It is the consequences of the pursuit of happiness that comprise the core of consumer theory. In this chapter, we will focus on two goods. In many cases, the generalization to an arbitrary number of goods is straightforward. Moreover, in most applications it won’t matter because we can view one of the g oods as a “composite good” reflecting consumption of a bunch of other goods.46 46 Thus, for example, savings for future consumption, or to provide for descendents, or to give to your alma mater, are all examples of consumption. Our consumer will, in the end, always spend all of her

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-140 As a starting point, suppose the two goods are X and Y To distinguish the quantity of the good from the good itself, we’ll use capi tal letters to indicate the good and a lower case letter to indicate the quantity consumed. If X is rutabagas, a consumer who ate three of them would have x =3. How can we represent preferences for this consumer? To fix ideas, suppose the consumer is both hungry and thirsty and the goods are beer and pizza. The consumer would like more of both, reflected in greater pleasure for greater consumption. Items one might consume are generally known as “bundles,” as in bundles of goods and services, and less freque ntly as “tuples,” a short-form for the “ntuple,” meaning a list of n quantities. Since we will focu s on two goods, both of these terms are strained in the application; a bu ndle because a bundle of two things isn’t much of a bundle, and a tuple because what we have here is a “two-tuple,” also known as a pair. But part of the job of studying econ omics is to learn the language of economics, and bundles it is. One might naturally consider measuring utili ty on some kind of physical basis – production of dopamine in the brain, for example – but it turns out that the actual quantities of utility don’t matter for the theo ry we develop. What matters is whether a bundle produces more than another, or less, or the same. Let u ( x y ) represent the utility a consumer gets from consuming x units of beer and y units of pizza. The function u guides the consumer’s choice, in the sense that, if the consumer can choose either ( x1, y1) or ( x2, y2), we expect him to choose ( x1, y1) if u ( x1, y1) > u ( x2, y2). But notice that a doubling of u would lead to the same choices, because u ( x1, y1) > u ( x2, y2) if and only if 2 u ( x1, y1) > 2 u ( x2, y2). Thus, doubling the utility doesn’t change the preferences of the consumer. But the situation is more extreme than this. Even exponentiating the utility doesn’t change the consumer’s preferences, because u ( x1, y1) > u ( x2, y2) if and only if eu(x1, y1)> eu(x2, y2). Another way to put this is that there are no natural units for utility, at least until such time as we are able to measure pleasure in the brain. It is possible to develop the theory of consu mer choice without suppo sing that a utility function exists at all. However, it is ex pedient to begin with uti lity, to simplify the analysis for introductory purposes. 5.1.1 Budget or Feasible Set Suppose a consumer has a fixed amount of money to spend, M There are two goods X and Y with associated prices pX and pY. The feasible choices the consumer can make satisfy M y p x pY X In addition, we will focus on consumption and rule out income, although this happens because we adopt a very broad notion of spending. In particular, savings are “future spending.”

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-141negative consumption, so x 0 and y 0. This gives a budget set or feasible set illustrated in Figure 5-1. Figure 5-1: Budget Set In this diagram, the feasible set of purchases that satisfy the budget constraint are illustrated with shading. If the consumer spends all her money on X she can consume the quantity x = Xp M Similarly, if she spends all of her money on Y she consumes Yp M units of Y The straight line between them, kn own as the budget line, represents the most of the goods she can consume. The slope of the budget line is Y Xp p An increase in the price of one good pivots or rotates the budget line. Thus, if the price of X increases, the endpoint Yp M remains the same, but Xp M falls. This is illustrated in Figure 5-2. Xp M x y Yp MBudget Line Budget Set

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-142 Figure 5-2: Effect of an Increase in Price on the Budget Figure 5-3: An Increase in Income x y Yp MBudget Line shifts out as M rises Xp M x y Yp MBudget Line Pivots Xp M

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-143The effect of increasing the available money M is to increase both Xp M and Yp M proportionately. This means an increase in M shifts the budget line out (away from the origin) in a parallel fashion, as in Figure 5-3. An increase in both prices by the same prop ortional factor has an effect identical to a decrease in income. Thus, one of the three financial values – the two prices and income – is redundant. That is, we can trace out all the possible budget lines with any two of the three parameters. This can prove useful; we can arbitrarily set pX to be the number one without affecting the generality of the analysis When setting a price to one, that related good is called the numeraire and essentially all prices ar e denominated with respect to that one good. A real world example of a numeraire occurred when the currency used was based on gold so that the prices of other goods are denominated in terms of the value of gold. Money is not necessarily the only constra int on the consumption of goods that a consumer faces. Time can be equally import ant. One can own all the compact discs in the world, but they are usele ss if one doesn’t actually have time to listen to them. Indeed, when we consider the supply of labo r, time will be a major issue – supplying labor (working) uses up time that could be us ed to consume goods. In this case there will be two kinds of budget constraints – a fina ncial one and a temporal one. At a fixed wage, time and money translate directly into one another and the existence of the time constraint won’t present significant challenges to the theory. The conventional way to handle the time constraint is to use as a baseline working “full out,” and then view leisure as a good which is purchased at a pr ice equal to the wage. Thus, if you earn $20/hour, we would set your budget at $480/d ay, reflecting 24 hours of work, but then permit you to buy leisure time, during whic h eating, sleeping, brushing teeth and every other non-work activity is accomplished at a price equal to $20 per hour. (Exercise) Graph the budget line for apples and oranges, with prices of $2 and $3 respectively and $60 to spend. Now in crease the price of apples from $2 to $4 and draw the budget line. (Exercise) Suppose that apples cost $1 each. Water can be purchased for 0.5 cents per gallon up to 20,000 gallons, an d 0.1 cent per gallon for each gallon beyond 20,000 gallons. Draw the budget constraint for a consumer who spends $200 per month on apples and water. (Exercise) Graph the budget line for apples and oranges, with prices of $2 and $3 respectively and $60 to spend. Now increase expenditure to $90 and draw the budget line. 5.1.2 Isoquants With two goods, we can graphically represen t utility by considering the contour map of utility. Utility contours are known as isoquants meaning “equal quantity,” and are also known as indifference curves since the consumer is indifferent between points on the line. We have met this idea already in th e description of production functions, where the curves represented input mixes th at produced a given output. The only difference

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-144here is that the output being produced is con sumer “utility” instead of a single good or service. Figure 5-4: Utility Isoquants Figure 5-4 provides an illustration of isoqua nts or indifference curves. Each curve represents one level of utility. Higher utiliti es occur to the northeast, further away from the origin. As with production isoquants, the slope of the indifference curves has the interpretation of the tradeoff between the two goods. The amount of Y that the consumer is willing to give up to obtain an extra bit of X is the slope of the indifference curve. Formally, the equation u ( x y ) = u0 defines an indifference curv e for the reference utility u0. Differentiating in such a way as to preserve the equality, we obtain the slope of the indifference curve: 0 dy y u dx x u or y u x u dx dyu u 0. This slope is known as the ma rginal rate of substitution and reflects the tradeoff, from the consumer’s perspective, between the goods. That is to say, the marginal rate of substitution (of Y for X ) is the amount of Y the consumer is willing to lose to obtain an extra unit of X An important assumption concerning isoquants is reflected in the diagram: “midpoints are preferred to extreme points.” Suppose the consumer is indifferent between ( x1, y1) 0 0.2 0.4 0.6 0.8 1 x 0 0.2 0.4 0.6 0.8 1 y u=1 u=2 u=3 u=4 u=5

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-145and ( x2, y2), that is, u ( x1, y1) = u ( x2, y2). Then we say preferences are convex if any point on the line segment connecting ( x1, y1) and ( x2, y2) is at least as good as the extremes. Formally, a point on the line segment connecting ( x1, y1) and ( x2, y2) comes in the form ( x1 + (1 – ) x2, y1 + (1 – ) y2), for between zero and one. This is also known as a “convex combination” between the two points. When is zero, the segment starts at ( x2, y2) and proceeds in a linear fashion to ( x1, y1) at equal to one. Preferences are convex if, for any between 0 and 1, u ( x1, y1) = u ( x2, y2) implies u ( x1 + (1 – ) x2, y1 + (1 – ) y2) u ( x1, y1). This property is illustrated in Figure 5-5. The line segment that connects two points on the indifference curve lies to the northeast of the indifference curve, which means the line segment involves strictly more consumpti on of both goods than some points on the indifference curve, which means that it is preferred to the indifference curve. Convex preferences mean that a consumer prefers a mi x to any two equally valuable extremes. Thus, if the consumer likes black coffee an d also likes drinking milk, the consumer prefers some of each (not necessarily mixed) to only drinking coffee or only drinking milk. This sounds more reasonable if you think of the consumer’s choices on a monthly basis; if you like drinking 60 cups of co ffee, and no milk, per month the same as 30 glasses of milk and no coffee, convex preferen ces entails preferring 30 cups of coffee and 15 glasses of milk to either extreme. Figure 5-5: Convex Preferences How does a consumer choose which bundle to select? The consumer is faced with the problem of maximizing u ( x y ) subject to M y p x pY X 0.2 0.4 0.6 0.8 1 x 0.2 0.4 0.6 0.8 1 y

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-146 We can derive the solution to the consumer’ s problem as follows. First, “solve” the budget constraint M y p x pY X for y to obtain .Y Xp x p M y If Y is a good, this constraint will be satisfied with equality and all the money will be spent. Thus, we can write the consumer’s utility as Y Xp x p M x u The first order condition for this problem, maximizing it over x has 0 y u p p x u p x p M x u dx dY X Y X This can be re-arranged to obtain the marginal rate of substitution (MRS). MRS dx dy y u x u p pu u Y X 0. The first order condition requires that the slope of the indifference curve equals the slope of the budget line, that is, there is a tangency between the indifference curve and the budget line. This is illustrated in Figure 5-6. Three indifference curves are drawn, two of which intersect the budget line, but are not tangent. At these intersections, it is possible to increase utility by moving “towa rd the center,” until th e highest of the three indifference curves is reached. At this point, further increases in utility are not feasible, because there is no intersection between the set of bundles th at produce a strictly higher utility and the budget set. Thus, the large black dot is the bundle that produces the highest utility for the consumer. It will later prove useful to also state the se cond order condition, although we won’t use this condition now: 2 2 2 2 2 2 2 2) ( ) ( ) ( 0 y u p p y x u p p x u p x p M x u dx dY X Y X Y X Note that the vector y u x u u u ) (2 1 is the gradient of u and the gradient points in the direction of steepest ascent of the function u Second, the equation which characterizes the optimum,

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-147 ) ( 0X Y Y Xp p y u x u x u p y u p Figure 5-6: Graphical Utility Maximization where is the “dot product” which multiplies th e components of vectors and then adds them, says that the vectors ( u1, u2) and (pY, pX) are perpendicular, and hence that the rate of steepest ascent of the utility func tion is perpendicular to the budget line. When does this tangency approach fail to solve the consumer’s problem? There are three ways it can fail. First, the utility migh t not be differentiable. We will set aside this kind of failure with the remark that fi xing points of non-differentiability is mathematically challenging but doesn’t lead to significant alterations in the theory. The second failure is that a tangency didn’t maximize utility. Figure 5-7 illustrates this case. Here, there is a tangency, but it doesn’t maximize utility. In Figure 5-7, the dotted indifference curve maximizes utility given the bu dget constraint (straight line). This is exactly the kind of failure that is ruled out by convex preferences. In Figure 5-7, preferences are not convex, because if we co nnect two points on th e indifference curves and look at a convex combination, we get something less preferred, with lower utility, not more preferred as convex preferences would require. x y Xp M Yp M

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-148 Figure 5-7: “Concave” Preferences, Prefer Boundaries The third failure is more fundamental: the deri vative might fail to be zero because we’ve hit the boundary of x =0 or y =0. This is a fundamental problem because in fact there are many goods that we do buy zero of, so zeros for some goods are not uncommon solutions to the problem of maximizing utilit y. We will take this problem up in a separate section, but we already have a major t ool to deal with it: convex preferences. As we shall see, convex preferences insure that the consumer’s maximization problem is “well-behaved.” 5.1.3 Examples The Cobb-Douglas utility function comes in the form 1, y x y x u Since utility is zero if either of the goods is zero, we see that a consumer with Cobb-Douglas preferences will always buy some of each good The marginal rate of substitution for Cobb-Douglas utility is ) 1 (0x y y u x u dx dyu u Thus, the consumer’s utility maximization problem yields ) 1 (0x y y u x u dx dy p pu u Y X Thus, using the budget constraint, ). ( ) 1 (X Y Xxp M yp xp x y

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-149This yields Y Xp M y p M x ) 1 ( Cobb-Douglas utility results in constant expenditure shares. No matter what the price of X or Y the expenditure xpX on X is M Similarly, the expenditure on Y is (1) M This makes the Cobb-Douglas utility very useful for computing examples and homework exercises. (Exercise) Consider a consumer with utility xy y x u ) (. If the consumer has $100 to spend, and the price of X is $5 and the price of Y is $2, graph the budget line, and then find the point that maximizes the consumer’s utility given the budget. Draw the utility isoquant through this point. What are the expenditure shares? (Exercise) Consider a consumer with utility xy y x u ) (. Calculate the slope of the isoquant directly, by solving 0) ( u y x u for y as a function of x and the utility level u0. What is the slope 0u udx dy ? Verify that it satisfies the formula given above. (Exercise) Consider a consumer with utility 2) ( ) ( xy y x u Calculate the slope of the isoquant directly, by solving 0) ( u y x u for y as a function of x and the utility level u0. What is the slope 0u udx dy ? Verify that the result is the same as in the previous exercise. Why is it the same? When two goods are perfect complements, th ey are consumed proportionately. The utility that gives rise to perf ect complements is in the form u ( x, y ) = min { x, y } for some constant (the Greek letter beta). First obse rve that with perfect complements, consumers will buy in such a way that x = y The reason is that, if x > y some expenditure on x is a waste since it brings in no addi tional utility, and the consumer gets higher utility by decreasing x and increasing y This lets us define a “composite good” which involves buying some amount y of Y and also buying y of X The price of this composite commodity is pX + pY, and it produces utility Y Xp p M u In this way, perfect complements boil down to a single good problem. (Exercise) The case of perfect substitutes arises when all that matters to the consumer is the sum of the products – e.g. red shirts and green shirts for a color-blind consumer. In this case, u ( x y ) = x + y Graph the isoquants for

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-150perfect substitutes. Show that the cons umer maximizes utility by spending their entire income on whichever product is cheaper. If the only two goods available in the worl d were pizza and beer, it is likely that satiation would set in at some point. How many pi zzas can you eat per month? How much beer can you drink? [Don’t answer that.] Figure 5-8: Isoquants for a Bliss Point What does satiation mean for isoquants? It means there is a point that maximizes utility, which economists call a bliss point An example is illustrated in Figure 5-8. Near the origin, the isoquants behave as before. Ho wever, as one gets full of pizza and beer, a point of maximum value is reached, illustrated by a large black dot. What does satiation mean for the theory? First, if the bliss poin t isn’t within reach, the theory behaves as before. With a bliss point within reach, co nsumption will stop at the bliss point. A feasible bliss point entails having a zero va lue of money. There may be people with a zero value of money, but even very wealth y people, who reach satiation in goods that they personally consume, often like to do ot her things with the weal th and appear not to have reached satiation overall. (Exercise) Suppose y x y x u for <1. Show X Y Xp p p M x 1 and Y X Yp p p M y 1 0 0.2 0.4 0.6 0.8 1 x 0 0.2 0.4 0.6 0.8 1 y u=100 u=50 u=40 u=30 u=20 u=10 u=120

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-1515.1.3.6 (Exercise) Suppose one consumer has the utility function u (which is always a positive number), and a second consumer has utility w Suppose, in addition, that for any x, y w ( x, y ) = ( u ( x, y ))2, that is, the second pe rson’s utility is the square of the first. Show that these co nsumers make the same choices – that is, u ( xa, ya) u ( xb, yb) if and only w ( xa, ya) w ( xb, yb). 5.1.4 Substitution Effects It would be a simpler world if an increase in the price of a good always entailed buying less of it. Alas, it isn’t so, as the following diagram illustrates. In this diagram, an increase in the price of Y causes the budget line to pivot around the intersection on the X axis, since the amount of X that can be purchased hasn’t changed. In this case, the quantity y of Y demanded rises. Figure 5-9: Substitution with an Increase in Price At first glance, this increase in the consumptio n of a good in response to a price increase sounds implausible, but there are examples where it makes sense. The primary example is leisure. As wages rise, the cost of leisur e (forgone wages) rises. But as people feel wealthier, they choose to work fewer hours. The other examples given, which are hotly debated in the “tempest in a teapot” kind of way, involve people subsisting on a good like potatoes but occasionally buying meat. When the price of potatoes rises, they can no longer afford meat and buy even more potatoes than before. Thus, the logical starting point on substituti on – what happens to the demand for a good when the price of that good increases? – does not lead to a useful theory. As a result, economists have devised an alternative a pproach, based on the following logic. An increase in the price of a good is really a co mposition of two effects: an increase in the relative price of the good, and a decrease in the purchasing power of money. As a result, it is useful to examine these two effects sepa rately. The substitution effect considers the x y

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-152change in the relative price, with a sufficien t change in income to keep the consumer on the same utility isoquant.47 The income effect changes only income. Figure 5-10: Substitution Effect To graphically illustrate the substitution effect, consider Figure 5-10. The starting point is the tangency between the isoquant and the budget line, denoted with a diamond shape and labeled “Initial Choice.” The price of Y rises, pivoting the budget line inward. The new budget line is illustrated with a heav y, dashed line. To find the substitution effect, increase income from the dashed line until the original isoquant is reached. Increases in income shift the budget line ou t in a fashion parallel to the original. We reach the original isoquant at a point labe led with a small circle, a point sometimes called the compensated demand, because we have compensated the consumer for the price increase by increasing income just enough to leave her unharmed, on the same isoquant. The substitution effect is just the difference between these points – the substitution in response to the price ch ange, holding constant the utility of the consumer. We can readily see that the substitution effect of a price increase in Y is to decrease the consumption of Y and increase the consumption of X .48 The income effect is the change in consumption resulting from the change in income. The effect of any change in price can be decomposed into the substitution effe ct, which holds utility constant and changes the relative prices, and the income effect, wh ich adjusts for the loss of purchasing power arising from the price increase. 47 Some authors instead change the income enough to make the old bundle affordable. This approach has the virtue of being readily computed, but the disadvantag e that the substitution effect winds up increasing the utility of the consumer. Overall the present approach is more economical for most purposes. 48 To construct a formal proof, first show that if pY rises and y rises, holding utility constant, the initial choice prior to the price increase is feasible after th e price increase. Use this to conclude that after the price increase it is possible to have strictly more of both goods, contradicting the hypothesis that utility was held constant. x y Initial Choice p Y Compensated Demand

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-153 Example (Cobb-Douglas): Recall that the Cobb-Douglas utility comes in the form 1, y x y x u Solving for x y we obtain Y Xp M y p M x ) 1 ( and ) 1 ( ,1 1 Y Xp p M y x u Thus, consider a multiplicative increase in pY, that is, multiplying pY by >1. In order to leave utility constant, M must rise by 1-. Thus, x rises by the factor 1and y falls, by the factor < 1. This is the substitution effect. What is the substitution effect of a small change in the price pY for any given utility function, not necessarily Cobb-Douglas? To address this question, it is helpful to introduce some notation. We will subscript th e utility to indicate partial derivative, that is, y u u x u u 2 1, Note that, by the definition of the substituti on effect, we are holding utility constant, so u ( x, y ) is being held constant. This means, locally, that dy u dx u du2 10 .49 In addition, we have y p x p MY X so Y Y X ydp dy p dx p dM Finally, we have the optimality condition y u x u p pY X which is convenient to write as 1 2u p u pY X Differentiating this equation, and letting 2 2 22 2 12 2 2 11) ( ) ( y u u and y x u u x u u we have ). ( ) (12 11 1 22 12dy u dx u p dp u dy u dx u pY Y X 49 Writing dx for an unknown infinitesimal change in x can be put on a formal basis. The easiest way to do so is to think of dx as representing the derivative of x with respect to a parameter, which will be pY.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-154For a given d pY, we now have three equations in three unknowns d x d y and d M However, dM only appears in one of the three. Thus, the effect of a price change on x and y can be solved by solving two equations: dy u dx u2 10 and ) ( ) (12 11 1 22 12dy u dx u p dp u dy u dx u pY Y X for the two unknowns dx and dy This is straightforward and yields: 22 2 12 11 2 12 u p u p p u p u p dp dxY Y X X Y Y and 22 2 12 11 2 22 u p u p p u p u p dp dyY Y X X Y Y These equations imply that x rises and y falls.50 We immediately see Y X Y Yp p u u dp dx dp dy 2 1. Thus, the change in ( x y ) follows the budget line locally. (This is purely a consequence of holding utility constant.) To complete the thought while we are embr oiled in these derivatives, note that 1 2u p u p Y X implies that 0 dy p dx p Y X Thus, the amount of money necessary to compensate the consumer for the price increase, keeping utility constant, can be calculated from our third equation: Y Y Y Xydp ydp dy p dx p dM The amount of income necessary to insure the consumer makes no losses from a price increase in Y is the amount that lets them buy the bundle they originally purchased, that is, the increase in the amount of money is precisely the amount needed to cover the increased price of y This shows that locally there is no difference from a substitution effect that keeps utility constant (which is what we explored) and one that provides sufficient income to permit purchasing th e previously purchased consumption bundle, at least when small changes in prices are contemplated. 50 This is a consequence of the fact that 0 222 2 12 11 2 u p u p p u pY Y X X, which follows from the already stated second order conditio n for a maximum of utility.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-1555.1.5 Income Effects Wealthy people buy more caviar than poor people. Wealthier people buy more land, medical services, cars, telephones, and comp uters than poorer people, because they have more money to spend on goods and services, and overall, buy more of them. But wealthier people also buy fewer of some g oods, too. Rich people buy fewer cigarettes and processed cheese food. You don’t see billionaires waiting in line at McDonald’s, and that probably isn’t because they have an assistant to wait for them. For most goods, at a sufficiently high income, the purchase tends to trail off as income rises. When an increase in income causes a consumer to buy more of a good that good is called a normal good for that consumer. When the consumer buys less, the good is called an inferior good, which is an example of sensible jarg on that is rare in any discipline. That is, an inferior good is any good whose quan tity demanded falls as incomes rise. At a sufficiently low income, almost all goods are normal goods, while at a sufficiently high income, most goods become inferior. Even a Ferrari is an inferior good against some alternatives, such as Lear jets. The curve that shows the path of consumption as incomes rise is known as an Engel curve.51 An Engel curve graphs ( x ( M ), y ( M )) as M varies, where x ( M ) is the amount of X chosen with income M and similarly y ( M ) is the amount of is the amount of Y An example of an Engel curve is illustrated in Figure 5-11. Figure 5-11: Engel Curve 51 The Engel curve is named for Ernst Engel (1821-18 96), a statistician, not for Friedrich Engels, who wrote with Karl Marx. x y

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-156Example (Cobb-Douglas): Since the equations Y Xp M y p M x ) 1 ( define the optimal consumption, the Engel curve is a st raight line through the origin with slope Y Xp p ) 1 ( (Exercise) Show that, in the case of perfect complements, the Engel curve does not depend on prices. An inferior good has the quantity fall as income s rise. Note that, with two goods, at least one is normal good – they can’t both be infe rior goods, for otherwise when income rose, less of both would be purchased. An example of an inferior good is illustrated in Figure 5-12. Here, as incomes rise, the consumption of x rises, reaches a maximum, then begins to decline. In the declining portion, X is an inferior good. The definition of the substitution effect no w permits us to decompose the effect of a price change into a substitution effect and an income effect. This is illustrated in Figure 5-13. What is the mathematical form of the in come effect? This is actually more straightforward to compute than the substituti on effect computed above. As with the substitution effect, we differentiate the conditions y p x p My x and 1 2u p u py x holding pX and pY constant, to obtain: dy p dx p dMY X and ) ( ) (12 11 22 12dy u dx u p dy u dx u pY X Figure 5-12: Backward Bending – Inferior Good x y

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-157 Figure 5-13: Income and Substitution Effects The second condition can also be written as 12 22 12 11u p u p u p u p dx dyY X X Y This equation alone defines the slope of th e Engel curve, without determining how large a change arises from a given change in M The two conditions together can be solved for the effects of M on X and Y The Engel curve is given by 12 22 22 2 12 11 22 u p u p u p u p u p dM dxY X X X Y and 12 11 22 2 12 11 22 u p u p u p u p u p dM dyX Y X X Y Note (from the second order condition) that good Y is inferior if 012 11 u p u pX Y, or if 02 12 1 11 u u u u or 2 1u u is increasing in x Since 2 1u u is locally constant when M increases, equaling the price ratio, and an increase in y increases 2 1u u (thanks to the second order condition), the only way to keep 2 1u u equal to the price ratio is for x to fall. This property characterizes an inferior good – an increase in the quantity of the good increases the marginal rate of substitution of that good for another good. x y Income Effect Substitution Effect

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-1585.1.5.2 (Exercise) Compute the substitution effect and income effect associated with a multiplicative price increase in pY, that is, multiplying pY by >1, for the case of Cobb-Douglas utility 1, y x y x u. 5.2 Additional Considerations Let us revisit the maximization problem consid ered in this chapter. The consumer can spend M on either or both of two goods. This yields a payoff of ) ( Y Xp x p M x u x h When is this problem well behaved? First, if h is a concave function of x which implies 0 ) ( x h ,52 then any solution to the first order co ndition is in fact a maximum. To see this, note that 0 ) ( x h entails ) ( x h decreasing. Moreover, if the point x* satisfies 0 *) ( x h then for x x* 0 ) ( x h and for x x* 0 ) ( x h because ) ( x h gets smaller as x gets larger, and 0 *) ( x h Now consider x x* Since 0 ) ( x h h is increasing as x gets larger. Similarly, for x x* 0 ) ( x h which means h gets smaller as x gets larger. Thus, h concave and 0 *) ( x h means that h is maximized at x* Thus, a sufficient condition for the first order condition to characterize the maximum of utility is that 0 ) ( x h, for all x pX, pY, and M Letting Y Xp p z this is equivalent to 0 222 2 12 11 u z zu u for all z >0. In turn, we can see th at this requires (i) u11 0 ( z =0) and (ii) u22 0 ( z ), and (iii) 012 22 11 u u u 22 11u u z In addition, since 12 22 11 2 22 11 22 2 12 112 2 u u u z u z u u z zu u (i), (ii) and (iii) are sufficient for 0 222 2 12 11 u z zu u Therefore, if (i) u11 0 and (ii) u22 0, and (iii) 012 22 11 u u u a solution to the first order conditions characterizes utility maximi zation for the consumer. We will assume that these conditions are met for the remainder of this chapter. 5.2.1 Corner Solutions When will a consumer specialize and consume zero of a good? A necessary condition for the choice of x to be zero is that the consumer doesn’t benefit from consuming a very small x that is, 0 ) 0 ( h This means 52 The definition of concavity is that h is concave if 0< a <1 and for all x, y h ( ax +(1a ) y ) ah ( x )+ (1a ) h ( y ). It is reasonably straightforward to show this implies the second derivative of h is negative, and if h is twice differentiable, the converse is true as well.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-159 0 0 0 ) 0 (2 1 Y X Y Yp p p M u p M u h or Y X Y Yp p p M u p M u 0 02 1. Moreover, if the concavity of h is met, as assumed above, th en this condition is sufficient to guarantee that the solution is zero. To see that, note that concavity of h implies h is decreasing. Combined with 0 ) 0 ( h that entails h maximized at 0. An important class of examples of this behavior are quasilinear utility. Quasilinear utility comes in the form u ( x, y ) = y + v ( x ), where v is a concave function (0 ) ( x v for all x ). (Exercise) Demonstrate that the quasilinea r consumer will consume zero X if and only if y xp p v ) 0 ( and that the consumer instead consumes zero Y if .y x Xp p p M v The quasilinear utility isoquants, for 3 0) 03 0 ( ) ( x x v are illustrated in Figure 5-14. Note that even though the isoquants curve, they are nonetheless parallel to each other Figure 5-14: Quasilinear Isoquants 0 0.2 0.4 0.6 0.8 1 x 0 0.2 0.4 0.6 0.8 1 y u=50 u=40 u=30 u=20 u=10 u=60 u=70 u=80

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-160The procedure for dealing with corners is gene rally this. First, ch eck concavity of the h function. If h is concave, we have a procedure to solve the problem; when h is not concave, an alternative strategy must be de vised. There are known strategies for some cases that are beyond the scope of this text. Given h concave, the next step is to check the endpoints, and verify that 0 ) 0 ( h (for otherwise x =0 maximizes the consumer’s utility) and that 0 Xp M h (for otherwise y =0 maximizes the consumer’s utility). Finally, at this point we seek the interior solution 0 ) ( x h With this procedure we can insure we find the actual maximum for the co nsumer, rather than a solution to the first order conditions that doesn’t ma ximize the consumer’s utility. 5.2.2 Labor Supply Consider a taxi driver who owns a car or co nvenience store owner, or anyone else who can set his own hours. Working has two effects on this consumer – more goods consumption, but less leisure cons umption. To model this, we let x be the goods consumption, L the amount of non-work time or leisure and working time T – L where T is the amount of time available for ac tivities of all kinds. The variable L includes a lot of activities that aren’t necessa rily fun, like trips to the dentist and haircuts and sleeping, but for which the consumer isn’t paid, and which represent choices. One could argue that sleeping isn’t really a choice, in the se nse that one can’t choos e zero sleep, but this can be handled by adjusting T to represent “time available for chosen behavior” so that T – L is work time and L the chosen non-work activities. We set L to be leisure rather than labor supply because it is leisure that is the good thing, whereas most of us view working as something we are willing to do provided we’re paid for it. Labor supply is different from other consum ption because the wage enters the budget constraint twice – first as the price of leisur e and second as income from working. One way of expressing this is to write the consumer’s budget constraint as p x + wL = M + wT. Here, M represents non-work income, such as gifts, government transfers, and interest income. We drop the subscript on the price of X and use w as the wage. Finally, we use a capital L for leisure because a small el looks like the number one. The somewhat Dickensian idea is that the consumer’s maximal budget entails working the total available hours T and any non-worked hours are purchased at the wage rate w Alternatively, one could express the budget cons traint so as to reflect that expenditures on goods px equals the total money, which is the sum of non-work income M and work income w ( T – L ), or p x = M + w ( T – L). These two formulations of the budget co nstraint are mathematically equivalent. The strategy for solving the problem is also equivalent to the standard formulation, although there is some expositional clarity used by employing the budget constraint to eliminate x That is, we write the utility u ( x L )

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-161 ) ( ) ( L p L T w M u L h As before, we obtain the first order condition 2 1*) ( 0 u p w u L h where the partial derivatives u1 and u2 are evaluated at *) ( L p L T w M Note that the first order condition is the same as the standard two-good theory developed already. This is because the effect so far is mere ly to require two components to income: M and wT both of which are constant. It is only when we evaluate the effect of a wage increase that we see a difference. To evaluate the effect of a wage increase, di fferentiate the first order condition to obtain p L T u p L T u p w p u dw dL u p w u p w u 12 11 1 22 12 2 112 0 Since 0 222 12 2 11 u p w u p w u by the standard second order condition, 0 dw dL if, and only if, 012 11 1 p L T u p L T u p w p u that is, these expressions are equivalent to one another. Simplifying the latter, we obtain 11 12 11 p u p L T u p L T u p w or, 1 ) (1 12 11 u u u p w L T or, ) ( 1 ) (1L T Log L L T u Log L or, 0 ) ( ) (1 L T Log L u Log L or,

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-162 0 ) (1 L T u Log L Since the logarithm is increasing, this is equivalent to ) (1L T u being an increasing function of L That is, L rises with an increase in wage s, and hours worked falls, if the marginal utility of goods times the hours worked is an increasing function of L holding constant everything else, but evaluate d at the optimal values. The value u1 is the marginal value of an additional good, while the value T-L is the hours worked. Thus, in particular, if goods and leisure are substitutes, so that an increase in L decreases the marginal value of goods, then an increase in the wage must decrease leisure, and labor supply increases in the wage. The case wh ere the goods are complements holds a hope for a decreasing labor supply, so we consid er first the extreme case of complements. Example (perfect complements): u ( x, L )= Min { x L } In this case, the consumer will make cons umption and leisure equal to maximize the utility, so *) ( L p L T w M or 1 w p wT M p w p wT M L Thus, L is increasing in the wage if pT > M that is, if M is sufficiently small that one can’t buy all one’s needs and not work at all. (Thi s is the only reasonable case for this utility function.) With strong complements between goods and leisure, an increase in the wage induces fewer hours worked. Example (Cobb-Douglas): ) ( ) (1 L p L T w M L h The first order condition gives L p L T w M p w L p L T w M L h ) ( ) 1 ( ) ( ) ( 01 1 or

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-163 p L T w M p w L ) ( ) 1 ( p wT M L p w ) 1 ( or T w M L ) 1 ( If M is high enough, the consum er doesn’t work but takes L = T ; otherwise, the equation gives the leisure, and labor supply is given by w M T Max L T ) 1 ( 0 Labor supply increases with the wage, no matter how high the wage goes. (Exercise) Show that an increase in the wa ge increases the consumption of goods, that is, x increases when the wage increases. The wage affects not just the price of leisure, but also the income level; this makes it possible that the income effect of a wage in crease dominates the substitution effect. Moreover, we saw that this is more likely when the consumption of goods takes time, that is, the goods and leisure are complements. 31 32 33 34 35 36 37 38 39 401964 1967 1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003YearHrs/wk Figure 5-15: Hours per Week As a practical matter, for most developed nati ons, increases in wages are associated with fewer hours worked. The average workweek prior to 1950 was 55 hours, which fell to 40 hours by the mid-1950s. The workweek has gradually declined since then, as Figure 5-15 illustrates.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-164 Thought Question: Does a beques t motive – the desire to give money to others – change the likelihood that goods and leisure are complements? 5.2.3 Compensating Differentials A number of physicists have changed career s, to become researchers in finance or financial economics. Research in finance pays substantially better than research in physics, and yet requires many of the same ma thematical skills like stochastic calculus. Physicists who see their former colleagues dr iving Porsches and buying summer houses are understandably annoyed that finance re search – which is intellectually no more difficult or challenging than physics – pays so much better. Indeed, some physicists say that other fields – finance, economics, an d law – “shouldn’t” pay more than physics. The difference in income between physics re searchers and finance researchers is an example of a compensating differential A compensating differential is income or costs that equalize different choices. There are individuals who could become either physicists or finance researchers. At equa l income, too many choose physics and too few choose finance, in the sense that there is a surplus of physicists, and a shortage of finance researchers. Finance salaries must exceed physics salaries in order to induce some of the researchers capable of doing either one to switch to finance, which compensates those individuals for doing the less desirable task. Jobs that are dangerous or unpleasant must pay more than jobs requiring similar skills but without the bad attributes. Thus, oil fi eld workers in Alaska’s North Slope, well above the Arctic Circle, earn a premium over wo rkers in similar jobs in Houston, Texas. The premium – or differential pay – must be such that the marginal worker is indifferent between the two choices – the extra pay compensates the worker for the adverse working conditions. This is why it is known in economics jargon by the phrase of a compensating differential. The high salaries earned by professional basketball players are not compensating differentials. These salaries are not created by a need to induce tall people to choose basketball over alternative jobs like painti ng ceilings, but instead are payments that reflect the rarity of the skills and abilitie s involved. Compensating differentials are determined by alternatives, not by direct sc arcity. Professional basketball players are well-paid for the same reason that Picasso’s paintings are expensive: there aren’t very many of them relative to demand. A compensating differential is a feature of othe r choices as well as career choices. For example, many people would like to live in California, for its weather and scenic beauty. Given the desirability of California over, say, Lincoln, Nebraska or Rochester, New York, there must be a compensating differential fo r living in Rochester, and two significant ones are air quality and housing prices. Ai r quality worsens as populations rise, thus tending to create a compensating differentia l. In addition, the increase in housing prices also tends to compensate – housin g is inexpensive in Rochester, at least compared to California.53 53 There are other compensations besides housing to li ving in Rochester – cross-country skiing, proximity to mountains and lakes. Generally employment is only a temporary factor that might compensate,

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-165 Housing prices also compensate for location wi thin a city. For most people, it is more convenient – both in commuting time and for services – to be located near the central business district than in the outlying suburb s. The main compensating differentials are school quality, crime rates, and housing prices We can illustrate the ideas with a simple model of a city. 5.2.4 Urban Real Estate Prices An important point to understand is that th e good in limited supply in cities is not physical structures like houses, but the land on which the houses sit. The cost of building a house in Los Angeles is quite si milar to the cost of building a house in Rochester, New York. The big difference is th e price of land. A $1 million house in Los Angeles might be a $400,000 house sitting on a $600,000 parcel of land. The same house in Rochester might be $500,000 – a $400,000 house on a $100,000 parcel of land. Usually, land is what fluctuates in value, rath er than the price of the house that sits on the land. When the newspaper re ports that house prices rose, in fact what rose was land prices, for the price of housing has changed only at a slow pace, reflecting increased wages of house builders and changes in the pr ice of lumber and other inputs. These do change, but historically the changes have been small compared to the price of land. We can construct a simple model of a city to illustrate the determination of land prices. Suppose the city is constructed in a flat plane. People work at the origin (0,0). This simplifying assumption is intended to captur e the fact that a relatively small, central portion of most cities involves business, with a large area given over to housing. The assumption is extreme, but not unreasonab le as a description of some cities. Suppose commuting times are proportional to distance from the origin. Let c ( t ) be the cost to the person of a commute of time t and let the time taken be t = r where r is the distance. The function c should reflect both the transportation costs and the value of time lost. The parameter accounts for the inverse of the speed in commuting, with a higher indicating slower commuting. In addi tion, we assume that people occupy a constant amount of land. This assumption is clearly wrong empirically, and we will consider making house size a choice variable later. A person choosing a house priced at p ( r ) at distance r thus pays c ( r ) + p ( r ) for the combination of housing and transportation People will choose the lowest cost alternative. If people have identical pr eferences about housing and commuting, then house prices p will depend on distance, and will be determined by c ( r ) + p ( r ) equal to a constant, so that people are indifferent to th e distance from the city center – decreased commute time is exactly compensated by increased house prices. because employment tends to be mobile, too, and move to the location the workers prefer, when that is possible. It is not possible on Alaska’s North Slope.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-166The remaining piece of the model is to figure out the constant. To do this, we need to figure out the area of the city. If the total population is N and people occupy an area of one per person, the city size rmax satisfies 2 maxr N and thus N rmax At the edge of the city, the value of land is given by some other use, like agriculture. From the perspective of the determinant of th e city’s prices, this value is approximately constant. As the city takes mo re land, the change in agricultural land is a very small portion of the total land used for agriculture. Let the value of agricultural land be v per housing unit size. Then the price of housing p ( rmax) = v because that is the value of land at the edge of the city. Th is lets us compute the price of all housing in the city: c ) c( ) p( ) c( p(r) ) c(max max maxv N v r r r r or ) c( c p(r) r v N This equation produces housing prices like those illustrated in Figure 5-16, where the peak is the city center. The height of the figure indicates the price of housing. Figure 5-16: House Price Gradient It is straightforward to verify that ho use prices increase in the population N and the commuting time parameter as one would expect. To quantify the predictions, we consider a city with a population of 1, 000,000, a population density of 10,000 per square mile, and an agricultural use value of $6 million per square mile. To translate these assumptions into the model’s structure, first note that a population density of 10,000 per square mile creates a fictitious “u nit of measure” of about 52.8 feet, which we’ll call a purlong, so that there is one pers on per square purlong (2788 square feet). Then the agricultural value of a property is v = $600 per square purlong. Note that this density requires a city of radius rmax equal to 564 purlongs, which is 5.64 miles.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-167The only remaining structure to identify in the model is the commuting cost c To simplify the calculations, let c be linear. Suppose that the daily cost of commuting is $2 per mile (roundtrip), so that the present valu e of daily commuting costs in perpetuity is about $10,000 per mile.54 This translates into a cost of commuting of $100.00 per purlong. Thus, we obtain 600 $ 100 $ ) c( c p(r) r N r v Nr 100 $ 000 57 $ Thus, the same 2788 square foot property at the city edge sells for $600, versus $57,000 less than six miles away at the city center With reasonable parameters, this model readily creates dramatic differences in land prices, based purely on commuting time. As constructed, a quadrupling of population approximately doubles the price of land in the central city. This probably understa tes the change, since a doubling of the population would likely increase road congestion, increasing and further increasing the price of central city real estate. As presented, the model contains three major unrealistic assumptions. First, everyone lives in an identically-sized piece of land. In fact, however, the amount of land used tends to fall as prices rise. At $53 per square foot, most of us buy a lot less land than at twenty cents per square foot. As a practical matter, the reduction of land per capita is accomplished both through smaller housing units and through taller buildings, which produce more housing floor space per acre of land. Second, people have distinct preferences, and the disutility of commuting as well as the value of increased space, vary with the individual. Third, congestion levels are generally endogenous – the more people that live between two points, the grea ter the traffic density and consequently the lower the level of The first two problems arise beca use of the simplistic nature of consumer preferences embedded in the model, while the third is an equilibrium issue requiring consideration of transportation choices. This model can readily be extended to inco rporate different types of people, different housing sizes, and endogenous congestion. To illustrate such generalizations, consider making the housing size endogenous. Suppose preferences are represented by the utility function: H r p r H u ) ( where H is the house size that the person chooses, and r is the distance they choose. This adaptation of the model reflects two issues First, the transport cost has been set to be linear in distance, for simplicity. Second the marginal value of housing decreases in the house size, but the value of housing doesn’ t depend on distance from the center. For 54 Figure 250 working days per year, fo r an annual cost of about $500 pe r mile, yielding a present value at 5% interest of $10,000. See Section 4.3.1. With a time value of $25 per hour, and an average speed of 40 mph (1.5 minutes per mile), the time cost is 62.5 cents per minute. Automobile costs (gas, car depreciation, insurance) are about 35-40 cents per mile. Thus the to tal is around $1 per mile, which doubles with roundtrips.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-168these preferences to make sense, <1 (otherwise either zero or an infinite house size emerges). A person with these preference s optimally would choose a house size of 1 1) ( r p H resulting in utility r r p u 1 1 1 1) ( Utility at every location is constant, so ) ( *1 1 1 1r p r u A valuable attribute of the form of the equation for p is that the general form depends on the equilibrium values only through the single number u *. This functional form produces the same qualitat ive shapes as in Figure 5-16. Using the form, we can solve for the housing size H ) ( 1 1 ) ( ) (1 1 1 1 1 1 1 1 1 1 1 r u r u r u r p r H The space in the interval [ r r + ] is (2 r + 2). In this interval, there are approximately 1 2 2) ( 1 ) 2 ( ) ( ) 2 ( r u r r H r people. Thus, the number of people within rmax of the city center is This equation, when combined with th e value of land on the periphery: N dr r u rr max0 1) ( 1 2

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-169 1 max 1 1 1 max* ) ( r u r p v jointly determine rmax and u *. (Exercise) For the case of = solve for the equilibrium values of u and rmax. When different people have different preferen ces, the people with the highest disutility of commuting will tend to live closer to the ci ty center. These tend to be people with the highest wages, since one of the costs of comm uting is time that could have been spent working. 5.2.5 Dynamic Choice The consumption of goods doesn’t take place in a single instance, but over time. How does time enter into choice? We’re going to simplify the problem a bit, and focus only on consumption and set aside working for the time being. Let x1 be consumption in the first period, x2 in the second period. Suppose the value of consumption is the same in each period, so that u ( x1, x2) = v ( x1) + v ( x2), where is called the rate of “pure” time preference. The consumer is expected to have income M1 in the first period and M2 in the second. There is a market for loaning and borrowing, which we assume has a common interest rate r The consumer’s budget constraint, then, can be written (1+ r )( M1 – x1) = x2 – M2. This equation says that the net savings in peri od 1, plus the interest on the net savings in period 1 equals the net expenditure in period 2. This is because whatever is saved in period 1 earns interest and can then be spen t in period 2; alternatively, whatever is borrowed in period 1 must be paid back wi th interest in period 2. Rewriting the constraint: (1+ r ) x1 + x2 = (1+ r ) M1 + M2. This equation is known as the intertemporal budget constraint It has two immediate consequences. First, 1+ r is the price of period 2 consumption in terms of period 1 consumption. Thus, the interest rate gives the relative prices. Second, the relevant income is “permanent income” rather than “current income.” That is, a change in incomes that leaves the present value of inco me the same should have no effect on the choice of consumption.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-170Once again, as with the labor supply, a change in the interest rate affects not just the price of consumption, but also the budget for consumption. Put another way, an increase in the interest rate represents an increase in budget for net savers, but a decrease in budget for net borrowers. As always, we rewrite the optimization prob lem to eliminate one of the variables, to obtain 2 1 1 1) )( 1 ( ) ( M x M r v x v u Thus the first order conditions yield 2 1) 1 ( ) ( 0 x v r x v This condition says that the marginal value of consumption in period 1, ) (1x v equals the marginal value of consumption in period 2, 2x v times the interest factor. That is, the marginal present values are equated. Note that the consumer’s private time preference, need not be related to the interest ra te. If the consumer values period 1 consumption more than does the market, so (1+ r ) < 1, then 2 1) ( x v x v that is, the consumer consumes more in period 1 than in period 2.55 Similarly, if the consumer’s discount of future consumption is exactly equal to the market discount, (1+ r ) = 1, the consumer will consume the same amount in both periods. Finally, if the consumer values period 1 consumption less than the market, (1+ r ) > 1, the consumer will consume more in period 2. In this case, th e consumer is more patient than the market. Figure 5-17: Borrowing and Lending 55 As usual, we are assuming that utility is concave, which in this instance means the second derivative of v is negative, which means the derivative of v is decreasing. In addition, to insure an interior solution, it is useful to require the Inada conditions: 0 ) 0 ( v v. x1x2 ( M1, M2) Period 1 Borrowing Repayment

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-171 Whether the consumer is a net lender or bo rrower depends not just on the preference for earlier versus later consumption, but also on incomes. This is illustrated in Figure 5-17. In this figure, the consumer’s income mostly comes in the second period. As a consequence, the consumer borrows in the firs t period, and repays in the second period. The effect of an interest rate increase is to pivot the budget constraint around the point ( M1, M2). Note that this point is always feasible – that is, it is feasible to consume one’s own endowment. The effect of an increase in the interest rate is going to depend on whether the consumer is a borrower or a lender. As Figure 5-18 illustrates, the net borrower borrows less in the first period – the price of first period consumption has risen and the borrower’s wealth has fallen. It is not clear whether the borrower consumes less in the second period because the price of second period consumption has fallen even though wealth has falle n, too, two conflicting effects. An increase in interest rates is a benefit to a net lender. The lender has more income, and the price of period 2 consumption has fa llen. Thus the lender must consume more in the second period, but only consumes more in the first period (lends less) if the income effect outweighs the substitution effect. This is illustrated in Figure 5-19. Figure 5-18: Interest Rate Change The government from time to time will reba te a portion of taxes to “stimulate” the economy. An important aspect of the effects of such a tax rebate is the effect to which consumers will spend the rebate, versus sa vings the rebate, because the stimulative effects of spending are thought to be larg er than the stimulative effects of savings.56 The 56 This belief shouldn’t be accepted as necessarily true ; it was based on a model that has since been widely rejected by the majority of economists. The general idea is that spending creates demand for goods, thus x1x2 ( M1, M2)

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-172theory suggests how people will react to a “o ne-time” or transitory tax rebate, compared to a permanent lowering of taxes. In particular, the budget constraint for the consumer spreads lifetime income over the lifetime. Thus, for an average consumer that might spend a present value of $750,000 over a life time, a $1,000 rebate is small potatoes. On the other hand, a $1,000/year reduction is worth $20,000 or so over the lifetime, which should have twenty times the effect of the tr ansitory change on the current expenditure. Tax rebates are not the only way we receive on e-time payments. Money can be found, or lost, and we can have unexpected costs or windfall gifts. From an intertemporal budget constraint perspective, these transitory e ffects have little significance, and thus the theory suggests people shouldn’t spend much of a windfall gain in the current year, nor cut back significantly when they have a moderately-sized unexpected cost. Figure 5-19: Interest Rate Increase on Lenders As a practical matter, most individuals can’t bo rrow at the same rate at which they lend. Many students borrow on credit cards at very high interest rates, and obtain a fraction of that in interest on savings. That is to say, borrowers and lenders face different interest rates. This situation is readily explored with a diagram like Figure 5-20. The cost of a first period loan is a relatively high loss of x2, and similarly the value of first period savings is a much more modest increa se in second period consumption. Such effects tend to favor “neither a borrower no r a lender be,” as Shakespeare recommends, although it is still possible for the consumer to optimally bo rrow in the first period (e.g. if M1=0) or in the second period (if M2 is small relative to M1). encouraging business investment in production. Ho wever, savings encourage investment by producing loanable funds, so it isn’t at all obvious whet her spending or savings have a larger effect. ( M1, M2) x1x2

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-173 Figure 5-20: Different Rates for Borrowing and Lending Differences in interest rates causes transito ry changes in income to have much larger effects than the intertemporal budget constr aint would suggest, and may go a long way to explaining why people don’t save mu ch of a windfall gain, and suffer a lot temporarily, rather than a little for a long time, when they have unexpected expenses. This is illustrated in Figure 5-21. Figure 5-21: The Effect of a Transitory Income Increase x1x2 ( M1, M2) x1x2 ( M1, M2)

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-1745.2.6 Risk There are many risks in life, even if one doesn’t add to these risks by intentionally buying lottery tickets. Gasoline prices go up and down, the demand for people trained in your major fluctuates, house prices chan ge. How do people value gambles? The starting point for the investigation is the von Neumann Morgenstern utility function. The idea of a von Neumann-Morgenstern utilit y function for a given person is that for each possible outcome x there is a value v ( x ) assigned by the person, and the average value of v is the value the person a ssigns to the risky outcome. This is a “state of the world” approach, in the sense that each of th e outcomes is associated with a state of the world, and the person maximizes the expected value of the various possible states of the world. Value here doesn’t mean a money value, but a psychic value or utility. To illustrate the assumption, consider equa l probabilities of winning $100 and winning $200. The expected outcome of this gamble is $150 – the average of $100 and $200. However, the expected value of the outcome could be anything between the value of $100 and the value of $200. The von Neumann-Morgenstern utility is v ($100) + v ($200). The von Neumann-Morgenstern formulation has certain advantages, including the logic that what matters is the averag e value of the outcome. On the other hand, in many tests, people behave in ways not consistent with the theory.57 Nevertheless, the von Neumann approach is the prevailing model of behavior under risk. To introduce the theory, we will consider on ly money outcomes, and mostly the case of two money outcomes. The person has a Neumann-Morgenstern utility function v of these outcomes. If th e possible outcomes are x1, x2, … xn and these occur with probability 1, 2, … n respectively, the consumer’s utility is n i i i n nx v x v x v x v u1 2 2 1 1) ( ) ( ... ) ( ) ( This is the meaning of “hav ing a von Neumann-Morgenstern utility function” – that utility can be written in this weighted sum form. The first insight that flows from this definition is that a individual dislikes risk if v is concave. To see this, note that the definition of concavity posits that v is concave if, for all in [0,1], and all values x1 and x2, ) ( ) 1 ( ) ( ) ) 1 ( (2 1 2 1x v x v x x v For smoothly differentiable functions, concavity is equivalent to a second derivative that is not positive. Using induction, the definiti on of concavity can be generalized to show: 57 For example, people tend to react more strongly to very unlikely events than is consistent with the theory.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-175) ( ... ) ( ) ( ) ... (2 2 1 1 2 2 1 1n n n nx v x v x v x x x v That is, a consumer with concave value fu nction prefers the average outcome to the random outcome. This is illustrated in Figure 5-22. There are two possible outcomes, x1 and x2. The value x1 occurs with probability and x2 with probability 1. This means the average or expected outcome is x1+(1)x2. The value v( x1+(1)x2) is the value at the expected outcome x1+(1)x2, while v (x1)+(1) v (x2) is the average of the value of the outcome. As is plainly visible in the picture, concavity makes the average outcome preferable to the random outcome. Peop le with concave von Neumann-Morganstern utility functions are known as risk averse people. Figure 5-22: Expected Utility and Certainty Equivalents A useful concept is the certainty equivalent of a gamble The certainty equivalent is an amount of money that provides equal utilit y to the random payoff of the gamble. The certainty equivalent is labeled CE in the diagram. Note that CE is less than the expected outcome, if the person is risk averse. This is because risk averse individuals prefer the expected outcome to the risky outcome. The risk premium is defined to be the difference between the expected payoff (in the graph, this is expressed as x1 + (1 – ) x2) and the certainty equivalent. This is the cost of risk – it is the amount of money an indivi dual would be willing to pay to avoid risk. This means as well that the risk premium is the value of insurance. How does the risk premium of a given gamble change when the ba se wealth is increased? It can be shown that the risk premium falls as wealth in creases for any gamble if, and only if,58 58 R. Preston McAfee and Daniel Vincent, The Price Decline Anomaly, Journal of Economic Theory 60, June, 1993, 191-212. x x1 x2x1+(1-)x2 v(x1+(1-)x2) v(x1)+(1-)v(x2) v CE

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-176 ) ( ) ( x v x v is decreasing. The measure ) ( ) ( ) ( x v x v x is known as the Arrow-Pratt59 measure of risk aversion, and also as the measure of absolute risk aversion To get an idea why this measure matters, consider a quadratic approximation to v Let be the expected value and 2 be the expected value of ( x – )2. Then we can approximate v (CE) two different ways. } ) )( ( ) )( ( ) ( { )} ( { ) ( ) )( ( ) (2 x v x v v E x v E CE v CE v v thus } ) )( ( ) )( ( ) ( { ) )( ( ) (2 x v x v v E CE v v. Canceling v () from both sides and noting that the average value of x is so E( x –)=0, we have 2) ( ) )( ( v CE v Then, dividing by ) ( x v 2 2) ( ) ( ) ( v v CE. That is, the risk premium, the difference be tween the average outcome and the certainty equivalent, is approximately equal to the Ar row-Pratt measure, times half the variance, at least when the variance is small. (Exercise) Use a quadratic approximation on both sides to sharpen the estimate of the risk premium. First, note ) ( ) )( ( ) )( ( ) (2CE v CE v CE v v } ) )( ( ) )( ( ) ( { )} ( {2 x v x v v E x v E Conclude that 1 1 12 2CE This approximation is exact to the second order. The translation of risk into dollars, by way of a risk premium, can be assessed even for large gambles if we are willing to make some technical assumptions. Suppose the utility 59 The measure was named after its discoverers No bel laureate Kenneth Arrow and John Pratt.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-177has constant absolute risk aversion or CARA, that is ) ( ) ( x v x v is a constant. This turns out to imply, after setting the utility of zero to zero, that ). 1 ( 1 ) (xe x v (This formulation is derived by setting v (0)=0 handling the case of =0 with appropriate limits.) Now also assume that the gamble x is normally distributed with mean and variance 2. Then the expected value of v(x) is 1 1 ) (22 e x Ev It is an immediate result from this formul a that the certainty equivalent, with CARA preferences and normal risks, is 22 Hence the risk premium of a normal distribution for a CARA individual is 22 This formulation will appear when we consider agency theory and the challenges of motivating a risk averse employee when outcomes have a substantial random component. An important aspect of CARA with normally distributed risks is that the preferences of the consumer are linear in the mean of the gamble and the variance. In fact, given a choice of gambles, the consumer sele cts the one with the highest value of 22 Such preferences are often called “mean variance preferences,” and they comprise the foundation of modern finance theory. (Exercise) Suppose u ( x ) = x0.95 for a consumer with a wealth level of $50,000. Consider a gamble with equal probability of winning $100 and losing $100 and compute the risk premium associated with the gamble. (Exercise) Suppose u ( x ) = x0.99 for a consumer with a wealth level of $100,000. A lottery ticket costs $1 an d pays $5,000,000 wi th the probability 000 000 10 1 Compute the certainty equiva lent of the lottery ticket. (Exercise) The return on U.S. government treasury investments is approximately 3%. Thus, a $1 investment returns $1.03 after one year. Treat this return as risk-free. The stock mark et (S&P 500) returns 7% on average and has a variance that is around 16% (the va riance of return on a $1 investment is $0.16). Compute the value of for a CARA individual. What is the risk

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-178premium associated equal probabilities of a $100 gain or loss given the value of ? 5.2.7 Search In most communities, every Wednesday groc ery stores advertise sale prices in a newspaper insert, and these prices vary from week to week. Prices can vary a lot from week to week and from store to store. The price of gasoline varies as much as fifteen cents per gallon in a one mile radius. Decide you want a specific Sony television, and you may see distinct prices at Best Buy, Circ uit City, and other electronics retailers. For many goods and services, there is substantia l variation in prices, which implies that there are gains for buyers to search for the best price. The theory of consumer search behavior is just a little bit arcane, but the basic insight will be intuitive enough. The general idea is that, from the perspective of a buyer, the price that is offered is random, and has a probability density function f ( p ). If a consumer faces a cost of search (e.g. if you ha ve to visit a store, in person, telephonically or virtually, the cost includes your time an d any other costs necessary to obtain a price quote), the consumer will set a reservation price which is a maximum price they will pay without visiting another store. Th at is, if a store offers a price below p *, the consumer will buy, and otherwis e they will visit another store, hoping for a better price. Call the reservation price p* and suppose that the cost of search is c Let J ( p* ) represent the expected total cost of purchase (including search costs). Then J must equal 0) ( *) ( ) ( *) (p pdp p f p J dp p pf c p J. This equation arises because the current draw (which costs c ) could either result in a price less than p* in which case observed price, with density f will determine the price paid p or the price will be too high, in which ca se the consumer is going to take another draw, at cost c and on average get the average price J ( p *). It is useful to introduce the cumulative distribution function F with xdp p f x F0) ( ) (. Note that something has to happen, so F ( )=1. We can solve the equality for J ( p* ), *) ( ) ( *) (* 0p F c dp p pf p Jp This expression has a simple inte rpretation. The expected price J ( p* ) is composed of two terms. The first is th e expected price, which is 0*) ( ) (pdp p F p f p This has the interpretation of the average price cond itional on that price being less than p *. This is

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-179because *) ( ) ( p F p f is in fact the density of the random variable which is the price given that the price is less than p* The second term is *) ( p F c This is the expected search costs, and it arises because *) ( 1 p F is the expected number of sear ches. This arises because the odds of getting a price low enough to be acceptable is F ( p *). There is a general statistical property underlying the number of sear ches. Consider a basketball player who successfully shoots a free throw with probability y How many throws on average must he throw to sink one basket? The answer is 1/ y To see this, note that the probability that exactly n throws are required is (1y )n-1 y This is because n are required means n -1 must fail (probability (1y )n-1) and then the remaining one go in, with probability y Thus, the expected number of throws is y y) (1 4 y y) 3(1 y)y 2(1 y3 2 = ) y) (1 4 y) 3(1 y) 2(1 y(13 2 = ) y) (1 y) (1 y) (1 (1 ) 1 ( ) y) (1 y) (1 y) (1 (1 y3 2 3 2 y + ... ) y) (1 y) (1 (1 ) 1 ( ) y) (1 y) (1 y) (1 (1 ) 1 (2 3 3 2 2 y y = 1 ... 1 ) 1 ( 1 ) 1 ( 1 ) 1 ( 1 y3 2y y y y y y y y Our problem has the same logic, where a su ccessful basketball throw corresponds to finding a price less than p* The expected total cost of purchase, given a reservation price p* is given by *) ( ) ( *) (* 0p F c dp p pf p Jp But what value of p* minimizes cost? Let’s st art by differentiating: 2 0*) ( ) ( *) ( *) ( *) ( *) ( p F c dp p pf p f p F p f p p Jp

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-180 *) ( *) ( *) ( *) ( ) ( *) ( *) (* 0p J p p F p f p F c dp p pf p p F p fp Thus, if p* < J ( p* ), J is decreasing, and it lowers cost to increase p* Similarly, if p* > J ( p* ), J is increasing in p* and it reduces cost to decrease p* Thus, minimization occurs at a point where p* = J ( p* ). Moreover, there is only one su ch solution to the equation p* = J ( p* ) in the range where f is positive. To see this, note that at any solution to the equation p* = J ( p* ), 0 *) ( p J and *) ( *) ( *) ( *) ( p J p p F p f dp d p J 0 *) ( *) ( *)) ( 1 ( *) ( *) ( *)) ( ( *) ( *) ( p F p f p J p F p f p J p p F p f dp d This means that J takes a minimum at this value, since its first derivative is zero and its second derivative is positive, and th at is true about any solution to p* = J ( p* ). Were there to be two such solutions, J would have to be both positive and negative on the interval between them, since J is increasing to the right of the first (lower) one, and decreasing to the left of the second (hig her) one. Consequently, the equation p* = J ( p* ) has a unique solution that minimizes the cost of purchase. Consumer search to minimize cost dictates setting a re servation price equal to the expected total cost of purchasing the good, and purchasing whenever the price offered is lower than that level. That is, it is not sens ible to “hold out” for a price lower than what you expect to pay on average, although this might be well useful in a bargaining context rather than in a store searching context. Example (Uniform): Suppose prices are un iformly distributed on the interval [ a b ]. For p* in this interval, a b a p c a b dp p p F c dp p pf p Jp a p *) ( ) ( *) (* 0 ) ( ) ( ) ( ) ( 2 2a p a b c a p a p a b c a p Thus, the first order conditio n for minimizing cost is

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-181 2) ( ) ( *) ( 0 a p a b c p J implying ) ( 2 a b c a p There are a couple of interesting observat ions about this solution. First, not surprisingly, as c0, p *a that is, as the search costs go to zero, one holds out for the lowest possible price. This is sensible in th e context of the model, but in the real search situations delay may also have a cost that isn’t modeled here. Second, p < b the maximum price, if 2 c <( b – a ). Put another way, if the most you can save by a search is twice the search cost, don’t se arch, because the expected gains from search will be half the maximum gains (thanks to the uniform di stribution) and the search unprofitable. The third observation, which is much more ge neral than the specific uniform example, is that the expected price is a concave function of the cost of search (second derivative negative). That is in fact true for any di stribution. To see this, define a function *) ( ) ( min *) ( min ) (* 0 *p F c dp p pf p J c Hp p p Since 0 *) ( p J, *) ( 1 *) ( ) ( p F p J c c H It then needs only a modest effort to show p* is increasing in c from which it follows that H is concave. This means that the effects of an increase in c are passed on at a decreasing rate. Moreover, it means that a consumer should rationally be risk averse about the cost of search. (Exercise) Suppose that there are two possible prices, 1 and 2, and that the probability of the lower price 1 is x Compute the consumer’s reservation price, which is the expected cost of searching, as a function of x and the cost of search c For what values of x and c should the consumer acce pt 2 on the first search, or continue searching until the lower price 1 is found? 5.2.8 Edgeworth Box The Edgeworth60 box considers a two person, two g ood “exchange economy.” That is, two people have utility functions of two good s and endowments (initial allocations) of the two goods. The Edgeworth box is a gr aphical representation of the exchange problem facing these people, and also perm its a straightforward solution to their exchange problem. 60 Francis Edgeworth, 1845-1926, introduced a vari ety of mathematical tools including calculus for considering economics and political issues, and was certainly among the first to use advanced mathematics for studying ethical problems.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-182The Edgeworth box is represented in Figure 5-23. Person 1 is “located” in the lower left (southwest) corner, and person 2 in the upper right (northeast). The X good is given on the horizontal axis, the Y good on the vertical. The dist ance between them is the total amount of the good they have between them. A point in the box gives the allocation of the good – the distance to the lower left to pe rson 1, remainder to person 2. Thus, for the point illustrated, person 1 obtains ( x1, y1), and person 2 obtains ( x2, y2). The total amount of each good available to the two people will be fixed. Figure 5-23: The Edgeworth Box What points are efficient? The economic noti on of efficiency is that an allocation is efficient if it is impossible to make one pers on better off without harming the other, that is, the only way to improve 1’ s utility is to harm 2, and vice versa. Otherwise, if the consumption is inefficient, there is a re-arra ngement that makes both parties better off, and the parties should prefer such a point. Now, there is no sense of fairness embedded in the notion, and there is an efficient point in which one person gets everything and the other nothing. That might be very unfair, bu t it could still be the case that improving 2 must necessarily harm 1. The allocation is ef ficient if there is no waste or slack in the system, even if it is wildly unfair. To dist inguish this economic notion, it is sometimes called Pareto Efficiency.61 We can find the Pareto-efficient points by fixi ng person 1’s utility and then asking what point, on the indifference isoquant of person 1, maximizes person 2’s utility? At that 61 Vilfredo Pareto, 1848-1923, was a pioneer in replacing concepts of utility with abstract preferences, which was later adopted by the economics pro fession and remains the modern approach. 1 2 y1 y2 x1 x2

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-183point, any increase in person 2’s utility must come at the expense of person 1, and vice versa, that is, the point is Pareto-efficient. An example is illustrated in Figure 5-24. Figure 5-24: An Efficient Point In Figure 5-24, the isoquant of person 1 is drawn wi th a dark thick line. This utility level is fixed. It acts like the “budget constraint” for person 2. Note that person 2’s isoquants face the opposite way because a movement southwest is good for 2, since it gives him more of both goods. Four isoquants are graphed for person 2, and the highest feasible isoquant, which leaves person 1 getting the fixed utility, has the Pareto-efficient point illustrated with a large dot. Such points occur at tangencies of the isoquants. This process, of identifying the points that are Pareto-efficient, can be carried out for every possible utility level for person 1. What results is the set of Pareto-efficient points, and this set is also known as the contract curve. This is illustrated with the thick line in Figure 5-25. Every point on this curve maximize s one person’s utility given the other, and they are characterized by th e tangencies in the isoquants. The contract curve need not have a simple shape, as Figure 5-25 illustrates. The main properties are that it is increasing and goes from person 1 consuming zero of both goods to person 2 consuming zero of both goods. 1 2 u1 u2

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-184 Figure 5-25: The Contract Curve Example: Suppose both people have Cobb-Doug las utility. Let the total endowment of each good be 1, so that x2 = 1 – x1. Then person 1’s utility can be written as u1 = x y1-, and 2’s utility is u2 = (1x ) (1y )1-. Then a point is Pareto-efficient if ) 1 )( 1 ( ) 1 ( ) 1 (2 2 1 1x y y u x u y u x u x y Thus, solving for y a point is on the contract curve if ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) ( ) 1 ( ) 1 ( x x x x x x x y Thus, the contract curve for the Cobb-Douglas case depends on a single parameter ) 1 ( ) 1 ( It is graphed for a variety of examples ( and ) in Figure 5-26. 1 2

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-185 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Figure 5-26: Contract Curves with Cobb-Douglas Utility (Exercise) If two individuals have the same utility function concerning goods, is the contract curve the di agonal? Why or why not? (Exercise) For two individuals with Cobb-Douglas preferences, when is the contract curve the diagonal? The contract curve provides the set of effi cient points. What point will actually be chosen? Let’s start with an endowment of th e goods. An endowment is just a point in the Edgeworth box, which gives the initial owne rship of both goods for both people. The endowment is marked with a triangle in Figure 5-27. Note that this point gives the endowment of both person 1 and 2, be cause it shows the shares of each. Figure 5-27 also shows isoquants for persons 1 and 2 going through the endowment. Note that the isoquant for 1 is concave toward the point labeled 1, and the isoquant for 2 is concave toward the point labeled 2. These utility isoquants define a reservation utility level for each person – the utility they co uld get alone, without exchange. This “no exchange” state is known as autarky. There are a variety of efficient points that give these people at least as much as they get under autarky, and those points are along the contract curve but have a darker line. In Figure 5-27, starting at the endowment, the ut ility of both players is increased by moving in the general direction of the southeas t, that is, down and to the right, until the contract curve is reached. This involves person 1 getting more X (movement to the right) in exchange for giving up some Y (movement down). Thus, we can view the increase in utility as a trade – 1 trades some of his Y for some of 2’s X.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-186 Figure 5-27: Individually Rational Efficient Points In principle, any of the darker points on th e contract curve, which give both people at least as much as they achieve under autarky, might result from trade. The two people get together and agree on exchange that puts them at any point along this segment of the curve, depending on the bargaining skills of the players. But there is a particular point, or possibly a set of points, that result from exchange using a price system. A price system involves a specific price for trading Y for X and vice versa, that is available to both parties. In this diagram, prices define a straight line, whose slope is the negative of the Y for X price (the X for Y price is the reciprocal). Figure 5-28 illustrates trade with a price system. The O in the center is the point on the contract curve connected to the endowment (trian gle) by a straight line (the price line), in such a way that the straight line is tangen t to both 1 and 2’s isoquants at the contract curve. This construction means that, if ea ch person took the price line as a budget constraint, they would maximize their utilit y function by choosing the point labeled O. That a price line that (i) goes through the endowment and (ii) goes through the contract curve at a point tangent to both people’s ut ility exists is relati vely easy to show. Consider lines that satisfy property (ii) and le t’s see if we can find one that goes through the endowment. Start on the contract curve at the point that maximizes 1’s utility given 2’s reservation utility, and you can easily see that the price line through that point passes above and to the right of the endowmen t. The similar price line maximizing 2’s utility given 1’s reservation utility passes be low and to the left of the endowment. These price lines are illustrated with dotted lines. Thus, by continuity, somewhere in between is a price line that passes through the endowment. 1 2

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-187 Figure 5-28: Equilibrium with a Price System The point marked with the O represents an equilibrium of the price system, in the sense that supply and demand are equated for both goods. Note that, given the endowment and given the price through the endowment, bo th parties maximize utility by going to the O. To see this, it may help to consid er a version of the picture that only shows person 2’s isoquants and the price line. Figure 5-29 removes player 1’s isoquants, leavin g only player 2’s isoquants and the price line through the endowment. The price li ne through the endowment is the budget facing each player at that price. Note that given this budget line, player 2, who gets more the less player 1 gets, maximizes utilit y at the middle isoquant, given the budget. That is, taking the price as given, play er 2 would choose the O given player 2’s endowment. The logic for player 1 is analogous. This shows that, if both players believe that they can buy or sell as much as they li ke at the tradeoff of the price through the O, both will trade to reach the O. This means th at, if the players accept the price, a balance of supply and demand emerges. In this se nse, we have found an equilibrium price. 1 2

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-188 Figure 5-29: Illustration of Price System Equilibrium In the Edgeworth box, we see that, given an endowment, it is possible to reach some Pareto-efficient point using a price system. Moreover, any point on the contract curve arises from as an equilibrium of the price system for some endowment. The proof of this proposition is startlingly easy. To show that a particular point on the contract curve is an equilibrium for some endowment, just start with an endowment equal to the point on the contract curve. No trade can occur be cause the starting point is Pareto-efficient – any gain by one party entails a loss by the other. Furthermore, if a point in the Edgeworth box represents an equilibrium using a price system (that is, if the quantity supplied equa ls the quantity demanded for both goods), it must be Pareto-efficient. At an equilibrium to the price system, each player’s isoquant is tangent to the price line, and hence tangent to each other. This implies the equilibrium is Pareto-efficient. Two of these three propositions – any equilibr ium of the price system is Pareto-efficient, any Pareto-efficient point is an equilibrium of the price system for some endowment, are known as the first and second welfare theorems of general equilibrium They have been demonstrated by Nobel laureates Ke nneth Arrow and Gerard Debreu, for an arbitrary number of people and goods. They also demonstrated the third proposition, that for any endowment, there exists an equi librium of the price system, with the same high level of generality. 5.2.9 General Equilibrium We will illustrate general equilibrium, fo r the case when all consumers have CobbDouglas utility in an exchange economy. An exchange economy is an economy where the supply of each good is just the tota l endowment of that good and there is no 1 2

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-189production. Suppose there are N people, indexed by n = 1, 2, …, N There are G goods, indexed by g = 1, 2, …, G Person n has Cobb-Douglas utility, which we can represent using exponents ( n,g ), so that the utility of person n can be represented as G g g ng n x1 ) () (, where x ( n,g ) is person n ’s consumption of good g Assume that ( n,g )0 for all n and g which amounts to assuming that the products are in fact goods. Without any loss of generality, we can require 1 ) (1 G gg n for each n (To see this, note that maximizing the function U is equivalent to maximizing the function U for any positive .) Let y ( n,g ) be person n ’s endowment of good g The goal of general equilibrium is to find prices p1, p2, …, pG for the goods, in such a way that demand for each good exactly equals supply of the good. The supply of good g is just the sum of th e endowments of that good. The prices yield a wealth for person n equal to G g g ng n y p W1) (. We will assume that 0 ) ( ) (1 N ni n y g n for every pair of goods g and i This assumption states that for any pair of goods, there is at least one ag ent that values good g and has an endowment of good i The assumption insures that there is always someone willing and able to trade if the pric e is sufficiently attractive. The assumption is much stronger than necessary but useful for exposition. The assumption also insures the endowment of each good is positive. Cobb-Douglas utility simplifies the analysis because of a feature that we already met in the case of two goods, but which holds in ge neral: the share of wealth for a consumer n on good g equals the exponent ( n,g ). Thus, the total demand for good g is N n g n gp W g n X1) ( The equilibrium conditions, then, can be expressed as saying supply (sum of the endowments) equals demand, or, for each good g N n g n g N np W g n X g n y1 1) ( ) (.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-190 We can rewrite this expression, provided pg>0 (and it must be for otherwise demand is infinite), to be 0 ) ( ) ( ) (1 1 1 N n N n G i i gg n y g n i n y p p Let B be the G G matrix whose ( g, i ) term is N n N n gig n y g n i n y b1 1) ( ) ( ) ( Let p be the vector of prices. Then we can write the equilibrium conditions as (I–B)p=0, where 0 is the zero vector. Thus, fo r an equilibrium (other than p=0) to exist, B must have an eigenvalue equal to 1, and a corresponding eigenvector p that is positive in each component. Moreover, if such an eigenvec tor, eigenvalue pair exists, it is an equilibrium, because demand is equal to supply for each good. The actual price vector is not completely identified, because if p is an equilibrium price vector, so is any po sitive scalar times p. Scaling prices doesn’t change the equilibrium because both prices and wealth (which is based on endowments) rise by the scalar factor. Usually economists assign one good to be a numeraire, which means all other goods are indexed in terms of that good, and th e numeraire’s price is artificially set to be 1. We will treat any scaling of a price vector as the same vector. The relevant theorem is the Perron-Frobenius theorem.62 It states that if B is a positive matrix (each component positive), then there is an eigenvalue >0 and an associated positive eigenvector p, and moreover is the largest (in absolute value) eigenvector of B.63 This conclusion does most of the wo rk of demonstrating the existence of an 62 Oskar Perron, 1880 1975 and Georg Frobenius, 1849 – 1917. 63 The Perron-Frobenius theorem, as usually stated, only assumes that B is non-negative and that B is irreducible. It turns out that a strictly positive matrix is irreducible, so this condition is sufficient to invoke the theorem. In addition, we can still apply the theorem even when B has some zeros in it, provided that it is irreducible. Irreducibility means that the economy can’t be divided into two economies, where the people in one economy can’t buy from the people in the second because they aren’t endowed with anything the people in the first value. If B is not irreducible, then some people may wind up consuming zero of things they value.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-191equilibrium. The only remaining condition to check is that the eigenvalue is in fact 1, so that (I–B)p=0. Suppose the eigenvalue is Then p = Bp. Thus for each g G i i N m N n gp g m y i n y g n p1 1 1) ( ) ( ) ( or G i i N n N m gp i n y g n g m y p11 1) ( ) ( ) (. Summing both sides over g G g G i i N n G g N m gp i n y g n g m y p111 11) ( ) ( ) ( G i i N n G i i N n G gp i n y p i n y g n11 111) ( ) ( ) (. Thus =1 as desired. The Perron-Frobenius theorem actually provid es two more useful conclusions. First, the equilibrium is unique. This is a feat ure of the Cobb-Douglas utility and does not necessarily occur for other utility functions. Moreover, the equilibrium is readily approximated. Denote by Bt the product of B with itself t times. Then for any positive vector v, p v B t tlim. While approximations are very useful for large systems (large numbers of goods), the system can readily be computed exactly with small numbers of goods, even with a large number of individuals. Moreover, the approximation can be interpreted in a potentially useful manner. Let v be a candidate for an equilibrium price vector. Use v to permit people to calculat e their wealth, which for person n is G i i ni n y v W1) ( Given the wealth levels, what pr ices clear the market? Demand for good g is G i N n i N n n gi n y g n v W g n v x11 1) ( ) ( ) ( ) (

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-192 and the market clears, given the wealth levels, if N n G i N n i gg n y i n y g n v p1 11) ( ) ( ) ( which is equivalent to p = Bv. This defines an iterative process. Start with an arbitrary price vector, compute wealth levels, then compute th e price vector that clears the market for the given wealth levels. Use this price to re calculate the wealth levels, and then compute a new market-clearing price vector for the new wealth levels. This process can be iterated, and in fact converges to the equilibrium price vector from any starting point. We finish this section by considering three spec ial cases. If there are two goods, we can let an = ( n 1), and then conclude ( n 2) = 1 – an. Then let N n gg n y Y1) ( be the endowment of good g Then the matrix B is N n n N n n N n n N n na n y Y a n y Y a n y Y a n y Y1 2 1 2 1 1 1 1) 1 )( 2 ( 1 ) 1 )( 1 ( 1 ) 2 ( 1 ) 1 ( 1 B N n n N n n N n n N n na n y Y a n y Y Y a n y Y a n y Y1 2 1 1 2 1 1 1 1) 2 ( 1 1 ) 1 ( 1 ) 2 ( 1 ) 1 ( 1 The relevant eigenvector of B is N n n N n na n y a n y1 1) 1 )( 1 ( ) 2 ( p. The overall level of prices is not pinned down – any scalar multiple of p is also an equilibrium price – so the relevant term is the price ratio, which is the price of good 1 in terms of good 2, or

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-193 N n n N n na n y a n y p p1 1 2 1) 1 )( 1 ( ) 2 (. We can readily see that an increase in the supp ly of good 1, or a decrease in the supply of good 2, decreases the price ratio. An increa se in the preference for good 1 increases the price of good 1. When people who value good 1 relatively highly are endowed with a lot of good 2, the correlation between preference for good 1 an and endowment of good 2 is higher. The higher the correlation the higher is the price ratio. Intuitively, if the people who have a lot of good 2 want a lot of good 1, the price of good 1 is going to be higher. Similarly, if the people who have a lot of good 1 want a lot of good 2, the price of good 1 is going to be lower. Thus, the correlati on between endowments and preferences also matters to the price ratio. In our second special case, we consider people with the same preferences, but who start with different endowments. Hypothesizin g identical preferences sets aside the correlation between endowments and preferen ces found in the two good case. Since people are the same, ( n, g ) = Ag for all n In this case, g i g N n N n giY Y A g n y g n i n y b 1 1) ( ) ( ) ( where as before N n gg n y Y1) ( is the total endowment of good g The matrix B has a special structure, and in this case, g g gY A p is the equilibrium price vector. Prices are proportional to the preference for the good divided by the total endowment for that good. Now consider a third special case, where no common structure is imposed on preferences, but endowments are proportional to each other, that is, the endowment of person n is a fraction wn of the total endowment. This implies that we can write y ( n, g ) = wn Yg, an equation assumed to hold for all people n and goods g Note that by construction, 11N n nw since the value wn represents n ’s share of the total endowment. In this case, we have

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McAfee: Introduction to Economic Analysis,, July 24, 2006 5-194 N n n g i N n N n gig n w Y Y g n y g n i n y b1 1 1) ( ) ( ) ( ) ( These matrices also have a special structur e, and it is readily verified that the equilibrium price vector satisfies N n n g gg n w Y p1) ( 1 This formula receives a similar interpretation – the price of good g is the strength of preference for good g where strength of preference is a wealth-weighted average of the individual preference, divided by the endowmen t of the good. Such an interpretation is guaranteed by the assumption of Cobb-Douglas preferences, since these imply that individuals spend a constant proportion of their wealth on each good. It also generalizes the conclusion found in the two good case to more goods, but with the restriction that the correlation is now between wealth and pref erences. The special case has the virtue that individual wealth, which is endogenous because it depends on prices, can be readily determined. (Exercise) Consider a consumer with Cobb-Douglas utility, G i a iix U1, where 11 G i ia and facing the budget constraint G i i ix p W1. Show that the consumer maximizes utility by choosing i i ip W a x for each good i Hint: Express the budget constraint as 1 11G i i i G Gx p W p x and thus utility as G ia G i i i G G i a ix p W p x U 1 1 1 11 This function can now be maximized in an unconstrained fashion. Verify that the resu lt of the maximization can be expressed as G G G i i ix p a a x p and thus G G G G i G G G i G i i ia x p x p a a x p W 1 1, which yields W a x pi i i

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-1956 Market Imperfections We have so far focused on unimpeded mark ets, and seen that markets may perform efficiently.64 In this chapter, we examine impediments to the efficiency of markets. Some of these impediments are imposed on otherwise efficiently functioning markets, as occurs with taxes. Others, such as mono poly or pollution, are problems that may arise in some circumstances, and may require correction by the government. 6.1 Taxes There are a variety of types of taxes, such as income taxes, property taxes, ad valorem (percentage of value) taxes, and excise taxes (taxes on a specific good like cigarettes or gasoline). Here, we are primarily concerned with sales taxes which are taxes on goods and services sold at retail. Our insights in to sales taxes translate naturally into some other taxes. 6.1.1 Effects of Taxes Consider first a fixed tax such as a twenty ce nt tax on gasoline. The tax could either be imposed on the buyer or the supplier. It is imposed on the buyer if the buyer pays a price for the good, and then also pays the tax on top of that. Similarly, if the tax is imposed on the seller, the price charged to th e buyer includes the tax. In the United States, sales taxes are generally imposed on th e buyer – the stated price does not include the tax – while in Canada, the sales tax is generally imposed on the seller. An important insight of supply and demand theo ry is that it doesn’t matter – to anyone – whether the tax is imposed on the supplier or the buyer. The reason is that ultimately the buyer cares only about the total price pa id, which is the amount the supplier gets plus the tax, and the supplier cares only abou t the net to the suppli er, which is the total amount the buyer pays minus the tax. Thus, wi th a twenty cent tax, a price of $2.00 to the buyer is a price of $1.80 to the seller. Whether the buyer pays $1.80 to a seller and additional twenty cents in tax, or pays $2 .00, produces the same outcome to both the buyer and seller. Similarly, from the selle r’s perspective, whether the sellers charge $2.00 and then pay twenty cents to the gove rnment, or charges $1.80 and pay no tax, leads to the same profit.65 64 The standard term for an unimpeded market is a free market, which is free in the sense of “free of external rules and constraints.” In this terminology, eBay is free market, even though it charges for the use of the market. 65 There are two minor issues here that won’t be consid ered further. First, the party who collects the tax has a legal responsibility and it could be that busine sses have an easier time complying with taxes than individual consumers. The transaction costs associated with collecting taxes could create a difference arising from who pays the tax. Such differences will be ignored in this book. Second, if the tax is percentage tax, it won’t matter to the outcome but the calculations are more complicated, because a ten percent tax on the seller at a seller’ s price of $1.80 is different from a ten percent tax on a buyer’s price of $2.00. Then the equivalence between taxes imposed on the seller and taxes imposed on the buyer requires different percentages that produce the same effective tax level. In addition, there is a political issue: imposing the tax on buyers makes the presenc e and size of taxes more transparent to voters.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-196 Figure 6-1: Effect of a Tax on Supply First, consider a tax imposed on the seller. At a given price p and tax t each seller obtains p t and thus supplies the amou nt associated with this net price. Taking the before tax supply to be SBefore Tax, the after tax supply is shifte d up by the amount of the tax. This is the amount that covers the margin al value of the last unit, plus providing for the tax. Another way of saying this is that at any lower price, the sellers would reduce the number of units offered. The change in supply is illustrated in Figure 6-1. Figure 6-2: Effect of a Tax on Demand Now consider the imposition of a tax on the buyer, illustrated in Figure 6-2. In this case, the buyer pays the price of the good, p plus the tax, t This reduces the willingness to p q DBefore Tax DAfter Tax Amount of Tax SAfter Tax p q SBefore Tax Amount of Tax

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-197pay for any given unit by the amount of the ta x, thus shifting down the demand curve by the amount of the tax. Figure 6-3: Effect of a Tax on Equilibrium In both cases, the effect of the tax on th e supply-demand equilibrium is to shift the quantity toward a point where the before ta x demand minus the before tax supply is the amount of the tax. This is illustrated in Figure 6-3. The quantity traded before a tax was imposed was qB* When the tax is imposed, the pric e that the buyer pays must exceed the price the sellers receive, by the amount eq ual to the tax. This pins down a unique quantity, denoted qA*. The price the buyer pays is denoted by pD* and the sellers receive that amount minus the tax, which is noted as pS*. The relevant quantities and prices are illustrated in Figure 6-3. Another thing notable from this picture is that the price that buyers pay rises, but generally by less than the tax. Similarly, the price the selle rs obtain falls, but by less than the tax. These ch anges are known as the incidence of the tax – is a tax mostly borne by buyers, in the form of higher prices or by sellers, in the form of lower prices net of taxation? There are two main effects of a tax: a fall in the quantity traded, and a diversion of revenue to the government. These are illustrated in Figure 6-4. First, the revenue is just the amount of the tax times the quantity traded, which is the area of the shaded rectangle. The tax raised of cou rse uses the after tax quantity qA* because this is the quantity traded once the tax is imposed. q A p q DBefore TaxSBefore Tax qB* pD* Tax pS*

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-198 Figure 6-4: Revenue and Dead weight Loss In addition, a tax reduces the quantity traded, thereby reducing some of the gains from trade. Consumer surplus falls because the pr ice to the buyer rises, and producer surplus (profit) falls because the price to the seller fa lls. Some of those losses are captured in the form of the tax, but there is a loss captured by no party – the value of the units that would have been exchanged were there no tax. The value of those units is given by the demand, and the marginal cost of the units is given by the supply. The difference, shaded in black in the diagram, is the lost ga ins from trade of units that aren’t traded because of the tax. These lost gains from trade are known as a dead weight loss. That is, the dead weight loss is the buyers’ values minus the sellers’ costs of units that are not economic to trade only because of a tax or other interference in the market. The net lost gains from trade, measured in dollars, of these lost units is illustrated by the black triangular region in the diagram. The dead weight loss is important because it represents a loss to society much the same as if resources were simply thrown away or lost. The dead weight loss is value that people don’t enjoy, and in this sense can be viewed as an opportunity cost of taxation. That is, to collect taxes, we have to take money away from people, but obtaining a dollar in tax revenue actually costs society more than a dollar. The costs of raising tax revenues include the money raised (which the taxpayers lose), the direct costs of collection like tax collectors and government agencies to administer tax collection, and the dead weight loss – the lost value create d by the incentive effects of taxes, which reduce the gains for trade. The dead weight loss is part of the overhead of collecting taxes. An interesting issue, to be considere d in the subsequent section, is the selection of activities and goods to tax in order to minimize the dead weight loss of taxation. Without more quantification, only a little more can be said about the effect of taxation. First, a small tax raises revenue approximatel y equal to the tax level times the quantity, or tq Second, the drop in quantity is also a pproximately proportional to the size of the q A Tax p q DBefore TaxSBefore Tax qB* Tax Revenue Dead Weight Loss

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-199tax. Third, this means the size of the dead weight loss is approximately proportional to the tax squared. Thus, small taxes have an almost zero dead weight loss per dollar of revenue raised, and the overhead of taxation, as a percentage of the taxes raised, grows when the tax level is increased. Consequently the cost of taxation tends to rise in the tax level. (Exercise) Suppose demand is given by qd( p ) = 1 – p and supply qs( p ) = p with prices in dollars. If sellers pay a 10 ce nt tax, what is the after tax supply? Compute the before tax equilibrium price and quantity, and the after tax equilibrium quantity, and buyer’s price and seller’s price. (Exercise) Suppose demand is given by qd( p ) = 1 – p and supply qs( p ) = p with prices in dollars. If buyers pay a 10 cent tax, what is the after tax demand? Do the same computations as the previous ex ercise and show that the outcomes are the same. (Exercise) Suppose demand is given by qd( p ) = 1 – p and supply qs( p ) = p with prices in dollars. Suppose a tax of t cents is imposed, t 1. What is the equilibrium quantity traded, as a function of t ? What is the revenue raised by the government, and for what level of taxation is it highest? 6.1.2 Incidence of Taxes How much does the quantity fall when a tax is imposed? How much does the buyer’s price rise and the price to the seller fall? The elasticities of supply and demand can be used to answer this question. To do so, we consider a percentage tax t and employ the methodology introduced in Chapter 2 and assume constant elasticity of both demand and supply. Let the equilibrium price to the seller be ps and the equilibrium price to the buyer be pb. As before, we will denote the demand function by qd( p )= apand supply function by qs( p )= bp. These prices are distinct because of the tax, and the tax determines the difference: pb = (1+ t ) ps. Equilibrium requires ) ( ) ( s s s b d dbp p q p q ap Thus, ) ( ) ( ) 1 ( s s s b d d sbp p q p q ap p t a This solves for ) 1 (1 t b a ps

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-200and ) 1 ( ) 1 ( ) ( t b a t b a b bp p q qs s s Finally, ) 1 ( ) 1 (1 t b a p t ps d Recall the approximation 1 1 rt tr Thus, a small proportional tax increases the price to the buyer by approximately t and decreases the price to the seller by t. The quantity falls by approximately t. Thus, the price effect is mostly on the “relativ ely inelastic party.” If demand is inelastic, is small, then the price decrease to the se ller will be small and the price increase to the buyer close to the entire tax. Simi larly, if demand is very elastic, is very large, and the price increase to the buyer will be small and the price decrease to the seller close to the entire tax. We can rewrite the quantity change as 1 1 t t. Thus the effect of a tax on quantity is small if either the demand or the supply is inelastic. To minimize the distortion in quantity, it is useful to impose taxes on goods that either have inelastic demand, or inelastic supply. For example, cigarettes are a product with ve ry inelastic demand and moderately elastic supply. Thus a tax increase will generally incr ease the price almost the entire amount of the tax. In contrast, travel tends to have re latively elastic demand, so taxes on travel – airport, hotel and rental car taxes – tend no t to increase the final prices so much, but have large quantity distortions. (Exercise) For the case of constant elasticity (of both supply and demand), what tax rate maximizes the government ’s revenue? How does the revenuemaximizing tax rate change when demand becomes more inelastic? 6.1.3 Excess Burden of Taxation The presence of the dead-weight loss implies th at raising $1 in taxes costs society more than $1. But how much more? This idea – that the cost of taxation exceeds the taxes raised – is known as the excess burden of taxation, or just the excess burden. We can quantify the excess burden with a remarkably sharp formula.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-201To start, we will denote the marginal cost of the quantity q by c ( q ) and the marginal value by v ( q ). The elasticities of demand an d supply are given by the standard formulae: ) ( ) ( q v q q v v dv q dq and ) ( ) ( q c q q c c dc q dq Consider an ad valorem tax that will be denoted by t If sellers are charging c ( q ), the ad valorem (at value) tax is tc ( q ), and the quantity q will satisfy v ( q *) = (1 + t ) c ( q *). From this equation, we immediately deduce ) 1 ( 1 1 ) 1 ( *) ( ) 1 ( *) ( *) ( *) ( ) 1 ( *) ( *) ( t q t q q q c t q q v q c q c t q v q c dt dq Tax revenue is given by *) ( q q tc Tax The effect on taxes collected, Tax, of an increase in the tax rate t is ) 1 ( 1 1 *) ( *) ( *) ( *) ( t q t q q c dt dq q c q q c t q q c dt dTax ) 1 ( ) 1 ( *) ( 1 ) 1 ( ) 1 ( *) ( t t q q c t t t q q c Thus, tax revenue is maximized when the tax rate is tmax, given by 1 1 1 ) 1 (max t The value 1 is the monopoly markup rate, which we will meet in Section 6.5. Here, it is applied to the sum of the inverse elasticities. The gains from trade (including the tax) is the difference between value and cost for the traded units, and thus is

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-202 0) ( ) (qdq q c q v GFT Thus, the change in the gains from trade as taxes increase is given by ) 1 ( ) 1 ( *) ( ) 1 ( *) ( *) ( ) 1 ( ) 1 ( *) ( *) ( *) ( t t q q c t q q c q v t t q q c dt dq q c q v t Tax t GFT dTax dGFT t t t t t t q c q tc max1 ) 1 ( ) 1 ( *) ( *) ( The value tmax is the value of the tax rate t that maximizes the to tal tax take. This remarkable formula permits the quantification of the cost of taxation. The minus sign indicates it is a loss – the dead weight loss of monopoly, as taxes are raised, and it is composed of two components. First, there is the term 1 which arises from the change in revenue as quantity is changed, thus measuring the responsiveness of revenue to a quantity change. The second term provides for the change in the size of the welfare loss triangle. The formula can readily be applie d in practice to assess the social cost of taxation, knowing only the tax rate and the elasticities of supply and demand. The formula for the excess burden is a local fo rmula – it calculates the increase in the dead weight loss associated with raising an extra dollar of tax revenue. All elasticities, including those in tmax, are evaluated locally around th e quantity associated with the current level of taxation. The calculated value of tmax is value given the local elasticities; if elasticities are not constant, this value will not necessarily be the actual value that maximizes the tax revenue. One can think of tmax as the projected value. It is sometimes more useful to express the formula directly in terms of elasticities rather than in terms of the projected value of tmax, in order to avoid the potential confusion between the projected (at current elasticities) and ac tual (at the elasticities relevant to tmax) value of tmax. This level can be read direct ly from the derivation above: t t dTax dGFT) 1 ( 6.2 Price Floors and Ceilings A price floor is a minimum price at which a product or service is permitted to sell. Many agricultural goods have price floors impose d by the government. For example, tobacco sold in the United States has historically b een subject to a quota and a price floor set by the Secretary of Agriculture. Unions may im pose price floors as well. For example, the Screen Actors Guild imposes minimum rates for guild members, generally pushing up the price paid for actors above that which would prevail in an unconstrained market.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-203(The wages of big name stars aren’t genera lly affected by SAG, because these are individually negotiated.) The most important example of a price floor is the minimum wage, which imposes a minimum amount that a worker can be paid per hour. A price ceiling is a maximum price that can be charged for a product or service. Rent control imposes a maximum price on apartments (usually set at the historical price plus an adjustment for inflation) in many U.S. cities. Taxi fares in New York, Washington, D.C. and other cities are subject to maximu m legal fares. During World War II, and again in the 1970s, the United States imposed price controls to limit inflation, imposing a maximum price for legal sale of many goods and services. For a long time, most U.S. states limited the legal interest rate that could be charged (these are called usury laws) and this is the reason so many credit ca rd companies are located in South Dakota. South Dakota was the first state to eliminate such laws. In addition, ticket prices for concerts and sporting events are often set below the equilibrium price. Laws prohibiting scalping then impose a price ceiling. Laws preventing scalping are usually remarkably ineffective in practice, of course. 6.2.1 Basic Theory The theory of price floors and ceilings is readily articula ted with simple supply and demand analysis. Consider a price floor – a minimum legal price. If the price floor is low enough – below the equilibrium price – ther e are no effects, because the same forces that tend to induce a price equal to the eq uilibrium price continue to operate. If the price floor is higher than the equilibrium price, there will be a surplus, because at the price floor, more units are supplied than are demanded. This surplus is illustrated in Figure 6-5. In Figure 6-5, the price floor is illustrated with a horizontal line and is above the equilibrium price. Consequently, at the price floor, a larger quantity is supplied than is demanded, leading to a surplus. There are un its that are socially efficient to trade but aren’t traded – because their value is less than the price floor. The gains from trade associated with these units, which is lost due to the price floor, represent the dead weight loss. The price increase created by a price floor will increase the total amount paid by buyers when the demand is inelastic, and otherwise will reduce the amount paid. Thus, if the price floor is imposed in order to be a benefi t to sellers, we would not expect to see the price increased to the point where demand becomes elastic, for otherwise the sellers receive less revenue. Thus, for example, if the minimum wage is imposed in order to increase the average wages to low-skilled work ers, then we would expect to see the total income of low-skilled workers rise. If, on the other hand, the motivation for the minimum wage is primarily to make low-ski lled workers a less effective substitute for union workers, and hence allow union workers to increase their wage demands, then we might observe a minimum wage which is in so me sense “too high” to be of benefit to low-skilled workers.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-204 Figure 6-5: A Price Floor Figure 6-6: Dead weight Loss of a Price Floor The dead weight loss illustrated in Figure 6-6 is the difference between the value of the units not traded, and value is given by the demand curve, and the cost of producing these units. The triangular shaped region representing the difference between value and cost is illustrated in the above diagram, in the shaded region. p Dead Weight Loss qD q D S qS Price Floor qD p q D S qS Price Floor Surplus

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-205 However, this is the minimum loss to society associated with a price floor. Generally there will be other losses. In particular, th e loss given above assumes that suppliers who don’t sell don’t produce. As a practical matte r, some suppliers who won’t in the end sell may still produce because they hope to sell. In this case additional costs are incurred and the dead weight loss will be larger to reflect these costs. Example: Suppose both supply and demand ar e linear, with the quan tity supplied equal to the price, and the quantity demanded equal to one minus the price. In this case, the equilibrium price, and the equilibrium qu antity, are both A price floor of p > induces a quantity demanded of 1–p. How many units will suppliers offer, if a supplier’s chance of trading is random? Suppose q 1–p units are offered. A supplier’s chance of selling is q p 1 Thus, the marginal supplier (who has a marginal cost of q by assumption) has a probability q p 1 of earning p, and a certainty of paying q. Exactly q units will be supplied when this is a breakeven proposition for the marginal supplier, that is, 0 1 q p q p, or ). 1 ( p p q The dead weight loss then includes not just the triangle illustrated in the previous picture, but also the cost of the ) 1 ( ) 1 ( p p p unsold units. (Exercise) In this example, show that the quantity produced is less than the equilibrium quantity, which is Co mpute the gains from trade, given the overproduction of suppliers. What is the dead weight loss of the price floor? (Exercise) Suppose that units aren’t produced until after a buyer has agreed to purchase, as typically occurs wi th services. What is the dead weight loss in this case? (Hint: what potential sellers will o ffer their services? What is the average cost of supply of this set of potential sellers?) The Screen Actors Guild, a union of actors, has some ability to impose minimum prices (a price floor) for work on regular Hollywood movies. If the Screen Actors Guild would like to maximize the total earnings of acto rs, what price should they set in the linear demand and supply example? The effects of a price floor include lost gains from trade, because too few units are traded (inefficient exchange), units produced that are never consumed (wasted production), and more costly units produced than necessary (inefficient production) A price ceiling is a maximum price. Analogous to a low price floor, a price ceiling that is larger than the equilibrium price has no effect Tell me that I can’t charge more than a billion dollars for this book (w hich is being given away free) and it won’t affect the price

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-206charged or the quantity traded. Thus, the im portant case of a price ceiling is a price ceiling less than the equilibrium price. In this case, which should now look familiar the price is forced below the equilibrium price, and too few units are supplied, while a larger number are demanded, leading to a shortage. The dead weight loss is illustrated in Figure 6-7, and again represents the loss associated with units that are valued mo re than they cost but aren’t produced. Figure 6-7: A Price Ceiling Analogous to the case of a price floor, there can be additional losses associated with a price ceiling. In particular, some lower value buyers may succeed in purchasing, denying the higher value buyers the ability to purchase. This effect results in buyers with high values failing to consu me, and hence their value is lost. (Exercise) Adapt the price floor example above to the case of a price ceiling, with p < and compute the lost gains from trade if buyers willing to purchase are all able to purchase with probability qS/qD. (Hint: Compute the value of qD units; the value realized by buyers co llectively will be that amount times the probability of trade.) In addition to the misallocation of resource s (too few units, and units not allocated to those who value them the most), price ceilings tend to encourage illegal trade as people attempt to exploit the prohibited gains from trade. For example, it became common practice in New York to attempt to bribe la ndlords to offer rent-controlled apartments, and such bribes could exceed $50,000. In addition, potential tenants expended a great deal of time searching for apartments, and a common strategy was to read the obituaries D S qS p q qD Price Ceiling Shortage

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-207late at night, when the New York Times had just come out, hoping to find an apartment that would be vacant and available for rent. An important and undesirable by-product of pric e ceilings is discrimination. In a free or unconstrained market, discrimination against a particular group, based on race, religion, or other factors, requires transacting not based on price but on another factor. Thus, in a free market, discrimination is costl y – discrimination entails, for instance, not renting an apartment to the highest bidder, bu t the highest bidder of the favored group. In contrast, with a price ceiling, there is a shortage, and sellers can discriminate at lower cost, or even at no cost. That is, if there are twice as many people seeking apartments as there are apartments at the price ceilin g, landlords can “pick and choose” among prospective tenants and still get the maximum legal rent. Thus a price ceiling has the undesirable by-product of reduci ng the cost of discrimination. 6.2.2 Longand Short-run Effects Both demand and supply tend to be more el astic in the long-run. This means that the quantity effects of price floors and ceilings tend to be larger over time. An extreme example of this is rent control, a maximum price imposed on apartments. Rent control is usually imposed in the followi ng way: as a prohibition or limitation on price increases. For example, New York Ci ty’s rent control, imposed during World War II, prevented landlords from increasing rent, even when their own costs increased, such as when property taxes increased. This law was softened in 1969 to be gradually replaced by a rent stabilization law that pe rmitted modest rent increases for existing tenants. Figure 6-8: Rent Control, Initial Effect Thus the nature of rent control is that it begins with at most minor effects because it doesn’t bind until the equilibrium rent incr eases. Moreover, the short-run supply of apartments tends to be extremely inelastic, because one doesn’t tear down an apartment D, initial LRS, initial p q Rent Control SRS

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-208or convert it to a condominium (there were limitations on this) or abandon it without a pretty significant change in price. Demand al so tends to be relatively inelastic, because one has to live somewhere and the alternatives to renting in the city are to live a long distance away or buy (which is relatively expensive), neither of which is a very good substitute for many consumers. Long-run demand and short-run demand are not very different and are treated as being identical. Finally, the long-run supply is much more elastic than the short-run supply, because in the long-run a price increase permits the creation of apartments from warehouses (lofts ), rooms rented in houses, etc. Thus, the apartment market in New York is characte rized by inelastic short-run supply, much more elastic long-run supply, and inelastic demand. This is illustrated in Figure 6-8. We start with a rent control law that has little or no immediate effect because it is set at current rents. Thus, in the near term, tena nts’ fears of price increases are eased and there is little change in the apartment rental market. This is not to say there is zero effect – some companies considering buil ding an apartment on the basis of an expectation of higher future rents may be deterred, and a few marginal apartments may be converted to other uses because the upside potential for the owner has been removed, but such effects are modest at best. Figure 6-9: Rent Control, Long-Run Effect Over time, however, the demand for apartm ents grows as the city population and incomes grow. Moreover, as th e costs of operating an apar tment rise due to property tax increases, wage increases and cost of main tenance increases, the supply is reduced. This has little effect on the short-run suppl y but a significant effect on the long-run supply. The supply reduction and demand incr eases cause a shortage, but results in few apartments being lost because the short-run supply is very inelastic. Over time, however, apartments are withdrawn from the market and the actual quantity falls, even D, initial LRS, initial p q Rent Control Shortage Old SRS LRS, later D, later New SRS

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-209as the demand rises, and the shortage ge ts worse and worse. These changes are illustrated in Figure 6-9. The old values of demand, short-run supply and long-run supply are illustrated in dashed grey lines. The new values, reflecting an increase in demand, a fall in long-run supply, and a reduction in the number available set of apartments (where the rent control covers th e long-run cost) are given in dark black lines. The shortage is created by two separate fact ors – demand is increasing as incomes and population rise, and supply is decreasing as costs rise. This reduces the quantity of available housing units supplied and increases the demand for those units. How serious is the threat that units will be withdrawn from the market? In New York City, over 200,000 apartment units were ab andoned by their owners, usually because the legal rent didn’t cover the property ta xes and legally mandated maintenance. In some cases, tenants continued to inhabit th e buildings even after the electricity and water were shut off. It is fair to say that rent control devastated large areas of New York City, such as the Bronx. So why would New York, and so many other communities, impose rent control on itself? 6.2.3 Political Motivations The politics of rent control are straightforw ard. First, rent control involves a money transfer from landlords to tenants, because tenants pay less than they would absent the law, and landlords obtain less revenue. In the short-run, due to the inelastic short-run supply, the effect on the quantity of apartments is small, so rent control is primarily just a transfer from landlords to tenants. In a city like New York, the majority of peop le rent. A tiny fraction of New Yorkers are landlords. Thus, it is easy to attract voters to support candidates who favor rent control – most renters will benefit, while landlords don’t. The numbers of course don’t tell the whole story, because while landlords are sm all in numbers, they are wealthier on average, and thus likely have political influe nce beyond the number of votes they cast. However, even with their larger economic infl uence, the political balance favors renters. In the 100ab zip codes of Manhattan (first three digits are 100), 80% of families were renters in the year 2000. Thus, a candidate who runs on a rent control platform appeals to large portion of the voters. Part of the attraction of rent control is th at there is little economic harm in the shortrun, and most of that harm falls on new residents to New York. As new residents generally haven’t yet voted in New York, potent ial harm to them has only a small effect on most existing New Yorkers, and thus isn’t a major impediment to getting voter support for rent control. The slow rate of harm to the city is important politically because the election cycle encourages a short ti me horizon – if successful at lower office, a politician hopes to move on to higher office and is unlikely to be blamed for the longrun damage to New York by rent control. Rent control is an example of a political situation sometimes called the tyranny of the majority, where a majority of people have an incentive to confiscate the wealth of a minority. But there is another kind of poli tical situation that is in some sense the

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-210reverse, where a small number of people ca re a great deal about something, and the majority are only slightly harmed on an indi vidual basis. No poli tical situation appears more extreme in this regard than that of refined sugar. There are few U.S. cane sugar producers (nine in 1997), yet the U.S. impose s quotas that raise domestic prices much higher than world prices, in some years tripling the price Americans pay for refined sugar. The domestic sugar producers be nefit, while consumers are harmed. But consumers are harmed by only a small amount each, perhaps twelve to fifteen cents per pound – which is not enough to build a con sensus to defeat politicians who accept donations from sugar producers. This is a case where concentrated benefits and diffused costs determine the political outcome. Because there aren’t many sugar producers, it is straightforward for them to act as a single force. In contrast, it is pretty hard for consumers to become passionate abou t twelve cents per pound increase in the domestic sugar price when they consume about 60 pounds per year of sugar. 6.2.4 Price Supports A price support is a combination of two prog rams – a minimum price or price floor, and government purchase of any surplus. Thus a price support is different from a price floor, because with a price fl oor, any excess production by sellers was a burden on the sellers. In contrast, with a price support, any excess production is a burden on the government. The U.S. Department of Agriculture operates a price support for cheese, and has possessed warehouses full of cheese in the past There are also price supports for milk and other agricultural products. Figure 6-10: Price Supports Figure 6-10 illustrates the effect of a support program. The government posts a price, called the support price, and purchases any excess production offered on the market. p q qD qS D S Support Price Gov’t Purchase Dead Weight Loss

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-211The government purchases, which are the di fference between the quantity supplied and quantity demanded, are illustrated on the diagram. The cost of the program to the government is the support price times this qu antity purchased, which is the area of the rectangle directly underneath the words “Gov’t Purchases.” There are two kinds of dead weight loss in a price support program. First, consumers who would like to buy at the equilibrium price are deterred by the higher prices, resulting in the usual dead weight loss, illust rated with the vertical shading. In addition, however, there are goods produced that are th en either destroyed or put in warehouses and not consumed, which means the costs of production of those goods is also lost, resulting in a second dead weight loss. That loss is the cost of production, which is given by the supply curve, and thus is the area under the supply curve, for the government purchases. It is shaded in a horizontal fashi on. The total dead weight loss of the price support is the sum of these two individual lo sses. Unlike the case of a price floor or ceiling, a price support creates no ambiguity about what units are produced, or which consumers are willing and able to buy, and th us the rationing aspect of a price floor or ceiling is not present for a price support, nor is the incentive to create a black market other than that created by selling the warehouse full of product. 6.2.5 Quantity Restrictions and Quotas The final common way that governments intervene in market transactions is to impose a quota. A quota is a maximal production quantity, usually set based on historical production. In tobacco, peanuts, hops, California oranges, and other products, producers have production quotas based on their historical production. Tobacco quotas were established in the 1930s and today a to bacco farmer’s quota is a percentage of the 1930s level of production. The percentage is set annually by the Secretary of Agriculture. Agricultural products are not th e only products with quotas. The right to drive a tax in New York requires a medallion issued by the city, and there are a limited number of medallions. This is a quota. Is it a restrictive quota? The current price of a New York taxi medallion – the right to drive a taxi legally in New York City – is $300,000 (2004 number). This adds approximately $30,000-$40,000 annually to the cost of operating a taxi in New York, using a risk adjusted interest rate. What are the effects of a quota? A quota restricts the quantity below that which would otherwise prevail, forcing the price up, which is illustrated in Figure 6-11. It works like a combination of a price floor and a prohibition on entry. Generally, the immediate effects of a quota involve a transfer of money from buyers to sellers. The inefficient production and surplus of the price floor are avoided because a production limitation created the price increa se. This transfer has an undesirable and somewhat insidious attribute. Because th e right to produce is a capital good, it maintains a value, which must be captured by the producer. For example, an individual who buys a taxi medallion today, and pays $300,000, makes no economic profits – he captures the foregone interest on the medalli on through higher prices but no more than that. The individuals who received the wind fall gain were those who were driving taxis and were grandfathered in to the system, an d issued free medallions. Those people – who were driving taxis 70 years ago and thus are mostly dead at this point – received a windfall gain from the establishment of th e system. Future generations pay for the

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-212program, which provides no net benefits to the current generation; all the benefits were captured by people long since retired. Figure 6-11: A Quota Does this mean it is harmless to eliminat e the medallion requirement? Unfortunately not. The current medallion owners, who if th ey bought recently paid a lot of money for their medallions, would see the value of thes e investments destroyed. Thus, elimination of the program would harm current medallion owners. If the right to produce is freely tradable the producers will remain the efficient producers, and the taxi medallions are an example of this. Taxi medallions can be bought and sold. Moreover, a medallion confe rs the right to operate a taxi, but doesn’t require that the owner of the medallion actu ally drive the taxi. Thus, a “medallion owning company” can lease the right to driv e a taxi to an efficient driver, thereby eliminating any inefficiency ass ociated with who drives a taxi. In contrast, because tobacco farming rights ar en’t legally tradable across county lines, tobacco is very inefficiently grown. The averag e size of a burley tobacco farm is less than five acres, so some are much smaller. Th ere are tobacco farms in Florida and Missouri, which only exist because of th e value of the quota – if they could trade their quota to a farm in North Carolina or Kentucky, which are much better suited to producing cigarette tobacco, it would pay to do so. In this case, the quota, which locked in production rights, also locked in production which gets progressively more inefficient as the years pass. Quotas based on historical production have the problem that they don’t evolve as production methods and technology evolve, thus tending to become progressively more p Dead Weight Loss Quota q D S

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-213inefficient. Tradable quotas eliminate this particular problem, but continue to have the problem that future generations are harmed with no benefits. (Exercise) Suppose demand for a product is qd = 1 – p, and the marginal cost of production is c. A quota at level Q 1 – c is imposed. What is the value of the quota, per unit of production? Use this to derive the demand for the quota as a function of the level of quota released to the market. If the government wishes to sell the quota, how much should they sell to maximize the revenue on the product? 6.3 Externalities When the person sitting next to you lights up a cigarette, he gets nicotine, and the cigarette company gets some of his money. You just suffer, with no compensation. If your neighbor’s house catches fire because he fell asleep with that cigarette burning in his hand, your house may burn to the ground. The neighbor on the other side who plays very loud music late into the night before yo ur big economics test enjoys the music, and the record company and stereo component compa nies get his money. You flunk out of college and wind up borrowing $300,000 to bu y a taxi medallion. Drunk drivers, cell phones ringing in movies, loud automobiles, polluted air, and rivers polluted to the point that they catch fire like Cleveland’ s Cuyahoga did, are all examples where a transaction between two parties harmed other people. These are “external effects.” But external effects are not necessarily nega tive. The neighbor who plants beautiful flowers in her yard brightens your day. Anot her’s purchase of an electric car reduces the smog you breathe. Your neighbor’s invest ment in making his home safe from fire conveys a safety advantage to you. Indeed, even your neighbor’s investment in her own education may provide an advantage to you – you may learn useful things from your neighbor. Inventions and creations, whether products or poetry, produce value for others. The creator of a poem, or a mathematic al theorem, provides a benefit to others. These effects are called external effects, or externalities. An externality is any effect on people not involved in a particular transactio n. Pollution is the classic example. When another person buys and smokes cigarettes, there is a transaction between the cigarette company and the smoker. But if you are sitting near the smoker, you are an affected party not directly compensated from the transa ction, at least before taxes were imposed on cigarettes. Similarly, yo u pay nothing for the benefits you get from viewing your neighbor’s flowers, nor is there a direct mechanism to reward your neighbor for her efforts. Externalities will generally cause competitive markets to behave inefficiently from a social perspective, absent a mechanism to invo lve all the affected parties. Without such a mechanism, the flower-planter will plant too few beautiful flowers, for she has no reason to take account of your preference s in her choices. The odious smoker will smoke too much, and too near others, and th e loud neighbor will play music much too late into the night. Externalities create a market failure, that is, a competitive market does not yield the socially efficient outcome.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-214Education is viewed as creating an importan t positive externality. Education generates many externalities, including more and better employment, less crime, and fewer negative externalities of other kinds. It is widely believed that educated voters elect better politicians.66 Educated individuals tend to make a society wealthy, an advantage to all of society’s members. As a consequen ce, most societies subsidize education, in order to promote it. A major source of externalities arises in communicable diseases. Your vaccination not only reduces the likelihood that you contract a disease, but also makes it less likely that you infect others with the disease. 6.3.1 Private and Social Value, Cost Let’s consider pollution as a typical example. A paper mill produces paper, and a bad smell is an unfortunate by-product of the pr ocess. Each ton of paper produced increases the amount of bad smells produced. The pa per mill incurs a marginal cost, associated with inputs like wood and chemicals and water. For the purposes of studying externalities, we will refer to the paper mill’s costs as a private cost, the cost to the paper mill itself. In addition, there are external costs, which are the costs borne by others, which arise in this case from the smell. Ad ding the private costs and the external costs yields the social costs. Thes e costs, in their marginal form, are illustrated in Figure 6-12. Figure 6-12: A Negative Externality In Figure 6-12, the demand has been labeled “mar ginal benefit,” for reasons that will become apparent, but it is at this point just the standard demand, the marginal value of 66 This is a logical proposition, but there is scant eviden ce in favor of it. There is evidence that educated voters are more likely to vote, but little evidence that they vote for better candidates. p q Marginal Benefit Private Marginal Cost Competitive Quantity Marginal Social Cost Socially Efficient Quantity Competitive Price Socially Efficient Price Marginal External Cost

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-215the product. The paper mill’s costs have been labeled marginal private cost to reflect the fact that these costs are only the mill’s costs and don’t incl ude the cost of the bad smell imposed on others. The marginal social cost is obtained by adding the marginal external cost to the marginal private cost. The ma rginal external cost isn’t graphed on the diagram, but the size of it is illustrated at one quantity, and it is generally the difference between marginal social cost and marginal private cost. Left to its own devices, the paper market wo uld equate the marginal private cost and the marginal benefit, to produce the competitive quantity sold at the competitive price. Some of these units – all of those beyond the quantity labeled “Socially Efficient Quantity,” are bad from a social perspectiv e – they cost more to society than they provide in benefits. This is because the soci al cost of these units includes pollution, but paper buyers have no reason to worry about po llution or even to know it is being created in the process of manufacturing paper. The dead weight loss of these units is a sh aded triangle. The loss arises because the marginal social cost of the units exceeds the benefit, and the difference between the social cost and the benefits yields the loss to society. This is a case where too much is produced because the market has no reason to account for all the costs; some of the costs are borne by others. Figure 6-13: External Costs and Benefits Generally, a negative externalit y like pollution creates a marginal social cost higher than the marginal private cost. Similarly, a positi ve externality like beautification creates a higher marginal social benefit than the marg inal private benefit (demand). These are to Competitive Quantity p q Marginal Private Benefit Private Marginal Cost Marginal Social Cost Socially Efficient Quantity Marginal Social Benefit

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-216some extent conventions – one could have incorporated a positive externality by a reduction in cost – but the convention remains. An example of a product that produces both positive and negative extern alities is illustrated in Figure 6-13. Street lights are an example of a product that produces both extern alities – most of us like lit streets, but they are terrible for astronomers. Simila rly, large highways produce benefits for commuters and harm to nearby residents. The marginal private benefit and the margin al private cost give the demand and supply of a competitive market, and hence the competitive quantity results from the intersection of these two. The marginal soci al benefit and the marginal social cost gives the value and cost from a social perspective; equating these two generates the socially efficient outcome. This can be either gr eater or less than th e competitive outcome depending on which externality is larger. Example (Tragedy of the Commons): Consider a town on a scenic bay filled with lobsters. The town members collect and eat lobsters, and over time the size of the lobsters collected falls, until they are hardly worth searching for. This situation persists indefinitely; few large lobsters are caught and it is barely worth one’ s time attempting to catch them. The tragedy of the commons is a problem with a common resource, in this case the lobster bay. Catching lobsters creates an ex ternality, by lowering the productivity of other lobster catchers. The externality leads to over-fishing, since individuals don’t take into account the negative effect they have on each other, ultimately leading to a nearly useless resource and potentially driving the lo bsters into extinction. As a consequence, the lobster catch is usually regulated. (Exercise) A child who is vaccinated against po lio is more likely to contract polio (from the vaccine) than an unvaccinate d child. Does this fact imply that programs forcing vaccination on schoolch ildren are ill-advised? Include with your answer with a diagram illustrati ng the negative marginal benefit of vaccination, and a horizontal axis representing the proportion of the population vaccinated. (Exercise) The total production from an oil field generally depends on the rate at which the oil is pumped, with faster rates leading to lower total production but earlier production. Suppose two di fferent producers can pump from the field. Illustrate, using an externality diagram where the horizontal axis is the rate of production for one of the produc ers, the difference between the socially efficient outcome and the equilibrium out come. Like many other states, Texas’ law requires that when multiple people own land over a single oil field, the output is shared among the owners, with each owner obtaining a share equal to proportion of the field under th eir land. This process is called unitization. Does it solve the problem of externalities in pumping and yield an efficient outcome? Why or why not?

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-2176.3.2 Pigouvian Taxes Arthur Cecil Pigou, 1877-1959, proposed a so lution to the problem of externalities that has become a standard approach. This simple id ea is to impose a per-unit tax on a good generating negative external ities equal to the marginal externality at the socially efficient quantity. Thus, if at the socially e fficient quantity, the marginal external cost is a dollar, then a one dollar per unit tax would lead to the right outcome. This is illustrated in Figure 6-14. The tax that is added is the difference, at the socially efficient quantity, between the marginal social cost and the marginal privat e cost, which equals the marginal external cost. The tax level need not equal the margin al external cost at other quantities, and the diagram reflects a marginal external cost th at is growing as the quantity grows. Nevertheless, the new supply curv e created by the addition of the tax intersects demand (the marginal benefit) at the socially effici ent quantity. As a result, the new competitive equilibrium, taking account of the tax, is efficient. (Exercise) Identify the tax revenue produced by a Pigouvian tax in Figure 6-14. What is the relationship between the tax revenue and the damage produced by the negative externality? Is the tax re venue sufficient to pay those damaged by the external effect an amount equal to their damage? Hint: Is the marginal external effect increasing or decreasing. The case of a positive externality is similar. In this case, a subsidy is needed to induce the efficient quantity. It is left as an exercise. Figure 6-14: The Pigouvian Tax p q Marginal Benefit Private Marginal Cost Marginal Social Cost Socially Efficient Quantity Socially Efficient Price Tax Private Marginal Cost + Tax

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-218 (Exercise) Identify on a diagram the Pigouvian subsidy needed to induce the efficient quantity in the case of a posi tive externality. When is the subsidy expended smaller than the total external benefit? (Exercise) Use the formulae for estimating the effect of a tax on quantity to deduce the size of the tax needed to adju st for an externality when the marginal social cost is twice the marginal private cost. Taxes and subsidies are fairly common inst ruments to control externalities. We subsidize higher education with state univer sities, and the federal government provides funds for research and limited funds for the arts. Taxes on cigarettes and alcoholic beverages are used to discourage these acti vities, perhaps because smoking and drinking alcoholic beverages create negative external ities. (Cigarettes and alcohol also have inelastic demands, which make them good cand idates for taxation since there is only a small distortion of the quantity.) However, while important in some arenas, taxes and subsidies are not the most common approach to regulation of externalities. 6.3.3 Quotas The Pigouvian tax and subsidy approach to dealing with externalities has several problems. First, it requires knowing the ma rginal value or cost of the external effect, and this may be a challenge to estimate. Seco nd, it requires the imposition of taxes and permits the payment of subsidies, which enco urages what might be politely termed as “misappropriation of funds.” That is, once a government agency is permitted to tax some activities and subsidize others, there will be a tendency to tax things people in the agency don’t like, and subsidize “pet” projects using the potential for externalities as an excuse rather than a real reason. U.S. po liticians have been es pecially quick to see positive externalities in oil, cattle and the family farm, externalities that haven’t been successfully articulated. (The Canadian gove rnment, in contrast, sees externalities in film-making and railroads.) An alternative to the Pigouvian tax or subsidy solution is to set a quota, which is a limit on the activity. Quotas can be maxima or minima, depending on whether the activity generates negative or positive externalities. We set maximum levels of many pollutants rather than tax them, and ban some activiti es, like lead in gasoline or paint, or chlorofluorocarbons (CFCs) outright (a quota equal to zero). We set maximum amounts of impurities, like rat feces, in foodstuffs We impose minimum educational attainment (eighth grade or age 16, whichever comes first), minimum age to drive, minimum amount of rest time for truck drivers and airl ine pilots. A large set of regulations govern electricity and plumbing, designed to promot e safety, and these tend to be “minimum standards.” Quotas are a much more commo n regulatory strategy for dealing with externalities than taxes and subsidies. The idea behind a quota is to limit the quanti ty to the efficient level. If a negative externality in pollution means our society pollutes too much, then impose a limit or quantity restriction on pollution. If th e positive externality of education means individuals in our society receive too little ed ucation from the social perspective, force them to go to school.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-219 As noted, quotas have the advantage that they address the problem without letting the government spend more money, limiting the go vernment’s ability to misuse funds. On the other hand, quotas have the problem of id entifying who should get the quota; quotas will often misallocate the resource. Indeed, a small number of power plants account for almost half of the man-made sulfur diox ide pollution emitted into the atmosphere, primarily because these plants historically em itted a lot of pollution and their pollution level was set by their historical levels. Qu otas tend to harm new entrants compared to existing firms, and discourage the adoption of new technology. Indeed, the biggest polluters must stay with old technology in order to maintain their right to pollute. (Exercise) If a quota is set to the socially effi cient level, how does the value of a quota right compare to the Pigouvian tax? (Exercise) Speeding (driving fast) creates externalities by increasing the likelihood and severity of automobile a ccidents, and most count ries put a limit on speed, but one could instead require fast drivers to buy a permit to speed. Discuss the advantages and disadv antages of “speeding permits.” 6.3.4 Tradable Permits and Auctions A solution to inefficiencies in the allocation of quota rights is to permit trading them. Tradable permits for pollution create a market in the right to pollute, and thereby create a tax on polluting: the emission of pollution requires the purchase of permits to pollute, and the price of these permits represents a tax on pollution. Thus, tradable permits represent a hybrid of a quota system an d a Pigouvian taxation system – a quota determines the overall quantity of polluti on as in a quota system, determining the supply of pollution rights, but the purchase of pollution rights acts like a tax on pollution, a tax whose level is determ ined by the quota supply and demand. Figure 6-15: SO2 Permit Prices

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-220 The United States has permitted the trading of permits for some pollutants, like sulfur dioxide. Figure 6-15 shows the price of sulfur dioxide permits over the past decade.67 Each permit conveys the right to emit one ton of sulfur dioxide into the air. The overall pollution level is being reduced over time, which accounts for some of the increase in prices. These prices represent significant ta xes on large polluters, as a coal-fired power plant, using coal with high sulfur content can annually produce as much as 200,000 tons of sulfur dioxide. The major advantage of a tradable permits syst em is that it creates the opportunity for efficient exchange – one potential polluter can buy permits from another, leaving the total amount of pollution constant. Such exchange is efficient because it uses the pollution in the manner creating the highes t value, eliminating a bias toward “old” sources. Indeed, a low value polluter might sell its permits and just shut down, if the price of pollution were high enough. A somewhat unexpected advantage of tradable permits was the purchase of permits by environmental groups like the Sierra Club. Environmental groups can buy permits and then not exercise them, as a way of cleaning the air. In this case, the purchase of the permits creates a major positive external ity on the rest of society, since the environmental group expends its own reso urces to reduce pollution of others. Tradable permits offer the advantages of a ta xation scheme – efficient use of pollution – without needing to estimate the social cost of pollution directly. This is especially valuable when the strategy is to set a quan tity equal to the current quantity, and then gradually reduce the quantity to reduce the e ffects of the pollution. The price of permits can be a very useful instrument is assessing the appropriate time to reduce the quantity, since high permit prices, relative to likely marginal external costs, suggests that the quantity of the quota is too lo w, while low prices suggest th at the quantity is too large and should be reduced. 6.3.5 Coasian Bargaining The negative externality of a neighbor playing loud music late at night is not ordinarily solved with a tax or with a quota, but instea d though an agreement. When there aren’t many individuals involved, the individuals may be able to solve the problem of externalities without involving a government, but through negotiation. This insight was developed by Nobel laureate Ronald Coase (1910 – ). Coase offered the example of a cattle ranch next to a farm. There is a negative externality, in that the cattle tend to wander over to the farm and eat the crops, rather than staying on the ranch. What happens next depends on property rights, which are the rights that come with ownership. One of three things might be efficient from a s ocial perspective. It might be efficient to erect a fence to keep the cows aw ay from the crops. It might be efficient to close down 67 Source: Environmental Protec tion Agency, July 22, 2004, ding/so2market/alprices.html

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-221the farm. Finally, it might be efficient to cl ose down the ranch, if the farm is valuable enough, and the fence costs more than the value of the ranch. If the farmer has a right not to have his cr ops eaten, and can confiscate the cows if they wander onto the farm, then the rancher will ha ve an incentive to erect a fence to keep the cows away, if that is the efficient solution. If the efficient solution is to close down the ranch, then the rancher will do that, sinc e the farmer can confiscate the cows if they go to the farm and it isn’t worth building the fence by hypothesis. Finally, if the efficient solution to the externality is to close down the farm, the rancher will have an incentive to buy the farm in order to purchase the farm ’s rights, so that he can keep the ranch in operation. Since it is efficient to close down the farm only if the farm is worth less than the ranch, there is enough value in operating the ranch to purchase the farm at its value and still have money left over – that is there are gains from trade from selling the farm to the rancher. In all three cases, if the farmer has the property rights, the efficient outcome is reached. Now suppose instead that the rancher has the rights, and that the farmer has no recourse if the cows eat his crops. If shutting down the farm is efficient, the farmer has no recourse but to shut down. Similarly, if bu ilding the fence is effi cient, the farmer will build the fence to protect his crops. Finally, if shutting down the ranch is efficient, the farmer will buy the ranch from the rancher, in order to be able to continue to operate the more valuable farm. In all cases, the efficient solution is reached through negotiation. Coase argued that bargaining can generally so lve problems of externalities, and that the real problem is ill-defined property rights. If the rancher and the farmer can’t transfer their property rights, then the efficient outc ome may not arise. In the Coasian view of externalities, if an individual owned the air, air pollution would not be a problem, because the owner would charge for the use and wouldn’t perm it an inefficient level of pollution. The case of air pollution demonstr ates some of the limitations of the Coasian approach, because ownership of the air, or ev en the more limited right to pollute into the air, would create an additional set of pr oblems, a case where the cure is likely worse than the disease. Bargaining to solve the problem of externalitie s is often feasible when a small number of people are involved. When a large number of people are potentially involved, as with air pollution, bargaining is unlikely to be successful in addressing the problem of externalities, and a different approach required. 6.3.6 Fishing and Extinction Consider an unregulated fishing market like the lobster market considered above, and let S be the stock of fish. The purpose of this example is illustrative of the logic, rather than an exact accounting of the biology of fish populations, but is not unreasonable. Let S be the stock of a particular species of fi sh. Our starting point is an environment without fishing: how does th e fish population change over time? Denote the change over time in the fish population by S (S is notation for the deri vative with respect to time, notation that dates back to Sir Isaac Ne wton.) We assume that population growth follows the logistic equation ). 1 (S rS S This equation reflects two underlying

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-222assumptions. First, mating and reproducti on is proportional to the stock of fish S. Second, survival is proportional to the amount of available resources 1-S, where 1 is set to be the maximum sustainable population. (Set the units of the number of fish so that 1 is the full population.) The dynamics of the number of fish is illustrated in Figure 6-16. On the horizontal axis is the number of fish, and on the vertical axis is the change in S. When S>0, S is increasing over time, and the arrows on the ho rizontal axis reflect this. Similarly, if S<0, S is decreasing. Absent fishing, the value 1 is a stable steady state of the fish population. It is a steady state because, if S=1, S=0, that is, there is no change in the fish population. It is stable because the effect of a small perturbation – S near but not exactly equal to 1 – is to return to 1. (In fact, the fish population is very nearly globally stable – start with any population other than zero and the population returns to 1.)68 Figure 6-16: Fish Population Dynamics Now we introduce a human population and tu rn to the economics of fishing. Suppose that a boat costs b to launch and operate, and that it captures a fixed fraction a of the total stock of fish S, that is, each boat catches aS. Fish sell for a price 1Q p, where the price arises from the demand curve, wh ich in this case has constant elasticity and Q is the quantity of fish offered for sale. Suppose there be n boats launched; then the quantity of fish caught is Q=naS. Fishers enter the market as long as profits are positive, which leads to zero profits for fishers, that is, ) (Q p n Q b This equation 68 It turns out that there is a closed form solution for the fish population: rte S S S t S )) 0 ( 1 ( ) 0 ( ) 0 ( ) (. 0.2 0.4 0.6 0.8 1 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 S S

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-223makes a company just indifferent to launchin g an additional boat, because the costs and revenues are balanced. These two equation s yield two equations in the two unknowns n and Q: 11 ) (Q b b Q Qp n, and Q=naS. These two equations solve for the number of fish caught: b aS Q and the number of boats 1 1 S b a n. Subtracting the capture by humans from th e growth in the fish population yields: ) 1 ( b aS S rS S Thus, a steady state satisfies ) 1 ( 0 b aS S rS S Figure 6-17: Fish Population Dynamics with Fishing Will human fishing drive the fi sh to extinction? Extinction must occur when the only stable solution to the stock of fish is zero. Consider first the case when demand is elastic (>1). In this case, for S near zero but positive, 0 rS S, because the other terms are S S S*

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-224small relative to the linear term. Thus, wi th elastic demand, ther e is always a steady state without extinction. (Extinction is also an equilibrium, too, but over-fishing won’t get the system there.) This equilibrium is illustrated in Figure 6-17. The dark curve represents S, and thus for S between 0 and the point labeled S*, S is positive and thus S is increasing over time. Similarly, to the right of S*, S is decreasing. Thus, S* is stable under small perturbations in the stock of fish and is an equilibrium. We see that if demand for fish is elastic, fishing will not drive th e fish to extinction. Even so, fishing will reduce the stock of fish below the efficient level, because individual fishers don’t take account of the externality they impose – their fishing reduces the stock for future generations. The leve l of fish in the sea converges to S* satisfying *) 1 ( 0 b aS S rS In contrast, if demand is inelastic, fishing may drive the fish to extinction. For example, if r=2 and a=b=1, and =0.7, extinction is necessary, as is illustrated in Figure 6-18. Figure 6-18: Fish Population Dynamics: Extinction Figure 6-18 shows that, for the given parameters, the net growth of the fish population is negative for every value of the stock S. Thus the population of fish consistently dwindles. This is a case when the fishing externality (overf ishing today reduces the stock of fish tomorrow) has particularly dire consequ ences. The reason why the elasticity of demand matters is that, with inelastic demand the fall in the stock of fish increases the price by a large amount (enough so that total revenue rises). This, in turn, increases the number of fishing boats, in spite of the fa ll in the catch. In contrast, with elastic demand, the number of fishing boats falls as the stock falls, reducing the proportion of fish caught, and thus preventing extinction. We see this for the equation for the number of fishing boats 0.1 0.2 0.3 0.4 0.5 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 S S

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-225 1 1 S b a n which reflects the fact that fishing effort rises as the stock falls if and only if demand is inelastic. It is possible, even with inelastic demand, fo r there to be a stable fish population: not all parameter values lead to extinction. Usin g the same parameters as before, but with =0.9, we obtain a stable outcome illustrated in Figure 6-19. Figure 6-19: Possibility of Multiple Equilibria In addition to the stable equilibrium outcome, there is an unstable steady state, which might either converge upward or downward. It is a feature of fishing with inelastic demand that there is a region where extinction is inevitable, for when the stock is near zero, the high demand price induced by inelas ticity forces sufficient fishing to insure extinction. As a consequence of the fishing externality, nations attempt to regulate fishing, both by extending their own reach 200 miles into the se a, and by treaties limiting fishing in the open sea. These regulatory attempts have met with only modest success at preventing over-fishing. What is the efficient stock of fish? This is a challenging mathematical problem, but some insight can be gleaned via a steady st ate analysis. A steady state arise when 0 S. If a constant amount Q is removed, a steady state in the stock must occur at Q S rS S ) 1 ( 0. This maximum catch then occurs at S = and Q = r. This is not the efficient level, for it neglects the cost of boats, and the efficient stock will actually be larger. More generally, it is never e fficient to send the population below the maximum point on the survival curve plotted in Figure 6-16. 0.1 0.2 0.3 0.4 -0.04 -0.02 0.02 S S Stable Equilibrium Unstable Equilibrium

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-226 Conceptually, fishing is an example of the tragedy of the commons externality already discussed. However, the threat of a perman ent extinction and alluring possibility of solving dynamic models make it a particularly dramatic example. (Exercise) Suppose = 1. For what parameter values are fish necessarily driven to extinction? Can you interpret this condition to say that the demand for caught fish exceeds the production via reproduction? 6.4 Public Goods A public good has two attributes: nonexcludability, which means the producer can’t prevent the use of the good by others, and nonrivalry, which means that many people can use the good simultaneously. 6.4.1 Examples Consider a company offering a fireworks displa y. Pretty much anyone nearby can watch the fireworks, and people with houses in the right place have a great view of them. The company that creates the fireworks can’t comp el those with nearby homes to pay for the fireworks, and so a lot of people get to wa tch them without paying This will make it difficult or impossible for the fireworks com pany to make a profit. In addition, fireworks offer nonrivalry, in that one pers on’s viewing of the di splay doesn’t impinge significantly on another’s viewing. Nonrivalry has the implication that the efficient price is zero, since the marginal cost of another viewer is zero. The classic example of a public good is nati onal defense. National defense is clearly non-excludable, for if we spend the resources necessary to defend our national borders, it isn’t going to be possible to defend everything except one apartment on the second floor of a three story apartment on East Mapl e Street. Once we have kept our enemies out of our borders, we’ve protected everyone within the borders. Similarly, the defense of the national borders exhibits a fair degr ee of nonrivalry, especially insofar as the strategy of defense is to deter an attack in the first place. That is, the same expenditure of resources protects all. It is theoretically possible to exclude some from the use of a poem, or a mathematical theorem, but exclusion is generally quite difficult. Both poems and theorems are nonrivalrous. Similarly, technological and software inventions are non-rivalrous, even though a patent grants the right to exclude th e use by others. Another good that permits exclusion at a cost is a highway. A toll high way shows that exclusion is possible on the highways. Exclusion is quite expensive, pa rtly because the tollbooths require staffing, but mainly because of the delays imposed on drivers associated with paying the tolls – the time costs of toll roads are high. Highwa ys are an intermediate case where exclusion is possible only at a significant cost, and thus should be avoided if possible. Highways are also rivalrous at high congestion levels, but nonrivalrous at low congestion levels. That is, the marginal cost of an additional us er is essentially zero for a sizeable number of users, but then marginal cost grows rapidly in the number of users. With fewer than 700 cars per lane per hour on a four lane highway, generally the flow of traffic is

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-227unimpeded.69 As congestion grows beyond this le vel, traffic slows down and congestion sets in. Thus, west Texas interstate hi ghways are usually nonrivalrous, while Los Angeles freeways are usually very rivalrous. Like highways, recreational parks are nonriv alrous at low use levels, becoming rivalrous as they become sufficiently crowded. Also like highways, it is possible but expensive to exclude potential users, since exclusion requ ires fences and a means for admitting some but not others. (Some exclusive parks provide keys to legitimate users, while others use gatekeepers to charge admission.) 6.4.2 Free-Riders Consider a neighborhood association which is considering buying land and building a park in the neighborhood. The value of the pa rk is going to depend on the size of the park, and we suppose for simplicity that th e value in dollars of the park to each household in the neighborhood is a bn S, where n is the number of park users, S is the size of the park and a and b a are parameters satisfying 0 a, because the gains from a large park exceed the congestion effects. That is, ther e is a scale advantage – doubling the number of people and the size of the park in creases each individual’s enjoyment. How much will selfish individuals voluntarily contribute to the building of the park? That of course depends on what they think others will contribute. Consider a single household, and suppose that househol d thinks the others will contribute S-1 to the building of the park. Should the househol d contribute, and if so, how much? If the household contributes s, the park will have size S = S-1 + s, which the household values at a bn s S ) (1. Thus, the net gain to a household that contributes s when the others contribute S-1 is s n s Sa b ) (1. (Exercise) Verify that individual residents gain from contributing to the park if b abn S 1 1) ( and gain from reducing their contributions if b abn S 1 1) (. The previous exercise shows that individu al residents gain from their marginal contribution if and only if the park is smaller than b abn S 1 1 0) (. Consequently, under voluntary contributions, the only equilibrium park size is S0. That is, for any park size smaller than S0, citizens will voluntarily contribute to make the park larger. For any larger size, no one is willing to contribute. 69 The effect of doubling the number of lanes from 2 to 4 is dramatic. A two lane highway generally flows at 60 mph or more provided there are fewer than 200 cars per lane per hour, while a four lane highway can accommodate 700 cars per lane per hour at the same speed.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-228Under voluntary contributions, as the neighborhood grows in number, the size of the park shrinks. This makes sense – the benefits of individual contributions to the park mostly accrue to others, which reduces the payoff to any one contributor. How large should the park be? The total value of the park of size S to the residents together is n times the individual value, wh ich gives a collective value of a bn S1, and the park costs S, so from a social perspective the park should be sized to maximizeS n Sa b1, which yields an optimal park of size b abn S 1 1 1) ( *. Thus, as the neighborhood grows, the park should grow but as we saw the park would shrink if the neighborhood has to rely on voluntary contributions. This is because people contribute individually as if they were bu ilding the park for themselves, and don’t account for the value they provide to thei r neighbors when they contribute. Under individual contributions, the hope that ot hers contribute leads individuals not to contribute. Moreover, use of the park by ot hers reduces the value of the park to each individual, so that the size of the park shrin ks as the population grows under individual contributions. In contrast, the park ought to grow faster than the number of residents grows, as the per capita park size is b a b bn b n S 1 1 1* which is an increasing function of n.70 The lack of incentive for individuals to con tribute to a social good is known as a freerider problem. The term refers to the individuals who don’t contribute, who are said to free-ride on the contributions of others. There are two aspects of the free-rider problem apparent in this simple mathematical mode l. First, the individual incentive to contribute to a public good is reduced by the contributions of others, and thus individual contributions tend to be smalle r when the group is larger. Put another way, the size of the free-rider problem grows as the community grows larger. Second, as the community grows larger, the optimal size of the public good grows. The market failure under voluntary contributions is greater the larger is the community. In the theory presented, the optimal size of the public good is b abn S 1 1 1) ( *, and the actual size under voluntary contributions is b abn S 1 1 0) (, a gap that gets very large as the number of people grows. The upshot is that people will voluntarily contr ibute too little from a social perspective, by free-riding on the contributions of others. A good example of the provision of public goods is a co-authored term paper. This is a public good because the grade given to the paper is the same for each author, and the quality of the paper depends on the sum of the efforts of the individual authors. Genera lly, with two authors, both work pretty hard on the manuscript in order to get a good grade. Add a third author and it is a virtual 70 Reminder: In making statements like should and ough t, there is no conflict in this model because every household agrees about the optimal size of the park, so that a change to a park size of S*, paid with equal contributions, maximizes every household’s utility.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-229certainty that two of the authors think the th ird didn’t work as hard and was a free-rider on the project. The term paper example also points to the li mitations of the theory. Many people are not as selfish as the theory assumed and wi ll contribute more than would be privately optimal. Moreover, with small numbers, bargaining between the contributors and the division of labor (each works on a section) may help reduce the free-rider problem. Nevertheless, even with these li mitations, the free-rider problem is very real and it gets worse the more people are involved. The theory shows that if some individuals contribute more than their share in an altr uistic way, the more selfish individuals contribute even less, undoing some of the good done by the altruists. (Exercise) For the model presented in this se ction, compute the elasticity of the optimal park size with respect to the number of residents, that is, the percent change in S* for a small percentage change in n. [Hint: use the linear approximation trick rr) 1 ( for near zero.] (Exercise) For the model of this section, show that an individual’s utility when the park is optimally sized and the expenses are shared equally among the n individuals is b a b b b bn b b u 1 1 1 1. Does this model predict an increase in utility from larger communities? (Exercise) Suppose two people, person 1 and person 2, want to produce a playground to share between them. The value of the playground of size S to each person is S, where S is the number of dollars spent building it. Show that under voluntary contributions, the si ze of the playground is and that the efficient size is 1. (Exercise) For the previous exercise, now su ppose person 1 offers “matching funds,” that is, offers to contribute an equal amount to the contributions of the person 2. How large a playground will person 2 choose? 6.4.3 Provision with Taxation Faced with the fact that voluntary contrib utions produce an inadequate park, the neighborhood turns to taxes. Many ne ighborhood associations or condominium associations have taxing authority, and can compel individuals to contribute. Clearly in the example from the previous section, and ind eed a solution is to require each resident to contribute the amount 1, resulting in a park that is optimally sized at n. Generally it is possible in principle to provide the correct size of the public good using taxes to fund it. However, it will be a challenge in practice which can be illustrated with a slight modification of the example. Let individuals have different strengths of preferences, so that individual i values the public good of size S at a b in S v in dollars. (It is useful to assume that no two people

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-230have the same v values to simplify arguments.) The optimal size of the park for the neighborhood is b a b b n i i b an v b v b n 1 1 1 1 1 1 1 1) ( where n i iv n v11 is the average value. Again, taxes can be a ssessed to pay for an optimally -sized park, but some people (those with small v values) will view that as a ba d deal, while others (with large v) view it as a good deal. What will the neighborhood choose? If there are an odd number of voters in the neighborhood, the prediction is that the park will serve the median voter the best.71 With equal taxes, an individual obtains n S n S va b i. If there are an odd number of people, n can be written as 2k+1. The median voter is the person for whom k have values vi larger than hers, and k have values smaller. Consider increasing S. If the median voter likes it, then so do all the people with higher v’s, and the proposition to increase S passes. Similarly, a proposal to decrease S will get a majority if the median vote r likes it. If the median voter likes reducing S, all the individuals with smaller vi will vote for it as well. Thus, we can see that voting maximizes the preferences of th e median voter, and simple calculus shows that entails b a b kn bv S 1 1 1 1) (. Unfortunately, voting does not result in an efficient outcome generally, and only does so when the average value equals the median va lue. On the other hand, voting generally performs much better than voluntary contributio ns. The park size can either be larger or smaller under median vo ting than is efficient.72 (Exercise) For the model of this section, show that, under voluntary contributions, only one person contributes and that person is the person with the largest vi. How much do they contribute? [Hint: which individual i is willing to contribute at the largest park size? Given the park this individual desires, can anyone else bene fit from contributing at all?] (Exercise) Show that if all individuals value the public good equally, voting on the size of the good results in the e fficient provision of the public good. 6.4.4 Local Public Goods The example in the previous section sho wed that there are challenges to a neighborhood’s provision of public goods created by differences in the preferences of the public good. Voting does not generally lead to the efficient provision of the public good, and does so only in special circumstances, like agreement of preferences. 71 The voting model used is that there is a status quo, which is a planned size of S. Anyone can propose to change the size of S, and the neighborhood votes yes or no. If an S exists such that no replacement gets a majority vote, that S is an equilibrium under majority voting. 72 The general principle here is that the median voting will do better when the distribution of values is such that the average of n values exceeds the median, which in turn exceeds the maximum divided by n. This is true for most empiri cally relevant distributions.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-231A different solution was proposed by Tiebout73 in 1956. This solution works only when the public goods are local in nature – pe ople living nearby may or may not be excludable, but people living further away can be excluded, and such goods are called “local public goods.” Schools are local – more distant people can readily be excluded. Parks are harder to exclude from, but are still local in nature; few people will drive 30 miles to use a park. Suppose that there are a variety of neighborh oods, some with high taxes, better schools, big parks, beautifully maintained trees on th e streets, frequent garbage pickup, a firstrate fire department, extensive police protec tion and spectacular fireworks displays, and others with lower taxes and more modest provis ion of public goods. People will tend to move to the neighborhood that fits their pref erences. The result is neighborhoods that are relatively homogeneous with respect to the desire for public goods. That homogeneity, in turn, makes voti ng work better. That is, th e ability of people to choose their neighborhoods to suit their preference s over taxes and public goods will make the neighborhood provision of public goods more efficient. The “Tiebo ut theory” shows that local public goods will tend to be efficientl y provided. In addition, even private goods like garbage collection and schools can be effi ciently provided publicly if they are local goods, and there are enough distinct localities to offer a broad range of services. (Exercise) Consider a baby-sitting cooperative, where parents rotate supervision of the children of several families. Suppose that, if the sitting service is available with frequency Y, the value placed by person i is vi Y and the costs of contribution y is ny2, where Y is the sum of the individual contributions and n is the number of families. Rank v1 v2 … vn. (i) What is the size of the service under voluntary contributions? (Hint: Let yi be the contribution of family i. Identify the payoff of family j as 2j j i i j jy n y y v What value of yj maximizes this expression?) (ii) What contributions maximize the total social value n 1 i 2 1 1j n j j n j jy n y v? [Hint: Are the values of yi different for different i?] (iii) Let n j jv n11 and 2 1 2) ( 1 n j jv n. Conclude that, under voluntary contributions, the total value ge nerated by the cooperative is 22 2 n (Hint: 73 Charles Tiebout, 1919-1962. His surname is pronounced “tee-boo.”

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-232It helps to know that 1 1 2 1 ) ( 12 1 2 1 2 1 1 2 2 1 2 n j j n j n j j n j j n j jv n n v n v n v n) 6.5 Monopoly We have spent a great deal of time on the competitive model, and we now turn to the polar opposite case, that of monopoly. A monopoly is a firm that faces a downward sloping demand, and has a choice about what price to charge – an increase in price doesn’t send most or all of the customers away to rivals. There are very few pure monopolies. The U.S. post office has a monopoly in first-class mail, but faces competition by FedEx and ot her express mail companies, as well as by faxes and email, in the broader “send documents to others” market. Microsoft has a great deal of market power, but a small pe rcentage of personal computer users choose Apple or Linux operating systems. There is only one U.S. manufacturer of aircraft carriers. However, there are many firms that have market power or monopoly power, which means that they can increase their price abov e marginal cost and sustain sales for a long period of time.74 The theory of monopoly is applicab le to such firms, although they may face an additional and important constraint: a price increase may affect the behavior of rivals. The behavior of rivals is the subject of the next chapter. A large market share is not a proof of monopo ly, nor is a small market share proof that a firm lacks monopoly power. U.S. Air do minated air traffic to Philadelphia and Pittsburgh, but still lost money. Porsche ha s a small share of the automobile market, or even the high-end automobile market, but still has monopoly power in that market. 6.5.1 Sources of Monopoly There are three basic sources of monopoly. The most common source is to be granted a monopoly by the government, either through patents, in which case the monopoly is temporary, or through a government franchise. Intelsat was a government franchise that was granted a monopoly on satellite co mmunications, a monopoly that ultimately proved lucrative indeed. Many cities and towns license a single cable TV company or taxi company, although usually basic rates an d fares are set by the terms of the license agreement. New drugs are granted patents that provide a monopoly for a period of time. (Patents generally last twenty years, but pharmace utical drugs have their own patent laws.) Copyright also confers a mo nopoly for a supposedly limited period of time. Thus, the Disney Corporation owns copyrights on Mickey Mouse, copyrights which by law should have expired, but have been granted an extension by Congress each time they were due to expire. Copyrights create monopoly power over music as well as cartoon characters, and Time-Warner owns th e rights to the song “Happy Birthday to 74 These terms are used somewhat differently by di fferent authors. Both require downward sloping demand, and usually some notion of sustainability of sales. Some distinguish the terms by whether they are “large” or not, others by how long the price increase can be sustained. We won’t need such distinctions here.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-233You,” and receives royalties every time it is played on the radio or other commercial venue.75 Many of the Beatles songs which Mc Cartney co-authored were purchased by Michael Jackson. This book is copyrigh ted under terms that expressly prohibit commercial use but permit most other uses. A second source of monopoly is a large econ omy of scale. The scale economy needs to be large relative to the size of demand. If the average cost when a si ngle firm serves the entire market is lower than when two or more firms serve the market, a monopoly can be the result. For example, long distance te lephone lines were expensive to install, and the first company to do so, A.T. & T., wound up being the only provider of long distance service in the United States. Similarly, scale economies in electricity generation meant that most communities had a single electric ity provider prior to the 1980s, when new technology made relatively smaller scale generation more efficient. The demand-side equivalent of an economy of scale is a network externality. A network externality arises when others’ use of a product makes it more valuable to each consumer. Standards are a common source of network externality. That AA batteries are standardized makes them more readily accessible, helps drive down their price through competition and economies of scal e, and thus makes the AA battery more valuable. AA batteries are available everyw here, unlike proprietary batteries. Fax machines are valuable only if others have simi lar machines. In addition to standards, a source of network externality is third-part y products. Choosing Microsoft Windows as a computer operating system means that ther e is more software available than for Macintosh or Linux, as the widespread adop tion of Windows has led a large variety of software to be written for it. The JVC Vide o Home System of VCRs came to dominate the Sony Beta system, primarily because there were more movies to rent in the VHS format than in the Beta format at the video rental store. In contrast, recordable DVD has been hobbled by incompatible standards of DVD+R and DVD-R, a conflict not resolved even as the next generation – 50GB discs such as Sony’s Blu-ray – start to reach the market. DVDs themselves were sl ow to be adopted by consumers, because few discs were available for rent at video rent al stores, which is a consequence of few adoptions of DVD players. As DVD players became more prevalent, and the number of discs for rent increased, the market tipped and DVDs came to dominate VHS. The third source of monopoly is control of an essential, or a sufficiently valuable, input to the production process. Such an input could be technology that confers a cost advantage. For example, software is run by a computer operating system, and needs to be designed to work well with the operatin g system. There have been a series of allegations that Microsoft kept secret some of the “application program interfaces” used by Word as a means of hobbling rivals. If so, access to the design of the operating system itself is an important input. 6.5.2 Basic Analysis Even a monopoly is constrained by demand. A monopoly would like to sell lots of units at very high prices, but higher prices necessari ly lead to a loss in sales. So how does a monopoly choose its price and quantity? 75 Fair use provisions protect individuals with non-commercial uses of copyrighted materials.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-234A monopoly can choose price, or a monopoly can choose quantity and let the demand dictate the price. It is slightly more con venient to formulate the theory in terms of quantity rather than price, because costs are a function of quantity. Thus, we let p(q) be the demand price associated with quantity q, and c(q) be the cost of producing q. The monopoly’s profits are ) ( ) (q c q q p The monopoly earns the revenue pq and pays the cost c. This leads to the first order condition, for the profit-maximizing quantity qm: ) ( ) ( ) ( 0m m m mq c q p q q p q The term ) ( ) (q p q q p is known as marginal revenue. It is the derivative of revenue pq with respect to quantity. Thus a monopoly choos es a quantity qm where marginal revenue equals marginal cost, and charges the maximum price p(qm) the market will bear at that quantity. Marginal revenue is below demand p(q) because demand is downward sloping. That is, ) ( ) ( ) (q p q p q q p Figure 6-20: Basic Monopoly Diagram (Exercise) If demand is linear, p(q)=a – bq, what is marginal revenue? Plot demand and marginal revenue, and total revenue qp(q) as a function of q. qm p q DMC qc Monopoly Profits Dead Weight Loss MR pm

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-2356.5.2.2 (Exercise) For the case of constant elasticity of demand, what is marginal revenue? (Exercise) If both demand and supply have constant elasticity, compute the monopoly quantity and price. The choice of monopoly quantity is illustrated in Figure 6-20. The key points of this diagram are, first, that marginal revenue lies below the demand curve. This occurs because marginal revenue is the demand p(q) plus a negative number. Second, the monopoly quantity equates marginal revenue and marginal cost, but the monopoly price is higher than the marginal cost. Third, ther e is a dead weight loss, for the same reason that taxes create a dead weight loss: the hi gher price of the monopoly prevents some units from being traded that are valued mo re highly than they cost. Fourth, the monopoly profits from the increase in price, and the monopoly profit is shaded. Fifth, since under competitive conditions supply equa ls marginal cost, the intersection of marginal cost and demand corresponds to th e competitive outcome. We see that the monopoly restricts output and charges a higher price than would prevail under competition. We can rearrange the monopoly pricing form ula to produce an additional insight. ) ( ) ( ) (m m m mq p q q c q p or 1 ) ( ) ( ) ( ) ( ) ( m m m m m mq p q p q q p q c q p The left hand side of this equation is known as the price-cost margin or Lerner Index.76 The right hand side is one over the elasti city of demand. This formula relates the markup over marginal cost to the elasticity of demand. It is important because perfect competition forces price to equal marginal cos t, so this formula provides a measure of the deviation from competition, and in pa rticular says that the deviation from competition is small when th e elasticity of demand is large, and vice versa. Marginal cost will always be at least zero or la rger. If marginal cost is less than zero, the least expensive way to produce a given quan tity is to produce more and throw some away. Thus, the price-cost margin is no greater than one, and as a result, a monopolist produces in the elastic portion of demand. One implication of this observation is that if demand is everywhere inelastic (e.g. aq q p) ( for a>1), the optimal monopoly quantity is essentially zero, and in any event would be no more than one molecule of the product. 76 Abba Lerner, 1903-1982. Note that ) ( ) ( 1 ) ( ) ( 1 p dp q dq q p q p q q p q p qm m m m m m which is used in the derivation.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-236 In addition, the effects of monopoly are related to the elasticity of demand. If demand is very elastic, the effect of monopoly on prices is quite limited. In contrast, if the demand is relatively inelastic, monopolies wi ll increase prices by a large margin. We can rewrite the formula to obtain ). ( 1 ) (m mq c q p Thus, a monopolist marks up marginal cost by the factor 1 at least when >1. This formula is sometimes used to justify a “fi xed markup policy,” which means a company adds a constant percentage markup to its prod ucts. This is an ill-advised policy not justified by the formula, because the form ula suggests a markup which depends on the demand for the product in question and thus not a fixed markup for all products a company produces. 6.5.3 Effect of Taxes A tax imposed on a seller with monopoly po wer performs differently than a tax imposed on a competitive industry. Ultimately a perf ectly competitive industry must pass on all of a tax to consumers, because in the long -run the competitive industry earns zero profits. In contrast, a monopolist might absorb some portion of a tax even in the longrun. To model the effect of taxes on a monopoly consider a monopolist who faces a tax rate t per unit of sales. This monopolist earns tq q c q q p ) ( ) (. The first order condition for profit maximization yields t q c q p q q p qm m m m ) ( ) ( ) ( 0 Viewing the monopoly quan tity as a function of t, we obtain: 0 ) ( ) ( ) ( 2 1 m m m m mq c q p q q p dt dq, with the sign following from the second orde r condition for profit maximization. In addition, the change in price satisfies 0 ) ( ) ( ) ( 2 ) ( ) ( m m m m m m mq c q p q q p q p dt dq q p

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-237Thus, a tax causes a monopoly to increase its price. In addition, the monopoly price rises by less than the tax if 1 ) ( dt dq q pm m, or 0 ) ( ) ( ) ( m m m mq c q p q q p This condition need not be true, but is a standard regularity condition imposed by assumption. It is true for linear demand and increasing marginal cost. It is false for constant elasticity of demand, >1 (which is the relevant case, for otherwise the second order conditions fail) and constant marginal cost. In the latter case (constant elasticity and marginal cost), a tax on a monopoly increa ses price by more than the amount of the tax. (Exercise) Use a revealed preference argument to show that a per unit tax imposed on a monopoly causes the quantity to fall. That is, hypothesize quantities qb before the tax, and qa after the tax, and show that two facts – the before tax monopoly preferred qb to qa and the taxed monopoly made higher profits from qb together imply the qb qa. (Exercise) When both demand and supply ha ve constant elasticity, use the results of (Exercise) to compute the effect of a proportional tax (i.e. a portion of the price paid to the government). 6.5.4 Price Discrimination Pharmaceutical drugs for sale in Mexico are generally priced substantially below their U.S. counterparts. Pharmaceutical drugs in Europe are also cheaper than in the U.S., although not as inexpensive as in Mexico, wi th Canadian prices usually between the U.S. and European prices. (The comparison is be tween identical drugs produced by the same manufacturer.) Pharmaceutical drugs differ in price across countries primarily because demand conditions vary. The formula ). ( 1 ) (m mq c q p shows that a monopoly seller wo uld like to charge a higher markup over marginal cost to customers with less elastic demand than to customers with more elastic demand, because 1 is a decreasing function of for >1. Charging differe nt prices for the same product to different customers is known as price discrimination. In business settings, it is sometimes known as value-based pricing, which is a more palatable term to tell to customers. Computer software vendors often sell a “stude nt” version of their software, usually at substantially reduced prices, and requiring pr oof of being a student to qualify for the lower price. Such student discounts are exam ples of price discrimination, and students

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-238have more elastic demand than business users. Similarly, the student and senior citizen discounts at movies and other venues sell th e same thing – a ticket to the show – for different prices, and thus qualify as price discrimination. In order for a seller to price-discri minate, the seller must be able to identify (approximately) the demand of groups of customers prevent arbitrage Arbitrage is also known as “buying low and selling high,” and represents the act of being an intermediary. Since price discrimination requires charging one group a higher price than another, there is potentially an opport unity for arbitrage, arising from members of the low price group buying at the low price an d selling at the high price. If the seller can’t prevent arbitrage, arbitrage essentially converts a two-price system to sales at the low price. Why offer student discounts at the movies? You already know th e answer to this – students have lower incomes on average than others, and lower incomes translate into a lower willingness to pay for normal goods. Consequently a discount to a student makes sense from a demand perspective. Arbitrag e can be mostly prevented by requiring a student identification card to be presented. Senior citizen discounts are a bit more subtle. Generally seniors aren’t poorer than other groups of customers (in the United States, at least). However, seniors have more free time, and thus are able to substitute to matinee showings77 or drive to more distant locations should those offer discounts. Thus seniors have relatively elastic demand more because of their ability to substitute than because of their income. Airlines commonly price discriminate, usin g “Saturday night stay-overs” and other devices. To see that such charges repres ent price discrimination, consider a passenger who lives in Dallas but needs to spend Mond ay through Thursday in Los Angeles two weeks in a row. This passenger could buy two round-trip tickets: Trip One: First Monday: Dallas Los Angeles First Friday: Los Angeles Dallas Trip Two: Second Monday: Dallas Los Angeles Second Friday: Los Angeles Dallas At the time of this writing, the approxim ate combined cost of these two flights was US$2,000. In contrast, another way of arrang ing exactly the same travel is to have two round-trips, one of which originates in Dallas, while the other originates in Los Angeles: 77 Matinee showings are those early in the day, whic h are usually discounted. These discounts are not price discrimination because a show at noon isn’t the same product as a show in the evening.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-239Trip One: First Monday: Dallas Los Angeles Second Friday: Los Angeles Dallas Trip Two: First Friday: Los Angeles Dallas Second Monday: Dallas Los Angeles This pair of round trips involves exactly the same travel as the firs t pair, but costs less than $500 for both (at the time of this writ ing). The difference is that the second pair involves staying over Saturday night for both legs, and that leads to a major discount for most U.S. airlines. (American Airlines quoted the fares.) How can airlines price discriminate? There are two major groups of customers: business travelers and leisure travelers. Bu siness travelers have the higher willingness to pay overall, and the nature of their trip s tends to be that they come home for the weekend. In contrast, a leisure traveler will usually want to be away for a weekend, so a weekend stay-over is an indicator of a leisure traveler. It doesn’t work perfectly as an indicator – some business travelers must be aw ay for the weekend – but it is sufficiently correlated with leisure travel that it is prof itable for the airline to price discriminate. These examples illustrate an important distin ction. Senior citizen and student discounts are based on the identity of the buyer, and qualifying for the discount requires showing an identity card. In contrast, airline price di scrimination is not based on the identity of the buyer but on the choices by the buyer. The former is known as direct price discrimination, while the latter is known as indirect price discrimination.78 Two common examples of indirect price di scrimination are coupons and quantity discounts. Coupons offer discounts for prod ucts and are especially common in grocery stores, where they are usually provided in a newspaper section available free at the front of the store. Coupons discriminate on the basis of the cost of time. It takes time to find the coupons for the products one is interest ed in buying, and thus those with a high value of time won’t find it worthwhile spendi ng twenty minutes to save $5 (effectively a $15 per hour return), while those with a low value of time will find that return worthwhile. Since those with a low value of time tend to be more price sensitive (more elastic demand), coupons offer a discount available to all but used primarily by customers with a more elastic demand, and thus increase the profits of the seller. Quantity discounts are discounts for buying mo re. Thus, the large size of milk, laundry detergent and other items often cost less per un it than smaller sizes, and the difference is greater than the savings on packaging costs In some cases, the larger sizes entail greater packaging costs; some manufacturers “ba nd together” individual units, incurring additional costs to create a larger size whic h is then discounted. Thus, the “twenty-four pack” of paper towels sells for less per roll than the individual rolls; such large volumes appeal primarily to large families, who are more price-sensitive on average. 78 The older and incoherent language for these concepts called direct price discrimination “third degree price discrimination,” while indirect price discriminati on was called second degree price discrimination. In the older language, first degree price discrimination meant perfect third degree price discrimination.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-2406.5.5 Welfare Effects Is price discrimination a good thing, or a bad thing? It turns out that there is no definitive answer to this question. Instead, it depends on circumstances. We illustrate this conclusion with a pair of exercises. (Exercise) Let marginal cost be zero for all quantities. Suppose there are two equal-sized groups of customers, group 1 with demand q(p)=12-p, group 2 with demand q(p)=8-p. Show that a non-discriminati ng monopolist charges a price of 5 and the discriminating monopolist charges group 1 the price 6 and group 2 the price 4. Then calculate the gains from trade, with discrimination and without, and show that price discrimina tion reduces the gains from trade. This exercise illustrates a much more gene ral proposition: if a price-discriminating monopolist produces less than a non-di scriminating monopolist, then price discrimination reduced welfare. This prop osition has an elementary proof. Consider the price discriminating monopo list’s sales, and then allo w arbitrage. The arbitrage increases the gains from trade, since every transaction has gains from trade. Arbitrage, however, leads to a common price like that ch arged by a non-discriminating monopolist. Thus, the only way price discrimination can incr ease welfare is if it leads a seller to sell more output than she would otherwise. This is possible, as the next exercise shows. (Exercise) Let marginal cost be zero for all quantities. Suppose there are two equal-sized groups of customers, group 1 with demand q(p)=12-p, group 2 with demand q(p)=4-p. Show that a non-discriminati ng monopolist charges a price of 6 and the discriminating monopolist charges group 1 the price 6 and group 2 the price 2. Then calculate the gains from trade, with discrimination and without, and show that price discrimina tion increases the gains from trade. In this exercise, we see that price discrimina tion that brings in a new group of customers may increase the gains from trade. In deed, this example involves a Pareto improvement: the seller and group 2 are bette r off, and group 1 no worse off, than without price discrimination. (A Pareto impr ovement requires that no one is worse off and at least one person is better off.) Whether price discrimination increases th e gains from trade overall depends on circumstances. However, it is worth rememb ering that people with lower incomes tend to have more elastic demand, and thus get lo wer prices under price discrimination than absent price discrimination. Consequently, a ban on price discrimination tends to hurt the poor and benefit the rich no matter what the overall effect. 6.5.6 Two-Part Pricing A common form of price discrimination is known as two-part pricing. Two-part pricing usually involves a fixed charge and a marginal charge, and thus offers an ability for a seller to capture a portion of the consumer surplus. For example, electricity often comes with a fixed price per month and then a pr ice per kilowatt-hour, which is two-part pricing. Similarly, long distance and cellu lar telephone companies charge a fixed fee per month, with a fixed number of “included” mi nutes, and a price per minute for additional

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-241minutes. Such contracts really involve three parts rather than two-parts, but are similar in spirit. From the seller’s perspective, the ideal two-pa rt price is to charge marginal cost plus a fixed charge equal to the customer’s consume r surplus, or perhaps a penny less. By setting price equal to marginal cost, the se ller maximizes the gains from trade. By setting the fixed fee equal to consumer surplus, the seller captures the entire gains from trade. This is illustrated in Figure 6-21. Figure 6-21: Two-Part Pricing 6.5.7 Natural Monopoly A natural monopoly arises when a single fi rm can efficiently serve the entire market because average costs are lower with one firm than with two firms. An example is illustrated in Figure 6-22. In this case, the average to tal cost of a single firm is lower than if two firms operate, splitting the outp ut between them. The monopolist would like to price at pm, which maximizes profits.79 79 The monopoly price may or may not be sustainable. A monopoly price is not sustainable if it would lead to entry, thereby undercutting the monopoly. The feasibility of entry, in turn, depends on whether the costs of entering are not recoverable (“sunk”), and how rapidly entry can occur. If the monopoly price is not sustainable, the monopoly may engage in limit pricing, which is jargon for pricing to deter (limit) entry. p q DMC qc Consumer Surplus = Fixed Fee Price=pc

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-242 Figure 6-22: Natural Monopoly Historically, the United States and other nati ons have regulated natu ral monopolies like those found in electricity, telephony and wa ter service. An immediate problem with regulation is that the efficient price, that is, the price that maximizes the gains from trade, requires a subsidy from outside the in dustry. We see the need for a subsidy in Figure 6-22 because the price that maximizes the gains from trade is p1, which sets the demand (marginal value) equal to the marginal cost. At this price, however, the average total cost exceeds the price, so that a firm wi th such a regulated price would lose money. There are two alternatives. The product cou ld be subsidized, and subsidies are used with postal service and passenger rail in th e United States, and historically for many more products in Canada and Europe including airlines and airplane manufacture. Alternatively, regulation could be impo sed that aims to limit the price to p2, the lowest break-even price. This is the more common strategy in the United States. There are two strategies toward limiting the price: price-cap regulation, which directly imposes a maximum price, and rate of return regulation, that limits the profitability of firms. Both of these approaches induce some inefficiency of production. In both cases, an increase in average cost may translate in to additional profits for the firm, causing regulated firms to engage in unnecessary activities. 6.5.8 Peak Load Pricing Fluctuations in demand often require holding capacity which is used only a fraction of the time. Hotels have off-seasons when most rooms are empty. Electric power plants are designed to handle peak demand, us ually hot summer days, with some of the capacity standing idle on other days. Demand for trans-Atlantic airline flights is much higher in the summer than the rest of the year All of these examples have the similarity that an amount of capacity – hotel space, ai rplane seats, electric generation capacity – will be used over and over, which means it is used in both high demand and low demand states. How should pricing be accomplishe d when demand fluctuates? This can be q ATC MC P D p1 pm p2

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-243thought of as a question of how to alloca te the cost of capacity across several time periods when demand systematically fluctuates. Consider a firm that experiences two kinds of costs – a capacity cost and a marginal cost. How should capacity be priced? This issue is applicable to a wide variety of industries, including pipelines, airlines, te lephone networks, construction, electricity, highways, and the internet. The basic peak-load pricing problem, pioneere d by Marcel Boiteux (1922 – ), considers two periods. The firm’s profits are given by ). ( } { max2 1 2 1 2 2 1 1q q mc q q q p q p Setting price equal to marginal costs is not su stainable, because a firm selling with price equal to marginal cost would not earn a retu rn on the capacity, and thus would lose money and go out of business. Consequent ly, a capacity charge is necessary. The question of peak load pricing is how the ca pacity charge should be allocated. This question is not trivial because some of th e capacity is used in both periods. For the sake of simplicity we will assume demands ar e independent, that is, q1 is independent of p2 and vice versa. This assumption is often unrealistic, and generalizing it actually doesn’t complicate the problem too much. The primary complication is in computing the social welfare when demands ar e functions of two prices. Independence is a convenient starting point. Social welfare is ). ( } { max ) ( ) (2 1 2 1 0 2 0 12 1q q mc q q dx x p dx x p Wq q The Ramsey problem is to maximize W subject to a minimum profit condition. A technique for accomplishing this maxi mization is to instead maximize L = W + By varying we vary the importance of profits to the maximization problem, which will increase the profit level in the solution as increases. Thus, the correct solution to the constrained maximization problem is the outcome of the maximization of L, for some value of A useful notation is 1A, which is known as the characteristic func tion of the set A. This is a function which is 1 when A is true, and zero otherwise. Using this notation, the first order condition for the maximization of L is: mc q p q q p mc q p q Lq q q q q q 2 1 2 11 ) ( ) ( 1 ) ( 01 1 1 1 1 1

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-244 or, 1 1 1 11 1 1 ) (2 1 p mc q pq q where 2 11q qis the characteristic function of the event q1 q2. Similarly, 2 2 2 21 1 1 ) (2 1 p mc q pq q Note as before that yields the monopoly solution. There are two potential types of solution. Let the demand for good 1 exceed the demand for good 2. Either q1>q2, or the two are equal. Case 1: q1>q2. 1 1 1 11 1 ) ( p mc q p and 2 2 2 21 1 ) ( p mc q p. In case 1, with all of the capacity charge a llocated to good 1, quantity for good 1 still exceeds quantity for good 2. Thus, the peak period for good 1 is an extreme peak. In contrast, case 2 arises when assigning the ca pacity charge to good 1 would reverse the peak – assigning all of the capacity charge to good 1 would make period 2 the peak. Case 2: q1=q2. The profit equation can be written p1(q) mc + p2(q) – mc = This equation determines q, and prices are determined from demand. The major conclusion from peak load pricing is that either the entire cost of capacity is allocated to the peak period, or there is no peak period in the sense that the two periods have the same quantity demanded given the pr ices. That is, either the prices equalize the quantity demanded, or the prices impose th e entire cost of capacity only on one peak period. Moreover, the price (or, more properly, the ma rkup over marginal cost) is proportional to the inverse of the elasticity, wh ich is known as Ramsey pricing.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-2456.6 Information An important advantage of the price system is that it economizes on information. A typical consumer needs to know only the prices of goods and their own personal preferences in order to make a sensible ch oice of purchases, and manufacturers only need to know the prices of goods in order to decide what to produce. Such economies of information are an advantage over centrally-planned economies, which attempt to direct production and consumption decision s using something other than prices, and centrally-planned economies typically expe rience chronic shortages and occasional surpluses. Shortages of important inputs to production may have dramatic effects and the shortages aren’t remedied by the price of the input rising in a centrally planned economy, and thus often persist for long periods of time. There are, however, circumstances where the prices are not the only necessary information required for firms and consumers to make good decisions. In such circumstances, information itself can lead to market failures. 6.6.1 Market for Lemons Nobel laureate George Akerlof (1940 – ) examined the market for used cars and considered a situation where the sellers are be tter informed than th e buyers. This is quite reasonable, as sellers have owned the car for a while and are likely to know its quirks and potential problems. Akerlof showed that this differential information may cause the used car market to collapse; that is, the information possessed by sellers of used cars destroys the market. To understand Akerlof’s insight, suppose that the quality of used cars lies on a 0 to 1 scale and that the population of used cars is uniformly distributed on the interval from 0 to 1. In addition, let that quality repres ent the value a seller places on the car, and suppose buyers put a value that is 50% higher than the selle r. Finally, the seller knows the actual quality, while the buyer does not. Can a buyer and seller trade in such a situation? First, note that trade is a good thing, because the buyer values the car more than th e seller. That is, both the buyer and seller know that they should trade. But can th ey agree on a price? Consider a price p. At this price, any seller who values the car less than p will be willing to trade. But because of our uniform distribution assumption, this me ans the distribution of qualities of cars offered for trade at price p will be uniform on the interval 0 to p. Consequently, the average quality of these cars will be p, and the buyer values these cars 50% more which yields p. Thus, the buyer is not willing to pay the price p for the average car offered at price p. The effect of the informed seller, and uninfo rmed buyer, produces a “lemons” problem. At any given price, all the lemons and only a few of the good cars are offered, and the buyer – not knowing the quality of the car – is n’t willing to pay as much as the actual value of a high value car offered for sale. Th is causes the market to collapse; and only the worthless cars trade at a price around zero. Economists call the differential information an informational asymmetry.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-246In the real world, of course, the market ha s found partial or imperfect solutions to the lemons problem identified by Akerlof. First, buyers can become informed and regularly hire their own mechanic to inspect a car th ey are considering. Inspections reduce the informational asymmetry but are costly in their own right. Second, intermediaries offer warranties and certification to mitigate the le mons problem. The existence of both of these solutions, which involve costs in their ow n right, is itself evidence that the lemons problem is a real and significant problem, even though competitive markets find ways to ameliorate the problems. An important example of the lemons problem is the inventor who creates an idea that is difficult or impossible to patent. Consider an innovation that would reduce the cost of manufacturing computers. The inventor would like to sell it to a computer company, but can’t tell the computer company what the innovation entails prior to price negotiations, because then the computer com pany could just copy the innovation. Similarly, the computer company can’t possib ly offer a price for the innovation in advance of knowing what the innovation is As a result, such innovations usually require the inventor to enter the computer ma nufacturing business, rather than selling to an existing manufacturer, entail ing many otherwise unnecessary costs. (Exercise) In Akerlof’s market for lemons mo del, suppose it is possible to certify cars, verifying that they are better than a particular quality q. Thus, a market for cars “at least as good as q” is possible. What price or prices are possible in this market? [Hint: sellers offer cars only if q quality p.] What quality maximizes the expected gains from trade? 6.6.2 Myerson-Satterthwaite Theorem The lemons problem is a situation where the buyers are relatively uninformed and care about the information held by sellers. Lemo ns problems are limited to situations where the buyer isn’t well-informed and can be miti gated by making information public. In many transactions, the buyer knows the qual ity of the product, so lemons concerns aren’t a significant issue. There can still be a market failure, however, if there are a limited number of buyers and sellers. Consider the case of one buyer and one seller bargaining over the sale of a good. The buyer knows his own value v for the good, but not the selle r’s cost. The seller knows her own cost c for the good, but not the buyer’s value. The buyer views the seller’s cost as uniformly distributed on the interval [0,1], and similarly the seller views the buyer’s value as uniformly distributed on [0,1].80 Can efficient trade take place? Efficient trade requires that trade occurs whenever v>c, and the remarkable answer is that it is impossible to arrange efficient trade if the buye r and seller are to trade voluntarily. This 80 The remarkable fact proved by Roger Myerson an d Mark Satterthwaite (Efficient Mechanisms for Bilateral Trade, Journal of Economic Theory, 28, 1983, 265-281) is that the distributions don’t matter; the failure of efficient trade is a fully general pr operty. Philip Reny and Preston McAfee (Correlated Information and Mechanism Design, Econometrica 60, No. 2, March 1992, 395-421) show the nature of the distribution of information matters, and Pres ton McAfee (Efficient Allocation with Continuous Quantities, Journal of Economic Theory 53, no. 1, February 1991: 51 -74.) showed that continuous quantities can overturn the Myerson-Satterthwaite theorem.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-247is true even if a third party is used to help arrange trade, provided the third party doesn’t subsidize the transaction. The total gains from trade under efficiency are 6 1 21 0 2 1 00 dv v dv dc c vv. A means of arranging trade, or a mechanism, asks the buyer and seller for their value and cost, respectively, and then orders trade if the value exceeds the cost, and dictates a payment p by the buyer to the seller. Buyers need not make honest reports to the mechanism, however, and the mechanisms must be designed to induce the buyer and seller to report honestly to the mechanism, so that efficient trades can be arranged.81 Consider a buyer who actually has value v but reports a value r. The buyer trades with the seller if the seller has a cost less than r, which occurs with probability r. u(r,v) = vr – Ecp(r, c). The buyer gets the actual value v with probability r, and makes a payment that depends on the buyer’s report and the seller’s repo rt, but we can take expectations over the seller’s report to eliminate it (from the buyer’s perspective), and this is denoted Ecp(r, c), which is just the expected payment given the report r. In order for the buyer to choose to be honest, u must be maximized at r=v for every v, for otherwise some buyers would lie and some trades would not be e fficiently arranged. Thus, we can conclude82 ) ( ) ( ) ( ) (2 2 1v r v v u v v u v v u v v u dv dv r The first equality is just the total derivative of u(v,v), because there are two terms; the second equality because u is maximized over the first argument r at r=v, and the first order condition insures u1 = 0. Finally, u2 is just r, and we are evaluating the derivative at the point r = v. A buyer who has a value v + but who reports v, trades with probability v and makes the payment Ecp(v, c). Such a buyer gets v more in utility than the buyer with value v. Thus a increase in value produces an increase in utility of at least v, showing that v v v u v v u ) ( ) ( and hence that v v v u dv d ) (. A similar argument consideri ng a buyer with value v who reports v + shows that equality occurs. 81 Inducing honesty is without loss of generality. Suppose that the buyer of type v reported the type z(v). Then we can add a stage to the mechanism, where th e buyer reports a type, which is converted via the function z to a report, and then that report given to th e original mechanism. In the new mechanism, reporting v is tantamount to reporting z(v) to the original mechanism. 82 We maintain an earlier notation that the subscript re fers to a partial derivative, so that if we have a function f, f1 is the partial derivative of f with respect to the first argument of f.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-248The value u(v,v) is the gain accruing to a buyer with value v who reports having value v. Since the buyer with value 0 gets zero, the to tal gain accruing to the average buyer can be computed by integrating by parts 6 1 ) 1 ( ) 1 ( ) ( ) 1 ( ) (1 0 1 0 1 0 1 0 vdv v dv dv du v v v u v dv v v uv. In the integration by parts, dv = d –(1–v) is used. The remarkable conclusion is that, if the buyer is induced to truthfully reveal the buyer’s value, the buyer must obtain the entire gains from trade! This is actually a quite general proposition. If you offer the entire gains from trade to a party, they are induced to maximize the gains from trade. Otherwise, they will want to distort away fr om maximizing the entire gains from trade, which will result in a failure of efficiency. The logic with respect to the seller is analogou s: the only way to get the seller to report her cost honestly is to offer her the entire gains from trade. (Exercise) Let h(r, c) be the gains of a seller who has cost c and reports r, h(r, c) = p(v, r) – (1-r)c. Noting that the highest cost seller (c=1) never sells and thus obtains zero profits, show that honesty by the seller implies the expected value of h is 6 1. The Myerson-Satterthwaite theorem shows that the gains from trade are insufficient to induce honesty by both parties. (Indeed, th ey are half the necessary amount!) Thus, any mechanism for arranging trades between the buyer and the seller must suffer some inefficiency. Generally this occurs because bu yers act like they value the good less than they do, and sellers act like their cos ts are higher than they truly are. It turns out that the worst case scenario is a single buyer and a single seller. As markets get “thick,” the per capita losses converge to zero, and markets become efficient. Thus, informational problems of this kind are a “small numbers” issue. However, many markets do in fact have small numbers of buye rs or sellers. In such markets, it seems likely that informational problems will be an impediment to efficient trade. 6.6.3 Signaling An interesting approach to solving informational problems involves signaling.83 Signaling, in economic jargon, means expend itures of time or money whose purpose is to convince others of something. Thus, peop le signal wealth by wearing Rolex watches, driving expensive cars or sailing in the Americ a’s Cup. They signal erudition by tossing out quotes from Kafka or Tacitus into convers ations. They signal being chic by wearing the right clothes and listening to cool music. Signaling is also rampant in the animal 83 Signaling was introduced by Nobel laureate Michae l Spence in his dissertation, part of which was reprinted in “Job Market Signaling,” Quarterly Journal of Economics 87, August 1973, 355-74.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-249world, from peacock feathers to elk battles and the subject of a vibrant and related research program. A university education serves not just to educat e, but also to signal the ability to learn. Businesses often desire employees who are able to adapt to changing circumstances, and who can easily and readily learn new strategi es and approaches. Education signals such abilities because it will easier for quick learne rs to perform well in university. A simple model suffices to illustrate the point. Su ppose there are two types of people. Type A has a low cost cA of learning, and type B has a higher cost cB of learning. It is difficult to determine from an interview whether someone is type A or not. Type A is worth more to businesses, and the competitive wage wA (expressed as a present value of lifetime earnings) for type A’s is higher than the wage wB for type B’s. A person can signal that they are a type A by taking a sufficient amount of education. Suppose the person devotes an amount of time x to learning in university, thus incurring the cost cA x. If x is large enough so that wA – cA x > wB > wA – cB x, it pays the type A to obtain the education, but not the type B, if education in fact signals that the student is type A. Thus, a level of education x in this case signals a trait (ease of learning) that is valued by business, and it does so by voluntary choice – those with a high cost of learning choose not to obtain the education, even though they could do it. This works as a signal because only type A would voluntarily obtain the education in return for being perceived to be a type A. There are several interesting aspects to this kind of signaling. First, the education embodied in x need not be valuable in itself; the student could be studying astronomy or ancient Greek, neither of which are very usef ul in most businesses, but are nevertheless strong signals of the ability to learn. Second, the best subject matter for signaling is that in which the difference in cost between th e type desired by employers and the less desirable type is greatest, that is, where cB – cA is greatest. Practical knowledge is somewhat unlikely to make this difference gr eat; instead, challenging abstract problemsolving may be a better separator. Clearly, it is desirable to have the subject matter be useful, if it can still do the signaling job. But interpreting long medieval poems could more readily signal the kind of flexible mind desired in management than studying accounting, not because the desirable type is g ood at it, or that it is useful, but because the less desirable type is so much worse at it. Third, one interprets signals by asking “wha t kinds of people woul d make this choice?” while understanding that the person makes the choice hoping to send the signal. Successful law firms have very fine offices, ge nerally much finer than the offices of their clients. Moreover, there are back rooms at mo st law firms, where much of the real work is done, that aren’t nearly so opulent. The purpose of the expensive offices is to signal success, essentially making the statement that “we couldn’t afford to waste money on such expensive offices if we weren’t very successful. Thus, you should believe we are successful.”

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McAfee: Introduction to Economic Analysis,, July 24, 2006 6-250 The law firm example is similar to the ed ucation example. Here, the cost of the expenditures on fancy offices is different fo r different law firms because more successful firms earn more money and thus value the marginal dollar less. Consequently, more successful firms have a lower cost of a given le vel of office luxury. What is interesting about signaling is that it is potentially qu ite wasteful. A student spends four years studying boring poems and dead languages in order to demonstrate a love of learning, and a law firm pays $75,000 for a conference table that it rarely uses and gets no pleasure out of, in order to convince a client that the firm is extremely successful. In both cases, it seems like a less costly solution should be available. The student can take standardized tests, and the law firm could show its win-loss record to the potential client. But standardized tests may measure test-taking skills rather than learning ability, especially if what matters is the lear ning ability over a long time horizon. Winloss records can be “massaged,” and in the majo rity of all legal disputes, the case settles and both sides consider themselves “the winner .” Consequently, statistics may not be a good indicator of success, and the expe nsive conference table a better guide.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-2517 Strategic Behavior Competitive theory studies price-taking consu mers and firms, that is, people who can’t individually affect the transaction prices. Th e assumption that market participants take prices as given is justified only when ther e are many competing participants. We have also examined monopoly, precisely because a monopoly by definition doesn’t have to worry about competitors. Strategic beha vior involves the examination of the intermediate case, where there are few enough participants that they take each other into account and their actions individually matter, and where the behavior of any one participant influences choices of the other participants. That is, participants are strategic in their choice of action recognizing that their choi ce will affect choices made by others. The right tool for the job of examining strategic behavior in economic circumstances is game theory, the study of how people play games. Game theory was pioneered by the mathematical genius John von Neumann (19031957). Game theory has also been very influential in the study of military strate gy, and indeed the strategy of the cold war between the United States and the U.S.S.R. was guided by game theoretic analyses.84 7.1 Games The theory of games provides a description of games that fits common games like poker or the board game “Monopoly” but will cover many other situations as well. In any game, there is a list of players. Games gene rally unfold over time; at each moment in time, players have information, possibly incom plete, about the current state of play, and a set of actions they can take. Both inform ation and actions may depend on the history of the game prior to that moment. Finally players have payoffs, and are assumed to play in such a way as to maximize their expected payoff, taking into account their expectations for the play of others. When the players, their information and available actions, and payoffs have been specified, we have a game. 7.1.1 Matrix Games The simplest game is called a matrix payoff ga me with two players. In a matrix payoff game, all actions are chosen simultaneously. It is conventional to describe a matrix payoff game as played by a row player and a column player. The row player chooses a row in a matrix; the column player simultaneously chooses a column. The outcome of the game is a pair of payoffs where the first entry is the payoff of the row player and the second is the payoff of the column player. Table 7-1 provides an example of a “2 2” matrix payoff game, the most famous ga me of all, which is known as the prisoner’s dilemma. 84 An important reference for game theory is Jo hn von Neumann (1903-1957) and Oskar Morgenstern (1902-1977), Theory of Games and Economic Behavior, Princeton: Princeton University Press, 1944. Important extensions were introduced by John Nash (1928 – ), the mathematician made famous by Sylvia Nasar’s delightful book A Beautiful Mind (Simon & Schuster, 1998). Finally, applications in the military arena were pioneered by Nobel Laureate Thomas Schelling (1921 – ), The Strategy of Conflict, Cambridge: Cambridge University Press, 1960.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-252Table 7-1: The Prisoner’s Dilemma Column Confess Don’t Confess (-10,-10) (0,-20) Row Don’t (-20,0) (-1,-1) In the prisoner’s dilemma, two criminals named Row and Column have been apprehended by the police and are being questi oned separately. They are jointly guilty of the crime. Each player can choose either to confess or not. If Row confesses, we are in the top row of the matrix (corresponding to the row labeled Confess). Similarly, if Column confesses, the payoff will be in the re levant column. In this case, if only one player confesses, that player goes free and th e other serves twenty years in jail. (The entries correspond to the number of years lost to prison. The first entry is always Row’s payoff, the second Column’s payoff.) Thus, for example, if Column confesses and Row does not, the relevant payoff is the first col umn and the second row, in reverse color in Table 7-2. Table 7-2: Solving the Prisoner's Dilemma Column Confess Don’t Confess (-10,-10) (0,-20) Row Don’t (-20,0) (-1,-1) If Column confesses and Row does not, Row loses twenty years, and Column loses no years, that is, goes free. This is th e payoff (-20,0) in reverse color in Table 7-2. If both confess, they are both convicted and neither goes free, but they only serve ten years each. Finally, if neither confesses, there is a ten percent chance they are convicted anyway (using evidence other than the confe ssion), in which case they average a year lost each. The prisoner’s dilemma is famous partly becaus e it is readily solvable. First, Row has a strict advantage to confessing, no matter what Column is going to do. If Column confesses, Row gets -10 from confessing, -20 from not, and thus is better off from confessing. Similarly, if Column doesn’t con fess, Row gets 0 from confessing, -1 from not confessing, and is better off confessing. Either way, no matter what Column does, Row should choose to confess.85 This is called a dominant strategy, a strategy that is optimal no matter what the other players do. The logic is exactly similar for Column: no matter what Row does, Column should choose to confess. That is, Column also has a dominant strategy, to confess. To establish this, first consider wh at Column’s best action is, when Column thinks Row will confess. Then consider Column’s best action when Column thinks Row won’t confess. 85 If Row and Column are friends are care about each other, that should be included as part of the payoffs. Here, there is no honor or friendship among thieves, and Row and Column only care about what they themselves will get.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-253Either way, Column gets a higher payoff (l ower number of years lost to prison) by confessing. The presence of a dominant strategy makes th e prisoner’s dilemma particularly easy to solve. Both players should con fess. Note that this gets them ten years each in prison, and thus isn’t a very good outcome from thei r perspective, but there is nothing they can do about it in the context of the game, beca use for each, the alternative to serving ten years is to serve twenty years. This outcome is referred to as (Confess, Confess), where the first entry is the row player’s choice, and the second entry is the column player’s choice. Consider an entry game, played by Micros oft (the row player) and Piuny (the column player), a small start-up company. Both Mi crosoft and Piuny are considering entering a new market for an online service. The payoff structure is Table 7-3: An Entry Game Piuny Enter Don’t Enter (2,-2) (5,0) MS Don’t (0,5) (0,0) In this case, if both companies enter, Micros oft ultimately wins the market, and earns 2, and Piuny loses 2. If either firm has the ma rket to itself, they get 5 and the other firm gets zero. If neither enters, both get zero. Microsoft has a dominant strategy to enter: it gets 2 when Piuny enters, 5 when Piuny does n’t, and in both cases does better than when Microsoft doesn’t enter. In contrast, Piuny does not have a dominant strategy: Piuny wants to enter when Microsoft doesn’t, and vice-versa. That is, Piuny’s optimal strategy depends on Microsoft’s action, or, more accurately, Piun y’s optimal strategy depends on what Piuny be lieves Microsoft will do. Piuny can understand Microsoft’s dominant strategy, if it knows the payoffs of Microsoft.86 Thus, Piuny can conclude that Microsoft is going to enter, and this means that Piuny should not enter. Thus, the equilibrium of the game is for MS to enter and Piuny not to enter. This equilibrium is arrived at by the iterated elimination of dominated strategies, which sounds like jargon but is ac tually plain speaking. First, we eliminated Microsoft’s dominated strategy in favor of its dominant strategy. Microsoft had a dominant strategy to enter, which mean s the strategy of not entering is dominated by the strategy of entering, so we eliminat ed the dominated strategy. That leaves a simplified game in which Microsoft enters: 86 It isn’t so obvious that one player will know the pa yoffs of another player, and that often causes players to try to signal that they are going to play a certain way, that is, to demonstrate commitment to a particular advantageous strategy. Such topics are taken up in business strategy and managerial economics.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-254Table 7-4; Eliminating a Dominated Strategy Piuny Enter Don’t MS Enter (2,-2) (5,0) In this simplified game, after the eliminat ion of Microsoft’s dominated strategy, Piuny also has a dominant strategy: not to enter. Thus, we iterate and eliminate dominated strategies again, this time eliminating Piun y’s dominated strategies, and wind up with a single outcome: Microsoft ente rs, and Piuny doesn’t. The iterated elimination of dominated strategies solves the game.87 Here is another game, with thr ee strategies for each player. Table 7-5: A 3 X 3 Game Column Left Center Right Top (-5,-1) (2,2) (3,3) Middle (1,-3) (1,2) (1,1) Row Bottom (0,10) (0,0) (0,-10) The process of iterated elimination of domina ted strategies is illustrated by actually eliminating the rows and colum ns, as follows. A reverse color (white writing on black background) indicates a dominated strategy. Middle dominates bottom for Row, yielding: Table 7-6: Eliminating a Dominated Strategy Column Left Center Right Top (-5,-1) (2,2) (3,3) Middle (1,-3) (1,2) (1,1) Row Bottom (0,10) (0,0) (0,-10) With bottom eliminated, Left is now dominated for Column by either Center or Right, which eliminates the left column. 87 A strategy may be dominated not by any particular alternate strategy but by a randomization over other strategies, which is an advanced topic not considered here.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-255Table 7-7: Eliminating Another Dominated Strategy Column Left Center Right Top (-5,-1) (2,2) (3,3) Middle (1,-3) (1,2) (1,1) Row Bottom (0,10) (0,0) (0,-10) With Left and Bottom eliminated, Top now dominates Middle for Row. Table 7-8: Eliminating a Third Dominated Strategy Column Left Center Right Top (-5,-1) (2,2) (3,3) Middle (1,-3) (1,2) (1,1) Row Bottom (0,10) (0,0) (0,-10) Finally, Column chooses Right over Center, yi elding a unique outcome after the iterated elimination of dominated strategies, which is (Top, Right). Table 7-9: Game Solved Column Left Center Right Top (-5,-1) (2,2) (3,3) Middle (1,-3) (1,2) (1,1) Row Bottom (0,10) (0,0) (0,-10) The iterated elimination of dominated strategies is a useful concept, and when it applies, the predicted outcome is usually quite reasonab le. Certainly it has the property that no player has an incentive to change their behavi or given the behavior of others. However, there are games where it doesn’t apply, an d these games require the machinery of a Nash equilibrium, named for Nobel laureate John Nash (1928 – ). 7.1.2 Nash Equilibrium In a Nash equilibrium, each player chooses the strategy that maximizes their expected payoff, given the strategies employed by others. For matrix payoff games with two players, a Nash equilibrium requires that the row chosen maximizes the row player’s payoff, given the column chosen by the column player, and the column, in turn, maximizes the column player’s payoff given the row selected by the row player. Let us consider first the prisoner’s dile mma, which we have already seen.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-256Table 7-10: Prisoner's Dilemma Again Column Confess Don’t Confess (-10,-10) (0,-20) Row Don’t (-20,0) (-1,-1) Given that the row player has chosen to confess, the column player also chooses confession because -10 is better than -20. Similarly, given that the column player chooses confession, the row player chooses con fession, because -10 is better than -20. Thus, for both players to confess is a Nash equilibrium. Now let us consider whether any other outcome is a Nash equilibrium. In any outcome, at least one player is not confessing. But that player could get a high er payoff by confessing, so no other outcome could be a Nash equilibrium. The logic of dominated strategies extends to Nash equilibrium, except possibly for ties. That is, if a strategy is strictly dominated, it can’t be part of a Nash equilibrium. On the other hand, if it involves a tied value, a st rategy may be dominate d but still part of a Nash equilibrium. The Nash equilibrium is justified as a solution concept for games as follows. First, if the players are playing a Nash equilibrium, no one has an incentive to change their play or re-think their strategy. Thus, the Nash equilibr ium has a “steady state” aspect in that no one wants to change their own strategy given the play of others. Second, other potential outcomes don’t have that property: if an outcome is not a Nash equilibrium, then at least one player does have an incentive to ch ange what they are doing. Outcomes that aren’t Nash equilibria involv e mistakes for at least one player. Thus, sophisticated, intelligent players may be able to deduce ea ch other’s play, and play a Nash equilibrium Do people actually play Nash equilibria? Th is is a controversial topic and mostly beyond the scope of this book, but we’ll consider two well-known games: Tic-Tac-Toe (see, e.g. and Chess. Tic-Tac-Toe is a relatively simple game, and the equilibrium is a tie. This equilibrium arises because each player has a strategy that prevents the other player from winning, so the outcome is a tie. Young children play Tic-Tac-Toe and even tually learn how to play equilibrium strategies, at which point the game ceases to be very interesting since it just repeats the same outcome. In contrast, it is known that Chess has an equilibrium, but no one knows what it is. Thus, at this po int we don’t know if the first mover (White) always wins, or the second mover (Black) always wins, or if th e outcome is a draw (neither is able to win). Chess is complicated because a strategy must specify what ac tions to take given the history of actions, and there are a very large number of potential histories of the game thirty or forty moves after the start. So we can be quite confident that people are not (yet) playing Nash equilibria to the game of Chess. The second most famous game in game theory is the battle of the sexes. The battle of the sexes involves a married couple who are going to meet each other after work, but haven’t decided where they are meeting. Thei r options are a baseball game or the ballet.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-257Both prefer to be with each other, but th e man prefers the baseball game and the woman prefers the ballet. This give s payoffs something like this: Table 7-11: The Battle of the Sexes Woman Baseball Ballet Baseball (3,2) (1,1) Man Ballet (0,0) (2,3) The man would rather that they both go to the baseball game, and the woman that they both go to the ballet. They each get 2 payo ff points for being with each other, and an additional point for being at their preferre d entertainment. In this game, iterated elimination of dominated strategies eliminat es nothing. You can readily verify that there are two Nash equilibria: one in which they both go to the baseball game, and one in which they both go to ballet. The logic is : if the man is going to the baseball game, the woman prefers the 2 points she gets at the baseball game to the single point she would get at the ballet. Similarly, if the woman is going to the baseball game, the man gets three points going there, versus zero at the ballet. Thus, for both to go to the baseball game is a Nash equilibrium. It is straightfo rward to show that for both to go to the ballet is also a Nash equilibrium, and finally that neither of the other two possibilities, involving not going to the same place, is an equilibrium. Now consider the game of matching pennies. In this game, both the row player and the column player choose heads or tails, and if they match, the row player gets the coins, while if they don’t match, the column player gets the coins. The payoffs are provided in the next table. Table 7-12: Matching Pennies Column Heads Tails Heads (1,-1) (-1,1) Row Tails (-1,1) (1,-1) You can readily verify that none of the four possibilities represents a Nash equilibrium. Any of the four involves one player getting -1; that player can convert -1 to 1 by changing his or her strategy. Thus, whatever the hy pothesized equilibrium, one player can do strictly better, contradicting the hypothesis of a Nash equilibrium. In this game, as every child who plays it knows, it pays to be unpredictable, and consequently players need to randomize. Random strategies are known as mixed strategies, because the players mix across various actions. 7.1.3 Mixed Strategies Let us consider the matching pennies game again.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-258Table 7-13: Matching Pennies Again Column Heads Tails Heads (1,-1) (-1,1) Row Tails (-1,1) (1,-1) Suppose that Row believes Column plays Heads with probability p. Then if Row plays Heads, Row gets 1 with probability p and -1 with probability (1-p), for an expected value of 2p – 1. Similarly, if Row plays Ta ils, Row gets -1 with probability p (when Column plays Heads), and 1 with probability (1-p), for an expected value of 1 – 2p. This is summarized in the next table. Table 7-14: Mixed Strategy in Matching Pennies Column Heads Tails Heads (1,-1) (-1,1) 1p + -1(1-p)=2p-1 Row Tails (-1,1) (1,-1) -1p + 1(1-p)=1-2p If 2p – 1 > 1 – 2p, then Row is better off on average playing Heads than Tails. Similarly, if 2p – 1 < 1 – 2p, Row is better off playing Tails than Heads. If, on the other hand, 2p – 1 = 1 – 2p, then Row gets the same payoff no matte r what Row does. In this case Row could play Heads, could play Tails, or could flip a coin and randomize Row’s play. A mixed strategy Nash equilibrium involves at least one pl ayer playing a randomized strategy, and no player being able to increase their expected payoff by playing an alternate strategy. A Nash equilibriu m without randomization is called a pure strategy Nash equilibrium. Note that that randomization requires equali ty of expected payoffs. If a player is supposed to randomize over strategy A or strategy B, then both of these strategies must produce the same expected payoff. Otherwise, the player would prefer one of them, and wouldn’t play the other. Computing a mixed strategy has one elemen t that often appears confusing. Suppose Row is going to randomize. Then Row’s payo ffs must be equal, fo r all strategies Row plays with positive probability. But that eq uality in Row’s payoffs doesn’t determine the probabilities with which Row plays the variou s rows. Instead, that equality in Row’s payoffs will determine the probabilities with which Column plays the various columns. The reason is that it is Co lumn’s probabilities that determine the expected payoff for Row; if Row is going to randomize, then Co lumn’s probabilities must be such that Row is willing to randomize. Thus, for example, we computed the payoff to Row of playing Heads, which was 2p – 1, where p was the probability Column played Head s. Similarly, the payoff to Row of playing Tails was 1 – 2p. Row is willing to randomize if these are equal, which solves for p =

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-259 (Exercise) Let q be the probability that Row play s Heads. Show that Column is willing to randomize if, and only if, q = (Hint: First compute Column’s expected payoff when Column plays Head s, and then Column’s expected payoff when Column plays Tails. These must be equal for Column to randomize.) Now let’s try a somewhat more challenging ex ample, and revisit the battle of the sexes. Table 7-15: Mixed Strategy in Battle of the Sexes Woman Baseball Ballet Baseball (3,2) (1,1) Man Ballet (0,0) (2,3) This game has two pure strate gy Nash equilibria: (Baseball,Baseball) and (Ballet,Ballet). Is there a mixed strategy? To compute a mixed strategy, let the Woman go to the baseball game with probability p, and the Man go to the baseball game with probability q. Table 7-16 contains the computation of the mi xed strategy payoffs for each player. Table 7-16: Full Computation of the Mixed Strategy Woman Baseball (p) Ballet (1-p) Man’s E Payoff Baseball (prob q) (3,2) (1,1) 3p + 1(1-p)=1+2p Man Ballet (prob 1-q) (0,0) (2,3) 0p + 2(1-p)=2-2p Woman’s E Payoff 2q + 0(1-q)=2q 1q + 3(1-q)=3-2q For example, if the Man (row player) goes to the baseball game, he gets 3 when the Woman goes to the baseball game (probability p) and otherwise gets 1, for an expected payoff of 3p + 1(1-p) = 1 + 2p. The other calculations are similar but you should definitely run through the logic and verify each calculation. A mixed strategy in the Battle of the Sexes game requires both parties to randomize (since a pure strategy by either party prev ents randomization by the other). The Man’s indifference between going to the baseball game and the ballet requires 1+2p = 2 – 2p, which yields p = That is, the Man will be wi lling to randomize which event he attends if the Woman is going to the ballet of the time, and otherwise to the baseball game. This makes the Man indifferent between the two events, because he prefers to be with the Woman, but he also likes to be at the baseball game; to make up for the advantage that the game holds for him, the woman has to be at the ballet more often. Similarly, in order for the Woman to rand omize, the Woman must get equal payoffs from going to the game and going to the ballet, which requires 2q = 3 – 2q, or q = Thus, the probability that the Man goes to the game is and he goes to the ballet of the time. These are independent probabilities, so to get the probability that both go to

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-260the game, we multiply the probabilities, which yields 16 3 The next table fills in the probabilities for all four possible outcomes. Table 7-17: Mixed Strategy Probabilities Woman Baseball Ballet Baseball 16 3 16 9 Man Ballet 16 1 16 3 Note that more than half the time, (Baseb all, Ballet) is the outcome of the mixed strategy, and the two people are not together. This lack of coordination is a feature of mixed strategy equilibria generally. The ex pected payoffs for both players are readily computed as well. The Man’s payoff was 1+2p = 2 – 2p, and since p = the Man obtained 1 A similar calculation shows the Woman’s payoff is the same. Thus, both do worse than coordinating on their less pref erred outcome. But this mixed strategy Nash equilibrium, undesirable as it may seem is a Nash equilibrium in the sense that neither party can improve their payoff, gi ven the behavior of the other party. In the Battle of the sexes, the mixed strate gy Nash equilibrium may seem unlikely, and we might expect the couple to coordinate more effectively. Indeed, a simple call on the telephone should rule out the mixed strategy. So let’s consider another game related to the Battle of the Sexes, where a failure of coor dination makes more sense. This is the game of “Chicken.” Chicken is played by two drivers driving toward each other, trying to convince the other to yield, which involves swervi ng into a ditch. If both swerve into the ditch, we’ll call the outcome a draw and both get zero. If one swerves and the other doesn’t, the swerver loses and the other wins, and we’ll give the winner one point.88 The only remaining question is wh at happens when both don’t yi eld, in which case a crash results. In this version, that has been set at four times the loss of swerving, but you can change the game and see what happens. Table 7-18: Chicken Column Swerve Don’t Swerve (0,0) (-1,1) Row Don’t (1,-1) (-4,-4) This game has two pure strategy equilibria: (Swerve, Don’t) and (Don’t, Swerve). In addition, it has a mixed strategy. Su ppose Column swerves with probability p. Then 88 Note that adding a constant to a player’s payoffs, or multiplying that player’s payoffs by a positive constant, doesn’t affect the Nash equilibria, pure or mixed. Therefore, we can always let one outcome for each player be zero, and another outcome be one.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-261Row gets 0p + -1(1-p) from swerving, 1p + (-4)(1-p) from not swerving, and Row will randomize if these are equal, which requires p = That is, the probability that Column swerves, in a mixed strategy equili brium is You can verify that the Row player has the same probability by setting the probability that Row swerves equal to q and computing Column’s expected payoffs. Thus, the probability of a collision is 16 1 in the mixed strategy equilibrium. The mixed strategy equilibrium is more likely in some sense in this game; if the players already knew which player would yield, they wouldn’t actually need to play the game. The whole point of the game is to find out who will yield, which means it isn’t known in advance, which means the mixed strategy equilibrium is in some sense the more reasonable equilibrium. Paper, Scissors, Rock is a child’s game in which tw o children simultaneously choose paper (hand held flat), scissors (hand with tw o fingers protruding to look like scissors) or rock (hand in a fist). The nature of th e payoffs is that paper beats rock, rock beats scissors, and scissors beat paper. This game has the structure Table 7-19: Paper, Scissors, Rock Column Paper Scissors Rock Paper (0,0) (-1,1) (1,-1) Scissors (1,-1) (0,0) (-1,1) Row Rock (-1,1) (1,-1) (0,0) (Exercise) Show that, in the Paper, Scissors Rock game, there are no pure strategy equilibria. Show that playing a ll three actions with equal likelihood is a mixed strategy equilibrium. (Exercise) Find all equilibria of the following games: 1 Column Left Right Up (3,2) (11,1) Row Down (4,5) (8,0) 2 Column Left Right Up (3,3) (0,0) Row Down (4,5) (8,0)

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-2623 Column Left Right Up (0,3) (3,0) Row Down (4,0) (0,4) 4 Column Left Right Up (7,2) (0,9) Row Down (8,7) (8,8) 5 Column Left Right Up (1,1) (2,4) Row Down (4,1) (3,2) 6 Column Left Right Up (4,2) (2,3) Row Down (3,8) (1,5) 7.1.4 Examples Our first example concerns public goods. In this game, each player can either contribute, or not. For example, two roommates can either clean their apartment, or not. If they both clean, the apartment is ni ce. If one cleans, that roommate does all the work and the other gets half of the benefits. Finally, if neither clean, neither is very happy. This suggests payoffs like: Table 7-20: Cleaning the Apartment Column Clean Don’t Clean (10,10) (0,15) Row Don’t (15,0) (2,2) You can verify that this game is similar to the prisoner’s dilemma, in that the only Nash equilibrium is the pure strategy in which neit her player cleans. This is a game theoretic version of the tragedy of the commons – ev en though the roommates would both be better off if both cleaned, neither do. As a practical matter, roommates do solve this problem, using strategies that we will in vestigate when we consider dynamic games.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-263 Table 7-21: Driving on the Right Column Left Right Left (1,1) (0,0) Row Right (0,0) (1,1) The important thing about the side of the road th e cars drive on is not that it is the right side but that it is the same side. This is captured in the Driving on the Right game above. If both players drive on the same si de, then they both ge t one point, otherwise they get zero. You can readily verify that there are two pure strategy equilibria, (Left,Left) and (Right,Right), and a mixed stra tegy equilibrium with equal probabilities. Is the mixed strategy reasonable? With auto mobiles, there is little randomization. On the other hand, people walking down hallwa ys often seem to randomize whether they pass on the left or the right, and sometimes do that little dance where they try to get past each other, one going left and the other going right, then both simultaneously reversing, unable to get out of each other’s way. That dance suggests that the mixed strategy equilibrium is not as unreasonable as it seems in the automobile application.89 Table 7-22: Bank Location Game NYC No Concession Tax Rebate No Concession (30,10) (10,20) LA Tax Rebate (20,10) (20,0) Consider a foreign bank that is looking to op en a main office and a smaller office in the United States. The bank narrows its choice fo r main office to either New York (NYC) or Los Angeles (LA), and is leaning toward Los An geles. If neither city does anything, LA will get $30 million in tax revenue and New Yo rk ten million. New York, however, could offer a $10 million rebate, which would swing the main office to New York, but now New York would only get a net of $20 M. The di scussions are carried on privately with the bank. LA could also offer the concession, which would bring the bank back to LA. 89 Continental Europe drove on the left until about the time of the French revolution. At that time, some individuals began driving on the right as a challenge to royalty who were on th e left, essentially playing the game of chicken with the ruling class. Driving on the right became a symbol of disrespect for royalty. The challengers won out, forcing a shift to driving on the right. Besides which side one drives on, another coordination game involves whether one stops or goes on red. In some locales, the tendency for a few extra cars to go as a light changes from green to yell ow to red forces those whose light changes to green to wait, and such a progression can lead to the opposite equilibrium, where one goes on red and stops on green. Under Mao Tse-tung, the Chinese considered changing the equilibrium to going on red and stopping on green (because ‘red is the color of pr ogress’) but wiser heads prevailed and the plan was scrapped.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-2647.1.4.1 (Exercise) Verify that the bank location game has no pure strategy equilibria, and that there is a mixed strategy equili brium where each city offers a rebate with probability Table 7-23: Political Mudslinging Republican Clean Mud Clean (3,1) (1,2) Dem Mud (2,1) (2,0) On the night before the election, a Democrat is leading the Wisconsin senatorial race. Absent any new developments, the Democrat will win, and the Republican will lose. This is worth 3 to the Democrat, and the Re publican, who loses honorably, values this outcome at one. The Republican could decide to run a series of negative advertisements (“throwing mud”) against the Democrat, and if so, the Republican wins although loses his honor, which he values at 1, and so only gets 2. If the Democrat runs negative ads, again the Democrat wins, but loses his honor, so only gets 2. These outcomes are represented in the Mudslinging game above. (Exercise) Show that the only Nash equilibriu m is a mixed strategy with equal probabilities of throwing mud and not throwing mud. (Exercise) Suppose that voters partially forgive a candidate for throwing mud when the rival throws mud, so that the (Mud, Mud) outcome has payoff (2.5,.5). How does the equilibrium change? You have probably had the experience of trying to avoid encountering someone, who we will call Rocky. In this instance, Rocky is actu ally trying to find you. The situation is that it is Saturday night and you are choosing which party, of two possible parties, to attend. You like party 1 better, and if Rocky goes to the other party, you get 20. If Rocky attends party 1, you are going to be uncomfortab le and get 5. Similarly, Party 2 is worth 15, unless Rocky attends, in which case it is worth 0. Rocky likes Party 2 better (these different preferences may be part of the reason you are avoiding him) but he is trying to see you. So he values Party 2 at 10, party 1 at 5 and your presence at the party he attends is worth 10. These values ar e reflected in the following table. Table 7-24: Avoiding Rocky Rocky Party 1 Party 2 Party 1 (5,15) (20,10) You Party 2 (15,5) (0,20)

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-2657.1.4.4 (Exercise) (i) Show there are no pure strategy Nash equilibria in this game. (ii) Find the mixed strategy Nash equilibr ia. (iii) Show that the probability you encounter Rocky is 12 7 Our final example involves two firms competing for customers. These firms can either price high or low. The most money is made if they both price high, but if one prices low, it can take most of the business away from the rival. If they both price low, they make modest profits. This description is reflected in the following table: Table 7-25: Price Cutting Game Firm 2 High Low High (15,15) (0,25) Firm 1 Low (25,0) (5,5) (Exercise) Show that the firms have a dominant strategy to price low, so that the only Nash equilibrium is (Low, Low). 7.1.5 Two Period Games So far, we have considered only games that are played simultaneously. Several of these games, notably the price cutting and apartment cleaning games, are actually played over and over again. Other games, like the bank location game, may only be played once but nevertheless are played over time. Recall the bank location game: Table 7-26; Bank Location Revisited NYC No Concession Tax Rebate No Concession (30,10) (10,20) LA Tax Rebate (20,10) (20,0) If neither city offered a rebate, then LA won the bidding. So suppose instead of the simultaneous move game, that first New Yo rk decided whether to offer a rebate, and then LA could decide to offer a rebate. This sequential structure leads to a game that looks like Figure 7-1: In this game, NYC makes the first move, and chooses Rebate (to the left) or No Rebate (to the right). If NYC chooses Rebate, LA can then choose Rebate or None. Similarly, if NYC chooses No Rebate, LA can choose Reba te or None. The payoffs (using the standard of (LA, NYC) ordering ) are written below the choices.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-266 Figure 7-1 Sequential Bank Location (NYC payoff listed first) What NYC would like to do depends on what NYC believes LA will do. What should NYC believe about LA? (Boy does that rhet orical question suggest a lot of facetious answers.) The natural belief is that LA will do what is in LA’s best interest. This idea – that each stage of a game is played in a maximizing way – is called subgame perfection. 7.1.6 Subgame Perfection Subgame perfection requires each player to act in its best interest, independent of the history of the game.90 This seems very sensible and in most contexts it is sensible. In some settings, it may be implausible. Even if I see a player make a particular mistake three times in a row, subgame perfection requ ires that I must continue to believe that player will not make the mistake again. Subgame perfection may be implausible in some circumstances, especially when it pays to be considered somewhat crazy. In the example, subgame perfection requires LA to offer a rebate when NYC does (since LA gets 20 by rebating versus 10), and not wh en NYC doesn’t. This is illustrated in the game using arrows to indicate LA’s choices. In addition, the actions that LA won’t choose have been re-colored in a light grey in Figure 7-2. Once LA’s subgame perfect choices are taken into account, NYC is presented with the choice of offering a rebate, in which case it ge ts 0, or not offering a rebate, in which case it gets 10. Clearly the optimal choice for NY C is to offer no rebate, in which case LA doesn’t either, and the result is 30 for LA, and 10 for NYC. Dynamic games are generally “solved backward” in this way. That is first establish what the last player does, then figure out based on the last player’s expected behavior, what the penultimate player does, and so on. 90 Subgame perfection was introduced by Nobel laureate Reinhart Selten (1930 – ). NYC Rebate No Rebate Rebate None Rebate None LA LA (20,0) (10,20) (20,10) (30,10)

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-267 Figure 7-2: Subgame Perfection We’ll consider one more application of subgame perfection. Suppose, in the game “Avoiding Rocky,” Rocky is actually stalking you, and can condition his choice on your choice. Then you might as well go to the pa rty you like best, because Rocky is going to follow you wherever you go. This is represented in Figure 7-3. Figure 7-3: Can’t Avoid Rocky Since Rocky’s optimal choice eliminates your best outcomes, you make the best of a bad situation by choosing Party 1. Here, Rocky has a second mover advantage: Rocky’s ability to condition on your choice mean t he does better than he would do in a simultaneous game. (Exercise) Formulate the battle of the sexes as a sequential game, letting the woman choose first. (This situation could arise if the woman can leave a message for the man about where she has go ne.) Show that there is only one You Party 1 Party 2 Party 1 Party 2 Party 1 Party 2 R R (5,15) (20,10) (15,5) (0,20) NYC Rebate No Rebate Rebate None Rebate None LA LA (20,0) (10,20) (20,10) (30,10)

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-268subgame perfect equilibrium, and that the woman enjoys a first-mover advantage over the man, and she ge ts her most preferred outcome. 7.1.7 Supergames Some situations, like the pricing game or th e apartment-cleaning game, are played over and over. Such situations are best modeled as a supergame.91 A supergame is a game played over and over again without end, wh ere the players discount the future. The game played each time is known as a stage game. Generally supergames are played in times 1, 2, 3, … Cooperation may be possible in supergames if the future is important enough. Consider the pricing game introduced above. Table 7-27: Price Cutting, Revisited Firm 2 High Low High (15,15) (0,25) Firm 1 Low (25,0) (5,5) The dominant strategy equilibrium to this ga me is (Low, Low). It is clearly a subgame perfect equilibrium for the players to just pl ay (Low, Low) over and over again, because if that is what Firm 1 thinks Firm 2 is doing, Firm 1 does best by pricing Low, and vice versa. But that is not the only equilibrium to the supergame. Consider the following strategy, called a grim trigger strategy. Price high, until you see your rival price low. After your rival has pric ed low, price low forever. This is called a trigger strategy because an action of the othe r player (pricing low) triggers a change in behavior. It is a grim strategy because it punishes forever. If your rival uses a grim trigger strategy what should you do? Basically, your only choice is when to price low, because once you price low, your rival will price low, and then your best choice is to also price low fr om then on. Thus, your strategy is to price high up until some point t – 1, and then price low from time t on. Your rival will price high through t, and price low from t + 1 on. This gives a payoff to you of 15 from period 1 through t – 1, 25 in period t, and then 5 in period t + 1 on. We can compute the payoff for a discount factor ...) ( 5 25 ) ... 1 ( 152 1 1 2 t t t t tV ) 20 10 ( 1 1 15 ) 5 ) 1 ( 25 15 ( 1 1 15 1 5 25 1 1 15 t t t t t. 91 The supergame was invented by Robert Aumann (1930 – ) in 1959.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-269If -10 + 20 < 0, it pays to price low immediately, at t=0, because it pays to price low and the earlier the higher the present value. If -10 + 20 > 0, it pays to wait forever to price low, that is, t = Thus, in particular, the grim trigger strategy is an optimal strategy for a player when the rival is playing the grim trigger strategy if In other words, cooperation in pricing is a subgame perf ect equilibrium if the future is important enough, that is, the discount factor is high enough. The logic of this example is that the promis e of future cooperation is valuable when the future itself is valuable, and that promise of future cooperation can be used to induce cooperation today. Thus, firm 1 doesn’t want to cut price today, because that would lead firm 2 to cut price for the indefinite future The grim trigger strategy punishes price cutting today with future low profits. Supergames offer more scope for cooperation th an is illustrated in the pricing game. First, more complex behavior is possible. For example, consider the following game: Table 7-28: A Variation of the Price Cutting Game Firm 2 High Low High (10,10) (0,25) Firm 1 Low (25,0) (5,5) Here, again, the unique equilibrium in the st age game is (Low, Low). But the difference between this game and the previous game is that the total profits of firms 1 and 2 are higher in either (High, Low) or (Low, High) than in (High, High). One solution is to alternate between (High, Low) and (Low, High). Such alternation can also be supported as an equilibrium, using the grim trigger strate gy – that is, if a firm does anything other than what is it supposed to in the alternatin g solution, the firms instead play (Low, Low) forever. (Exercise) Consider the game in Table 7-28, and consider a strategy in which firm 1 prices high in odd numbered peri ods, and low in even numbered periods, while 2 prices high in even numbered peri ods, low in odd numbered periods. If either deviate from these strategies, both price low from then on. Let be the discount factor. Show that these firms have a payoff of 21 25 or 21 25 depending on which period it is. Then show that the alternating strategy is sustainable if 1 25 1 5 102 This, in turn, is equivalent to 2 6 7.1.8 The Folk Theorem The folk theorem says that if the value of th e future is high enough, any outcome that is individually rational can be supported as an equilibrium to the supergame. Individual rationality for a player in this context mean s that the outcome offers a present value of profits at least as high as that offered in the worst equilibrium in the stage game from that player’s perspective. Thus, in the pr icing game, the worst equilibrium of the stage

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-270game offered each player 5, so an outcome ca n be supported if it offers each player at least a running average of 5. The simple logic of the folk theorem is this First, any infinite repetition of an equilibrium of the stage game is itself a subgame perfect equilibrium. If everyone expects this repetition of the stage game equilibrium, no one can do better than to play their role in the stage game equilibrium ever y period. Second, any other plan of action can be turned into a subgame perfect equili brium merely by threatening any agent who deviates from that plan with an infinite re petition of the worst stage game equilibrium from that agent’s perspective. That threat is credible because the repetition of the stage game equilibrium is itself a subgame perfect eq uilibrium. Given such a grim trigger type threat, no one wants to deviate from the intended plan. The folk theorem is a powerful result, and shows that there are equilibria to supergames that achieve very good outcomes. The kinds of coordination failures we saw in the battle of the sexes, and the failure to cooperate in the prisoner’s dilemma, need not arise, and cooperative solutions are possible if the future is sufficiently valuable. However, it is worth noting some assumptions that have b een made in our descriptions of these games, assumptions that matter and are unlikely to be true in practice. First, the players know their own pa yoffs. Second, they know their rival’s payoffs. They possess a complete description of the av ailable strategies and can calculate the consequences of these strategies, not just fo r themselves, but for their rivals. Third, each player maximizes his or her expected payo ff, and they know that their rivals do the same, and they know that their rivals know that everyone maximizes, and so on. The economic language for this is the structure of the game and the player’s preferences are common knowledge. Few real world games will sati sfy these assumptions exactly. Since the success of the grim trigger strategy (and other strategies we haven’t discussed) generally depends on such knowledge, informational considerations may cause cooperation to break down. Finally, the folk theorem shows us that there are lots of equilibria to supergames, and provides no gu idance on which one will be played. These assumptions can be relaxed, although they ma y lead to wars on the equilibrium path “by accident,” and a need to recover from such wars, so that the gr im trigger strategy becomes sub-optimal. 7.2 Cournot Oligopoly The Cournot92 oligopoly model is the most popular model of imperfect competition. It is a model in which the number of firms matters and represents one way of thinking about what happens when the world is neithe r perfectly competitive, nor a monopoly. In the Cournot model, there are n firms, who choose quantities. We denote a typical firm as firm i and number the firms from i = 1 to i = n. Firm i chooses a quantity qi 0 to sell and this quantity costs ci(qi). The sum of the quantities produced is denoted by Q. The price that emerges from the competition among the firms is p(Q) and this is the same price for each firm. It is probably best to think of the quantity as really 92 Augustus Cournot, 1801-1877.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-271representing a capacity, and competition in prices by the firms determining a market price given the market capacity. The profit that a firm i obtains is ) ( ) (i i i iq c q Q p 7.2.1 Equilibrium Each firm chooses qi to maximize profit. The first order conditions93 give: ) ( ) ( ) ( 0i i i i iq c q Q p Q p q This equation holds with equality provided qi > 0. A simple thing that can be done with the first order conditions is to rewrite them to obtain the average value of the price-cost margin: ) ( ) ( ) ( ) ( ) ( ) ( ) ( i i i i is Q q Q p Q p Q Q p q Q p Q p q c Q p Here Q q si i is firm i’s market share. Multiplying this equation by the market share and summing over all firms i = 1, … n yields HHI s s Q p q c Q pn i i n i i i i1 2 11 ) ( ) ( ) ( where n i is HHI1 2 is the Hirschman-Herfindahl Index.94 The HHI has the property that if the firms are identical, so that si = 1/n for all i, then the HHI is also 1/n. For this reason, antitrust economists will sometimes use 1/HHI as a proxy for the number of firms, and describe an industry with “2 firms,” meaning an HHI of 0.4.95 We can draw several inferences from these eq uations. First, larger firms, those with larger market shares, have a larger deviatio n from competitive behavior (price equal to marginal cost). Small firms are approxim ately competitive (price nearly equals marginal cost) while large firms reduce output to keep the price higher, and the amount of the reduction, in price/cost terms, is pr oportional to market share. Second, the HHI reflects the deviation from perfect competition on average, that is, it gives the average 93 Bear in mind that Q is the sum of the firms’ quantities, so that when firm i increases its output slightly, Q goes up by the same amount. 94 Named for Albert Hirschman (1918 – 1972), who inve nted it in 1945, and Orri s Herfindahl (1915 – ), who invented it independently in 1950. 95 To make matters more confusing, antitrust economists tend to state the HHI using shares in percent, so that the HHI is on a 0 to 10,000 scale.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-272proportion by which price equal to marginal cost is violated. Third, the equation generalizes the “inverse elasticity result” proved for monopoly, which showed that the price – cost margin was the inverse of the el asticity of demand. The generalization states that the weighted average of the price – cost margins is the HHI over the elasticity of demand. Since the price – cost margin reflects the de viation from competition, the HHI provides a measure of how large a deviation from competi tion is present in an industry. A large HHI means the industry “looks like monopoly .” In contrast, a small HHI looks like perfect competition, holding const ant the elasticity of demand. The case of a symmetric (identical cost function s) industry is especially enlightening. In this case, the equation for the first order condition can be restated as n Q c n Q Q p Q p) ( ) ( 0 or n Q c n n Q p1 ) (. Thus, in the symmetric model, competition le ads to pricing as if demand were more elastic, and indeed is a substitute for elasticity as a determinant of price. 7.2.2 Industry Performance How does the Cournot industry perform? Let us return to the more general model, that doesn’t require identical cost functions. We already have one answer to this question: the average price – cost margin is the HHI divi ded by the elasticity of demand. Thus, if we have an estimate of the demand elasti city, we know how much the price deviates from the perfect competition benchmark. The general Cournot industry actually has two sources of inefficiency. First, price is above marginal cost, so there is the dead we ight loss associated with unexploited gains from trade. Second, there is the inefficiency associated with different marginal costs. This is inefficient because a re-arrangemen t of production, keeping total output the same, from the firm with high marginal cost to the firm with low marginal cost, would reduce the cost of production. That is, not only is too little output produced, but what output is produced is inefficiently produced, unless the firms are identical. To assess the productive inefficiency, we let 1c be the lowest marginal cost. The average deviation from the lowest marginal cost, then, is n i i i n i i i n i i ic p s c p c p c p s c c s1 1 1 1 1 1) ( )) ( ( ) (

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-273 HHI p c p s p c p p c p s p c pn i i n i i i 1 1 2 1 1 1) ( Thus, while a large HHI means a large deviation from price equal to marginal cost and hence a large level of monopoly power (holdi ng constant the elasticity of demand), a large HHI also tends to indicate greater pr oductive efficiency, that is, less output produced by high cost producers. Intuitively, a monopoly produces efficiently, even if it has a greater reduction in total outp ut than other industry structures. There are a number of caveats worth mentioning in the assessment of industry performance. First, the analysis has held co nstant the elasticity of demand, which could easily fail to be correct in an application. Second, fixed costs have not been considered. An industry with large economie s of scale, relative to dema nd, must have very few firms to perform efficiently and small numbers sh ould not necessarily indicate the market performs poorly even if price – cost margin s are high. Third, it could be that entry determines the number of firms, and that the firms have no long-run market power, just short-run market power. Thus, entry and fixed costs could lead the firms to have approximately zero profits, in spite of price above marginal cost. (Exercise) Suppose the inverse demand curve is p(Q) = 1 – Q, and that there are n Cournot firms, each with constant marginal cost c, selling in the market. (i) Show that the Cournot equilibrium quantity and price are 1 ) 1 ( n c n Q and 1 1 ) ( n nc Q p. (ii) Show each firm’s gross profits are 21 1 n c. Continuing with (Exercise), suppose there is a fixed cost F to become a firm. The number of firms n should be such that firms are able to cover their fixed costs, but add one more and they can’t. This gives us a condition determining the number of firms n: 2 22 1 1 1 n c F n c. Thus, each firm’s net profits are 2 2 2 2 2 2) 2 ( ) 1 ( ) 1 )( 3 2 ( 2 1 1 1 1 1 n n c n n c n c F n c. Note that the monopoly profits m are (1-c)2. Thus, with free entry, net profits are less than mn n n 2 2) 2 ( ) 1 ( 4 ) 3 2 ( and industry net profits are less than mn n n n 2 2) 2 ( ) 1 ( 4 ) 3 2 ( Table 7-29 shows the performance of the con stant cost, linear demand Cournot industry, when fixed costs are taken into accoun t, and when they aren’t. With two firms, gross industry profits are 8/9ths of the mono poly profits, not substantially different from monopoly. But when fixed costs sufficien t to insure that only two firms enter are

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-274considered, the industry profits are at most 39% of the monopoly profits. This number – 39% -is large because fixed costs could be “r elatively” low, so that the third firm is just deterred from entering. That still leaves the two firms with significant profits, even though the third firm can’t prof itably enter. As the number of firms rises, gross industry profits fall slowly toward zero. The net industry profits, on the other hand, fall dramatically rapidly to zero. With ten firms, the gross profits are still about a third of the monopoly level, but the net profits are only at most 5% of the monopoly level. Table 7-29: Industry Profits as a Fraction of Monopoly Profits Number of Firms Gross Industry Profits (%) Net Industry Profits (%) 2 88.9 39.0 3 75.0 27.0 4 64.0 19.6 5 55.6 14.7 10 33.1 5.3 15 23.4 2.7 20 18.1 1.6 The Cournot model gives a useful model of imperfect competition, a model that readily permits assessing the deviation from perfec t competition. The Cournot model embodies two kinds of inefficiency: the exercise of mo nopoly power, and technical inefficiency in production. In settings involving entry and fi xed costs, care must be taken in applying the Cournot model. 7.3 Search and Price Dispersion Decades ago, economists used to make a bi g deal about the “Law of One Price,” which states that identical goods sell at the same pr ice. The argument in favor of the law of one price is theoretical. Well-informed consumers will buy identical goods from the lowest price seller. Consequently, the only seller to make any sales is the low-price seller. This kind of consumer behavior forc es all sellers to sell at the same price. There are few markets where the law of one price is actually observed to hold. Organized exchanges, like stock, bond and commodity markets, will satisfy the law of one price. In addition, gas stations across the street from each other will often offer identical prices, but often is not always. Many economists considered that the internet would force prices of standardized goods – DVD players, digital cameras, MP3 players – to a uniform, and uniformly low, price. However, this has not occurred. Moreover, it probably can’t occur, in the sense that pure price competition would put the firms out of business, and hence can’t represent equilibrium behavior. There are many markets where prices appear unpredictable to consumers. The price of airline tickets is notorious for unpredictability. The price of milk, soft drinks, paper

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-275towels and canned tuna varies 50% or more depending on whether the store has an advertised sale of the item or not. Pric es of goods sold on the internet varies substantially from day to day.96 Such variation is known as price dispersion by economists. It is different from price discri mination, in that price dispersion entails a given store quoting the same price to all cust omers; the variation is across stores, while price discrimination is across customers. Why are prices so unpredictable? 7.3.1 Simplest Theory To understand price dispersion, we divide consumers into two types: shoppers and loyal customers. Loyal customers won’t pay more than a price pm for the good, but they only consult a particular store; if that store has the good for less than the price pm, the loyal customer buys, and otherwise not. In contr ast, the shoppers buy only from the store offering the lowest price; shoppers are inform ed about the prices offered by all stores. We let the proportion of shoppers be s. The loyal customers are allocated to the other stores equally, so that, if there are n stores, each store gets a fraction (1 – s)/n of the customers. Let the marginal cost of the good be c, and assume c < pm. Both kinds of customers buy only one unit. For the purposes of this anal ysis, we will assume that pric es can be chosen from the continuum. This makes the analysis more st raightforward, but there is an alternate version of the analysis (not developed here) that makes the more reasonable assumption of prices that are an integer number of pennies. First note that there is no pure strategy eq uilibrium. To see this, consider the lowest price p charged by any firm. If that price is c, the firm makes no money, so would do better by raising its price to pm and selling only to the loyal customers. Thus, the lowest price p exceeds c. If there is a tie at p, it pays to break the tie by charging a billionth of a cent less than p, and thereby capturing all the shoppers rather than sharing them with the other firm charging p. So there can’t be a tie. But no tie at p means the next lowest firm is charging something strictly greater than p, which means the lowest price firm can increa se price somewhat and not suffer any loss of sales. This contradicts profit maximization for that firm. The con clusion is that firms must randomize and no pure strategy equilibrium exists. But how do they randomize? We are going to l ook for a distribution of prices. Each firm will choose a price from the continuous distribution F, where F(x) is the probability the firm charges a price less than x. What must F look like? We use the logic of mixed strategies: the firm must get the same prof its for all prices that might actually be charged under the mixed strategy, for otherwis e it would not be willing to randomize. A firm that charges price p pm always sells to its captive cust omers. In addition, it sells to the shoppers if the other firms have hi gher prices, which occurs with probability 1)) ( 1 (np F. Thus, the firm’s profits are 96 It is often very challenging to assess internet prices because of variation in shipping charges.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-276 1)) ( 1 ( 1 ) ( ) (np F s n s c p p. On each sale, the firm earns p – c. The firm always sells to its loyal customers, and in addition captures the shoppers if the other firms price higher. Since no firm will exceed pm, the profits must be the same as the level arising from charging pm, and this gives n s c p p F s n s c p pm n 1 ) ( )) ( 1 ( 1 ) ( ) (1. This equation is readily solved for F: ) ( ) 1 )( ( 1 ) (1 1 n mn c p s s p p p F The lower bound of prices arises at the point L where F(L)=0, or 1 1 ) (s n s n s c p c Lm These two equations provide a continuous dist ribution of prices charged by each firm which is an equilibrium to the pricing game. That is, each firm randomizes over the interval [L, pm], according to the continuous distribution F. Any price in the interval [L,pm] produces the same profits for each firm, so the firms are willing to randomize over this interval. The loyal customers get a price chosen randomly from F, so we immediately see that the shoppers make life better for the loyal cu stomers, pushing average price down. (An increase in s directly increases F, which means prices fall – recall that F gives the probability that prices are below a given level, so an increase in F is an increase in the probability of low prices.) Similarly loyal customers make life worse for shoppers, increasing prices on average to shoppers. The distribution of prices facing shoppers is actually the distribution of the minimum price. Since all firms charge a price exceeding p with probability (1 – F(p))n, at least one charges a price less than p with probability 1 – (1 – F(p))n, and this is the distribution of prices facing shoppers. That is, the distribution of prices charged to shoppers is 1) ( ) 1 )( ( 1 )) ( 1 ( 1 n n m nn c p s s p p p F.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-277 7.3.2 Industry Performance How does a price dispersed industry perform? First, average industry profits are ). 1 )( ( ) (s c p p nm An interesting aspect of this equation is that it doesn’t depend on the number of firms, only on the number of loyal customers. E ssentially, the industry profits are the same that it would earn as if the shoppers paid marginal cost and the loyal customers paid the monopoly price, although that isn’t what happe ns in the industry, except in the limit as the number of firms goes to infinity. Note th at this formula for industry profits does not work for a monopoly. In order to capture monopoly, one must set s=0, because shoppers have no alternative under monopoly. Figure 7-4: Expected Prices in Search Equilibrium As the number of firms gets large, the pric e charged by any one firm converges to the monopoly price pm. However, the lowest price offered by any firm actually converges to c, marginal cost. Thus, in the limit as the number of firms get large, shoppers obtain price equal to marginal cost and loyal firms pay the monopoly price. The average price charged to shoppers and no n-shoppers is a complicated object, so we consider the case where there are n firms, s = pm =1 and c = 0. Then the expected prices for shoppers and loyal customers are given in Figure 7-4, letting the number of firms vary. Thus, with many firms, most of the gains created by the shoppers flow to shoppers. In contrast, with few firms, a si gnificant fraction of the gains created by shoppers goes instead to the loyal customers. 20 40 60 80 100 n 0.2 0.4 0.6 0.8 1 Price Expected price for loyal customers Ex p ected p rice for sho pp ers

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-278Similarly, we can examine the average prices for loyal customers and shoppers when the proportion of shoppers varies. Increasing th e proportion of shoppers has two effects. First, it makes low prices more attractive, thus encouraging price competition, because capturing the shoppers is more valuable. Second, it lowers industry profits, because the set of loyal customers is reduced. Figure 7-5 plots the average price for loyal customers and shoppers, as the proportion of shoppers ra nges from zero to one, when there are five firms, pm = 1 and c = 0. Figure 7-5: Expected Prices ( s= Proportion of Shoppers) People who are price-sensitive and shop around convey a positive externality on other buyers by encouraging price competition. Si milarly, people who are less price sensitive and don’t shop around convey a negative exte rnality on the other buyers. In markets with dispersed information about the best pr ices, where some buyers are informed and some are not, randomized prices are a natu ral outcome. That is, randomization of prices, and the failure of the law of one pric e, is just a reflection of the different willingness or ability to search on the part of consumers. This difference in the willingness to search could arise simply because search is itself costly. That is, the shoppers could be determ ined by their choice to shop, in such a way that the cost of shopping just balances the expected gains from searching. The proportion of shoppers may adjust endogenously to insure that the gains from searching exactly equal the costs of searching. In this way, a cost of shopping is translated into a randomized price equilibrium in which th ere is a benefit from shopping and all consumers get the same total cost of purchase on average. 0.2 0.4 0.6 0.8 1 s 0.2 0.4 0.6 0.8 1 Price Expected price for loyal customers Expected price for shoppers

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-2797.4 Hotelling Model Breakfast cereals range from indigestible, unprocessed whole grains to cereals that are almost entirely sugar with only the odd mole cule or two of grain. Such cereals are hardly good substitutes for each other. Ye t similar cereals are viewed by consumers as good substitutes, and the standard model of this kind of situation is the Hotelling model.97 Hotelling was the first to use a line se gment to represent both the product that is sold and the preferences of the consume rs who are buying th e products. In the Hotelling model, there is a line, and prefer ences of each consumer is represented by a point on this line. The same line is used to represent products. For example, movie customers are differentiated by age, and we can represent moviegoers by their ages. Movies, too, are designed to be enjoyed by particular ages. Thus a “pre-teen” movie is unlikely to appeal very much to a six year old or to a nineteen year old, while a Disney movie appeals to a six year old, but less to a fifteen year old. That is, movies have a target age, and customers have ages, an d these are graphed on the same line. Figure 7-6: Hotelling Model for Breakfast Cereals Breakfast cereal is a classic application of the Hotelling line, and this application is illustrated in Figure 7-6. Breakfast cereals are primarily distinguished by their sugar content, which ranges on the Hotelling line fr om low on the left to high on the right. Similarly, the preferences of consumers also fall on the same line. Each consumer has a “most desired point,” and prefers cereals closer to that point than those at more distant points. 7.4.1 Types of Differentiation There are two main types of differentiation each of which can be modeled using the Hotelling line. These types are quality and va riety. Quality refers to a situation where consumers agree what product is better; the disagreement among consumers concerns whether higher quality is worth the cost. In automobiles, faster acceleration, better braking, higher gas mileage, more cargo ca pacity, more legroom, and greater durability are all good things. In computers, faster pr ocessing, brighter screens, higher resolution screens, lower heat, greater durability, more megabytes of RAM and more gigabytes of hard drive space are all good things. In con trast, varieties are the elements about which there is not widespread agreement. Colors and shapes are usually varietal rather than quality differentiators. Some people like almond colored appliances, others choose white, with blue a distant third. Food flavors are varieties, and while the quality of ingredients is a quality differentiator, the type of food is usually a varietal differentiator. Differences in music would primarily be varietal. Quality is often called vertical differentiation, while variety is horizontal differentiation. 97 Hotelling Theory is named for Harold Hotelling, 1895-1973. Sugar Content High Fiber Adult CerealsKid Cereals

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-2807.4.2 The Standard Model The standard Hotelling model fits two ice cr eam vendors on a beach. The vendors sell the identical product, and moreover they can choose to locate wherever they wish. For the time being, suppose the price they charge for ice cream is fixed at $1. Potential customers are also spread randomly along the beach. We let the beach span an interval from 0 to 1. People desiring ice cream will walk to the closest vendor, since the price is the sa me. Thus, if one vendor locates at x and the other at y, and x < y, those located between 0 and (x + y) go to the left vendor, while the rest go to the right vendor. This is illustrated in Figure 7-7. Figure 7-7: Sharing the Hotelling Market Note that the vendor at x sells more by moving toward y, and vice versa. Such logic forces profit maximizing vendors to both loca te in the middle! The one on the left sells to everyone left of while the one on the right sells to the rest. Neither can capture more of the market, so equilibrium locations have been found. (To complete the description of an equilibrium, we need to let the two “share” a point and still have one on the right side, one on the left side of that point.) This solution is commonly used as an expl anation of why U.S. political parties often seem very similar to each other – they have met in the middle in the process of chasing the most voters. Political parties can’t directly buy votes, so the “price” is fixed; the only thing parties can do is locate their platform close to voters’ preferred platform, on a scale of “left” to “right.” But the same lo gic that a party can grab the middle, without losing the ends, by moving closer to the other party will tend to force the parties to share the same “middle of the road” platform. (Exercise) Suppose there are four ice cream vendors on the beach, and customers are distributed uniformly. Show that it is a Nash equilibrium for two to locate at and two at The model with constant prices is unrealistic for the study of the behavior of firms. Moreover, the two-firm model on the beac h is complicated to solve and has the undesirable property that it matters signific antly whether the number of firms is odd or even. As a result, we will consider a Hotelling model on a circle, and let the firms choose their prices. 7.4.3 The Circle Model In this model, there are n firms evenly spaced around th e circle whose circumference is one. Thus, the distance between any firm and each of its closest neighbors is 1/n. 0 x ( x + y )y1 Buy at x Buy at y

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-281Consumers care about two things: how distant the firm they buy from is, and how much they pay for the good, and they mini mize the sum of the price paid and t times the distance between the consumer’s location (a lso on the circle) and the firm. Each consumer’s preference is uniformly distributed around the circle. The locations of firms are illustrated in Figure 7-8. Figure 7-8: A Segment of the Circle Model We conjecture a Nash equilibrium in which all firms charge the price p. To identify p, we look for what p must be to make any one firm choose to charge p, given that the others all charge p. So suppose the firm in the middle of Figure 7-8 charges an alternate price r, but every other firm charges p. A consumer who is x units away from the firm pays the price r + tx from buying at the firm, or p + t(1/n – x) from buying from the rival. The consumer is just indifferent between the nearby firms if these are equal, that is, r + tx* = p + t(1/n – x*) where x* is the location of the consumer who is indifferent. t r p n t r n t p x2 2 1 2 Thus, consumers who are closer than x* to the firm charging r buy from that firm, and consumers who are further away than x* buy from the alternative firm. Demand for the firm charging r is twice x* (because the firm sells to both sides), so profits are price minus marginal cost times two x*, that is, t r p n c r x c r1 ) ( 2 ) ( The first order condition98 for profit maximization is 1 1 ) ( 0t c r t r p n t r p n c r r 98 Since profit is quadratic in r, we will find a global maximum. Firm length 1/n FirmFirm length 1/n

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-282We could solve the first order condition for r. But remember that the concern is when is p a Nash equilibrium price? The price p is an equilibrium price if the firm wants to choose r = p. Thus, we can conclude that p is a Nash equilibrium price when .n t c p This value of p insures that a firm facing rivals who charge p also chooses to charge p. Thus, in the Hotelling model, price exceeds marginal cost by an amount equal to the value of the average distance between the firms, since the average distance is 1/n and the value to a consumer of traveling that distance is t. The profit level of each firm is 2n t, so industry profits are n t. How many firms will enter the mark et? Suppose the fixed cost is F. We are going to take a slightly unusual approach and assume that the number of firms can adjust in a continuous fashion, in which case the number of firms is determined by the zero profit condition 2n t F or F t n What is the socially efficient number of firm s? The socially efficient number of firms minimizes the total costs, which are the sum of the transportation costs and the fixed costs. With n firms, the average distance a consumer travels is n n n xdx n dx x nn n n4 1 2 1 2 | |2 2 1 0 2 1 2 1 Thus, the socially efficient number of firms minimizes the transport costs plus the entry costs nF n t 4 This occurs at F t n 2 1 The socially efficient number of firms is half the level that enter with free entry! Too many firms enter in the Hotelling circle model. This extra entry arises because efficient entry is determined by the cost of entry and the average distance of consumers, while prices are determined by the marginal di stance of consumers, or the distance of the marginal consumer. That is, competing fi rms’ prices are determined by the most distant customer, and that leads to prices that are too high relative to the efficient level; free entry then drives net profits to zero only by excess entry. The Hotelling model is sometimes used to just ify an assertion that firms will advertise too much, or engage in too much R&D, as a means of differentiating themselves and creating profits.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-2837.5 Agency Theory An agent is a person who works for, or on behalf of, another. Thus, an employee is an agent of a company. But agency extends be yond employee relationships. Independent contractors are also agents. Advertising fi rms, lawyers and accountants are agents of their clients. The CEO of a company is an agent of the board of directors of the company. A grocery store is an agent of the ma nufacturer of corn chips sold in the store. Thus, the agency relationship extends be yond the employee into many different economic relationships. The entity – person or corporation – on whose behalf an agent works is called a principal Agency theory is the study of incentives pr ovided to agents. Incentives are an issue because agents need not have the same intere sts and goals as the principal. Employees spend billions of hours every year browsing the web, emailing friends, and playing computer games while they are supposedly working. Attorneys hired to defend a corporation in a lawsuit have an incentive no t to settle, to keep the billing flowing. (Such behavior would violate the attorneys’ ethics requirements.) Automobile repair shops have been known to use cheap or used replacement parts and bill for new, high quality parts. These are all examples of a co nflict in the incentives of the agent and the goals of the principal. Agency theory focuses on the cost of prov iding incentives. When you rent a car, an agency relationship is created. Even though a car rental co mpany is called an agency, it is most useful to look at the renter as the agen t, because it is the renter’s behavior that is an issue. The company would like the agent to treat the car as if it were their own car. The renter, in contrast, knows it isn’t th eir own car, and often drives accordingly. "[T]here's a lot of debate on this su bject---about what kind of car handles best. Some say a front-engined car; some say a rear-engined car. I say a rented car. Nothing handles better than a rented car. You can go faster, turn corners sharper, and put the transmission into reverse while going forward at a higher rate of speed in a rented car than in any other kind." 99 How can the car rental company insure that you don’t put their car into reverse while going forward at a high rate of speed? Th ey could monitor your behavior, perhaps by putting a company representative in the car wi th you. That would be a very expensive and unpleasant solution to the problem of incentives. Instead, the company uses outcomes – if damage is done, the driver has to pay for it. That is also an imperfect solution, because some drivers who abuse the cars get off scot-free and others who don’t abuse the car still have cars that break do wn, and are then mired in paperwork while they try to prove their good behavior. That is, a rule that penalizes drivers based on outcomes imposes risk on the drivers. Modern technology is improving monitoring with GPS tracking. 99 P. J. O'Rourke, Republican Party Re ptile, Atlantic Monthly Press, 1987.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-2847.5.1 Simple Model To model the cost of provision of incentives, we consider an agent like a door-to-door encyclopedia salesperson. The agent will vi sit houses, and sell encyclopedias to some proportion of the households; the more work the agent does, the more sales that are made. We let x represent the average dollar value of sales for a given level of effort; x is a choice the agent makes. However, x will come with risk to the agent, which we model using the variance 2. The firm will pay the agent a share s of the money generated by the sales. In addition, the firm will pay the agent a salary y which is fixed independently of sales. This scheme – a combination of salary and commission – covers many different situations. Real estate agents receive a mix of salary and commission. Authors receive an advance and a royalty, which works like a salary and commission. The monetary compensation of the agent is sx + y In addition, the agent has a cost of effort, which we take to be a x 22. Here a represents the ability of the agent: more able agents, who have a higher value of a have a lower cost of effort. Finally, there is a cost of risk. The actual risk imposed on the agent is proportional to the degree they share in the proceeds; if s is small, the agent faces almost no monetary risk, while if s is high, most of the risk is imposed on the agent. We use the “linear cost of risk” model developed earlier, to impose a cost of risk which is s 2. Here, 2 is the variance of the monetary risk, defines the agent’s attitude or cost of risk, and s is the share of the risk imposed on the agent. This results in a payoff to the agent of 2 22 s a x y sx u. The first two terms, sx + y are the payments made to the agent. The next term is the cost of generating that level of x The final term is the cost of risk imposed on the agent by the contract. The agency game works as follows. First, th e principal offers a contract, which involves a commission s and a salary y The agent can either accept or reject the contract and accepts if he obtains at least u0 units of utility, the value of his next best offer. Then the agent decides how much effort to expend, that is, the agent chooses x As with all subgame perfect equilibria, we work backwards to first figure out what x an agent would choose. Because our assumptions make u quadratic in x this is a straightforward exercise, and we conclude x=sa. This can be embedded into u and we obtain the agent’s optimized utility, u *, is 2 2 2 2 2 2 ) ( s a s y s a sa y a s u.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-285 Incentive Compensation: A Percentage of What? Most companies compensate their sales force based on the revenue generated. However, maximizing revenue need not be the same thing as maximizing profit, which is generally the goal of the company. In what follows, Steve Bisset discusses the difference. “Many years ago I was CEO of a company called Megatest, founded by Howard Marshall and myself in 1975. Around 1987 we were selling test systems in the $1M+ price range. Every Monday morning we would have a meeting with sales management and product marketing, mediated by myself. Individual salesmen came in to make their cases for how they just had to have a huge discount, else they would lose a particular sale. The meeting was part circus, with great performances, and part dogfight. “I could visualize the sales guys spending their time in the car or the shower using their substantial creative powers to dream up good justifications for their next plea for a huge discount. They knew that if we were willing to bleed enough we could usually win the sale. I wanted to solve both the resultant profitability problem and the problem of the unpleasant meeting. “Commissions were traditionally a percentage of bookings (net of discount), with part held back until cash was received. The percentage increased after a salesman met his individual quota for the quarter (the performances at quota-setting time to sandbag the quota were even more impressive). The fact that a discount reduced commission did not affect a salesman's behavior, because the difference in commission was small. Better to buy the order by giving a big discount and then move on quickly to the next sale. “Salesmen are "coin operated", and will figure out how to maximize their total commission. Most salesmen I have met are quite good at math. Further, they have learned to “watch the hips, not the lips” – in other words, they judge management intentions by actions rather than words. They reason – and I agree with them – that if management really wanted them to maximize margins rather than revenues, then they would pay them more if they maximize margins. “We decided to try a new scheme, against some howling from the sales department. We set a base "cost" for each product, approximately representing the incremental cost to manufacture and support the product. Then we offered commission on the amount that the net sales price exceeded this base cost. The base cost may have been as much as 80% of the list price (it was a very competitive market at the time). Then we increased the commission rate by a factor of about six, so that if the salesman brought in an order at a price near list, then his commission was somewhat higher than before. If he started discounting, his commission dropped sharply. We gave broad discretion to sales management to approve discounts. “We still had sales guys claiming that a sale was massively strategic and had to be sold at or below cost, and that they needed their commission anyway to justify the effort. In some cases we approved this, but mostly we said that if it's strategic then you'll get your commission on the follow-on sales. While salesmen have a strong preference for immediate cash, they will act so as to maximize income over time, and will think and act strategically if financially rewarded for such. “The results were that our margins increased significantly. Revenues were increasing too. It's hard to attribute the revenue gain to the new commission plan, given the number of other variables, but I like to think that it was a factor. Now our salesmen spent more of their creative energies devising ways to sell our customers on the value of our products and company, instead of conspiring with sales management as to the best tale of woe to present to marketing at the Monday pricing meeting. “The Monday meetings became shorter and more pleasant, focused on truly creative ways to make each sale. There certainly were steep discounts given in some cases, but they reflected the competitive situation and future sales potential at each account much more accurately.” (Source: private correspondence, quotation permission received)

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-286The agent won’t accept employment unless u u0, the reservation utility. The principal can minimize the cost of employing the agent by setting the salary such that u *= u0, which results in 2 2 0 s a s u y. Observe that the salary has to be higher, the greater is the risk 2. That is, the principal has to cover the cost of risk in the salary term. 7.5.2 Cost of Providing Incentives The principal obtains profits which are the remainder of the value after paying the agent, and minus the salary: 2 2 0 ) ( ) 1 ( ) 1 (2 2 0 s a s u sa s a s u sa s y x s Note that the principal gets the entire output x = sa minus all the costs – the reservation utility of the agent u0, the cost of providing effort, and th e risk cost on the agent. That is, the principal obtains the full gains from trade – the value of production minus the total cost of production. However, the fact that the principal obtains the full gains from trade doesn’t mean the principal induces the agent to work extremely hard, because there is no mechanism for the principal to induce the agent to work hard without imposing more risk on the agent, and this ri sk is costly to the principal. Agents are induced to work hard by tying their pay to their performance, and such a link necessarily imposes risk on the agent, and risk is costly.100 We take the principal to be risk neutral. This is reasonable when the principal is “economically large” relative to the agent, so that the risks faced by the agent are small to the principal. For example, the risks a ssociated with any one car are small to a car rental company. The principal who maximizes expected profits chooses s to maximize which yields 21 a s. This formula is interesting for several reasons. First, if the agent is neutral to risk, which means =0, then s is 1. That is, the agent gets 100% of the marginal return to effort, and the principal just collects a lump sum. This is remini scent of some tenancy contracts used by landlords and peasants; th e peasant paid a lump sum for the right to farm the land and then kept all of the crop s grown. Since these peasants were unlikely to be risk neutral, while the landlord was re latively neutral to risk, such a contract was 100 There is a technical requirement that the principal’s return must be positive, for otherwise the principal would rather not contract at all. This amounts to an assumption that u0 is not too large. Moreover, if s comes out less than zero, the model falls apart, and in this case, the actual solution is s=0.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-287unlikely to be optimal. The contract with s =1 is known as “selling the agency” since the principal sells the agency to the agen t for a lump sum payment. (Here, y will generally be negative – the principal gets a payment rather than paying a salary.) The more common contract, however, had the landowner and the tenant farmer share the proceeds of farming, which gives rise to the name sharecropper Second, more risk or more risk aversion on the part of the agent decreases the share of the proceeds accruing to the agent. Thus, when the cost of risk or the amount of risk is high, the best contract imposes less risk on the agent. Total output sa falls as the risk costs rise. Third, more able agents (higher a ) get higher commissions. That is, the principal imposes more risk on the more able agent becau se the returns to imposition of risk – in the form of higher output – are greater, and thus worth the cost in terms of added risk. Most real estate agencies operate on a mi x of salary and commi ssion, with commissions paid to agents averaging about 50%. Th e agency RE/MAX, however, pays commissions close to 100%, collecting a fixed monthly f ee that covers agency expenses from the agents. RE/MAX claims that their formula is appropriate fo r better agents. The theory developed suggests that more able agents sh ould obtain higher commissions. But in addition, RE/MAX’s formula also tends to a ttract more able agents, since able agents earn a higher wage under the high commission formula. (There is a potential downside to the RE/MAX formula, that it di scourages agency-wide cooperation.) 7.5.3 Selection of Agent Consider what contracts attract what kinds of agents. For a fixed salary y and commission s the agent’s utility, optimizing over x is 2 2 s a s y u. The agent’s utility is increasing in a and decreasing in Thus, more able agents get higher utility, and less risk averse agents get higher utility. How do the terms of the contract affect th e pool of applicants? Let us suppose two contracts are offered, one with a salary y1 and commission s1, the other with salary y2 and commission s2. We suppose y2 < y1 and s2 > s1. What kind of agent prefers contract 2, the high commission, low salary contract, over contract 1? 2 1 2 1 1 2 2 2 2 2 s a s y s a s y, or, equivalently, 2 1 2 1 2 2 1 2 2) ( y y s s s s a Thus, agents with high ability a or low level of risk aversion prefers the high commission, low salary contract. A company th at puts more of the compensation in the form of commission tends to attract more able agents, and agents less averse to risk.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-288The former is a desirable feature of the incentive scheme, since more able agents produce more. The latter, the attraction of less risk averse agents, may or may not be desirable but is probably neutral overall. One important consideration is that agents wh o overestimate their ability will react the same as people who have higher ability. Thus, the contract equally attracts those with high ability and those who ov erestimate their ability. Agency theory provides a characterization of the cost of providing incentives. The source of the cost is the link between ince ntives and risk. Incentives link pay and performance; when performance is subject to random fluctuations, linking pay and performance also links pay and the random fluctuations. Thus the provision of incentives necessarily imposes risk on the agen t, and if the agent is risk averse, this is costly. In addition, the extent to which pay is linked to performance will tend to affect the type of agent who is willing to work for the pr incipal. Thus, a principal must not only consider the incentive to work hard created by the commission and salary structure, but also the type of agent who would choose to accept such a contract. 7.5.4 Multi-tasking Multi-tasking refers to performing several activities simultaneously. All of us multitask. We study while drinking a caffeinated beverage; we think about things in the shower; we talk all too much on cell phones and eat French fries while driving. In the context of employees, an indi vidual employee will be assigned a variety of tasks and responsibilities, and the employee must divide their time and efforts among the tasks. Incentives provided to the employee must direct not only the total efforts of the employee, but also the allocation of time and effort across activities. An important aspect of multi-tasking is the interaction of incentives provided to employees, and the effects of changes in one incentive on the be havior of the employee over many different dimensions. In this section, we will establi sh conditions under which the problem of an employer disaggregates, that is to say, the in centives on each individual task can be set independently of the incentives applied to the others. This section is relatively challenging and invo lves a number of pieces. To simplify the presentation, some of the piec es are set aside as claims. To begin the analysis, we consider a person who has n tasks or jobs. For convenience we will index these activities with the natural numbers 1, 2, … n The level of activity, which may also be thought of as an action, in task i will be denoted by xi. It will prove convenient to denote the vector of actions by ) ( 1nx x x. We suppose the agent bears a cost c (x) of undertaking the vector of actions x. We make four assumptions on c : c is increasing in each xi, c has a continuous second derivative c is strictly convex, and

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-289 c is homogeneous101 of degree r For example, if there are two tasks ( n =2), then all four of these assumptions are met by the cost function 2 1 2 2 2 1 2 1x ) ( x x x x x c This function is increasing in x1 and x2, has continuous derivatives, is strictly convex (more about this below) and is homogeneous of degree 2. It is assumed that c is increasing to identify the act ivities as costly. Continuity of derivatives is used for convenience. Convexity of c will insure that a solution to the first order conditions is actually an optimum for the employee. Formally, it means that for any vectors x y and scalar between zero and one (0 1), ) ) 1 ( ( ) ( ) 1 ( ) (y x y x c c c One way of interpreting this requirement is that it is less costly to do the average of two things than the average of the costs of the things. Intuitively, convexity requires that doing a medium thing is less costly than the average of two extremes This is plausible when extremes tend to be very costly. It also means the set of vectors which cost less than a fixed amount, {x | c (x) b }, is a convex set. Thus, if two points cost less than a given budget, the line segment connecting th em does, too. Convexity of the cost function insures that the agent’s optimization problem is concave, and thus that the first order-conditions describe a maximum. When the inequality is strict for satisfying 0 < < 1, we refer to convexity as strict convexity. The assumption of homogeneity dictates th at scale works in a particularly simple manner. Scaling up activities increases costs at a fixed rate r Homogeneity has very strong implications that are probably unre asonable in many settings. Nevertheless, homogeneity leads to an elegant and useful theo ry, as we shall see. Recall the definition of a homogeneous function: c is homogeneous of degree r means that for any > 0, ) ( ) (x xc cr Claim: strict convexity implies that r > 1. Proof of Claim: Fix any x and consider the two points x and x. By convexity, for 0 < <1, ) ( ) 1 ( ) ( ) ( ) r ) 1 ( (x x x c c c ) ( r ) ) 1 ( ( )) ) 1 ( (x x xc c which implies r ) ) 1 ( ( ) r ) 1 ( ( Define a function k which is the left hand side minus the right hand side: 101 Homogeneous functions were defined in (Exercise).

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-290r r) ) 1 ( ( ) 1 ( ) ( k. Note that k (0)= k (1)=0. Moreover, 2 2 r) 1 ( ) ) 1 ( )( 1 ( ) ( r r k. It is readily checked that if a convex function of one variable is twice differe ntiable, then the second derivative is greater than zero. If r 1, 0 ) ( k implying that k is convex, and hence, if 0 < < 1, 0 ) 1 ( ) 0 ( ) 1 ( ) 1 0 ) 1 (( ) ( k k k k Similarly, if r > 1, k is concave and k ( ) > 0. This completes the proof, showing that r 1 is not compatible with the strict convexity of c How should our person behave? Consider li near incentives, which are also known as piece rates. With piece rates, the employee gets a payment pi for each unit of xi produced. The person then chooses x to maximize ) ( ) (1x x p xc c x p un i i i Here is the dot product, which is the su m of the products of the components. The agent chooses x to maximize u resulting in n first order conditions ) ( ) (x xi i i i ic p x c p x u where ci is the partial derivative of c with respect to the ith argument xi. This first order condition can be expressed more compactly as ) ( 0x pc where ) (xc is the vector of partial derivatives of c Convexity of c insures that any solution to this problem is a global utility maximum, since the function u is concave, and strict convexity insures that there is at most one solution to the first order conditions.102 One very useful implication of homogeneity is that incentives scale. Homogeneity has the effect of turning a very complicated op timization problem into a problem that is readily solved, thanks to this very scaling. Claim: If all incentives rise by a scalar factor then x rises by 1 1r. 102 This description is slightly inadequate, because we haven’t considered boundary conditions. Often a requirement like xi0 is also needed. In this case, the first order conditions may not hold with equality for those choices where xi=0 is optimal.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-291Proof of Claim: Note that differentiating ) ( ) (x xc cr with respect to xi yields ) ( ) (x xi r ic c and thus ) ( ) (1x x c cr That is, if c is homogeneous of degree r c is homogeneous of degree r – 1. Consequently, if ) ( 0x pc ) ( 01 1x p rc. Thus, if the incentiv es are scaled up by the efforts rise by the scalar factor 1 1 r. Now consider an employer with an agent engaging in n activities. The employer values the i th activity at vi, and thus wishes to maximize n i i i i n i i i ix c v x p v1 1)) ( ( ) (x. This equation embodies a standard trick in agency theory. Th ink of the principal (employer) not as choosing the incentives p, but instead as choosing the effort levels x, with the incentives as a constraint. That is, the principal can be thought of choosing x and then choosing the p that implements this x. The principal’s expected profit is readily differentiated with respect to each xj, yielding n i i ij j jx c c v1)) ( ) ( 0x x. However, since cj(x) is homogeneous of degree r – 1, ) ( ) 1 ( ) ( ) ( )) (1 1 1 1x x x xj j r j n i i ijc r c d d c d d x c and thus ) ( )) ( ) ( 01x x xj j n i i ij j jrc v x c c v This expression proves the main result of this section. Under the maintained hypotheses (convexity and homogeneity), an employer of a multi-tasking agent uses incentives which are a constant proportion of value, that is, r v pj j where r is the degree of homogeneity of the agent’s costs. Recalling that r >1, the principal uses a sharing rule sharing a fixed proportion of value with the agent.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-292When agents have a homogeneous cost functi on, the principal has a very simple optimal incentive scheme, requiring quite limited knowledge of the agent’s cost function (just the degree of homogeneity). Moreover, the incentive scheme works through a somewhat surprising mechanism. Note that if the value of one activity, say activity 1, rises, p1 rises and all the other payment rates stay constant. The agent responds by increasing x1, but the other activities may rise or fall depending on how complementary they are to activity 1. Overall, the agent’ s substitution across activities given the new incentive level on activity 1 implements the desi red effort levels on other activities. The remarkable implication of homogeneity is that, although the principal desires different effort levels for all activities, only the incentive on activity 1 must change! 7.5.5 Multi-tasking without Homogeneity In the previous section we saw, for example, that if the agent has quadratic costs, the principal pays the agent half the value of each activity. Moreover, the more rapidly costs rise in scale, the lower the payments to the agent. This remarkable theorem has several limitati ons. The requirement of homogeneity is itself an important limitation, although this assumption is reasonable in some settings. More serious is the assumption that all of the incentives are set optimally for the employer. Suppose, instead, that one of the incent ives is set “too high,” at least from the employer’s perspective. Th is might arise if, for example, the agent acquired all the benefits of one of the activities. An increase in the power of one incentive will then tend to “spill over” to the other actions, increasing for complements and decreasing for substitutes. When the efforts are substitutes, an increase in the power of one incentive will cause others to optimally rise, to comp ensate for the reduced supply of efforts of that type.103 We can illustrate the effects of cost functi ons that aren’t homogeneous in a relatively straightforward way. Suppose the cost de pends on the sum of the squared activity levels: ) ( ) (1 2x x x g x g cn i i. This is a situation where vector notation (dot-products) dramatically simplifies the expressions. You may find it useful to work through the notation on a separate sheet, or in the margin, using summation notation to ve rify each step. At the moment, we won’t be concerned with the exact specification of g but instead use the first order conditions to characterize the solution. The agent maximizes ) ( x x x p g u. 103 Multi-tasking, and agency theory more generally, is a rich theory with many implications not discussed here. For a challenging and importan t analysis, see Bengt Holmstrom and Paul Milgrom, "The Firm as an Incentive System,” American Economic Review, Vol. 84, No. 4 (Sep., 1994), pp. 972-991.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-293This gives a first order condition x x x p ) ( 2 0 g It turns out that a sufficient condition for this equation to characterize the agent’s utility maximization is that g is both increasing and convex (increasing second derivative). This is a particularly simple expression, because the vector of efforts, x, points in the same direction as the incentive payments p. The scalar that gives the overall effort levels, however, is not necessarily a cons tant, as occurs with homogeneous cost functions. Indeed, we can readily see that xx is the solution to ) ( )) ( 2 (2x x x x p p g Because xx is a number, it is worth introducing notation for it: S = xx. Then S is the solution to 2)) ( ( 4 S g S p p. The principal or employer chooses p to maximize ) )( ( 2 x x x x x v x p x v g. This gives the first order condition x x x x x x x v 0 ) ( ) ( ) ( 4 g g. Thus, the principal’s choice of p is such that x is proportional to v, with constant of proportionality ) ( ) ( x x x x x x g g. Using the same trick (dotting each side of the first order condition x x x x x x x v ) ( ) ( 4 g g with itself), we obtain: *) ( *) ( 162S S g S S g v v, which gives the level of xx= S induced by the principal. Given S *, p is given by v v x x x p *) ( *) ( 1 1 2 1 *)) ( *) ( ( 4 *) ( 2 ) ( 2 S g S g S S g S S g S g g Note that this expression gives the right an swer when costs are homogeneous. In this case, g ( S ) must be in the form Sr/2, and the formula gives

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-294 r r v v p 1 1 1 2 1 as we already established. The natural assumption to impose on the function g is that S S g S S g2) ( ) ( is an increasing function of S This assumption implies that as the value of effort rises, the total effort also rises. Suppose ) ( ) ( S g S g S is increasing in S Then an increase in vi increases S decreasing pj for j i That is, when one item becomes more va luable, the incentives on the others are reduced. Moreover, since 2)) ( ( 4 S g S p p an increase in S only occurs if p p increases. These equations together imply that an increase in any one vi increases the total effort (as measured by S = x x ), increases the total incentives as measured by p p and decreases the incentives on all activities other than activity i In contrast, if ) ( ) ( S g S g S is a decreasing function of S then an increase in any one vi causes all the incentives to rise. Intuitively, the increase in vi directly causes pi to rise, since xi is more valuable. This causes the agent to substitute toward activity i This causes the relative cost of total activity to fall (since ) ( ) ( S g S g S decreases), which induces a desire to increase the other activity levels, which is accomplished by increase in the incentives on the other activities. This conclusion generalizes readil y and powerfully. Suppose that c ( x ) = g ( h ( x )), where h is homogeneous of degree r and g is increasing. In the case just considered, h ( x )= x x Then the same conclusion, that the sign of j idv dp is determined by the derivative of ) ( ) (S g S g S holds. In the generalization, S now stands for h( x ).

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-295 7.6 Auctions When we think of auctions, we tend to think of movies where people scratch their ear and accidentally pu rchase a Faberge egg, like the one pictured at left.104 However, stock exchanges, bond markets and commodities markets are organized as auctions, too, and because of such exchanges, auctions are the most co mmon means of establishing prices. Auctions are one of the oldest transactions means recorded in human history, and were used by the Babylonians. The word auction comes from the Latin auctio, meaning to increase. Auctions have been used to sell a larg e variety of things. Internet auction house eBay is most famous for weird items th at have been auctioned (e.g. one person’s attempt to sell their soul), but in additi on, many of the purchases of the U.S. government are made by auction. The U.S. pu rchases everything from fighter aircraft to French fries by auction, and the U.S. govern ment is the world’s largest purchaser of French fries. In addition, corporations are occasionally sold by auction. Items that are usually sold by auction include prize bulls, tobacco, used cars, race horses, coins, stamps, antiques, and fine art. Information plays a significant role in bidding in auctions. The two major extremes in information, which lead to distinct theories, are private values, which means bidders know their own value, and common values, in which bidders don’t know their own value, but have some indication or signal about the value. In the private values situation, a bidder may be outbid by anothe r bidder, but doesn’t learn anything from another bidder’s willingness to pay. The ca se of private values arises when the good being sold has a quality apparent to all bidde rs, no hidden attributes, and no possibility of resale. In contrast, the case of common values arises when bidders don’t know the value of the item for sale, but that value is common to all. The quintessential example is an off-shore oil lease. No one knows for sure how much oil can be extracted from an offshore lease, and companies have estimates of the amount of oil. The estimates are closely guarded because rivals could learn from them. Similarly, when antiques dealers bid on an antique, the value they place on it is primarily the resale value. Knowing rivals’ estimates of the resale value would in fluence the value each bidder placed on the item. The private values environment is unrealistic in most settings, because items for sale usually have some element of common value. However, some situations approximate the private values environment and thes e are the most readily understood. 7.6.1 English Auction An English auction is the common auction form used for selling antiques, art, used cars and cattle. The auctioneer starts low, and ca lls out prices until no bidder is willing to bid higher than the current high price. Th e most common procedure is for a low price to 104 Photo courtesy of Paris Jewelers, 107 East Ridgewood Ave. Ridgewood, New Jersey 07450.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-296be called out, and a bidder accept it. A high er price is called out, and a different bidder accepts it. When several accept simultaneo usly, the auctioneer accepts the first one spotted. This process continues until a price is called out that no one accepts. At that point the auction ends, and the highest bidder wins. In a private values setting, a very simple bi dding strategy is optimal for bidders: a bidder should keep bidding until the price exceeds the value a bidder places on it, at which point the bidder should stop. That is, bidders should drop out of the bidding when the price exceeds their value, because at that point, winning entails a loss. Every bidder should be willing to continue to bid and not le t the item sell to someone else if the price is less than their value. If you have a value v and another bidder is about to win at a price pa < v, you might as well accept a price pb between the two, pa < pb < v, since a purchase at this price would provide profits. This strategy is a dominant strategy for each private values bidder, because no matt er what strategy other bidders adopt, bidding up to value is the strategy th at maximizes the profits of a bidder. The presence of a dominant strategy makes it straightforward to bid in the private values environment. In addition, it makes an analysis of the outcome of the English auction relatively simple. Most auctioneers use a somewhat flexible system of bid increments. A bid increment is the difference between successive price requests The theory is simplest when the bid increment, which we will denote as is very small. In this case, the bidder with the highest value will win, and the price will be no more than the second-highest value, but at least the second-highest value minus since if the price was less than this level, the bidder with the second-highest value would submit another bid. If we denote the second-highest value with the somewhat obscure (but standard) notation v(2), the final price p satisfies ) 2 ( ) 2 (v p v As the bid increment gets small, this nails do wn the price. The conclusion is that, when bid increments are small and bidders have private values, the bidder with the highest value wins the bidding at a price equal to the second-highest value. The notation for the highest value is v(1), and thus the seller obtains v(2), and the winning bidder obtains profits of v(1) – v(2). 7.6.2 Sealed-bid Auction In a sealed-bid auction, each bidder submit s a bid in an envelope. These are opened simultaneously, and the highest bidder wins th e item and pays his or her bid. Sealedbid auctions are used to sell off-shore oil le ases, and used by governments to purchase a wide variety of items. In a purc hase situation, known often as a tender, the lowest bidder wins and is paid the bid. The analysis of the sealed-bid auction is more challenging because the bidders don’t have a dominant strategy. Indeed, the best bid depends on what the other bidders are bidding. The bidder with the highest value would like to bid a penny more than the next highest bidder’s bid, whatever that might be.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-297 To pursue an analysis of the sealed-bid au ction, we are going to make a variety of simplifying assumptions. These assumptions aren’t necessary to the analysis but are made to simplify the ma thematical presentation. We suppose there are n bidders, and label the bidders 1,…,n. Bidder i has a private value vi which is a draw from the uniform distributi on on the interval [0,1]. That is, if 1 0 b a, the probability that bidder i’s value is in the interval [a, b] is b – a. An important attribute of this assumption is symmetry – the bidders all have the same distribution. In addition, the formulation has assumed independence – the value one bidder places on the object for sale is statis tically independent from the value placed by others. In addition, each bidder knows their own value, but doesn’t know the other bidders’ values. Each bidder is assumed to bi d in such a way as to maximize expected profit, and we will look for a Nash equi librium of the bidding game. Bidders are permitted to submit any bid not less than zero. To find an equilibrium, it is helpful to restri ct attention to linear rules, in which a bidder bids a proportion of their value. Thus, we suppose each bidder bids v when their value is v, and examine under what conditions this is in fact Nash equilibrium behavior. We have an equilibrium if, when all other bidders bid v when their value is v, the remaining bidder will, too. So fix a bidder and suppose that bidder’s value is vi. What bid should the bidder choose? A bid of b wins the bidding if all other bidders bid less than b. Since the other bidders, by hypothesis, bid v when their value is v, our bidder wins when jv b for each other bidder j. This occurs when jv b for each other bidder j, and this in turn occurs with probability b.105 Thus, our bidder with value vi who bids b wins with probability 1 nb, since the bidder must beat all n-1 other bidders. That creates expected profits for the bidder of 1) ( n ib b v. The bidder chooses b to maximize expected profits. The first order condition requires 1 2 1) 1 )( ( 0 n n i nb n b v b. The first order condition solves for v n n b 1 105 If b then in fact the probability is one. You can show that no bidder would ever bid more than

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-298 But this is a linear rule! Thus, if n n 1 we have a Nash equilibrium. The nature of this equilibrium is that each bidder bids a fraction n n 1 of their value, and the highest value bidder wins at a pric e equal to that fraction of their value. In some cases, the sealed-bid auction produces regret. Regret means that a bidder wishes she had bid differently. Recall our notation for values: v(1) is the highest value and v(2) is the second-highest value. Since the price in a sealed-bid auction is ) 1 (1 v n n the second-highest bidder will regret her bid when ) 1 ( ) 2 (1 v n n v In this case, the bidder with the second-highest value could ha ve bid higher and won, if the bidder had known the winning bidder’s bid. In contrast, the English auction is regret-free, in that the price rises to the point that the bidder with the second-highest value won’t pay. How do the two auctions compare in prices? It turns out that statistical independence of private values implies revenue equivalence, which means the two auctions produce the same prices on average. Given the highest value v(1), the second-highest value has distribution 1 ) 1 ( ) 2 ( nv v, since this is the probability that all n-1 other bidders have values less than v(2). But this gives an expected value of v(2) of ) 1 ( ) 2 ( 0 1 ) 1 ( 2 ) 2 ( ) 2 ( ) 2 (1 ) 1 () 1 (v n n dv v v n v Evv n n Thus, the average price paid in the sealed-bid auction is the same as the average price in the English auction. 7.6.3 Dutch Auction The Dutch auction is like an English auctio n, except that prices start high and are successively dropped until a bi dder accepts the going price, at which point the auction ends. The Dutch auction is so named because it is used to sell cut flowers in Holland, in the enormous flower auctions. A strategy in a Dutch auction is a price at which the bidder bids. Each bidder watches the price decline, until such a point that either the bidder bids, or a rival bids, and the auction ends. Note that a bidder could revise their bid in the course of the auction, but there isn’t any point. For example, suppose the price starts at $1,000, and a bidder decides to bid when the price reaches $400. Once the price gets to $450, the bidder could decide to revise and wait until $350 However, no new information has become available and there is no reason to revise. In order for the price to reach the original

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-299planned bid of $400, it had to reach $450, meaning that no one bid prior to a price of $450. In order for a bid of $400 to wins, the price had to reach $450; if the price reaching $450 means that a bid of $350 is optimal, than the original bid of $400 wasn’t optimal.106 What is interesting about the Dutch auction is that it has exactly the same possible strategies and outcomes as the sealed-bid auct ion. In both cases, a strategy for a bidder is a bid, no bidder sees the others’ bids un til after their own bid is formulated, and the winning bidder is the one with the highest bid. This is called strategic equivalence. Both games – the Dutch auction and the sealed-bid auction – offer identical strategies to the bidders, and given the strategies chosen by all bidders, produce the same payoff. Such games should produce the same outcomes. The strategic equivalence of the Dutch auct ion and the sealed-bid auction is a very general result, which doesn’t depend on the na ture of the values of the bidders (private versus common) or the distribution of info rmation (independent versus correlated). Indeed, the prediction that the two games should produce the same outcome doesn’t even depend on risk aversion, although th at is more challenging to demonstrate. 7.6.4 Vickrey Auction The strategic equivalence of the Dutch and sealed-bid auction suggests another fact: there may be more than one way of implementing a given kind of auction. Such logic led Nobel laureate William Vickre y (1914-1996) to design what has become known as the Vickrey auction, which is a “second-price sealed-bid” auction. This auction is most familiar because it is the foundation of eBay ’s auction design. The Vickrey auction is a sealed-bid auction, but with a twist: the hi gh bidder wins, but pays the second-highest bid. This is why the Vickrey auction is calle d a second-price auction: the price is not the highest bid, but the second-highest bid. The Vickrey auction underlies the eBay outc ome because when a bidder submits a bid in the eBay auction, the current “going” price is not the highest bid, but the second-highest bid, plus a bid increment. Thus, up to the granularity of the bid increment, the basic eBay auction is a Vickrey auction run over time. As in the English auction, bidders with private values in a Vickrey auction have a dominant strategy. Fi x a bidder, with value v, and let p be the highest bid of the other bidders. If the bidder bids b, the bidder earns profits of p b if p v p b if0. 106 Of course, a bidder who thinks losing is likely may wait for a lower price to formulate the bid, a consideration ignored here. In addition, because the Dutch auction unfolds over time, bidders who discount the future will bid slightly higher in a Dutc h auction as a way of speeding it along, another small effect that is ignored for simplicity.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-300It is profitable for the bidder to win if v > p, and lose if v < p. To win when v > p, and lose if v < p, can be assured by bidding v. Essentially, there is no gain to bidding less than your value, because your bid doesn’t affect the price, only the likelihood of winning. Bidding less than value causes the bidder to lose when the highest rival bid falls between the bid and the value, which is a circumstan ce that the bidder would like to win. Similarly, bidding more than value only crea tes a chance of winning when the price is higher than the bidder’s value, in whic h case the bidder would prefer to lose. Thus, bidders in a Vickrey auction have a domi nant strategy to bid their value. This produces the same outcome as the English au ction, however, because the payment made is the second-highest value, which was the price in the English auction. Thus, the Vickrey auction is a sealed-bid implementa tion of the English auction when bidders have private values, producing the same outc ome, which is that the highest value bidder wins, but pays the second-highest value. Because the Vickrey auction induces bidders to bid their value, it is said to be demand revealing. Unlike the English auction, in wh ich the bidding stops when the price reaches the second-highest value and thus do esn’t reveal the highest value, the Vickrey auction reveals the highest value. In a cont rolled, laboratory setting, demand revelation is useful, especially when the goal is to id entify buyer values. Despite its theoretical niceties, the Vickrey auction can be politica lly disastrous. Indeed, New Zealand sold radio spectrum with the Vickrey auction on th e basis of advice by a nave economist, and the Vickrey auction created a political ni ghtmare when a nationwide cellular license received a high bid of $110 million, and a second-highest bid of $11 million. The political problem was that the demand revelation showed th at the government received only about 10% of the value of the license, making the public quite irate and dominating news coverage at the time.107 Some smaller licenses sold for tenths of a percent of the highest bid. In a private values setting, the Vickrey auctio n, or the English auction, are much easier on bidders than a regular sealed-bid auctio n, because of the dominant strategy. The sealed-bid auction requires bidders to forecast their rivals’ likely bids, and produces the risks of either bidding more than necessary, or losing th e bidding. Thus, the regular sealed-bid auction has undesirable properti es. Moreover, bidders in the sealed-bid auction have an incentive to bribe the auction eer to reveal the best bid by rivals, because that is useful information in formulating a bid. Such (illegal) bribery occurs from time to time in government contracting. On the other hand, the regular sealed-bid auction has an advantage over the other two that it makes price-fixing more difficult. A bidder can cheat on a conspiracy and not be detected until after the current auction is complete. Another disadvantage of the sealed-bid auction is that it is easier to make certain kinds of bidding errors. In the U.S. PCS auctions in which rights to use the radio spectrum 107 The Vickrey auction generally produces higher prices than regular sealed-bid auctions if bidders are symmetric (share the same distribution of values), but is a poor choice of auction format when bidders are not symmetric. Since the incumbent telephone company was expected to have a higher value than others, the Vickrey auction was a poor choice for that reason as well.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-301for cellular phones was sold for around $20 billion, one bidder, intending to bid $200,000, inadvertently bid $200,000,000. Such an error isn’t possible in an English auction, because prices rise at a measured pa ce. Such errors have little consequence in a Vickrey auction, since getting the price wrong by an order of magnitude requires two bidders to make such errors. 7.6.5 Winner’s Curse "I paid too much for it, but it's worth it." -Sam Goldwyn The analysis so far has been conducted un der the restrictive assumption of private values. In most contexts, bidders are not sure of the actual value of the item being sold, and information held by others is relevant to the valuation of the item. If I estimate an antique to be worth $5,000, but no one else is willing to bid more than $1,000, I might revise my estimate of the value down. This revision leads bidders to learn from the auction itself what the item is worth. The early bidders in the sale of oil lease rights in the Gulf of Mexico (the outer continental shelf) were often observed to pay more than the rights were worth. This phenomenon came to be known as the winner’s curse. The winner's curse is the fact that the bidder who most over estimates the value of th e object wins the bidding. Nave bidders, who don’t adjust for the winner’s curs e, will tend to lose money because they only win the bidding when they’ve bid too high. Figure 7-9: Normally Distributed Estimates Auctions, by their nature, select optimistic bidders. Consider the case of an oil lease (right to drill for and pump oil) that has an unknown value v. Different bidders will obtain different estimates of the value, and we may view these estima tes as draws from a normal distribution illustrated in Figure 7-9. The estimates are correct on average, which is represented by the fa ct that the distribution is centered on the true value v. Thus a randomly chosen bidder will have an esti mate that is too high as often as it is too -4 -2 2 4 0.1 0.2 0.3 0.4 v v+2 v+4 v – 2 v – 4

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-302low, and the average estimate of a randomly selected bidder will be correct. However, the winner of an auction will tend to be bidde r with the highest estimate, not a randomly chosen bidder. The highest of five bidders will have an estimate that is too large 97% of the time. The only way the hi ghest estimate is not too large is if all the estimates are below the true value. With ten bidders, the high est estimate is larger than the true value with probability 99.9%, because the odds that all the estimates are less than the true value is ()10 = 0.1%. This phen omenon – that auctions tend to select the bidder with the highest estimate, and the highest estimate is larger than the true value most of the time – is characteristic of the winner’s curse. A savvy bidder corrects for the winner’s curse. Such a correction is actually quite straightforward when a few facts are availabl e, and here a simplified presentation is given. Suppose there are n bidders for a common value good, and the bidders receive normally distributed estimates that are correct on average. Let be the standard deviation of the estimates.108 Finally, suppose that no prior information is given about the likely value of the good. In this case, it is a straightforward matter to compute a correction for the winner’s curse. Because the winning bidder will generally be the bidder with the highest estimate of value, the winner’s curse correct ion should be the expected amount by which the highest value exceeds the average value. This can be looked up in a table for the normal distribution. The values are given for selected numbers n in Table 7-30. This shows, as a function of the number of bidders, how mu ch each bidder should reduce their estimate of value to correct for the fact that auctio ns select optimistic bidders. The units are standard deviations. Table 7-30: Winner's Curse Correction n 1 2 3 4 5 10 15 WCC ( ) 0 .56 .85 1.03 1.16 1.54 1.74 n 20 25 50 100 500 1000 10,000 WCC ( ) 1.87 1.97 2.25 2.51 3.04 3.24 3.85 For example, with one bidder, there is no correction, since it was supposed that the estimates are right on average. With two bidders, the winner’s curse correction is the amount that the higher of two will be above the mean, which turns out to be 0.56 a little more than half a standard deviation. This is the amount which should be subtracted from the estimate to insure that when the bidder wins, the estimated value is on average correct. With four bidders, th e highest is a bit over a whole standard deviation. As is apparent from the ta ble, the winner’s curse correction increases relatively slowly after ten or fifteen bi dders. With a million bidders, it is 4.86 108 The standard deviation is a measure of the dispersion of the distribution, and is the square root of the average of the square of the difference of the random value and its mean. The estimates are also assumed to be independently distributed around the true value. Note that estimating the mean adds an additional layer of complexity.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-303The standard deviation measures how much randomness or noise there is in the estimates. It is a measure of the averag e difference between the true value and the estimated value, and thus the average level of error. Oil companies know from their history of estimation how much error arises in the company estimates. Thus, they can correct their estimates to account for th e winner’s curse using their historical inaccuracies. Bidders who are imperfectly informed about the value of an item for sale are subject to losses arising from the way auctions select the winning bidder. The winning bidder is usually the bidder with the high est estimate, and that estimate is too high on average. The difference between the high est estimate and the average estimate is known as the winner’s curse correction. The size of the wi nner’s curse correction is larger the more bidders there are but tends to grow slowly beyond a dozen or so bidders. If the bidders have the same information on a common value item, they will generally not earn profits on it. Indeed, there is a general principle that it is the privacy of information, rather than the accuracy of inform ation, that leads to profits. Bidders earn profits on the information that they hold th at is not available to others. Information held by others will be built into the bi d price and therefore not lead to profits. 7.6.6 Linkage The U.S. Department of the Interior, when se lling off-shore oil leases, not only takes an upfront payment (the winning bid) but also takes 1/6 of the oil that is eventually pumped. Such a royalty scheme links the pa yment made to the outcome, and in a way, shares risk, since the payment is higher when there is more oil. Similarly, a book contract provides an author with an up front payment and a royalty. Many U.S. Department of Defense purchases of majo r weapons systems involve cost-sharing, where the payments made pick up a portion of the cost. Purchases of ships, for example, generally involve 50% to 70% cost sharing, which means the DOD pays a portion of cost overruns. The contract for U.S. television broadcast rights for the summer Olympics in Seoul, South Korea, involved payments that depended on the size of the U.S. audience. Royalties, cost-sharing and contingent paym ents generally link the actual payment to the actual value, which is unknown at the ti me of the auction. Linkage shares risk, which is a topic already considered in Section 7.5. But linkage does something else, too. Linkage reduces the importance of estimates in the auction, replacing the estimates with actual values. That is, the price a bidder pays for an object, when fully linked to the true value, is just the true value. Thus, linkage reduces the importance of estimation in the auction by taking the price out of th e bidder’s hands, at least partially. The linkage principle109 states that, in auctions where bidders are buyers, the expected price rises the more the price is linked to the actual value. (In a parallel fashion, the expected price in an auction where bidders are selling falls.) Thus, linking price to value generally improves the performance of auctio ns. While this is a mathematically deep result, an extreme case is straightforward to understand. Suppose the government is purchasing by auction a contract for delivery of 10,000 gallons of gasoline each week for 109 The linkage principle, and much of modern auction theory, was developed by Paul Milgrom (1948 – ).

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-304the next year. Suppliers face risk in the form of gasoline prices; if the government buys at a fixed price, the suppliers’ bids will buil d in a cushion to compensate for the risk, and for the winner’s curse. In addition, becaus e their estimates of future oil prices will generally vary, they will earn profits based on their private information about the value. In contrast, if the government buys only de livery and then pays for the cost of the gasoline, whatever it might be, any profits th at the bidders earned based on their ability to estimate gasoline prices evaporates. The overall profit level of bidders falls, and the overall cost of the gasoline supply can fall. Of course, paying the cost of the gasoline reduces the incentive of the supplier to shop around for the best price, and that agency incentive effect must be balanced agains t the reduction in bidder profits from the auction to select a supplier. 7.6.7 Auction Design We saw above that the English auction tends to reduce regret relative to sealed-bid auctions, and that the linkage principle suggest s tying payments to value where possible. These are examples of auction design, in which auctions are designed to satisfy objectives of the auction designer. Proper auction design should match the rules of the auction to the circumstances of the bidders and the goal of the seller. Some of the principles of auction design include: Impose an appropriate rese rve price or minimum bid Use ascending price (English) auct ions rather than sealed-bid Reveal information about the value of the item Conceal information about the extent of competition Handicap bidders with a known advantage However, many of these precepts change if the se ller is facing a cartel. For example, it is easier for bidders to collude in a sealed-bid auction than in an English auction; and reserve prices should be made relatively high. Reserve prices (minimum bid) have several effects. They tend to force marginal bidders to bid a bit higher, which increases bids of all bidders, reducing bidder profits. However, reserve prices also lead to a failur e to sell on occasion, and the optimal reserve trades off this failure to sell against the high er prices. In addition, reserve prices may reduce the incentive of bidders to investigate the sale, reducing participation, which is an additional negative consideration for a high reserve price. Ascending price auctions like the English auction have several advantages. Such auctions reduce the complexity of the bi dder’s problem, because bidder’s can stretch their calculations out over time, and because bi dders can react to the behavior of others and not plan for every contingency in advanc e. In addition, because bidders in an English auction can see the behavior of others there is a linkage created – the price paid by the winning bidder is influenced not just by that bidder’s information but also by the information held by others, tending to drive up the price, which is an advantage for the seller. One caveat to the selection of the English auct ion is that risk aversion doesn’t affect the outcome in the private values case. In contra st, in a sealed-bid auction, risk aversion

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-305works to the advantage of the seller, becaus e bidders bid a little bit higher than they would have otherwise, to reduce the risk of losing. Thus, in the private values case, risk averse bidders will bid higher in the sealed -bid auction than in the English auction. When considering the revelation of information, there is always an issue of lying and misleading. In the long-run, lying and mislea ding is found out, and thus the standard approach is to ignore the possibility of lyin g. Making misleading statements is, in the long-run, the same thing as silence, sinc e those who repeatedly lie or mislead are eventually discovered, and then not believed. Thus, in the long-run, a repeat seller has a choice of being truthful or silent. Becaus e of the linkage principle, the policy of revealing truthful information about the value of the good for sale dominates the policy of concealing information, because the reve lation of information links the payment to the actual outcome. In contrast, revealing information about the extent of competition may not increase the prices. Consider the case where occasion ally there are three bidders, and sometimes only one. If the extent of competition is concealed, bidders will bid without knowing the extent of competition. If the bidders are ri sk neutral, it turns out that the revelation doesn’t matter and the outcomes are the same on average. If, in contrast, bidders are risk averse, the concealment of information te nds to increase the bid prices, because the risk created by the uncertainty about the extent of competition works to the advantage of the seller. Of course, it may be difficult to conceal the extent of competition in the English auction, suggesting a sealed-bid auction instead. Bidders with a large, known advantage have several deleterious effects. For example, incumbent telephone companies generally are wi lling to pay more for spectrum in their areas than outsiders are. Advantaged bidders discourage participation of others, since the others are likely to lose. This can resu lt in a bidder with an advantage facing no competition and picking up the good cheapl y. Second, rivals don’t present much competition to the advantaged bidder, even if the rivals do participate. Consequently, when a bidder has a large advantage over rivals, it is advantageous to handicap the advantaged bidder, favoring the rivals. This handicapping encourages participation and levels the playing field, forcing the advantaged bidder to bid more competitively to win. A common means of favoring disa dvantaged bidders is by the use of bidder credits. For example, with a 20% bidder credit for di sadvantaged bidders, a disadvantaged bidder only has to pay 80% of the face amount of th e bid. This lets such a bidder bid 25% more (since a $100 payment corresponds to a $125 bid) than they would have otherwise, which makes the bidder a more formidable competitor. Generally, the ideal bidder credit is less than the actual advantage of the advantaged bidder. Auction design is an exciting development in applied industrial organization, in which economic theory and experience is used to improve the performance of markets. The U.S. Federal Communications Commissions auctions of spec trum, were the first major instance of auction design in an important practical setting, and the auction design was credited with increasing the revenue ra ised by the government substantially.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-3067.7 Antitrust In somewhat archaic language, a trust was a gr oup of firms acting in concert, which is now known as a cartel. The antitrust laws made such trusts illegal, and were intended to protect competition. In the United States, th ese laws are enforced by the Department of Justice’s Antitrust Division, and by the Fe deral Trade Commission. The United States began passing laws during a time when some European na tions were actually passing laws forcing firms to join industry cartels. By and large, however, the rest of the world has since copied the U.S. antitrust laws in one version or another. 7.7.1 Sherman Act The Sherman Act, passed in 1890, is the first si gnificant piece of antitrust legislation. It has two main requirements: Section 1. Trusts, etc., in restraint of trade illegal; penalty Every contract, combination in the form of trust or otherwise, or conspiracy, in restraint of trade or commerce among the several States, or with fore ign nations, is declared to be illegal. Every person who shall make any contract or engage in any combination or conspiracy hereby declared to be illegal shall be deemed guilty of a felony, and, on conviction thereof, shall be punished by fine not exceeding $10,000,000 if a corporation, or, if any other person, $350,000, or by imprisonment not exceeding three years, or by bo th said punishments, in the discretion of the court. Section 2. Monopolizing trade a felony; penalty Every person who shall monopolize, or attempt to monopolize, or combine or conspire with any other person or persons, to monopolize any pa rt of the trade or commerce among the several States, or with foreign nations, shall be deemed guil ty of a felony, and, on conviction thereof, shall be punished by fine not exceeding $10,000,000 if a corporation, or, if any other person, $350,000, or by imprisonment not exceeding three years, or by both said punishments, in the discretion of the court.110 The phrase “in restraint of trade” is challeng ing to interpret. Early enforcement of the Sherman Act followed the “Peckham Rule,” named for noted Justice Rufus Peckham, which interpreted the Sherman Act to prohibit contracts that reduced output or raised prices, while permitting contracts that would increase output or lower prices. In one of the most famous antitrust cases ever, the United States sued Standard Oil, which had monopolized the transportation of oil from Pennsylvania to the east coast cities of the United States, in 1911. The exact meaning of the Sherman Act had not been settled at the time of the Standard Oil case. Indeed, Supreme Court Justice Edward White suggested that, because contracts by their nature set the terms of tr ade and thus restrain trade to those terms and Section 1 makes contracts restraining trade illegal, one could read the Sherman Act to imply all contracts were illegal. Chie f Justice White concluded that, since Congress couldn’t have intended to make all contracts illegal, the intent must have been to make unreasonable contracts illegal, and therefor e concluded that judicial discretion is necessary in applying the antitrust laws. In addition, Chief Justice White noted that the 110 The current fines were instituted in 1974; the original fines were $5,000, with a maximum imprisonment of one year. The Sherman Act is 15 U.S.C. § 1.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-307act makes monopolizing illegal, but doesn’t make havi ng a monopoly illegal. Thus, Chief Justice White interpreted the act to prohibit certain acts leading to monopoly, but not monopoly itself. The legality of monopoly was further clarified through a series of cases, starting with the 1945 Alcoa case, in which the United States sued to break up the aluminum monopoly Alcoa. The modern approach involves a two-part test. First, does the firm have monopoly power in a market? If not, no mo nopolization has occurred and there is no issue for the court. Second, if so, did the firm use illegal tactics to extend or maintain that monopoly power? In the language of a later decision, did the firm engage in “the willful acquisition or maintenance of that power as distinguished from growth or development as a consequence of superior product, business acumen or historic accident?” (U.S. v. Grinnell, 1966.) There are several important points that are widely misunderstood and even misreported in the press. First, the Sherman Act does not make having a monopoly illegal. Indeed, three legal ways of obtaining a monopoly – a better product, running a better business, or luck – are spelled out in one decision. It is illegal to leverage that existing monopoly into new products or services, or to engage in anticompetitive tactics to maintain the monopoly. Moreover, you must have monopoly power currently to be found guilty of illegal tactics. When the Department of Justice sued Micros oft over the incorporation of the browser into the operating system and other acts (including contracts with manufacturers prohibiting the installation of Netscape), the allegation was not that Windows was an illegal monopoly. The DOJ alleged Microsof t was trying to use its Windows monopoly to monopolize another market, the internet browser market. Microsoft’s defense was two-fold. First, it claimed not to be a mono poly, citing the 5% share of Apple. (Linux had a negligible share at the time.) Seco nd, it alleged a browser was not a separate market but an integrated product necessary fo r the functioning of th e operating system. This defense follows the standard “two-part test.” Microsoft’s defense brings up the question of “what is a mo nopoly?” The simple answer to this question depends on whether there are good substitutes in the minds of consumers – will they substitu te to an alternate product in the event of some bad behavior by the seller? By this test, Micr osoft had an operating system monopoly in spite of the fact that there was a rival, because for most consumers, Microsoft could increase the price, tie the browser and MP3 player to the operating system, or even disable Word Perfect, and the consumers wo uld not switch to the competing operating system. However, Microsoft’s second defe nse, that the browser wasn’t a separate market, is a much more challenging defense to assess. The Sherman Act provides criminal penalt ies, which are commonly applied in pricefixing cases, that is, when groups of firms join together and collude to raise prices. Seven executives of General Electric and We stinghouse, who colluded in the late 1950s to set the prices of electrical turbines, spent several years in jail each, and there was over $100 million in fines. In addition, Archer Da niels Midland executives went to jail after their 1996 conviction for fixing the price of lysine, which approximately doubled the

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-308price of this common additive to animal fe ed. When highway contractors are convicted of bid-rigging, generally the conviction is under the Sherman Act, for monopolizing their market. 7.7.2 Clayton Act Critics of the Sherman Act, including fa mous “trust-buster” and President Teddy Roosevelt, felt the ambiguity of the Sherman Act was an impediment to its use and that the United States needed a more detailed law setting out a list of illegal activities. The Clayton Act, 15 U.S.C. §§ 12-27, was passed in 1914 and it adds detail to the Sherman Act. The same year, the FTC Act was pass ed, creating the Federal Trade Commission, which has authority to enforce the Clayton Ac t, as well as engage in other consumer protection activities. The Clayton Act does not have criminal penalt ies, but does allow for monetary penalties that are three times as large as the da mage created by the illegal behavior. Consequently, a firm, motivated by the possib ility of obtaining a large damage award, may sue another firm for infringement of the Clayton Act. A plaint iff must be directly harmed to bring such a suit. Thus, customers who paid higher prices, or firms driven out of business by exclusionary practices ar e permitted to sue under the Clayton Act. When Archer Daniels Midland raised the pr ice of lysine, pork producers who bought lysine would have standing to sue, while fi nal pork consumers who paid higher prices for pork, but who didn’t dire ctly buy lysine, would not. Highlights of the Clayton Act include: Section 2, which prohibits price di scrimination that would lessen competition, Section 3, which prohibits exclusionary practices that lessen competition, such as tying, exclusive dealing and predatory pricing, Section 7, which prohibits share acquisition or merger that would lessen competition or create a monopoly The language “lessen competition” is genera lly understood to mean that a significant price increase becomes possible – that is, co mpetition has been harmed if the firms in the industry can successfully increase prices. Section 2 is also known as ‘Robinson-Patm an’ because of a 1936 amendment by that name. It prohibits price discrimination that lessens competition. Thus, price discrimination to final cons umers is legal under the Clayton Act; the only way price discrimination can lessen competition is if one charges different prices to different businesses. The logic of the law was articulated in the 1948 Morton Salt decision, which concluded that lower prices to large chain stor es gave an advantage to those stores, thus injuring competition in the grocery market. The discounts in that case were not costbased, and it is permissible to char ge different prices based on costs. Section 3 rules out practices that lessen comp etition. A manufacturer who also offers service for the goods it sells may be pr ohibited from favoring its own service organization. Generally manufacturers may not require the use of the manufacturer’s

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-309own service. Moreover, an automobile manufacturer can’t require the use of replacement parts made by the manufacturer and many car manufacturers have lost lawsuits on this basis. In an entertaining example, Mercedes prohibited Mercedes dealers from buying Bosch parts directly from Bosch, even though Mercedes itself was selling Bosch parts to the dealers. This prac tice was ruled illegal because the quality of the parts was the same as Mercedes (indeed, identical), so Mercedes’ action lessened competition. Predatory pricing involves pricing below cost in order to drive a rival out of business. It is relatively difficult for a firm to engage in predation, simply because it only makes sense if, once the rival is eliminated, the pred atory firm can then increase its prices and recoup the losses incurred. The problem is that once the prices go up, entry becomes attractive; what keeps other potential entr ants away? One answ er is reputation: a reputation for a willingness to lose money in order to dominate market could deter potential entrants. Like various rare diseases that happen more ofte n on TV than in the real world (e.g. Tourette’s syndrome), predat ory pricing probably happens more often in textbooks than in the real world.111 The Federal Trade Commission also has author ity to regulate mergers that would lessen competition. As a practical matter, the Department of Justice and the Federal Trade Commission divide respon sibility for evaluating mergers. In addition, other agencies may also have jurisdiction over mergers and business tactics. The Department of Defense has oversight of defense contractor s, using a threat of “we’re your only customer.” The Federal Communications Commission has statutory authority over telephone and television companies. The Federal Reserve Bank has authority over national and most other banks. Most states have antitrust laws as well, and can challenge mergers that would affect commerce in the state. In addition, attorn eys general of many states may join the Department of Justice or the Federal Trade Co mmission is suing to block a merger or in other antitrust actions, or sue independentl y. For example, many states joined the Department of Justice in its lawsuit against Microsoft. Forty-two states jointly sued the major record companies over their “minimum advertised prices” policies, which the states argued resulted in higher compac t disc prices. The “MAP” case settlement resulted in a modest payment to compact disc purchasers. The Federal Trade Commission had earlier extracted an agreement to stop the practice. 7.7.3 Price-Fixing Price-fixing, which is called bid-rigging in a bidding context, involves a group of firms agreeing to increase the prices they charge and restrict competition against each other. The most famous example of price-fixing is probably the “Great Electrical Conspiracy” in which GE and Westinghouse (and a smal ler firm, Allis-Chalmers and many others) fixed the prices of turbines used for electr icity generation. Generally these turbines were the subject of competitive (or in this case not-so-competitive) bidding, and one way that the companies set the prices was to have a designated winner for each bidding situation, and using a price book to provide id entical bids by all companies. An amusing 111 Economists have argued that American Tobacco, St andard Oil and A.T. & T. each engaged in predation in their respective industries.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-310element of the price-fixing scheme was the means by which the companies identified the winner in any given competition: it used the phase of the moon. Th e phase of the moon determined the winner and each company knew what to bid based on the phase of the moon. Executives from the companies met of ten to discuss terms of the price-fixing arrangement, and the Department of Justice ac quired a great deal of physical evidence in the process of preparing its 1960 case. Se ven executives went to jail and hundreds of millions of dollars in fines were paid. Most convicted price-fixers are from small fi rms. The turbine conspiracy and the Archer Daniels Midland lysine conspiracy are unusual. (There is evidence that large vitamins manufacturers conspired in fixing the price of vitamins in many nations of the world.) Far more common conspiracies involve hi ghway and street construction firms, electricians, water and sewer construction companies or other “owner operated” businesses. Price-fixing seems most common when owners are also managers and there are a small number of competitors in a given region. As a theoretical matter, it should be difficult for a large firm to motivate a manager to engage in price-fixing. The problem is that the firm can’t write a contract promising the manager extraordinary returns for successfully fixing prices because such a contract itself would be evidence and moreover implicate higher management. Indeed, Archer Daniels Midland executives paid personal fi nes of $350,000 as well as each serving two years in jail. Thus, it is difficult to offer a substantial portion of the rewards of pricefixing to managers, in exchange for the pe rsonal risks the managers would face from engaging in price-fixing. Most of the gains of price-fixing accrue to shareholders of large companies, while large risks and costs fall on executives. In contrast, for smaller businesses in which the owner is the manage r, the risks and rewards are borne by the same person, and thus the personal risk more likely to be justified by the personal return. We developed earlier a simple theory of coop eration, in which the grim trigger strategy was used to induce cooperation. Let us appl y that theory to price-fixing. Suppose that there are n firms, and that they sh are the monopoly profits m equally if they collude. If one firm cheats, that firm can obtain the enti re monopoly profits until the others react. This is clearly the most the firm could get from cheating. Once the others react, the collusion breaks down and the firms earn zero profits (the competitive level) from then on. The cartel is feasible if 1/n of the monopoly profits forever is better than the whole monopoly profits for a short period of ti me. Thus, cooperation is sustainable if: m m mn n ...) 1 ( ) 1 (2. The left hand side gives the profits from cooperating – the present value of the 1/n share of the monopoly profits. In contrast, if a fi rm chooses to cheat, it can take at most the monopoly profits, but only temporarily. How many firms will this sustain? The inequality simplifies to 1 1 n. Suppose the annual inte rest rate is 5% and the reaction time is 1 week – th at is, a firm that cheats on the cooperative agreement

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-311sustains profits for a week, after which time pric es fall to the competitive level. In this case, 1is a week’s worth of interest ( is the value of money received in a week) and therefore 999014 95 052 1 According to standard th eory, the industry with a week-long reaction time should be able to support cooperation with up to a thousand firms! There are a large variety of reasons why this theory fails to work very well empirically, including that some people are actually honest and don’t break the law, but we will focus on one game-theoretic reason here. The cooperative equilibrium is not the only equilibrium, and there are good reasons to think that full cooperation is unlikely to persist. The problem is the prisoner’s dile mma itself: generally the first participant to turn in the conspiracy can avoid jail. Thus if one member of a cartel is uncertain whether the other members of a price-fixing conspiracy are contacting the Department of Justice, that member may race to the DOJ – the threat of one confession may cause them all to confess in a hurry. A majority of the conspiraci es that are prosecuted arise because someone – a member who feels guilty a disgruntled ex-spouse of a member, or perhaps a member who thinks another member is suffering pangs of conscience – turns them in. Lack of confidence in the other members creates a self-fulfilling prophecy. Moreover, cartel members should lack confid ence in the other cartel members who are, after all, criminals. On average, prosecuted conspiracies were ab out seven years old when they were caught. Thus, there is about a 15% chance annually of a breakdown of a conspiracy, at least among those that are eventually caught. 7.7.4 Mergers The U.S. Department of Justice and the Fe deral Trade Commission share responsibility for evaluating mergers. Firms with more th an $50 million in assets are required under the Hart-Scott-Rodino Act to file an inte ntion to merge with the government. The government then has a limited amount of time to either approve the merger or request more information (called a second request). Once the firms have complied with the second request, the government again has a limited amount of time before it either approves the merger or sues to block it. Th e agencies themselves don’t stop the merger, but instead sue to block the merger, asking a federal judge to prevent the merger as a violation of one of the antitrust laws. Me rgers are distinct from other violations, because they have not yet occurred at the time the lawsuit is brought, so there is no threat of damages or criminal penalties; the only potential penalty imposed on the merging parties is that the proposed merger may be blocked. Many proposed mergers result in settlements. As part of the settlement associated with GE’s purchase of RCA in 1986, a small appl iance division was sold to Black & Decker, thereby maintaining competition in the small kitchen appliance market. In the 1999 merger of oil companies Exxon and Mobil, a Ca lifornia refinery, shares in oil pipelines connecting the gulf with the northeast, and th ousands of gas stations were sold to other companies. The 1996 merger of KimberleyClark and Scott Paper would have resulted in a single company with over 50% of the fa cial tissue and baby wipes markets, and in both cases divestitures of production capa city and the “Scotties” brand name preserved

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-312competition in the markets. Large bank mergers, oil company mergers and other large companies usually present some competitive concerns, and the majority of these cases are solved by divestiture of business units to preserve competition. A horizontal merger is a merger of competitors, such as Exxon and Mobil or two banks located in the same city. In contrast, a vertical merger is a merger between an input supplier and input buyer. The attempt by bo ok retailer Barnes and Noble to purchase the intermediary Ingram, a company that bu ys books from publishers and sells to retailers but didn’t directly sell to the public, would have resulted in a vertical merger. Similarly, Disney is a company that sells pr ograms to television stations (among other activities), so its purchase of TV network ABC was a vertical merger. The AOL--Time Warner merger involved several vertical rela tionships. For example, Time Warner is a large cable company, and cable represents a way for AOL to offer broadband services. In addition, Time Warner is a content provider, and AOL delivers content to internet subscribers. Vertical mergers raise two related problems: foreclosure and raising rivals’ costs. Foreclosure refers to denying access to necessary inputs. Thus, the AOL--Time Warner merger threatened rivals to AOL internet se rvice (like EarthLink) with an inability to offer broadband services to consumers with Ti me Warner cable. This potentially injures competition in the internet service market, fo rcing Time Warner customers to use AOL. In addition, by bundling Time Warner content and AOL internet service, users could be forced to purchase AOL internet service in or der to have access to Time Warner content. Both of these threaten foreclosure of rivals, and both were resolved to the government’s satisfaction by promises that the merged firm would offer equal access to rivals. Raising rivals’ costs is a softer version of foreclosure. Rather than deny access to content, AOL--Time Warner could instead make the content available under disadvantageous terms. For example, American Airlines developed the Sabre computerized reservation system, which was us ed by about 40% of travel agents. This system charged airlines, rather than trav el agents, for bookings. Consequently, American Airlines had a mechanism for increasi ng the costs of its rivals, by increasing the price of bookings on the Sabre system. The advantage to American was not just increased revenue of the Sabre system but also the hobbling of airline rivals. Similarly, banks offer free use of their own automated teller machines (ATMs), but charge the customers of other banks. Such charges ra ise the costs of customers of other banks, thus making other banks less attractive, and hence providing an advantage in the competition for bank customers. The Department of Justice and the Federal Trade Commission periodically issue horizontal merger guidelines, which set out ho w mergers will be evaluated. This is a three step procedure for each product that the merging companies have in common. The procedure starts by identifying product markets. To identify a product market, start with a product or products produced by both companies. Then ask if the merged parties can profitably raise price by a small but significant and no n-transitory increase in price, also known as a “SSNIP,” pronounced ‘sni p.’ A SSNIP is often taken to be a 5% price increase, which must prevail for severa l years. If the companies can profitably

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-313increase price by a SSNIP, then they are ju dged to have monopoly power and consumers will be directly harmed by the merger. (This is known as a unilateral effect, because the merging parties can increase price unilateral ly after the merger is consummated.) If they can’t increase prices, then an addition al product has to be added to the group; generally the best substitute is added. As k whether a hypothetical monopoly seller of these three products can profitably raise pric e. If so, an antitrust market has been identified; if not, yet another substitute product must be added. The process stops adding products when enough su bstitutes have been identified which, if controlled by a hypothetical monopoly, would have th eir prices significantly increased. The logic of product market definition is that if a monopoly wouldn’t increase price in a meaningful way, that there is no threat to consumers – any price increase won’t be large or won’t last. The market is defined by the smallest set of products for which consumers can be harmed. The test is also known as the hypothetical monopoly test. The second step is to identify a geographic market. The process starts with an area in which both companies sell, and asks if the me rged company has an incentive to increase price by a SSNIP. If so, that geographic area is a geographic market. If not, it is because of buyers substituting outside the area to bu y cheaply, and the area must be expanded. For example, owning all the gas stations on a corner doesn’t let one increase price profitably because an increase in price lead s to substitution to stations a few blocks away. If one company owned all the stations in a half mile radius, would it be profitable to increase price? Probably not, as there wo uld still be significant substitution to more distant stations. Suppose, instead, that one owned all the stations for a 15 mile radius. Then an increase in price in the center of the area is not going to be thwarted by too much substitution outside the area, and the likely outcome is that prices would be increased by such a hypothetical monopoly. In this case, a geographic market has been identified. Again, parallel to the product ma rket definition, a geographic market is the smallest area in which competitive concerns would be raised by a hypothetical monopoly. In any smaller area attempts to increase price are defeated by substitution to sellers outside the area. The product and geographic markets together are known as a relevant antitrust market, relevant for the purposes of analyzing the merger. The third and last step of the procedure is to identify the level of concentration in each relevant antitrust market. The Hirschman-Herf indahl index, or HHI, is used for this purpose. The HHI is the sum of the squared market shares of the firms in the relevant antitrust market, and is justified because it measures the price – cost margin in the Cournot model. Generally in practice the shares in percent are used, which makes the scale range from 0 to 10,000. For example, if one firm has 40%, one 30%, one 20% and the remaining firm 10%, the HHI is 402 + 302 + 202 + 102 = 3,000. Usually, anything over 1800 is considered “v ery concentrated,” and anything over 1000 is “concentrated.”

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McAfee: Introduction to Economic Analysis,, July 24, 2006 7-314Suppose firms with shares x and y merge, and nothing in the industry changes besides the combining of those shares. Then the HHI goes up by (x + y)2 – x2 – y2 = 2xy. This is referred to as the change in the HHI. Th e merger guidelines suggest the government will likely challenge mergers with (i) a chan ge of 100 and a concentrated post-merger HHI, or (ii) a change of 50 and a very concentrated post-merger HHI. It is more accurate to understand the merger guidelines to say that the government likely won’t challenge unless either (i) or (ii) is met. Even if the post-merger HHI suggests a very concentrated industry, the government is unli kely to challenge is the change in the HHI is less than 50. Several additional factors affect the govern ment’s decision. First, if the firms are already engaging in price discrimination the government may define quite small geographic markets, and possibly as small as a single customer. Second, if one firm is very small (less than a percent) and the othe r not too large (less than 35%) the merger may escape scrutiny because the effect on compet ition is likely small. Third, if one firm is going out of business, the merger may be allowed as a means of keeping the assets in the industry. Such was the case with Greyho und’s takeover of Trailways, a merger to monopoly of the only intercity bu s companies in the United States. Antitrust originated in the United States and the United States remains the most vigorous enforcer of antitrust laws. Howeve r, the European Union has recently taken a more aggressive antitrust stance and in fa ct blocked mergers that obtained tentative U.S. approval, such as General Electric and Honeywell. Antitrust is, in some sense, the applied arm of oligopoly theory. Because real situations are so complex, the application of oligopol y theory to antitrust analysis is often challenging, and we have only scratched the su rface of many of the more subtle issues of law and economics in this text. For example, intellectual property, patents and standards all have their own distinct antitrust issues.

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McAfee: Introduction to Economic Analysis,, July 24, 2006 8-3158 Index 8.1 List of Figures FIGURE 2-1: THE DEMAND CURVE............................................................................................................................... ....................................2-9 FIGURE 2-2: CONSUMER SURPLUS............................................................................................................................... ..................................2-11 FIGURE 2-3: AN INCREASE IN DEMAND............................................................................................................................... ........................2-12 FIGURE 2-4: THE SUPPLY CURVE............................................................................................................................... .....................................2-14 FIGURE 2-5: SUPPLIER PROFITS............................................................................................................................... .......................................2-15 FIGURE 2-6: AN INCREASE IN SUPPLY............................................................................................................................... ...........................2-16 FIGURE 2-7: MARKET DEMAND............................................................................................................................... ........................................2-19 FIGURE 2-8: EQUILIBRATION............................................................................................................................... ............................................2-21 FIGURE 2-9: AN INCREASE IN DEMAND............................................................................................................................... ........................2-23 FIGURE 2-10: AN INCREASE IN SUPPLY............................................................................................................................... .........................2-24 FIGURE 2-11: PRICE OF STORAGE............................................................................................................................... .....................................2-25 FIGURE 2-12: ELASTICITIES FOR LINEAR DEMAND............................................................................................................................... ...2-29 FIGURE 2-13: THE PRODUCTION POSSIBILITIES FRONTIER.................................................................................................................2-33 FIGURE 2-14: TWO PRODUCTION POSSIBILITIES FRONTIERS..............................................................................................................2-34 FIGURE 2-15: JOINT PPF............................................................................................................................... ......................................................2-35 FIGURE 3-1: US RESIDENT POPULATION............................................................................................................................... .......................3-41 FIGURE 3-2: US URBAN AND WHITE POPULATION............................................................................................................................... ....3-42 FIGURE 3-3: POPULATION PROPORTIONS BY AGE GROUP....................................................................................................................3-42 FIGURE 3-4: PROPORTION OF POPULATION UNDER AGE FIVE...........................................................................................................3-43 FIGURE 3-5: US LIFE EXPECTANCY AT BIRTH............................................................................................................................... ..............3-44 FIGURE 3-6: US IMMIGRANT POPULATION, IN PERCENT, BY CONTINENT OF ORIGIN................................................................3-44 FIGURE 3-7: NATIONAL ORIGIN OF IMMIGRANTS, 1900-2000..............................................................................................................3-45 FIGURE 3-8: MALE MARITAL STATUS (PERCENTAGES)...........................................................................................................................3-45 FIGURE 3-9: FEMALE MARITAL STATUS (PERCENT)............................................................................................................................... .3-46 FIGURE 3-10: PERCENT OF BIRTHS TO UNWED MOTHERS....................................................................................................................3-46 FIGURE 3-11: PERCENT OF BIRTHS TO WOMEN AGE 19 OR LESS.........................................................................................................3-47 FIGURE 3-12: EDUCATIONAL ATTAINMENT IN YEARS (PERCENT OF POPULATION)....................................................................3-47 FIGURE 3-13: GRADUATION RATES............................................................................................................................... .................................3-48 FIGURE 3-14: HOUSEHOLD OCCUPANCY............................................................................................................................... .......................3-49 FIGURE 3-15: PROPORTION OF HOUSEHOLDS BY TYPE..........................................................................................................................3-50 FIGURE 3-16: PERCENTAGE OF INCARCERATED RESIDENTS................................................................................................................3-50 FIGURE 3-17: INCOME SHARES FOR THREE QUINTILES.........................................................................................................................3-51 FIGURE 3-18: FAMILY INCOME............................................................................................................................... ..........................................3-51 FIGURE 3-19: FAMILY INCOME, CUMULATIVE PERCENTAGE CHANGE.............................................................................................3-52 FIGURE 3-20: BLACK FAMILY INCOME AS A PERCENTAGE OF WHITE INCOME.............................................................................3-53 FIGURE 3-21: CONSUMER PRICE INDEX (1982 = 100)............................................................................................................................... 3-53 FIGURE 3-22: CPI PERCENT CHANGES............................................................................................................................... ...........................3-54 FIGURE 3-23: FOOD EXPENDITURE AS PERCENT OF INCOME, AND PROPORTION SPENT OUT................................................3-55 FIGURE 3-24: AFTER TAX INCOME SHARES............................................................................................................................... .................3-55 FIGURE 3-25: OUTPUT, CONSUMPTION, INVESTMENT AND GOVERNMENT...................................................................................3-57 FIGURE 3-26: MAJOR GDP COMPONENTS IN LOG SCALE.......................................................................................................................3-58 FIGURE 3-27: PER CAPITA INCOME AND CONSUMPTION.......................................................................................................................3-58 FIGURE 3-28: CONSUMPTION, INVESTME NT AND GOVERNMENT (% GDP).....................................................................................3-59 FIGURE 3-29: US AGRICULTURAL OUTP UT, 1982 CONSTANT DOLLARS............................................................................................3-60 FIGURE 3-30: AGRICULTURAL OUTPUT, TOTAL AND PER WORKER (1982 $, LOG SCALE)...........................................................3-60 FIGURE 3-31: MAJOR NON-AGRICULTURAL SECTOR S OF US ECONOMY, PERCENT OF GDP......................................................3-61 FIGURE 3-32: AIR TRAVEL PER CAPITA............................................................................................................................... ..........................3-62 FIGURE 3-33: ELECTRICITY PRODUCTION (M KWH)............................................................................................................................... .3-62 FIGURE 3-34: ENERGY USE (QUADRILLION BTUS)............................................................................................................................... ....3-63 FIGURE 3-35: CARS PER THOUSAND POPULATION AND MILES DRIVEN PER CAPITA..................................................................3-63 FIGURE 3-36: PERCENTAGE OF POPULATION EMPLOYED (MILITARY & PRISONERS EXCLUDED)..........................................3-64 FIGURE 3-37: LABOR FORCE PARTICIPATION RATES, ALL WOMEN AND MARRIED WOMEN....................................................3-64 FIGURE 3-38: DEFENSE AS A PERCENTAGE OF GDP............................................................................................................................... ..3-65 FIGURE 3-39: FEDERAL EXPENDITURES AND REVENUES (PERCENT OF GDP)...............................................................................3-66 FIGURE 3-40: FEDERAL, STATE & LOCAL AND TO TAL GOVERNMENT RECEIPTS (% GDP)..........................................................3-66 FIGURE 3-41: FEDERAL, REGIONAL AND TOTAL EXPENDITURES AS A PERCENT OF GDP..........................................................3-67 FIGURE 3-42: FEDERAL EXPENDITURES, ON AND OFF BUDGET, PERCENT OF GDP.....................................................................3-67 FIGURE 3-43: FEDERAL AND REGIONAL GOVERNMENT EMPLOYMENT (000S).............................................................................3-68 FIGURE 3-44: MAJOR EXPENDITURES OF THE FEDERAL GOVERNMENT........................................................................................3-68 FIGURE 3-45: MAJOR TRANSFER PAYMEN TS (% OF FEDERAL BUDGET)...........................................................................................3-69 FIGURE 3-46: SOCIAL SECURITY REVENUE AND EXPENDITURE, $ MILLIONS...............................................................................3-69 FIGURE 3-47: FEDERAL DEBT, TOTAL AND PERCENT OF GDP..............................................................................................................3-70 FIGURE 3-48: FEDERAL SPENDING ON R&D, AS A PERCENT OF GDP.................................................................................................3-72 FIGURE 3-49: SOURCES OF FEDERAL GOVERNMENT REVENUE..........................................................................................................3-73 FIGURE 3-50: TOTAL IMPORTS AND EXPORT S AS A PROPORTION OF GDP......................................................................................3-74

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McAfee: Introduction to Economic Analysis,, July 24, 2006 8-316FIGURE 3-51: US TRADE IN GOODS AND SERVICES............................................................................................................................... ...3-74 FIGURE 3-52: INCOME AND PAYMENTS AS A PERCENT OF GDP..........................................................................................................3-75 FIGURE 3-53: POSTWAR INDUSTRIAL PRODUCTION AND RECESSIONS...........................................................................................3-77 FIGURE 3-54: PERCENTAGE OF THE POPULATION EMPLOYED...........................................................................................................3-77 FIGURE 3-55: INDUSTRIAL FACTORY CAPACITY UTILITZATION (SOURCE: FRED)........................................................................3-78 FIGURE 4-1: COBB-DOUGLAS ISOQUANTS............................................................................................................................... ....................4-82 FIGURE 4-2: THE PRODUCTION FUNCTION............................................................................................................................... .................4-82 FIGURE 4-3: FIXED PROPORTIONS............................................................................................................................... .................................4-83 FIGURE 4-4: PERFECT SUBSTITUTES............................................................................................................................... .............................4-84 FIGURE 4-5: PROFIT-MAXIMIZING LABOR INPUT............................................................................................................................... .....4-86 FIGURE 4-6: TANGENCY AND ISOQUANTS............................................................................................................................... ...................4-94 FIGURE 4-7: SHORT-RUN SUPPLY............................................................................................................................... ...................................4-98 FIGURE 4-8: AVERAGE AND MARGINAL COSTS............................................................................................................................... ..........4-99 FIGURE 4-9: INCREASED PLANT SIZE............................................................................................................................... ..........................4100 FIGURE 4-10: LONG-RUN EQUILIBRIUM............................................................................................................................... ....................4105 FIGURE 4-11: A SHIFT IN DEMAND................................................................................................ .................................................................4106 FIGURE 4-12: RETURN TO LONG-RUN EQUILIBRIUM...........................................................................................................................4-10 6 FIGURE 4-13: A DECREASE IN DEMAND............................................................................................................................... ......................4107 FIGURE 4-14: A BIG DECREASE IN DEMAND............................................................................................................................... ..............4-108 FIGURE 4-15: A DE CREASE IN SUPPLY............................................................................................. .............................................................4109 FIGURE 4-16: EQUILIBRIUM WITH EXTERNAL SCALE ECONOMY....................................................................................................4110 FIGURE 4-17: DECREASE IN DEMAND............................................................................................................................... ..........................4111 FIGURE 4-18: LONG-RUN AFTER A DECREASE IN DEMAND................................................................................................................4111 FIGURE 4-19: DRAM MARKET............................................................................................................................... .........................................4113 FIGURE 4-20: DRAM REVENUE CYCLE............................................................................................................................... ........................4113 FIGURE 4-21: INVESTMENT STRIKE PRICE GIVEN INTEREST RATE R IN PERCENT....................................................................4123 FIGURE 4-22 INVESTMENT STRIKE PRICE GIVEN INTEREST RATE R IN PERCENT....................................................................4124 FIGURE 4-23: OPTIMAL SOLUTION FOR T............................................................................................................................... ..................4129 FIGURE 4-24: THE PORSCHE SPEEDSTER............................................................................................................................... ...................4130 FIGURE 4-25: $500 CONFEDERATE STATES BILL............................................................................................................................... .....4-134 FIGURE 4-26: WESTERN ELECTRIC MODEL 500 TELEPHONE............................................................................................................4135 FIGURE 4-27: PRICES OVER A CYCLE FOR SEASONAL COMMODITIES.............................................................................................4137 FIGURE 4-28: LOG OF PRICE OF GOLD OVER TIME............................................................................................................................... .4-138 FIGURE 5-1: BUDGET SET............................................................................................................................... .................................................5141 FIGURE 5-2: EFFECT OF AN INCREASE IN PRICE ON THE BUDGET..................................................................................................5142 FIGURE 5-3: AN INCREASE IN INCOME............................................................................................................................... .......................5142 FIGURE 5-4: UTILITY ISOQUANTS............................................................................................................................... .................................5144 FIGURE 5-5: CONVEX PREFERENCES............................................................................................................................... ...........................5145 FIGURE 5-6: GRAPHICAL UTILITY MAXIMIZATION............................................................................................................................... 5-147 FIGURE 5-7: “CONCAVE” PREFERENCES, PREFER BOUNDARIES......................................................................................................5148 FIGURE 5-8: ISOQUANTS FOR A BLISS POINT............................................................................................................................... ...........5-150 FIGURE 5-9: SUBSTITUTION WITH AN INCREASE IN PRICE...............................................................................................................5151 FIGURE 5-10: SUBSTITUTION EFFECT............................................................................................................................... .........................5152 FIGURE 5-11: ENGEL CURVE............................................................................................................................... ............................................5155 FIGURE 5-12: BACKWARD BENDING – INFERIOR GOOD......................................................................................................................5156 FIGURE 5-13: INCOME AND SUBSTITUTION EFFECTS...........................................................................................................................5-15 7 FIGURE 5-14: QUASILINEAR ISOQUANTS............................................................................................................................... ....................5159 FIGURE 5-15: HOURS PER WEEK............................................................................................................................... ....................................5163 FIGURE 5-16: HOUSE PRICE GRADIENT............................................................................................................................... ......................5166 FIGURE 5-17: BORROWING AND LENDING............................................................................................................................... .................5170 FIGURE 5-18: INTEREST RATE CHANGE............................................................................................................................... ......................5171 FIGURE 5-19: INTEREST RATE INCREASE ON LENDERS.......................................................................................................................5172 FIGURE 5-20: DIFFERENT RATES FOR BORROWING AND LENDING...............................................................................................5173 FIGURE 5-21: THE EFFECT OF A TRANSITORY INCOME INCREASE..................................................................................................5173 FIGURE 5-22: EXPECTED UTILITY AND CERTAINTY EQUIVALENTS.................................................................................................5175 FIGURE 5-23: THE EDGEWORTH BOX............................................................................................................................... ..........................5182 FIGURE 5-24: AN EFFICIENT POINT............................................................................................................................... ..............................5183 FIGURE 5-25: THE CONTRACT CURVE............................................................................................................................... .........................5184 FIGURE 5-26: CONTRACT CURVES WITH COBB-DOUGLAS UTILITY.................................................................................................5185 FIGURE 5-27: INDIVIDUALLY RATIONAL EFFICIENT POINTS.............................................................................................................5186 FIGURE 5-28: EQUILIBRIUM WITH A PRICE SYSTEM............................................................................................................................5-1 87 FIGURE 5-29: ILLUSTRATION OF PRICE SYSTEM EQUILIBRIUM.......................................................................................................5188 FIGURE 6-1: EFFECT OF A TAX ON SUPPLY............................................................................................................................... ................6196 FIGURE 6-2: EFFECT OF A TAX ON DEMAND............................................................................................................................... .............6-196 FIGURE 6-3: EFFECT OF A TAX ON EQUILIBRIUM............................................................................................................................... ...6-197 FIGURE 6-4: REVENUE AND DEAD WEIGHT LOSS............................................................................................................................... ...6-198 FIGURE 6-5: A PRICE FLOOR............................................................................................................................... ...........................................6204 FIGURE 6-6: DEAD WEIGHT LOSS OF A PRICE FLOOR..........................................................................................................................6-204 FIGURE 6-7: A PRICE CEILING............................................................................................................................... ........................................6206 FIGURE 6-8: RENT CONTROL, INITIAL EFFECT............................................................................................................................... ........6-207 FIGURE 6-9: RENT CONTROL, LONG-RUN EFFECT............................................................................................................................... ..6-208 FIGURE 6-10: PRICE SUPPORTS............................................................................................................................... ......................................6210

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McAfee: Introduction to Economic Analysis,, July 24, 2006 8-317FIGURE 6-11: A QUOTA............................................................................................................................... .......................................................6212 FIGURE 6-12: A NEGATIVE EXTERNALITY............................................................................................................................... ...................6214 FIGURE 6-13: EXTERNAL COSTS AND BENEFITS............................................................................................................................... .......6-215 FIGURE 6-14: THE PIGOUVIAN TAX............................................................................................................................... ...............................6217 FIGURE 6-15: SO2 PERMIT PRICES............................................................................................................................... ..................................6219 FIGURE 6-16: FISH POPULATION DYNAMICS............................................................................................................................... .............6-222 FIGURE 6-17: FISH POPULATION DYNAMICS WITH FISHING..............................................................................................................6223 FIGURE 6-18: FISH POPULATION DYNAMICS: EXTINCTION................................................................................................................6224 FIGURE 6-19: POSSIBILITY OF MULTIPLE EQUILIBRIA..........................................................................................................................6-225 FIGURE 6-20: BASIC MONOPOLY DIAGRAM............................................................................................................................... ...............6-234 FIGURE 6-21: TWO-PART PRICING............................................................................................................................... .................................6241 FIGURE 6-22: NATURAL MONOPOLY............................................................................................................................... ............................6242 FIGURE 7-1 SEQUENTIAL BANK LOCATION (NYC PAYOFF LISTED FIRST).......................................................................................7266 FIGURE 7-2: SUBGAME PERFECTION............................................................................................................................... ............................7267 FIGURE 7-3: CAN’T AVOID ROCKY............................................................................................................................... ..................................7267 FIGURE 7-4: EXPECTED PRICES IN SEARCH EQUILIBRIUM.................................................................................................................7277 FIGURE 7-5: EXPECTED PRICES (S=PROPORTION OF SHOPPERS).....................................................................................................7278 FIGURE 7-6: HOTELLING MODEL FOR BREAKFAST CEREALS.............................................................................................................7279 FIGURE 7-7: SHARING THE HOTELLING MARKET............................................................................................................................... ....7-280 FIGURE 7-8: A SEGMENT OF THE CIRCLE MODEL............................................................................................................................... ....7-281 FIGURE 7-9: NORMALLY DISTRIBUTED ESTIMATES..............................................................................................................................7 -301 8.2 Index A ABC, 7-312 ADM, 4-79, 7-307, 7-308, 7-310 Agency, 3-72, 5-177, 6-218, 7-283, 7-284, 7-286, 7287, 7-288, 7-291, 7-292, 7-293, 7-304 Akerlof, George, 6-245, 6-246 Alcoa, 7-307 Allocation, 1-1, 2-35, 2-37, 2-38, 5-182, 6-219, 7288 Ambiguity, 2-37, 6-211, 7-308 American Airlines, 6-239, 7-312 Analysis Normative, 1-2 Positive, 1-2 Antitrust, 1-2, 7-271, 7-306, 7-309, 7-311, 7-313, 7314 Clayton Act, 7-308 FTC Act, 7-308 Market Power, 6-232, 7-273 Monopoly Power, 6-232, 6-236, 7-273, 7-274, 7307, 7-313 Peckham Rule, 7-306 Predatory Pricing, 7-308, 7-309 Robinson Patman, 7-308 Sherman Act, 7-306, 7-307, 7-308 SSNIP, 7-312, 7-313 Tying, 7-286, 7-304, 7-308 Vertical Merger, 7-312 AOL, 7-312 Arbitrage, 4-114, 4-115, 4-126, 4-136, 6-238, 6-240 Arbitrage Condition, 4-126, 4-136 Attitude, 7-284 Auction, 1-1, 7-295, 7-298, 7-299, 7-300, 7-301, 7302, 7-303, 7-304, 7-305 Bid-Increment, 7-296, 7-299 Common Values, 7-295, 7-302, 7-303 English, 7-295, 7-296, 7-298, 7-299, 7-300, 7301, 7-304, 7-305 Linkage Principle, 7-303, 7-304, 7-305 Sealed-Bid, 7-296, 7-297, 7-298, 7-299, 7-300, 7-304, 7-305 Vickrey, 7-299, 7-300, 7-301 Autarky, 5-185, 5-186 Availability, 4-126 B Bargaining, 5-180, 5-186, 6-221, 6-229, 6-246 Basketball, 4-79, 5-164, 5-179 Beer, 2-9, 2-12, 5-140, 5-150 Benefit Concentrated, 6-210 Bliss Point, 5-150 Boeing, 4-101 Bosch, 7-309 Brands, 4-127, 7-311 British Petroleum, 1-2 Budget Line, 5-141, 5-143, 5-146, 5-147, 5-149, 5151, 5-152, 5-154, 5-187 Budget Set, 5-141, 5-146 C Certainty Equivalent, 1-4, 5-175, 5-176, 5-177 Ceteris Paribus, 1-5 Coase, Ronald, 6-220, 6-221 Cobb-Douglas Production Function, 4-82, 4-84, 485, 4-90, 4-91, 4-92, 4-93, 4-96, 4-104, 5-148, 5-149, 5-153, 5-156, 5-158, 5-162, 5-184, 5-185, 5-188, 5-189, 5-191, 5-194 Compact Disc, 5-143, 7-309 Comparative Advantage, 2-36, 2-37, 2-38, 2-39, 240 Comparative Statics, 1-7, 2-30, 4-87, 4-92 Compensating Differential, 5-164, 5-165

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McAfee: Introduction to Economic Analysis,, July 24, 2006 8-318Competition Imperfect, 7-270, 7-272, 7-273, 7-274, 7-313 Competitive Equilibrium, 6-217 Complement, 2-12, 2-15, 2-16, 2-17, 2-24, 2-25, 230, 4-83, 4-89, 4-90, 4-91, 5-149, 5-156, 5-162, 5-163, 5-164, 7-292 Concave, 2-33, 5-158, 5-159, 5-160, 5-170, 5-174, 5-175, 5-181, 5-185, 7-289, 7-290 Consols, 4-115 Consumer Surplus, 2-8, 2-10, 2-11, 2-13, 2-14, 2-19, 2-21, 2-30, 6-240, 6-241 Consumption, 2-8, 2-9, 2-10, 2-12, 3-56, 3-58, 3-59, 3-78, 4-126, 4-137, 5-139, 5-140, 5-143, 5-145, 5-150, 5-151, 5-152, 5-154, 5-155, 5-156, 5-160, 5-162, 5-163, 5-169, 5-170, 5-171, 5-172, 5-182, 5-189, 6-245 Contract Curve, 5-183, 5-184, 5-185, 5-186, 5-188 Contracts, 2-15, 2-38, 6-241, 7-286, 7-287, 7-306, 7-307 Convexity, 7-289, 7-290, 7-291 Cooperation, 7-269, 7-270, 7-287, 7-310, 7-311 Cartel, 2-26, 7-304, 7-306, 7-310, 7-311 Tacit Collusion, 7-310 Coordination, 4-81, 7-260, 7-263, 7-270 Corporate Finance, 4-121 Cost, 1-2, 1-3, 1-4, 1-5, 1-6, 1-7, 2-8, 2-9, 2-14, 2-15, 2-16, 2-17, 2-18, 2-22, 2-23, 2-26, 2-34, 2-35, 2-36, 2-37, 2-38, 3-53, 3-54, 3-56, 3-61, 3-63, 3-70, 4-80, 4-84, 4-85, 4-86, 4-93, 4-94, 4-95, 4-96, 4-97, 4-98, 4-99, 4-100, 4-101, 4-102, 4103, 4-104, 4-105, 4-107, 4-108, 4-109, 4-112, 4-114, 4-118, 4-120, 4-121, 4-123, 4-124, 4-126, 4-130, 4-131, 4-136, 5-143, 5-151, 5-165, 5-167, 5-172, 5-175, 5-178, 5-179, 5-180, 5-181, 6-199, 6-200, 6-201, 6-202, 6-204, 6-205, 6-206, 6207, 6-208, 6-211, 6-214, 6-215, 6-216, 6-217, 6-218, 6-220, 6-225, 6-226, 6-233, 6-234, 6235, 6-237, 6-238, 6-239, 6-241, 6-242, 6-243, 6-244, 6-246, 6-247, 6-248, 6-249, 6-250, 7-271, 7-272, 7-273, 7-277, 7-278, 7-279, 7-282, 7-283, 7-284, 7-286, 7-287, 7-288, 7-289, 7-292, 7293, 7-294, 7-303, 7-304, 7-308, 7-309 Fixed, 4-96, 4-112, 7-273, 7-274, 7-282 Long-Run, 4-96, 6-209 Marginal, 1-7, 2-14, 2-15, 2-16, 2-18, 2-23, 2-24, 2-37, 2-38, 4-95, 4-96, 4-97, 4-98, 4-99, 4100, 4-104, 6-198, 6-201, 6-205, 6-213, 6214, 6-226, 6-232, 6-234, 6-235, 6-236, 6237, 6-240, 6-241, 6-242, 6-243, 6-244, 7271, 7-272, 7-273, 7-275, 7-277, 7-281, 7-282 Opportunity, 1-3, 1-4, 2-33, 2-34, 2-36, 2-37, 6198 Short-Run, 4-95, 4-96 Variable, 4-96, 4-97, 4-98, 4-99, 4-104, 4-107 Cost-Benefit Analysis, 1-2 Cournot, 7-270, 7-272, 7-273, 7-274, 7-313 D Damages, 7-311 Dead Weight Loss, 6-198, 6-199, 6-202, 6-203, 6204, 6-205, 6-206, 6-211, 6-215, 6-235, 7-272 Demand Compensated, 5-152 Constant Elasticity, 2-29, 2-30, 4-130, 6-235, 6237 Elasticity, 2-27, 2-29, 2-30, 4-127, 4-135, 6-200, 6-208, 6-218, 6-224, 6-225, 6-235, 6-236, 6-237, 6-238, 6-239, 6-240, 7-272, 7-273 Market, 2-18, 2-19 Perfect Substitute, 4-83, 5-149 Depreciation, 4-131, 4-135, 4-136, 4-137, 4-138, 5167 Differentiation, 7-279 Horizontal, 7-279 Vertical, 7-279 Diminishing Marginal Returns, 2-33, 2-37 Diminishing Marginal Value, 2-9 Discounting, 4-123, 4-127, 4-128, 4-129, 4-131, 4133, 4-136, 4-138, 7-268, 7-269 Disney, 6-232, 7-279, 7-312 Distribution, 3-50, 4-122, 5-178, 5-181, 6-230, 6245, 6-246, 7-275, 7-276, 7-297, 7-298, 7-299, 7-300, 7-301, 7-302 Dominant strategy, 7-252, 7-253, 7-254, 7-265, 7268, 7-296, 7-299, 7-300 DRAM, 2-8, 2-23, 4-112, 4-113 DuPont, 4-81 Durable Good, 3-78, 4-130 Dynamic Optimization, 5-169, 5-172, 5-173 E eBay, 1-1, 2-18, 6-195, 7-295, 7-299 Economy of Scale, 4-100, 4-101, 4-102, 4-103, 4104, 4-109, 4-110, 4-112, 4-113, 4-114, 6-233, 7273 Economy of Scope, 4-101 Edgeworth Box, 5-181, 5-182, 5-185, 5-188 Education, 1-3, 2-28, 3-48, 4-120, 6-213, 6-214, 6218, 6-249, 6-250 Efficiency, 2-22, 2-26, 2-37, 4-93, 4-109, 4-130, 5182, 5-183, 5-185, 5-188, 6-195, 6-203, 6-212, 6-213, 6-216, 6-217, 6-218, 6-219, 6-220, 6221, 6-224, 6-225, 6-226, 6-227, 6-229, 6-230, 6-231, 6-233, 6-242, 6-246, 6-247, 6-248, 7273, 7-282 Elasticity, 2-29, 4-109, 4-110, 4-133, 4-135, 6-200, 6-203, 6-207, 6-208, 6-223, 6-224, 6-235, 6236, 6-237, 6-238, 6-239, 6-240, 7-272 Demand, 2-27, 2-29, 2-30, 4-127, 6-224, 6-235, 6-236, 7-272, 7-273 Supply, 2-30 Unitary, 2-29 Engel Curve, 5-155, 5-156, 5-157

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McAfee: Introduction to Economic Analysis,, July 24, 2006 8-319Entrepreneur, 2-17, 2-18, 4-85, 4-86, 4-87, 4-88, 4-89, 4-91, 4-92, 4-93, 4-97 Entry, 3-64, 4-102, 6-211, 6-241, 7-251, 7-252, 7253, 7-273, 7-274, 7-282, 7-309 Equilibrium, 2-8, 2-20, 2-21, 2-22, 2-30, 4-93, 4104, 4-105, 4-107, 4-108, 4-109, 4-110, 4-112, 4-114, 4-133, 5-167, 5-168, 5-169, 5-187, 5-188, 5-189, 5-190, 5-191, 5-192, 5-193, 5-194, 6-197, 6-199, 6-203, 6-205, 6-206, 6-207, 6-211, 6216, 6-224, 6-225, 6-227, 6-230, 7-253, 7-255, 7-256, 7-257, 7-258, 7-260, 7-261, 7-262, 7263, 7-264, 7-265, 7-268, 7-269, 7-270, 7-273, 7-274, 7-275, 7-276, 7-278, 7-280, 7-281, 7-282, 7-297, 7-298, 7-311 Exclusive Dealing, 7-308 Expenditure Shares, 5-149 Externality, 4-112, 6-213, 6-214, 6-215, 6-216, 6217, 6-218, 6-219, 6-220, 6-221, 6-224, 6-225, 6-226, 6-233, 7-278 Common Resource, 6-216 Commons, 4-127, 6-216, 6-226, 7-262 Network, 4-112, 6-233 Extinction, 6-216, 6-223, 6-224, 6-225, 6-226 Exxon-Mobil, 4-81, 7-311, 7-312 F Factor of Production, 2-40 Factor Price Equalization, 2-40 Factors of Production, 2-38, 2-39 FedEx, 6-232 Firm, 4-79, 4-97, 7-265, 7-268, 7-269, 7-270, 7292 Competitive, 4-97, 4-98, 4-104 Corporation, 1-5, 4-79, 4-80, 4-81, 7-283, 7-306 Non-Profit, 4-80 Partnership, 4-79, 4-80 Proprietorship, 4-79, 4-80 Fixed-Proportions, 4-83 Fluctuations, 3-77, 3-78, 7-288 Ford, 1-1 Foreclosure, 1-2, 7-312 Free Market, 6-195, 6-207 Free-Rider, 6-228, 6-229 G Gains from Trade, 2-10, 2-22, 2-37, 2-39, 4-131, 6198, 6-201, 6-202, 6-203, 6-205, 6-206, 6-221, 6-240, 6-241, 6-242, 6-246, 6-247, 6-248, 7272, 7-286 Game Theory, 7-251, 7-256 Battle of the Sexes, 7-256, 7-259, 7-267, 7-270 Common Knowledge, 7-270 Elimination of Dominated Strategies, 7-253, 7254, 7-255, 7-257 Folk Theorem, 7-269, 7-270 Grim Trigger Strategy, 7-268, 7-269, 7-270, 7310 Mixed Strategy, 7-258, 7-259, 7-260, 7-261, 7263, 7-264, 7-265, 7-275 Pure Strategy, 7-258, 7-259, 7-260, 7-261, 7262, 7-263, 7-264, 7-265, 7-275 Second-Mover Advantage, 7-267 Strategic Behavior, 7-251 Subgame Perfection, 7-266, 7-267, 7-268, 7-269, 7-270, 7-284 Gasoline, 1-2, 2-8, 2-11, 2-16, 2-24, 2-26, 4-81, 4108, 4-109, 5-178, 6-195, 6-218, 7-303 General Electric, 7-309, 7-311 General Equilibrium, 5-188, 5-189 Welfare Theorems, 5-188 General Motors, 4-81 Chevrolet, 2-28 Goldwyn, Samuel, 7-301 Government, 1-1, 1-2, 1-5, 3-48, 3-56, 3-58, 3-59, 3-65, 3-66, 3-67, 3-68, 3-69, 3-70, 3-71, 3-72, 4-80, 4-114, 4-115, 5-160, 5-171, 5-177, 6-195, 6-197, 6-198, 6-199, 6-200, 6-202, 6-210, 6211, 6-213, 6-218, 6-219, 6-220, 6-232, 6-237, 7-295, 7-296, 7-300, 7-303, 7-305, 7-311, 7-312, 7-314 Greyhound, 7-314 Gross Domestic Product, 3-56, 3-58, 3-59, 3-61, 365, 3-66, 3-67, 3-70, 3-71, 3-72, 3-73, 3-74, 375 H Holmstrom, Bengt, 7-292 Homo Economicus, 1-5 Homogeneity, 6-231, 7-289, 7-290, 7-291, 7-292 Homogeneous Function, 4-104, 7-289, 7-291, 7-294 Hotelling Model, 7-279, 7-280, 7-282 Hotelling, Harold, 7-279 Hypothesis, 5-152, 6-221, 7-257, 7-297 Hysteresis, 2-26 I IBM, 4-101 Identification, 6-238 Income, 1-3, 2-11, 2-13, 3-50, 3-51, 3-52, 3-53, 354, 3-55, 3-56, 3-58, 3-59, 3-64, 3-71, 3-72, 373, 4-105, 5-139, 5-143, 5-150, 5-152, 5-154, 5155, 5-156, 5-158, 5-160, 5-161, 5-163, 5-164, 5169, 5-171, 5-172, 5-173, 6-195, 6-203, 6-238 Income Effect, 5-152, 5-156, 5-158, 5-163, 5-171 Indifference Curve, 5-143, 5-144, 5-145, 5-146, 5147 Inferior Good, 2-11, 5-155, 5-156, 5-157 Information, 2-23, 3-56, 3-57, 4-101, 4-120, 6-245, 6-246, 7-251, 7-278, 7-295, 7-298, 7-299, 7300, 7-301, 7-302, 7-303, 7-304, 7-305, 7-311

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McAfee: Introduction to Economic Analysis,, July 24, 2006 8-320Intertemporal Budget Constraint, 5-169, 5-172, 5173 Inventory, 2-21, 4-136, 4-137 Isocost, 4-94 Isoquant, 4-82, 4-94, 5-149, 5-152, 5-182, 5-183, 5-185, 5-187, 5-188 M Managers, 4-80, 4-101, 7-310 Marginal Product, 4-84, 4-89, 4-103, 4-104 Value of, 4-84, 4-85, 4-86, 4-92, 4-103 Marginal Rate of Substitution, 5-144, 5-146, 5-148, 5-157 Marginal Rate of Technical Substitution, 4-94 Marginal Returns, Diminishing, 2-33, 2-37 Marginal Revenue, 6-234, 6-235 Marginal Value, Diminishing, 2-9 Market for Lemons, 6-246 Marketing, 4-81 Markets, 1-1, 1-2, 1-6, 2-21, 2-24, 2-25, 4-113, 4126, 6-195, 6-213, 6-246, 6-248, 7-274, 7-278, 7-295, 7-305, 7-311, 7-312, 7-313, 7-314 Maximum Sustainable Yield, 4-129, 4-130 McAfee, R. Preston, i, iii, 4-81, 5-175, 6-246 Mercedes, 7-309 Merger, 7-308, 7-309, 7-311, 7-312, 7-313, 7-314 Horizontal, 7-312 Microsoft, 1-2, 2-24, 6-232, 6-233, 7-253, 7-254, 7307, 7-309 Milgrom, Paul, 7-292, 7-303 Monopoly, 6-195, 6-201, 6-202, 6-232, 6-233, 6234, 6-235, 6-236, 6-237, 6-241, 6-244, 7-251, 7-270, 7-272, 7-273, 7-274, 7-277, 7-307, 7308, 7-310, 7-313, 7-314 Monopoly Power, 6-232, 6-236, 7-273, 7-274, 7307, 7-313 Morgenstern, Oskar, 7-251 Mortgage, 4-115, 4-116, 4-124, 4-125 Multi-Tasking, 7-288, 7-291 Myerson, Roger, 6-246, 6-248 Myerson-Satterthwaite Theorem, 6-246, 6-248 N Nash, John, 7-251, 7-255 Netscape, 1-2, 7-307 Network, 4-112, 6-233, 7-312 Newton, Sir Isaac, 6-221 Nonexcludability, 6-226 Nonrivalry, 6-226 Normal Good, 5-155, 5-156, 6-238 Northwest, 2-38, 2-40 NPV, 4-117, 4-118, 4-119, 4-120, 4-121, 4-122, 4123, 4-124, 4-125 Numeraire, 5-143, 5-190 O Ohlin, Bertil, 2-40 Oil, 1-2, 2-8, 2-16, 2-17, 2-25, 2-26, 3-60, 3-72, 481, 4-108, 4-118, 4-120, 4-121, 4-124, 4-125, 4126, 4-127, 4-135, 5-164, 6-216, 6-218, 7-295, 7-296, 7-301, 7-303, 7-304, 7-306, 7-311 Oligopoly, 7-270, 7-314 OPEC, 2-26 Opportunity Cost, 1-3, 1-4, 2-33, 2-34, 2-36, 2-37, 6-198 Option Value of Investment, 4-121 Options, 1-4, 4-121, 4-122, 4-124, 4-125 O'Rourke, P. J., 7-283 P Pareto Efficiency, 5-182, 5-183, 5-184, 5-188 Partnership, 4-79, 4-80 Patent, 4-85, 6-232, 6-246 Perron-Frobenius Theorem, 5-190, 5-191 Peter Principle, 4-101 Pigou, Arthur Cecil, 6-217 Porsche, 2-11, 4-130, 4-131, 6-232 Post Office, US, 6-232 Preferences, 1-2, 2-13, 4-125, 5-139, 5-140, 5-145, 5-147, 5-148, 5-165, 5-167, 5-168, 5-169, 5-177, 5-182, 5-185, 5-193, 5-194, 6-213, 6-229, 6230, 6-231, 6-245, 7-264, 7-270, 7-279 Present Value, 4-114, 4-115, 4-116, 4-117, 4-118, 4119, 4-120, 4-126, 4-127, 4-133, 4-136, 5-167, 5169, 5-170, 5-172, 6-249, 7-269, 7-310 Price Ceiling, 6-203, 6-205, 6-206, 6-207 Hedonic, 1-4 Law of One, 7-274, 7-278 Peak Load, 6-243, 6-244 Reservation, 2-13, 5-178, 5-179, 5-180, 5-181 Price – Cost Margin, 7-272, 7-273, 7-313 Price Discrimination, 6-237, 6-238, 6-239, 6-240, 7-275, 7-308, 7-314 Coupons, 6-239 Direct, 6-239 Indirect, 6-239 Quantity Discount, 6-239 Price Dispersion, 7-275 Price Floor, 6-202, 6-203, 6-205, 6-206, 6-207, 6210, 6-211 Price Support, 6-210, 6-211 Price System, 2-37, 2-38, 5-186, 5-187, 5-188, 6238, 6-245 Price-Cost Margin, 6-235, 7-271 Pricing Ramsey, 4-125, 4-126, 6-244 Private Values, 7-295, 7-296, 7-297, 7-298, 7-299, 7-300, 7-301, 7-304 Production Possibilities Frontier, 2-32, 2-33, 2-34, 2-35, 2-36, 2-37, 2-38 Property Rights, 6-220, 6-221

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McAfee: Introduction to Economic Analysis,, July 24, 2006 8-321Proprietorship, 4-79, 4-80 Public Good, 6-226, 6-228, 6-229, 6-230, 6-231, 7262 Q Quintiles, 3-50, 3-51 R Raising Rivals’ Cost, 7-312 Ramsey Pricing, 4-125, 6-244 Ramsey, Frank, 4-125 RE/MAX, 7-287 Reagan, Ronald, 3-65 Regulation Price-Cap, 6-242 Rent Control, 6-207, 6-208, 6-209 Rents, 4-85, 6-208 Research and Development, 3-72, 4-101, 4-102, 7282 Reservation Price, 2-13, 5-178, 5-179, 5-180, 5-181 Revealed Preference, 4-87, 4-88, 4-89, 4-90, 6-237 Revenue equivalence, 7-298 Risk, 1-4, 2-8, 4-85, 4-101, 4-114, 4-118, 4-119, 4120, 4-121, 4-125, 4-138, 5-174, 5-175, 5-176, 5177, 5-181, 6-211, 7-283, 7-284, 7-286, 7-287, 7288, 7-299, 7-303, 7-304, 7-305, 7-310 Certainty Equivalent, 1-4, 5-175, 5-176, 5-177 Risk Management, 1-4, 4-101 Risk Aversion, 5-176, 5-177, 7-287, 7-299, 7-304 Risk Premium, 1-4, 5-175, 5-176, 5-177, 5-178 Rivalry, 4-81 Roosevelt, Teddy, 7-308 S S&P 500, 5-177 Samuelson, Paul, 2-40 Satiation, 5-150 Satterthwaite, Mark, 6-246 Saving, 2-26, 4-115, 4-127 Scale Economy, 4-102, 6-233 Schelling, Thomas, 7-251 Schwarzenegger, Arnold, 5-139 Sears, 4-81 Second-Price Auction, 7-299 Self-Interest, 1-5, 1-6 Selten, Reinhart, 7-266 Service, 2-20, 2-37, 3-63, 3-68, 4-85, 5-144, 6202, 6-203, 6-231, 6-233, 6-242, 7-253, 7-308, 7-312 Shadow Value, 4-91, 4-92 Sharecropper, 7-287 Shortage, 2-8, 2-20, 2-21, 4-126, 5-164, 6-206, 6207, 6-208, 6-209 Signaling, 6-248, 6-249, 6-250 Smith, Adam, 2-35, 2-36 Sony, 1-1, 5-178, 6-233 Spence, Michael, 6-248 Standard Oil, 4-81, 7-306, 7-309 Standardization, 4-79, 6-233, 6-250, 7-274 Statistics, 3-41, 3-57, 3-76, 6-250 Correlation, 5-193, 5-194 Normal Distribution, 5-177, 7-301, 7-302 Steady State, 2-20, 6-222, 6-223, 6-224, 6-225, 7256 Strategy, 4-97, 4-121, 4-122, 4-123, 4-133, 5-160, 6-206, 6-218, 6-220, 6-226, 6-242, 7-251, 7252, 7-253, 7-254, 7-255, 7-256, 7-257, 7-258, 7-259, 7-260, 7-261, 7-263, 7-268, 7-269, 7270, 7-296, 7-298, 7-299, 7-310 Substitute, 2-11, 2-12, 2-17, 2-24, 2-25, 4-89, 4110, 6-203, 6-208, 6-238, 7-272, 7-294, 7-307, 7-313 Supergames, 7-268, 7-269 Supply, 1-6, 2-8, 2-13, 2-14, 2-15, 2-16, 2-17, 2-18, 2-19, 2-20, 2-21, 2-22, 2-23, 2-24, 2-25, 2-26, 2-30, 2-31, 2-32, 2-37, 3-61, 4-103, 4-104, 4105, 4-107, 4-108, 4-109, 4-110, 4-112, 4-113, 4114, 4-126, 4-129, 4-130, 4-131, 4-137, 5-139, 5143, 5-160, 5-162, 5-163, 5-165, 5-170, 5-187, 5188, 5-189, 5-190, 5-193, 6-195, 6-196, 6-197, 6198, 6-199, 6-200, 6-201, 6-202, 6-203, 6-205, 6-207, 6-208, 6-209, 6-211, 6-216, 6-217, 6219, 6-235, 6-237, 7-292, 7-304 Complement, 2-16, 2-17, 2-25 Constant Elasticity, 2-30 Elasticity, 2-30, 6-200 Increase in, 2-16, 2-23, 2-25 Market, 2-19, 4-104 Substitute, 2-17, 2-25 Surplus, 2-8, 2-10, 2-13, 2-20, 2-21, 3-73, 5-164, 6198, 6-203, 6-210, 6-211, 6-241 T Tax Ad Valorem, 6-195, 6-201 Excess Burden of, 6-200, 6-202 Excise, 3-72, 3-73, 6-195 Income, 3-72, 3-73, 6-195 Sales, 1-3, 3-73, 4-80, 6-195 Time preference, 5-169, 5-170 Trailways, 7-314 Transaction Costs, 4-114, 6-195 Truman, Harry, 1-5 Tyranny of the Majority, 6-209 U U. S. Federal Trade Commission, 7-306, 7-308, 7309, 7-311, 7-312 U.S. Department of Justice, 1-2, 7-306, 7-307, 7309, 7-310, 7-311, 7-312 Uncertainty, 2-26, 4-120, 4-121, 4-136, 7-305 Unitization, 6-216

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McAfee: Introduction to Economic Analysis,, July 24, 2006 8-322Utility, 5-139, 5-140, 5-143, 5-144, 5-146, 5-147, 5148, 5-149, 5-150, 5-151, 5-152, 5-153, 5-154, 5158, 5-159, 5-160, 5-162, 5-167, 5-168, 5-170, 5174, 5-175, 5-176, 5-181, 5-182, 5-183, 5-184, 5185, 5-186, 5-187, 5-188, 5-189, 5-191, 5-194, 6228, 6-229, 6-247, 7-284, 7-286, 7-287, 7-290, 7-293 Compensated Demand, 5-152 Convex Preferences, 5-145, 5-147, 5-148 Indifference Curve, 5-143, 5-144, 5-145, 5-146, 5147 Quasilinear, 5-159 Satiation, 5-150 von Neumann-Morgenstern, 5-174 V Value Marginal, 1-7, 2-9, 2-10, 2-9, 2-11, 2-14, 2-15, 489, 4-131, 4-132, 4-133, 5-162, 5-167, 5-170, 6-196, 6-201, 6-214, 6-218, 6-242 Vickrey, William, 7-299 Video Home System, 2-23, 6-233 Vincent, Daniel, 5-175 von Neumann, John, 7-251 Voting Median Voter, 6-230 W Warranties, 6-246 Welfare Analysis, 1-2 Western Electric, 4-134, 4-135 Willingness To Pay, 1-2, 2-8, 2-11, 2-13, 6-197, 6238, 6-239, 7-295