THE ECONOMICS OF MUNICIPAL
ROBERT LEE GREENE
A DISSERTATION PRESENTED TO HE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
It is impossible to acknowledge everyone who has contributed to this
dissertation. However, there are several individuals to whom I must pay
I am deeply indebted to Dr. Milton Z. Kafoglis, my chairman, who gave
freely of his time, patience, and knowledge in suggesting revisions in this
work. I am particularly grateful to Dr. Kafoglis for his help in the de-
velopment of several models including: the broader application of vertical
summation, the methods of eliminating excess profits which lead to a quantity
maximization conclusion, and the suburban tax problem.
Dr. Donald R. Escarraz devoted many of his evenings to helping me de-
velop the interpretations of the economic models. These interpretations
(particularly the Davidson and Hirshleifer interpretations) are my views,
and not necessarily those of my chairman or my committee.
Appreciation is given to Dr. Ralph H. Blodgett and Dr. Clayton C.
Curtis who, as members of my committee, gave me the encouragement needed
for this undertaking.
I am indebted to the Institute of Government and the Department of
Finance at the University of Georgia for making available the time necessary
to complete this dissertation.
I also wish to acknowledge the loyalty of my wife Heather, and our two
children, Rob and Tracy.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS . . . . iii
LIST OF TABLES .... . ...... .... vii
LIST OF FIGURES . . . .. . . viii
I. INTRODUCTION .. ...... ..... .. 1
The Problem ....... ......... 1
Approach of the Study . . . . . . 4
Importance of the Study . . . . . 9
Method and Outline of the Study . . 10
II. MARGINAL COST PRICING AND UTILITY RATE THEORY . 14
Introduction .... ..... ...... 14
Development of the Marginal Cost
Pricing Principle ...... ...... 15
Marginal Cost Pricing as a Policy . . 20
Limitations to Marginal Cost Pricing . . . 24
Imperfect Competition ........... 24
Externalities . . . . . . . 25
Decreasing Costs ........ . . . 26
Interpersonal Comparisons and Equity . 29
Joint Supply and Indivisibilities . . 33
An Alternative Solution to Jointness,
Externality, Decreasing Costs, and
Indivisibility ......... ...... 35
Interdependent Demands . . . . . 39
Conclusions . . . . . . . 40
III. THE THEORETICAL MODELS . . . . . . 43
Introduction . . . . . . . 43
The Hotelling Model . . . . . 44
The Steiner Model . . . . . . 46
The Hirshleifer Model . . . . . 49
The Williamson Model . . . . . 54
The Davidson Model . . . . . . 60
The Buchanan Model . . . . . . 67
TABLE OF CONTENTS (Continued)
The Underlying Assumptions . . . . .. 70
Joint Supply . . . . . . . 71
Administrative Problems: Price Stability
and Technological Deficiencies . . 74
Cost Functions . . . . . . .. 77
The Equity Implications . . . . .. 80
The Peak Problem . . . . . .. 80
Cost Versus Ability to Pay . . . .. 84
Elasticity of Demand and Resource Impact . 84
Conclusions .... . . . . .. 87
IV. PRACTICAL DESIGN OF WATER RATES . . . . 89
Introduction .. . . . . . 89
The Cost Allocation Technique . . . .. 90
The Bias Towards Costs . . . . 90
The Patterson Allocation . . . .. 95
Types of Charges .. . . . . . 98
Fixed Charges . . . . . . .. 99
Variable Charges ............. 103
An Evaluation of Current Practices . . .. 104
The Concepts of Equity and Efficiency . .. 104
An Evaluation of the Cost Allocation
Technique ............... 107
Summary of Cost Allocation Evaluation . .. 115
An Evaluation of Water Charges . . .. 116
The Lack of Zone Pricing . . . .. 122
Conclusions .. . . . . . 123
V. APPLIES MODELS . . . . . 126
Introduction . . . . . . .. 126
Extension of the Model . . . . 128
The Economics of Block Pricing . . .. 133
Models which Interpret Present Practices . 139
A Further Modification . . . . 144
TABLE OF CONTENTS (Continued)
Administrative Problems . . . . 146
A Fixed Charge Model . . . . . . 149
Zone Pricing . . . . . . . 153
Conclusions .. ............. 163
VI. DISTRIBUTIVE JUDGMENTS AND
ECONOMIC EFFICIENCY . . . . . . 166
Introduction . . . . . . 166
Water Rates and Tax Policy . . . . . 167
Tax Efficiency .. ............. 168
Block Pricing and Taxation . . . . 173
Problems Associated with the Use of a
"Water-Rate Tax" .......... ... 178
Further Tax Efficiency Matters . . . . 180
The Suburban Tax Problem . . . . 182
Community Growth and Development Policy . 186
Industrial Location . . . . . 186
Rate Differentials and Community Development 189
Conclusions ...... . . . . . 192
VII. CONCLUSIONS AND RECOMMENDATIONS . . . 195
Conclusions . . . . . . . . 195
Recommendations . . . . . . . 199
BIBLIOGRAPHY . . . ..... 204
LIST OF TABLES
1. Zone Share of Capacity Costs . . . . . .. 156
LIST OF FIGURES
I. A Firm with Decreasing Costs . . . . .. 27
II. Vertical Summation . . . . . . . 36
III. The Hotelling Model . . . . . . . 45
IV. The Steiner Model . . . . . . . 47
V. Hirshleifer's Continuous Cost Model . . . .. 51
VI. Hirshleifer's Discontinuous Cost Formulation . . 53
VII. The Williamson Model . . . . . . .. 56
VIII. The Williamson Model Assuming Indivisibility . . 58
IX. The Davidson Model . . . . . . 61
X. An Interpretation of the Davidson Solution . . 62
XI. A Total Cost Interpretation of the Davidson Model 63
XII. The Average Cost Interpretation of the
Davidson Model . . . . . . . 65
XIII. The Optimal Solution in the Absence of Time
Jointness . . . . . . . . . 72
XIV. The Vertical Summation of Intra-Cycle Demands .. 73
XV. Traditional Efficiency Pricing with Interdependent
Demands . . . . . . . . . . 75
XVI. Cost Functions with a Capacity Constraint . . 78
XVII. An Hourly Load Chart for Three Users . . .. 81
XVIII. An Hourly Load Chart . . . . . . 83
XIX. Elasticity and the Impact of Price . . . .. 85
XX. The Combined Benefits Derived from Water Capacity . 100
LIST OF FIGURES (Continued)
XXI. Long-Run Costs of Monopoly Firm Producing
Water with Constant Factor Prices . . .. 106
XXII. An Hourly Load Chart . . . . . . .. 109
XXIII. Determination of the Peak and Off-Peak Demands . 113
XXIV. The Williamson Model . . . . . . .. 129
XXV. Non-Discriminating Monopoly . . . . .. 130
XXVI. First Degree Price Discrimination . . .. 133
XXVII. A Block-Rate Model . . . . . . 136
XXVIII. Williamson's Model with Block Pricing . . 137
XXIX. A Monopoly under Different Pricing Alternatives 141
XXX. An Interpretation of Utility Pricing and
Investment Practices . . . . . .. 145
XXXI. Fixed Charge Based upon the Ability to Congest
the Water System ....... ....... 150
XXXII. Capacity Costs and Use Characteristics of
Three Zones . . . . . . . . 155
XXXIII. The Marginal Solution for Serving Three Zones . 161
XXXIV. A Municipal Water Utility with a Profit Restraint. 169
XXXV. A Per-Unit Tax versus an Ad Valorem Tax . . 171
XXXVI. A Five-Block-Rate Schedule . . . . .. 175
XXXVII. Supply and Demand for Public Goods . . .. 184
In the last decade, there has been growing concern at the policy
level about the rate practices of municipally owned water utilities.
To an increasing extent, this concern has been reflected in the applied
literature and has centered on the relation of municipal water rate
structures to the efficiency of resource use and to various criteria of
equity or "reasonableness." Thus, writers have generated two basic
questions: (1) do present municipal water rate practices contribute to
economic efficiency?; and (2) are present water rate practices equi-
Economic efficiency is concerned with the attainment of that allo-
cation of resources, or "input-output mix," which maximizes the satis-
factions of the consumers in the economy. This efficiency criterion
requires each water user to pay a price which reflects the marginal costs
1Jack Hirshleifer, James C. DeHaven, and J.W. Milliman, Water
Supply (Chicago: University of Chicago Press, 1960), pp. 161-162; J.C.
Bonbright, "Fully Distributed Costs in Utility Rate Making," American
Economic Review, LI (May, 1961), p. 312; Irving K. Fox and Orric C.
Herfindahl, "Attainment of Efficiency in Satisfying Demands for Water
Resources," American Economic Review, LIV (May, 1964), p. 205; Gordon
P. Fisher, "New Look at Resources Policy," Journal of the American
Water Works Association, LVII (March, 1965), p. 359.
he imposes, assuming there are no complications stemming from joint
supply, interdependent demands, externalities, and distributional ob-
jectives in conflict with economic efficiency.
While economic efficiency reflects the orientation of economists,
water engineers seem to have a different set of criteria. Their major
concern centers around the ability of the rate structure to recover
total costs. Thus, water rates are established so that total revenue
is at least equal to total costs. This criterion of total cost re-
covery is combined with a distributional criterion which requires each
user to pay the "full" costs he imposes upon the utility.2
Utility engineers appear to pay little attention to the formal
distinction between the efficiency and the distributional aspects of
the rate structures they develop. It seems that water engineers equate
a dubious concept of economic efficiency with distributional equity
and try to solve both problems simultaneously. Thus, rate structures
which yield total revenue equal to total cost are considered both equi-
table and efficient. Utility engineers do not appear to consider the
2For articles reflecting these different definitions and criteria
see the following: William G. Shepherd, "Marginal Cost Pricing in
American Utilities," Southern Economic Journal, XXIII (July, 1966), p.
60; Hirshleifer, et al., Water Supply, p. 162; Staff Report, "The Water
Utility Industry in the United States," Journal of the American Water
Works Association, LVIII (July, 1966), p. 772; E.D. Bonine, "Making a
Water Utility Solvent," Journal of the American Water Works Association,
XLV (May, 1953), p. 457; Bonbright, "Fully Distributed Costs in Utility
Rate Making," pp. 305-12; J.C. Bonbright, "Two Partly Conflicting
Standards of Reasonable Utility Rates," American Economic Review, XLVIII
(May, 1957), pp. 386-93.
possibility that a utility might have to operate with profits or losses
to satisfy the criteria of economic efficiency, nor do they seem to
really appreciate the significance of marginal analysis. In short,
they employ a full cost allocation as a criterion of both efficiency
and equity. This study examines the possibility of applying the cri-
teria suggested by modern welfare economists in the hope that some
measure of contribution might be made toward the development of improved
water rate structures.
There is little in the present literature which evaluates the al-
ternative criteria a municipally owned water utility might use in de-
termining water rate practices. Consequently, there is no analysis of
the economic implications of alternative criteria. A modest contribu-
tion to the field entails recognition of the various alternative criteria
and an analysis of their impact upon economic efficiency. The welfare
economist is in a position to provide such analysis, as ably stated by
H. Thomas Koplin:
It is true that equity and environment are sub-
jective factors, that they rest on personal values,
and the economist has no claim to superior values.
But it is equally true that both are as important
See: Louis R. Howson, "Review of Ratemaking Theories," Journal
of the American Water Works Association, LVIII (July, 1966), p. 855;
William L. Patterson, "Practical Water Rate Determination," Journal
of the American Water Works Association, LIV (August, 1962), p. 906;
Jerome W. Milliman, "The New Price Policies for Municipal Water Ser-
vice," Journal of the American Water Works Association, LVI (Febru-
ary, 1964), p. 127.
in determining policy, and human satisfactions,
as is efficiency. It is therefore not only proper
but essential that the economist incorporate them
in his policy analysis. In doing so he will simply
be catching up with regulators, not to mention the
Welfare economics lends itself to this type of policy analysis.
Approach of the Study
The problem of maximizing welfare is one of combining economic
efficiency with distributional equity in such a manner that the well-
being of the individuals in the community is maximized. In the stan-
dard Paretian sense, economic efficiency is achieved when resources are
allocated so that it is impossible to increase the welfare of one
individual without decreasing the well-being of some other individual.
In other words, a situation is "Pareto inefficient" so long as it is
possible to move to another situation and in the process make at least
one person better off without making some other person worse off.
Efficiency, thus defined, is achieved when (1) the consumer equates the
exchange value of the last unit purchased with the production value of
that unit, and (2) the exchange value of the last unit purchased is equal
H. Thomas Koplin, "Discussion," American Economic Review, LI
(May, 1961), p. 336.
5For the use of this criterion in the Paretian context see:
William J. Baumol, Welfare Economics and the Theory of the State
(Cambridge: Harvard University Press, 1965), pp. 163-79.
for all consumers. These criteria are fulfilled when price is equal
to marginal cost. However, for policy prescription, it is also necessary
that the income distribution which results from marginal cost pricing be
deemed desirable. If the resulting distribution of income is considered
undesirable, then a departure from the marginal cost pricing policy may
be required.7 Any departure, however, entails distributional judgments
which will have an impact upon the attainment of economic efficiency as
the allocation fails to satisfy the two basic criteria.
6For the presentation of these criteria see: Abba P. Lerner, The
Economics of Control (New York: The Macmillan Co., 1944), pp. 7-136;
William J. Baumol, Economic Theory and Operations Analysis (2nd ed.;
Englewood Cliffs: Prentice-Hall, 1965), pp. 355-63; Abram Bergson,
Essays in Normative Economics (Cambridge: Harvard University Press,
1966), pp. 78-90; A. Reder, Studies in the Theory of Welfare Eco-
nomics (New York: Columbia University Press, 1947), Chapter 2; J.
Hirshleifer and J.W. Milliman, "Urban Water Supply: A Second Look,"
American Economic Review, LVII (May, 1967), pp. 169-78; Oliver E.
Williamson, "Peak-Load Pricing and Optimal Capacity under Indivisibility
Constraint," American Economic Review, LVI (September, 1966), p. 812;
Harvey Averch and Leland L. Johnson, "Behavior of the Firm under Regu-
latory Constraint," American Economic Review, LII (December, 1962), p.
1052; F.P. Linaweaver and John C. Geyer, "Use of Peak Demands in Deter-
mination of Residential Rates," Journal of the American Water Works
Association, LVI (April, 1964), p. 413; Hirshleifer, et al., Water Supply,
7See the following works: Paul A. Samuelson, Foundations of Eco-
nomic Analysis (New York: Atheneum, 1965), p. 253; Hirshleifer, et al.,
Water Supply, p. 90; Linaweaver and Geyer, "Use of Peak Demands in
Determination of Residential Rates," p.417; Shepherd, "Marginal Cost
Pricing in American Utilities," pp. 59-60.
Some economists, notably John R. Hicks, maintain that judgments
concerning the distribution of income do not have to be made even for
a policy prescription. Since any change from an inefficient to an
efficient solution has the "potential" of making everyone better off,
the distributional problem may, according to this view, be avoided.8
However, "actual" welfare and "potential" welfare are different things,
and Samuelson, Little, and others,maintain that the distributional
problem cannot be avoided at the policy level. The writer, at least
for purposes of this applied study, holds to the view that a policy
recommendation should consider both tests.
A municipal water utility is embroiled in decisions which entail
distributional or equity judgments. The water utility, as part of the
community's available revenue sources, is manipulated by local officials
to create tax equity (or inequity), to acquire additional general
revenue, and to carry out community growth and development programs.
What these officials may fail to recognize is the conflict that some-
times arises between economic efficiency and distributional equity. Rate
structures which satisfy the criteria for economic efficiency are not
8John R. Hicks, "The Foundations of Welfare Economics," Economic
Journal, XLIX (December, 1939), pp. 696-712; Nicholas Kaldor, "Welfare
Propositions of Economics and Interpersonal Comparisons of Utility,"
Economic Journal, XLIX (September, 1939), pp. 549-52.
Paul A. Samuelson, "Welfare Economics and International Trade,"
American Economic Review, XXVIII (June, 1938), pp. 261-66; Samuelson,
Foundations of Economic Analysis, pp. 249-52; I.M.D. Little, A
Critique of Welfare Economics (Oxford: Oxford University Press, 1958),
always the most equitable. On the other hand, rates considered to be
equitable may not satisfy the criteria for economic efficiency. The
problem facing the municipal water utility is one of combining eco-
nomic efficiency with distributional equity in such a manner as to
maximize the welfare of the people in the community.10 The problem
becomes one of finding an optimal size plant and an optimal rate
structure without impeding, and hopefully encouraging, improvements
in the distribution of income.
However, when economic efficiency and distributional equity con-
flict in the establishment of municipal water rate practices, a trade-
off between the two becomes necessary. The existence of such conflict
is evidenced by the presence of conflicting practices in present water
rate structures which represent failures to resolve the conflict. For
example, a utility may offer a discount for water used during the off-
peak period in an attempt to reduce the peak load problem. At the same
time, the rate structure will incorporate a low promotional rate which
may encourage peak use. Obviously,the two are in conflict.
Efficiency and equity also come into conflict when a utility em-
ploys a declining block-rate structure with a rate below marginal cost
in the last block. This practice is often used to promote industrial
10A more precise definition of welfare maximization is presented
in Chapter II.
location.1 Since low rates benefit large users, their effect is
greater capacity costs for the utility. The utility's costs increase
as the ratio of peak use to off-peak use increases. Therefore, a
decision to sell at below marginal cost results in both economic in-
efficiency--expansion of capacity to sell a greater output at below
marginal cost--and distributional effects, since the over expansion
must be paid for by someone.
Marginal cost pricing, as a single pricing policy, however, is not
always applicable to a municipally owned utility because the assumptions
underlying the marginal cost principle are not always satisfied. The
water utility is faced with a time jointness problem that arises be-
cause of peak and off-peak water use.2 Attempts to resolve this prob-
lem are further complicated because peak and off-peak demands may be
interdependent, particularly on an hourly basis. In addition, there are
externalities,such as public health and sanitation,which must be taken
into account.13 When these conditions arise, a strict adherence to
11Low promotional rates are also used to generate greater water de-
mand from present water users. These low rates lead to greater lawn
sprinkling and uncontrolled air conditioning. The low rates, therefore,
conflict with water scarcities arising during the summer months and the
overall effect is to put greater demand upon the system capacity.
12For an adequate discussion of the problems of joint supply see:
Donald H. Wallace, "Joint Supply and Overhead Costs and Railway Rate Policy,"
Quarterly Journal of Economics, XLVIII (August, 1934), pp. 583-619; A.C.
Pigou, The Economics of Welfare (4th ed.; London: The Macmillan and Co.,
Ltd., 1932), pp. 297-99, 300-301.
1A classification of externalities is given by: Tibor Scitovsky,
"Two Types of Externalities," Journal of Political Economy, LXII (April,
1954), pp. 143-51; Milton Z. Kafoglis, Welfare Economics and Subsidy Pro-
grams (Gainesville: University of Florida Press, 1961), pp. 16-38.
marginal cost pricing is not feasible or desirable. Indeed, these com-
plications shall occupy the bulk of the attention of this study.
Importance of the Study
This study recognizes a need for additional analysis of municipal
water rate practices in relation to economic criteria for determining
plant capacity and rate structures.
The need for additional analysis is especially urgent in the light
of the current development of congestion of most public facilities, and
the problems faced by local governments such as the need for increased
general revenues, tax equity, and community growth and development.
Since these problems require solutions involving distributional judg-
ments, traditional theory puts them aside, consequently failing to pro-
vide a complete analysis of some of the more important factors deter-
mining municipal water rate practices. Distributional decisions which
are incorporated into the rate structures must be isolated and, if
possible, evaluated if their impacts are to be revealed and taken into
account in the development of policy.
In determining their water rate practices, municipally owned
utilities do not seem to follow any consistent criteria. Distributional
judgments are built into the rate structures in a haphazard, ad hoc,
manner.14 Some of the more common criteria by which rates are evaluated
14James M. Buchanan, "Peak Loads and Efficient Pricing: Comment,"
Quarterly Journal of Economics, LXXX (August, 1966), pp. 463-71.
are (1) average costs,5 (2) rates which will maximize revenue with
least resistance from members of the community,16 (3) rate structures
developed by neighboring communities,17 and (4) rates which will generate
industrial location. What is lacking is a series of broad pricing
guidelines that municipally owned utilities can follow in the deter-
mination of rate structures and plant size which embody the effects
of distributional decisions. The intent of this study is not to deter-
mine specific rate structures under alternative conditions. Rather,
the intent is to arrive at a set of guidelines which will achieve
efficiency, with equity considerations temporarily set aside. Rate
practices based upon these guidelines will enable those who make rate
decisions to realize the efficiency implications of their distributional
judgments when equity and efficiency come into conflict. The need is to
highlight and delineate the nature of the conflict in the hope that this
might encourage superior rate policies.
Method and Outline of the Study
The hypothesis of this study is that additional economic analysis
of water rate practices can provide a basis for policy changes which
15This criterion seems to be the most common. See: Hirshleifer,
et al.,Water Supply, p. 111; Howson, "Review of Ratemaking Theories,"
16Samuel S. Baxter, "Principles of Rate Making for Publically fsic]
Owned Utilities," Journal of the American Water Works Association, LII
(October, 1960), p. 1227.
17Raymond J. Faust, "The Needs of Water Utilities," Journal of the
American Water Works Association, LI (June, 1959), p. 703.
will make rates not only more efficient, but also more equitable.
The study is an applied theoretical analysis which can be divided
into four basic parts. The first part, which entails Chapter II, is
an evaluation of marginal cost pricing as a basis for rate and capacity
determination for municipally owned water utilities. This part con-
trasts the Paretian framework with the "old" welfare economics of
Marshall, Pigou, Lerner,and others. This section also examines the
assumptions underlying the marginal cost pricing principle--independent
demands, the absence of externalities, the absence of joint supply, and
the distributional aspect--to determine the extent to which it can serve
as the basis for improved municipal water rate practices.
The second section, which includes Chapter III, is an evaluation of
the theoretical models developed by recent writers including Buchanan,
Davidson, Hirshleifer, Hotelling, Steiner, and Williamson, which have
appeared in the current literature. This evaluation seeks to determine
the validity of these models as useful guidelines for municipal water
rate practice. These models are found to be lacking and in need of fur-
ther modification and development if they are to be useful at the applied
18Buchanan, "Peak Loads and Efficient Pricing: Comment," pp. 463-
71; Jack Hirshleifer, "Peak Loads and Efficient Pricing: Comment,"
Quarterly Journal of Economics, LXXII (August, 1958), pp. 451-62;
Ralph K. Davidson, Price Discrimination in Selling Gas and Electricity
(Baltimore: Johns Hopkins University Press, 1955); Harold Hotelling,
"The General Welfare in Relation to Problems of Taxation and of Railway
and Utility Rates," Econometrica, VI (July, 1938), pp. 242-69; Peter O.
Steiner, "Peak Loads and Efficient Pricing," Quarterly Journal of Eco-
nomics, LXXI (November, 1957), pp. 585-610; Williamson, "Peak-Load
Pricing and Optimal Capacity under Indivisibility Constraint," pp. 810-
level. Modifications are suggested by relaxing some of the underlying
assumptions, and then examining and developing the implications of the
Chapter IV constitutes the third section which is an evaluation
of water rate practice as it is used and advocated in the day-to-day
operations of water utilities. A model developed by William L. Patter-
son is examined closely since it represents a typical approach to the
allocation of costs advocated by water rate analysts.19 This section
probes the various cost classifications and rate classifications used
by water utilities to discern their limitations as well as their
efficiency and equity implications. It also critically examines the
various types of charges employed by water utilities.
The fourth part, which includes Chapters V and VI, is a synthesis
of present theory and practice, which leads to some policy recommen-
dations concerning the type of water rate practices that might be used.
The intent is to derive practical water rate guidelines using economic
analysis under different sets of assumptions. Chapter V examines the
welfare implications of alternative models such as simple and dis-
criminating monopoly, block-rate schedules, quantity maximization and
out-of-pocket cost pricing, and zone pricing. Chapter VI analyzes and
evaluates the efficiency and equity implications of the various
19Patterson, "Practical Water Rate Determination," p. 909.
alternatives used in water rate structures to achieve such objectives
as tax equity, greater general fund revenues, and community growth and
MARGINAL COST PRICING AND UTILITY RATE THEORY
Modern public utility rate theory may be contrasted sharply with
traditional rate practice. The essence of traditional rate practice has
been the protection of consumers from monopoly exploitation through pub-
lic regulation, and the emphasis has been placed on the attainment of
reasonableness or equity in the relationship between buyer and seller.
This emphasis also characterized the writings of early economists who
emphasized the reasonableness of both rates and profits. Thus, utility
rate theory developed as a separate compartment of applied economic
theory and revolved around the valuation of the rate base and unjust rate
discrimination. In recent years, there has been a marked shift in the
emphasis of both theory and practice from standards of equity and reason-
ableness to more or less objective standards of efficiency in the develop-
ment of utility rate structures. This change has brought public utility
rate theory into a much closer relationship with orthodox Marshallian
This chapter briefly summarizes the historical development and con-
ceptual basis of marginal cost pricing. In addition, it organizes some
of the complex situations--indivisibility, joint supply, externality--
which create difficulties at both the theoretical and applied levels.
1Adam Smith, The Wealth of Nations, Modern Library (New York:
Random House, Inc., 1937), pp. 681-716.
Although each of these complexities has been discussed separately by
various writers in relation to specific problems, it is felt that a
modest effort to pull the strands together will provide a more complete
picture of the problems associated with marginal cost pricing and
utility rate theory--both theoretical and applied--than exists at the
Development of the Marginal Cost
Alfred Marshall was the first economist to develop a sophisticated
model for the analysis of economic efficiency. Marshall advocated the
use of marginal analysis as a means of analyzing economic efficiency and
suggested consumers' surplus as a normative criterion.2 According to
Marshall, the equality of supply and demand characterized efficiency in
resource use. Moreover, this equality led to the maximization of con-
sumers' surplus. Marshall's discussion of increasing and decreasing
costs is a classic example of the use of marginal analysis in evaluating
the welfare effects of alternative situations.3 Employing the maximization
2Although the welfare significance of consumers' surplus is still
debated, it seems to have better standing today among many writers than
even Marshall attributed to it. See: David M. Winch, "Consumer's Sur-
plus and the Compensation Principle," American Economic Review, LV
(June, 1965), pp. 395-423; John R. Hicks, "The Rehabilitation of Con-
sumers' Surplus," Review of Economic Studies, VIII (February, 1941),
For Marshall's discussion see: Alfred Marshall, Principles of
Economics (8th ed.; London: Macmillan and Co., Ltd., 1920), pp. 390-94.
of consumers' surplus as a criterion, Marshall analyzed the policy of
taxing firms experiencing decreasing returns (increasing costs) and
paying subsidies to firms experiencing increasing returns (decreasing
costs). Marshall proposed that the output of firms operating under
increasing costs be contracted because at a profit-maximizing industry
equilibrium, marginal cost exceeds average cost. On the other hand,
the output of firms operating under decreasing costs should be ex-
panded.4 Marshall's classic tax-bounty analysis is recognized as the
original marginal cost theory of pricing.
The concept of consumers' surplus developed by Marshall implied
an interpersonal comparison of utility. Marshall himself recognized
that the "surplus," measured under a market demand function, measures
satisfaction only if one passes over the possibility that a given sum
of money gives different degrees of satisfaction to different people.5
Following in Marshall's footsteps, Pigou formalized the conditions
necessary for the maximization of welfare.6 Pigou, substituting social
net product or the national dividend in place of consumer's surplus
For a summary of this analysis see: Milton Z. Kafoglis, Wel-
fare Economics and Subsidy Programs (Gainesville: University of Florida
Press, 1961), pp. 9-10.
5This point is well developed by: Nancy Ruggles, "The Welfare Basis
of the Marginal Cost Pricing Principle," Review of Economic Studies, XVII
(1949-1950), p. 31.
A.C. Pigou, The Economics of Welfare (4th ed.; London: Macmillan
and Co., Ltd., 1932), p. 290.
as the measure of welfare, established a set of marginal conditions
which must be satisfied to achieve a welfare maximum. Nancy Ruggles
Pigou in effect implied a set of marginal
conditions of production when he said that
the marginal net social product of resources
in each use must be equal in order to maxi-
mize the national dividend. His proof is
the demonstration that if the marginal con-
ditions are not met, the aggregate national
dividend could be increased by removing re-
sources from uses in which the marginal social
net product is lower and employing them in
uses with higher social net products.7
By virtue of using the national dividend as the measure of welfare,
Pigou also built interpersonal utility comparisons into his theory be-
cause the value of the national dividend depends upon the distribution
of income. That is, any single bundle of goods will be valued dif-
ferently depending upon the distribution of income. Pigou recognized
the problem and,accordingly, employed both the size and the distribution
of the national dividend as coordinate measures of welfare which have to
be traded off.8 According to Pigou:
On this basis, it is desired, if possible, to
establish some connection between changes in
the distribution of the national dividend and
changes in economic welfare, corresponding to
the connection established in the preceding
chapter between changes in the size of the
7Ruggles, "The Welfare Basis of the Marginal Cost Pricing Principle,"
8Ibid., p. 32.
national dividend and changes in economic
The problem of interpersonal comparisons was eventually formalized
rigorously in the works of Pareto. Through the use of ordinalist
assumptions and marginal analysis, Pareto developed a welfare criterion
which avoided cardinal utility as well as the necessity of interpersonal
comparisons of utility. Pareto defined the welfare maximum as an
arrangement where it is not possible to make everyone better off by any
movement within the system.0
Throughout the works of the early theorists, there was a gradual
development of the concept of welfare along with the marginal conditions
necessary to achieve the welfare maximum. Marshall, Pigou, Lerner,
9Pigou, The Economics of Welfare, p. 89.
10Ruggles, "The Welfare Basis of the Marginal Cost Pricing Prin-
ciple," p. 32.
11The seven marginal conditions are: (1) the marginal rates of sub-
stitution between two commodities should be equal for any two consumers;
(2) the marginal rates of transformation between two commodities are
equal for all producers; (3) the marginal physical product of a given
factor for a given commodity is equal for all commodity producers; (4)
the marginal rates of equal-product substitution of two factors are the
same for all producers; (5) the marginal rate of indifferent substi-
tution of any consumer for two products is equal to the marginal rate
of transformation of these two commodities in production; (6) the mar-
ginal reward of a factor equals the marginal rate of substitution of
reward for use; and (7) the marginal rates of time substitution of two
individuals for a given asset are equal. This listing and the impli-
cations are given by: Kenneth E. Boulding, "Welfare Economics," A Sur-
vey of Contemporary Economics, ed. by Bernard F. Haley, I (Homewood:
Richard D. Irwin, Inc., 1952), pp. 1-34. It should be noted that satis-
faction of the seven marginal conditions does not imply that welfare is
maximized. The conditions can define a minimum as well as a maximum.
Moreover, the maximum they do define can only be a relative maximum and
and others, made use of interpersonal comparisons, while Pareto and his
followers avoided such essentially subjective valuations. Both schools
of thought developed identical marginal conditions, but there was, and
still is, a serious difference of opinion as to their welfare signifi-
cancer. Nevertheless, the marginal conditions which evolved are not a
source of controversy and are a familiar part of the work of all welfare
economists. Moreover, the development of these conditions has provided
the basis for a distinction between economic efficiency and distri-
butional equity. Basically, this distinction is the difference between
the "old" and the "new" welfare economics. The choice between the two
approaches does not involve a choice between equity and efficiency as the
proper objective of economic analysis, but does involve a choice re-
lating to the validity of the distinction between the two. By isolating
the difficult distributional question, the "new," or Paretian, welfare
economists have developed a single pricing principle--price equal to
not an absolute maximum. The conditions do, however, have to be satis-
fied to maximize total welfare. They are necessary, but not sufficient,
conditions which must be combined with the so-called "total" conditions
which remain poorly defined. See: John R. Hicks, Value and Capital
(2nd ed.; Oxford: Oxford University Press, 1946), pp. 62-77; George J.
Stigler, The Theory of Price (rev. ed.; New York: The Macmillan Co.,
1962), pp. 42-95.
12For discussions of these points see: Ruggles, "The Welfare Basis
of the Marginal Cost Pricing Principle," pp. 29-40; I.M.D. Little, A
Critique of Welfare Economics (Oxford: Oxford University Press, 1958),
pp. 67-216; William J. Baumol, Economic Theory and Operations Analysis
(2nd ed.; Englewood Cliffs: Prentice-Hall, 1965), pp. 355-85.
marginal cost--as a policy guide. Obviously, this approach is very
limited where difficult distributional problems exist and must, in
most instances, be supplemented by some sort of distributional judg-
Marginal Cost Pricing as a Policy
Pigou was the first to develop the utility rate theory implied in
the marginal conditions, and he applied the criteria to the railroad
industry.3 Pigou concluded that price equal to marginal cost is a
completely general guide to economic efficiency and went on to evalu-
ate the implications of divergences between price and marginal cost.4
In his analysis, Pigou struggled with the problem of applying marginal
cost pricing to decreasing cost industries. In these instances, the
difficulty of applying the principle is that prices equal to marginal
cost will fail to recover the total costs of operation. Losses arise
because average costs are higher than marginal costs which have been
13Many of the important points arose during the controversy be-
tween Pigou and Taussig in 1913 over the question of joint supply.
See the following: A.C. Pigou, The Economics of Welfare, pp. 290-
317; F.W. Taussig, "Railway Rates and Joint Costs Once More," Quarterly
Journal of Economics, XXVII (February, 1913), pp. 378-84; A.C. Pigou,
"Railway Rates and Joint Costs," Quarterly Journal of Economics, XXVII
(May, 1913), pp. 535-36 and the rebuttal immediately following by F.W.
Taussig, Quarterly Journal of Economics, XXVII (May, 1913), pp. 536-
38; A.C. Pigou, "Railway Rates and Joint Costs," Quarterly Journal of
Economics, XXVII (August, 1913), pp. 687-92 and the rebuttal immediately
following by F.W. Taussig, Quarterly Journal of Economics, XXVII
(August, 1913), pp. 692-93.
14Pigou, The Economics of Welfare, pp. 381-408.
equated with price. If the firm does not receive some type of bounty,
or negative tax, services cannot be provided in the long-run. In
this case, if the community wants the commodity to be produced in opti-
mal quantities, it will have to subsidize the producer, or take over
the operation of the firm.
Much of the early work on marginal cost pricing revolved around
the problems of welfare concepts, joint costs, decreasing costs, and
distributional equity. The early works developed terminology such as
Marshall's surplus, Pigou's social net product, and the marginal
qualities which characterized a welfare maximum.
In 1938, the appearance of Harold Hotelling's article "The General
Welfare in Relation to Problems of Taxation and of Railway and Utility
Rates" marked a major breakthrough in the use of marginal cost pricing
as a policy prescription to be applied to public utilities.5 Along
with Hotelling's article, a controversy arose concerning the use of
marginal cost pricing as a practical basis for public utility regulation.
In a broader sense, the "marginal cost controversy,' which arose during
the late 1930's,represented the response of neo-classical welfare
economics to the problems of unemployment, excess capacity, and income
distribution that developed during the Great Depression. Although the
controversy exposed and illuminated many theoretical and applied issues,
15Harold Hotelling, "The General Welfare in Relation to Problems
of Taxation and of Railway and Utility Rates," Econometrica, VI (July,
1938), pp. 242-69.
the center of the stage, during the period 1940-1960, was held by
Keynesian economics; the problems of efficient pricing remained in the
background. The publication of Arrow's impossibility theorem in 1949
further weakened the theoretical case for marginal cost pricing.6
Rehabilitation began in 1952 with the publication of Baumol's Welfare
Economics and the Theory of the State, which developed the concept of
externality in a Paretian framework. Recent contributions by Oliver E.
Williamson, James M. Buchanan, and others, have added impetus to the
development of an applied welfare economics which combines the concept
of externality with the marginal cost criterion to provide a reasonably
rigorous framework for policy prescriptions. It is important to note
Hotelling's significant conclusions, and the controversy which arose
after the article appeared.
Hotelling set forth and rigorously defended the proposition that
marginal cost pricing is the proper policy to follow in the determi-
nation of utility rates.7 Marginal cost pricing was now set forth as
a definite policy rigorously derived from highly formal welfare con-
ditions. Like Pigou, Hotelling discussed the problem of applying mar-
ginal cost pricing to firms operating with decreasing costs. Hotelling's
position was that such firms should receive compensation from the
16Kenneth J. Arrow, Social Choice and Individual Values (New York:
John Wiley and Sons, 1951), p. 26.
1Hotelling, "The General Welfare in Relation to Problems of
Taxation and of Railway and Utility Rates," p. 242.
community. He recommended that the compensation (subsidy) be raised
through a general income tax.1
The tax criteria developed by Hotelling demonstrated that, for a
community to maximize welfare, it was necessary to tax income rather
than commodities. Although the contribution Hotelling made to utility
theory is a matter of degree and sophistication in relation to the
contribution of Marshall and Pigou, he succeeded in converting marginal
cost pricing into a definite policy criterion, whereas the earlier
writers recognized marginal cost pricing as an attractive possibility.
Hotelling converted subsidy into a definite policy recommendation and
set forth the type of taxes which should be used to raise the subsidy.
In a brief overview, marginal cost pricing is an outgrowth of wel-
fare economics. The policy of price equal to marginal cost is an
outgrowth of the seven marginal conditions that characterize a welfare
maximum defined either as (a) the maximization of consumers' surplus
(Marshall) or (b) a situation in which no one can be made better off
without making someone else worse off (Pareto).
The controversy that began with Hotelling's article pertains to the
practical use of marginal cost pricing, a controversy that has continued
18Ibid., p. 242.
19A very excellent discussion on the evolution of marginal cost
pricing is found in the following article: Ruggles, "The Welfare Basis
of the Marginal Cost Pricing Principle," pp. 29-46.
for the past thirty years and remains unresolved. It appears that
this long-standing controversy over marginal cost pricing prevented
economic theory from providing an acceptable general rate theory. How-
ever, within the past decade, the issues, although still unresolved,
are in a perspective which has contributed to progress at the applied
Limitations to Marginal Cost Pricing
Although marginal cost pricing has become a policy recommendation
to be used when the private market mechanism fails to achieve the most
efficient use of resources and the ideal output, there are several
instances where the use of marginal cost pricing is limited. Marginal
cost pricing maximizes welfare under the assumptions that there are no
joint costs, no relevant externalities, no distributional problems, in-
dependent demand functions, and infinite divisibility.21 When these
conditions are not satisfied, the solution derived from the application
of marginal cost pricing must be modified.
Marginal cost pricing is limited in its application when elements
20For detailed discussions see: William J. Baumol, Welfare Eco-
nomics and the Theory of the State (2nd ed.; Cambridge: Harvard Uni-
versity Press, 1965), p. 107; Paul A. Samuelson, Foundations of Eco-
nomic Analysis (New York: Atheneum, 1965), p. 247.
21Kafoglis, Welfare Economics and Subsidy Programs, p. 6.
of monopoly develop in the market. Monopolistic tendencies are found
particularly on the selling side of the product market. In the case
of monopoly, or imperfect competition, the firm maximizes profits by
producing the output indicated by the equation of marginal revenue with
marginal cost. The price exceeds marginal cost by an amount depending
upon the elasticity of the demand function. Since price does not equal
marginal cost when demand has any degree of inelasticity, the conditions
necessary for the attainment of the optimum allocation of resources are
not satisfied. If there are elements of monopoly in one or more sectors
of the economy, the marginal conditions are not satisfied in any sector
of the economy.22 When one firm is required to follow a price equal to
marginal cost policy, and the equality is not satisfied elsewhere, the
firm tends to over produce with respect to the output of all other com-
modities. The basic problem resolves into one of achieving partial
equality through marginal cost pricing and, therefore, attaining an
output which more closely approximates the ideal output than that out-
put produced when none of the marginal conditions is satisfied.23
A condition which further complicates the attainment of efficiency
through marginal cost pricing, or any form of market pricing for that
22Little, A Critique of Welfare Economics, p. 185.
23This argument is summarized in the following: Little, A Critique
of Welfare Economics, pp. 162-65; Abba P. Lerner, The Economics of Con-
trol (New York: The Macmillan Co., 1944), pp. 134-36.
matter, is the existence of externalities-spillovers--which are not
internalized in the free market. In these instances, positive govern-
ment policy may encourage the attainment of the ideal output. The
externality can take the form of either an external benefit or an ex-
ternal cost.24 When significant external benefits are associated with
the consumption or production of a commodity, the market, even if fully
competitive, tends to produce too little of the commodity. When sig-
nificant external costs exist, there is a tendency in the market to
produce too much of the commodity. To achieve an optimal allocation
of resources, it is desirable to expand the production of those com-
modities which have external benefits associated with them and contract
the production of those commodities which have external costs connected
with them. The means for implementing such policies are many and varied,
but they are beyond the immediate scope of this study.
When a firm operates with decreasing costs, the use of marginal
cost pricing converts a profit-maximizing firm into one incurring losses.
The problem is demonstrated in Figure I below. If left to its own
24An external benefit is a benefit received from a user's consump-
tion of a commodity by individuals other than the direct user. Everyone
benefits from a vaccination received by any one member of the community
since everyone's risk of contracting small pox is reduced. An external
cost is a cost incurred from a user's consumption of a commodity and
borne by individuals other than the direct user. An example is the
social costs of caring for alcoholics resulting from the sale and use of
A Firm with Decreasing Costs
devices, the firm produces output OX1 where marginal revenue (MR) is
equal to marginal cost (MC), and the selling price is OP Profits
are equal to the area PBCD. When marginal cost pricing is imposed,
the firm is forced to equate marginal cost with price. To meet this
requirement, the firm now produces output OX2 and sells the output at
the price OP2. The firm now has losses equal to P2KEF. The losses
have been generated because the selling price (OP2) is less than average
cost (X2E). With these losses, the firm will discontinue production in
the long-run unless it receives some form of subsidy.
It is important to note two factors which are relevant in deter-
mining whether or not a subsidy is to be paid. The first factor is
consumer demand as reflected in a collective context. Consumers must
reveal a preference for the commodity to be produced and a willingness
to pay the subsidy. The preference and willingness to pay criterion
requires showing that the total benefits exceed the total costs to the
community. When benefits exceed costs, consumer preferences should re-
veal a willingness of the consumers to pay the subsidy. The second
factor is the cause of the losses. In developing a system of efficiency
taxes and prices, the nature of the losses becomes a relevant factor.
What might appear to be a firm operating with decreasing costs might
actually be a non-optimum sized firm with increasing costs. In this in-
stance, the losses result from improper decision-making by the owners of
the firm. If losses arise because improper business decisions have led to
excessive investment which created decreasing costs in the relevant range
of output, a correct price structure would lead to overall losses. How-
ever, the losses are a correct penalty and should continue until the in-
vestment is brought down to proper size.25 There is no need for a subsidy.
For many reasons, this particular dictate has not been followed by public
policy, especially in the case of railroads where externality, distribu-
tion, and politics have played an important role.
25William G. Shepherd, "Marginal Cost Pricing in American Utilities,"
Southern Economic Journal, XXIII (July, 1966), p. 60.
Interpersonal Comparisons and Equity
Marginal cost pricing has been attacked on the grounds that it
fails to come to grips with the distributional problem.26 Critics
maintain that any decision about pricing has distributional impli-
cations and,therefore, imposes an interpersonal comparison of utility.
Thus, it is maintained that marginal cost pricing assumes (explicitly
or implicitly) that the income distribution generated by its appli-
cation is superior, or at least not inferior, to the distribution that
existed before it was applied. It can be demonstrated that the gains
which accrue as a result of marginal cost pricing always exceed the
losses, i.e., the gainers can more than compensate the losers. However,
marginal cost pricing coupled with compensation seems impractical and
would involve, for example, compensation to a monopolist upon the in-
stitution of policies designed to reduce his price to marginal cost.
The critics feel that the use of marginal cost pricing assumes that
individuals who gain are more important, have more relevant tastes, or
should be given more weight than the individuals who lose from the
change. If it is in some sense "better" for the gainers to gain than
it is for the losers to lose, a decision to adopt marginal cost pricing
would seem to be in order. However, such a decision requires an inter-
personal comparison. Such comparisons may seem reasonable if it is
26See: J. deV. Graaff, Theoretical Welfare Economics (Cambridge:
Cambridge University Press, 1963), pp. 154-55.
established that the gainers have a greater propensity to enjoy than
do the losers. The basis for making this judgment can be either eco-
nomic or political.
Any standard for the attainment of economic
efficiency necessarily assumes the existence
of some process for the measurement and aggre-
gation of individual costs and utilities. If
we define social welfare as a function of in-
dividual welfare, this process cannot be autho-
ritarian but must reflect the voluntary choice
of individuals. These choices can be expressed
either through the market or through the polls.
One gives weight to purchasing power, the other
to political power.
Advocates of marginal cost pricing usually avoid the problem of
distributional decisions and interpersonal comparisons by assuming
that the net gains are distributed in a manner that either adequately
compensates the losers or does not lead to an inferior distribution of
income. There is also the unlikely possibility that the gains are dis-
tributed randomly, leaving the distribution of income unchanged but at
a higher general level. This latter possibility hinges on the assumption
that there is no consistent institutional bias which favors some groups
more than others in the distribution of gains--an assumption which seems
27Kafoglis, Welfare Economics and Subsidy Programs, p. 7.
28For the development of this point see: Irving K. Fox and Orric
C. Herfindahl, "Attainment of Efficiency in Satisfying Demands for
Water Resources," American Economic Review, LIV (May, 1964), pp. 198-
In the case of distributional equity, the criticism misses the
central point of the marginal cost pricing principle. It is true that
marginal cost pricing generates an income distribution different from
that which existed before its application. If the distribution is to
be maintained, however, and the problem of redistribution is to be
avoided, the principle of compensation can be used.29 To maintain the
original distribution of income, although possibly impractical, the
gainers can always compensate the losers for the change (assuming away
such problems as the monopolist who is compensated for reducing his
price). When the sum of the benefits exceeds the losses, and compen-
sation is paid to the losers, the aggregate effect is a net increase in
welfare with no one worse off, and the change is consistent with the
Paretian welfare criterion.
There are other adjustments which can be made to retain distribu-
tional equity. If the income.distribution created through marginal
cost pricing is not considered "desirable," an adjustment can be made
29For discussions of the compensation principle see: Abram
Bergson, "A Reformulation of Certain Aspects of Welfare Economics,"
Quarterly Journal of Economics, LII (February, 1938), pp. 310-34;
John R. Hicks, "The Foundations of Welfare Economics," Economic
Journal, XLIX (December, 1939), pp. 696-712; Nicholas Kaldor, "Wel-
fare Propositions of Economics and Interpersonal Comparisons of
Utility," Economic Journal, XLIX (September, 1939), pp. 549-52;
Tibor Scitovsky, "A Note on Welfare Propositions in Economics,"
Review of Economic Studies, IX (1941-42), pp. 77-88.
through a lump-sum tax. The lump-sum tax can be imposed to acquire
the more desirable distribution of income without destroying the
equality of the marginal conditions created through the use of mar-
ginal cost pricing. The proceeds from the lump-sum tax can be used
as the source of revenue for compensation. The fixed plant costs can
be paid by the state from revenues derived from the tax. The use of
income or excise taxes, which impinge on resource margins, will destroy
qualities in the marginal relationships. Income taxes will alter the
equilibrium between work and leisure, and excise taxes the equilibrium
of the marginal rates of substitution between commodities.31 In her
discussion of marginal cost pricing, Nancy Ruggles points to the key
factor in determining the means of financing subsidies:
Introducing the compensation would in effect mean
that the revenue for subsidizing any given product
would have to be derived from the people who con-
sumed the product, and not from anyone else. To
do this without violating the marginal conditions,
the levy would have to fall on the consumers' sur-
plus derived by the purchasers from the consumption
of that specific product. It could not bear upon
30The first to suggest a lump-sum tax as a policy was Hotelling,
"The General Welfare in Relation to Problems of Taxation and of Rail-
way and Utility Rates," pp. 242-69. A multi-part tariff was also
suggested by: R.H. Coase, "The Marginal Cost Controversy," Econo-
mica, XIII (August, 1946), pp. 169-82.
31These points are well developed by Hotelling, "The General
Welfare in Relation to Problems of Taxation and of Railway and Utility
Rates," pp. 242-69; Richard A. Musgrave, The Theory of Public Finance
(New York: McGraw-Hill Book Co., 1959), pp. 136-54; Nancy Ruggles,
"Recent Developments in the Theory of Marginal Cost Pricing," Review
of Economic Studies, XVII (1949-1950), pp. 107-26.
the marginal unit purchased by any consumer,
so any form of per-unit tax would be inad-
missable. A tax that must fall upon a spe-
cific product, but not upon the marginal unit,
would of necessity yield a form of price dis-
By combining marginal cost pricing with lump-sum taxes paid by the
gainers, it is possible to maintain economic efficiency, compensate
losers, and avoid interpersonal comparisons of utility.33 Notwith-
standing these possibilities, the distribution of gains due to policy
changes ultimately involves the political process and hinges more on
the distribution of political power than on the distribution of market
Joint Supply and Indivisibilities
In the case where a commodity is supplied within a capacity con-
straint, and/or in jointness with another commodity, marginal cost
pricing can be difficult to apply.
When indivisibility of some input exists, marginal cost pricing can
lead to an allocation of resources other than the optimal allocation.
32Ruggles, "Recent Developments in the Theory of Marginal Cost
Pricing," p. 121.
33In support of marginal cost pricing and this point see: J.C.
Bonbright, "Major Controversies as to the Criteria of Reasonable Public
Utility Rates," American Economic Review, XXX (May, 1940), pp. 379-89;
Emory Troxel, "Incremental Cost Determination of Utility Rates," Jour-
nal of Land and Public Utility Economics, XVIII (1942), pp. 458-67.
However, it should be noted that lump-sum taxes provide an alternative
means of correcting the income distribution although it may not always
be a practical alternative.
In such instances where there is input-indivisibility, the use of
marginal cost pricing can lead to profits or losses rather than to a
breakeven solution as a result of the firm's being under built or over
built with respect to demand. The specific solution is indeterminate.3
The actual financial position of the firm is determined by the demand
at the time the additional capacity is added. A plant which is under
built will yield profits under marginal cost pricing. However, after
additional capacity is added, the firm may operate with losses due to
excess capacity, but there is a net gain in surplus. This result is
caused by the indivisibility of the capital input. The indivisibility
can make the sum of the prices less than marginal costs, and the firm
will require a subsidy. To avoid the use of subsidies, the firm may be
permitted to practice price discrimination, but this type of pricing be-
comes a question of distributional equity and violates efficiency cri-
If the commodity is supplied in jointness with another, such as
beef and hides, or peak and off-peak water,35 the use of marginal cost
pricing is complicated by the inability to identify the marginal costs
of the separate products. Marginal cost of the product is no longer a
34Oliver E. Williamson, "Peak-Load Pricing and Optimal Capacity
under Indivisiblity Constraint," American Economic Review, LVI (Sep-
tember, 1966), p. 824.
35In these cases, the concept of jointness is not used in the tra-
ditional sense as the beef and hides case. The use of joint supply in
the case of peak load problems is employed in the sense that the verti-
cal summation of demand functions is necessary to determine an efficient
usable criterion for determining prices.
An Alternative Solution to Jointness, Externality,
Decreasing Costs, and Indivisibility
The problems of externalities, decreasing costs, jointness, and
indivisibilities have been mentioned as factors which impose re-
strictions upon the use of marginal cost pricing. The orientation of
this study is toward municipal water utilities,and these problems have
direct bearing upon an economic analysis of the rate practices of these
utilities. A municipal water utility, in the process of providing
water service, creates externalities in the form of improved public
health and increased property values, and develops indivisibility in its
capital plant, including jointness with respect to providing peak and
off-peak water. Subsequently, these problems have an important bearing
upon the solution which might be applied to municipal water rates.
A tool which has recently been developed in economic analysis is
the vertical summation of demand functions. This tool has been applied
separately to the problems of externalities and jointness. However,
the power and general applicability of vertical summation as a geomet-
ric tool has not been fully recognized. This type of construction
applies to the classic joint supply situation, to time jointness, and
to external economies and diseconomies of consumption. Some of the
possible interpretations are demonstrated in Figure II below where D1
represents the demand for beef, D2 the demand for hides, and S the
supply function of cows. The optimal number of cows, beef, and hides is
OX,1 and the Pareto optimal price is OP2 for beef and P2P for hides.
P2 plus P2P1 equals OP1 which, in turn, is equal to marginal cost. When
two commodities are supplied jointly, the marginal cost of each com-
modity is not separable,and the price is based upon the demands for
each of the two commodities. The price of each of the commodities does
not equal the marginal cost of supplying it, but the sum of the prices
is equated to the marginal cost of the joint output. Through the verti-
cal summation of demands, the optimal output of both products can be
obtained where the sum of the prices is equal to marginal cost. This
solution apparently satisfies the criteria for economic efficiency.37
It is noteworthy, however, that a redistribution of income which alters
the demands will also alter the price relationships of the joint pro-
ducts, even under constant costs. This phenomenon will not occur in the
case of separable products,and, thus, it places joint-cost pricing in a
vaguely defined "in-between" area with respect to the distinction be-
tween equity and efficiency.
The analysis in Figure II can also be applied to time jointness if
the diagram is reinterpreted so that D1 is a peak demand, D2 is an off-
peak demand, and S is the cost of the facility which is available to
serve both demands. The optimal quantity is OX1 with a peak price of
OP2 and an off-peak price of P2P1. Again, at the output OX1, the sum of
the prices is equal to the marginal cost of providing the combined ser-
vices to meet the peak and off-peak demands.38
Howard R. Bowen, Toward Social Economy (New York: Rinehart and
Co., Inc., 1948), pp. 177-80.
37Kafoglis, Welfare Economics and Subsidy Programs, pp. 21-33.
38The development of time jointness has been notable in the theory
of public utility rate structures because of the inability of the
utility to adjust its capacity to meet the peak and off-peak demands.
If the utility could adjust its plant size neatly to these demands, a
joint cost problem would not exist. However, it has been noted that
the capacity required to produce the output for one demand automatically
provides the capacity to produce for the other demand. The capacity to
produce water at four o'clock is the same capacity used to produce water
at eight o'clock. Although the outputs in the two periods are different,
the capacity is the same, and this fixed proportion of capacity between
the two periods is the basic requirement for a joint cost problem. See:
If D2 is interpreted as the value of the "spillover" at the margin
to individual A as a result of individual B's consumption, A is an in-
direct beneficiary. Optimal consumption for B is not the quantity OX2
at price OP3, the amount determined by the market; it is OX1, the amount
forthcoming only if some means is found to reduce the price to individual
B below the supply price (marginal cost). This interpretation applies
to vaccines and other quasi-collective goods, where it is necessary for
individual A to guarantee his own health by subsidizing individual B's
consumption through public subsidy or other means requiring governmental
The same type of analysis is sometimes applied in the cases of ex-
cess capacity and indivisibility where marginal cost pricing leads to
losses. The overhead (the loss) is considered a collective, or joint, in-
put which applies to both demands. However, a fixed overhead allocated
Donald H. Wallace, "Joint Supply and Overhead Costs and Railway Rate
Policy," Quarterly Journal of Economics, XLVIII (August, 1934), pp. 583-
616. The joint supply problem is a difficult one to resolve as evi-
denced by the "Pigou-Taussig controversy," and many of the issues still
remain unclear. The inability to adjust capacity to the peak and off-
peak demands in the same time period has been treated as a joint cost
problem by several notable writers. See: James M. Buchanan, "Peak
Loads and Efficient Pricing: Comment," Quarterly Journal of Economics,
LXXX (August, 1966), pp. 463-471; M.A. Crew, "Peak-Load Pricing and
Optimal Capacity: Comment," American Economic Review, LVIII (March,
1968), pp. 168-70; Peter O. Steiner, "Peak Loads and Efficient Pricing,"
Quarterly Journal of Economics, LXXI (November, 1957), pp. 585-610;
Williamson, "Peak-Load Pricing and Optimal Capacity under Indivisibility
Constraint," pp. 810-27; William S. Vickery, Microstatics (New York:
Harcourt, Brace and World, Inc., 1964), pp. 225-44.
39This analysis assumes that all individuals will reveal their
through vertical summation, in these instances, does not lead to
a Pareto efficient solution. In the case of common costs, the de-
mands must be summed horizontally, and each user pays the same price
and contributes to the overhead in proportion to the total output that
The type of analysis embodied in Figure II may be applied to the
entire range of congestion problems, multi-product problems, and
quasi-collective goods. It is possible to analyze many problems
through a simultaneous application of collective demands (vertical
summation) and the marginal cost standard. These constructions are
crucial to the application presented in Chapters V and VI.
Marginal cost pricing assumes that individual demands are inde-
pendent of each other. When demands are not independent, marginal cost
pricing weakens. For example, in the case of water, the hourly peak
demand is a partial function of the hourly off-peak price and the off-
peak demand is a partial function of the peak price. The interdepen-
dence of demands leads to several complex problems such as a shifting
peak and the need for continual price adjustments when there is a price
differential between the peak and off-peak periods (This point is de-
veloped further in the following chapter.). These problems are both ad-
ministrative and theoretical. Consequently, they involve a highly
40See: Wallace, "Joint Supply and Overhead Costs and Railway Rate
Policy," pp. 583-616.
technical type of analysis to determine their solution. However, a
combination of fixed charges and marginal cost pricing provides a
means of overcoming these problems. The solution is developed in
Many of the objections raised about marginal cost pricing have
rested upon theoretical grounds. In general, the criticisms have
been aimed at the underlying assumptions of the marginal cost pricing
principle. To the extent that the assumptions are not fulfilled, the
marginal cost pricing principle is limited as a single pricing guide.
However, these limitations do not mean that marginal cost pricing has
no validity as a useful pricing guideline. It has been demonstrated
that marginal cost pricing, combined with vertical summation of demand
curves, permits the determination of a Pareto efficient solution in
cases of joint products, joint costs, externalities, and indivisibilities.
Marginal cost pricing provides a take-off point as a basis for deter-
mining the rate structure and the size of plant necessary to achieve the
Some of the most damaging criticism of marginal cost pricing has been
practical. In many respects, the practical difficulties have been the
41William S. Vickery, "Some Implications of Marginal Cost Pricing for
Public Utilities," American Economic Review, XLV (May, 1955), pp. 605-20.
deterrents which have kept economic theory from making a generally
acceptable contribution to utility rate theory until the last decade.42
Some critics maintain that industrialists do not think in terms of
marginal costs. This point has not been established. The industrialist
may well have his own terminology for what the economist labels marginal
cost. The inconsistency in terminology between the economist and the
industrialist is not a valid basis for discarding marginal cost pricing.4
The other practical objection to marginal cost pricing relates to
cases of joint supply. Because of the difficulty in identifying marginal
costs, some critics advocate the use of average cost pricing as an al-
ternative. This argument throws the baby out with the water. If it is
difficult to identify marginal costs, it is equally difficult to identify
average cost. In either case, the costs must be allocated between the
42Kenneth E. Boulding, Economic Analysis: Microeconomics (New York:
Harper and Row, Publishers, 1966), pp. 498-99.
4For a discussion of this problem see: Fritz Machlup, "Theories
of the Firm: Marginalist, Behavioral, Managerial," American Economic
Review, LVII (March, 1967), pp. 1-33.
4Boulding, Economic Analysis: Microeconomics, pp. 498-99; Wallace,
"Joint Supply and Overhead Costs and Railway Rate Policy," pp. 583-616.
commodities,and average cost pricing does not eliminate the problem.45
45For excellent discussions of all the major aspects of marginal
cost pricing and the controversy over its usefulness see: J. deV. Graaff,
Theoretical Welfare Economics, pp. 142-55; Little, A Critique of Wel-
fare Economics, pp. 185-216; Ruggles, "Recent Developments in the Theory
of Marginal Cost Pricing," pp. 107-26; William S. Vickery, "Some Ob-
jections to Marginal Cost Pricing," Journal of Political Economy, LVI
(June, 1948), pp. 218-238.
THE THEORETICAL MODELS
During the past decade, many writers have advocated a return to the
marginal cost pricing principle as a basic policy recommendation in the
field of public utilities, and significant contributions to utility rate
theory have been made. One of the more significant contributions of re-
cent writers is their handling of the difficult peak load problem which
entails the problem of time jointness with respect to supply. Taken in
their entirety, these contributions form a relatively complete analysis
of the problems faced in determining optimal capacity and optimal rate
structures. The purpose of this chapter is to evaluate these models in
terms of their analyses, assumptions, and distributional implications in
order to determine their applicability to the special problems of muni-
cipally owned water utilities.
1The models included in this chapter are: James M. Buchanan, "Peak
Loads and Efficient Pricing: Comment," Quarterly Journal of Economics,
LXXX (August, 1966), pp. 463-71; Ralph K. Davidson, Price Discrimi-
nation in Selling Gas and Electricity (Baltimore: Johns Hopkins Univer-
sity Press, 1955); Jack Hirshleifer, "Peak Loads and Efficient Pricing:
Comment," Quarterly Journal of Economics, LXXII (August, 1958), pp.
451-62; Harold Hotelling, "The General Welfare in Relation to Problems
of Taxation and of Railway and Utility Rates," Econometrica, VI (July,
1938), pp. 242-69; Peter 0. Steiner, "Peak Loads and Efficient Pricing,"
Quarterly Journal of Economics, LXXI (November, 1957), pp. 585-610;
Oliver E. Williamson, "Peak-Load Pricing and Optimal Capacity under
Indivisibility Constraint," American Economic Review, LVI (September,
1966), pp. 810-27.
The Hotelling Model
One of the first rigorous solutions to advocate marginal cost
pricing was that presented by Harold Hotelling. Hotelling attempted
to develop pricing policies which would lead to maximum consumers'
surplus. Through the use of mathematics, Hotelling rigorously demon-
strated that the optimum for the general welfare corresponds to the
sale of everything at marginal cost prices. He refutes the position
taken by utility engineers that commodities produced by industry must
be sold at a price high enough to cover full costs. Indeed, he states
that this policy leads to economic inefficiency and to a loss of wel-
fare as measured by consumers' surplus. The essence of Hotelling's
analysis is summarized in Figure III below. Assuming constant short-
run marginal costs, a capacity constraint at output OX and demand as
shown by Dl, the welfare maximizing firm supplies the output OX1, at
the price OP1, which is equal to short-run marginal cost (b). As de-
mand increases, the firm expands output toward OX at a constant price
(OP ) until output OX is reached. As demand continues to increase,
the firm cannot expand output beyond OXo, except in the long-run. In
the short-run, the price moves upward along the vertical segment NR.
At price OP2, with demand now at D2, the firm is charging a price which
2Hotelling, "The General Welfare in Relation to Problems of Tax-
ation and of Railway and Utility Rates," pp. 242-69.
Ibid., p. 242,
4Ibid., p. 242.
The Hotelling Model
exceeds short-run marginal production costs by an amount equal to NT.
The surplus NT of price over marginal cost is the effect of rationing
a fixed supply through the use of efficiency pricing. Hotelling's
position on this surplus is twofold. At one point he defines the sur-
plus as a rental charge not unlike the site rental of land. Such a
rental charge, according to Hotelling, becomes a source of revenue to
the state and can be taxed away without affecting the allocation of re-
sources. However, the charge is also needed to reflect the "social
costs" of congestion. As the quantity demanded exceeds the capacity
5bid., p. 249.
to produce, the utility becomes overcrowded, thus reducing the quality
of service to all users, and Hotelling feels that the price should be
high enough to reflect these social costs of congestion. The inter-
pretation Hotelling gives to the vertical segment NR is a marginal
opportunity and/or a marginal social cost; not a marginal money cost of
production as the segment bN reflects. Prices OP1 and OP2 are welfare
The basic elements of this model provided the background and the
framework upon which more recent writers have based their analyses. The
change in the price policy from a situation such as that defined by
point K in Figure III to that defined by point T provided the basis for
the eventual development of sophisticated solutions to the peak load
The Steiner Model
One of the first general solutions to the problem of defining an
optimal plant and rate structure for firms facing a peak load problem
was presented by Peter 0. Steiner. Subsequent models have been vari-
ations of and improvements upon Steiner's analysis. The proper policy,
Steiner, "Peak Loads and Efficient Pricing," pp. 585-610. At the
time the Steiner article appeared, a similar solution appeared in
France. See: Marceo Boiteux, "La Tarification des Demandes en Pointe:
Application de la Theorie de la Vent au Cout Marginal," trans. by H.W.
Izzard, Marginal Cost Pricing in Practice, ed. by James R. Nelson
(Englewood Cliffs: Prentice-Hall, 1964), pp. 59-89. It should be noted
that the nature of the solution was recognized by transportation eco-
nomists many years ago. See: Michael R. Bonavia, The Economics of
Transport (New York: Pitman Publishing Corp., 1936), pp. 103-11.
The Steiner Model
^, --- ^ ^------ ^
according to Steiner, is one which achieves a social optimum, i.e.,
the policy which maximizes the excess of expressed consumer satisfaction
over the cost of the resources used in production. Steiner's model
attempts to formulate a price policy which leads to the optimal amount
of physical capacity and which is consistent with marginal social costs.
Using a vertical summation of peak and off-peak demand curves,
Steiner presents the framework shown in Figures IVA and IVB where (b)
--the horizontal axis-- is the short-run marginal operating cost (as-
sumed to be zero), and bB represents the long-run marginal capacity
costs. In Figure IVA, there is excess capacity (X2X1) during the off-
peak period (D2), and the off-peak user places no demand on the sys-
tem capacity. The off-peak user pays a price equal to the short-run
marginal operating costs (b). In Figure IVB, the solution entails no
excess capacity during the off-peak period. Both the peak user and the
off-peak user make a contribution to the capacity costs. The off-peak
user pays the price bP2, the peak user pays the price bP1, and the sum
of the two prices is equal to the marginal capacity costs (bB).
Steiner's general argument is that a unit of capacity can be added if
the costs can be covered by the sole demand of any one period (D in
Figure IVA) or by the combined demands of two or more periods (D1 plus
D2 in Figure IVB).8
7Steiner, "Peak Loads and Efficient Pricing," pp. 585-87.
8Ibid., p. 589.
Upon arriving at the solution in Figure IVB, Steiner concludes
that an efficient pricing solution entails discrimination. In Figure
IVB, for example, the off-peak and peak outputs are equal, but the
prices are unequal. Steiner states:
If demand curves are different, at a given output,
the prices are unequal and since this is truly a
case of joint costs, unequal prices in the face of
equal output and joint costs means discriminatory
This statement reflects Steiner's recognition that in the case of a
firm having both a peak load problem and a capacity constraint (short-
run) the peak load problem becomes one of joint costs. Prices do not
equal short-run marginal operating costs, but they maintain long-run
optimal capacity. The strength of the relative demands determines each
period's users' share of the utility's capacity costs. Although the
prices are discriminatory in the usual sense, the pricing solution is
efficient in the Paretian sense and satisfies the criteria of welfare
maximization. The sum of the prices (P1 plus P2) is equal to the long-
run marginal costs (Bb). Therefore, Steiner's formulation does satisfy
the criteria for welfare maximization.
The Hirshleifer Model
An alternative formulation of the peak load problem has been de-
veloped by Jack Hirshleifer, who employs a different concept of marginal
Ibid., p. 590.
cost than Steiner, which leads him to reject Steiner's description of the
optimal solution as one which involves price discrimination. The dif-
ference between the peak and off-peak prices is explained in terms of
marginal opportunity costs.
Hirshleifer uses the same optimizing criteria as Steiner, but mar-
ginal cost is defined as a marginal opportunity cost when the utility
faces a capacity constraint. The marginal opportunity cost is the value
set upon the resources in the most valuable alternative use being sac-
rificed. Hirshleifer divides constant long-run marginal costs be-
tween joint and separable long-run cost elements. The joint long-run
marginal cost is the cost per-combined-unit of production--a variable
cost for each of the two periods plus a capacity cost. The separable
long-run marginal cost is the cost of increasing the output of one of
the two periods, the output for the other period being held constant.2
Hirshleifer's solution is based upon two different short-run mar-
ginal cost functions. One solution assumes a continuous cost function,
the other a discontinuous cost function. Hirshleifer's model, based up-
on the assumption of constant long-run costs and a continuous short-run
marginal cost function, is shown in Figure V below. Each period's price
is equal to short-run marginal cost. The off-peak price is OC and the
10Hirshleifer, "Peak Loads and Efficient Pricing: Comment," pp.
11bid., p. 451.
12Ibid., p. 455.
Hirshleifer's Continuous Cost Model
quantity supplied is OX2. The peak price is OD and the quantity
supplied is OX1. The short-run conditions for welfare maximization
are satisfied. Since the short-run marginal cost function is continu-
ous and each period's demand is equated with this cost function, there
is no discrimination in the pricing solution. Each period's price is
equal to short-run marginal production costs.
The long-run conditions for welfare maximization are also satis-
fied. The sum of the prices (OC plus OD) is equal to long-run marginal
costs (2b plus B). If the sum of the prices is not equal to long-run
marginal cost, total surplus can be increased by restoring the equality.
If the sum of the prices is greater than long-run marginal costs, an
addition to capacity increases consumers' surplus by more than pro-
ducers' surplus is reduced. If the sum of the prices is less than
long-run marginal costs, producers' surplus can be increased by a
greater amount than consumers' surplus is reduced by a contraction
Steiner's conclusion about discriminatory prices was based upon
the assumption of joint supply combined with a discontinuous short-
run marginal cost function. Hirshleifer also recognizes the problem
of a vertical short-run marginal cost function, but he does not use
the joint cost assumption. According to Hirshleifer, the vertical seg-
ment leads to an indeterminate marginal cost as a cash outlay concept.
Once the maximum output is reached, short-run marginal cost becomes in-
determinate. Hirshleifer's formulation on the assumption of a dis-
continuous marginal cost function is shown in Figure VI. In the frame-
work below, X1T is the off-peak price, and X1R is the peak price. The
sum of the prices (X1T plus X1R) is equal to long-run marginal cost (b
plus B). The quantity OX1 is supplied during both the peak and off-
peak periods. Once output OX1 is reached, marginal cost increases from
Ob (operating costs) to Ob plus B (long-run marginal cost). Short-run
marginal cost is indeterminate along the vertical segment beginning at
point N. Although two different prices are charged for the same out-
put, Hirshleifer explains the difference in terms of costs. He departs
from the marginal money cost concept and explains the difference in terms
of the differences in the marginal opportunity costs between the two
periods; therefore, his solution described above does not entail price
DD "---------------------- ------ '-" '"-
discrimination.13 This approach puts Hirshleifer in a position
13Crucial to his analysis is Hirshleifer's explanation as to why
this argument does not explain away all forms of discrimination.
Hirshleifer explains that in the textbook cases of discrimination
"...the market is divided artificially. The commodity being the same,
at the profit-maximizing solution, the marginal customers in each class,
while paying different prices, are being served at the same opportunity
cost--the value of the first unit of unsatisfied demand in the higher
priced market is the most valuable alternative foregone. Therefore, no
price difference is justified on opportunity cost grounds. In the case
under consideration, the market division is not artificial--taking a
unit away from the off-peak does not make it possible to supply a unit
on-peak, so the higher on-peak value is not the relevant alternative
social opportunity cost of the off-peak service," ibid., p. 459.
different from that of both Steiner and Hotelling. As mentioned
earlier, Hotelling interprets the price differentials in terms of the
marginal social congestion costs and the ensuing deterioration of
service resulting from congestion. Steiner treats the problem as a
joint cost problem. Hirshleifer, on the other hand, defines the price
differentials in terms of marginal opportunity costs.
Hirshleifer concludes that,when the marginal cost function is con-
tinuous, prices are not discriminatory since they are equal to short-
run marginal money costs. When the marginal cost function is discon-
tinuous, Hirshleifer concludes that prices are not discriminatory in
the Steiner sense since prices are equal to marginal opportunity costs.
In this context, Hirshleifer's definition of marginal cost is not the
traditional money outlay concept of marginal cost, but the opportunity
cost interpretation, and the marginal value of the last unit is equal
to the value of the first unit of unsatisfied demand in each of the
periods. Therefore, prices equated to opportunity costs are deter-
mined by the relative strength of the demand functions in each of the
periods. With the exception of interpretation, both solutions are, for
all practical purposes, identical.
The Williamson Model
A recent development in the peak load pricing problem has been the
framework developed by Oliver E. Williamson.4 The Williamson model was
14Williamson, "Peak-Load Pricing and Optimal Capacity under Indi-
visibility Constraint," pp. 810-27.
presented to improve upon the geometry developed by Steiner and the con-
cepts employed by Hirshleifer. Williamson recognizes that, for electric
utilities (equally valid for water utilities), the peak period is of
longer duration than the off-peak period. When the two periods are of
unequal duration, an adjustment must be made in the vertical summation
since only an assumption of equal time periods permits straightforward
summation such as the Steiner summation. This adjustment is the cent-
ral point of Williamson's geometric model. When the two periods are of
unequal duration, a weighted vertical summation of the two demands is
Williamson's model, shown in Figure VII below, assumes constant
costs, a peak period of sixteen hours, and an off-peak period of eight
hours. The measure of welfare is total surplus as developed by Marshall
under an assumption of constant costs. Total welfare is equal to (total
revenue + consumers' surplus) minus total costs.15 In Figure VII, the
short-run solution equates each period's price with the short-run mar-
ginal cost. The peak price is OP and the off-peak price is OP2. The
long-run solution equates long-run marginal cost with the effective de-
mand curve DE, which is the weighted sum of the individual demands.
Since it is assumed that the peak lasts for sixteen hours and the off-
15For the technical application of consumers' and producers' sur-
plus see: John R. Hicks, Value and Capital (2nd ed.; Oxford: Oxford
University Press, 1946), pp. 38-41; John R. Hicks, Revision of Demand
Theory (London: Oxford University Press, 1956), pp. 67-106; John R. Hicks,
rThe Four Consumer Surpluses," Review of Economic Studies, XI (1943), pp.
68-74; John R. Hicks, "The Generalized Theory of Consumer's Surplus,"
Review of Economic Studies, XIII (1945-46), pp. 68-74.
The Williamson Model
peak for eight hours, the peak demand is given a weight of 2/3 and the
off-peak a weight of 1/3. The weighted curve reflects the average price
the utility receives over the entire cycle which can be applied toward
the capacity costs. It is basically a "long-run" demand curve.
Using Williamson's model, the basic conclusions are threefold. First,
optimal price in every subperiod is given by the intersection of the
short-run marginal cost and the subperiod demand. Secondly, plant size
is given by the intersection of the effective demand for capacity curve
and the long-run marginal cost function. Thirdly, in a fully adjusted,
continuously utilized system with only two period loads, (a) peak load
price always exceeds long-run marginal cost, (b) off-peak price is al-
ways below long-run marginal cost, and (c) only when the off-peak fails
to utilize capacity when priced at short-run marginal cost does the
peak load bear the entire burden of the capacity costs.16
As in the case of the Steiner and Hirshleifer models, the distri-
bution of the capacity costs depends upon the relative strengths of the
two demands when prices are equated with short-run marginal cost. When
plant is divisible, making optimal capacity possible, the utility operates
with zero net revenues in the long-run. The surplus generated during the
peak period exactly equals the deficit experienced during the off-peak
Williamson modifies the assumption of a completely divisible plant
in his model (an assumption implied in the other models) by constructing
a solution based upon the alternative assumption of indivisible plant.
In this model, there is no guarantee that revenues will be sufficient to
cover costs. Both the peak and off-peak prices can be below long-run
marginal cost because of the indivisibility of the capital stock. Since
prices are not sufficient to cover the long-run marginal costs, the firm
16Williamson, "Peak-Load Pricing and Optimal Capacity under Indi-
visibility Constraint," pp. 821-22.
operates with negative net revenues. The solution is shown in Figure
VIII below, where the peak price is OPI, and the off-peak price is OP2.
It can be seen that the off-peak price and the peak price are both less
than long-run marginal cost, and the firm is operating with total losses
equal to (P2CEb+B) during the off-peak period plus (P JEb+B) during the
The Williamson Model
Although the firm is operating with losses, the plant size is opti-
mal from a welfare standpoint because the effective demand and the long-
run marginal cost function are the co-determinants of plant size. When
indivisibility is present, the firm operates with zero net revenues only
accidentally. In Figure VIII,assume output OX1 is the present capacity,
and D1 and D2 are the peak and off-peak demands respectively. Based up-
on these demands, output and capacity should be that defined by point K
where the effective demand intersects long-run marginal costs. Because
of the indivisibility, the firm must add capacity which is capable of
producing output OX2. The addition to capacity is warranted, however,
because the gain in consumers' surplus exceeds the loss in producers'
surplus, as shown by the relationship between the two triangles RKT and
EKH. The triangle RKT is larger than EKH so there is an increase in
total welfare from the expansion of capacity from OX1 to OX2, with the
net increase equal to the difference between these two triangles. It is
evident that there is no guarantee that the number of units of output
capable of being produced is the exact number needed to put the firm in
a position to realize zero net revenues when indivisibility exists. The
result might be profits, losses, or breakeven.
The general conclusion of Williamson's model is that in the case of
divisibility, a plant size is optimal if "...an increase in scale leads
to a decrease in producers' surplus that exceeds the gain in consumers'
surplus; a decrease in scale yields an increase in producers' surplus that
is less than the loss of consumers' surplus."17
17Ibid., p. 820.
The Davidson Model
Another work representing a variation on the former models, but
which came earlier in time, is the model formalized by Ralph K. David-
son.18 Davidson's study was a significant and comprehensive contri-
bution to the practical application of discriminatory prices. In the
course of a more general analysis, Davidson derives a solution to the
peak load problem. The significant feature of this solution is the use
of the long-run as the relevant time period for policy determination.
Davidson argues that prices equated with long-run marginal costs are
more relevant for policy purposes, whereas the other models equate price
with short-run marginal cost. Davidson also concludes that a rate
schedule should not be discriminatory, and that all rate differentials
should be based upon costs. Although Davidson's work is frequently cited
in other works on utility rate theory, it can be demonstrated that, in
reaching his conclusions, Davidson ultimately develops an average cost
Davidson's basic analysis is summarized in Figure IX below. Each
period's price is equated to the long-run marginal cost of supplying the
period. The difference between the long-run marginal cost functions re-
flects the cost differences of serving the two periods. As plant is ex-
panded to meet the peak demand (D1), the firm has greater operating and
capacity costs. To expand production during the off-peak (D2), the
18Davidson, Price Discrimination in Selling Gas and Electricity
(Baltimore: Johns Hopkins University Press, 1955).
utility does not have to expand capacity. The only additional costs
are operating costs. According to Davidson, it is this solution which
maximizes total surplus.
The Davidson Model
It is difficult to understand why a firm's long-run marginal cost
function can be high or low depending upon which period's production
the firm attempts to expand. Davidson has implicitly separated the
time periods into separate markets. The off-peak period is one market
and the peak period is the other. Each of the markets has its own long-
run marginal cost function. The analysis is presented in Figure X below.
An Interpretation of
the Davidson Solution
A (Peak) P B (Off-Peak)
Plant size is sufficient to meet the peak demand (the determinant of
plant capacity),and the off-peak production is,therefore, always less
Davidson fails to consider that a single firm is producing both the
peak and the off-peak outputs with the same basic plant capacity. Figure
XI below demonstrates the implicit analysis of the Davidson model. The
capacity of the plant is determined by the expected peak demand (Dl).
To achieve economic efficiency, the plant operates at the minimum point
of the short-run average cost function (given Davidson's assumption of
A Total Cost Interpretation
of the Davidson Model
P\ B (Off-Peak)
constant long-run average and marginal costs). During the peak, the
utility produces output OX1 with total costs of X A. The peak price
is X1A divided by OX1. Peak production is carried to the point where
short-run average cost equals short-run marginal cost equals long-run
marginal cost. It appears that Davidson selects long-run marginal
cost rather than short-run average cost to equate with price since they
are equal to each other. Therefore, price is also equal to short-run
average cost. Off-peak price is determined by dividing total variable
costs of X2B by the off-peak output of OX2. Since capacity costs have
been recovered during the peak period, variable (operating) costs are
the only relevant costs for the off-peak period.
The Davidson solution can also be presented by using average cost
curves shown in Figure XII. Expected peak demand determines the size
of plant. If the firm is operating under constant cost conditions,
economic efficiency is achieved when short-run average cost is a minimum
at the peak output. The peak price is OPl,and all the capacity costs
(the difference between SAC and SAVC) are recovered from the peak period
output. Variable costs are the relevant costs for the off-peak period.
With an off-peak demand of D2, the off-peak price is OP2, and the only
costs included in this price are the firm's variable costs. Davidson's
pricing technique is entirely a cost-of-service based pricing solution,
and it is the same, in most respects, as the solutions proposed by
utility engineers and managers.
The Average Cost Interpretation
of the Davidson Model
That Davidson uses average cost pricing is further demonstrated by
his definition of marginal cost.1 He takes the position that the
utilities must cover their long-run marginal costs if they are to have
successful financial operations without having to resort to price
19It would appear that part of Davidson's misinterpretation also
stems from his apparent failure to either recognize or acknowledge the
jointness between the peak and off-peak periods.
discrimination.20 Davidson cites the tendency of people in the utility
field to use the term "total increment costs" to refer to long-run
marginal costs. Davidson uses this utility concept of incremental
cost and calls it long-run marginal cost. The incremental cost in-
terpretation permits the utility to have different long-run marginal
cost functions depending upon the time period being served. Incre-
mental costs are different depending upon whether output is expanded
during the peak or the off-peak period. Davidson's concept of marginal
cost is different from the economic concept of long-run marginal cost.
The latter designates the change in costs associated with changes in pro-
duction when all inputs are changed. Davidson's use of long-run mar-
ginal cost as incremental costs in which either all or only part of the
inputs are changed depending upon the time period in question is evi-
denced by the following statement:
The relevant cost concept consists of long-run marginal
customer costs, and long-run marginal output costs,
which always includes energy costs and may or may not
include capacity costs depending upon the time of day 22
and season of the year when the marginal unit is used.
All inputs are changed during the peak period, and,during the off-peak
period, the capital input is held constant. The use of incremental
costs can lead to an average cost pricing solution.
20Davidson, Price Discrimination in Selling Gas and Electricity, p. 72.
21Ibid., p. 72.
22Ibid., p. 72.
The Buchanan Model
James M. Buchanan provides an analysis of the peak load problem,
which represents a further sophistication and modification of the pre-
vious models.23 Buchanan's contribution is the introduction of first
degree discrimination into the standard models which destroys their
apparent determinacy. According to Buchanan, the determinacy is
"...produced only by the implicit adoption of unjustified assumptions
concerning the uniformity of marginal price over quantity."24 Buchanan
maintains that there is no reason for a utility to charge the same
price for all quantities demanded in each period by the same buyer.
Buchanan's solution takes into consideration the block-rate structures
that are actually used,and his position is summarized in the following
If the model should be restricted to goods, and not
applied to services that are consumed as purchased,
then the possibility of interpersonal resale within
periods might tend to insure against the possibility
of "price discrimination" over quantities sold to
single buyers. However, it seems plausible to ex-
pect that "price discrimination" over quantities
sold to single buyers will accompany "price dis-
crimination" among separate buyers.
The utility can charge different prices for different units sold to
a single buyer. The price paid for the marginal unit is affected by
23Buchanan, "Peak Loads and Efficient Pricing: Comment," pp. 463-71.
24Ibid., p. 462.
25Ibid., p. 465.
the prices charged for the infra-marginal units. In this event, it
becomes necessary to distinguish between marginal and average prices.
For each marginal price, there may exist several quantities demanded
as the price of the entire offer is altered. In these instances, no
precise demand curve can be derived unless the buyer is presented with
all the possible price offers. When a precise demand cannot be identi-
fied, vertical summation is impossible,and the solution becomes in-
Buchanan reiterates the condition necessary to achieve Pareto opti-
mality in joint cost situations: the sum of the prices at the margin
must equal marginal cost. The use of discriminatory pricing in the
infra-marginal units, however, affects the distribution of the cost
shares among the various users. Buchanan maintains that the distribution
of the cost shares over the infra-marginal units and the income effects
of the discrimination can affect the location of the margin which, in
turn, determines the marginal capacity and the marginal prices.27
Through the manipulation of discriminatory rate structures, the utility
can arrive at a set of marginal prices where the sum is equal to long-
run marginal cost,and there is no excess capacity during the off-peak
period. Since the rate structure only affects the cost shares, there is
no significant effect on Pareto optimality. There are, however,
26Ibid., pp. 465-66.
27Ibid., p. 466.
significant distributive effects. Buchanan states that "...the choice
among different price offer sets finally rests on the decision maker's
evaluation of different distributions of consumers' surplus among the
separate period's demanders."28
Buchanan has set up a unique model which reconciles price discri-
mination, distributional effects, and Pareto optimality. Once the price
offer is given, the margin is located, and the plant size becomes deter-
minate. The selection of the rate is made by the utility manager whose
decision determines the cost share of each user. The cost share "..
will influence the location of equilibrium via income effect feedbacks
on demand."29 The solution is now determinate since the utility manager
has selected one of an infinite number of possible price offers which
could be made. However, its selection entails a distributive judgment
on the part of the utility manager.
The type of model described by Buchanan has particular significance
to the practices of municipally owned water utilities. Buchanan's con-
clusions rest on the assumption of first degree price discrimination in
infra-marginal units. In municipalities, there is considerable variation
of rates between some types of users. The rate variations can be used
for taxation or for community development. The community leaders make
the selection of the price offers, and,hence, the distributive judgment
about the basis for rates. Buchanan states that the efficiency and
28Ibid., p. 466.
29Ibid., p. 466.
distributive aspects cannot be separated. In Buchanan's words:
Choosing a specific distribution of the total cost
among separate-period demanders will, of course,
determine a specific allocation of resources and
a specific set of marginal prices that must be
obtained if Pareto efficiency is to be achieved.
On the other hand, within certain limits, choosing
a specific investment in system capacity will de-
termine the distribution of total costs along with
the set of marginal prices that must be present if
the Pareto conditions are to be satisfied. In
either case, the limits of economic analysis are 30
reached sooner than...the Steiner...analysis implies.
The position taken by Buchanan further reinforces the view that
additional economic analysis is needed to guide municipal water utility
managers to make efficient decisions with respect to plant size and rate
structures. There is an even greater need for economic evaluation of
the various distributive judgments these same managers make, and to which
Buchanan explicitly draws attention. Previous economic models have
failed to provide this evaluation.
The Underlying Assumptions
The various solutions to the peak load problem all contain assump-
tions which either enhance or limit their applicability to municipal
water utility problems. In other respects, the authors have failed to
make explicit the nature of their assumptions. It becomes important to
examine these assumptions,since the assumptions affect the validity of
30Buchanan, "Peak Loads and Efficient Pricing: Comment," p. 471.
Buchanan refers the reader to: R.H. Strotz, "Two Propositions Related
to Public Goods," Review of Economics and Statistics, XL (November,
1958), pp. 329-31.
the model in terms of its applicability to the problems faced by
municipal water utilities.
The first solution to the peak load problem was the Steiner solution
which included the vertical summation of the peak and off-peak demands
to solve for optimal capacity and the optimal rate structure. The use
of vertical summation--a tool used also by Williamson and Buchanan, but
not Hirshleifer--recognizes the joint nature of the peak and off-peak
periods in which the capacity used to provide output in one period is
the same capacity used to provide output in the other period. In using
vertical summation, Steiner employs the straightforward summation which
is valid when the time periods are of equal duration, whereas William-
son, recognizing that the time periods are not of equal duration, in-
troduces a weighted summation to allow for the difference. Williamson
weights each demand by the proportion of the time it represents of the
A significant aspect of these solutions is their recognition of the
joint supply problem. However, the authors leave their readers with the
impression that the output produced during the off-peak period is pro-
duced jointly with the output produced during the peak period, but such
is not the true joint nature of the problem. The jointness is in re-
spect to time and capacity. The jointness arises because of the firm's
inability to vary neatly the size of its plant between the peak and off-
If capacity could be changed between the two periods, plant size
would vary in accordance with the illustrations in Figures XIIIA and
XIIIB. The off-peak case, shown in Figure XIIIA, yields a capacity
The Optimal Solution in the
Absence of Time Jointness
sufficient to produce output OX2. As the peak demand becomes the
effective demand, the utility expands its capacity to produce output
OX1 shown in Figure XIIIB. As the peak demand declines, and the off-
peak demand becomes the effective demand, plant capacity is reduced
back to that shown in Figure XIIIA. The inability to vary the amount
of capacity in this manner over the short-run creates the joint supply
The Vertical Summation
of Intra-Cycle Demands
(,I s .)
condition since the firm is faced with two demands in the same time
horizon and is unable to adjust its capacity neatly to each of these
demands. Consequently, the capacities between the two subperiods are
provided in fixed proportions to each other. The proper price and out-
put decisions are, therefore, based upon a vertical summation of the
two subperiod demands. The vertical sum shows the average price the
firm receives over the entire time horizon under consideration. The
weighted summation, shown in Figure XIV as the demand function DE under
the assumption of periods of equal duration, is applicable to a water
utility since this type of firm must produce to meet seasonal, daily,
and hourly demands which can exceed'the average demand by as much as
1500 per cent. The inability to adjust plant capacity to these demands
generates a true case of time jointness in the operation of a water
Administrative Problems: Price Stability
and Technological Deficiencies
There are several problems a water rate analyst may encounter in
attempts to administer prices based on an efficiency criterion. One of
these problems entails the possibility of continually altering prices
31Storage might be a feasible alternative for meeting the hourly
peak demand. Storage tanks and reservoirs may be filled during the
off-peak period to aid in meeting the heavy demand occurring during
the peak period. The use of storage facilities enables the utility to
partially adjust capacity between the peak and off-peak periods.
However, storage is not a perfect substitute for production and pumping
capacity. Therefore, the existence of storage will modify but will not
drastically change the conclusions.
when the peak and off-peak demands are interdependent, and price differ-
entials cause a shifting peak. If the utility charges a single price for
providing both peak and off-peak water service, the rate structure clearly
fails to achieve Pareto optimality. Figure XV demonstrates that an attempt
to use efficiency pricing, when demands are interdependent, might
p Figure XV
Traditional Efficiency Pricing
with Interdependent Demands
p5 L- -_ ii-D,
0 X. X1 Q
conceivably lead to difficulties in making the price adjustment. In
Figure XV, the original price is OP (the single price), the quantity
taken during the off-peak is OX2, and the quantity taken during the peak
is OX1. At this level, price is less than the peak short-run marginal
cost and greater than the off-peak short-run marginal cost, and the
Paretian criterion is not satisfied. If each period's price is to
equal short-run marginal cost, the off-peak price should be lowered to
OP2 and the peak price increased to OP However, as the off-peak price
is lowered, and the peak price is increased, the users will shift their
demand from the peak hours to take advantage of the lower off-peak rate.
This shifting causes the peak demand (D1) to move downward and to the
left, and the off-peak demand (D2) upward and to the right--assume to
D' and D' respectively. To maintain prices equal to short-run marginal
cost, the peak price must be lowered to OP1 and the off-peak price in-
creased to OP However, as the differential between the peak and off-
peak prices is reduced, the demands will move back toward D1 and D2 re-
spectively, as the users now shift part of their demand away from the
off-peak hours and back to the peak hours. This movement requires raising
the peak price and lowering the off-peak price to keep prices equal to
short-run marginal cost. Gross adjustments of this sort might lead to a
fruitless back and forth adjustment of prices. However, small adjust-
ments, taken one at a time, may lead to a converging cobweb.
The problem of interdependent demands combined with shifting peaks
becomes involved in a highly theoretical analysis of dynamic economic
equilibrium entailing such factors as the nature of price stability, the
existence of a single equilibrium versus multiple equilibria, and the
problem of price determinacy. The area should be recognized as a problem
area worthy of further investigation, but the investigation is beyond the
immediate scope of this study. In any case, interdependence complicates
the pricing problem, especially in the case of shifting peaks.
Another administrative problem associated with prices based upon
rigorous efficiency criteria is the technological deficiencies present
in today's water meters. Assume the utility is able to vary the peak
and off-peak prices to reflect cost differences in satisfying the hourly
water demands without the complexities of interdependent demands and
shifting peaks. To arrive at a proper pricing policy, water use must be
measured on an hourly basis. However, at the present time, the metering
equipment necessary to accomplish this task is far from being perfected
to the point which could make this a relevant alternative. The present
cost of such metering equipment is so high that any gain through a rigorous
price policy probably would be offset by increases in costs which would
accompany the installation of the metering equipment. Thus, flat monthly
rates, which seem not to conform to the marginal cost criterion, indeed,
may represent the most efficient pricing arrangement. Efficiency, in any
real sense, must encompass all costs including administrative costs.
These administrative problems associated with a price policy rigor-
ously tied to marginal cost pricing may require some alternative form of
pricing such as the use of fixed charges. These alternatives are developed
in Chapter V.
All the models assume constant returns to scale as the utility expands,
an assumption retained, in part, in the remainder of this study. There is,
however, a definitional point which requires clarification. There appears
to be some ambiguity arising from the use of various interpretations of
short-run marginal costs when dealing with a straight line discontinuous
marginal cost function. To show a capacity constraint, the models assume
a cost relationship such as that illustrated in Figure XVI. At output
OX, the firm is unable to produce additional output,and the short-run
marginal cost function becomes a vertical line. The implication is
that output OX defines the point where the short-run marginal cost
function becomes infinite. To achieve economic efficiency, the models
Cost Functions with a
call for prices equal to the short-run marginal costs.32 The horizontal
32One exception is Davidson whose solution equates price with long-
run marginal cost. Davidson, Price Discrimination in Selling Gas and
segment (bN) of the short-run marginal cost function is the relevant
segment from the standpoint of production costs. This part represents
the short-run marginal operating (variable) costs. The vertical seg-
ment beginning at point N, although labeled short-run marginal cost, is
not an operating cost, and, thus, not short-run marginal cost. The
additional neo-classical concept of the marginal cost of production is
the change in total costs associated with a change in output.3 Once
output OX is reached, the firm cannot increase output without adding to
its capacity. All inputs become variable, and an attempt to increase
output without adding to capacity would be non-economic behavior since
it cannot be done. For additional output beyond OX, the relevant cost
function becomes the long-run marginal cost function and not short-run
marginal costs. To refer to the vertical segment as either opportunity
costs or social costs is to vary the definition of marginal cost along the
same cost function, a practice which easily leads to confusion.
The models tend to employ three different concepts of short-run mar-
ginal cost. The horizontal segment represents short-run operating costs.
When the firm comes up against the capacity constraint, Hirshleifer changes
the definition to one of marginal opportunity cost while Hotelling uses the
marginal social cost of congestion concept. Steiner, Williamson, and
Buchanan do not resort to the marginal opportunity or social costs concept
since they are treating the problem as a joint cost problem and use the
For the traditional concept of marginal costs see: George J.
Stigler, The Theory of Price (rev. ed.: New York: The Macmillan Co.,
1962), pp. 96-97.
vertical summation of demands.34 It can be seen that several concepts
of marginal cost are used in the models, and it is important that these
different concepts be made explicit,as they make an important distinction
between the approaches of the various models.
The Equity Implications
The basic solutions to the peak load problem presented by the eco-
nomic models have been those that require equating price with marginal
costs. All price differences are explained by costs, with any differ-
ences in costs reflected in the prices. With the exception of Buchanan,
all distributive judgments have been assumed away either implicitly or
explicitly. The solutions are based upon the criteria of economic effi-
ciency, but these efficiency solutions have distributional implications.
The Peak Problem
All the solutions advocate that the peak users pay all the capacity
costs when the off-peak users fail to utilize all the capacity. The con-
clusion is based upon a view that only the peak user imposes capacity
3It was this point that led Steiner to the conclusion that prices
set along the vertical short-run marginal cost function are discrimi-
natory. Steiner, "Peak Loads and Efficient Pricing," p. 590. The sig-
nificant point Steiner made was that, although the prices are discrimi-
natory since they are based upon demand and not cost, the solution is
consistent with Pareto optimality since the sum of the prices is equal
to long-run marginal cost. The other writers either ignored the point
made by Steiner or, as did Hirshleifer, redefined the vertical segment
in terms of some other form of costs. Hirshleifer, "Peak Loads and
Efficient Pricing: Comment," pp. 458-59.
costs.35 A question of equity arises in the case of the off-peak user,
who appears to go scot-free. Figure XVII shows the hourly load chart
for a water utility serving three users (A, B, and C). User A takes 30
gallons per minute (gpm) every hour between 12:00 A.M. and 4:00 P.M.
User B takes 30 gpm during each hour between 12:00 A.M. and 4:00 P.M. plus
60 gpm for each hour from 4:00 P.M. to 12:00 A.M. User C takes 30 gpm
for each hour from 8:00 P.M. to 4:00 P.M. and 60 gpm during each hour from
4:00 P.M. to 8:00 P.M. Using the capacity cost allocation advocated by
An Hourly Load Chart for Three Users
p ~* ~
r.M. P.M A.M
35A conclusion reached by: Steiner, "Peak Loads and Efficient
Pricing," pp. 585-610; Hirshleifer, "Peak Loads and Efficient Pricing:
Comment," pp. 451-62; Williamson, "Peak-Load Pricing and Optimal Capa-
city under Indivisibility Constraint," pp. 810-27; Davidson, Price Dis-
crimination in Selling Gas and Electricity. Williamson and Davidson are
the most explicit on this point.
the theoretical models, total capacity costs are recovered during the four-
hour period, 4:00-8:00 P.M., and, in this example, only users B and C pay
capacity costs while user A pays only the variable costs of his service and
makes no payment toward the capacity costs. The conclusion is that, if
user A reduces his consumption, there is no change in the capacity costs
of the system. User A, however, uses part of the capacity during the off-
peak period, and, if users B and C were to reduce their consumption (or
cease their consumption), all capacity costs would not be eliminated. Part
of the capacity would have to be retained to continue serving user A. If
the peak user pays all the capacity costs, user A becomes a free rider.
The general conclusion of the models is based upon the assumption of
jointness, and the distribution of the cost shares is based upon the
relative strength of the various demands. However, to let some users
acquire water service without making a contribution to the capacity
costs might be deemed "inequitable."
A second distributional problem arises when the peak hours or months
do not contain a level load. Pushing the models' cost distribution to
their limits, all capacity costs would be recovered in the price of the
water used during one second or one minute of the year. The problem is
demonstrated in Figure XVIII. Given this load distribution, all the
36The exceptions to this generality are the Hirshleifer and Davidson
models. Hirshleifer treats the problem as one involving opportunity
cost. Davidson's analysis appears to be erroneous and uses average cost
pricing rather than marginal analysis. Hotelling treats the problem as
one involving social congestion costs.
capacity costs are recovered in prices charged for water used at precisely
6:00 P.M. Putting all the capacity costs into the charge for water used
at one moment in time makes the charge exhorbitant. The high price, along
with the high costs of measuring water use during each minute in the year,
makes this alternative impractical as well as one that is possibly inequi-
An Hourly Load Chart
table and inefficient (since costs of administration must be considered
in any efficiency situation). The most feasible alternative is to average
the water quantities used over the four-hour period and spread the capa-
city costs over the entire peak period. If this alternative is used,
there is a series of redistributive effects. The water user at 6:00 P.M.
pays less capacity costs than he does under the pure economic solution.
An individual user taking water at 5:45 and 6:15 P.M., but not at 6:00 P.M.,
now pays part of the capacity costs, but makes no contribution to capa-
city, according to the economic solutions. The water users at 6:00 P.M.
receive their service at below cost since they pay less than the capacity
costs they impose upon the system.
Costs Versus Ability to Pay
The requirement that users pay in accordance to the marginal costs
they impose becomes tangled in a distributional problem. The use of a price
policy consistent with marginal costs may be in conflict with the ability
to pay on the part of the user. People using water early in the morning
and during the early evening hours are the users who must bear the full
capacity costs. Users who have to use water during the peak hours because
of their work habits, and, hence, must pay the capacity costs, might not
be the same users who have the ability to pay the capacity costs; they may
be unduly burdened. The imposition of the greatest part of the capacity
costs on these users might result in a distributional effect the community
considers to be undesirable.
Elasticity of Demand and Resource Impact
It is significant to note that elasticity of demand plays an im-
portant role in determining the effects of price changes on the total
level of resource use. The impact on resource use varies inversely with
elasticity of demand. This impact is illustrated in Figure XIX where a
change in price from OPo to OP is shown to have differential effects on
the quantity produced and the size of plant. If prices are to be manipu-
lated, resource impacts are minimized when the manipulations relate to
the inelastic demand. If D1 (Figure XIX) represents the demand for
electricity, while D2 represents the demand for water, a community which
controls the prices of both services may elect to derive general revenue
from its water operations so as to minimize the negative adjustment in
resource use. It is even possible that the water utility may subsidize
Elasticity and the Impact of Price
the electric utility, subsequently leading to a net increase in the level
of resource use in the community. The same general relationships, of
course, apply in the case of the customer classifications of each utility.
Thus, price discrimination may be structured so as to increase resource
use in the local economy, or to raise general revenue with a minimum de-
crease in resource use.
Since the demand for water probably is more inelastic than the demand
for electricity, one would predict greater reliance on water utilities for
general funds than on municipally owned electric plants. Data for such
comparisons are not available. However, tradition generally favors pro-
fitable electricity operations and frowns on the sale of water under a
value-of-service principle. These practices probably are explained better
in terms of historical patterns in the institutional development of the
two types of utilities.
It should also be noted that although resource impacts are small, in
the presence of demand inelasticity, income effects may be large. Thus, a
serious equity problem arises when the utility takes the easy way out by
raising the price of the commodity which is in inelastic demand. If those
users who have inelastic demands also happen to have low incomes, a policy
justified in terms of minimizing resource effects may lead to unacceptable
This chapter has attempted to demonstrate that the economic models
have made significant inroads into the problems of pricing and invest-
ment in the cases of firms faced with a peak load problem. Some of the
solutions have combined marginal cost pricing with the vertical sum-
mation of demand functions to handle the problem of time jointness be-
tween the peak and off-peak demands. However, these theoretical models
have been weak on some points such as the use of discontinuous marginal
cost functions, which, in the case of Hotelling and Hirshleifer, leads to
a shifting back and forth between operating costs, opportunity costs, and
congestion costs along the same cost function. The strict application
of capacity costs to the peak user leads to rate structures which can
either be too high to be practical, or else permit one user to acquire
water service at prices below marginal cost while other users pay rates
in excess of marginal costs. It is significant to note that the Davidson
solution, which was intended to be a marginal cost pricing solution,
appears to be a full cost distribution resulting in average cost pricing.
The basic problem in the models is the elimination of the distri-
butional problem. Although the rates suggested may conform to efficiency
criteria, the models have developed rate structures which might have
undesirable distributive effects. The models do not take into considera-
tion the distributional aspects of their efficiency solutions. The
impact becomes significant because a conflict between efficiency pricing
and distributional equity requires trade-offs. The failure to evaluate
the distributional impacts requires more analysis to make the theoretical
models directly applicable to the problems faced by a municipally owned
PRACTICAL DESIGN OF WATER RATES
This chapter describes, highlights, and evaluates those concepts
and techniques that are currently employed in the practical design of
water rates. It assesses the implications of current practice for the
efficient utilization and development of water facilities. A develop-
ment of models which attempt to describe these water utility practices
is presented in Chapter V.
The major criteria employed in the practical design of utility
rates are (a) the cost-of-service, (b) the value-of-service, and (c)
competition. Properly interpreted, these criteria represent an effort
to determine rates in some relation to the forces of supply and demand.
Since there is no near substitute for water, at least within any realis-
tic range of prices, it is generally conceded that water supply must be
provided under monopoly conditions in order to avoid duplication and to
capture economies of scale. Therefore, the influence of competition on
demand elasticity is not a significant rate-making factor. However, com-
petition may play a significant role in determining prices at the exten-
sive geographical margin in the case of disputed or overlapping territories
1For an excellent discussion of these and other rate-making criteria
see: Eli Clemens, Economics and Public Utilities (New York: Appleton-
Century, 1960), pp. 247-369; D. Philip Locklin, Economics of Transpor-
tation (6th ed.; Homewood: Richard D. Irwin, Inc., 1966), pp. 130-57.
and, in some instances, where wells provide an alternative source of
supply. Notwithstanding these possibilities, competition is not a
standard factor in deliberations concerning the design of water rates.
Similarly, value-of-service seems to play a very limited role in the
design of water rates although it may be very important in actual prac-
tice as price discrimination takes place. High rates charged on the
basis of value-of-service violate the tradition that the price of water
should not reach a monopoly level. Low rates to certain users are
usually justified in terms of an "out-of-pocket" cost criterion.
Regardless of the rationale by which competition and value-of-ser-
vice are excluded, the literature on water rate design is almost entirely
devoted to a "fully allocated cost-of-service" criterion. In this re-
spect, the criteria for water rate design differ from those used in the
case of most utility services. It can also be argued that the models
developed by utility practitioners fail to satisfy their own criteria
for equity and efficiency.
The Cost Allocation Technique
The Bias Towards Costs
The bias towards full cost-of-service criteria in water utility rate-
making is related to a tradition which views cost as a criterion of both
efficiency and equity. According to the literature, a rate structure is
"equitable" if each user pays in accordance with the costs assignable to
The total revenue requirements imposed upon a municipally owned
utility are oriented toward the recovery of all costs. Indeed, the
major criterion of efficiency and success relates to the ability of
the rate structure to recover costs. To be financially viable and
"efficient" (according to current thinking), the municipally owned
utility should recover operating and maintenance costs, interest and
amortization of the public investment, reserves to provide distribution
mains, meters and meter servicing, and a payment to the community's
The general fund contribution is usually viewed as a cost payment
to the city in lieu of taxes. Obviously, the revenue requirements are
entirely cost oriented. Demand factors are simply not discussed in the
2William L. Patterson, "Practical Water Rate Determination," Jour-
nal of the American Water Works Association, LIV (August, 1962), pp.
905-6; Louis R. Howson, "Review of Ratemaking Theories," Journal of
the American Water Works Association, LVIII (July, 1966), p. 855;
Charles W. Keller, "Design of Water Rates," Journal of the American
Water Works Association, LVIII (March, 1966), p. 296; Jerome W. Milli-
man, "The New Price Policies for Municipal Water Service," Journal of
the American Water Works Association, LVI (February, 1964), p. 127;
Staff Report, "The Water Utility Industry in the United States,"
Journal of the American Water Works Association, LVIII (July, 1966),
E.D. Bonine, "Making a Water Utility Solvent," Journal of the Ameri-
can Water Works Association, XLV (May, 1953), p. 457.
This breakdown appears to be the generally accepted criteria for a
utility's revenue requirements. See: Patterson, "Practical Water Rate
Determination," p. 904; Howson, "Review of Ratemaking Theories," p. 850.