ECONOMICS
OF 'WATER RESOURCES
PLANNING
McGRAWHILL SERIES IN WATER RESOURCES
AND ENVIRONMENTAL ENGINEERING
Ven T. Chow, Rolf Eliassen, and Ray K. Linsley, Consulting Editors
GRAF Hydraulics of Sediment Transport
HALL AND DRACUP Water Resources Systems Engineering
JAMES AND LEE Economics of Water Resources Planning
WALTON Groundwater Resource Evaluation
ECONOMICS
OF WATER RESOURCES
PLANNING
L. DOUGLAS JAMES
Associate Professor in the Environmental Resources Center
Georgia Institute of Technology
ROBERT R. LEE
Director, Idaho Water Resource Board
McGRAWHILL BOOK COMPANY
New York St. Louis San Francisco Diisseldorf
Johannesburg Kuala Lumpur London Mexico Montreal New Delhi
Panama Rio de Janeiro Singapore Sydney Toronto
ECONOMICS OF WATER RESOURCES PLANNING
Copyright @ 1971 by McGrawHill, Inc. All rights reserved.
Printed in the United States of America. No part of this
publication may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic,
mechanical, photocopying, recording, or otherwise, without
the prior written permission of the publisher.
Library of Congress Catalog Card Number 79115146
070322635
67890 KPKP 7987
PREFACE
In a world where an expanding population and an even more rapidly
expanding urbanindustrial development are intensifying the pressures
for a better planned water resources management program, the engineer
involved in water resources planning recognizes he must maintain a
planning methodology capable of producing a viable resource develop
ment program. At the same time, he wonders how he can do it. How can
he better structure his design to meet current human needs? How can he
make his design flexible enough to accommodate future human needs,
the nature of which he can scarcely anticipate? How can he devise a
management system for adequately sensing changes in human need as
they occur and quickly adjusting management policy and even system
design as is necessary?
The hydrologic cycle is the vast natural water resources system.
Water falls on the earth, travels downward, over, or under the surface of
the ground, reaches the ocean, and returns to the atmosphere through
evaporation induced by solar energy. But nature picks its own times and
places. The water resources planner seeks to modify the natural cycle by
structural measures that force the movement of water to times and places
better meeting known human needs. He also seeks to modify, by non
structural measures, the activities of man, so as to better conform to
known movement patterns. He considers a flood control channel and
flood plain management. He considers water supply systems to existing
cities and the development of new cities closer to available supplies. The
best management selects the best possible combination of measures.
The concept of the optimum program is changing as men recognize
the wisdom, if not the necessity, of taking additional factors into con
sideration. Pressures from many groups and disciplines have contri
buted to the expanding awareness of relevant considerations. Business
management has long been concerned with decision making to maximize
returns to the firm. Engineering economy provides the procedures for
the cost analysis of alternatives for the purpose of finding the leastcost
approach, irrespective of viewpoint. Studies in microeconomics examine
VI PREFACE
benefits produced as well as costs incurred, and provide rules for maxi
imizing benefits minus costs as a step in optimization to enhance the wel
fare of the general public. The other social sciences provide further insight
into how the welfare of man, as an individual being and as part of a
group, can be improved. The biological sciences extend the analysis to all
life systems. The institutions responsible for actual resource development
draw from all of these sources in formulating their plans, and improvise
to supplement the available procedures where action cannot wait for
research. As the state of the art now stands, the concepts and procedures
required for planning are scattered through the literature of many
disciplines, some in journal publications but many in otherwise un
published research reports and conference proceedings. The purpose of
this book is to consolidate into a single volume the basic economic
concepts required in water resources planning.
In one companion volume, "Water Resources Engineering,"'
Linsley and Franzini present the basic physical system and survey the
available structural measures for engineered water resources development.
In another volume, "Water Resources Systems Engineering,"2 Hall and
Dracup present the procedures for analysis of how water resources
systems may be designed to function together as a whole to better achieve
specific objectives. In this book, we are seeking to examine how relevant
objectives can be specified as well as the reasoning needed to apply rather
abstract concepts of social welfare to specific design choices.
The reader should approach the material contained in this book from
the viewpoint of developing a philosophy of planning. The detailed
procedural steps as presented are intended to illustrate basic concepts,
rather than to finalize a method to be followed by rote. These concepts
and the written material describing them have been presented by the
authors in teaching courses on water resources planning and have been
applied by the authors in their planning experience.
The material in this book has been used as the basic text for a one
year course sequence dealing with the economic, social, and institutional
issues involved in water resources management. Parts 1, 2, and 6 plus
Chapters 8 and 9 can be adopted to a onesemester senior or onesemester
firstyear graduate course in public works economics (appropriate within
programs in transportation, air pollution, and civil engineering manage
ment as well as programs in water resources). The remaining portions can
then be covered in a secondsemester course for those specifically inter
ested in water resources. Although the material is covered in a manner
L Ray K. Linsley and Joseph B. Franzini, "Water Resources Engineering" (New York: McGraw
Hill Book Company, 1964).
SWarren A. Hall and John A. Dracup, "Water Resources Systems Engineering" (New York:
McGrawHill Book Company, 1970).
PREFACE VIl
requiring no specific prerequisites, owing to the diversity of background
among students interested in resource development, some background
in one or more of the areas of engineering economy, sophomore micro
economics, hydrology, water resources engineering, and systems analysis
may add depth to the understanding of selected sections.
The authors gratefully acknowledge the contributions made in
directly to the book through discussion with numerous colleagues.
Particular thanks are extended to Professors Ray K. Linsley and Eugene
L. Grant of Stanford University and Dr. Charles W. Howe of Resources
for the Future for their review of and contribution to various parts of the
manuscript. Mrs. Betty Bradshaw and Mrs. Alice Taylor spent many
hours typing preliminary drafts, and Miss Pat Miller typed much of the
final manuscript.
L. DOUGLAS JAMES
ROBERT R. LEE
CONTENTS
PREFACE v
1 ENGINEERING ECONOMY
1 PRINCIPLES OF ENGINEERING ECONOMICS 3
11 Equivalence of Kind 3
12 Equivalence of Time 4
13 Whose Viewpoint3 5
14 Sunk Cost 6
15 Incremental Cost 7
16 Intangible Values 7
17 Predictive Uncertainty 8
18 Planning Horizons 9
19 Structuring Alternatives 11
2 MATHEMATICS OF ECONOMIC ANALYSIS 15
Formulating the Analysis
21 Defining the Alternatives 15
22 Physical Consequences 16
23 Cash Flow Diagram 16
Discounting Factors
24 Singlepayment Factors 18
25 Uniformannualseries Factors 19
26 Uniformgradientseries Factors 21
27 Nonuniformgradientseries Factors 24
28 Other Cash Flow Patterns 26
Discounting Techniques
29 Presentworth Method 28
210 Rateofreturn Method 30
211 BenefitCost Ratio Method 32
212 Annualcost Method 33
213 Evaluation of Discounting Techniques 34
Other Approaches
214 Unreliable Techniques 35
2 MICROECONOMICS AND EFFICIENT
RESOURCE ALLOCATION
3 PRICE THEORY AND RESOURCE ALLOCATION 43
Introduction to Microeconomics
31 The Market Economy 43
32 Pure Competition 46
X CONTENTS
33 Market Demand 47
34 Price Elasticity 49
35 Market Supply 50
36 Market Price Determination 51
37 Results of Shifts in Demand and Supply 51
38 Significance of Market Equilibrium 52
Consumer Demand
39 Indifference Curves 54
310 Maximization of Satisfaction 55
311 Consumerdemand Curves 56
312 Aggregatedemand Curves 57
4 CONDITIONS OF PROJECT OPTIMALITY 61
Production Theory
41 Input and Output 61
42 The Production Function 63
43 The Objective Function 64
44! Cost and Benefit Curves 64
45 Optimality Conditions 66
Geometric Derivation of Basic Rules
46 Optimality Condition 1: Combination of Inputs 67
47 Optimality Condition 2: Combination of Outputs 69
48 Optimality Condition 3: Level of Output 72
Mathematical Derivation of Basic Rules
49 Lagrange Multipliers 74
410 Application of the Lagrange Multiplier 75
411 Three Basic Optimality Conditions 76
412 Application of Optimality Conditions 77
Market Allocation under Pure Competition
413 Very Short Run Analysis 82
414 Shortrun Analysis 82
415 Longrun Analysis 85
Applications
416 Use of Supply and Demand Curves in Project Evaluation 86
417 Market Allocation under Pure Monopoly 88
5 WELFARE ECONOMICS 95
Economic Analysis and Social Objectives
51 The Role of Welfare Economics 95
52 The Social Welfare Function 96
53 Firstorder (Social) Efficiency 97
54 The Question of Sovereignty 98
55 The Question of Goals 101
56 Basic Optimality Criteria 102
57 Secondorder (Economic) Efliciency 103
Adjustments Required by Imperfect Markets
58 Adjustment Procedure 105
59 Public Wants 106
CONTENTS Xi
510 External Effects 107
511 Natural Monopoly 110
Adjustments Required by Multiple Goals
512 Stabilization of the Economy 112
513 Income Redistribution 114
514 Regional Development 115
515 Environmental Quality 116
6 DISCOUNT RATE 119
61 Capital Formation 119
62 Discount Rate vs. Interest Rate 120
63 Opportunity Cost 121
64 The Capital Market 121
65 Deficiencies in the Capital Market 124
66 Basic Approaches for Dealing with Market Deficiencies 126
67 Specific Approaches for Picking a Discount Rate 126
68 Current Discounting Practice 129
69 Consequences of Discounting at a Low Rate 130
3 PLANNING IN A REAL WORLD SETTING
7 THE INSTITUTIONAL FRAMEWORK 137
Organizational Considerations in Economic Analysis
71 The Division of Responsibility 137
72 Interagency Coordination 141
73 Senate Document 97 142
74 The Water Resources Planning Act of 1965 143
75 Steps in Project Development 145
Legal Consideration in Economic Analysis
76 The Legal Hierarchy 14~8
77 Surfacewater Law 149
78 Groundwater Law 151
79 Law on Diffuse Surface Waters 152
710 Other Miscellaneous Laws 152
711 The Federal Government and Water Law 153
712 Constitutional Basis for Federal Project Development 153
713 Government Liability 156
8 BENEFITCOST ANALYSIS 161
Project Evaluation
81 Feasibility Tests 161
82 The Need for Testing Economic Feasibility 163
83 Defining Benefits and Costs 163
84 BenefitCost Categories 164
BenefitCost Measurement
85 Direct Primary Benefits 169
86 Indirect Primary Benefits 171
87 Landenhancement Benefits 171
88 Secondary Benefits 171
xii CONTENTS
89 Employment Benefits 173
810 Incomeredistribution Benefits 174
811 Other Public Benefits 178
812 Intangible Benefits 178
813 Associated and Induced Costs 178
814 Project Installation Cost 179
815 Operation, Maintenance, and Replacement Costs 181
BenefitCost Variation
816 The Probabilistic Approach 181
817 Sensitivity Analysis 186
The Value of BenefitCost Analysis
818 Project Formulation 187
819 Adequacy of Measurement 189
820 A Critique of BenefitCost Analysis 191
9 THE DYNAMICS OF PROJECT ANALYSIS 197
91 The Role of Projections 197
92 Population Projection 198
93 InputOutput Analysis 200
94 Consumer Preference Projection 204
95 Conservation Dynamics and Supply Projections 205
96 Price Indices 207
97 Normalized Prices 209
98 Inflation 209
99 Stages of Project Life 210
910 Optimum Construction Date 211
911 Stage Construction 215
912 Project Formulation and Ranking under Capital Rationing 216
4 ECONOMIC PLANNING BY PROJECT PURPOSE
10 FLOOD CONTROL 229
The Planning Context
101 Definition 229
102 Historical Development 229
103 Flood Hydrology 231
Developing the Supply
104 Structural Measures 235
105 Land Treatment 238
106 Flood Proofing 239
107 Land Use Adjustment 240
108 Flood Insurance 242
109 The Supply Curve 243
Estimating the Demand
1010 Flood Severity 250
1011 Flood Damages 250
1012 Flood Protection Benefits 257
CONTENTS X111
Project Feasibility
1013 Economic Feasibility 258
1014 Evaluation of Alternative Measures 261
1015 Optimum Level of Protection 261
1016 Financial Feasibility 262
11 DRAINAGE 267
The Planning Context
111 Definition 267
112 Historical Development 268
Developing the Supply
113 Storm Drainage 268
114 Highway Drainage 270
115 Land Drainage 271
Estimating the Demand
116 Storm Drainage 274
117 Highway Drainage 274
118 Land Drainage 277
Project Feasibility
119 Economic Feasibility 281
1110 Financial Feasibility 281
12 WATER SUPPLY 285
The Planning Context
121 Definition 285
122 Historical Development 285
Developing the Supply
123 Fresh Water 287
124 Saline Water 295
125 Land Treatment 296
126 Distribution System 297
127 User Distribution 299
128 Return Flows 299
Estimating Irrigation Demand
129 Cropping Pattern 301
1210 Crop Income 302
1211 Crop Water Requirement 302
1212 Monthly Crop Water Requirement 306
1213 Distribution Losses 308
1214 Other Water Requirements 309
1215 Demand Curve 310
Estimating Urban Demand
1216 Urban Use at Base Price 312
1217 Demand Curve 314
1218 Benefits fromn Urban Water Supply 316
XIV CONTENTS
Project Feasibility
1219 Economic Feasibility 319
1220 Financial Feasibility 320
13 HYDROELECTRIC POWER 325
The Planning Context
131 Definition 325
132 Historical Development 325
133 Selecting the Method of Generation 326
Developing the Supply
134 Hydroelectric Plant Terminology 329
135 Hydroelectric Plant Technology 330
136 Hydroelectric Plant Efficiency 331
137 Power Transmission 331
138 Power Systems 332
139 Hydroelectric Plant Hydrology 333
1310 The Supply Curve 340
Estimating the Demand
1311 Powerdemand Curves 341
1312 The Powermarket Survey 341
1313 The Alternativecost Approach 342
1314 Project Benefits 344
Project Feasibility
1315 Economic Feasibility 344
1316 Government Regulation 347
1317 Government Ownership 347
1318 Financial Feasibility 349
14 NAVIGATION 353
The Planning Context
141 Definition 353
142 Historical Development 353
143 Types of Facilities 354
Developing the Supply
144 Trial Design 356
145 Harbor Capacity 357
146 Harbor Cost 357
147 Waterway Improvement Alternatives 358
148 Waterway Capacity 360
149 Waterway and Vessel Cost 361
Estimating the Demand
1410 Navigation Benefit 364
1411 Transportation Cost Analysis 365
1412 Commodities Shipped 366
1413 Demand for Waterway Facilities 367
1414 Demand for Harbor Facilities 367
CONTENTS XV
1415 Demand for Combined Harbor and Waterway
Improvement 368
1416 Recreation Demand 368
Project Feasibility
1417 Economic Feasibility 368
1418 Financial Feasibility 369
15 WATER QUALITY CONTROL 373
The Planning Context
151 Definition 373
152 Types of Pollution 373
153 Coordination with Other Project Purposes 375
154 Historical Development 378
Developing the Supply
155 Wastewater Treatment 379
156 Wastewater Dilution 381
157 Wastewater Separation 381
158 Industrial Process Modification 382
159 The Supply Curve 382
Estimating the Demand
1510 Damage Evaluation 383
1511 Problems in Damage Measurement 386
1512 Benefit Measurement 387
System Management
1513 Economic Optimization 388
1514 Coordinated System Management 389
1515 Establishment of a Management System 391
16 RECREATION 395
The Planning Context
161 Definition 395
162 Historical Development' 395
Developing the Supply
163 Facility Design 396
164 Estimating Recreation Capacity 398
165 The Activity Composite 399
166 The Capacity Coeflcients 400
167 User Days and Activity Days 400
168 The Marginalcost Curve 401
Estimating the Demand
169 Time Distribution of Reservoir Visitation 402
1610 Overall Utilization of Reservoir Capacity 405
1611 Factors Determining Potential Visitation 407
1612 Estimating Recreation Benefits 408
1613 Relating Recreation Visitation to Travel Distance 410
1614 Expressing Travel Distance in Economic Units 411
XVI CONTENTS
1615 Developing the Demand Curve 412
1616 Totalbenefit Curve 416
Project Feasibility
1617 Economic Feasibility 417
1618 Financial Feasibility 417
17 FISH AND WILDLIFE ENHANCEMENT 421
The Planning Context
171 Definition 421
172 Historical Development 422
Developing the Supply
173 Warmwater Fish 423
174 Coldwater Fish 424
175 Anadromous Fish 424
176 Waterfowl 428
177 Insect Control 428
178 The Supply Curve 429
Estimating the Demand
179 Recreation Demand 430
1710 Commercial Demand 431
1711 Environmental Conservation 431
1712 Insect Control 431
Project Feasibility
1713 Financial Feasibility 432
5 MULTIPURPOSE WATER RESOURCES
DEVELOPMENT
18 GRAPHIC OPTIMIZATION TECHNIQUES 437
The Graphic Approach
181 The Method 437
182 Data for Siample Study 438
Optimum Singlepurpose Projects
183 Irrigation 441
184 Flood Control 443
185 Water Quality Control 443
Optimum Dualpurpose Projects
186 Irrigation and Flood Control 444
187 Flood Control and Water Quality 447
188 Irrigation and Water Quality 448
Overall Evaluation
189 Optimum Triplepurpose Project 449
1810 Summary 452
19 ANALYTICAL OPTIMIZATION TECHNIQUES 453
191 The Analytic Approach 453
192 Linear Programming 454
CONTENTS XV11
193 Dynamic Programming 457
194 Queuing Theory 458
195 Criticalpath Method 459
196 Harvard Studies 460
197 Summary 461
20 OPTIMIZATION BY SIMULATION 463
The Simulation Approach
201 Simulation Models 463
202 Simulation Parameters 466
Operating Procedure
203 Deriving Operating Rules 469
204 Use of Flood Storage 470
205 Use of Total Storage 471
206 Release of Stored Water 472
207 Release by Reservoir 475
208 Use of Available Water 475
209 Release Elevation 476
Specifying Uncontrolled Events
2010 Economic Event Sequences 477
2011 Hydrologic Event Sequences 481
The Simulation Procedure
2012 State Variables 485
2013 System Yield 486
2014 Searching for the Optimum System 486
2015 Method Evaluation 488
21 PLANNING IMPLEMENTATION 493
The Planning Procedure
211 The Planning Ideal 493
212 Levels of Decision Making 495
213 The Total Water Resources Program 497
214 Program Coordination 499
215 Human Factors in Planning 499
Interactions with Other Resources
216 Land Use and Project Output 501
217 Land Use and Project Demand 502
218 Urban Land Use Planning 503
219 Water and Transportation Planning 504
2110 Water and Weather Planning 505
6 FINANCIAL ANALYSIS
22 FINANCIAL FEASIBILITY 511
Financial Studies
221 The Role of Financial Analysis 511
222 Distinctions from Economic Analysis 511
xviii CONTENTB
Federal Financing
223 Sma~llProjects 513
224 Large Projects 515
State and Local Financing
225 Funding Options 517
226 Bond Financing 520
227 General Obligation Bonds 521
228 Revenue Bonds 522
229 Assessment Bonds 523
Planning the Debt Structurb
2210 Coordinating the Debt Service 523
2211 Jointventure Financing 525
23 COST ALLOCATION 527
231 The Need for Cost Allocation 527
232 Separable and Nonseparable Costs 527
233 Allocation Rules 528
234 Allocation Consequences 529
235 Cost Center Classification 530
236 Definition of Cost 530
237 Separable Cost Identification 531
238 Allocation Vehicle 532
239 Method Matrix 532
2310 Evaluation of Alternative Methods 533
2311 Summary 537
24 WATER RESOURCES PRICING 541
241 Charging Functions 541
242 Charging Dilemma 542
243 Reconciling Economic and Financial Requirements 544
244 Total Charges 545
245 Charging Vehicle 546
246 Utility Rates 547
APPENDIXES
A Compoundinterest Factors 551
B Gradientseries Factors 573
C Accelerated, Normal, Deferredseries Factors 595
INDEX 607
ECONOMICS
OF WATER RESOURCES
PLANNING
ENGINEERING
ECONOMY
Engineering economy is the science of applying economic criteria to
select the best of a group of alternative engineering designs. A design if
implemented will produce a time pattern of consequences which must be
predicted, evaluated, and compared. The method of comparison might
more appropriately be called decision economics since the principles may
just as well be used to select among the choices available to other dis
ciplines. For example, the same type of analysis could be called business
economics, education economics, or medical economics depending on the
skills needed to define the choices. Although the approach emphasizes
comparison in economic units, it also includes identification, for com
parison to the fullest extent possible, of those consequences which do not
relate to economic goals or cannot be expressed in money terms.
Arthur MI/. Wellington pioneered modern engineering economics
through its application to the analysis of alternative railway locations in
1877.' Wellington was prompted by the neglect of economic factors in
selecting railway location at a time when capital investment in railroads
exceeded that in all manufacturing endeavors. The approach has been
more thoroughly developed and applied to many other kinds of choices
through the years, and current techniques are presented in a number of
recent works.2
SArthur M. Wellington, "The Economic Theory of the Location of Railways," 1st ed. (New York:
John Wiley & Sons, Inc., 1877).
For example, see Eugene L. Grant and Grant Ireson, "Principles of Engineering Economy," 5th ed.
(New York: The Ronald Press Company, 1970); and E. P. DeGarmo, "Engineering Economy,"
4th ed. (New York: The Macmillan Company, 1967).
2 ENGINEERING ECONOMY
The principles and techniques of engineering economics are not
always clearly understood nor correctly applied in water resources
planning. In separate books, Eckstein and McKean examined federal
practices for analyzing water resources projects and suggested numerous
revisions.' Hirshleifer, DeHaven, and Milliman received two large water
resources projects, one in New York and the other in California, and
found serious conceptual errors in official economic feasibility studies.2
Lee produced similar findings in an examination of procedures used in
analysing water projects on the local government level.3 These failures
demonstrate the need for the water resources planner to thoroughly under
stand the principles and techniques of engineering economics. Part 1
reviews the basic principles of engineering economics as applied to public
works in general and water resources development more specifically and
presents the mathematics required in their application. Part 2 builds on
this foundation by examining some of the more knotty conceptual
problems in economic analysis.
Otto Eckstein, "Water Resource Development: The Economics of Project Evaluation" (Cambridge,
Mass.: Harvard University Press, 1958); Roland McKean, "Efficiency in Government through
Systems Analysis" (New York: John W~iley & Bons, Inc., 1958).
Jack Hirshleifer, James C. DeHaven, and Jerome W. Milliman, "Water Supply: Economies, Tech
nology, and Policy" (Chicago: The University of Chicago Prees, 1960).
SRobert R. Lee, "Local Government Public Works DecisionMaking" (Stanford, Calif.: Stanford
University Institute in Engineering Economic Systems, 1964).
CHAPTER
ONE
PRINCIPLES
OF ENGINEERING
ECONOMICS
The principles of engineering economics guide the structuring of alter
natives so they may be compared to determine which should be selected.
The evaluation process requires prediction of the consequences expected
to result from picking the alternative, estimation of the magnitude of
each consequence, and conversion of each consequence magnitude into
commensurable units. The purpose of this chapter is to review the con
ceptual problems and basic principles involved in the process.
11 EQUIVALENCE OF KIND The major obstacles to expressing
the consequences of alternative courses of action in commensurable units
are differences in kind and differences in time. The two differences may
be illustrated through the example of comparing two alternative irriga
tion projects. One project provides irrigation water for peaches. The
second provides water for cotton. Construction of the first project will
produce x tons of peaches. Construction of the second will produce y
bales of cotton. If the two projects can be constructed for equal cost,
selection depends on whether x tons of peaches or y bales of cotton is
more valuable.
As long as the two outputs are expressed in these diverse units, the
projects cannot be compared. Only when common units are used is com
parison possible. The first step in economic analysis must be to find a
common value unit. One might use tons of peaches. A farmer selling y
bales of cotton might receive the same price as if he had sold y' tons of
peaches. The decision could then be based on whether x or y' was the
larger. One might use bales of cotton as the common unit, express the
4 ENGINEERING ECONOMY
value of x tons of peaches as x' bales of cotton, and make the decision.
As a third approach, farmers grow both peaches and cotton to buy bread
for their families. A farmer could sell x tons of peaches and buy x" loaves
of bread. A farmer could sell y bales of cotton and buy y" loaves of bread.
The decision could be based on a comparison between x" and y".
However, such approaches are uncommon because society has estab
lished a system of units for comparing relative value. Tons of peaches,
bales of cotton, and loaves of bread may all be evaluated in monetary
units. The use of monetary units in economy studies is based solely on
convenience and does not imply a materialistic approach of considering
only monetary profit while ignoring the many values of life, health, and
happiness which can not be expressed in money terms. Handling intan
gible values will be discussed later (Sec. 16).
The simple fact is that diverse values are understood by more people
when expressed in monetary terms than when any other kind of unit is
used. Far more people can visualize worth in dollars than in tons of
peaches, bales of cotton, or loaves of bread. The proper approach for
comparing the two irrigation projects is to convert both tons of peaches
and bales of cotton into dollars, compare the dollar totals, and (provided
intangible values do not indicate otherwise) select the project producing
the greater total.
16 EQUIVALENCE OF TIME An irrigation project will provide
water for many years. In evaluating the example project, should peaches
produced this year be reckoned as having equal value to those expected
to be produced 30 years from now? Most people would be more inclined
to invest a dollar to produce 5 lb of peaches now than to invest the dollar
to produce 5 lb of peaches 30 years from now.
An earlier realization of investment returns is desirable for the in
vestor because it gives him greater flexibility for future action. If the
returns are needed for consumption, they become available with less wait
ing. If they are to be reinvested, an earlier reinvestment will speed sub
sequent returns and result in a more rapid expansion of capital. To fail to
differentiate returns by date is to say all economic expansion rates are
equally desirable.
In order to make realistic investment decisions, each monetary value
must be identified by both amount and time. Amounts at different times
should not be directly compared or combined. They are not in common
units. Amounts in different time periods may be made equivalent by
multiplying future amounts by a factor becoming progressively smaller
into the more distant future. The discount rate is the time rate of decrease
PRINCIPLES OF ENGINEERING ECONOMIcs 5
in this factor expressed in percent per time period. An investment of a
dollar at an annual rate of return of 5 percent would yield $1.05 a year
hence. Similarly, $1.05 available a year from now is equivalent to $61.00
now when discounted at 5 percent.
The discount rate used has a great influence on the project selected.
Future benefits and costs receive less weight with higher, and more weight
with lower, discount rates. High discount rates favor projects with little
initial investment, while low discount rates favor capital intensive proj ects.
Determination of the proper discount rate for water resources planning is
discussed in detail in Chap. 6.
13 WHOSE VIEWPOINTP Monetary value depends on the view
point taken in the evaluation. The grower who produces a ton of peaches
will equate value with sale price. The community will add to this the
gains accruing to food processors, farm workers, farm suppliers, and other
individuals who profit indirectly. However, from the national viewpoint,
committing resources to one community to grow more peaches may deny
investment capital to another. Furthermore, peaches grown in different
communities are competing goods.
The above description thus pinpoints the three viewpoints possible
in an engineering economy study.'
1 That of the group sponsoring or financing the project. Consider only
consequences affecting this group.
2 That of all the people in a specific area such as a state, county, or
special district. Consider only consequences affecting those living in
this defined area.
8 That of the entire nation. Consider all consequences to whomsoever
they may accrue.
Viewpoint 1 is based on the premise that the sponsoring group should
promote its own welfare. It is a legitimate viewpoint for private enterprise,
but one of the primary justifications for action by government is to avoid
the adverse consequences of individuals' putting personal above public
welfare. Therefore, there appears to be no justification for a public
agency's taking anything less than the public viewpoint. When a planning
group ignores conflicting viewpoints, a higher level of government must
bring about any adjustments necessary to protect the public interest.
SEugene L. Grant and Grant Ireson, "Principles of Engineering Economy," 4th ed. (New York:
The Ronald Press Company, 1960), p. 445. For a parallel discussion, see Tillo Kuhn, "Public Enter
prise Economies and Transport Problems" (Berkeley: University of California Press, 1962), pp.
1316.
6 ENGINEERING ECONOMY
Practical realities may restrict the freedom of a local government to
take the national viewpoint.' First, the cost of tracing the consequences of
proposed alternatives beyond its jurisdiction may be excessive. Secondly,
a local government is subj ect to much political pressure from the taxpayers
who support it but little from those living outside its jurisdiction. The
tendency is to ignore these outside consequences. Higher levels of govern
ment must be responsible for making sure local planners adequately con
sider project consequences occurring in other areas.
Viewpoint 3 should, in principle, be taken by every level of govern
ment to maximize aggregate national welfare in the long run. Where
federal programs, such as the reclamation of the arid West or the economic
development of Appalachia, are designed to achieve regional goals, project
consequences should be evaluated from both the national and regional
viewpoints. Regional interests may try to influence federal agencies to
select projects producing regional benefit, where they must repay only a
fraction of project costs.2 The decision maker needs to know if such a
project can be justified from the national viewpoint and weigh the national
sacrifice required to achieve local goals.
14 SUNK COST The justification for following a course of action
depends on the events occurring with it being better than those occurring
without it, by an amount exceeding its implementation cost. An engineer
ing economy study need analyze only differences between alternatives
and differences between resulting consequences. All costs and benefits
unaffected by which alternative is chosen should be disregarded. Obviously,
past events have already occurred and cannot be retracted by future
action. Past expenditure, or sunk costs, are past events and thus should
have no influence on deciding among alternatives except as they affect
future cash flows.
Despite their economic irrelevance, sunk costs have often been allowed
to influence decisions for two main reasons. The decision makers may have
a psychological, political, or even a legal commitment to continue a past
policy so that past efforts are not wasted. Secondly, accounting records
indicating an undepreciated book value for assets having no economic
worth may restrict freedom to make new investment. However, in no case
are past mistakes a legitimate excuse for continuing a policy which cannot
be justified by future benefits.
I Roland N. McKean, Costs and Benefits from Different Viewpoints, "Public Expenditure Decisions
in the Urban Community" (Washington: Resources for the Future, 1962), pp. 148151.
see Kubn, op. cit., p. 18. Kuhn would have the decisionmaking authority set at the highest level so
that the broadest view of the public interest is observed. McKean, "Costs and Benefits from Differ
ent Viewpoints," p. 147, sees central planning as too often leading "to planning of the people, by the
few, and for the few."
PRINCIPLES OF ENGINEERING ECONOMICS 7
The sunkcost principle is illustrated in the following example. Sup
pose $b5 million have been spent on a hydropower installation ultimately
costing $b10 million. A steam plant costing an estimated $b3 million is
subsequently found to be capable of supplying the same energy. Which
facility should be selected, assuming all other future costs to be the same?
The $F5 million already spent on the hydropower facility is a sunk cost,
hence is irrelevant. Since the cost of the steam plant is less than the
remaining cost of the hydropower installation, the steam plant should be
selected. Continuing the initial project is not in the economic interest of
the public.
15 INCREMENTAL COST According to the incrementalcost prin
ciple, the change in benefits and the change in costs resulting from a given
decision determine the merit of that decision. Each project segment should
be judged on its own merits. The decision to enlarge a project should be
justified by the enlargement's increasing benefits more than it increases
cost.
For illustration, consider a 10,000 acreft reservoir which a city has
determined to build for $61 million. Before construction begins, increasing
the storage to 20,000 acreft and the cost to $1,500,000 is found to achieve
$b600,000 in downstream flood control benefits. The incorrect averagecost
approach would preclude flood control on the basis that half the storage
means half the cost, and $750,000 exceeds $b600,000. The correct incre
mentalcost approach would include flood control because the additional
expenditure of $500,000 is exceeded by the benefit of $b600,000.
By the same token, an element costing $b50,000 but producing only
$20,000 in benefits should not be justified by inclusion in a large project
with costs of $2 million and benefits of $3 million. The maximum net
benefit is achieved with that element excluded.
16 INTANGIBLE VrALUES Even though an economy study seeks
to evaluate all consequences in commensurable monetary units, many
values defy such quantification. Unique or extremely rare items such as
species of plant or animal life or sights of unusual beauty have no acknowl
edged money value. Neither have direct effects on human beings physically
through loss of health or life, emotionally through loss of national prestige
or personal integrity, or psychologically through environmental changes.
Nor do monetary values serve to measure the achievement of such extra
economic goals as income redistribution, increased economic stability,
or improved environmental quality (Sec. 51). Each value which cannot be
expressed in monetary terms is called an intangible or irreducible.
8 ENGINEERING ECONOMY
Inability to express a value in economic units does not necessarily
preclude evaluation in other units. All intangible values should be quanti
fied as precisely as possible. Vague statements on threat to human life are
not nearly as helpful as a precise statement on the number of lives expected
to be lost. In weighing whether a given sacrifice in economic value is
worthwhile to achieve a goal, the decision maker should have access to the
best possible information on the nature of the intangible consequences as
well as the magnitude of the economic consequences.
17 PREDICTIVE UNCERTAINTY Because economic analysis
compares future consequences of engineering alternatives, the reliability
of each conclusion depends on the ability to predict future events. A
project may only appear to be economically feasible because of incorrect
predictions. No matter how much data or experience one has, predicting
the future is inherently uncertain.
Uncertainty with respect to water resources project evaluation has
been described by M~cKean as "inherent in the nature of things and is not
necessarily evidence of lazy or careless estimation."' He gives five
classifications :
1 Uncertainty about objectives. Even though planning objectives as cur
rently conceived may be perfectly clear, future developments may
significantly alter social goals.
d Uncertainty about constraints on the system. It is computationally in
feasible to plan all economic decisions simultaneously. A particular
analysis must be performed in the context of constraints imposed by
outside events. The price of steel may be taken as given in an economy
study without attempting to determine an optimum price through
industry analysis. However, future developments may produce un
predictable price changes.
8 Uncertainty about public response. Even though a thorough analysis
may indicate the need for projectproduced services, public inertia
against learning new ways or psychological commitment to established
procedures will affect their use in an often unpredictable manner.
SUncertainty about technological change. Even though a project currently
produces a needed output at low cost, innovations or technological
changes may cause the output to be no longer needed or introduce an
even less costly production process.
5 Uncertainty about the chance element in recurring events. Even when the
probability of occurrence of random events can be established statisti
SRoland N. McKean, "Efliciency in Government through Systems Analysis" (New York: John
Wiley &r Sons, Inc. 1958), pp. 6568.
PRINCIPLES OF ENGINEERING ECONOMIcs 9
cally, the precise time of occurrence (of flood peaks, for example) is
never known in advance. Furthermore, the many random elements in
any system cumulatively increase overall uncertainty.
Widely used approaches to treating uncertainty include (1) applying
preselected percentages to increase costs or reduce benefits, (2) limiting
the period of analysis, or (3) adding a risk increment to the discount rate.
However, because each of these approaches requires selection of a numeri
cal factor without providing any help on how a specific value is to be
selected, Eckstein has well argued:
These crude adjustments are intellectually not very satisfying and one
should try to derive better adj ustments from explicit objective functions and
from the probabilistic nature of benefits.'
A more satisfactory approach is to recognize explicitly that project
effects should not be predicted as single fixed values but rather as variables
having some probability distribution of possible values. A more detailed
description of specific approaches is found following Sec. 816, and decision
theory techniques are presented in a number of works by other authors.2
18 PLANNING HORIZONS The planning horizon is the most dis
tant future time considered in an engineering economy study. The inherent
uncertainty of predicting the more distant future favors short planning
periods, but the need for analysis of the longrun effects of plans to meet
immediate needs favors a longer period. Actually, four different periods
of time must be considered in any economic analysis: (1) the economic
life, (2) the physical life, (3) the period of analysis, and (4) the construe
tion horizon.
The economic life ends when the incremental benefits from continued
use no longer exceed the incremental cost of continued operation. Economic
life is usually shorter for such project elements as pumps and canal linings
than for a water resources project as a whole.
The physical life ends when a facility can no longer physically perform
its intended function. While the economic life never exceeds the physical
life, it may be shorter because of obsolescence and changing demands for
services. As an example, electric generation by nuclear power may become
x Otto Eckstein, A Survey of the Theory of Public Expenditure Criteria, in National Bureau of
Economic Research, "Public Finances: Needs, Sources, and Utilization" (Princeton, N.J.: Princeton
University Press, 1961), p,. 470.
See H. Chernoff and L. Moses, "Elementary Decision Theory" (New York: John Wiley & sons, Inc.,
1954).
10 ENGINEERING ECONOMY
so inexpensive as to make electric generation by fossil fuels uneconomical
while such plants still function perfectly well.
The period of analysis is the length of time over which project con
sequences occurring are included in a particular study. The period of
analysis for comparing alternative project designs has the project economic
life as its upper limit but may be shortened arbitrarily to exclude the
highly uncertain events of the very distant future.
The construction horizon is reached when the constructed facilities
are no longer expected to satisfy the future demands. For example, the
water supply alternatives for a community may be studied for a period of
analysis of 40 years even though the original facilities may be planned
large enough to supply the water usage predicted for only 20 years. The
longer period of analysis helps integrate present action into the longrun
solution. The shorter construction horizon adds flexibility to deal with
unforeseen changes.
Regular maintenance and periodic replacement of worn parts may
extend the life of a water resources project almost indefinitely, but a period
of analysis of 50 or 100 years is generally used.' The optimum construction
horizon for individual project components is often a shorter period and
may be determined by economic analysis (Sec. 910). For example, tunnels
may be economically built to maximum capacity because of the high cost
of subsequent enlargement, whereas channels may be economically en
larged in 10 or 20year stages.
When alternative schemes of water resources development are being
compared, all must be evaluated over the same period of analysis. If a
short economic life causes some alternatives to require periodic replace
ment, the most common assumption is that each cost will be repeated in a
fixed cycle over a series of economic lives until the total project life is
reached. However, this assumption should not be used automatically
without considering, with respect to the cost or desirability of cyclic
replacement, the effects of differential inflation (Sec. 98), the development
of new production techniques through technological advance, and the
changing nature of demand with time. Uncertainty with respect to any of
these tends to favor shortlived alternatives.
If the period of analysis is not an even multiple of element lives, an
adjustment must be made through a negative cash flow or salvage value
equal to the value of the element at the end of the period of analysis. A
refined value estimate is seldom warranted because of the relatively small
present worth and the difficulty of predicting cash flows in the distant
I The President's Water Resources Council, Policies, Standards, and Procedures in the Fomuzlation,
Evaluation, and Reaiewo of Plans for Use and Daeveopment of Water and Related Land Resources,
87th Cong. 2d Sess., Sen. Doc. 071, 1962.
PRINCIPLES OF ENGINEERING ECONOMICS 11
future. Straightline depreciation may be used for a quick estimate of the
value of unused life as
8 = 1 X (11)
where X is the years of unused life, L is the years of total life, and K is the
initial value (Ex. 11).
EXAMPLE 11
A certain type of pump is estimated to require replacement every 20 years
and is to be used in a project where the economy study is based on a
50year period of analysis. What salvage value should be used if the initial
cost is $b15,000?
The third pump will be installed in year 40 and thus will have 10
years of useful life remaining at the end of the period of analysis. Thus,
X = 10 years, L = 20 years, and K = $b15,000. From Eq. (11),
S =( 10 1 rv $15,00 = 7,0
19 STRUCTURING ALTERNATIVES Recognition of the full
spectrum of potential alternatives for analysis is of paramount importance
if the most efficient course of action is not to be omitted at the outset. All
reasonable possibilities should be considered. The analyst must be imagina
tive in defining courses of action which will attain designated objectives.
One of the most useful treatises on structuring and handling alternatives
is found in the pioneering work of E. L. Grant' and is used as a basis for
summarizing this chapter with the following points:
1 All alternatives physically capable of achieving the design objective
should be clearly defined. One alternative is to "do nothing" if none
of the other proposals is economically feasible. Limitations of time
and funds often prevent a complete analysis of all alternatives. Before
extending the study, the costs of additional information must be com
pared with the potential savings from better project selection.
d The physical consequences of each alternative should be identified and
evaluated in money units. Benefits and costs which cannot be evaluated
in monetary terms should be explicitly identified.
8 The difference between alternatives should be the basis for comparison.
Sunk costs are irrelevant in choosing between alternatives except as
1 Eugene L. Orant, Concepts and Applications of Engineering Economy, in "Special Report 56,
Economic Analysis in H~ighway Programming, Location, and Design" (Washington: Highway
Research Board, 1960), pp. 814.
12 ENGINEERING ECONOMY
they affect the future. Allocated costs or average costs should not be
used in economy studies; incremental or marginal costs should be used.
Each separable increment of investment must return at least an equal
increment of benefits in order to be justified.
4 Weight should be given to differences in intangibles as well as to
differences in market consequences when comparing alternatives. Arbi
trary monetary values should not be placed on intangibles since they
distort the economic analysis. Economic analysis should not be ignored
even if decisions must be based largely on intangibles. The decision
maker should be aware of the cost of achieving other values when
projects are justified on extraeconomic grounds.
5 The alternatives should be compared on a uniform basis. Such values
as discount rates, period of analysis, and unit costs must be the same.
SELECTED REFERENCES
American Telephone and Telegraph Company: "LEngineering Economy"
(New York: 1963).
Chernoff, H., and L. Mloses: "Elementary Decision Theory" (New York:
John Wiley & Sons, Inc., 1954).
DeGarmo, E. P.: "Engineering Economy," 3d ed. (New York: The
Mac~iillan Company, 1960).
Grant, Eugene L., and Grant Ireson: "Principles of Engineering Econ
omy," 5th ed. (New York: The Ronald Press Company, 1970).
Kuhn, Tillo: "Public Enterprise Economics and Transport Problems"
(Berkeley: University of California Press, 1962).
Wellington, Arthur M/.: "The Economic Theory of the Location of Rail
ways," 1st ed. (New York: John Wiley & Sons, Inc., 1877).
PROBLEMS
11 Costs and revenues for a particular project having alternate possible
levels of investment have been estimated on an equivalent basis and
found to be
Cost: 39 83 117 155 194
Revenue: 100 150 175 185 190
Which project level should be selected?
16 To develop a new water supply, an industry will have to spend
$1 million. The resulting increased production is predicted to increase
net income to the company from sales by $6900,000. Associated
economic development will benefit the community by $6400,000 and
other nearby communities in the same state by $6250,000. However,
PRINCIPLES OF ENGINEERING ECONOMIcs 13
$b500,000 of the increased state income represents transfers from
other states. The river on which the industry is located flows into
another country. The new industrial development is expected to
deteriorate water quality sufficiently to cause $g250,000 worth of
damage downstream from the border.
a Would the project be economically justified from the viewpoint of
the industry?
b The community?
c The state?
d The nation?
e What kinds of intangible factors might be weighed by each of the
four viewpoints?
f Should the project be built from the overall viewpoint?
18 A community has spent $b50,000 developing a new well and has not
yet obtained water. The geological consultant estimates another
$b50,000 will be required to guarantee a good supply but admits
sufficient water may be obtained after spending only $610,000. As an
alternative a spring exists several miles away from which an equiva
lent supply could be pumped for $b40,000. What course of action
would you recommend and why?
CHAPTER
MATHEMATICS
OF ECONOMIC ANALYSIS
Formulating the Analysis
Economic analysis is performed in a series of steps. Each alternative
must be explicitly defined and the resulting physical consequences must be
predicted. A monetary value must be placed on each physical consequence.
A discount rate must be selected and applied to convert the predicted time
stream of monetary values into an equivalent single number. Only then
can the alternatives be directly compared. Each step is developed in the
following pages.
1DEFINING THE ALTERNATIVES An engineering alternative
is a course of action physically capable of achieving the design objective.
Structural alternatives (a dam, for example) characteristically involve a
large first cost for project construction to produce benefits throughout the
project life. Nonstructural alternatives (floodplain zoning, for example)
involve benefits and costs which are both fairly well distributed over
project life. A properly defined alternative must be specified by the engi
neer with sufficient clarity for its economic and intangible consequences
to be evaluated and its nature understood by those responsible for the
final selection. Properly defined alternatives are an evidence of clear
thinking and a necessity for adequate consequence prediction.
A properly formulated set of engineering alternatives includes all
possibilities for action (including taking no action at all) which have a
realistic chance of proving optimum. Special care is necessary to include
16 ENGINEERING ECONOMY
nonstructural alternatives with which engineers may be less familiar. The
alternatives are called mutually exclusive if only one of a set can be selected.
Alternatives may be mutually exclusive because of conflicting space re
quirements, limited financial resources, limited resource inputs (water, for
example), or limited demand or need for resulting output. At other times,
it may be practical to implement two or more of the alternatives.
d PH YSICA L CONSEQUENCES Each engineering alternative will
if implemented produce a series of physical consequences occurring at
various times into the future. For example, a project built to irrigate
peaches, tomatoes, and alfalfa will produce a number of results. The
project will have to be constructed. After construction is completed, the
project will have to be maintained. Certain elements may wear out and
require periodic replacement. Each such costassociated event needs to be
predicted by nature and date.
The water delivered by the project will be used to irrigate peaches,
tomatoes, and alfalfa. The first year water is delivered, the acreage and
increased yield of each crop can be used to predict a project output of X
tons of peaches, Y tons of tomatoes, and Z tons of alfalfa. In a similar
manner, X, Y, and Z may be predicted for each subsequent year of project
life. The outputs can be expected to increase steadily in the early years as
more and more land is irrigated. Later, they may be expected to fluctuate
with changing weather and other factors which influence crop yield.
8 CASH FLOW DIAGRAM Having identified the physical conse
quences of each alternative, it is necessary to decide which ones are relevant
to the analysis. Some may not be because of the viewpoint taken in the
study, a neutral effect which is neither desirable nor undesirable, a tenuous
connection to the project, their small magnitude, or some other reason.
Other consequences may be dropped from further evaluation because they
are identical for each alternative and an economy study is concerned only
with differences (incremental costs). The relevant consequences can be
separated into two groups. Some can be assigned a reasonable monetary
value. The others may have some monetary value but also require sup
plemental determination of the intangible factors (Sec. 16).
The assignment of a monetary value to physical consequences is a
very complicated process having many ramifications which will be dis
cussed throughout the remainder of the book. However, for the time being
we will assume that meaningful monetary values can be assigned to the
MATHEMATICS OF ECONOMIC ANALYSIS 17
major project consequences. Economic analysis becomes a less reliable
guide to decision making as more consequences fall in the intangible class.
For the sample project, the cost of installation, the cost of maintenance
in each year, replacement cost for each shortlived item, and the benefit
resulting from the increased yield of each crop in each year would have
to be determined.
The graphic presentation of each value plotted by time is called a
cash flow diagram. The standard representation for a cash flow diagram is
that receipts (benefits) are represented by arrows pointing upward, while
costs are represented by arrows pointing downward. Arrows pointing
toward the centerline indicate cash flows which may be taken either way
in a general diagram (see Fig. 22). The length of the arrow is made pro
portional to the cost or benefit. The horizontal axis denotes time. For
convenience in analysis and with little loss in accuracy for longlived
projects, all cash flows during a year are by convention combined into
lump sums occurring at the end of the year. Figure 21 is a cash flow dia
gram which might be predicted for our hypothetical irrigation project.
Annual benefits and costs will not in fact be constant every year but will
vary around average values in an almost random fashion with crop pro
duction and maintenance needs. However, only expected average values
are normally predicted in advance, even though the random component
could conceivably be introduced through simulation (Sec. 2010). Drawing
of the cash flow diagram can be greatly simplified by use of envelope curves
as a substitute for the many arrows.
Incrersn 0 ndsi Benefit from average crop production
to new land
Annual operation and maintenance cost
with periodic larger replacement cost
Envelope curves
Large expenditure during period
of initial project construction
FIGURE 1 Cash flow diagram for hypothetical irrigal
tion project.
18 ENGINEERING ECONOMY
Discounting Factorsl
a4 SINGLEPA YMENT FACTORS In applying discounting to con
vert cas h fows to a single number suitable for use in comparing alterna
tives, the basic objective is to convert a value at one date to an equivalent
value at another date. Two singlepayment factors are available for this
purpose (Fig. 22).
Singlepayment Compoundamount Factor The singlepayment com
poundamount factor indicates the number of dollars which will have
accumulated after N years for every dollar initially invested at a rate of
return of i percent. The functional notation is (F/P,i%,N),2 Where F
implies a future and P a present amount. If one were to deposit P dollars
initially, after 1 year
F = P(1 + i) (21)
Each year the amount must again be multiplied by 1 + i to account for
that year's interest; therefore after N years
F = P(1 + i)N (22)
The desired factor becomes
/PiN =(+i) = P (23
Singlepayment Presentworth Factor The singlepayment presentworth
factor indicates the number of dollars one must initially invest at i percent
to have $61 after N years. It will be abbreviated by (P/F,ib,N). The
factor is the inverse of the previous factor, or
F' %' (1 + i)N F(24
SStandardized notation for discounting formulas has been suggested by the Ad Hoe Committee for
Study of Standardization of Engineering Economy Notation, Eng. Economist, vol. 12 (Summer,
1967), pp. 253263. Committee recommendations are followed for the most part in the subsequent
development.
t The alternative mnemonic notation has been widely used, but it creates a group of rather artificial
symbols which make it more difficult to learn and follow.
SFIGURE 99 Singlepayment factors. (a)
/= discount rate Singlepayment compound
P amount factor = F/P; (b)
N years  singlepayment present
7 worth factor = P/F.
MATHEMATICS OF ECONOMIC ANALYSIS 19
Interest Tables Selected values of both singlepayment factors are pre
sented in Table A in the Appendix. When a discounting factor is needed
for a combination of N and i not found in the tables, an approximate
value may be found by interpolation. The error from interpolation becomes
increasingly severe with higher discount rates. For precise values or values
outside the range covered by the tables, one must substitute values for i
and N in the appropriate formula.
d5 UNIFORMA NN UALSERIES FA CTORS All discounting prob
lems can be solved by applying the two singlepayment factors. However,
additional factors can be developed to greatly reduce the required work.
As an example, one may take the irrigation project of Fig. 21. If it were
to produce crops having equal value for each of 50 years, fifty separate
singlepayment presentworth factors would have to be applied to find the
present worth of this uniform annual cash flow. The task is made much
shorter by developing uniformannualseries factors.
Uniformannualseries factors indicate equivalence between the value
at an earlier date, P, and equal amounts A at the end of each of the N
years or between the N equal values of A and an accumulated amount
F (Fig. 23).
Sinkingfund Factor The sinkingfund factor indicates the number of
dollars one must invest in uniform amounts at i percent interest at the
end of each of N years to accumulate $1l. The functional notation is
(A/F,i%,N). If one were to apply the singlepayment compoundamount
factor individually to each of the N values of A in Fig. 23 and sum the
/= discount rate F
N years
1c lyear
Figure  Uniform series factors. (a) Sinkingfund
factor = A/F; (b) compoundamount fac
tor = F/A; (c) capitalrecovery factor =
A/P; (d) presentworth factor = P/A.
20 ENGINEERING ECONOMY
results to obtain F, the result would be
F = A [1 + (1 + i) + (1 + i) 2 . 1+iN ] (25)
where the last value of A accumulates no interest because it is withdrawn
immediately upon deposit and the first value of A accumulates interest
for N 1 years. Multiplying both sides of Eq. (25) by 1 + i gives
(1 +i)F = A[(1 +i) +(1 +i)2 + 1+i3
+ + (1 + i)N] (26)
The relationship may now be converted from a series to an explicit
expression through termbyterm subtraction of Eq. (25) from Eq. (26)
to give
iF = A[(1 + i)N 1] (27)
The desired factor becomes
(A %, (28)
Capitalrecovery Factor The capitalrecovery factor indicates the number
of dollars one can withdraw in equal amounts at the end of each of N
years if $1 is initially deposited at i percent interest. The functional nota
tion is (A/P,i%,N). Because
A AF
P FP (29)
One may substitute Eqs. (28) and (23) in Eq. (29) to get
(A i(1 + i)N
7, iN 1 i" (210)
Series Compoundamount Factor The series compoundamount factor
indicates the number of dollars which will accumulate if exactly $b1 is
invested at i percent interest at the end of each of N years. The functional
notation is (F/A,i%/,N). The factor is the inverse of the sinkingfund
factor, or
(F .~ (1+iAs, (211)
Series Presentworth Factor The series presentworth factor indicates the
number of dollars one must initially invest at i percent interest to with
draw $b1 at the end of each of N years. The factor (P/A,i%,N) is the
MATHEMATICS OF ECONOMIC ANALYSIs 21
inverse of the capitalrecovery factor or
(P .(1 +i~sN == (212)
A i(1 + i) A
Interest Tables Values for all four uniformannualseries factors for se
lected values of i and N are tabulated in Table A in the Appendix.
6 UNIFORMGRADIENTSERIES FACTORS The uniforman
nualseries factors can be applied to an equal cash flow in each year.
Often cash flows will not be equal but will follow some definite pattern.
The simplest pattern is the uniformly increasing gradient series, a series
in which the cash flow increases by some constant amount between each
pair of years.
Uniformgradientseries Presentworth Factor The uniformgradientseries
presentworth factor indicates the number of dollars one must initially
invest at i percent interest to withdraw $1 one year later, $b2 two years
later, to N dollars N years later. The functional notation is (P/G,i%,N).
If one were to apply the singlepayment compoundamount factor
individually to each value, beginning with the last, in the gradient series of
Fig. 24 and sum to obtain the accumulated amount just after the last
deposit, the result would be
F = G [N + (N 1) (1 + i) +
+ 2(1 + i)"2 + (1 + i)N1] (213)
FIGURES4 Gradientseries presentworth factor = P/G.
22 ENGINEERING ECONOMY
Multiplying both sides by 1 + i gives
(1 + i)F = G'[N(1 + i) + (N 1)(1 + i)2 .
+ 2(1 + i)N' + (1 + i)N] (214)
Termbyterm subtraction of Eq. (213) from Eq. (214) gives
iF = G {N + (1 + i) + + (1 + i)N) (215)
Multiplication of both sides by 1 + i gives
(1 +i)iF = G[N(1 +i)+t(1 +i)2 +
+ (1 + i)N+'] (216)
Termbyterm subtraction of Eq. (215) from Eq. (216) then gives
iZF = G'[N N(1 + i) (1 + i) + (1 + i)N+'] (217)
Rearranged, Eq. (217) becomes
F (1 + i)N+' (1 + Ni + i)
(218)
G i2
When Eq. (218) is combined with Eq. (24) to convert from F to P, the
final result is
(P.~ (1 +i)N+1 _(1 +Ni +i)
G, z%,N = ~i2(1 + i)" 29
Interest Tables Values of the uniformgradientseries presentworth factor
for selected values of i and N are tabulated in Table B in the Appendix.
Conversion from Present Worth Whenever the uniform gradient series
needs to be converted to an equivalent uniform annual series, Eqs. (210)
and (219) can be combined to give
G G, i%,N P, i%,N) (220)
Similarly, the uniform gradient series may be converted to a single lump
sum at the end [as an alternate to direct substitution in Eq. (218)] by
combining Eqs. (23) and (219) to get
G ,i, ,i, (221)
Uniformly Decreasing Series Conversions The gradient series used to
derive the expression for (P/G,i%/,N) increases in value from year to
year. When the present worth of a gradient series that decreases in value
from year to year is needed, it may be determined by subtracting a uni
formly increasing gradient series from a uniform annual series in the
manner shown in Ex. 21.
30
20
io
I I I~I I I I I ( I I I _
MATHEMATICS OF ECONOMIC ANALYSIs 23
EXAMPLE d1
An individual invested the following amounts of money at 4 percent
interest. How much would he have at the end of year 25?
Year
1
2
3
4
5
6
7
Investment
5
10
15
20
25
30
35
Year Investment
8 40
9 45
10 50
11 45
12 40
13 35
14 30
Year Investment
15 25
16 20
17 15
18 10
19 5
2025 0
The present worth of the pyramidshaped series can be found by
subdividing it into three portions to which factors from the tables can
be directly applied. (Cf. Fig. 25.)
F= 911.99
Enelp curve etabished
0 5 10) (
10
20 25
t
nn
FIGURE 5 Cash flow diagram for Ex. 21.
1.Present worth of series 5, 10, . ., 45, 50 in years 1 through 10.
5 \i, 4%o,10 = 5(41.99225) = $209.96
24 ENGINEERING ECONOMY
9.Plus present worth of series of 45 per year in years 11 through 19,
45 ,iP 4%,9 4%,10 45o(7a.43533)(.67556)= $226.04
8. Minus present worth of series 5, 10, . ., 35, 40 in years 12 through
19,
5 ,G 4%.l ,8, 4%,011 = 5(28.91333)(0.64958)= $93S.92
The three values sum to a $342.08 present worth, which can be converted
to a value in year 25 by
342.08 \, 4%,25 = (342.08) (2.66584) = $911.99
97 NON UNIFORMGRADIENTSERIES FA CTORS Proj ect plan
ning often requires determination of the present worth of some mono
tonically but not uniformly increasing time stream of benefits. Typical
situations involve benefits increasing by a uniform annual percentage,
benefits increasing rapidly in the early years of project life but more
slowly later, benefits increasing most rapidly near the middle of project
life, and benefits increasing most rapidly toward the end of project life.
Uniformpercentagegradientseries Presentworth Factor This factor indi
cates the present worth at i percent interest of investment of $1l at the
end of the first year and an amount increasing by j percent from year to
year until the N years are completed. While there is no standard functional
notation for this factor, the notation (Pj,i%,N) will be used.
As shown in Fig. 26a, the deposit at the end of the last year would
amount to (1 + j)N 1. Since this last value is withdrawn immediately
after deposit, no interest is added to it. The next to last deposit would be
smaller by an amount found by dividing by 1 +j, but accumulated
interest would increase its worth by the factor 1 + i. Summing each term
backward through the series of Fig. 26 gives
F (1 +j) x1 1+ i+ N (222)
1+j 1+j
Multiplication of both sides by (1 + i)/(1 + j) gives
1+ +i 1+ 2+
F=(+) i+ + +(
(223)
MATHEMATICS OF ECONOMIC ANALYSIs 25
/= discount rate
N years I
n years
FIGURE 126 Nonuniformgradientseries presentworth
factors. (a) Percentagegradientseries
presentworth factor (P,,i%,N); j=
growth rate, Y1 = 1.00, Y, = (1 + j)"'
(b) Acceleratedgrowthcurve present
worth factor (P/F,,i%,N); N = 50 years,
F. = YN, Yn = (ln (n + 1)/3.93183]F,.
(c) Normnalgrowthcurve presentworth
factor (P/F,,,i%,N); N = 50 years, F,,
Yu, Yn = (0.0012n2 0.000016n3)F,. (d)
Deferredgrowthcurve presentworth fac
tor (P/Fd,i%,N); N = 50 years, Fd
YN, Yn = (0.0004n2)Fd.
Termbyterm subtraction of Eq. (222) from Eq. (223) produces
F=(1+}) 1 (224)
wherein the lefthand term may be transformed to [(i j)/(1 + j)]F.
Substituting P(1 + i)" for F [Eq. (22)] and simplifying gives
(1 + i) (1 + j) "
(Pi,i%,N) = (225)
The righthand side of Eq. (225), the uniformpercentagegradientseries
presentworth factor, is tabulated for selected values of i, N, and i in
Table B in the Appendix.' Factors for negative growth rates can be found
by substituting negative values of j in Eq. (225).
Acceleratedgrowth Presentworth Factor This factor indicates the present
worth at i percent interest of an annual investment pattern in which
deposits begin by increasing very rapidly, but increase at a progressively
slower rate in later years. Such a series is represented by an equation
suggested by the Corps of Engineers2 and indicated in Fig. 26b. The
SIn the special case where i = j, Eq. (225) is indeterminate, but (Pi,i%,N) = N/(1 + j).
U.S. Army Corps of Engineers, Eng. Manual EM 11202118, (Washington, June, 1960), app. 2,
change 1.
26 ENGINEERING ECONOMY
presentworth factors (P/F,,i%/,N) found by summing the present worths
of the individual yearly values are tabulated in Table C in the Appendix.
Each tabulated value should be multiplied by the fiftieth value in the
series F, to get the present worth. For example, the 10year factor gives
the present worth of the first 10 values in the series as a multiple of the
fiftieth value.
Normalgrowth Presentworth Factor This factor indicates the present
worth at i percent interest of a series of deposits which increase at a
progressively faster rate until the midpoint of proj ect life and then increase
progressively more slowly through the later years. Its typical use would be
in finding the present worth of a benefit series realized in an area where
development is expected to be most rapid 20 to 25 years after project
construction. Such a series is represented by a curve suggested by the
Corps of Engineers' and depicted in Fig. 26c. The present worth factors
(P/F,,i%,N) are found in Table C in the Appendix.
Deferredgrowth Presentworth Factor This factor indicates the present
worth at i percent interest of a series of deposits which increase slowly
throughout most of the project life only to increase very rapidly in the last
few years. Such a series is represented by another curve suggested by the
Corps of Engineers2 and shown in Fig. 26d. The presentworth factors
(P/Fa,i%,N) are also found in Table C in the Appendix.
98 OTHER CASH FLOW PA TTERNS Cash flow patterns for econ
omy studies are based on projected future events, cannot be known with
any real certainty, and thus can normally be approximated with sufficient
accuracy by one of the above patterns. Sometimes it may be better to
approximate a future cash flow pattern by using different gradients over
different time periods as illustrated in Ex. 22.
EXAMPLE dd
A particular water resources project produces benefits which amount to
$12,000 in year 1 and increase on a uniform gradient to $b120,000 in year 10.
Thereafter, they increase on another uniform gradient of $5,000 per year to
$b200,000 in year 26, at which point they remain constant at $200,000 each
year until the end of project life in year 50. What is the present worth of
these benefits at a 4 percent discount rate?
The present worth of the given benefit series can be found by sub
l Iid.
SIbid.
MATHEMATICS OF ECONOMIC ANALYSIs 27
dividing it into four portions to which factors from the tables can be
directly applied.
1. Present worth of 12, 24, .. 108, 120 in years 1 through 10
12,000 \, 4%/,l10 = 12,000 X 41.99225 = $g503,900
d. Present worth of 120 per year in years 11 through 25
120,000 4%,15) 4%0,10 = 120,000 X 11.11839 X 0.67556
= $901,400
8. Present worth of 5, 10, . ., 70, 75 in years 11 through 25
5,000 4%,1 4,10 = 5,000 X 80.85389 X 0.67556
= $6273,100
4. Present worth of 200 per year in years 26 through 50
200,000 4%,25r~ 4%,25Y = 200,000 X 15.62208 X 0.37512
= $b1,172,000
The total present worth is the sum of these four values, or $b2,850,400.
If use of more complicated series is justified, an exact solution may
be obtained by individual application of singlepayment factors. For
approximate results, a graphic solution may be used.'
Discounting Techniques
The procedure in which discounting factors may be systematically applied
to compare alternatives (either different projects or different sizes of the
same project) is called a discounting technique. The four conceptually
correct discounting techniques are (1) the presentworth method, (2) the
rateofreturn method, (3) the benefitcost ratio method, and (4) the
annualcost method. Each method, if used correctly, leads to the same
evaluation of the relative merit. However, each has advantages and
disadvantages.
1 George E. Ribble, Graphical Methods for Discounting Future Benefits in Feasibility Studies,
Civil Eng., vol. 35, no. 11 (November, 1965), pp. 8087.
28 ENGINEERING ECONOMY
d9 PRESENTW~OR TH ME THOD The presentworth method selects
the project with the largest present worth PW of the discounted algebraic
sum of benefits minus costs over its life.
PW= ,it(B C,) (226)
where Ct is the cost and B, the benefit in the subscripted year, n is the
period of analysis in years, and i is discount rate. When the annual net
benefits B = B, Ct are constant over the project life except for the
initial first cost K, the formula may be simplified to
PW = K + B ~, i%,n) (227)
When the net benefits vary according to some regular gradient, the appro
priate gradient factor should be used.
Calculation of present worth from a cash flow diagram is a purely
mechanical process. However, certain rules must be followed in comparing
the calculated present worths to make correct choices.
RULE 1 Figure all present worths to the same time base. Whether or not
alternatives are to be initiated at the same time, each present worth must
be discounted to the same base year (1970, for example) because sums of
money at different times are different economic goods.
RULE 2 Figure all present worths by using the same discount rate. Whether
or not alternatives are to be financed from the same funds, each must
be discounted at the same rate if the result is to be an index of intrinsic
project merit.
RULE 3 Base all present worths on the same period of analysis. Whether
or not alternatives have a common economic life, the comparison must
be based on a service provided over a common period of time. This may
be done either by evaluating the cost of extending the service past the
termination of the shorterlived alternatives or by calculating the value of
the unused life of the longerlived alternatives (Ex. 11).
RULE 4 Calculate the present worth of each alternative. Choose all alterna
tives having a positive present worth. Reject the rest. This ends the procedure
if no sets of mutually exclusive alternatives are involved. The choice
among alternatives in such a set is made by Rule 5.
RULE 5 Choose the alternative in a set of mutually exclusive alternatives
having the greatest present worth.
RULE 6 If the alternatives in the set of mutually exclusive alternatives have
benefits which cannot be quantified but are approximately equal, choose the
alternative having least cost.
MATHEMATICS OF ECONOMIC ANALYSIs 29
A single example based on the two mutually exclusive alternative
water supply projects described in Table 21 will be used to illustrate all
four discounting techniques. Project A provides an initial investment large
enough to meet the demands for water for 40 years, and project B uses
investment in two stages to meet the same demand. The present worths
are calculated to be
PW of A = $40,000,000 $160,000 5% iV,40I
+ $2,500V,000 \, 5%,40)
= 8b40,000,000 $b160,000(17.159) + $b2,500,000(17.159)
= $b153,000
PW of B = $25,000,000 $30,000,000F 5%,20)
 $100,000 \, 5%O,20 220,000 5%,20) 5,%,20)
S$2u,500v,000 \, 5%,40,v
= $25,000,000 $30,000,000(0.377) $6100,000(12.462)
$8220,000 (12.462) (0.377) + $b2,500,000 (17.159)
= $4,308,000
Therefore we should choose project B since its present worth is greater.
If the rule of analyzing only differences were strictly applied, the equal
annual benefits could be deleted from the evaluation of each alternative to
provide the same conclusion with less work.
Proj ect B would appear even more favorable were an adj ustment made
to account for the economic life of the second stage lasting 20 years past the
period of analysis. The adjustment according to Eq. (11) would add a $15
TABLE 9 Data for Satmple Problem
Project A Project B
Construction cost $40,000,000 $25,000,000, 1st stage
$30,000,000, 2d stage
Operations and $160,000 per year for $100,000 per year for 1st 20 years
maintenance 40 years $220,000 per year for 2d 20 years
Economic life 40 years 40 years for each stage
Period of analysis 40 years 40 years
Annual benefits $2,500,000 $2,500,000
Discount rate 5 percent 5 percent
30 ENGINEERING ECONOMY
million salvage value in year 40 or $15 million (P/F,5%,40) = 9$2,130,000
to the present worth.
If the time value of money is neglected, A seems preferable to B
because of its smaller total cost. However, at a 5 percent discount rate, B
is definitely preferable. Sensitivity analysis shows the cost of B's second
stage could increase to $641 million, and B would still be preferable! This
example dramatically illustrates the desirability of postponing costs until
further investment is actually needed so as to free capital for alternative
productive investment.
Capitalizedl worth is defined as the present worth of perpetual service.
The present worth may be converted to a capitalized worth by assuming
an equivalent reinvestment at the end of each economic life and multiply
ing by the ratio of the capitalrecovery factor to the discount rate. The
multiplier is close to 1 with long lives or high discount rates. Appropriate
discount factors may be used to estimate capitalized worth where cash
flows for reinvestment are expected to differ from those for initial invest
ment. The decision rules used for present worth also apply for capitalized
worth.
b10 RATEOFR~ETUIRN METHOD The rate of return is the dis
count rate at which the present worth as defined by Eq. (226) equals
zero as found by trial and error. Other decision rules apply when comparing
alternatives by the rateofreturn method.
RULE 1 Compare all alternatives over the same period of analysis. Rates of
return over different economic lives cannot be meaningfully compared
because investment opportunity for the returns from the shorterlived
alternatives must be considered in determining whether capital should
remain committed to the longerlived alternative.
RULE 2 Calculate the rate of return for each alternative. Choose all alterna
tives having a rate of return exceedinzg the minimum acceptable value. Reject
the rest. If sets of mutually exclusive alternatives are involved, proceed
to Rule 3.
RULE 3 Rank the alternatives in the set of mutually exclusive alternatives
in order of increasing cost. Calculate the rate of return on the incremental cost
and incremental benefits of the next alternative above the least costly alternative.
Choose the more costly alternative if the incremental rate of return exceeds the
minimum acceptable discount rate. Otherwise choose the less costly alternative.
Continue the analysis by considering the alternatives in order of increased
costliness, the alternative on the less costly side of each increment being the
most costly project chosen thus far.
The rateofreturn method will not lead to the same decisions as the
presentworth method unless the incremental analysis of Rule 3 is used in
place of selecting the mutually exclusive alternative with the highest rate
of return. The rateofreturn method must be applied with caution because
more than one rate of return exists when annual costs exceed annual
benefits in years after annual benefits first exceed annual costs, but Heebink
has shown that the rateofreturn method using Rule 3 still gives consistent
answers even when dual solutions exist.' The water resources planner
needs to be alert to this problem in comparing stage construction or non
structural alternatives by the rateofreturn method.
In the example of two alternative water supply projects, each has
been found to have a positive present worth when discounted at 5 percent
and thus must have a rate of return exceeding the minimum acceptable
value. Therefore, the difference between alternatives (A B in Table 22)
is used to compute the incremental rate of return as directed by Rule 3.
The procedure is to assume discount rates until the present worth, or
PW =$15,W000000 30,000,000 i%,20) + $6,000 i%,.20)
$6,000 ~, i%,20) (, i%,20)
equals zero. For i = 5%, PW = $64,155,000 indicates the trial discount
rate to be too high and the extra cost of A over B to be not justified at a
minimum acceptable rate of return of 5 percent. Therefore, project B is
chosen. Had the present worth at 5 percent been negative, the incremental
rate would have been greater than 5 percent. A complete solution provides
SDavid Heebink. "A Critique of CompoundInterest Models Used in Decisionmaking for Capital
Budgets," Ph.D. thesis, stanford University, stanford, Calif., 1960, app. B, pp. 8794.
TABLE 28 Incremental Data for Sample Problem
IN MILLIONS OF DOLLARS
Project A Project B A B
First cost 40.0 25.0, first stage +15.0
30.0, second stage 30.0
Operations and 0 16 per yr 0 .10 per yr, +0 .06 per yr,
maintenance first 20 yr first 20 yr
0.22 per yr, 0.06 per yr,
second 20 yr second 20 yr
Benefits 2.5 per yr 2.5 per yr
MATHEMATICS OF ECONOMIC ANALYSIS 31
32 ENGINEERING ECONOMY
an incremental rate of return of 3.39 percent and indicates project B to be
favored only as long as the minimum acceptable rate of return exceeds
3.39 percent.
11 BENEFITCOST RATIO METHOD The benefitcost ratio
PWb/PW, is the present worth of the benefits PWa divided by the present
worth of the costs PW,. Annual values can alternatively be used without
affecting the ratio. The present worth PWb Of annual benefits Be is
Pe=I , .i%, B, (228)
The present worth PW, of the costs C1 is
PW=~=l 7, i%,i C, (229)
Series discounting factors may be used in either summation as appropriate.
The decision on whether particular cash flows should be considered
costs or negative benefits is sometimes arbitrary (Sec. 84) and affects
the benefitcost ratio. While it does not affect project selection by the
procedure described below, it is important to recognize that the best
project has the greatest net benefits, not the largest benefitcost ratio.
Several authors have suggested that the benefitcost ratio method leads to
different decisions than the other techniques do.' However, this conflict
only occurs when the incrementalcost principle of Rule 4 is neglected.
Four rules must be followed to apply the method correctly.
RULE 1 Figure all benefitcost ratios by using the same discount rate.
RULE 2 Compare all alternatives over the same period of analysis.
RULE 3 Calculate the benefitcost ratio for each alternative. Choose all alterna
tives having a benefitcost ratio exceeding unity. Reject the rest. If sets of
mutually exclusive alternatives are involved, proceed to Rule 4.
RULE 4 Ranke the alternatives in the set of mutually exclusive alternatives
in order of increasing cost. Calculate the benefitcost ratio by using the in
cremental cost and incremental benefit of the next alternative above the least
costly alternatives. Choose the more costly alternative if the incremental benefit
cost ratio exceeds unity. Otherwise, choose the less costly alternative. Continue
the analysis by considering the alternatives in order of increased costliness,
the alternative on the less costly side of each increment being the most costly
project chosen thus far.
SRoland N. McKean, "Efficiency in Government through Systems Analysis" (New York: John
Wiley and Sons, Inc., 1958), pp. 108112; and Otto Eckstein, "Water Resource Development: The
Economics of Project Evaluation" (Cambridge, Mass.: Harvard University Press, 1958), pp. 5354.
MATHEMATICS OF ECONOMIC ANALYSIS 33
From our previous calculations on our sample problem, we know that
each project has a positive present worth at a 5 percent discount rate;
therefore, each project has a benefitcost ratio greater than 1 and Rule 3 is
met. As with the rateofreturn method, differences between alternatives
(Table 22) are taken to see if the incremental costs are justified. The
incremental net cost found in Sec. 210 when coupled with the zero
incremental benefit indicates a zero incremental benefitcost ratio. There
fore, project B is chosen.
While project B has the higher overall benefitcost ratio (1.11 instead
of 1.00), the preferred project sometimes has a lower one. This may be
illustrated by considering a project whose benefits equal 3 and whose
costs equal 1 and which has an increment of investing an additional 4 to
increase benefits to 10. The smaller project has a benefitcost ratio of 3,
while the larger one has a ratio of 2. Because the incremental ratio is 1.75,
the larger investment should be chosen even though it has a smaller
benefitcost ratio.
E19 ~ANNUALCOST METHOD The annualcost method converts
all benefits and costs into equivalent uniform annual figures. Decision
rules resemble those for the presentworth method because each annual
cost is a present worth times a constant capitalrecovery factor.
RULE 1 Figure all annual costs by using the same discount rate.
RULE 2 Base all annual costs on the same period of analysis.
RULE 3 Calculate the net annual benefit of each alternative. Choose all
alternatives having a positive net annual benefit. Reject the rest. If sets of
mutually exclusive alternatives are involved, proceed to Rule 4.
RULE 4 Choose the alternative in a set of mutually exclusive alternatives,
having the greatest net annual benefit.
TABLE 93 Summation of Annualcost Method
Project A Project B
Present worth of benefits $42,898,000 $42,898,000
Present worth of costs 42,745,000 38,590,000
Net present worth 153,000 4,308,000
Capitalrecovery factor 0.05828 0.05828
(A /P,5%,40)
Annual benefits $ 2,500,000 $ 2,500,000
Annual cost 2,491,200 2,249,000
Net annual worth 8,800 251,000
34 ENGINEERING ECONOMY
RULE 5 If the alternatives in the set of mutually exclusive alternatives have
benefits which cannot be quantified but are approximately equal, choose the
alternative having the least annual cost.
Since the present worths for projects A and B are calculated in
Sec. 29, they may be multiplied by the appropriate capitalrecovery
factor to get the equivalent annual figures shown in Table 23. Rule 4
says to choose project B as having the greater annual worth. Since the
benefits are the same for each project, Rule 5 could be used to find the
project accomplishing this benefit at least annual cost.
218 EVALUATION OF DISCOUNTING TECHNIQUES Each of
the four discounting methods will when used correctly select the same
proj ect, given the same data. However, each technique has advantages and
disadvantages associated with ease of calculation or presentation and
understanding of the results. These need to be considered in selecting the
method to apply in a given analysis.
Because it does not require an additional set of computations to apply
the incrementalcost principle, the presentworth technique has been
described as "simpler, safer, easier, and more direct."' Others have said
this method is "logically prior to others, and we recommend its use."2
The simple, direct expression of net present worth is conceptually straight
forward and easily presented. However, one is working with larger numbers
which may be harder to visualize and lead more frequently to numerical
errors. Furthermore, the presentworth method cannot be used to
rank projects in order of economic desirability unless all require equal
investment.
The rateofreturn technique has been recommended because it does
not require a preselected discount rate, rates of return are intuitively
meaningful to many investors, and the resulting rates can be compared
with those for many other types of investment.3 On the other hand, it has
been criticized (1) as giving ambiguous answers because of dual solutions,
(2) because of the necessity of calculating incremental rate of return for
interdependent projects, (3) the danger of people's accepting overall as
contrasted with incremental rates of return as indicators of rank, and (4)
the complexity of the required trialanderror solutions.4 Some have gone
SH. Bierman and S. Smidt, "The Capital Budgeting Decision" (New York: The MacMillan Company,
1960), p. 46.
r Jack Hirshleifer, James C. DeHaven, and Jerome W'. Milliman, "Waster Supply: Economics, Tech
nology and Policy" (Chicago: The University of Chicago Press, 1960), p. 152.
r C. H. Oglesby and Eugene L. Grant, Economic Analysisthe Fundamental Approach to Decisions
in Highway Planning and Design, Highw~ay Res. Board Proc., vol. 37 (1958), pp. 4849.
SBierman and Smidt, loc. cit.
MATHEMATICS OF ECONOMIC ANALYSIs 35
so far as to suggest the technique never be used.' However, the cited
advantages are important enough to make the rateofreturn method a
valuable analytic tool.
The benefitcost ratio method is almost universally used by federal
and state water resource agencies and can be expected to remain in this
position into the indefinite future. Moreover, Krutilla and Eckstein2 base
their analysis on benefitcost methods, and Marglin's work shows it to be
consistent with economic theory.3 On the other hand, the use of the benefit
cost ratio without applying the required incremental benefitcost analysis
can lead to serious errors. Interdependent projects cannot be ranked ac
cording to their benefitcost ratios because each enlargement must pass
the incremental benefitcost ratio test. Nevertheless, the fact remains that
the benefitcost ratio method can lead to the same results as other correct
discounting techniques.
The annualcost method uses constant multiples of the presentworth
method and has the same advantages and disadvantages (except for the
use of smaller numbers). However, the annual cost is sometimes preferred
because more people are accustomed to thinking in terms of annual costs
than of present worths.
Which method should be used? The answer depends primarily on the
purpose of the analysis. Where benefits cannot be evaluated, it is not
possible to use benefitcost or rateofreturn techniques. Costs alone must
be compared by using the presentworth or annualcost method. There
are more calculations for the rateofreturn or benefitcost ratio methods
and more opportunities for errors of interpretation, but computational
work is never more than a minor part of the total analysis.
Other Approaches
4 UNRELIABLE TECHNIQUES Of the many other decision
criteria in use, which do not give consistent, reliable results, the three most
commonly found in the analysis of water resources projects are urgency
ratings, standards, and least total costs.
The urgencyrating technique rates proposals on their postponability,
those being least postponable getting priority. Since this method is highly
subjective, the selection process tends toward a political content because
SHirehleifer et al., op. cit., p. 156.
s John V. Kirutilla and Otto Eckstein, "Multiple Purpose River Development" (Baltimore: The Johns
Hopkins Press, 1958), pp. 7677.
i Stephen A. Marglin, Objectives of Waterresource Development, in Arthur Manes et al., "Design of
Waterresource Systems" (Cambridge, Mass.: Harvard University Press, 1962), pp. 1787.
36 ENGINEERING ECONOMY
no firm figures are available to assess relative merit. If projects are truly
nonpostponable, this will be reflected in the efficiency calculations.
Standards are expressions of minimum acceptable project quality
often made prior to, and thus without the benefit of, economic analysis.
Engineers are familiar with standards for structural design, water quality,
street widths, design freeboard, etc. No matter what standard is used,
it should be based on economic analysis unless intangible factors can be
demonstrated to be overriding. Unfortunately, standards which reflect
the ultimate goals of professional groups rather than the relative needs
of the local community are sometimes taken as a valid representation of
community needs to the neglect of other important services. No standard
can be achieved without cost, and costs incurred should be commensurate
with utility achieved. Standards are a poor substitute for a searching
appraisal to obtain a balanced level of public services.
Leastcost methods are used when the benefits are estimated to be
the same. Two common variations are (1) the leasttotalcost method and
(2) the leasttotalannualcost method. The leasttotalcost method merely
sums the estimated investment, operations, and maintenance costs over
the life of the project and thus obviously ignores the timing of costs
required by the discounting concept. The leasttotalannualcost method
adds an interest cost to the total cost. Those using this method confuse
financial analysis with economic analysis by including interest as a cost
without determining time equivalence by discounting specific cash flows.
SELECTED REFERENCES
Bierman, H., and S. Smidt: "'The Capital Budgeting Decision" (New
York: The MacMillan Company, 1960).
Brown, W. H., Jr., and C. E. Gilbert: "Planning Municipal Investment"
(Philadelphia: University of Pennsylvania Press, 1961).
Grant, Eugene L., and Grant Ireson:"LPrinciples of Engineering Economy"
(New York: The Ronald Press Company, 1970).
Savage, L. J., and J. H. Lorie: Three Problems in Rationing Capital,
J. Business (October, 1955).
Smith, Gerald W.: "Engineering Economy: Analysis of Capital Expendi
tures" (Ames: The Iowa State University Press, 1968).
Thuesen, H. G., and W. J. Fabrycky: "Engineering Economy" (Englewood
Cliffs, N.J.: PrenticeHall, Inc., 1964).
PROBLEMS
1A writer on the subject of the determination of the costs of public
hydroelectric power projects included the following items as costs:
(1) interest on the first cost of the project; (2) depreciation by the
MATHEMATICS OF ECONOMIC ANALYSIS 37
straightline method based on the estimated life of the project; (3) an
annual deposit in an amortization sinking fund sufficient to amount
to the first cost of the project at the end of 50 years (or at the end of
the life of the project if that should be less than 50 years); (4) where
money is borrowed, the annual disbursements for bond interest and
bond repayment; (5) all actual annual disbursements for operation
and maintenance of the project.
Do you believe that annual cost should properly be considered
as the sum of these items? Explain your answer.'
d3 A project to be evaluated at a 4.25 percent discount rate cost $1
million and has a $20,000 annual cost. Project benefits are expected
to be $20,000 in the first year, increase to $100,000 in the fiftieth
year following an accelerated growth curve, remain constant at
$100,000 annually until the ninetieth year, and then decline on a
uniform gradient to nothing in the hundredth year. What is the
benefitcost ratio?
8 An industry which requires 10 percent return on its capital has an
opportunity to invest in a business estimated to be profitable for
10 years. Alternative levels of investment and alternative net annual
returns by level of investment are:
Annual Annual
Investment benefits O d' 2M cost
$1,000 $160 $10
1,500 265 15
2,000 340 20
2,500 445 25
3,000 535 30
3,500 610 35
4,000 665 40
a How large an investment should be made?
b How large would the minimum attractive rate of return of the
industry have to be to prevent any of the above investments from
being made?
c What minimum attractive rate of return would lead to a decision
to invest $4,000?
4A certain project has a first cost of $b100,000 and an annual main
tenance cost of $b2,500 each year over a 50year life. Benefits realized
increase from $4,000 in the year immediately after construction to
$10,000 in the last year of project life.
a At 4 percent interest, what is the annual project cost?
x Problem taken from Eugene L. Grant and Grant Ireson, "Principles of Engineering Economy,"
4th ed., rev. ptg. (New York: The Ronald Press Company, 1964), p. 435.
38 ENGINEERING ECONOMY
b At 4 percent interest and with a straightline gradient, what is the
annual project benefit?
c What is the benefitcost ratio?
d What is the annual benefit if benefits increase in an accelerated
growth pattern?
e What is the annual benefit if benefits increase in a deferred growth
pattern?
f What is the internal rate of return of the project using a straight
line gradient?
65 An investor has $20,000 and the four investment opportunities de
scribed below:
Initial Net cash proceeds in year
cost 1 9
A $10, 000 $10, 000 $1, 000 $1, 000
B 10, 000 4, 400 4, 400 4, 400
C 10,000 2,000 3,000 9,000
D 10, 000 1, 000 2, 000 12, 000
a In which two projects should a private investor invest his money
if he uses the rateofreturn method?
b In which two projects should a public agency invest its money if it
uses a social discount rate of 3 percent?
c What should the private investor do if he has no alternative in
vestments this year, but starting next year (year 1), he can invest
his money at a guaranteed return of 20 percent?
26 The three alternatives described below are available for supplying a
community water supply for the next 50 years when all economic
lives as well as the period of analysis terminates.
Project A Project B Project C
Construction cost
Year 0 $20,000,000 510,000,000 )15,000,000
Year 20 0 10,000,000 12,000,000
Year 35 0 10,000,000 0
0 and M cost
Years 120 70, 000 40, 000 60, 000
Years 2135 80, 000 70, 000 80, 000
Years 3650 90, 000 90, 000 90, 000
Using a 4.5 percent discount rate where applicable, compare the
projects by:
a The presentworth method
b The rateofreturn method
c The benefitcost ratio method
d The annualcost method
MATHEMATICS OF ECONOMIC ANALYSIS 3Q
d7 A decision must be made between two alternative investments which
perform equally well. Investment A has a life of 5 years, first cost of
$62,000, annual maintenance cost of $625, and salvage value of $250.
Investment B has a life of 10 years, first cost of $4,000, annual
maintenance cost of $630, and salvage value of $1,000.
a Which alternative is to be preferred at a minimum acceptable rate
of return of 8 percent?
b Investment A employs a scarce material which is expected to
increase greatly in price during the next 5 years. How much would
the cost of replacing A in 5 years have to be in current dollars to
make B more economical in the present decision? Assume all other
costs are unchanged.
68 Already $620,000 has been spent on a $6200,000 project when it is
learned that a research breakthrough may soon develop a substitute
having a cost of $6135,000. The substitute has an annual operations
and maintenance cost of $4,000 instead of 965,000 with construction
in the originally planned manner. Annual benefits are projected to
follow an accelerated growth curve from 0 to $650,000 in year 50.
The discount rate is 6 percent, and the period of analysis is 50 years.
a Compute the benefitcost ratio for the proj ect as initially conceived.
b Compute the benefitcost ratio for implementing the substitute
project if it could be done immediately.
c If the breakthrough is delayed 5 years and an interim measure
is to be considered, what is the maximum uniform annual cost
one could afford to pay to achieve the benefit during the inter
vening period rather than build the initial project? Neglect the
salvage value of the substitute at the end of the period of analysis.
99 Two mutually exclusive investment alternatives which provide the
identical service may be described as follows:
First cost Annual cost Salvage value Life
A $610, 000 $62, 000 $1, 000 10
B 25, 000 1, 500 5, 000 20
Based on a minimum attractive rate of return of 5 percent:
a Which alternative has the lower annual cost?
b What is the incremental annual cost of going from the less to the
more expensive alternative?
c Select the optimum alternative by the presentworth method.
d What is the rate of return on the incremental investment of B?
e What first cost of replacing A after 10 years would make the two
alternatives equivalent, assuming none of the other costs change?
MI CRO ECONO MIC S
AND EFFICIENT
RESOURCE ALLOCATION
An economy study assigns a value to each predicted physical consequence
of each alternative and proceeds through a series of mathematical opera
tions to condense these values into a scalar index of aggregate worth. A
complex set of alternatives is reduced to a group of numbers which can be
ranked in order of magnitude for deciding relative merit. The assignment
of value is thus the critical step in the procedure. If done improperly, no
meaningful conclusion can result from the calculations described in
Chap. 2.
Part 2 seeks to provide the framework for evaluating these physical
effects. Chapter 3 presents the competitive market under conditions of
pure competition as providing the framework for establishing economic
value. Chapter 4 shows how values once assigned can be analyzed to
indicate an optimum or economically efficient design. Chapter 5 discusses
how goals other than economic efficiency can be introduced to achieve
desirable social objectives. Chapter 6 presents the implications of these
objectives on discount rate selection. Microeconomics provides the tools
for designing projects which in the aggregate will allocate to best use the
total supply of available resources.
CHAPTER
THREE
PRICE THEORY
AND RESOURCE ALLOCATION
Introduction to Microeconomics
Price exerts a major influence on individual decisions of whether or not to
use a particular economic good, and these many little decisions aggregate
to allocate resources by use. In an economy based on the private ownership
of property, or what may be called capitalism or the free enterprise system,
economic forces interact to determine price. Under ideal conditions (pure
competition), economic forces produce a firstorder approximation of a
normative system. Thus, analysis of these forces can be used to provide
the values needed in engineering economy studies (Sec. 23). Price theory
provides the framework for systematic study of these forces. It provides
a foundation for production theory, the study of how a firm should
operate to maximize profits. The result is an analogy indicating how a
public works project should be designed to maximize benefits. The study
of price theory guides the decision on whether a particular market price
is a fair measure of true public worth for use in an economy study. It
provides the tools needed for generating a shadow price for use where the
market price is not fair or where none has been established. Price theory
provides the analytic framework for establishing benefits and costs.
81 THE MARKET ECONOMY Because price theory analyzes the
activities of individual participants in a market economy, it is a micro
economic approach. Study of the cumulative effect of all the many
individual decisions on the national economy is a macroeconomic approach.
The normative framework traditionally used for establishing value in
44 MrICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
economy studies is based on the principles of microeconomics under the
assumption that water resources projects represent too small a portion
of the total national productive capital for individual design decisions to
have significant macroeconomic effects. The initial assumption is a
macroeconomic setting of a stable economy and full resource employment.
A free enterprise economy reacts to economic decisions of individuals.
Within limits, consumers are free to choose from a variety of goods and
services, enterprisers are free to produce what they desire, and resource
owners are free to sell to whatever buyer may be found. Voluntary ex
changes occur in the marketplace whenever it is mutually advantageous
to participants. Although profits are made as enterprisers correctly antic
ipate consumer demands and produce efficiently, losses occur if opposite
conditions hold.
The market provides a link between consumers and producers and
permits the exchange of goods and services. Some are geographically
small, others are worldwide. A market may have few or many buyers and
sellers. One product or many products may be offered. Government con
trol may override economic forces in certain instances. In a market sys
tem, prices are the basic signals that direct production and distribution.
To the degree that the goods exchanged are owned by many individuals
free to buy and sell as they wish, prices are determined by impersonal
market forces.
Cash flows within a market system can be classified by the use of
Fig. 31. The owners of productive resources (landowners, laborers, and
capitalists) sell them to enterprisers (firms) in the productive resources
market. The money the resource owners receive is spent to buy the prod
Money
payments
Money
payments
FIGURE 81 Model of a free enterprise economy
(spending and production).
PRICE THEORY AND RESOURCE ALLOCATION 45
ucts of the enterprisers in the market for goods and services. In turn, the
money that the enterprisers receive from consumers is used to buy addi
tional productive resources. Thus, consumers and enterprisers operate in
both the productive resources market and the goods and services market.
This simplified model of a free enterprise economy neglects government
action, interfirm transactions, and income from gifts or charity. It does
not distinguish transactions occurring in the market or private sector of
the economy from those occurring in the governmental or public sector.
A free enterprise system determines what, how, and for whom in
the following manner:'
1 What is to be produced is determined by the dollar votes of consumers
(the demand) cast each day for commodities purchased in the
marketplace.
d How things are to be produced is determined as individual firms are
required to adopt the most efficient (least costly) methods of produc
tion to stay in business.
3 For whom things are produced is determined by the number of market
place votes (income) an individual has. Incomes are determined as
supply and demand in markets for productive services set wage rates,
profits, land rents, and interest payments.
The absence of a costly regulatory structure is one of the greatest
strengths of the market system. No central planning authority makes the
myriad of economic decisions necessary to supply the goods and services
needed for a city such as Chicago. Yet, instead of chaos, an order exists
which supplies the variety of food, clothing, sundries, and entertainment
that a cosmopolitan city demands. As Samuelson has stated:
A competitive system is an elaborate mechanism for unconscious coordi
nation through a system of prices and markets, a communication device for
pooling the knowledge and actions of millions of diverse individuals. Without
a central intelligence, it solves one of the most complex problems imaginable,
involving thousands of unknown variables and relations. Nobody designed it,
it just evolved, and like human nature, it is changing; but at least it meets
the first test of any social organizationit is able to survive.2
How does the price system achieve an efficient allocation of produc
tive resources? Allocations must be made at three levels: among industries,
among firms in each industry, and within each firm. If an industry's
products are in great demand, they can be sold for high prices. The
SMuch of this paragraph is condensed from Paul A. Samuelson, "Economics: An Introductory Analy
sis," 5th ed. (New York: McGrawHill Book Company, 1962), pp. 4142.
Ibid., pp. 3830.
46 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
industry is able to pay high prices for productive resources and bid them
away from industries whose products are less highly valued by consumers.
Firms within an industry which produce a given output at a lower cost
can pay more for productive resources and expand relative to inefficient
firms. Lastly, the individual enterpriser strives to produce a given product
in the least expensive manner from the cheapest combination of productive
resources. According to Stigler:
A competitive enterprise system allocates resources with maximum
efficiency. If resources are used where they obtain the highest rates of renum
eration, if they are employed efficiently in these industries, and if they pro
duce the commodities that consumers most desire, output is as large as
possible.'
Ebenstein says,
The economy justification of competition is that it keeps everybody
worker, businessman, investoron his toes, constantly alert to changes in
the market, and constantly on the outlook for ways to increase his efficiency
and thereby improve his chances in the market. By increasing his own effi
ciency, the individual worker or entrepreneur proportionately increases the
efficiency and productivity of the whole market. Better products, lower
prices, better services and ultimately higher living standards for all result
from the constant incentive to keep up with, and if possible outdo, one's
competitors.2
82 PURE COMPETITION A market economy will automatically
maximize production from a given set of resources and thus be economically
efficient under the conditions of pure competition. Competition as defined
in economics does not necessarily denote rivalry. In fact, under pure com
petition, there is no rivalry among individual sellers or buyers. The con
ditions necessary for pure competition include:"
1Consumers must be consistent and independent. A consistent con
sumer gets more satisfaction from a larger amount of a given com
modity than from a smaller amount. The satisfaction gained by one
consumer must be independent of purchases by others.
d Producers must operate with the goal of profit maximization. The
SGeorge J. Stigler, "The Theory of Price" (New York: The Maemillan Company, 1961), p. 9.
SWilliam Ebenstein, "Today's Isms," 4th ed. (Englewood Cliffs, N.J.: PrenticeHall, Inc., 1964), p.
166.
Otto Eckstein, "Water Resource Development: The Economics of Project Evalutsion" (Cambridge,
Mass.: Harvard University Press, 1958), pp. 2530.
PRICE THEORY AND RESOURCE ALLOCATION 47
production processes of the firms must be independent so that one
firm's costs are not borne by others.
3 The transactions by each buyer or seller must be too small in relation
to the market to affect prices paid or received.
4 No price regulation or rationing or other artificial constraints by
government, labor, business, or other institutions are placed on the
demand and supply of goods and resources or on their prices.
5 Goods and services and resources must be mobile. This requires free
entry by firms into any industry and goods and labor free to move
from one local market to another to seek the best price.
6 Buyers and sellers must be aware of prices throughout the economy.
When buyers and sellers receive such information instantaneously,
we have what is known as perfect competition. The closest approxima
tion to this condition is on the New York Stock Exchange where
information on stock prices is transmitted continuously to all parts
of the nation.
7 Commodities must be sufficiently divisible so that sellers can withhold
all or part of the product from individual buyers who do not pay the
market price.
8 The existing income distribution must be considered equitable for the
dollar votes of the individual participants to be weighted equally.
9 All resources must be fully employed. When unemployment persists,
prices do not reflect opportunity costs or returns from the viewpoint
of the nation.
Even though pure competition does not exist in real markets, the
model provides an ideal for judging the efficiency of actual markets and
guidelines to help develop criteria for establishing value when its condi
tions are not met.
88 MARKET DEMAND Experience tells us that people will buy
less at higher prices provided income, tastes, and prices of substitutes
remain constant. Obversely, people buy more at lower prices. The demand
for a good is the quantity per unit time that people within a defined area
will buy as a function of all possible prices, all other factors remaining
constant.
One way to indicate demand is by a demand schedule, a list of the
different quantities of a good that people will take within a particular
time period at various prices. A hypothetical demand schedule for Idaho
potatoes is shown by the first two columns of Table 31. A demand curve
is the plot of the demand schedule. The vertical axis indicates the price
per unit, and the horizontal axis indicates the quantity of the good pur
8
1 2 34 5 67 8 910 1II2 13
Quantity, in millions /years
FIGURE 3d Hypothetical demand curve for Idatho
potatoes, given for 100lb sacks.
chased per unit time. Figure 32 is a hypothetical demand curve for Idaho
potatoes.
The demand curve slopes downward to the right because lower prices
increase sales, a principle designated as the law of downwardsloping
demand~. Two reasons for this increase are that (1) lowered prices attract
new buyers and (2) lowered prices induce extra purchases by former users.
Lower water prices would cause some to abandon more expensive alternate
sources of supply and become new buyers. Old buyers would use more.
When water is very expensive, a person only buys enough to drink. As
the price lowers progressively, he buys some for personal cleanliness, then
for household cleaning, and finally for yard watering.
It is important to distinguish between movement along a given
TA BLE 81 Supply and Demand Schedules for Idaho Potatoes, given for 100lb sacks
Price, in Quantity demanded, Quantity supplied, Price Market
dollars in millions per year in millions per year tendency conditions
7 1 10 Fall Surplus
6 2 9 Fall Surplus
5 4 8 Fall Surplus
4 6 6 Neutral Equilibrium
3 9 4 Rise Shortage
2 13 1 Rise Shortage
48 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
PRICE THEORY AND RESOURCE ALLOCATION 49
FIGURE 83 Shifts in demand curves
(caused by changes in
demand). O
demand curve and a shift of the demand curve caused by a change in
demand. Movement along a demand curve occurs with a change in the
price of the good. Shifting of the demand curve is caused by changes in
(1) consumer preferences, (2) the number of consumers, (3) consumer
incomes, (4) the prices of related goods, and (5) the range of goods avail
able.' When price changes from P1 to P2, We Should have movement along
the demand curve as the quantity of potatoes purchased changed from
Qx to Qs (Fig. 32). If there were an increase in consumer preferences for
Idaho potatoes, more consumers, greater consumer incomes, higher prices
for Maine potatoes, or fewer alternative foods available for purchase, the
demand curve would shift from DD to DID1 (Fig. 33). Opposite changes
would shift the demand curve to D2Dz.
84 PRICE ELASTICITY One of the most important relationships
expressed by a demand curve is the change in sales resulting from a given
change in price. This change could be measured by the slope of the demand
curve, but the general usefulness of the answer is limited by its units
(sacks per dollar in Fig. 32). A different slope on a curve plotted in different
units (bushels per cent) would indicate the identical relationship between
price and demand. Economists avoid this difficulty in units by use of the
price elasticity of demand defined as
AQ/Q aQ P
E or (31)
AP/P ap Q
The negative sign is introduced because Q increases as P decreases.
SRichard H. Leftwich, "The Price System and Resource Allocation" (New York: Holt, Rinehart and
Winston, Inc., 1964), p. 27.
50 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
0 5
(E= OFIGURE 84 Price elasticity of de
o mand.
Computation of the price elasticity of demand is illustrated by Fig.
34. Applying Eq. (31) to points A, B, and C, respectively, gives elas
ticities of 9, 1, and M. Even though the demand curve at all three points
has the same slope, the price elasticity of demand varies from infinity
along the vertical axis to zero along the horizontal axis.
A value of infinity for E indicates a perfectly elastic good which no
one at all will buy if the price is raised. It is represented by a horizontal
demand curve. Goods become perfectly elastic at the price that they are
completely priced out of the market.
As the price is reduced, elasticity drops. Eventually it reaches unity,
and the good is no longer said to be elastic. This point would provide the
supplier the largest gross revenue; the PQ product is a maximum. Until
the elasticity reaches unity, additional sales more than offset lower prices
and revenue increases. If the price is reduced past the point of unit elas
ticity, sales no longer increase fast enough to offset the lowering price
and revenue declines. The good is said to be inelastic. A value of zero for
E indicates a perfectly inelastic good or one for which price has no effect
on demand. It is represented by a vertical demand curve. Goods become
perfectly inelastic as the price becomes too low to remain a factor deter
mining the amount purchased. The same good is inelastic at low prices
and elastic at high prices.
85 MARKET SUPPLY On the sellers' side of the market, the supply
schedule and supply curve indicate the amounts that producers are willing
to sell at various prices, other things being equal. The first and third
column of Table 31 show the supply schedule for Idaho potatoes, and
Fig. 35 is the corresponding supply curve.
The supply ourve slopes upward to the right since old sellers will
produce more goods for sale and new sellers will enter the market as the
PRICE THEORY AND RESOURCE ALLOCATION 51
FIGURE 85 Hypothetical supply
curve for Idaho potatoes, OI234567891 I1
given for 100lb sacks. Quantity, in millions/year
price increases. The supply curve shifts (in contrast with movement along
the curve) with a change in the supply produced when the price remains
unchanged. Shifts to the right may be caused by technological advance,
favorable production conditions, or lower prices for the input factors of
production. Shifts to the left are caused by opposite conditions.
86 MARKET PRICE DETERMINATION The demand curve and
the supply curve combine to establish the equilibrium market price. The
combined demand and supply schedules in Table 31 illustrate the tendency
toward an equilibrium price.
If the initial price is above $84 per 100 lb, more will be supplied than
the quantity demanded. Potatoes will be in oversupply, and sellers will
cut prices in order to sell their crops. Also, if the initial price is below
$4 per 100 lb, less will be supplied than the quantity demanded and con
sumers will bid the price up. Only at the equilibrium price of $4 per 100
lb will the demand equal the supply. Figure 36 shows the same result
graphically. This equilibrium price is the minimum under conditions of
pure competition that each individual buyer must pay for each 100 lb
of potatoes purchased and the maximum that each farmer can receive
for each 100 lb of potatoes sold.
87 RESULTS OF SHIFTS IN DEMAND AND SUPPLY If the
demand for potatoes increases while the supply remains fixed, Fig. 37
shows how a shift in demand from DD to D1D1 causes an increase in price
from P to P1. This happens because at P there is now a shortage of potatoes,
the price will be bid up, and sellers will be induced to place more on the
market. Just the opposite happens with a decrease in demand: a surplus
creates pressure to lower prices.
When the supply curve shifts and the demand curve is fixed, equilib
52 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
,,, Supply curve
6 Surplus
.C 4 ~  + ~f Equiblbrtum point
a 3 7; A Demand curve
a. 2 Sh~rtage;
O 1 2 3 4 5 6 7 8 9 10 11 12 13
FIGURE S6 Hypothetical mar
ket demand and sup
ply curves for Idaho
potatoes, given for
100lb sacks.
rium prices and quantities are also affected. Suppose higher labor costs
were to increase the cost of growing potatoes. In Fig. 38, this would cause
a shift from SS to S1S1. At the original price P, there would be a shortage
and the price would be bid up to P1. The opposite will happen if lower
production costs should shift the supply curve to the right. There will
be a surplus and a downward pressure on prices to a new equilibrium
point.
88 SIGNIFICANCE OF MARKET EQUILIBRIUM Supply and
demand curves provide additional background for understanding the
automatic way the market system handles the allocation of goods or
answers the basic economic questions of what, how, and for whom. For
rohom is partially determined by individual willingness to pay. If you have
the money and wish to eat Idaho potatoes three times a day, you merely
pay the market price. On the other hand, you may not like Idaho potatoes
Quantity
FIGURE 87 Shift in demand curve.
PRICE THEORY AND RESOURCE ALLOCATION 53
s, s
D
FIGURE 38 Shift in supply curve. Quantity
and spend your entire income on other goods. What is partially determined
by the price for potatoes' determining the degree to which farmers shift
productive resources from the production of other crops. How is also
partially answered because the price for potatoes determines the money
farmers have to invest in potato processing equipment, sprinkler systems,
and fertilizer. A given pair of supply and demand curves only partially
answers these questions because events in other food markets and resource
markets also affect economic equilibrium. Therefore, the partial equilibrium
solution for Idaho potatoes plays only a small part in determining the
total price structure.
Consumer Demand
The above bird'seye view of the interaction between supply and demand
to achieve equilibrium prices provides the background for a more thorough
discussion of the economic principles governing demand. Two approaches
have been used to derive a theory of consumer demand. For years, classical
economics has used the utility approach based on values assigned in
absolute units. More recently, the indifferencecurve approach has been
developed because it avoids the problem of evaluation in absolute units
by using relative values. For brevity, we shall only explain the more
recent indifferencecurve approach.'
SLeftwich, "The Price System and Resource Allocation," gives both approaches in chaps. 4 and 5.
54 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
7
O 1 2 34 56 7 8910
Potatoes, in Ib/week FIGURE 89 Indifference curves.
89 INDIFFERENCE CURVrES An indifference curve (sometimes
called the equalutility contour) shows the consumption combinations
which give a consumer equal satisfaction. It is called an indifference
curve because the consumer is equally satisfied with any of the com
binations depicted by the curve. An indifference curve is theoretically
obtained by asking a consumer which combinations of goods yield equal
satisfaction, and its development assumes he can order his preferences.
The indifference curves described below will be two dimensional so they
can be graphically presented. However, real consumers must choose in
hundreds of dimensions, one for each good consumed. The true indifference
curve is a multidimensional indifference surface.
If a consumer buys only two goods with his income, meat and
potatoes, the combinations in Table 32 might give equal satisfaction.
More meat compensates for giving up potatoes and vice versa. The plot
of these combinations gives a single indifference curve (Fig. 39). Greater
TABLE 8I Combinations of
Meat and Potatoes Giving
Equal Satisfatction
Meat, Potatoes,
lb wk' lb wk'
1 10
2 6
3 3
4 2
6 1.5
8 1
PRICE THEORY AND RESOURCE ALLOCATION 55
quantities of both meat and potatoes will give the consumer with an
inadequate diet a greater total value since greater quantities of the goods
bring more satisfaction. A second indifference curve can be drawn to
indicate the combinations of meat and potatoes that provide this higher
level of satisfaction. Thus, there are an infinite number of indifference
curves, each one indicating a separate level of satisfaction the consumer
may experience. This system of indifference curves is called an indiference
map. The indifference curves may be viewed as contour lines of increasing
elevation as one moves upward to the right.
Some of the properties of indifference curves are:
1 They cannot intersect since it is impossible for a single combination
of goods to yield two levels of satisfaction simultaneously.
A They slope downward to the right. As the amount of one good is
increased, the other must decrease if equal satisfaction is to be main
tained. The slope of the indifference curve is called the marginal rate
of substitution.
3 They tend to approach the axes asymptotically because as less and
less of a good is consumed, the sacrifice of parting with an additional
unit becomes greater. Many more units of the other good must be
substituted to bring equal satisfaction.
S10 MAXIMIZATION OF SATISFACTION A consumer maxi
mizes his satisfaction by picking the highest indifference curve available
to him as determined by his income and the prices of the two goods.
These two consumer's opportunity factors determine what may be called
a line of attainable combinations. It intercepts each axis at a value equal
to the income divided by the price. For example, one intercept would be
the amount of potatoes which could be purchased were the entire income
spent for that purpose. The other intercept indicates the amount of meat
which could be purchased by the entire income. Points on a straight line
between these intercepts indicate the combinations of meat and potatoes
open to the consumer. The highest attainable indifference curve is the
one tangent to the line of attainable combinations (Fig. 310).
The total income I spent on two goods y, and yb with respective
prices per unit of P,, and P~s equals
I = P,,Y, + Payb b ~2)
The number of units of y, which can be purchased thus equals
Pa I
X = ya + (33)
56 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
~ ~ ~f v4u
uus
FIGURE 810 Income allocation.
This is the equation of a straight line (the line of attainable combinations)
with a slope of Pyb ya*.
The consumer is able to attain points A, B, C, D, and E in Fig. 310.
If he chooses A, he is on U1, which gives less satisfaction than U2 Or Ua.
Point C gives maximum satisfaction. All the other points are on lower
indifference curves. The income constraint prevents achievement of Uq.
At point C, the slopes of the indifference curves and the line of attainable
combinations are equal. The marginal rate of substitution (the slope of
the indifference curve) equals the ratio of the prices (the slope of the line
of attainable combinations),
MRS,,,s= 34
where MRS,,ab is read as the marginal rate of substitution of y, for ye.
81CONSUMERDEMAND CURVrES Consumerdemand curves
are derived by varying the price of one good, P,,, while keeping constant
the income, consumer preferences (position and shape of indifference
curves), and the price of the other good, Pyb, Or in the general case, of all
other goods. By changing the price of y,, the slope of the line of attainable
combinations changes to become tangent to a different indifference curve.
Suppose the initial line of attainable combinations (AB in Fig. 311) is
tangent to Un, which means that for an income of I, an amount of y,
will be purchased at P ~, and an amount yb at P ,. If P,, increases to
P ~, the new line of attainable combinations AC will have a steeper slope
than AB because fewer units of y, can be purchased when all of I is spent
PRICE THEORY AND RESOURCE ALLOCATION 57
B U2
v6 6 Ioy6~ y
FIGURE 811 Derivattion of priceconsumption curve.
on y,. The line of attainable combinations pivots about A because I/P,b
is constant. The price increase lowers the level of satisfaction the con
sumer on a fixed income can attain. He moves to a lower indifference
curve and now would allocate I so as to purchase y,' unt o ,an
units of yb. If Pva is assigned a series of values, the line connecting the
points of tangency is called the priceconsumption curve. The demand
curve of a specific individual for y, can be found by plotting corresponding
values of y, and P,, taken from the priceconsumption curve (Fig. 312).
819 AGGREGATEDEMAND CURVES Individual demand curves
for Idaho potatoes may be used to develop a combined demand curve
(Fig. 313). At each price, the demand by each individual may be added
dv
Dollars/yo
Pyb tI
v6uv~
FIGURE 819 Consumerdemand curve.
58 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
Jones' demand
%0.70 .0
0.60 d4 060
S0.50 d ; 1 0.50
0.40 0.40
o .so I o0.0
S0.20 I 0.20
O.1O I d40.10
O 1 2 3 4 5 O
Quantity
dZ Black's demand
~
d
 yf = 2
I2 4
Qu I it
,mn uv
Combined demand
S0.70
S0.60
=0.50
0.40
E
0 0.30
0.20
S0.10
Market de
O 1 2 3 4 5 6 7 8 9 10
Quantity
FIGURE 818
Summation of consumer
market demand curve.
demands to obtain
to get the combined demand. This combination occurs automatically in
the market because the demand by both individuals must be met from
the same supply. Such horizontal addition is thus characteristic of market
goods.
Note that each consumer is free to decide his relative preference for
different goods according to his own tastes. Since people have different
tastes, individual demand curves differ. However, the horizontal addition
of the demand integrates different individual tastes into the market
demand curve.
SELECTED REFERENCES
Ebenstein, William: "LToday's Isms," 4th ed. (Englewood Cliffs, N.J.:
PrenticeHall, Inc., 1964).
PRICE THEORY AND RESOURCE ALLOCATION 59
Eckstein, Otto: "LWater Resource Development: The Economics of Proj ect
Evaluation" (Cambridge, Mass.: Harvard University Press, 1958).
Leftwich, Richard H.: "The Price System and Resource Allocation"
(New York: Holt, Rinehart and Winston, Inc., 1964).
Samuelson, Paul A.: "Economics: An Introductory Analysis," 5th ed.
(New York: McGratwHill Book Company, 1962).
Stigler, George J.: "LThe Theory of Price" (New York: The M~acmillan
Company, 1961).
PROBLEMS
31 The supply and demand schedules for fidwots is:
Price, 8: 1 2 3 4 5 6 7
Demand: 41 30 21 14 9 6 5
Supply: 12 14 18 24 32 42 55
a Plot the supply and demand curves.
b What is the equilibrium market price for fidwots?
c What is the price elasticity of demand at a price of $2? Of $b6?
d At what price is there unitary elasticity?
e What equilibrium price would result from a doubling of demand?
8b Lines of consumer indifference between commodities A and B are
represented by the indifference map represented by the equation
0.1A2B = V, where V is a scalar measure of satisfaction.
a Construct an indifference map covering the region A < 10, B < 50.
b What level of satisfaction is gained at the point A = 5, B = 20?
What is the marginal rate of substitution, dA/dB, at this point?
c A consumer has an income of 20 to spend in a market where
PAl = 2 and Pg = 0.5. Plot the line of attainable combinations.
What is the maximum level of satisfaction the consumer can
reach? What amounts of A and B does he purchase to obtain this
satisfaction?
d Based on values of Pg = 1.0, 2.0, 4.0, and 10.0, plot the price
consumption curve on the indifference map and then the con
sumerdemmand curve for B.
83 A second consumer is in a situation identical with that of the con
sumer in Prob. 32c except his available income is 5.
a Plot a consumerdemand curve for this second man.
b Plot a marketdemand curve, assuming these are the only two
consumers.
CHAPTER
FOUR
CONDITIONS
OF PROJECT OPTIMALITY
Production Theory
Economic forces act within the private sector of the economy to deter
mine the supply curves of individual productive units (firms) and integrate
them into a marketsupply curve. Production theory attempts to explain
the operation of these forces in ordering private sector production and
thereby determining:
1 The total expenditure on inputs (raw material, machinery, labor, etc.)
SThe division of this expenditure among individual inputs
S The way inputs are combined to produce each type of output
C The amount of each output produced
Water resources development is a production process. In planning a
production process for the public sector, many valuable insights can be
gained from analyzing how economic forces would act to order production
under ideal conditions.
41 INPUT AND OUTPUT The basic purpose of production is to
convert material (input) into a more useful form (output). A water
resources project is constructed to produce such desired output as irriga
tion water, reduced flood damage, a navigable channel, or electric power
from a set of such inputs as earth, concrete, steel, and natural stream
flow. As expressions of varied composition, both input and output are
vector quantities. Each coordinate of the vector represents a specific
input or output item.
62 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
The input vector consists of the sum of all the individual inputs.
Thus,
X = xx + x, + * + x,. (41)
where X is the total input vector and the x are m individual inputs.
The inputs fall by nature into two types. One type is the earth, concrete,
and steel which go into a construction project, the required production
capital. The other type is the natural streamflow from which the output
is produced, the required raw material.
Evaluation of the input vector is complicated by the fact that the
timing and magnitude of future streamflow~s cannot be predicted in
advance and vary over wide ranges. Streamflow may be expressed as a
continuous hydrograph, a running plot of stream discharge throughout
the life of the project, as a probability distribution, or as such distribu
tion moments as the mean and the standard deviation. Furthermore, one
must keep in mind that all the other input and output vectors also have
probability distributions even though their variance may be much smaller
because one cannot predict any future event with absolute certainty.
The concept of the input vector can be simplified by defining its
coordinates as the intermediate products of reservoirs, channel improve
ments, or powerhouses rather than the construction items of earth, con
crete, and steel. The optimum combination of construction items in build
ing a given intermediate product (say a dam of specified size) is best
found from an engineering economy study seeking to minimize total
cost. The first phase or the economy study determines how much earth,
concrete, and steel should go into a dam of given size. The second phase
or optimality analysis determines how big the dam should be and when
and where it should be built.
The output vector consists of the sum of all the individual outputs.
Thus,
Y = y, + vb + +y (42)
where Y is the total output vector and the y are n individual outputs.
Again, each output can only be predicted during planning and has a
probability distribution expressing the range of possible values. The
individual outputs are distinguished by type (irrigation from flood control)
and by location (the area served.)
The composition of both vectors varies with time. Input contains
investment in original construction, periodic replacement, and regular
operation and maintenance. The output occurs in a time stream lasting
the life of the proj ect and varying from year to year but normally increasing
with the general growth of the economy.
CONDITIONS OF PROJECT OPTIMALITY 63
Locus of ef ficient points Combinations of output produced
( production function) frorn a fixed quantity of input
XC
Technologically
~Q B infeasible region
Technologically
feasible region
Output yo
FIGURE 41 Production function.
42 THE PRODUCTION FUNCTION Economists have traditionally
expressed the ability of a production process to produce an output vector
from an input vector by a production function. As a simple illustration,
one might have a production process in which two outputs are produced
(Fig. 41). Engineering analysis may show that the combination of out
puts represented by point A could physically be produced and the com
bination of point C would not be physically possible to produce. Continued
analysis of alternative production possibilities will show which output
vectors can and which output vectors cannot be produced. All those that
can be produced are said to fall in the technologically feasible region.
Some points in the technologically feasible region are efficient, while
others are inefficient. For example, it would be physically possible to
dispose of the input vector without producing anything, but this would
be wasteful and inefficient. The production represented by point A is
inefficient because with the same input, the output vector could be
increased to point B. For an inefficient point, the output vector can be
unambiguously increased without increasing the input vector or the same
output vector can be produced after an unambiguous decrease in the
input vector. An unambiguous increase of a vector means some coordinates
are increased without any being decreased. An unambiguous decrease
means some coordinates are decreased without any being increased. The
locus of efficient points is the outer limit of the technologically feasible
region. The production function is the mathematical representation of
this line. It is related to the entire input and output vectors and, by putting
64 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
all the terms on the lefthand side, can be represented by the expression
f(X, Y) = (43)
48 THE OBJECTIVE FUNCTION Selection of the best point on
the production function requires some criteria for evaluating worth. The
criteria must assign a scalar value to each point on the production fune
tion. A scalar value is needed because vectors cannot be unambiguously
ranked according to magnitude as one may be larger in one coordinate
but smaller in the other. The objective function necessarily depends on
both the input and output vectors.
U = u(X, Y) (44)
Where n outputs are produced from mz inputs,
U = Byy; Ox< (45)
j=a i=1
where the Bi refer to the unit benefits associated with the corresponding
coordinates of the output vector, and the Ci refer to the unit costs asso
ciated with the corresponding coordinates of the input vector. Con
ceptually, both benefits and costs may be either measured in monetary
units or defined with respect to some broader based social welfare func
tion (Sec. 52) without affecting the optimality criteria derived below.
44 COST AND BENEFIT CURVES Economic evaluation of pro
duction alternatives is based on the variation in total production cost
with level of production output (called the totalcost curve) and the varia
tion in the resulting benefit with level of production output (called the
totalbenefet curve). The totalcost curve is developed by summing the
required input costs for a series of levels of output. The totalbenefit
curve is developed by summing values received by output users.
The total cost includes fixed cost and variable cost. Fixed costs
remain constant regardless of output. They include capital recovery
charges on the production facilities and other overhead costs. Variable
costs vary with level of output. They include the costs of labor and
material which can be added or deleted according to the level of produc
tion. Variable costs are marginal costs and are used to determine the
optimum level of production according to the incrementalcost principle.
Fixed costs are not marginal and thus have no influence on the optimum
level of production, but they do influence whether or not total benefits
exceed total costs or whether the project should be constructed at all.
Averagecost or benefit curves are developed from totalcost or
CONDITIONS OF PROJECT OPTIMALITY 65
500
450
400
350
8300
P~250
I / 'Total cost curve
J r, T lf)v jf'( Y) dY
a 200
150
Marginal cost curve
df(Y)
100 dv
Average cost curve
50
O 1 2 3 4 5 6 7 8 9 10
Level of output = Y
( for a project with only one output)
FIGURE 48 Representative totatl, average, and marginal
cost curves.
benefit curves by dividing the total value by the level of output (Fig.
42). Averagecost curves are usually Ushaped. They decrease at first
because of economies of scale, savings in production cost per unit stemming
from increases in size of plant and output. Economies of scale result
from (1) specialization of labor and (2) advanced technology. Specializa
tion of labor may be impossible in a small plant because no single task
requires the fulltime effort of any individual worker, but as the size of
plant and production increases, assembly line techniques become practical
and reduce unit production costs. Advanced technology may be available,
but its fixed cost may be too large to be warranted for a small plant. How
ever, increased output spreads fixed cost over more units to make advanced
technology economical. In reservoir construction, much of the fixed cost
stems from providing sediment storage space, spillways to pass rare floods,
access roads, and other minimum structural requirements. Average cost
66 MICROECoONMICS AND EFFICIENT RESOURCE ALLOCATION
increases again as production becomes very large because of diseconomies
of scale caused by difficulties in managing, controlling, and coordinating
a very large firm or by inducing increases in the price of inputs. In reservoir
construction, diseconomies of scale are largely caused by a decreasing
yield and increasing dam size per acrefoot of storage as reservoir size
gets too large.
Marginalcost or benefit curves present the slope of totalcost or
benefit curves. The slope represents the change in total cost or benefit
associated with a oneunit change in output. Because a firm will not
produce an extra unit unless the price exceeds the marginal cost, the
rising limb of the marginalcost curve is a supply curve. It indicates the
price necessary to induce an extra unit of production. Because a buyer
will not make a purchase unless it provides a value exceeding the cost,
the marginalbenefit curve is a demand curve. It indicates the maximum
price that a firm which uses the output produced can afford to spend to
acquire an extra unit.
Total values may be established from marginal curves because the
area under the marginal curve to the left of an abscissa equals the ordinate
of the total curve. The total cost of a group of items equals the sum of
their individual costs. At the low point on the average curve, the marginal
values equals the average value because, otherwise, adding the marginal
increment would change the average. A marginal curve must plot below
a falling average curve to cause average values to drop. Obversely, a
marginal curve must plot above a rising average curve. Therefore, mar
ginal curves, like average curves, are generally Ushaped, but justified more
to the left.
45 OPTIMALITY CONDITIONS One approach to determining the
optimum production process would be systematic trialanderror evalua
tion of the net benefit [Eq. (45)] for each point on the production func
tion [Eq. (43)], which will lead to the point having maximum value.
The search is made easier by being able to recognize the characteristics
peculiar to such a point.
A mathematician distinguishes necessary from sufficient conditions
when seeking such characteristics. Necessary conditions are characteristics
the solution must have, but they do not guarantee that a point having
them is the solution. Sufficient conditions guarantee a solution. A simple
example applying this distinction to project optimization is found in the
production of one output from two inputs. If the two inputs are plotted
on a horizontal plane, the maximum output which can be produced from
each combination of the two can be plotted vertically to produce what
CONDITIONS OF PROJECT OPTIMALITY 67
resembles a topographic contour map and is called a productionn or a
response surface. A necessary condition to show that the maximum output
has been produced from these two inputs is that the surface slope down
ward in all directions. However, this condition is not sufficient because
there might be a higher hilltop on the other side of a valley. For greater
numbers of inputs the surface is multidimensional, but the principle is
the same.
Proof that a given peak is the maximum may follow one of two lines.
The height of each peak may be computed to show which is highest.
Evidence may be presented to show onlly one peak exists. A single peak
will in fact be the case if all second derivatives of the objective function
are continuously negative, a situation occurring under conditions of
diminishing marginal utility as supply curves slope continuously upward
and demand curves slope continuously downward to the right. The
second approach is more frequently used in water resources planning.
It needs to be emphasized that the conditions of project optimality to
be developed below are necessary but not sufficient conditions. Proof
of absolute optimality requires evidence that no higher peak exists.
For a water resources project, the goal is to find an alternative
having maximum value of u(X, Y) with the constraint that only alter
natives contained on the production function f(X, Y) = 0 need be con
sidered. The conditions necessary to having a maximum value may be
determined by (1) the geometrical approach or (2) the mathematical
approach. The geometrical approach follows immediately, while the
mathematical approach is presented later.
Geometric Derivation of Basic Rules
a6 OPTIMALITY CONDITION 1: COMBINATION OF INPUTS
The optimal production process must use the least costly combination
of inputs able to produce any given level of output. For example, the
sizes of the two dams to provide flood control must be selected to achieve
the desired level of flood reduction at minimum cost. The leastcost
combination of inputs can be found geometrically by the use of isoquant
lines and isocost lines. If the problem is reduced to two dimensions for
practical presentation, isoquant lines (Fig. 43) show different combina
tions of two inputs which can produce equal amounts of a single output.
Isoquants are analogous to indifference curves and have analogous
characteristics :
68 MICROECONOlMICS AND EFFICIENT RESOURCE ALLOCATION
1 Two isoquants cannot intersect. An intersection would require the
maximum output which could be produced with the same input to
be two different amounts.
d Isoquants slope downward to the right because increased use of one
input reduces the quantity of another required to obtain a given level
of output. Channel improvement can be substituted for reservoirs to
provide flood control.
8 Isoquants are convex to the origin because of the decreasing ability
of one input to be substituted for another to obtain a given level of
output. As more channel improvement and less reservoir storage
are used to produce a given level of flood control, the larger is the
incremental channel improvement required to effect a unit reduction
in flood storage. This is called the principle of diminishing marginal
rate of substitution.
In Fig. 43, the isoquant for output y, shows the possible combina
tions of xx and x2 which could be used in its production. The most efficient
combination depends on the unit prices of the inputs, just as the unit
prices of goods guide spending to maximize consumer satisfaction (Sec.
310).
Isocost lines (Fig. 44) indicate the input combinations that can
be purchased by a given production budget and are analogous to the line
of attainable combinationsused in demand analysis. If the production
budget is T, the price of xx is P,,, and the price of x2 18 P,,;
T = P,,xx + P,,x2(46
which is the equation of a straight line with a slope of P,,/P,2'
x;  7  _Output yo
FIGURE 48 Isoquatnts (lines
X1' XI* x, of equal out
Input put).
CONDITIONS OF PROJECT OPTIMALITY C)9
Budget T
T/Px, xr
Input
FIGURE 44 Isocost lines (lines of equal cost).
Production of a given level of output with the leastcost combination
of resources occurs where an isocost line (slope of Ps,/P,,) is tangent to
the isoquant (slope of MRS,,,,). Therefore,
P,
MRS,,,, =(47)
Figure 45 shows a number of isoquants with tangent isocosts. The line
AB joining the points of tangency is called the expansion path and is the
locus of the leastcost combinations of inputs for varying levels of total
output.
An example will clarify the procedure for combining inputs. Suppose
the axes in Fig. 45 represent two singlepurpose flood control reservoirs.
We can calculate combinations of storage capacity in the two reservoirs
providing a fixed level of flood peak reduction (isoquant) and the storage
combinations that can be constructed with a fixed budget (isocost). As
many isoquants and isocosts as needed may be calculated and drawn.
Each point of tangency represents the leastcost combination of reservoirs
to provide a given level of flood reduction.
47 OPTIMALITY CONDITION 9: COMBINATION OF OUTPUTS
With two outputs, such as municipal water supply and hydropower, total
production must be divided between the two to maximize benefits. One
may begin the analysis by plotting on coordinate axes representing two
70 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
Isocost line
N
,~
Expansion path
Least cost combination of
x, and xy to supply output Y,
S,Output Ye
Input
FIGURE 45 Determination of leastcost combina
tion of inputs.
outputs y, and Yb (Fig. 46) each of the family of curves showing com
binations of outputs that can be produced at a given cost. Each curve
indicates all combinations of outputs y, and yb that can be produced for
the indicated sum and is called a producttransformation. curve because to
move along it, one output must be increased while the other is reduced.
Isorevenue hine
Producttransformation curve
Optimum combination of
outputs yo and y6 for
given level of benefits
FIGURE 46 Optimum com
bination of out
puts.
Output
CONDITIONS OF PROJECT OPTIMALITY 71
The slope of the producttransformation curve is called the marginal rate
of transformation.
A family of parallel lines called isorevenue lines may also be drawn in
Fig. 46. The slopes of these lines are the ratios of the market prices of the
two outputs. Each isorevenue line shows the different combinations of
outputs that would sell for the same amount of gross revenue or would
produce a given benefit.
The optimum mix of outputs achieves a given level of benefits at
least cost, or put another way, the maximum level of benefits for a given
level of costs. In Fig. 46, the optimum combinations are located at the
points of tangency of the isorevenue lines (slope of P,,/P,,) and product
transformation curves (slope of MR T,,,,). Therefore,
P,
MR T,,,, (48)
P,,
Producttransformation curves are concave to the origin (Fig. 46)
if the outputs are joint products of the same productive process or if the
production of one is facilitated by production of the other. But if the
production of one output hinders production of another, the product
transformation curves are convex to the origin. In this case, benefits are
maximized by producing only one of the two outputs, a boundary solution
for which Eq. (48) does not apply (Fig. 47). The producttransformation
curve reaches the highest isorevenue line on the y, axis; therefore, only
Isorevenue line
4 Producttransformation curve
Optimum output
for given level
of benefits
Output ye
FIGURE 47 Optimum output for concave prod
ucttransformation curves.
72 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
1/ [/ FIGURE 48 Determination of opti
Output Fo / ear mum level of output.
y, would be produced. Note the isorevenue line is not tangent to the
producttransformatio n curve at this point.
Optimality condition 2 may be summarized by saying production
should be divided between two outputs so that the marginal benefit of
any input in the production of one equals the marginal benefit of the input
in production of the other. Otherwise production could be shifted be
tween the outputs to increase benefits.
48 OPTIMALITY CONDITION 8: LEVEL OF OUTPUT Opti
mality condition 3 determines the optimum level of output, on the assump
tion that conditions 1 and 2 have already been met. It states that benefit
is maximized if output is increased up to the point where the marginal
Marginal cost curve
sl\ / /Average cost curve
Average variable cost curve
Marginot benefit curve
~I j Average fixed cost curve
IOptimum level of output yo
Output yo /year
FIGURE 49 Optimum level of output by using margi
nalcost and marginalbenefit curves.
CONDITIONS OF PROJECT OPTIMALITY 73
p/\ ~ ,Expansion path
Annual benefits
FIGURE 410 Optimum construction \ Annual costs
costs and combination of
outputs. Annual output Yo
costs equal the marginal benefits, or in engineeringeconomy terminology,
incremental costs equal incremental benefits. The two marginal values are
equal where the slopes of the totalvalue curves are equal or where the
distance between them is maximum (see Fig. 411).
For the twoinput oneoutput case, optimality condition 1 provides
the basis for calculating the minimum cost of attaining different levels of
output. The results may be plotted in a totalcost curve (Fig. 48.) The
totalbenefit curve may be plotted by multiplying the unit price of the
output times the quantity of output. Under the conditions of pure com
petition, the unit price is constant; hence the totalbenefit line is straight.
For only one output, optimality condition 2 does not apply, and we can
Present worth of
total benefits
c/ Present worth of
Mxmum construction costs
fl / net
BI benefits
Otimum level of construction costs and output
Construction costs
FIGURE 411 Determination of optimum level of con
struction costs and output.
74 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
go directly to optimality condition 3. As can be seen from Fig. 48, net
benefits are maximized where the slopes of the totalbenefit and totalcost
curves are equal. The same optimum level of output can be expressed on
marginal curves as the point where marginal cost equals marginal benefit
(supply equals demand) as seen in Fig. 49. For the oneinput twooutput
case, only optimality conditions 2 and 3 apply since there is no problem
of combining inputs.
The benefits and costs associated with output combinations on the
expansion path (Fig. 410) may be plotted (Fig. 411) to determine the
optimal level of construction cost. This cost then is divided between
producing outputs y, and yb by referring back to the corresponding point
on the expansion path. For the multipleinput multipleoutput case, total
cost and totalbenefit curves must be used as described in the next chapter.
Mathematical Derivation of Basic Rules
The goal of project optimization is to maximize the objective function
u(X, Y) by choosing the best alternative on the production function
f(X, Y) = 0, where X is an m coordinate and Y is an n coordinate vector.
49 LAGRANGE MULTIPLIERS Differential calculus can be used
to find such a maximum by differentiating the objective function with
respect to each of the (n + m) vector components, setting each differential
equal to zero, and solving the resulting equations. However, the solution
can only be constrained to alternatives contained on the production func
tion by including it in the equations being solved. This introduces one
more equation than unknown, and the problem becomes overdetermined.
One way out is to introduce an artificial unknown called a Lagrange
multiplier' as a coefficient of the production function and add the product
to the objective function. Thus,
L = u(X, Y) + Xf(X, Y) (49)
where X is the Lagrange multiplier and L is the variable to be maximized.
Equation (49) is based on the principle that if the constraint is satisfied,
the production function will equal zero. By differentiating with respect to
X, setting the differential (which will be the production function) equal to
SLagrange multipliers are described in much greater detail in William J. Baumol, "Economic Theory
and Operations Analysis" (Englewood Cliffs, N.J.: PrenticeHall, Inc., 1961) pp. 5450.
CONDITIONS OF PROJECT OPTIMALITY 75
zero, and including it in the set of equations to be solved, one incorporates
the production function into the solution. The approach is illustrated by
Ex. 41.
EXAMPLE 41
In a, constrained maximization problem, the objective function is
Y = 10ab, and the constraint is 5a + b = 200. In other words, we are
seeking the values for a and b which maximize the first expression without
exceeding the upper limit of the second expression. Placed in the format of
Eq. (49), the constraint expression 5a + b 200 = 0 may be added to
the objective function without changing its value, to give
Y = 10ab + x (5a + b 200)
By partial differentiation with respect to each of the unknowns a, b, and X
and by setting each differential equal to zero,
aY
10b + 5X = 0
aY
10a + X= 0
aY
5a + b 200 = 0
Solution of the three equations gives a = 20, b = 100, and X = 200.
When these values are substituted in the objective function, the maximum
value is found to be Y = 20,000.
The economic significance of X is that if the number on the right
hand side of the constraint equation had been 201 instead of 200, the
optimum value of Y would have been 20,000 + X, or 20,200.
Lagrange multipliers permit constrained maximization by introducing
as many artificial unknowns as there are constraints to make the number
of unknowns and equations equal. The problem could be solved without
using Lagrange multipliers by substituting the constraint expression in
the objective function before using the differential calculus approach,
but for many expressions the algebra makes this approach difficult if not
impossible.
410 APPLICATION OF THE LAGRANGE MULTIPLIER In
order to find the maximum, Eq. (49) must be differentiated individually
with respect to each coordinate of the two vectors as well as X; and each
76 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
differential must be set equal to zero. Thus,
Bu(X,Y) af(X,Y)
Ai=1 2 (410)
and
au(X,Y) af(X,Y)
x j = a, b,. ., n (411)
and
au(X, Y)
f (X, Y) (412)
By dividing Eq. (411) into Eq. (410) and pairs of Eqs. (410) and (411)
into each other, one obtains
au(X, Y)/ax, af(X, Y)/axe
(413)
au(X, Y)/ayj af(X, Y)/ayi
Bu(X, Y)/axx af(X, Y)/axx
(414)
au(X,Y)/8x2 af(X,Y)/ax,
au(X, Y)/0ye af(X, Y)/ayn
(415)
au(X, Y)/aya af(X, Y)/ays
Since f(X, Y) must equal zero, an increase in one element must be offset
by a decrease in another. Therefore,
af(X,Y)/ax, yi
(416)
af(X,Y)/ay, Ox;
af(X, Y)/axt al
(417)
af(X,Y)/8x2 a 1
af(X, Y)/8y, y>
(418)
af(X, Y) /ays y
By combining Eqs. (413) and (416), (414) and (417), and (415) and
(418), one finally achieves
au(X, Y)/axe y;
(419)
au(X, Y)/ay; ax<
au(X,Y)/8xt ax2
(420)
au(X, Y)/8x2 1z
au(X,Y)/ay, yb
(421)
du(X, Y)/aya ay,
411 THREE BASIC OPTIMAIJTY CONDITIONS Analysis of the
meaning of the terms in Eq. (419) reveals au(X, Y)/8xe to equal the
CONDITIONS OF PROJECT OPTILMALITY 77
marginal cost of input i, or MCi, and au(X,Y)/8yi to equal the marginal
benefit from output j, or MBi. The term on the righthand side of the
expression, 8yi/Dze, is what economists call the marginal physical
product, or the additional output which can be produced per unit of in
crease in input. The negative sign results from the opposite nature of
inputs and outputs. Thus
MCi
MPPii (422)
MBi
where MPPii is read as the marginal physical productivity of the ith
input when devoted to the jth output.' Similar analysis of Eq. (420)
reveals MC1 and MC2. The marginal rate of substitution was defined in
Sec. 46 as the marginal rate at which quantities of the second input need
to be substituted for a unit reduction in the first input while holding the
level of production constant, ax,/axx. Thus
MC1
SMRS21 (423)
MC2
Equation (421) contains MB,, MB,, and the marginal rate of transforma
tion (Sec. 47), or the marginal rate at which production can be shifted
from the second output to the first to effect a unit change in the first
without changing the input. Thus
MB,
SMRThe (424)
MBa
410 APPLICATION OF OPTIMALITY CONDITIONS The three
equations of the last section may be used to answer the four questions
fundamental to structuring production. The application may be illustrated
by a water resources project which produces two outputs, flood control
y, and irrigation ye, from two inputs, reservoir storage xx and channel
improvement xs.
The first fundamental question is: How should the inputs be com
bined to produce a given output? The answer is found in Eq. (423). The
marginal cost of an input is its unit price. If unit price varies with amount
purchased, the marginal price at the input actually used should be applied.
Equation (423) says the inputs should be combined in such amounts that
the ratio of their prices equals the marginal rate at which one input can be
substituted for another with all other components of the production fune
tion constant. With the other inputs and outputs constant, and based on
a typical production function, one may evaluate xt as a function of x,
to obtain the data in Table 41 and the curve in Fig. 412. The curve is
SThis notation is simplified from that previously used, which would be MPPa.i,
78 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
0 5 10 15 20
FIGURE 419 Optimum combination of inputs.
concave toward the origin because of an increasing inability to substitute
a second input for the first as one approaches the point where the first
is not used at all. Our rule says to substitute x2 for xx until the marginal
rate of substitution MRS of x2 for xx equals the ratio of marginal costs
MC. This would give the point xx = 5, x, = 2.5. The totalcost TC column
in Table 41 and the price line in Fig. 412 show how this is also the point
of minimum total cost.
The second fundamental question is: How should total production
be divided among specific outputs? The answer is found in Eq. (424). The
rule says that the outputs should be produced in such amounts that the
ratio of their unit benefits equals the marginal rate at which production
can be shifted from one output to another with all other inputs and outputs
TAlBLE 41 Optimum Ratte of Substitution
xx x2 MRS TC
10 1 60
0.25
6 2 50
0.5
4 3 50 MC1 = $5
2 MCs = $10
3 5 65 Optimum MRSn1 = 4io = 0.5
2 10 110
1 20
CONDITIONS OF PROJECT OPTIMALITY 79
FIGURE 418 Optimum division of out o 6 1
puts. v4
held constant. For this condition, one may evaluate y, as a function of ye
to obtain the data in Table 42 and the curve in Fig. 413. The curve
is concave away from the origin because the first units of additional
output can normally be produced at less marginal cost than later ones.
Our rule says to substitute yb for ya until the marginal rate of transforma
tion MR T of ya for y, equals the ratio of their marginal benefits MB.
This corresponds to the point y, = 5.5, yb = 7. The totalbenefit TB
column in Table 42 and the income line in Fig. 413 show how this is also
the point of maximum total benefit.
The third fundamental question is: How much of a specified input
should be devoted to the production of a specified output? The answer
is found in Eq. (422). The rule says that the input should be utilized
in such amount that the ratio of marginal input cost to marginal output
TABLE 48 Optimum Rate of Transformation
y. y6 MRT TB
10 0 20
9 3 27
8 5 31 MB. = $2
1 MBb = $3
7 6 32 Optimum MR Ta = % = 0.67
0.67
4 8 32
0 10
80 MICROECONOMICS AND EFFICIENT RESOURCE ALLOCATION
'fo J //7 x2
0 1 2 3 4 FIGURE 4614 Optimum input to use in pro
x1 during a specified output.
benefit equals the ratio of marginal physical output to marginal physical
input. For this condition, one may evaluate y. as a function ofxt1 to obtain
the data in Table 43 and the curve in Fig. 414. This is only one of four
possible curves of this type for our twoinput twooutput example, one
for each combination of output and input. Our rule says to increase the
amount of xx used in producing y, until the marginal increase in y, for
a unit of xx equals the ratio of their marginal unit values. This corre
sponds to the point y, = 7, xx = 2. The netbenefit B C column in
Table 43 and the distance between the two lines in Fig. 414 show this to
be also the point of maximum excess of benefits over costs.
The fourth fundamental question is: How large should the total
project be? The answer is found by shifting the approach demonstrated
by Fig. 414 from coordinate pairs to the total input and output vectors.
However, the analysis requires the selection and evaluation of trial vectors
as no straightforward solution of the type used in answering the first three
questions is possible. A trial output vector is selected. The first rule is
TABLE 48 Optimum Physical Product
sl y, MPPI, B C
1 4 3
3 MC = 5
2 7 4 MB = 2
2 Optimum MPP1. = 2.5
3 9 3
4 10
