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ESSAYS ON ENDOGENOUS TECHNOLOGICAL PROGRESS By IORDANIS PETSAS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2002 Copyright 2002 by Iordanis Petsas This dissertation is dedicated to my mother, Stavroula. Without her love and support I would not have started my studies. During my graduate studies, while seriously ill, she always supported me in an extraordinary way and encouraged me to finish my dissertation. I dedicate this work to her with all my love. ACKNOWLEDGMENTS My biggest heartfelt thanks go to my mother, Stavroula, who provided me with continuous encouragement. To my deepest sorrow, my mother passed away while I was working on this dissertation. 1 am extremely sorry that she can not have the chance to see my achievement. In writing this dissertation, I have benefited from the guidance, encouragement, and knowledge of many people and I offer them my thanks. I would like to thank my parents Stavroula and Nikos Petsas and the rest of my family, Panagiotis and Maria, for supporting my studying abroad and patiently waiting during the long spells of separation. I am deeply indebted and grateful to Elias Dinopoulos for his guidance and encouragement at every stage of the dissertation. Elias has contributed immensely to my development as an individual, teacher, and researcher. I am very grateful for the role that he has played so well as my great motivator and advisor. I also wish to recognize David Figlio, Douglas Waldo, and Steven Slutsky for their helpful comments and advice. Their keen eyes for detail and their astute comments always came at the right time. I would also like to thank Panos Hatzipanayotou, my former undergraduate teacher and advisor at the Aristotle University of Thessaloniki, who made me realize the excitement of macroeconomics. He assured me that it is truly worthwhile to pursue a Ph.D. in economics. I am also grateful to Manolis Loukakis, George Papachristou, Nikos Varsakelis, and Chrisoula Zacharopoulou for their dedicated teaching. My life in Gainesville would have been miserable had I not had wonderful friends. I greatly acknowledge the friendship of Huseyin Yildirim and HsiuChuan Yeh. I also thank Sofia Vidalis for her support and for sharing my path despite her own challenges as a Ph.D. student. This dissertation is dedicated to my mother, who has been the best teacher of my life. TABLE OF CONTENTS page ACKNOW LEDGM ENTS ............................................................... ...................... iv A B ST R A C T ................................................................... ............. ............................ x CHAPTER I IN T R O D U C T IO N ........................................................................ .......................... 1.1 Process versus Product Innovation and Firm Size......................... ............ 1 1.2 General Purpose Technologies and Schumpeterian Growth ................................. 3 1.3 Sustained Comparative Advantage and Relative Wages....................................... 6 2 PROCESS VERSUS PRODUCT INNOVATION IN MULTIPRODUCT FIRMS.......9 2.1 Introduction........................................................................................................ 9 2.2 A Model of Process versus Product Innovation................... .................... .. 13 2.2.1 Demand for Differentiated Products...................... ...........................13 2.2.2 The Duopoly M odel....................................................... ............................. 15 2.3 Model of Product versus Process Innovation Considering the R&D Costs .......... 19 2.4 Implications and Extensions ............................... ... ....................... 24 3 THE DYNAMIC EFFECTS OF GENERAL PURPOSE TECHNOLOGIES ON SCHUMPETERIAN GROWTH............................. .... ................ .....30 3.1 Introduction ...................................................... ............................................... 30 3 .2 T he M odel ............................................................................... .............. .......... 39 3.2.1 Industry Structure.......................... ........ .... .....................39 3.2.2 Diffusion of a New GPT...............................................40 3.2.3 H ouseholds............................................ .......................................... 42 3.2.4 Product M markets ...................................................... ....................... 44 3.2.5 R & D R aces ............................................................... ....................... 45 3.2.6 Labor M market ........................................................... ........................ 49 3.3 LongRun Equilibrium .............................. ......................................................... 49 3.4 Transitional Dynamics.................... ................................... 54 3.4.1 Stock Market Behavior ................... ...................... 57 3.4.2 Aggregate Investment ...................... .... ....................... 61 3.5 Concluding Rem arks....................................................................... ............. 62 4 SUSTAINED COMPARATIVE ADVANTAGE IN A MODEL OF SCHUMPETERIAN GROWTH WITHOUT SCALE EFFECTS..............................70 4.1 Introduction ...................................................... ............................................... 70 4.2 T he M odel ............................. ..... ......................................................... 78 4.2.1 Household Behavior........................... ........................... 79 4.2.2 Product M markets .................... ................................... 81 4.2.3 R & D R aces ............................................... ....................................... 83 4.2.4 Labor M arkets............................................................. ..................... 89 4.3 SteadyState Equilibrium Under the PEG Specification .................................. 90 4.4 Comparative SteadyState Analysis Under the PEG Specification.................... 94 4.5 SteadyState Equilibrium Under the TEG Specification.................................. 99 4.6 Comparative SteadyState Analysis Under the TEG Specification.................. 102 4.7 Conclusions............................. .. .............. ...................................... 106 5 C O N C LU SIO N .................................................................... ..................................122 APPENDIX A PROOFS OF PROPOSITIONS IN CHAPTER 2...........................................127 B SUMMARY OF THE HISTORICAL EVOLUTION OF SEMICONDUCTOR TECHNOLOGY AND PROOFS OF PROPOSITIONS IN CHAPTER 3 ...............134 C PROOFS OF PROPOSITIONS IN CHAPTER 4............................................. 149 R EFE R EN C E S ................................................................................ ....................164 BIOGRA PHICAL SKETCH ........................................................ ....................... 169 LIST OF FIGURES Figure page 21 Marginal Profits from Process and Product Innovation Considering R&D Costs as Sunk.................... ... .................... ..........................26 22 Net Marginal Profits from Process and from Product Innovations ..................27 23 Effect of an Increase in R&D Cost from Product Innovation on n*k...................28 24 Effect of an Increase in R&D Cost from Process Innovation on n*k.................29 31 SteadyState Equilibria................. .... ........... .................. 64 32 Stability of the BalancedGrowth Equilibrium..........................................65 33 Time Path of the Per Capita Consumption Expenditure After a GPT Arrives in the Economy ....... ........... .... .............. .... ... .......................66 34 Time Path of the Market Interest Rate After a GPT Arrives in the Economy........66 35 Evolution of the Aggregate Investment During the Diffusion Path.....................67 36 Effects of a GPT on the Schumpeterian Growth Rate ..................................68 37 Effects of a GPT on the Stock M market ........... ..... .......... ... .................. 69 41 SteadyState Equilibrium Under the PEG Specification............................... 11 42 Ranking of the Global R&D Investment, I, Between Home and Foreign, Under the PEG Specification.................................. ...............112 43 Effects of an Increase in the Foreign County's Relative Size or in the Population Growth Rate (Under the PEG Specification)..............................113 44 Effects of an Increase in the R&D Difficulty Parameter, k, in the Unit R&D Labor Requirement or in the Consumer's Subjective Discount Rate (Under the PEG Specification).........................................114 45 Effects of an Increase in the Foreign's Unit Labor Requirement in Manufacturing (Under the PEG Specification) .................................... 115 46 Effects of an Increase in Home's Unit Labor Requirement in Manufacturing (Under the PEG Specification)........................................116 47 SteadyState Equilibrium Under the TEG Specification............................117 48 Effects of an Increase in the Foreign County's Relative Size in the Consumer's Subjective Discount Rate or in the R&D Difficulty Growth Parameter (Under the TEG Specification).................................. 118 49 Effects of an Increase in the Size of Innovations or in the Population Growth Rate (Under the TEG Specification)........................................119 410 Effects of an Increase in Foreign's Unit Labor Requirement in Manufacturing (Under the TEG Specification)................ ..... ...................120 411 Effects of an Increase in Home's Unit Labor Requirement in Manufacturing (Under the TEG Specification)......... ................................ ................121 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ESSAYS ON ENDOGENOUS TECHNOLOGICAL PROGRESS By lordanis Petsas August 2002 Chair: Elias Dinopoulos Major Department: Economics This dissertation analyzes several issues involving the economics of endogenous technological progress: * The effects of firm size on the choice of R&D effort between process and product innovation * The transitional and longrun dynamic effects of generalpurpose technologies on Schumpeterian growth in a closed economy setting * The interactions among endogenous technological change, the pattern of trade, and income distribution within each country Chapter 2 develops a differentiatedgoods duopoly model in which firms engage in CoumotNash quantity competition. I showed that an increase in a firm's efforts devoted to product innovation (given that it is in the product R&D regime) increases its incentives to switch from product to process innovation. Once the firm is in the process R&D regime, it will perform process R&D indefinitely. Chapter 3 builds a qualityladders model of scaleinvariant Schumpeterian growth. The introduction of a new generalpurpose technology increases Schumpeterian growth and percapita R&D investment in the final steadystate. During the transition path from the initial to the final steadystate equilibrium, the measure of industries that adopt the new generalpurpose technology increases, consumption per capital falls, and the interest rate increases. The growth rate of the stock market depends negatively on the rate of the generalpurpose technology diffusion process and the magnitude of the generalpurpose technologyridden R&D productivity gains; and positively on the rate of population growth. In addition, the model generates transitional growth cycles of percapita GNP. Chapter 4 constructs a twocountry (Home and Foreign) general equilibrium model of Schumpeterian growth without scale effects. The scale effects property is removed by introducing two distinct specifications in the knowledge production function: the permanent effect on growth (PEG) specification, which allows policy effects on longrun growth; and the temporary effects on growth (TEG) specification, which generates exogenous longrun economic growth. Under the PEG specification, an increase in the rate of population growth raises Home's relative wage and decreases its range of goods exported to Foreign. Under the TEG specification, an increase in the rate of population growth rate lowers Home's relative wage and increases its range of goods exported to Foreign. CHAPTER 1 INTRODUCTION Economic historians have placed great weight on technology as a force of change. The importance of technological progress has been increased dramatically in the last decade. Many macroeconomic studies place technological progress at the center of the growth process. This trend has been triggered by theoretical developments that allow microeconomic aspects of the innovation process to be linked with macroeconomic outcomes. The search for causes and effects of technological progress has dominated the recent literature in growth and international trade. In this dissertation, I consider these causes and effects in the context of dynamic growth models. Chapter 2 provides an analysis of two types of technological change, (process versus product innovations) and examines the effects of the firm size on the choice of R&D effort between process and product innovation. Chapter 3 analyzes the impacts of a certain type of drastic innovations, termed generalpurpose technologies (GPTs) on Schumpeterian growth. Chapter 4 investigates interactions among endogenous technological change, the pattern of trade, and income distribution within each country. Chapter 5 presents general conclusions. The rest of Chapter 1 briefly describes the motivation, scope, and main findings for each chapter of the dissertation. 1.1 Process versus Product Innovation and Firm Size Numerous studies have tested the hypothesis about the advantages of size for innovative activity. Kamien and Schwartz (1972), Baldwin and Scott (1987) and Scherer (1980) reviewed the findings concerning the relationship within industries between firm size and R&D effort. Cohen and Klepper (1996) summarized these findings in the form of stylized facts that characterize this relationship within industries between firm R&D effort and firm size. Some of their key findings are as follows: * Among R&D performing firms, the number of patents and innovations per dollar of R&D decreases with firm size or level of R&D. * Among all firms, smaller firms account for a disproportionately large number of patents and innovations relative to their size. Scherer (1991) found that among manufacturing business units considered as a whole, process R&D increases relative to product R&D as the size of the firm increases, with each tenfold increase in business unit sales associated with a highly significant tenpoint increase in the percentage of R&D expenditures devoted to process innovation. Cohen and Klepper (1996) proposed and tested a theory of how firm size conditions the relative amount of process and product innovation undertaken by firms. using patent data. They found that the share of process R&D undertaken by firms indeed rises with firm size within most industries. The motivation behind Chapter 2 was to examine the findings mentioned in the studies above using a differentiatedgoods duopoly model in which firms engage in CournotNash quantity competition. Firms optimally choose to engage either in product or in process innovation. The analysis generates several insights. An increase in the number of goods produced by a firm and thus an increase in its size causes the firm to perform process R&D regardless of the R&D regime in which it was located originally. Given that the firm starts with product R&D, as the number of goods produced increases, the firm's incentives to switch from product to process innovation increase. Once the firm is in the process R&D regime, it continues to perform process R&D indefinitely. This constitutes then a costbased mechanism that links firm size to the type of R&D activity the firm develops, which is not clear in the models mentioned earlier. 1.2 General Purpose Technologies and Schumpeterian Growth The unusual combination of more rapid growth and slower inflation in the 1990s initiated a strenuous debate among economists about whether improvements in America's economic performance can be sustained. Many believe that the behavior of information technology provides the key to the recent surge in economic growth. Jorgenson (2001) provides plenty of evidence that the rise in structural productivity growth in the late 1990s in the United States can be traced to the introduction of personal computers and semiconductors, which constitutes the necessary building blocks for the information technology revolution. His estimates indicate that the importance of information technology has steadily increased in the United States over the last 50 years. He estimates that the contribution of information technology to the average growth rate of 3.46% for the last 50 years is 0.4%. The informationtechnology products contribute 0.5 percentage points to total factor productivity growth for 1995 to 1999, compared to 0.25 percentage points for 1990 to 1995. This reflects the accelerating decline in relative price changes resulting from shortening the product cycle for semiconductors. Information technology is considered a drastic innovation. In recent years, the literature on these types of drastic innovations has been initiated by Bresnahan and Trajtenberg (1995), who coined the term, "General Purpose Technologies" (GPTs henceforth). A drastic innovation qualifies as a GPT if it has the potential for pervasive use in a wide range of sectors in ways that drastically change their modes of operation. To quote from Bresnahan and Trajtenberg (1995): [Most GPTs play the role of "enabling technologies," opening up new opportunities rather than offering complete, final solutions. For example, the productivity gains associated with the introduction of electric motors in manufacturing were not limited to a reduction in energy costs. The new energy sources fostered the more efficient design of factories taking advantage of the newfound flexibility of electric power. Similarly, the users of microelectronics benefit from the surging power of silicon by wrapping around the integrated circuits their own technical advances. This phenomenon involves what we call innovationall complementarities,"....These complementarities magnify the effects of innovation in the GPT, and help propagate them throughout the economy.] Bresnahan and Trajtenberg (1995, p. 84). Even though the distinction between a drastic technological innovation and an incremental one is needed to understand the proper roles of technological innovations as engines of growth, economists have paid relatively less attention to the former. Several empirical studies have documented the crossindustry pattern of diffusion for a number of GPTs (For example, Helpman and Trajtenberg, 1998b). In addition, a strand of empirical literature has established that the crossindustry diffusion pattern of GPTs is similar to the diffusion process of productspecific innovations and it is governed by standard Scurve dynamics (Griliches, 1957; Jovanovic and Rousseau, 2001). These studies imply that the internalinfluence epidemic model can provide an empirically relevant framework to analyze the dynamic effects of a GPT. During this diffusion process, these drastic innovations could generate growth fluctuations and even business cycles (Schumpeter, 1939, 1950). Building an analytical model that incorporates all these key characteristics of GPTs and examining their dynamic effects on Schumpeterian growth constitutes one motivation of Chapter 3. The other motivation is to formalize relationships among GPTs, stock market, and R&D investment that were empirically investigated by Jovanovic and Rousseau (2001) using 114 years of U.S. stock market data. Therefore, I incorporate the presence of a GPT into the standard qualityladder framework of Schumpeterian growth without scale effects that was developed by Dinopoulos and Segerstrom (1999), and explore the impacts of GPTs on Schumpeterian growth, the stock market, and R&D investment. It is a dynamic model of endogenous growth with endogenous determination of innovation. Firms engage in R&D to discover higherquality products. In the presence of a GPT, the model has two steadystate equilibria: the initial steadystate in which no industry has adopted the new GPT, and the final steadystate in which all industries have adopted the new GPT and its diffusion process has been completed. At the final steadystate relative to the initial steadystate: the longrun growth rate is higher even though the per capital consumption expenditure is lower, due to higher aggregate investment. There are two additional findings of the model. First, there exists a unique globally stablesaddlepath along which the measure of industries that adopt the new GPT increases, the per capital consumption expenditure decreases, the market interest rate increases, and the innovation rate of the industries that have adopted the new GPT decreases at a higher rate than that of those that have not adopted the new GPT. Second, the model exhibits transitional growth cycles of per capital GNP as a consequence of Scurve dynamics. The introduction of positive population growth in Aghion and Howitt's (1998b) model of GPTs will make these growth cycles shorter and shorter as the size of the economy increases, and in the longrun the GPTinduced cycles disappear. In the model developed in Chapter 3, the fall in output comes from the reduction in per capital consumption expenditure on final goods and the rise in the per capital R&D investment. As the size of the economy increases (as a result of positive population growth) the duration of the per capital GNP cycle remains the same. When all industries have adopted the new GPT and the diffusion process has been completed, the economy experiences a higher per capital income constant growth rate. In addition, the growth rate of the stock market depends negatively on the rate of GPT diffusion process and the magnitude of the GPTridden R&D productivity gains, and positively on the rate of population growth. Finally, the effect of the GPT diffusion on the aggregate investment during the adoption process is ambiguous. In the initial stages of the diffusion process, only a limited number of industries adopt the new GPT. These industries are called the early adopters. As more industries adopt the new GPT, the aggregate investment increases. 1.3 Sustained Comparative Advantage and Relative Wages Many models of endogenous growth and trade emphasize the role of continual product innovation based on R&D investment in determining the pattern of trade between countries (Grossman and Helpman, (1991a, b, c,), Taylor (1993)). All these studies exhibit the scale effect property: if one incorporates population growth in these models, then the size of the economy (scale) increases exponentially over time, R&D resources grow exponentially, and so does the longrun growth rate of percapita real output. The scale effects property is a consequence of the assumption that the growth rate of knowledge is directly proportional to the level of resources devoted to R&D. Jones (1995a) argued that the scale effects property of earlier endogenous growth models is inconsistent with postwar time series evidence from all major advanced countries that shows an exponential increase in R&D resources and a moreorless constant rate of percapita GDP growth. Building a qualityladders model of scaleinvariant Schumpeterian growth and examining interactions among endogenous technological change, the pattern of trade, and income distribution within each country constitutes the motivation of Chapter 4. It constructs a twocountry (Home and Foreign) general equilibrium model. The scale effects property is removed by introducing two distinct specifications in the knowledge production function: The permanent effects on growth (PEG) specification which allows policy effects on longrun growth; and the temporary effects on growth (TEG) specification, which generates exogenous longrun economic growth. My approach borrows from Taylor's work (1993) in that industries differ in production technologies. In his model, industries also differ in research technologies and in the set of technological opportunities available for each industry. The absence of heterogeneity in research technologies in my model makes the removal of scale effects more tractable, but eliminates the need for trade in R&D services between countries. The main point of Chapter 4 is that the removal of scale effects affects the comparative static properties of the model. Furthermore, the various comparative steadystate results depend on the way that the scale effects property is removed. Under the PEG specification, changes in the size of innovations do not affect Home's comparative advantage and its relative wage, while under the TEG specification, an increase in the size of innovations reduces Home's relative wage and raises its comparative advantage. Under the PEG specification, changes in the size of innovations do not affect relative wages and comparative advantage, while under the TEG specification, an 8 increase in the size of innovations reduces Home's relative wage and raises its comparative advantage. In addition, under the PEG specification, an increase in the rate of population growth raises Home's relative wage and decreases its range of goods exported to Foreign. On the other hand, under the TEG specification, an increase in the rate of population growth rate lowers Home's relative wage and increases its range of goods exported to Foreign. CHAPTER 2 PROCESS VERSUS PRODUCT INNOVATION IN MULTIPRODUCT FIRMS 2.1 Introduction There is heterogeneity among firms in the degree of product and process innovation in which they engage. The percentage of total R&D dedicated to different types of innovative activity differs greatly across industries. For example, in petroleum refining, almost threequarters of total R&D is dedicated to process innovation, whereas less than onequarter of pharmaceutical R&D is dedicated to process innovation. A large part of such differences is due to differences in exogenous industrylevel conditions that systematically differentiate the returns to one sort of innovative activity versus another. Link (1982), for example, finds that greater product complexity increases the fraction of effort dedicated to process innovation. There is an extensive empirical literature in industrial organization, investigating at the industry level the relationship between firm size and the composition of R&D effort, and hence, the nature of innovation. However, inadequate attention seems to have been paid to modeling these two types of R&D activities and trying to support the empirical papers. The findings of Link (1982) all suggest that within industries, firm size, and, thus, across industries, market structure, may also influence the composition of R&D. Scherer (1991) finds that among manufacturing business units considered as a whole, process R&D increases relative to product R&D as the size of the firm increases, with each tenfold increase in business unit sales associated with a highly significant ten point increase in the percentage of R&D expenditures devoted to process innovation. Cohen and Klepper (1996) proposed and tested a theory of how firm size conditions the relative amount of process and product innovation undertaken by firms. Their theory explains the close and often proportional relationship within industries between firm size and innovative activity. Their model generates predictions about the relationship between firm size and the share of process R&D. They also tested these predictions using patent data that distinguish between process and product innovation and business unit sales data from the Federal Trade Commission's Line of Business Program. They found that the share of process R&D undertaken by firms indeed rises with firm size within most industries. One critical question arises about how technologically progressive industries evolve from birth through maturity. When industries are new, there is a lot of entry, firms offer many different versions of the industry's product, the rate of product innovation is high, and market shares change rapidly. Despite continued market growth, subsequently entry slows, exit overtakes entry and there is a shakeout in the number of producers, the rate of product innovation and the diversity of competing versions of the product decline, increasing effort is devoted to improving the production process, and market shares stabilize. This evolutionary pattern has come to be known as the product life cycle (PLC). While numerous papers have contributed to this description, perhaps the most influential one has been that of Abemathy and Utterback's (1978). In their paper, they stress that when a product is introduced, there is considerable uncertainty about user preferences and the technological means of satisfying them. As a result, many firms producing different variants of the product enter the market and competition focuses on product innovation. As users experiment with the alternative versions of the product and producers learn about how to improve the product, opportunities to improve the product are depleted and a defacto product standard, dubbed a dominant design, emerges. Producers who are unable to produce efficiently the dominant design exit, contributing to a shakeout in the number of producers. The depletion of opportunities to improve the product coupled with lockedin of the dominant design leads to a decrease in product innovation. This in turn reduces producers' fears that investments in the production process will be rendered obsolete by technological change in the product. Consequently, they increase their attention to the production process and invest more in capitalintensive methods of production, which reinforces the shakeout of producers by increasing the minimum efficient size firm. Klepper (1996) summarized regularities concerning how entry, exit, market structure, and innovation vary from the birth of technologically progressive industries through maturity. He developed a model emphasizing differences in firm innovative capabilities and the importance of firm size in appropriating the returns from innovation. His model predicts that over time firms devote more effort to process innovation but the number of firms and the rate and diversity of product innovation eventually wither. One of the goals in this chapter is to examine these findings (within industries the fraction of total R&D a firm devotes to process R&D will be an increasing function of the firm's size) using a differentiatedgoods duopoly model in which firms engage in CourotNash quantity competition (in contrast to Klepper's model (1996), in which all firms produce a standard product). Firms optimally choose to engage either in product or in process innovation. The present model differs from the ones in the studies mentioned above in a number of respects. First, I consider multiproduct firms that produce a number of differentiated goods in a duopoly setting, and investigate the relationship between firm size and R&D activity based on demand and cost functions. Second, labor is the only primary factor of production: it can be used to produce the differentiated goods and R&D services. R&D services result in discoveries of better production techniques, which enhance the productivity of labor employed in the manufacturing of the differentiated goods. R&D product services result in discoveries of new goods (adding or improving product features). The analysis generates several insights. For example, an increase in the number of goods produced by a firm and thus an increase in its size (since in my model, the firm size is measured by the firm's sales and the firm's sales are proportional to the number of goods produced) causes the firm to perform process R&D regardless the R&D regime in which it is located originally. Given that the firm starts with product R&D, as the number of goods produced increases, the firm's incentives to switch from product to process innovation increase. Once the firm is in the process R&D regime, it continues to perform process R&D indefinitely. This constitutes a costbased mechanism that links firm size to the type of R&D activity the firm develops, which is not clear in the models mentioned earlier. This chapter is organized as follows. Section 2.2 develops the model, and shows how it can explain Klepper's finding. Section 2.3 investigates the relationship between firm size and innovative activity in different regimes. Section 2.4 considers the implications of the analysis for the relationship between firm size and the types of innovative activities undertaken by firms within industries as well as extensions of the model are considered. The algebraic details and proofs of propositions in this chapter are relegated to Appendix A. 2.2 A Model of Process versus Product Innovation 2.2.1 Demand for Differentiated Products In this section, I develop a model to explain the influence of firm size on the effort devoted to process relative to product innovation. I imagine an identifiable sectoral structure of commodities. Thus, a pencil is a welldefined object and so are a refrigerator, a personal computer, a restaurant meal, and a haircut. Each one of these goods is a differentiated product, however, in the sense that there are many varieties of it available in the market and many more varieties that could potentially be produced. There are red and yellow pencils, soft and hard pencils, white and green refrigerators, 16MB memory personal computers and 128 MB memory personal computers and so on. Since products can be differentiated in many dimensions, one way to introduce preferences for differentiated products is to assume that there are commodities that individuals like to consume in many varieties, so that variety is valued in its own right. The tastes of a representative individual are represented by the following utility function (the economy is able to produce a large number of products, all of which enter symmetrically into demand): N U = D, +xo, 0< a < 1, (1) where D, is consumption of the i1h product, N is the number of available products and x0 is an outside good.' The demand for any individual product i can be derived by solving the following maximization problem: max U D 1D2,. ., N subject to D,P, =I, where I is the individual's income. By forming the lagrangian function (L), the above problem is equivalent to: maxL = max D, +Xo+ I DP,xo , D,,D,.D D D.D ..T )1 where X is the lagrangian multiplier. The first order conditions to the above problem are the following: aL  0 aDl P, =0, Vii=,...N. aD, oL N = 0 X, =IJPiDi , aL =0a =l. ax, Since k is fixed, the inverse demand functions for the differentiated goods are2: See more on this in Dixit and Stiglitz (1977). 2In the monopolistic competition models, X is taken as fixed because it is assumed that the number of goods produced is large and thus each firm's pricing policy has a negligible effect on the marginal utility of income. Pi =aD,, Vi = ,...N. (2) where D, is the quantity demanded per capital. As the total number of consumers is fixed, I can set population at 1 without loss of generality. In this case, I do not have to distinguish between total and percapita quantities, so I let D, (i = l,...N)denote the respective (total or per capital) quantities of the differentiated goods. 2.2.2 The Duopoly Model I consider a duopoly in which each firm produces a range of differentiated goods. Good i is produced with the following production function: Xik =l"kL,k, Vi=l,...n, ifk=l, and Vi =n, +1,...N if k =2, (3) where k denotes firm, Xik denotes the output of good i produced by firm k, and Lik denotes the amount of labor used in the production of good i by firm k. Thus, production functions are the same for all product varieties within each firm. The number of potential varieties is assumed to be countably infinite, so that only a finite subset of the range is actually produced. The parameter A (>1) is the quality increment per innovation, whereas the parameters q, and q2 represent the number of innovations undertaken by firm I and 2 respectively (q e {0,1,2,....}) The total number of goods available in the economy is n, + n, = N. Production of each product variety in such an industry will be undertaken by only one producer, since the other firm can always do better by introducing a new product variety than by sharing in the production of an existing product type. Using labor as the numeraire, I can normalize the wage rate to unity. Then, the marginal cost (the cost per unit) of good X,, is 1/pIq (Vi = l,...n,), when firm I knows how to produce X, with the q, th process. The marginal cost of good X, is 1/(Ui (Vi= 1,...n2), when firm 2 knows how to produce X,, with theq2 th process. I assume that both firms produce the quantities demanded of each good. Since each firm is a multiproduct firm, I measure the firm size by their sales. For example, sales for firm 1 are given by: PiDi = nPD. The above equation implies that the prices are the same for all goods produced within each firm and the quantities produced are the same within each firm. Since the sales of each firm are proportional to the number of goods produced by that firm, I can use that number to measure the firm size. In this section, I consider R&D costs as sunk (that is they already have been incurred and they are fixed). The profit function for firm 1 for producing n, goods is as follows: , ~D, D, (4) In the same way, the profit function for firm 2 for producing n, goods, is given by: N IN n, = YP,D D (5) By differentiating equations (4) and (5), with respect to the quantities (D,), I obtain the quantities produced of each good and then I can obtain the value profit functions. Since the value profit functions are symmetric for both firms, I can generalize them and use k to denote either firm (k = 1,2). By doing some comparative statics with respect to n k and qk, I obtain the following two equations and Proposition 1: Ak n*, kn+ 1 q kg n >0, (6) where An ,k is the marginal profit of firm k from performing product innovation. A, k,q =k(qk +l, nk) Kk(qk nk)=nk I ) ( 1)>0, (7) where Ackqk is the marginal profit of firm k from performing process innovation and where k denotes firm (k= 1,2). Proposition 1. Total profits for firm k (=1,2) depend positively on the number of goods produced and the number ofprocess innovations it performs. Proof. See Appendix A. Equations (6) and (7) provide a basis for explaining why the firm size tends to increase the marginal returns from process R&D, but not the marginal returns from product R&D. In equation (6), the marginal returns to product R&D are independent of the firm size, whereas in equation (7) the marginal returns to process R&D depend positively on the firm size. I can now solve explicitly for the number of goods (nk), at which the firm is indifferent between performing one more product R&D and one more process R&D. By equalizing equations (6) and (7), I obtain the following proposition: Proposition 2. If n (1/11q )" '(, ..*. ( 1)), then firm k is indifferent between performing one more product innovation and performing one more process innovation. Proof. See Appendix A. Proposition 2 provides a rationale for Scherer's (1991) finding that larger firms devote a greater fraction of their R&D to process innovation. The intuition behind this result is straightforward. The marginal returns to process R&D rise with the firm's size (as it is measured with nk) while the marginal returns to product R&D are constant (independent of nk). If n. > (l/ts ; (', t1 '" (. '1)), then it is more profitable for firm k (=1,2) to switch from product to process innovation. Figure 21 shows the relationship between the marginal profits from process and from product innovation considering the R&D costs as sunk costs. The marginal profit from process innovation is an increasing function of the number of goods produced, while the marginal profit from product innovation is independent of the number of goods produced. Point A in Figure 21 reflects a situation, in which the firm is indifferent between performing one more process innovation and one more product innovation. For a small number of goods produced (any point to the left of point A), the firm performs only product R&D. The reason is that, when a firm develops a product innovation, it reaches new buyers as well as raise price given some degree of transient monopoly power. For a large number of goods produced (any point to the right of point A), the firm performs only process innovation. The reason is that, when a firm develops a process innovation, it lowers its average cost. Thus, the firm will increase its profits by the decrease in average cost times the level of output of each good produced. Since, in the model, whenever a firm develops a process innovation, it applies to all goods, the larger the number of goods produced the greater the increase in the firm's profits. Next, I consider the case, where the firm k is in the product R&D regime (any point to the left of point A in Figure 21). I examine the effect of developing one more product innovation on the incentives to switch from product to process innovation. Firm k is in the product R&D regime if and only if Akqk An knk . / (n(tk / ))<0, (8) or n < '' ). By differentiating equation (8) with respect to nk, and after rearranging, I obtain: A(A4.,a A1t )  )i ) >0, (9) since i > 1. Proposition 3. Given that the firm k (=1,2) is in the product R&D regime, as each firm produces more goods (develops more product innovations), its incentives to switch from product to process innovations increase. Proof. See Appendix A. Proposition 3 explains the relationship between the firm size and the type of innovative activity. The intuition behind this result is straightforward. The returns to process R&D rise proportionally with the firm size while the returns to product R&D are constant with the firm size. Consequently, an increase in the number of goods produced must have a positive effect on process than product R&D, causing the firm to switch from product to process R&D. 2.3 Model of Product versus Process Innovation Considering the R&D Costs In this section, I examine how firms choose optimally their R&D effort between process and product innovation. I assume for simplicity that firms develop either product or process R&D, but not both of them at the same time. There are three regimes: product R&D regime (in the area to the left of point A in Figure 21), process R&D regime (in the area to the right of point A in Figure 21) and neutral R&D regime (point A in Figure 2 1). Firms start with product R&D because of monopoly power and then they switch to process R&D to exploit economies of scale. In this section, I examine the decision of the firm to perform one more product innovation or to perform one more process innovation for a given number of goods produced and processes developed and given that the firm is in the process R&D regime (any point to the right of point A in Figure 21). This decision is based on the difference between the net marginal profit from product innovations and the net marginal profit from process innovations. Net marginal profit from process innovation includes the R&D cost for process innovation and net marginal profit from product innovation includes the R&D cost for product innovation. If the net marginal profit from process is larger than the net marginal profit from product innovations, then it is more profitable for the firm to switch from product to process innovations. I assume that firms spend money on product R&D (R ) and process R&D (Rc). Let's denote with Rpik the amount spent on product R&D for good i by firm k and Rck the amount spent on process R&D for all goods by firm k. I assume that process R&D applies to all goods, that is it reduces equally the average cost of all goods produced. I further assume that the amount spent on product R&D is the same for all goods produced within the firm. Utilizing the assumptions mentioned above, I arrive at the following proposition: Proposition 4. (a+1) aqk a If nk / 0 ) ( g(R kR (1) ( ) ) then firm k is indifferent between performing one more product innovation and performing one more process innovation. Proof. See Appendix A. Figure 22 depicts the net marginal profits from process and from product innovations. At point A, firm k produces n~, and it is indifferent between performing one more product innovation and performing one more process innovation. Now, by differentiating n in Proposition 3 with respect to Rpk and Rk, I obtain the following propositions: Proposition 5. The number of goods produced byfirm k is a decreasing function of the firm's R&D cost on product innovation. Proof. See Appendix A. Intuitively, as the firm's R&D cost on product innovation is higher, the net marginal profit from product R&D decreases. Thus, at the original number of goods produced at which the firm is indifferent between performing one more product and one more process innovation, the net marginal profit from process innovation is greater than the net marginal profit from product innovation causing the firm to produce fewer goods. Figure 23 shows the effect of an increase in the R&D cost from product innovation (Rpk) on n When Rpk increases, the net marginal profit from product R&D line shifts down and the equilibrium moves from point A to point B. At point B, the number of goods produced at which the firm is indifferent between developing one more product innovation and developing one more process innovation is less than that of point A (n*k < n'k). Proposition 6. The number of goods produced byfirm k is an increasing function of the firm's R&D cost on process innovation. Proof. See Appendix A. As the firm's R&D cost on process innovation increases, the net marginal profit from process innovation decreases. Thus, at the original number of goods produced at which the firm is indifferent between performing one more product and one more process innovation, the net marginal profit from process innovation is lower than the net marginal profit from product innovation causing the firm to produce more goods. Figure 24 shows the effect of the increase in the R&D cost from process innovation (Rck) on n When Rek increases, the net marginal profit from process innovation shifts to the right and the equilibrium moves from point A to point B. At point B, the number of goods produced at which the firm is indifferent between developing one more product innovation and developing one more process innovation is higher than that of point A (n*k > n*k). Next, I consider the case, where the firm is at a point to the right of point A in the Figures, so that it performs only process innovations, since the net marginal profit from process is greater than the nets marginal profit from product innovations in this area. The firm k is in the process R&D regime if and only if Atkqk kAl.nk_ )aqkl(nk/') 1 0. This last expression implies the following: [n;(a/"a 1)1] > 0. (10) Suppose the firm has already developed nk product innovations and qk process innovations. Given (nk ,qk), and given the fact that the firm is performing only process innovations, I examine if it is more profitable for the firm to develop one more process innovation (qk +1) or to develop one more product innovation (n k +1). By developing one more process innovation, the difference between the marginal profit from process and the marginal profit from product innovation increases by: 1a .%,,(] ) I A(A:.,, An*,,), = ( ) ) (V '' lI [nk(/ )l], (11) where A(A7n, An[,,, ), is the change in the difference between the marginal profit from process and the marginal profit from product innovation due to change in the number of process innovations by one. If firm k is in the process R&D regime, equation (10) holds and implies that the sign of equation (11) is positive. That is, the firm has an incentive to continue performing process R&D. By developing one more product innovation, the difference between the marginal profit from process and the marginal profit from product innovation increases by: A(A7t. A7t, ), (.o )(Li 0 l) > o. (12) The signs of equations (11) and (12) imply: Proposition 7. Given that firm k (=1,2) is in the process R&D regime, it will continue to perform process R&D indefinitely. Proof. See Appendix A. This finding suggests that large firms have no incentives to do product R&D, reinforcing the conclusion of earlier studies on R&D effort and firm size (i.e., it supports the idea of R&D cost spreading). The fraction of process R&D versus product R&D rises monotonically with firm size. There is a critical point where the firm enters the process R&D regime and its incentives to remain in this regime are high. Once the firm performs only process R&D, its incentives to switch to product R&D disappear. That is, the firm will find more profitable to find ways to produce its goods cheaper than to create higher quality goods or more variety. 2.4 Implications and Extensions The notion that the firm size and the choice between process and product innovation follow a common pattern has become part of the folklore. My findings support the basic idea that larger firms have an advantage in R&D because of the larger output over which they can apply the resultsand thus spread the costsof their R&D (Cohen and Klepper (1996)). Note, however, that once a firm switches from product to process, it continues to develop only process (since the size of process innovation in my model is greater than one). If the size of process innovation were small, by performing one more product R&D in the process R&D regime would increase the incentives to switch from process to product innovation. Indeed, if the size of process innovation is less than one, process innovation is not profitable. Two limitations of the model need to be brought to the forefront. First, my conclusions depend on and are limited by the functional forms and the assumptions of my model. Another production function, for example, can make the model conform better to the real world. That is, once a firm is in the process R&D regime, after a critical number of process innovations, it can be possible to switch from process to product innovations. 25 Second, the assumption that firms will not attend to the production process until product innovation has slowed sufficiently is also restrictive. Yet the history of the automobile industry and others, such as tires and antibiotics, indicates that great improvements were made in the production process well before the emergence of any kind of dominant design. Indeed, many of these improvements were based on human and physical improvements that were not rendered obsolete by subsequent major product innovations. One possible extension of the model is to relax this assumption and explore the implications of this change on firm's optimal decision on choosing the fraction of process and product innovation. This relaxation would change some of the results, but it would also complicate the analysis of the model. A7M *k,q R&D Regime _r *k, n 0 n*k nk Figure 21. Marginal Profits from Process and Product Innovation Considering R&D Costs as Sunk AT*nk ,Rpk An *k,. Rk An,kq R,, *kn Rpk Figure 22. Net Marginal Profits from Process and from Product Innovations An*, Rpk A *k, Rck An*kk Rck A An *k,n R'pk An *k,%n R2pk R n*k' n*k nk Figure 23. Effect of an Increase in R&D Cost from Product Innovation on n*k k,n Rpk, Ant,, R'ck An*kq k /R At*k, R2k A B An*k,n Rpk R k nk *k" nk R ck Figure 24. Effect of an Increase in R&D Cost from Process Innovation on n*k CHAPTER 3 THE DYNAMIC EFFECTS OF GENERAL PURPOSE TECHNOLOGIES ON SCHUMPETERIAN GROWTH 3.1 Introduction In any given economic "era" there are major technological innovations, such as electricity, the transistor, and the Intemet, that have farreaching and prolonged impact. These drastic innovations induce a series of secondary incremental innovations. The introduction of the transistor, for example, triggered a sequence of secondary innovations, such as the development of the integrated circuit and the microprocessor, which are also considered drastic innovations.3 These main technological innovations are used in a wide range of different sectors inducing further innovations. For example, microprocessors are now used in many everyday products like telephones, cars, personal computers, and so forth. In general, drastic innovations have three key characteristics. The first feature refers to the generality of purpose, i.e., drastic innovations affect a wide range of industries and activities within each industry. Consequently, Bresnahan and Trajtenberg (1995) christened these types of drastic innovations "General Purpose Technologies" (GPTs henceforth). Several empirical studies have documented the crossindustry pattern SIn Appendix B, I provide a summary of the historical evolution of semiconductor technology, from the birth of the transistor in 1947 invented by the Bell Labs to the development of the microprocessor in 1971 by Intel. As a special case, I elaborate on the computer and the Intemet to show how these important drastic innovations are based on a previous drastic innovation, the transistor. of diffusion for a number of GPTs.4 In addition, a strand of empirical literature has established that the crossindustry diffusion pattern of GPTs is similar to the diffusion process of productspecific innovations and it is governed by standard Scurve dynamics.5 In other words, the internalinfluence epidemic model can provide an empiricallyrelevant framework to analyze the dynamic effects of a GPT. During this diffusion process, these drastic innovations could generate growth fluctuations and even business cycles. Schumpeter provides details on the connection between drastic innovations and business cycles: 4 For example, Helpman and Trajtenberg (1998a) provide evidence for the diffusion of the transistor. They state that transistors were first adopted by the hearing aids industry. Later, transistors were used in radios followed by the computer industry. These three industries are known as early adapters of the transistor GPT. The fourth sector to adopt the transistor was the automobile industry, followed by the telecommunication sector. 5 For example, Griliches (1957) studied the diffusion of hybrid seed corn in 31 states and 132 cropreporting areas among farmers. His empirical model generates an Scurve diffusion path. Andersen (1999) confirmed the Sshaped growth path for the diffusion of entrepreneurial activity, using corporate and individual patents granted in the U.S. between 1890 and 1990. Jovanovic and Rousseau (2001) provide more evidence for an Sshaped curve diffusion process by matching the spread of electricity with that of personal computer use by consumers. 6 As Paul Saffo, founder and director of the Institute for the Future, in Palo Alto, California, in his interview, Surfing the "S" Curve, in the News Link 1s Quarter (2001) edition (Volume 9, No. 1), describes the Scurve dynamics for Internet: "In terms of the Internet, we're at the bottom of the Scurve. Everything we've seen so far is just the beginning." The vertical rise in the Scurve will depend on new technologies that are going to advance in the future such as fiber optics, 3D chip arrays, DNA computing, nanocomputers, and much more. For more information on emerging technologies see "Taking Advantage of Technological Acceleration: Tracking Emerging Technologies and Trends" by Manyworlds (2001, www.manyworlds.com) and Appendix B. "It is by no means farfetched or paradoxical to say that "progress" destabilizes the economic world, or that it is by virtue of its mechanism a cyclical process." Schumpeter (1939, p. 138). "These revolutions periodically reshape the existing structure of industry by introducing new methods of productionthe mechanized factory, the electrified factory, chemical synthesis and the like; new commodities, such as railroad service, motorcars, electrical appliances; new forms of organization." Schumpeter (1950, p. 68). "Times of innovation...are times of effort and sacrifice, of work for the future, while the harvest comes after...The harvest is gathered under recessive symptoms and with more anxiety than rejoicing... [During] recession...much dead wood disappears." Schumpeter (1939, p. 143). Second, the dynamic effects of these GPTs take a long period of time to materialize. For instance, David (1990) argues that it may take several decades before major technological innovations can have significant impact on macroeconomic activity.7 Third, these GPTs act as "engines of growth". As a better GPT becomes available, it gets adopted by an increasing number of user industries and fosters complementary advances that raise the industry's productivity growth. As the use of a GPT spreads throughout the economy, its effects become significant at the aggregate level, thus affecting overall productivity growth. In his presidential address to the American Economic Association, Jorgenson (2001) documents the role of information technology in the resurgence of U.S. growth in the late 1990s.8 There is plenty of evidence that the rise in structural 7 David (1990) describes a phase of twentyfive years in the case of the electric dynamo. He argues that the observed productivity slowdown in the earlier stage of electrification and computerization was due to the adjustment process associated with the adoption of a new GPT. 8 At the aggregate level, information technology is identified with the output of computers, communications equipment, and software. These products appear in the GDP as investments by businesses, households, and governments along with net exports to the rest of the world. productivity growth in the late 1990s can be traced to the introduction of personal computers and the acceleration in the price reduction of semiconductors, which constituted the necessary building blocks for the information technology revolution.9 The growth effects of GPTs have been analyzed formally by Helpman and Trajtenberg (1998b). In their model GPTs require complementary inputs before they can be applied profitable in the production process. Complementary inputs developed for previous GPTs are not suited for use with a newly arrived GPT. The sequential arrival of GPTs generates business cycles. A typical cycle consists of two phases, a phase where firms produce final goods with the old GPT and components are being developed for the new GPT, and a second phase where final goods producers switch to the new GPT and the development of components for that GPT continues. 1 Output declines in the first phase of a cycle as workers switch from production to research to invent new inputs and increases again in the second phase once the new technology is implemented. 9 Another study from OECD documents that U.S. investment in information processing equipment and software increased from 29% in 1987 to 52% in 1999. The diffusion of information and communication equipment accelerated after 1995 as a new wave of information and communication equipment, based on applications such as the World Wide Web and the browser, spread rapidly throughout the economy. 10 In the same paper, the authors also provide an extension of their model in which they introduce a continuum of finalgood sectors with different degrees of productivity gains from GPTs and with the intermediate inputs being common to all sectors. As a result, a GPT spreads gradually across the economy, first to sectors that stand to gain most from its adoption and last to sectors that stand to gain least. Concerning output growth, it drops right at the beginning of the cycle (as in the base case), but then it exhibits positive growth during phase 1. " There is a growing literature with this approach. See, for example, Helpman and Rangel (1998), Aghion and Howitt (1998b) and the volume edited by Helpman (1998). In this chapter, I analyze formally the effects of a GPT within a stateoftheart model of Schumpeterian growth without scale effects. Schumpeterian (R&Dbased) growth is a type of growth that is generated through the endogenous introduction of new goods or processes based on Schumpeter's (1934) process of creative destruction, as opposed to physical or humancapital accumulation.2 Earlier models of Schumpeterian growth assumed that the growth rate of technological change depends positively on the level of R&D resources devoted to innovation at each instant in time. As population growth causes the size of the economy (scale) to increase exponentially over time, R&D resources also grow exponentially, and so does the longrun growth rate of percapita real output. In other words, longrun Schumpeterian growth in these models exhibits scale effects. Two influential papers by Jones ( 1995a, b) provided time series evidence for the absence of these scale effects. This evidence led theorists to construct Schumpeterian growth models that exclude scale effects. 3 My approach to modeling GPTs has the following features. First, the model abstracts from scale effects and generates longrun growth, which is consistent with the timeseries evidence presented by Jones (1995a). Second, I take into consideration the above mentioned evidence on long diffusion lags associated with the adoption of a new GPT. I therefore analyze both the transitional dynamics and longrun effects of a new 12 There are two classes of scale invariant Schumpeterian growth models; endogenous and exogenous. Endogenous [exogenous] Schumpeterian growth models are those in which longrun growth can [cannot] be affected by permanent policy changes. 13 Dinopoulos and Thompson (1999) provide a survey of the empirical evidence on scale and growth, and describe recent attempts to develop models that generate growth without scale effects. GPT. Third, I assume that a GPT is beneficial to all firms in each industry. Thus, when a GPT is implemented in an industry, it affects the productivity of R&D workers, the size of all future innovations in that industry and its growth rate. Finally, I assume that although a GPT's rate of diffusion is exogenous, its diffusion path across a continuum of industries is governed by Scurve dynamics.14 I incorporate the presence of a GPT into the standard qualityladder framework of Schumpeterian growth without scale effects that was developed by Dinopoulos and Segerstrom (1999). In the model, there is positive population growth and one factor of production, labor. Final consumption goods are produced by a continuum of structurally identical industries. Labor in each industry can be allocated between two economic activities, manufacturing of highquality goods, and R&D services that are used to discover new products of higher quality. The arrival of innovations in each industry is governed by a memoryless Poisson process whose intensity depends positively on R&D investments and negatively on the rate of difficulty of conducting R&D. Following Dinopoulos and Segerstrom (1999), I assume that R&D becomes more difficult over time in each industry. Specifically, I assume that the productivity of R&D workers declines as the size of the market (measured by the level of population) increases. This assumption captures the notion that is more difficult to introduce new products and replace old ones in a larger market.15 14 This assumption can be justified by the empirical observation provided by David (1991) that the measure of industries that adopted the dynamo and the number of consumers that have adopted computers follow an Scurve. is Several authors have developed microfoundations for this assumption. Young (1998), Dinopoulos and Thompson (1998), and Aghion and Howitt (1998a, Chapter 12) have combined tastes for horizontal and vertical product differentiation to generate models in which absolute levels of R&D drive productivity growth at the firmlevel, but aggregate For any given measure of industries with the new GPT (i.e., in the absence of Scurve dynamic diffusion of technology), the model generates a unique steadystate equilibrium. In this equilibrium, per capital consumption expenditure is constant over time, the aggregate stock value increases at the same rate as the constant rate of population growth, and the longrun Schumpeterian growth is endogenous (Proposition 1). In the absence a new GPT, the economy does not exhibit zero longrun growth as in previous models of GPTs (see Helpman and Trajtenberg (1998b), Aghion and Howitt (1998b), and Eriksson and Lindh (2000) among others). That is, the longrun growth rate depends positively on the rate of innovation (which equals per capital R&D) and thus, any policy that affects per capital R&D investment has longrun growth effects. In addition, the removal of scale effects allows me to analyze the effects of changes in the rate of growth of population that is absent from earlier models of GPTs. In the presence of a GPT, the model has two steadystate equilibria: the initial steadystate in which no industry has adopted the new GPT, and the final steadystate in R&D in larger economies is diffused over a larger number of product lines or industries. At the steady state, the number of varieties is proportional to the level of population. As population grows, the number of varieties increases and aggregate R&D is diffused over a larger number of product lines or industries, making R&D more difficult. Dinopoulos and Syropoulos (2000) have provided microfoundations for this specification in a model of Schumpeterian growth, where the discovery of higher quality products is modeled as an R&D contest (as opposed to an R&D race) in which challengers engage in R&D and incumbent firms allocate resources to rentprotecting activities. Rentprotecting activities are defined as costly attempts of incumbent firms to safeguard the monopoly rents from their past innovations. These activities can delay the innovation of better products by reducing the flow of knowledge spillovers from incumbents to potential challengers, and/or increase the costs of copying existing products. Their model postulates that R&D may become more difficult as the size of the economy grows because incumbent firms may allocate more resources to rentprotecting activities. which all industries have adopted the new GPT and its diffusion process has been completed. At the final steadystate relative to the initial steadystate: the longrun growth rate is higher, the aggregate investment is higher, the per capital consumption expenditure is lower, and the market interest rate is equal to the subjective discount rate (Proposition 2). I use Mulligan and SalaiMartin's (1992) timeelimination method to study the transitional dynamics of the model. This analysis generates several additional findings. First, there exists a unique globally stablesaddlepath along which the measure of industries that adopt the new GPT increases, the per capital consumption expenditure decreases, the market interest rate increases, and the innovation rate of the industries that have adopted the new GPT decreases at a higher rate than that of those that have not adopted the new GPT. Second, the model exhibits transitional growth cycles of per capital GNP as a consequence of Scurve dynamics (Proposition 3). The introduction of positive population growth in Aghion and Howitt's (1998b) model of GPTs will make these growth cycles shorter and shorter as the size of the economy increases, and in the longrun the GPTinduced cycles disappear. In the present model, the fall in output comes from the reduction in per capital consumption expenditure on final goods and the rise in the per capital R&D investment. As the size of the economy increases (as a result of positive population growth) the duration of the per capital GNP cycle remains the same. When all industries have adopted the new GPT and the diffusion process has been completed, the economy experiences a higher per capital income constant growth rate. I also analyze the effects of the adoption of the new GPT on the stock market. The growth rate of the stock market depends negatively on the rate of GPT diffusion process and the magnitude of the GPTridden R&D productivity gains, and positively on the rate of population growth. It also follows a Ushaped path during the diffusion process of the new GPT (Proposition 4). During the transition period, there are two types of industries in the economy: one that has adopted the new GPT and one that has not adopted it yet. The former type of industries is more innovative in terms of discovering higher quality products than the latter type of industries. In the initial stages of a GPT's diffusion, the aggregate stock value decreases, since most of the industries belong to the latter type of industries. As more industries switch to the new GPT, the aggregate stock value rises. This result is consistent with the previous GPTbased models.16 However, the mechanism that links the growth rate of the stock market with the GPT differs from these models. This mechanism allows the model to identify important factors, such as the rate of the population growth, that affect the stock market, which otherwise would have been ignored. In Helpman and Trajtenberg's (1998b) model, for 16 Jovanovic and Rousseau (2001) document empirically how technology has affected the U.S. economy over the past century, using 114 years of U.S. stock market data. Their estimates reveal evidence that entries to the stock market as a percentage of firms listed in each year, were proportionately largest between 1915 and 1929, and that these levels were not again approached until the mid1980s. About half of American households and most businesses were connected to electricity in 1920, and about one half of the households and most businesses today own or use computers. Both expansions, therefore, coincide with periods during which electricity and information technology saw widespread adoption. During times of rapid technological change, the new entrants of the stock market will grab the most value from previous entrants because the incumbents will find hard to keep up. The downward trend in the starting values of the vintages reflects a slowing down in the growth of the stock market. example, during the first phase, the components of both the best practice GPT and of the previous one have positive value. When the economy is in the second phase of a typical cycle, only components of the best practice GPT are valuable because at that time it is known that no component of older technologies will ever be used. Thus, the introduction of a new GPT brings a sharp decline in the real value of the stock market during a substantial part of phase one, but it picks up toward the end of the phase. In the second phase, the stock market rises. The effect of the GPT diffusion on the aggregate investment during the adoption process is ambiguous (Proposition 5). In the initial stages of the diffusion process, only a limited number of industries adopt the new GPT. These industries are called the early adopters. As more industries adopt the new GPT, the aggregate investment increases. The rest of this chapter consists of four parts. Section 3.2 develops the structure of the model. Section 3.3 analyzes the longrun properties of the model and Section 3.4 deals with the transitional dynamics. Section 3.5 summarizes the model's key findings and suggests possible extensions. Appendix B describes the evolution of semiconductor technology. The algebraic details and proofs of propositions are also relegated to Appendix B. 3.2 The Model 3.2.1 Industry Structure I consider an economy with a continuum of industries indexed by 0 E [0, 1]. In each industry 0, firms are distinguished by the quality j of the products they produce. Higher values ofj denote higher quality and j is restricted to taking on integer values. At time t = 0, the stateoftheart quality product in each industry is j = 0, that is, some firm in each industry knows how to produce aj = 0 quality product and no firm knows how to produce any higher quality product. To learn how to produce higher quality products, firms in each industry engage in R&D races. In general, when the stateoftheart quality in an industry is j, the next winner of an R&D race becomes the sole producer of aj+l quality product. Thus, over time, products improve as innovations push each industry up its "quality ladder", as in Grossman and Helpman (1991c). 3.2.2 Diffusion of a New GPT The diffusion path of a new GPT is modeled as follows: The economy has achieved a steadystate equilibrium, manufacturing final consumption goods with an old GPT. I begin the analysis at time t = to, when a new GPT arrives unexpectedly. Firms in each industry start adopting the new GPT at an exogenous rate.7 At each point in time, a fraction of industries, co, uses the new GPT and a fraction of industries, (lo), does not use the new GPT. For example, if the old GPT is the steam power and the new GPT is electricity, c industries use electricity in their production and 17 Aghion and Howitt (1998b) model the spread of GPTs using a continuum of sectors. In their model, the innovation process involves three stages. First, the GPT is discovered. Then each sector discovers a "template" on which research can be based. Finally, that sector implements the GPT when its research results in a successful innovation. They have computed paths of the fraction of sectors experimenting with the new GPT and the fraction using the new GPT and found that the time path of the later follows a logistic curve (Scurve). Thus, they analyze the transitional dynamics of GPTs by endogenizing the arrival rate of GPTs and running simulations. Endogenizing the rate of diffusion in my model would be an interesting avenue for further research, although it would make the analysis of the transitional dynamics complicated without adding much value on the results. (1co) industries use steam power in their production.18 I use the epidemic model to describe the diffusion of a new GPT across the continuum of industries.19 Its form can be described by the following differential equation, 6=8(1CO), (1) O) where 6) = 8ao/t denotes the rate of change in the fraction of industries that use the new GPT and 8 > 0 is the rate of diffusion. Equation (1) states that the number of new adoptions during the time interval dt, 6), is equal to the number of remaining potential adopters, (l(o), multiplied by the probability of adoption, which is the product of the fraction of industries that have already adopted the new GPT, C, and the parameter 6, which depends upon factors such as the attractiveness of the innovation and the frequency of adoption, both of which are assumed to be exogenous. 18 Devine (1983) provides an excellent historical perspective on electrification where he documents the transformation from shafts to wires. He states: "Until late in the nineteenth century, production machines were connected by a direct mechanical link to the power sources that drove them. In most factories, a single centrally located prime mover, such as a water wheel or steam engine, turned iron or steel "line shafts" via pulleys and leather belts...By the early 1890s then, direct current motors had become common in manufacturing, but were far from universal. Mechanical drive was first electrified in industries such as clothing and textile manufacturing and printing, where cleanliness, steady power and speed, and ease of control were critical". Helpman and Trajtenberg (1998a) explore the adoption of the transistor, an important semiconductor GPT by a number of industries. As they state: The early user sectors were hearing aids and computers. The prominent laggards were telecommunications and automobiles. These examples indicate that the timing of adopting a new GPT differs across industries. 19 See Thirtly and Ruttan (1987, pp.7789) for various applications of the epidemic model to the diffusion of technology. The solution to equation (1) expresses the measure of industries that have adopted the new GPT as a function of time and yields the equation of the sigmoid (Sshaped) logistic curve as follows: CO (2) [I + e("' ]' where y is the constant of integration. Notice that for t oo, equation (2) implies that all industries have adopted the new GPT.20 3.2.3 Households The economy is populated by a continuum of identical dynastic families that provide labor services in exchange for wages, and save by holding assets of firms engaged in R&D. Each individual member of a household is endowed with one unit of labor, which is inelastically supplied. The number of members in each family grows over time at the exogenous rate g, > 0. I normalize the measure of families in the economy at time 0 to equal unity. Then the population of workers in the economy at time t is N(t) = e8'. Each household is modeled as a dynastic family2, which maximizes the discounted utility U= o e'"" logu(t)dt, (3) 20 Whent oo, then =0. If one assumes that the new GPT arrives at time t=0, then co > 0. That is, the new GPT is introduced in the economy by a given fraction of industries co (i.e., the industry or industries that developed this particular GPT). 21 Barro and SalaiMartin (1995, Ch. 2) provide more details on this formulation of the household's behavior within the context of the Ramsey model of growth. where p > 0 is the constant subjective discount rate. In order for U to be bounded, I assume that the effective discount rate is positive (i.e., p g, > 0). Expression log u(t) captures the per capital utility at time t, which is defined as follows: logu(t) f log[Xh(0)Jq(j,6,t)]d0. (4) J In equation (4), q (j, 0, t) denotes the quantity consumed of a final product ofqualityj in industry 0 E [0, 1] at time t. Parameter X(0) measures the size of quality improvements and is equal to I if 0 e[0, m] 1(9) = (5) ,) = 1X0 if 0 E [o, 1], (5) where > 0 > 1. At each point in time t, each household allocates its income to maximize equation (4) given the prevailing market prices. Solving this optimal control problem yields a unit elastic demand function for the product in each industry with the lowest qualityadjusted price q(j, 0, t) c(t)N(t) p(jq(j t)(6) where c(t) is per capital consumption expenditure, and p(j, 0, t) is the market price of the good considered. The quantity demanded of all other goods is zero. Given this static demand behavior, the inter temporal maximization problem of the representative household is equivalent to max o e ('**" logc(t)dt, (7) (') subject to the inter temporal budget constraint a(t) = r(t)a(t) + w(t) c(t) g a, where a(t) denotes the per capital financial assets, w(t) is the wage income of the representative household member, and r(t) is the instantaneous rate of return. The solution to this maximization problem obeys the wellknown differential equation c(t) r(t) p, (8) c(t) According to equation (8), per capital consumption expenditure would increase over time if the instantaneous interest rate exceeded the consumer's subjective discount rate p. 3.2.4 Product Markets Every firm in each industry 0 uses labor L(0, t) as the sole input in its production according to the following production function Q(,t) L(t) (9) ao where aQ is the unit labor requirement. The monopolist engages in limit pricing, i.e., it charges a price equal to unit cost of manufacturing a product times the quality increment P = X()aQw. (10) At each instant in time, the incumbent monopolist produces the stateoftheart quality product and earns a flow of profits (,t)= (0) Ic(t)N(t) (11) X (o) ) 3.2.5 R&D Races Labor is the only input used to do R&D in any industry. Each firm in each industry 0 produces R&D services by employing labor L R (, t) under the constant returns to scale production function22 R(O,t) (0) LR(0,t), (12) aR In equation (12), aR/I(0) is the unitlabor requirement in the production of R&D services and ji(0) is equal to IT = if 0E[0, o] p(0) = (13) o = 1 if O [co, l], where > 1. A firm k that engages in R&D discovers the next higherquality product with instantaneous probability Ikdt, where dt is an infinitesimal interval of time and Ik(0, t) =R( (14) X(t) R (0, t) is firm k's R&D outlays and X(t) captures the difficulty of R&D in a typical industry. I assume that the returns to R&D investments are independently distributed across challengers, across industries, and over time. Therefore, the industrywide 22 The empirical evidence on returns to scale of R&D expenditure is inconclusive. Diminishing returns would make the analysis of the transitional dynamics more complicated. Segerstrom and Zolnierek (1999) among others developed a model where they allow for diminishing returns to R&D effort at the firm level and industry leaders have R&D cost advantages over follower firms. In their model, when there are diminishing returns to R&D and the government does not intervene both industry leaders and follower firms invest in R&D. probability of innovation can be obtained from equation (14) by summing up the levels of R&D across all challengers. That is, 1(0, t) = Ik (0, t) R (15) X(t) where and R(6, t) denotes total R&D services in industry 0. Variable I(0, t) is the effective R&D.23 The arrival of innovations follows a memoryless Poisson process with intensity I, for the industries that have adopted the new GPT and I, for industries that have not adopted the new GPT. Early models of Schumpeterian growth considered X(t) to be constant over time. This implied that the rates of innovation and the longrun growth increase exponentially as the scale of the economy grows exponentially. This scaleeffects property is inconsistent with postwar timeseries evidence presented in Jones (1995a). A recent body of theoretical literature has developed models of Schumpeterian growth without scale effects.24 Two approaches have offered possible solutions to the scaleeffects problem. The first generates exogenous longrun Schumpeterian growth models.25 The second approach generates models that exhibit endogenous longrun 23 The variable 1(0, t) is the intensity of the Poisson process that governs the arrivals of innovations in industry 0. 24 See Dinopoulos and Thompson (1999) for an overview of these models. 25 Jones (1995b), and Segerstrom (1998) have removed scale effects by assuming that R&D becomes more difficult over time because "the most obvious ideas are discovered first." The model that results from their specification is called the temporary effects of growth (TEG) model. In these models, the growth rate does not depend on any measure of scale. Increases in the steadystate level of R&D raise technology and income per capital at any point in time, but they do not raise the growth rate. Schumpeterian growth.26 Here I adopt the second approach and remove the scaleeffects property by assuming that the level of R&D difficulty is proportional to the market size measured by the level of population, X(t)=kN(t), (16) where k > 0 is a parameter.27 Consumer savings are channeled to firms engaging in R&D through the stock market. The assumption of a continuum of industries allows consumers to diversify the industryspecific risk completely and earn the market interest rate. At each instant in time, each challenger issues a flow of securities that promise to pay the flow of monopoly profits defined in (11) if the firm wins the R&D race and zero otherwise. Consider now the stockmarket valuation of the incumbent firm in each industry. Let V(t) denote the expected discounted profits of a successful innovator at time t when the monopolist charges a price p for the stateoftheart quality product. Because each quality leader is targeted by challengers who engage in R&D to discover the next higherquality product, a shareholder faces a capital loss V(t) if further innovation occurs. The event that the next innovation will arrive occurs with instantaneous probability Idt, whereas the event that no innovation will arrive occurs with instantaneous probability 26 Young (1998), Aghion and Howitt (1998a, Ch..12), Dinopoulos and Thompson (1998), Peretto (1998), and Peretto and Smulders (1998) remove the scale effects property by essentially the same mechanism as the one developed by exogenous Schumpeterian growth models. They introduced the concept of localized intertemporal R&D spillovers. Dinopoulos and Syropoulos (2000) proposed a novel mechanism based on the notion of innovationblocking activities that removes the scaleeffects property and generates endogenous longrun Schumpeterian growth. Their model offers a novel explanation to the observation that the difficulty of conducting R&D has been increasing over time. 27 Informational, organizational, marketing, and transportation costs can readily account for this difficulty. Arroyo, et al. (1995) have proposed this specification under the name of the permanent effects of growth (PEG) model, and have provided timeseries evidence for its empirical relevance. 1Idt. Over a time interval dt, the shareholder of an incumbent's stock receives a dividend 7t(t)dt and the value of the incumbent appreciates by dV(t) = [aV(t)/Ot]dt = V(t)dt. The absence of profitable arbitrage opportunities requires the expected rate of return on stock issued by a successful innovator to be equal to the riskless rate of return r; that is, V(, [1 I(, t)dt]dt + ( dt t)0] I(0, t)dt = rdt. (17) V(9, t) V(9, t) V(0, t) Taking limits in equation (17) as dt ) 0 and rearranging terms appropriately gives the following expression for the value of monopoly profits n(6, t) V(, t)= t) (18) V((,, t) =t) r(t) + I(, t) , V(0, t) Consider now the maximization problem of a typical challenger k. This firm chooses the level of R&D investment Rk (0, t) to maximize the expected discounted profits V(, t)Rk(t) dtw RRk(0, t)dt, (19) X(t) () where Ikdt = [Rk (0, t)/X(t)]dt is the instantaneous probability it will discover the next higherquality product and waRRk(0, t)/u(0) is the R&D cost of challenger k. Free entry into each R&D race drives the expected discounted profits of each challenger down to zero and yields the following equilibrium condition as follows: waRkN(t) V(0, t) = wakN(t) (20) ko)e 3.2.6 Labor Market All workers are employed by firms in either production or R&D activities. Taking into account that each industry leader charges the same price p and that consumers only buy goods from industry leaders in equilibrium, it follows from (9) that total employment of labor in production is J Q(, t)dO. Solving (12) for each industry leader's R&D employment LR (0, t) and then integrating across industries, total R&D employment by industry leaders is j [R(0, t)g /p.(0)]dO. Thus, the full employment of labor condition for the economy at time t is N(t)= Q(,t)QdO+ d)d. (21) Equation (21) completes the description of the model. 3.3 LongRun Equilibrium The dynamic behavior of the economy is governed by two equations that determine the evolution of the per capital consumption expenditure, c, and the number of industries that adopt the new GPT, o. To facilitate the interpretation and understanding of my results, I begin by deriving expressions for longrun per capital real output and longrun growth. Following the standard practice of Schumpeterian growth models, one can obtain the following deterministic expression for subutility u(t), which is appropriately weighted consumption index and corresponds to real per capital income28 log u(t) = log c log a, + I(, t)t log X(0) log (0) (22) 28 See Dinopoulos (1994) for an overview on Schumpeterian growth theory. The economy's longrun Schumpeterian growth is defined as the rate of growth of subutility u(t), g, = u(t)/u(t). By differentiating equation (22) with respect to time, I obtain the following: g, = I(0,t)log ( ), (23) u(t) which is a standard expression for longrun growth in qualityladders growth models. Because the size of each innovation becomes larger (i.e., Xi > Xo) after all industries have adopted the new GPT (i.e., the diffusion process has been completed), longrun growth, g,, can be affected not only through changes in the rate of innovation, but also through the diffusion of the new GPT. After substitution of equations (6) and (10) into the first integral of equation (21), the demand for manufacturing labor is given by the following expression: cN(t) ( ( _ ) cN tl ( l (24) Substituting equation (12) into the second integral of equation (21), the demand for R&D labor is given by: oa R, +(1)aRR. (25) At this point it is useful to choose labor as the numeraire of the model and setting w = 1. (26) Combining equations (24) and (25) with equations (15) and (16) and taking into account (26) yields the resource condition l= c ( +kaR ,+(lm)I, (27) ^, \, [f(9 which defines a negative linear relationship between per capital consumption expenditure, c, and the effective R&D, I. The above resource condition holds at each instant in time because by assumption factor markets clear instantaneously. I now derive the differential equation that determines the growth rate of per capital consumption expenditure, c/c, as a function of its level and the rate of innovation. Equation (20) holds at each instant in time, so it yields 'V(, t)/V(0, t) = X(t)/X(t)= g,. In other words, the values of expected discounted profits, V(t), and the level of R&D difficulty, X(t), grow at the constant rate of population growth, gN, Combining equations (18) and (20) (after substituting equation (11) into equation (18) and equation (26) into equation (20)), I obtain )(0)aRk c= ) ][ p+I(0,t)g ], (28) (h(0) l)u(0) which defines a positive linear relationship between per capital consumption expenditure, c, and the effective R&D, I. It also implies the familiar condition that r = p, which means that the market interest rate must be equal to the subjective discount rate in the steady state equilibrium. This property is shared by all Schumpeterian models where growth is generated by the introduction of final consumption (as opposed to intermediate production) goods. Let a hat "A" over variables denote their market value in a steadystate equilibrium. The resource condition (27) and the equilibrium R&D condition (28) determine simultaneously the longrun equilibrium values of per capital consumption expenditure, E, and the rates of innovation, i, and I,. Figure 31 illustrates the two steadystate equilibria: the initial steadystate (point A) where no industry has adopted the new GPT (i.e., o = 0) and the final steadystate (point B) where all industries have adopted the new GPT (i.e., o = 1). When co = 0 the balancedgrowth resource condition is 1= +knaRl, (29) and the balancedgrowth R&D condition is given by equation (28) (when co = 0). The vertical axis measures consumption per capital, c, and the horizontal axis measures the rate of innovation, I. The resource condition is reflected by the negativelysloped line NoNo and the R&D equilibrium condition is represented by the positivelysloped line RoRo. Their unique intersection at point A determines the longrun values c(0) and Io (0), where c(0) denotes the percapita consumption expenditure evaluated at o = 0 and i0 (0) denotes the innovation rate for industries that have adopted the new GPT evaluated at co = 0. Therefore, I arrive at: Proposition 1. For a given c e [0, I], where co is the measure of industries with a new GPT, there exists a unique steadystate equilibrium such that the longrun Schumpeterian growth, k, is endogenous and does not exhibit scale effects: it depends positively on policies that affect the size of innovations, A, the labor productivity in R&D services, pu()/a, and the rate of population growth, g.; it depends negatively on the consumer's subjective discount rate, p. At each steadystate equilibrium, consumption expenditure per capital, c, is constant, the interest rate, i(t), is equal to the constant subjective discount rate, p, and the aggregate stock value, V, increases at the same rate as the constant rate of population growth, g,. Proof. See Appendix B. The removal of scale effects from the longrun growth rate, g,, depends on the assumption that the level ofR&D difficulty is proportional to the market size. At the steadystate equilibrium, the level of R&D difficulty, X(t), increases exponentially at the rate of population growth g, (i.e., X(t)/X(t) = g ) as can be seen from equation (16). The absence of a new GPT does not result in zero longrun growth rate as in the Helpman and Trajtenberg (1998b), Aghion and Howitt (1998b), and Eriksson and Lindh (2000) models. That is, the longrun growth rate depends positively on per capital R&D and thus, any policy that affects this variable has longrun growth effects. The following proposition describes the longrun properties of the economy: Proposition 2. If o is governed by Scurve dynamics, there are only two steady state equilibria: the initial steadystate equilibrium arises before the adoption of the new GPT, where o = 0, and the final steadystate equilibrium is reached after the diffusion process of the new GPT has been completed, where to = 1. At the final steadystate equilibrium: aggregate investment is higher, I(1) > 1(0), longrun growth rate is higher, g, (1) > g, (0), per capital consumption expenditure is lower, c(l) < C(0), per capital stock market valuation of the incumbent in each industry is lower, V(1)/N < V(O)/N, relative to the initial steadystate equilibrium. In both steady states the market interest rate is equal to the subjective discount rate, P = p. Proof. See Appendix B. These comparative steadystate properties can be illustrated with the help of Figure 31. Before the introduction of the new GPT, the economy is in a steady state (point A) where o = 0, with per capital consumption expenditure Z(0), and with innovation rate i0. An increase in the measure of industries that adopt the new GPT makes the R&D condition in Figure 31 shift downward from RoRa (where o = 0) to RIRI (where o = 1) and the resource condition shift upward from NoNo to NINI, resulting in higher longrun rate of innovation and in lower longrun consumption per capital. In other words, when all industries have adopted the new GPT, the longrun Schumpeterian growth rate increases. The new steady state is at point B, where o = 1, with per capital consumption expenditure 8(1), and innovation rate i,. 3.4 Transitional Dynamics I analyze the transitional dynamics of the model by adapting the timeelimination method described by Mulligan and SalaiMartin (1992).29 The timeelimination method enables me to construct a system of two differential equations that govern the evolution of c and o. Since equation (28) holds at each instant in time (when the subjective discount rate, p, is replaced by the interest rate, r), I can solve for the rates of innovation for the two types of industries, Io and I,. After substituting these rates into the resource condition (27), which holds at each instant in time, I can solve for the market interest rate along any path and obtain p(cl1) r) +gg. (30) Substituting (30) into (8) yields the following differential equation: =rp= +gs P. (31) c ckaR[(l) 1 29 See also Mulligan and SalaiMartin (1991) for more details on this method. Equations (31) and (1) determine the evolution of the two endogenous variables of the model, per capital consumption expenditure, c, and the number of industries that have adopted the new GPT, o. Since the righthand side of equation (31) is decreasing in o, c = 0 defines the downwardsloping curve in Figure 32. Starting from any point on this curve, an increase in co leads to c > 0 and a decrease in to leads to c < 0. The right hand side of equation (1) is independent of c, and therefore the 6 = 0 locus is a vertical line. Starting from any point on this line, decrease in to leads to 6 > 0. The area to the left of the vertical line (i.e., locus 6 = 0) identifies a region in which the potential number of adopters is greater than one. Therefore, this region is not feasible. There exists a downwardsloping saddle path going through the unique balancedgrowth equilibrium point B. Thus, I arrive at: Proposition 3. Assume that 5 > (g, p). Then, there exists a unique negativesloping globally stablesaddlepath going through the final unique balancedgrowth equilibrium point B. Along the saddle path, the measure of industries that adopt the new GPT, co, increases, the per capital consumption expenditure, c, decreases, the market interest rate, r, increases, the innovation rate of the industries that have adopted the new GPT, I,, decreases at a higher rate than that of those that have not adopted the new GPT, 10. In addition, there exist transitional growth cycles ofpercapita GNP. Proof. See Appendix B. The analysis is predicated on the assumption of perfect foresight.30 When the new GPT arrives, per capital consumption expenditure, c, jumps down instantaneously to (point A' in Figure 32). This per capital consumption expenditure jump lowers the interest rate to 7 (Figure 34) since there are more savings available. The downward jumps on the per capital consumption expenditure and on the interest rate imply an upward jump on the innovation rates of both types of industries; those that have adopted the new GPT and those that have not adopted the new GPT ( I and I, in Figure 35). Figure 31 illustrates that the R&D line RoRo will shift downwards and the resource line NoNo will shift upwards with the arrival of the new GPT resulting in lower percapita consumption expenditure. Going back to Figure 32, the instantaneous decrease in c is reflected by a movement from point A to point A'. The decrease in percapita consumption expenditure leads to a decrease in the market interest rate r (from equation (30), which always hold). When the market interest rate r is lower than the subjective rate p, per capital consumption expenditure decreases even further, until the market interest rate approaches the subjective discount rate at the new steady state (point B in Figures 3.1 and 3.2). During the transition dynamics (i.e., as the equilibrium moves from point A' to point B in Figure 32), the interest rate increases leading to more savings and a decrease in per capital consumption expenditure. At point B in Figure 32, all industries have adopted the new GPT. 30 There also exists a degenerate equilibrium where the adoption of the new GPT is not completed. Suppose that when a new GPT arrives, every potential consumer expects that no one will decrease their consumption expenditure, in order to finance innovation. As a result, it does not pay to decrease consumption expenditure of a single consumer, because the new GPT will never be fully adopted. In this event, the pessimistic expectations are selffulfilling, and no new GPTs are fully adopted. I do not discuss these types of equilibria in what follows. Along the transition path, the aggregate investment may increase or decrease. One possible path of the aggregate investment is shown in Figure 35 by the dotted curve. There is an upward jump in the innovation rate of industries that have not adopted the new GPT (from point Bo to point B in Figure 35). Figures 33 and 34 show the time paths of per capital consumption expenditure and the market interest rate (where to indicates the time when the new GPT arrives in the economy and to indicates the time when all industries in the economy have adopted the new GPT). Figure 36 shows the effect of a GPT on the Schumpeterian growth rate. The adoption of the new GPT entails cyclical growth patterns.3 The growth rate decreases in the initial stages of the adoption of the new GPT. There exist transitional growth cycles. 3.4.1 Stock Market Behavior The fact that the adoption of a new GPT affects positively the productivity of R&D together with free entry into each R&D race are the key factors in explaining the behavior of the stock market. The probability of discovering the next higher quality product in each industry increases with the adoption of the new GPT and so does the probability that the incumbent in each industry will be replaced by a follower firm (i.e., the hazard rate). This link between the GPT adoption and higher risk for incumbent firms captures the effects of creative destruction on the stock market valuation of monopoly profits. In other words, during the diffusion of a GPT per capital consumption declines, the market interest rate rises, and the hazard rate increases. These changes lower the 31 Earlier contributions on this issue include the macroeconomic model of Cheng and Dinopoulos (1996) in which Schumpeterian waves obtain as a unique nonsteadystate equilibrium solution and the current flow of monopoly profits follows a cyclical evolution. percapita expected discounted profits of the successful innovator and drives down its per capital stock market valuation.32 Furthermore, the larger the productivity gains associated with the new GPT, the larger the slump of the stock market. For example, it may be that the productivity gains generated by the introduction of the new GPT are large not because the new GPT is technologically very advanced at that initial stage, but because the previous GPTs are particularly inadequate for the needs of these sectors.33 However, the size of the slump in the stock market is more severe, when the new GPT is diffused at a higher rate. The aggregate stock value is given by the following equation: V1 V5 V=[+o +(1 o) ]N(t), (32) N(t) N(t) where V, and V. are given by equation (20) after taking account equation (26). After substitution of these values into equation (32) and taking logs and derivatives with respect to time, I obtain the growth rate of the aggregate stock value: 32 Hobijn and Jovanovic (2001) argue that U.S. stock market decline in the early 1970s is due to the arrival of information technology and the fact that the stockmarket incumbents were not ready to implement it. They state "Instead, new firms would bring in the new technology after the mid1980s. Investors foresaw this in the early 1970s and stock prices fell right away." The U.S. stock market value relative to GDP plummeted to 0.4 in 1973,just after Intel had developed the microprocessor in late 1971. The decrease of the stock market value relative to GDP did not recover until the mid1980s, and it then rose sharply. Leading OECD countries also experienced similar movements in their stock markets, following a Ushaped path. 33 This was clearly the case for early computers, where even that valves had been getting smaller for over a decade prior to the arrival of the transistor, the transistor was still an order of magnitude smaller. V (1)o08(l o) gv = =gN (33) V [CO+(1 )C] At the initial steadystate, where o = 0 and at the final steadystate, where o = 1, the growth rate of the aggregate stock value is equal to the rate of the population growth. That is, v =g, (34) The effects of a GPT on the stock market valuation of monopoly profits are summarized in the following proposition: Proposition 4. The growth rate of the stock market, g,, depends negatively on the rate of GPT diffitsion process, 3, and the magnitude of the GPTridden R&D productivity gains, p, and positively on the rate ofpopulation growth, g,. It also follows a Ushaped path relative to the population growth rate during the diffusion process of the new GPT. Proof. See Appendix B. These comparative properties, which differentiate the model from several others in its class, can be illustrated with the help of Figure 37, which shows the growth rate of stock market as a function of the measure of industries that have adopted the new GPT. The initial adoption of the new GPT decreases the growth rate of the stock market below the rate of the population growth. In the later stages of the adoption of the new GPT, the growth rate of the stock market increases. When the diffusion process of the new GPT has been completed, the growth rate of the stock market is equal to the rate of the population growth. That is, it follows a Ushaped path relative to the population growth rate during the diffusion process of the new GPT. This last result can be seen from the second term of the right hand side in equation (33).34 The free entry condition in each R&D race (equation 20) implies that the per capital stock value in any industry 0 ( V /N(t)) is constant over time. It jumps down instantaneously with the adoption of the new GPT, and it remains constant thereafter. The aggregate stock value, which increases exponentially at the rate of the population growth, jumps down with the arrival of the new GPT, and then increases again at the population growth rate. The slump in the aggregate stock value is due to the realization of the R&D productivity gain associated with the new GPT. The higher these R&D productivity gains are, the higher is the jump in the per capital industry and aggregate stock value at the time of the adoption of the new GPT.35 An increase in the GPT diffusion rate, 8, increases the economywide resources devoted to R&D. Thus, the probability that the incumbent firm will be replaced by a follower firm increases. This can be seen from equation (18), which gives the value of monopoly profits. In other words, when the GPT diffusion process accelerates, the decrease in per capital consumption expenditure is more severe, and the per capital R&D investment increases. In this case, the Ushaped path of the growth rate of the stock market sags down (this is shown by the dottedshaped curve in Figure 37). That is, the 34 The numerator in equation (33), which is positive and it reflects the slope of a truncated Scurve, is equal to 6r times a positive fraction that depends on the magnitude of the GPTridden R&D productivity gains and on the number of the industries that have adopted the new GPT. 35 This can be seen from equation (33), where the first term in the righthand side gets smaller when each industry adopts the new GPT relative to the second term of the right hand side of the same equation. slope of the growth of the stock market gets steeper at the initial stages of the diffusion process of the new GPT and it gets flatter at the final stages of this process.36 An increase in the productivity gains generated by the new GPT, 11, lowers the cost of discovering the next higher quality product. This, in turn, will affect negatively the stock market valuation of the incumbent firm (see equation (20)). An increase in the rate of population growth, gs, shifts the Ushaped curve in Figure 37 upwards and increases the growth rate of the stock market. 3.4.2 Aggregate Investment Proposition 5. The effect of the GPT diffusion on the aggregate investment during the adoption process is ambiguous. Proof. See Appendix B. The initial steadystate equilibrium is at point A in Figure 35. There is an upward jump in the aggregate investment with the introduction of the new GPT (from I to 1). Along the diffusion path, both innovation rates (I, and I,) decrease until the economy reaches at the final steadystate equilibrium point B, where the aggregate investment is higher relative to the initial steadystate equilibrium point A. There is an upward jump on 36 The first OPEC shock may also explain a part of the drop in the stock market in the early 1970s, as well as a part of the productivity slowdown. Hobijn and Jovanovic (2001) argue that there are several problems associated with the oilshock explanation. One problem is that a rise in oil prices should have lowered current profits more than future profits, because of the greater ease of finding substitutes for oil on the longrun, perhaps current output more than future output and, therefore, should have produced a rise in the ratio of market capitalization to GDP, not a fall. This scenario also implies a constant entry in the stock market, something that contradicts their evidence. Another problem that is associated with the oilpriceshock explanation for the stockmarket drop is that the energyintensive sectors did not experience the largest drop in value in 19731974. Their evidence supports that the informationtechnologyintensive sectors experienced the largest drop in 19731974. the innovation rate of the industries that have not adopted the new GPT at the final steady state (from point Bo to B). One possible picture of how the aggregate investment behaves along the diffusion of the new GPT is shown in Figure 35. Along the transition path, the aggregate investment decreases and then increases. In the initial stages of the diffusion process, only a limited number of industries adopt the new GPT (see equation (1)). These industries are called the early adopters. As more industries adopt the new GPT, the aggregate investment increases. 3.5 Concluding Remarks Previous models that have analyzed GPTs exhibit the scale effects property. The present paper analyzed the effects of a GPT on shortrun and longrun Schumpeterian growth without scale effects. The absence of growth scale effects and the modeling of the diffusion process through Scurve dynamics generate several novel and interesting results. First, the longrun growth rate of the economy depends positively on the magnitude of quality innovations. Any policy that affects this magnitude has longrun growth effects. However, the absence of the arrival of a new GPT in the economy does not reduce the longrun growth rate to zero as in the previous GPTsbased growth models. All the previous R&Dbased models that analyze the effects of GPTs exhibit scale effects. The assumption that the diffusion of the new GPT follows an Scurve generates two steadystate equilibria: one is the initial steadystate before the adoption of the new GPT begins and the other is the final steadystate after the diffusion process of the new GPT has been completed. At the final steadystate relative to the initial steadystate: the longrun growth rate is higher, the aggregate investment is higher, the percapita consumption expenditure is lower, and the market interest rate is equal to the subjective discount rate. The growth rate of the stock market depends negatively on the rate of GPT diffusion process, and the magnitude of the GPTridden R&D productivity gains, and positively on the rate of population growth. It also follows a Ushaped path relative to the population growth rate during the diffusion process of the new GPT. This is consistent with the empirical evidence provided by Jovanovic and Rousseau (2001) who empirically document that during times of rapid technological change the growth of the stock market slows down, since the new entrants of the stock market will grab the most value from previous entrants because the incumbents will find hard to keep up. One could also develop a dynamic general equilibrium model to study the effects of a GPT diffusion on a global economy that exhibits endogenous Schumpeterian growth. As in this model, the adoption of a GPT by a particular industry can generate an increase in the productivity of R&D workers and the magnitude of all future innovations and its diffusion across industries can be governed by Scurve dynamics. The diffusion of the GPT within an industry from one country to the other can occur with a time lag. Under this framework, it would be interesting to analyze the longrun and transitional dynamic effects of a new GPT on trade patterns, product cycles and (transitional) divergence in percapita growth rates between the two countries. This is a fruitful direction for future research. Initial R&D Condition / Ro Final R&D Condition R, Initial Resource Condition 0 I(0) 1(1) No NI I Figure 31. SteadyState Equilibria: Point A: No industry has adopted the new GPT. Point B: All industries have adopted the new GPT. do =(1 )o =0 dt B dc 0 dt=0 dt 0 1 ( Figure 32. Stability of the BalancedGrowth Equilibrium do) =0 dt c a(0) . C )  to to Time Figure 33. Time Path of the Per Capita Consumption Expenditure After a GPT Arrives in the Economy r to t, Time Figure 34. Time Path of the Market Interest Rate After a GPT Arrives in the Economy II \ ~ ~ ~  ':,~= i() A  Bo 0 1 Figure 35. Evolution of the Aggregate Investment During the Diffusion Path: The initial steadystate equilibrium is at point A. There is an upward jump in the aggregate investment with the introduction of the new GPT. One possible path of the aggregate investment, along the diffusion path, is depicted in the Figure by the dotted curve IB. IllogXl   Iologo to t Time Figure 36. Effects of a GPT on the Schumpeterian Growth Rate: When all industries have adopted the new GPT, the economy experiences higher steadystate Schumpeterian growth. There also exist transitional growth cycles of per capital GNP. 9N1 I 0o 0/(i+ i1 Figure 37. Effects of a GPT on the Stock Market: The growth rate of the stock market depends on the rate of GPT diffusion process, the magnitude of the GPTridden R&D productivity gains, and the rate of population growth. It also follows a Ushaped path relative to the population growth. CHAPTER 4 SUSTAINED COMPARATIVE ADVANTAGE IN A MODEL OF SCHUMPETERIAN GROWTH WITHOUT SCALE EFFECTS 4.1 Introduction Many models of endogenous growth and trade emphasize the role of continual product innovation based on R&D investment in determining the pattern of trade between countries. Grossman and Helpman (1991a, b, c) have developed models where innovations lead to either improvements in the quality of existing products ("quality ladders" models) or increase in the variety of the goods ("love for variety" models). Taylor (1993) has extended the continuum Ricardian model of Dombusch et al. (1977) based on the "quality ladders" approach by Grossman and Helpman. All these studies exhibit the scale effect property: if one incorporates population growth in these models, then the size of the economy (scale) increases exponentially over time, R&D resources grow exponentially, and so does the longrun growth rate of percapita real output. The scale effects property is a consequence of the assumption that the growth rate of knowledge is directly proportional to the level of resources devoted to R&D. Jones (1995a) has argued that the scale effects property of earlier endogenous growth models is inconsistent with postwar time series evidence from all major advanced countries that shows an exponential increase in R&D resources and a moreorless constant rate of per capita GDP growth. Jones's criticism has stimulated the development of a new class of models that generate growth without scale effects.37 However, the theoretical literature on 37 See Dinopoulos and Thompson (1999) for more details on this issue. trade and growth without scale effects has focused either on closed economy models or on structurally identical economies engaging in trade with each other.38 This chapter develops a twocountry general equilibrium framework without scale effects to determine the equilibrium relative wages and the pattern of trade between countries. My approach borrows from Taylor's work (1993) in that industries differ in production technologies. In his model, industries also differ in research technologies and in the set of technological opportunities available for each industry. In the presence of heterogeneous research technologies (captured by different productivity in R&D services), the pattern of R&D production and the pattern of goods production within each country can differ. As a result, there is a case for trade between countries in R&D services. The absence of heterogeneity in research technologies in my model makes the removal of scale effects more tractable, but eliminates the need for trade in R&D services between countries. In the present model, there are two countries that may differ in relative size: Home and Foreign. The population in each country grows at a common positive and exogenously given rate and labor is the only factor of production. There is a continuum of industries producing final consumption goods. Labor in each industry can be allocated between the two economic activities, manufacturing of highquality goods and R&D services, which are used to discover new products of higher quality. As in Grossman and Helpman 's (199 Ic) version of the qualityladders growth model, the quality of each final 8Dinopoulos and Syropoulos (2001) have recently developed a twocountry general equilibrium model of endogenous Schumpeterian (R&D based) growth without scale effects to examine the effect of globalization on economic growth when countries differ in population size and relative factor endowments. good can be improved through endogenous innovation. The arrival of innovations in each industry is governed by a memoryless Poisson process whose intensity depends positively on R&D investments and negatively on the rate of difficulty of conducting R&D. I consider two alternative specifications regarding the difficulty of conducting R&D in order to remove the scale effects property. The first specification is called the permanent effects of growth (PEG) and it has been proposed by Dinopoulos and Thompson (1996). According to this specification, R&D becomes more difficult over time and the degree of R&D difficulty is proportional to the size of the world market. The second specification is called the temporary effects of growth (TEG) and it has been proposed by Segerstrom (1998). With this specification, R&D also becomes more difficult over time but the degree of difficulty is an increasing function of cumulative R&D effort in each industry. The removal of scale effects property under either specification regarding the R&D difficulty leaves some predictions in Taylor's (1993) model robust. A uniform increase in Home's unit production costs reduces its relative wage and reduces its range of goods that Home produces. A uniform increase in Foreign's unit production costs increases Home's relative wage and increases Home's comparative advantage. In addition, Home's relative wage and its comparative advantage are left unaffected by equiproportionate changes in Home and Foreign production costs. Finally, factor price equalization is not a generic property of the steadystate equilibrium. However, several comparativesteadystate results in Taylor's (1993) model change with the removal of the scale effects property. First, in his model, the direction of the effect of the size of innovations (which can vary across industries) on the pattern of goods production, R&D production, the pattern of trade, and the relative wage depends on the assumption that the size of innovations is heterogeneous. Under the heterogeneity assumption, the increase in the inventive step creates a deficit in the balance of payments for Home because it raises the royalties' payments that Home has to pay for using the frontline technology.39 Balance of payments is maintained through two adjustments; Home raises its goods trade balance by increasing the range of goods produced at Home and it reduces its reliance on imported R&D by conducting more itself. Removing part of this heterogeneity in his model, by eliminating Home's relative advantage in goods versus R&D, results in zero trade in R&D and no effect of the size of innovations on the pattern of trade and Home's relative wage.40 On contrast, in the present model, the direction of the effect of the size of innovations on the pattern of trade and Home's relative wage depends on the way in which the scale effects property is removed. Under the PEG specification, changes in the size of innovations do not affect Home's comparative advantage and its relative wage, while under the TEG specification, an increase in the size of innovations reduces Home's relative wage and raises its comparative advantage. Under the PEG specification, the increase in the size of innovations increases the profitability of R&D activity. As a result, more firms enter into R&D races until the 39 Taylor (1993) divides the world's available technologies into two sets: the set of front line technologies and the set of backward technologies. Frontline technologies are those that are minimum cost given the prevailing wage rate. He further assumes that when an innovator located in Foreign succeeds in the global R&D races and discovers the front line technology, it has two options: it can either implement this improvement on the foreign technology or it can go multinational and carry the innovation abroad to a wholly owned subsidiary. This subsidiary would then pay the foreign firm a royalty. 40 Eliminating the across country heterogeneity in his model, results in factor price equalization and indeterminate pattern of trade in both goods and R&D. expected profits are driven back to zero. The adjustment to equilibrium requires an increase in the level of global R&D investment, while it leaves Home's relative wage and its comparative advantage unchanged. On contrast, the TEG specification implies that the global level of R&D investment depends positively on the rate of population growth and negatively on the R&D difficulty parameter, since the R&D difficulty grows at the constant rate of population growth rate at the steadystate equilibrium. Therefore, an increase in the size of innovations raises the rewards of innovating and thus it induces more firms to engage in R&D. However, since the longrun level of R&D activity is fixed by the unchanged parameters of the model, to maintain the equilibrium, the relative wage has to decrease. The decrease in Home's relative wage raises its comparative advantage. Second, the direction of the effect of a change in R&D costs in Taylor's (1993) model depends on the heterogeneous research assumption. A uniform increase in Home's R&D costs reduces Home's relative wage and raises the measure of Home industries active in production. A uniform increase in Foreign's costs increases Home's relative wage and reduces the measure of Home industries active in production. However, if one considers an equiproportionate changes in R&D costs in his model, then the effect of these changes on Home's relative wage and its comparative advantage is ambiguous.41 41 This ambiguity arises because R&D technologies play two roles in Taylor's (1993) model. First, R&D technologies determine the competitive margin for R&D production within each country and the division of R&D across countries. Second, R&D technologies determine, via the free entry process into the R&D, the value of a patent in any industry. Equiproportionate changes in Home and Foreign unit R&D labor requirements increase the value of patents and thus raise world expenditure levels. The effect of the increased world expenditure on Home produced goods is ambiguous without making further assumptions. Eliminating the heterogeneous research technologies assumption, results in factor price equalization in any equilibrium with both countries conducting R&D. In this case, an equiproportionate changes in R&D costs in his model, has no effect on Home's relative wage and its comparative advantage. Furthermore, if one assumes the absence of heterogeneous size of innovations and unity budget share for products in Taylor's model, then an equiproportionate changes in R&D costs makes Home conduct less R&D. On contrast, in the present model, the direction of the effect of an equiproportionate change of the unit R&D labor requirement depends on the way in which the scale effects property is removed. Under the PEG specification, an increase in R&D costs reduces Home's relative wage and raises its comparative advantage, while under the TEG specification, changes in R&D costs do not affect Home's relative wage and its comparative advantage. Under the PEG specification, the increase in R&D costs decreases the profitability of R&D activity. As a result, fewer firms enter into R&D races until the expected profits are driven back to zero. The adjustment to equilibrium requires a decrease in the level of Home's relative wage. The decrease in Home's relative wage raises its comparative advantage. On contrast, under the TEG specification, since the longrun level of R&D activity is fixed by the unchanged parameters of the model, an increase in R&D costs, increases the populationadjusted difficulty of R&D and thus leaving Home's relative wage and its comparative advantage unchanged. The analysis in the present model generates new additional findings. Under both specifications, the model generates a unique steadystate equilibrium in which there is complete specialization in both goods and R&D production within each industry. Trade between the two countries occurs only in goods and not in R&D services. In contrast to the work of Grossman and Helpman (1991c), factor price equalization does not hold in the steadystate equilibrium under either specification (Propositions 1 and 5). Under the PEG specification, Home's relative wage depends positively on the Foreign's relative size, Foreign's unit labor requirement in manufacturing, and the population growth rate. It depends negatively on the R&D difficulty parameter, the unit R&D labor requirement, the consumer's subjective discount rate, and Home's unit labor requirement in manufacturing (Proposition 2). In contrast to previous models (Grossman and Helpman (1991c), and Taylor (1993)), these results highlight the effects of population growth and the R&D difficulty on relative wages. Under the PEG specification, the range of goods produced in Home and exported depends positively on the R&D difficulty parameter, the unit R&D labor requirement, the consumer's subjective discount rate, and Foreign's unit labor requirement in manufacturing. It depends negatively on Foreign's relative size, the population growth rate, and Home's unit labor requirement in manufacturing (Proposition 3). These results also highlight the importance of population growth and the R&D difficulty parameter on the pattern of goods and R&D production, and the pattern of trade between the two countries. In addition, when the scale effects property is removed via the PEG specification, the global innovation rate depends positively on Home's relative size measured by its share of world's population, Foreign's relative size measured by its share of world's population, and the population growth rate. It depends negatively on the R&D difficulty parameter, the consumer's subjective discount rate, and the unit R&D labor requirement (Proposition 4). Some results that hold under the PEG specification analysis are reversed when the TEG specification is introduced in the model. Home's relative wage depends positively on the consumer's subjective discount rate and the R&D difficulty growth parameter. It depends negatively on the population growth rate (Proposition 6). The range of goods Home produces and exports depends positively on the population growth rate. It depends negatively on the consumer's subjective discount rate and the R&D difficulty growth parameter (Proposition 7). The global level of R&D investment, under the TEG specification is completely determined by the exogenous rate of population growth and the R&D difficulty growth parameter. Specifically, the global innovation rate is higher when the population of consumers grows faster or when R&D difficulty increases more slowly over time (Proposition 8). This chapter is organized as follows. Section 4.2 outlines the features of the model. Section 4.3 describes the steady state equilibrium of the model under the PEG specification and section 4.4 presents the comparative steady state results under the PEG specification. Section 4.5 analyzes the steady state equilibrium of the model under the TEG specification and section 4.6 presents the comparative steady state results under the TEG specification. Section 4.7 concludes this chapter by summarizing the key findings and suggesting possible extensions. The algebraic details and proofs of propositions in this chapter are relegated to Appendix C. 4.2 The Model This section develops a twocountry, dynamic, generalequilibrium model with the following features. Each country engages in two activities: the production of final consumption goods and research and development. Each of the two economies is populated by a continuum of industries indexed by 0 E [0, 1]. A single primary factor, labor, is used in both goods and R&D production for any industry. In each industry 0 firms are distinguished by the quality j of the products they produce. Higher values ofj denote higher quality and j is restricted to taking on integer values. At time t =0, the stateoftheart quality product in each industry is j=0, that is, some firm in each industry knows how to produce a j=0 quality product and no firm knows how to produce any higher quality product. The firm that knows how to produce the stateofthe art quality product in each industry is the global leader for that particular industry. At the same time, challengers in both countries engage in R&D to discover the next higher quality product that would replace the global leader in each industry. If the stateofthe art quality in an industry is j, then the next winner of an R&D race becomes the sole global producer of aj+ quality product. Thus, over time, products improve as innovations such as push each industry up its "quality ladder," as in Grossman and Helpman (1991c). I assume for simplicity, that all firms in the global economy know how to produce all products that are at least one step below the stateoftheart quality product in each industry. This assumption, which is standard in most qualityladders growth models, prevents the incumbent monopolist from engaging in further R&D, which is standard assumption in most qualityladder models. For clarity, I adopt the following conventions regarding notation. Henceforth, superscripts "h" and "f" identify functions and variables of "Home" and "Foreign" countries, respectively. Functions and variables without superscripts are related to the global economy, while functions and variables with subscripts are related to activities and firms within an industry. 4.2.1 Household Behavior Let N'(t) be country i's population at time t. I assume that each country' s population is growing at a common constant, exogenously given rate g, = N'(t)/N'(t) > 0. In each country there is a continuum of identical dynastic families that provide labor services in exchange for wages, and save by holding assets of firms engaged in R&D. Each individual member of a household is endowed with one unit of labor, which is inelastically supplied. I normalize the measure of families in each country at time 0 to equal unity. Thus, the population of workers at time t in country i is N(t) = e"' Each household in country i maximizes the discounted utility42 U= e '"" logu(t)dt, (1) where p > 0 is the constant subjective discount rate. In order for U to be bounded, I assume that the effective discount rate is positive (i.e., p g, > 0). Expression log u(t) captures the per capital utility at time t, which is defined as follows: logu(t) log[ 'q(j,0, t)]dO. (2) 42 Barro and SalaiMartin (1995, Ch. 2) provide more details on this formulation of the household's behavior within the context of the Ramsey model of growth. In equation (2), q (j, 0, t) denotes the quantity consumed of a final product of quality (i.e., the product that has experienced j quality improvements) in industry 09 [0,1] at time t. Parameter X > 1 measures the size of quality improvements (i.e., the size of innovations). At each point in time t, each household allocates its income to maximize (2) given the pre' all.r mrrrkci prices. Solving this optimal control problem yields a unit elastic demand function for the product in each industry with the lowest qualityadjusted price c'(t)N'(t) q'(j, t)= (t (3) p'(t) where c (t) is country i's per capital consumption expenditure, and p(t) is the market price of the good considered. Because goods within each industry adjusted for quality are by assumption identical, only the good with the lowest qualityadjusted price in each industry is consumed. The quantity demanded of all other goods is zero. The global demand for a particular product is given by aggregating equation (3) across the two countries to obtain q(j,a0, t)= ,q'(j,0,t). (4) i=h.r Given this static demand behavior, the intertemporal maximization problem of country i's representative household is equivalent to max je "(g, logc'(t)dt, (5) subject to the intertemporal budget constraint (t)= r (t)a(t) + w'(t) c' (t) ga', where a'(t) denotes the per capital financial assets in country i, w'(t) is the wage income of the representative household member in country i, and r'(t) is country i's instantaneous rate of return at time t. The solution to this maximization problem obeys the wellknown differential equation (t) r'(t) p, (6) c'(t) Equation (6) implies that a constant percapita consumption expenditure is optimal when the instantaneous interest rate in each country equals the consumer's subjective discount rate p. 4.2.2 Product Markets In each country firms can hire labor to produce any final consumption good 0 e[0,1]. Let L'(6, t) and Q'(6,t) respectively denote the amounts of labor devoted in manufacturing of final consumption good 0 in country i and the output of final consumption good 0 in country i. Then the production function of the final consumption good 0 in country i is given by the following equation Q'(, t)= L(,) (7) where a'(0) is the unit labor requirement associated with the final consumption good 0 in country i. I assume that each vertically differentiated good must be manufactured in the country in which the most recent product improvement has taken place. That is, I rule out international licensing and multinational corporations.43 43 Taylor (1993) incorporates multinational corporations in a model of endogenous growth and trade. In his model, innovations are always implemented on front line production technologies (i.e., technologies that are minimum cost given the prevailing wage rates) and when innovation and implementation occur at different countries, the resulting transactions are considered as imports and exports of R&D. Following Dombusch et al. (1977), the relative labor unit requirement for each good 0 is given by A(0)= A'(0) <0. (A.1) al(0) The relative unit labor requirement function in (A. ) is by assumption continuous, and decreasing in 0. The assumptions that goods within an industry are identical when adjusted for quality and Bertrand price competition in product markets imply that the monopolist in each industry engages in limit pricing. The assumption that the technology of all inferior quality products is public knowledge imply that the quality leader charges a single price, which is times the lowest manufacturing cost between the two countries: p =min a (0) wh,a (0)w ). (8) I choose the wage of foreign labor, w', as the numeraire of the model by setting: w'l. (9) I also assume that the wage of home labor, w h, which is also Home's relative wage, o, is greater than one44 w =CO >1. (10) Assumption (10) implies that the price of every top quality good is equal to p=ka(09). (11) It follows that the stream of profits of the incumbent monopolist that produces the state oftheart quality product in Home will be equal to 44 In Propositions I and 5, I provide sufficient conditions under which this assumption holds. ,(9,t)=[,, ., ; l j) E(t), (12) while the stream of profits of the incumbent monopolist that produces the stateoftheart quality product in Foreign will be equal to 7'(0, t) = [a (0)a (0)]q = E(t), (13) where E(t)[ch (t)N(t) + c (t)N (t)] is the world expenditure on final consumption goods. 4.2.3 R&D Races Labor is the only input engaged in R&D in any industry 0 e[0,1]. Let L' (0, t) and R'(0, t) respectively denote the amounts of labor devoted in R&D services in industry 0 in country i and the output of R&D services in industry 0 in country i. The production function of R&D services in industry 0 in country i exhibits constant returns and is given by the following equation45 R'(0,t) = L ) (14) ,R where aR is the unit labor requirement in the production of R&D services. Note that the absence of a superscript and the absence of the industry index 0 in the unit labor requirement imply that they are the same across countries, industries and goods of 45 The empirical evidence on returns to scale of R&D expenditure is inconclusive. Segerstrom and Zolnierek (1999) among others developed a model where they allow for diminishing returns to R&D effort at the firm level and industry leaders have R&D cost advantages over follower firms. In their model, when there are diminishing returns to R&D and the government does not intervene both industry leaders and follower firms invest in R&D. different quality levels. The absence of heterogeneous research technologies allows me to focus on the implications of assumption (A. 1) on the properties of the model.46 In each industry 0 there are global, sequential and stochastic R&D races that result in the discovery of higherquality final products. A challenger firm k that is located in country i E {h,f }targeting a quality leader in country i E {h,f} engages in R&D in industry 0 and discovers the next higherquality product with instantaneous probability 1k (0, t)dt, where dt is an infinitesimal interval of time and Ik'(6,t) R(, (15) X(t) where R, (0, t) denotes firm k's R&D outlays and X(t) captures the difficulty of R&D in industry 0 at time t. I assume that the returns to R&D investments are independently distributed across challengers, countries, industries, and over time. Therefore, the industrywide probability of innovation can be obtained from equation (14) by summing up the levels of R&D across all challengers in that country. That is, I'(, t)= I ;( (0 t) (16) k X(t) where R' (, t) denotes total R&D services in industry 0 in country i. Variable I'(0, t) is the effective R&D.47 The arrival of innovations in each industry follows a memoryless 46 Taylor (1993) has introduced heterogeneity in the research technologies and in the technological opportunity for improvements in technologies. The presence of heterogeneous research technologies makes trade in R&D services between countries possible. The absence of heterogeneous research technologies in the present model, makes the removal of scale effects more tractable, but eliminates the possibility of trade in R&D services between the two countries. 47 The variable I'(0, t) is the intensity of the Poisson process that governs the arrivals of innovations in industry 0 in country i. Dinopoulos and Syropoulos (2001) model the Poisson process with intensity I(0,t)= ZR'(0,t)/X(t)which equals the global rate of innovation in a typical industry. The function X(t) has been introduced in the endogenous growth literature after Jone's (1995a) empirical criticism of R&D based growth models generating scale effects. A recent body of theoretical literature has developed models of Schumpeterian growth without scale effects.48 Two alternative specifications have offered possible solutions to the scaleeffects property. The first specification proposed by Dinopoulos and Thompson (1996) removes the scaleeffects property by assuming that the level of R&D difficulty is proportional to the market size measured by the level of population, X(t)= kN(t), (17) where k > 0 is a parameter. This specification captures the notion that it is more difficult to introduce new products and replace old ones in a larger market. The model that results form this specification is called the permanent effects of growth (PEG) model because policies such as an R&D subsidy and tariffs can alter the percapita longrun growth rate.49 The second specification proposed by Segerstrom (1998) removes the scale strategic interactions between a typical incumbent and its challengers as a differential game for Poisson jump processes and derive the equilibrium conditions that govern the solution to a typical R&D contest. They also provide an informal and intuitive derivation of these conditions. In the present model, I follow their informal derivation to derive my results. 48 See Dinopoulos and Thompson (1999) for an overview of these models. 49 Dinopoulos and Thompson (1998) provide micro foundations for this specification in the context of a model with horizontal and vertical product differentiation. effects property by assuming that R&D becomes more difficult over time because "the most obvious ideas are discovered first." The model that results from this specification is called the temporary effects of growth (TEG) model. In this model, the longrun growth rate is proportional to the exogenous rate of population growth and it is not affected by any standard policy instruments. Under the TEG specification, R&D starts being equally difficult in all industries (X(0,0)= 1 for all ), and the level of R&D difficulty grows according to X(t) =_[lh(0,t)+If('i, i ]= =I(0,t), (18) X(t) where t > 0 is a constant. In subsequent sections I will consider each specification separately to analyze the steadystate equilibria and to derive their comparativestatic properties. Consider now the stockmarket valuation of temporary monopoly profits. Consumer savings are channeled to firms engaging in R&D through the stock market. The assumption of a continuum of industries allows consumers to diversify the industry specific risk completely and earn the market interest rate. At each instant in time, each challenger issues a flow of securities that promise to pay the flow of monopoly profits if the firm wins the R&D race and zero otherwise.50 Consider now the stockmarket valuation of the incumbent firm in each industry. Let V'(t) denote the expected global discounted profits of a successful innovator at time t in country i, when the global monopolist charges a price p for the stateoftheart quality product. Because each global 50 If the monopolist is located in Home, the monopoly profits are define by equation (12) and if the monopolist is located in Foreign the monopoly profits are defined by equation (13). quality leader is targeted by challengers from both countries who engage in R&D to discover the next higherquality product, a shareholder faces a capital loss V'(t) if further innovation occurs. The event that the next innovation will arrive occurs with instantaneous probability Idt, whereas the event that no innovation will arrive occurs with instantaneous probability 1Idt. Over a time interval dt, the shareholder of an incumbent's stock receives a dividend xt(t)dt and the value of the incumbent appreciates by dV (t) = [aV'(t)/5t]dt = V (t)dt. Perfect international capital mobility implies that rh = r' = r. The absence of profitable arbitrage opportunities requires the expected rate of return on stock issued by a successful innovator to be equal to the riskless rate of return r; that is, V'( ,tdttdt (6,t)d~ [V'(6, t)0] t[1 I(0, t)dt]dt + dt t) 1(0, t)dt = rdt. (19) v'(0,t) v'(0, t) v'(O, t) Taking limits in equation (19) as dt > 0 and rearranging terms appropriately gives the following expression for the value of monopoly profits VV ((6,,t) r(t)+ I(0, t) V(0, V'(0, t) A typical challenger k located in country i chooses the level of R&D investment Rk (0, t) to maximize the following expected discounted profits: R (O, t) V'(, t) k tdtw'aR (, t)dt, (21) X(t) where I dt = [Rk(0, t)/X(t)]dt is the instantaneous probability of discovering the next higherquality product and w'agRR (0,t) is the R&D cost of challenger k located in country i. Free entry into each R&D race drives the expected discounted profits of each challenger down to zero and yields the following zero profit condition: V'(t)= w'aRX(t). (22) The pattern of R&D production across the two countries can be determined by utilizing equations (20) and (22). Evaluating these equations on the competitive margin in R&D production, I can obtain the R&D schedule (i.e., the schedule of relative labor productivities in goods) as follows co=RD(0)= (23) a,(6) where RD(6) is continuous and decreasing in 6 For low values of 0, Home has higher relative labor productivity than Foreign, and thus it earns higher wage. Therefore, Home has comparative advantage in producing and conducting R&D the final goods with lower 0 and Foreign has comparative advantage in producing and conducting R&D the final goods with higher 0. The R&D schedule can be depicted in Figure 41. Lemma 1. Under assumption (A.1) and for any given value of the relative wage, ro e (, i',l/a (1), a (0)/ao (0)), there exists an industry 0 defined by equation (23) such that (a) firms are indifferent between conducting R&D in Foreign or in Home, (b) for each industry 0 e [0, ), only Home conducts R&D, (c) for each industry 0 e (, 1], only Foreign conducts R&D. Proof. See Appendix C. One can find the results from Lemma 1 in Dombusch et al. (1977). However, the derivation of Lemma 1 differs between the present model and the one in Dombusch et al. (1977). In their model, the results from Lemma 1 come from the assumption of perfect competition in all markets. In the present model, the intuition behind Lemma 1 results from the zero profit conditions regarding R&D. If in industry 0, R&D is undertaken by Home, then the zero profit conditions for R&D imply that Foreign has negative profits in this particular industry (see equations (20) and (22)). Thus, for all industries that Home undertakes R&D, Foreign has negative profits and does not engage in R&D in these industries. The reverse is true for those industries that Foreign undertakes R&D. Home has negative profits in these industries, so it does not engage in R&D in those industries. Thus, both countries sustain their comparative advantage. 4.2.4 Labor Markets Consider first the Home labor market. All workers are employed by firms in either production or R&D activities. Taking into account that each industry leader charges the same price p and that consumers only buy goods from industry leaders in equilibrium, it follows from (7) that total employment of labor in production in Home is [I Q'(O,t)axd9. Solving equation (14) for each industry leader's R&D employment L' (0, t) and then integrating across industries, total R&D employment by industry leaders in Home is E R"(O,t)aRdO. Thus, the full employment of labor condition for Home at time t is given by N"(t)= J Q' .' I 1" ,idi+ EJ Rh(0,t)ad. (24) 
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