,tus
Environmental Protection
Office of Health and
Ecological Effects
Washington DC 20460
EPA-600/5-79-001d
February 1979
Research and Development
SEPA
Methods Development
for Assessing Air
Pollution Control
Benefits
Volume IV,
Studies on Partial Equilibrium
Approaches to Valuation of
Environmental Amenities
-------
OTHER VOLUMES OF THIS STUDY
Volume I, Experiments in the Economics of Air Pollution Epidemiology,
EPA-600/5-79-00la.
This volume employs the analytical and empirical methods of economics to
develop hypotheses on disease etiologies and to value labor productivity and
consumer losses due to air pollution-induced mortality and morbidity.
Volume II, Experiments in Valuing Non-Market Goods; A Case Study of
Alternative Benefit Measures of Air Pollution Control in the South
Coast Air Basin of Southern California, EPA-600/5-79-001b.
This volume includes the empirical results obtained from two experiments
to measure the health and aesthetic benefits of air pollution control in the
South Coast Air Basin of Southern California.
Volume III, A Preliminary Assessment of Air Pollution Damages for
Selected Crops within Southern California. EPA-600/5-79-001c.
This volume investigates the economic benefits that would accrue from
reductions in oxidant/ozone air pollution-induced damages to 14 annual
vegetable and field crops in southern California.
Volume V, Executive Summary, EPA-600/5-79-001e.
This volume provides a 23 page summary of the findings of the first four
volumes of the study.
This document is available to the public through the National Technical
Information Service, Springfield, Virginia 22161.
-------
EPA-600/5-79-001d
February 1979
METHODS DEVELOPMENT FOR ASSESSING
AIR POLLUTION CONTROL BENEFITS
Volume IV
Studies on Partial Equilibrium Approaches to
Valuation of Environmental Amenities
by
Maureen L. Cropper, William R. Porter and Berton J. Hansen
University of California
Riverside, California 92502
Robert A. Jones and John G. Riley
University of California
Los Angeles, California 90024
USEPA Grant #R805059010
Project Officer
Dr. Alan Carlin
Office of Health and Ecological Effects
Office of Research and Development
U.S. Environmental Protection Agency
Washington, D.C. 20460
OFFICE OF HEALTH AND ECOLOGICAL EFFECTS
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY'
WASHINGTON, D.C. 20460
EpA-RTF LIBRARY
-------
DISCLAIMER
This report has been reviewed by the Office of Health and Ecological
Effects, Office of Research and Development, U.S. Environmental Protection
Agency, and approved for publication. Approval does not signify that the
contents necessarily reflect the views and policies of the U.S. Environmental
Protection Agency, nor does mention of trade names or commercial products
constitute endorsement or recommendation for use.
11
-------
PREFACE
The research studies presented in this volume emphasize some factors
that are not completely treated in previous volumes. Most of the indepen-
dent studies presented here tend to qualify the results of the experimental
procedures set forth in earlier volumes. Each of them is therefore worthy
of detailed attention.
111
-------
ABSTRACT
The research presented in this volume explores various facets of the
two central project objectives (the development of new experimental tech-
niques for measuring the value of impovements in environmental amenities;
the use of microeconomic methods to develop hypotheses on disease etiologies,
and to value labor productivity and consumer losses due to air pollution-
induced mortality and morbidity that have not been given adequate attention
in the previous volumes. The valuations developed in these volumes have all
been based on a partial equilibrium framework. W.R. Porter considers the
adjustments and changes in underlying assumptions these values would require
if they were to be derived in a general equilibrium framework. In a second
purely theoretical paper, Robert Jones and John Riley examine the impact
upon the aforementioned partial equilibrium valuations under variation in
consumer uncertainty about the health hazards associated with various forms
of consumption.
Two empirical efforts conclude the volume. M.L. Cropper employs and
empirically tests a new model of the variations in wages for assorted
occupations across cities in order to establish an estimate of willingness
to pay for environmental amenities. The valuation she obtains for a 30
percent reduction in air pollution concentrations accords very closely with
the valuations reported in earlier volumes.
The volume concludes with a report of a small experiment by W.R. Porter
and B.J. Hansen intended to test a particular way to remove any biases that
bidding game respondents have to distort their true valuations.
All of these studies tend to qualify the results of the experimental
procedures discussed in earlier volumes. Further research will require:
(1) an adequate specification of the mobility decision in response to de-
graded air quality; (2) consideration of relative price changes not directly
related to air pollution as set forth in Chapter II and verified by Porter;
and (3) how consumers evaluate a multitude of risks simultaneously, both in
eating habits and pollution exposures where their economic and physical
looses are uncertain.
IV
-------
CONTENTS
Abstract iv
Figures . . . vii
Tables viii
Chapter I Introduction 1
Chapter II Public Goods Decisions Within the Context of a General
Competitive Economy 2
2.1 Partial Equilibrium Procedure 12
2.2 Bidding Procedure for a General Economy 16
2.3 Conclusions 22
2.4 Recommendations 22
Footnotes 24
References 25
Chapter III The Value of learning about Consumption Hazards ... 26
3.1 The Value of Information 26
3.2 Marshallian Analysis 28
3.3 Logarithmic Utility Functions 31
3.4 General Utility Functions 33
3.5 The Value of Imperfect Information 36
3.6 Information and Price Adjustment 39
3.7 Information About Product Quality with Negative Social
Value 40
3.8 Urban Location and Land Values with Environmental
Hazards 43
3.9 Optimal Urban Location 48
3.10 Uncertain Environmental Quality and the Prospect of
Better Information 51
3.11 Precautionary Response to the Prospect of Information 53
3.12 When Learning Prospects Do Not Affect Current Actions 54
3.13 Utility Affected by Accumulated Consumption 55
Footnotes 62
References 63
Chapter IV The Valuation of Locational Amenities: An Alternative
To the Hedonic Price Approach 64
4.1 An Equilibrium Model of Urban Location 65
4.2 Empirical Specification and Estimation of the Model . 72
-------
4.3 Empirical Results 75
4.4 Conclusion 84
Chapter V Valuation Revealing Guesses: A Report on the
Experimental Testing of a Non-Market Valuation . . 86
5.1 Description of the Experiment-in-Guessing 86
5.2 Incentive Structure of the Proposed Public Good
Guessing Procedure 91
VI
-------
FIGURES
Number Page
2.1 The Non-Binding Bid . 19
3.1 The Value of Information 30
3.2 Urban Location and Land Values 45
3.3 Homo the tic Preference Case 46
3.4 Uncertainty and Better Information 52
3.5 The Case of No Speculation 52
-------
TABLES
Number Page
3.1 The Value of Perfect Information as a Percentage of
Income ..... 34
4.1 Estimated Labor Supply Functions . 76,77
4.2 Labor Supply Functions, of Blue-Collar Workers 80
4.3 Labor Supply Functions of .White-Collar Workers 81
4.4 Valuations of Environmental Amenities 83
5.1 Selected Values of R (a) 89
vnx
-------
CHAPTER I
INTRODUCTION TO VOLUME IV
The research presented in this volume explores various facets of the
two central project objectives (the development of new experimental tech-
niques for measuring the value of improvements in environmental amenities;
the use of microeconomic methods to develop hypotheses on disease etiologies,
and to value labor productivity and consumer losses due to air pollution-
induced mortality and morbidity that have not been given adequate attention
in the previous volumes. The valuations developed in these volumes have all
been based on a partial equilibrium framework. W.R. Porter considers the
adjustments and changes in underlying assumptions these values would require
if they were to be derived in a general equilibrium framework. In a second
purely theoretical paper, Robert Jones and John Riley examine the impact
upon the aforementioned partial equilibrium valuations under variations in
consumer uncertainty about the health hazards associated with various forms
of consumption.
Two empirical efforts conclude the volume. M.L. Cropper employs and
empirically tests a new model of the variations in wages for assorted occu-
pations across cities in order to establish an estimate of willingness to
pay for environmental amenities. The valuation she obtains for a 30 percent
reduction in air pollution concentrations accords very closely with the val-
uations reported in earlier volumes.
The volume concludes Xi>ith a report of a small experiment by W.R. Porter
and B.J. Hansen intended to test a particular way to remove any biases that
bidding game respondents have to distort their true valuations.
All of these studies tend to qualify the results of the experimental
procedures discussed in earlier volumes. Further research will require:
(1) an adequate specification of the mobility decision in response to de-
graded air quality; (2) consideration of relative price changes not directly
related to air pollution as set forth in Chapter II and verified by Porter;
and (3) how consumers evaluate a multitude of risks simultaneously, both in
eating habits and pollution exposures where their economic and physical
losses are uncertain.
-------
CHAPTER II
PUBLIC GOODS DECISIONS WITHIN THE CONTEXT
OF A GENERAL COMPETITIVE ECONOMY
by
William R. Porter
The purpose of this paper is to analyze the problem of public goods de-
cision-making within the context of a general competitive economy for pri-
vate goods. It is related to, but quite different from, recent works on the
theory of value in economies with public goods.!_/ The. focal point of those
works is the theoretical relationship between a Lindahl equilibrium and the
core or Pareto optimum. Here we deal with the more mundane matter of what
is involved in making a public goods production decision that will move the
economy from its current equilibrium allocation to one that is Pareto super-
ior. The theoretical techniques used are similar to allocation techniques
for a planned economy,^/ however, the situation differs because private goods
allocation he;re is accomplished in competitive markets.
There are two major types of problems involved in public goods decisions
that are not encountered in private goods decisions. The first is to deter-
mine the proper concept of public good valuation, since the market does not
provide one as it does in the case of private goods. The second is to ob-
tain correct information about people's preferences concerning public goods
in order to use the chosen valuation concept. Again the market normally does
not provide this information, and the individuals usually have strong incen-
tives to conceal or misrepresent their preferences.
The two problems are present when dealing with any public good (whether
it is air pollution, public health, or national defense), therefore, although
we are primarily interested in questions of environmental quality, the ana-
lysis and discussion will be presented in terms of an abstract public good.
The two problems are examined separately beginning with the determina-
tion of an appropriate valuation concept and a method of using that concept
for decisionmaking when there is no problem of incorrect revelation of pref-
erences. The framework for analysis is a general competitive economy model
with public goods, but the ultimate object is to obtain results that will be
useful in making real decisions on public goods allocation.
Many of the currently used concepts and methods of applied cost-benefit
analysis have their theoretical foundations in partial equilibrium models.
Therefore, it is quite possible that their use in a general economy having
interactions among markets can lead to misallocation problems.
It has long been recognized by practitioners of cost-benefit analysis
that the public good decision will have secondary effects on related markets
-------
therefore rendering the partial equilibrium methods inappropriate. However,
this has not led to the development of general equilibrium methods for sev-
eral reasons.
1. Many of the public good projects are small compared with the size
of the overall economy, and therefore the secondary effects are
thought to be small by comparison.
2. The possible complexity of a method that would try to model all
the general equilibrium interactions would be unmanageable for
applied work.
3. The tendency to separate the calculation of project benefits from
those of project costs makes it seem that public good decisions
deal more with the production of a sealer called net surplus ra-
ther than with the redistribution of vectors of commodities.
4. And among economists who have been interested in general economies
with public goods and externalities, there has been an almost ex-
clusive interest in the problems of existence of a competitive
[Lindahl] equilibrium and its optimality properties, rather than
in the problems facing the public decisiqnmaker of how to move
from a non-optimal equilibrium to one that is Pareto superior.
This study uses the theoretical framework of a general competitive econ-
omy with public goods, however, the ultimate purpose is to obtain implica-
tions that will be useful in applications to real-world decision problems.
We will look for ways in which the use of a general economy approach will
yield results that are superior to the partial equilibrium methods. There-
fore, efforts will be made to identify the types of errors that can arise
when strictly partial equilibrium valuation methods are used in a general
equilibrium economy. We will also propose ways in which the partial equili-
brium methods can be modified in order to minimize the errors that are pro-
duced due to general equilibrium adjustments in the economy.
Before beginning the development of the basic model, we present the
following example to illustrate the type of misallocation that can result
from using partial equilibrium valuation measures in a general equilibrium
context.
In a city plagued with air pollution, the property values in areas that
are relatively free from pollution are quite high. The city government is
considering a project that will uniformly reduce the average pollution levels
throughout the city. It bases its acceptance of the project on whether the
sum of people's valuations of the proposed pollution reduction exceeds the
known cost of the project. The project is accepted, and the air pollution is
reduced. After the pollution has been cleaned up, there is a general read-
justment in property values resulting in large losses for the owners of the
property that was previously "relatively free from pollution." These areas
now have lower levels of pollution than before but they are not relatively
so desirable. In view of the property value losses, these owners wish that
the project had not been approved. If they could have anticipated the price
changes that have occurred then their valuations would have been much lower
and the project may not have been accepted.
-------
The problem of unanticipated price changes due to the public good de-
cision is more troublesome than is generally recognized for the folloxving
reasons.
1. It might be thought that the individuals could take the possibil- -
ity of price changes into consideration when they evaluate the
proposed public good project, however, there is really no way for
the individual to do this since the new equilibrium prices after
the project is completed depend on complex interaction of produc-
tion technology and consumers' preferences which cannot be known
by all individuals. Each person may be able to make a rough guess
concerning the new prices, and that might reduce, but certainly
would not eliminate, the possibility of misallocation due to im-
perfect price anticipation.
2. It is tempting to think that the problem is simply one of distri-
bution where the losses of some are more than offset by the gains
of others, and if the net surplus were appropriately redistributed
then everyone would be better off than before. Unfortunately,
movements from one general equilibrium to another are not so nicely
behaved. It is entirely possible that even though the total ap-
parent net surplus of the project, measured at the old equilibrium,
is positive, the realized net surplus after the new equilibrium is
reached is negative. Indeed, it is possible that everyone over-
valued the public good project by assuming he could trade at the
old prices.
3. The problem is not just one of using local measures of valuation
for discrete changes. The; difficulty is present even when dis-
crete valuation measures are used. On the other hand, if the pro-
posed public project is infinitesimal in size then the problem
disappears.
In this air pollution example, it is important to note that the problem
cannot be taken care of by using an estimate of the demand function for pro-
perty. The property price change is simply used as an example, and it is
important to realize that many other prices will change in a general adjust-
ment. Furthermore, the estimate of the demand for property function will
normally use data from a single equilibrium (in a cross-sectional study)
which cannot reveal information about changes from one equilibrium to an-
other.
To illustrate the problems of determining the proper level of public
good production we examine a competitive market economy having two private
goods and one public good. There are I consumers i =1, . . .,1, who each
have constant endowment flows u> = (w , to ) of the two private goods and
i il ±2
strictly quasi-concave utility functions u^(x.»z) defined on their own con-
sumption of private goods x. = (x ,x ) and the amount available z of the
public good. The level of public good z is produced according to the pro-
duction function z = f(y), where y is input of good 1.
-------
Initially we assume that the government has perfect knowledge of the
current market prices of private goods and the preferences of the individual
consumers and is charged with the task of collecting the input of good 1
from the consumers in order to produce the proper level of the public good.
(Note that the government's problem here is different than that of a central
planner in that the private goods prices are determined in the market and
are taken as given by the government).
We assume that the government's problem begins at a general equilibrium
[p>(x.),z]. Even though the level of the public good is not market deter-
mined and would not normally be thought of as a component of the general
equilibrium, we include it here since it will be changing along with changes
in the equilibrium prices p and allocation of private goods (x ). The ob-
i
ject is to specify a decision procedure that will use the collection of in-
puts of good 1 from consumers (taxation) and the production of the public
good to bring about movement along a Pareto improving path toward a Pareto
optimum. (Note that the tax used here is simply a flow of good 1 that is
taken from each consumer independent of his own actions. In that sense it is
a lump-sum tax) .
A Continuous Path Method
In this simple model having only a single public good, the government's
decision will deal only with the taxation problem since all of the proceeds
of taxation must go into the single activity of public good production. The
government's decision will be based on the individual marginal valuations of
the public good defined as follows. At the equilibrium [p,(x.),z], person
i's marginal valuation of the public good in terms of good 1 Is:
V.(x.,z)=
11
(x. ,z)
? - i - = MRS
i , . z for 1
i(Vz)
(2.1)
The marginal social valuation of the public good is defined as:
V(z) =
(2.2)
The social cost of z units of the public good is:
C(z) = f (z), where f denotes the inverse
function of f.
The marginal social cost of the public good is:
C'(z) = [f'^z)]'
(2.3)
(2.4)
Let s. denote the total tax, in units of good 1, that person i is
charged, and let Y. be a non-negative weight that is assigned to person i,
where Zy . = 1- The rate of change in the level of the public good is based
1 1
on the magnitude of [V(z) - C'(z)], which is called the net marginal social
5
-------
valuation of the public good. The rate of change is given by:
z = = a[V(z) - C'(z)], where a > 0. (2.5)
at
Each person i's tax share is changed in such a way that he receives the
share y. °f the net social surplus resulting frcm the change. Therefore,
.
ds.
.
- = v - y [V(z) - C'(z)], where y > 0 for all i,
,
dz i i i
and I = 1. (2.6)
i
Summing over all individuals, we see that the sum of the tax changes is just
sufficient to provide the necessary input C'(z) of good 1.
ds.
L - - = Ev - [V(z) - C'(z)]Zy
.dz . i . i
11 i
= V(z) - V(z) + C'(z) = C'(z). (2.7)
No person is made worse off by the change, since each person's tax change is
less than his own marginal valuation. Therefore, the procedure is contin-
uously Pareto improving as long as the net marginal social valuation is non-
zero.
The time rate of change in person i's tax is:
ds.
= ^ . ££ = afv.[V(z) - C'(z)] - Y.[V(z) - C'(z)J ]. (2.8)
i d z d t i . i
Equations (2.5) and (2.8) completely describe the time path of govern-
ment action with respect to allocation in the economy. However, other real-
location is continuously occuring outside the domain of the government. As
the level of the public good changes and taxes change, the consumers have
incentive to adjust their private goods bundles through trade. Therefore,
the government's actions are accompanied by continuously changing private
goods prices. This fact is extremely important because if we think of an
economy where private goods trading does not occur as the government changes
taxes and the public good level, then the economy would not, in general, be
at a Pareto optimum once the reallocation defined by (2.5) and (2.8) was
complete.
The method of continuous government allocation in a three good economy
can be easily generalized to more complicated economies having more private
and public goods and a more general type of public good production function.
However, the model just described is adequate to illustrate the main fea-
tures involved in an optimal procedure of public good production and finan-
cing.
The continuous procedure summarized in equations (2.5) and (2.8) repre-
sents an extreme theoretical form for which we can guarantee that the econ-
omy will move in a continuously Pareto improving direction, but the model is
very far from being applicable even in a real 3-good economy. It is impor-
tant to note the massive informational and decisionmaking demands on both
-------
the government and the consumers in order to carry out the procedure.
a. The government must have continuous perfect information about each
person's marginal valuation of the public good and about the mar-
ginal productivity of the public good production function.
b. The consumers must be continually in the private goods market of-
fering and trading in order that the market can continuously find
its new equilibrium. They must also be kept continuously up to
date on their latest tax assessment so that they will know how
much they have to trade.
The object is to develop procedures that are more applicable, but that
will retain the optimality properties of the foregoing procedure. We will
continue to use the model of a 3-good economy with public good production in
order to examine the general equilibrium and Pareto optimality features of
the problem. (It is clear that the Pareto optimality feature of public good
production cannot be dealt with in a partial equilibrium framework, even.
though writers often use the terminology of general welfare economics when
dealing with benefit-cost in partial equilibrium analysis).
The first step toward making the procedure applicable is to discretize
the decision steps, since no real world decision procedure in economics can
be carried out in a truly continuous fashion. In order to focus on the pro-
blems that are strictly associated with the discreteness of the procedure we
will retain the assumption that the government has perfectly knowledge of
people's valuations.
The use of a discrete decision procedure requires some additional defi-
nitions as follows. Beginning at some economy equilibrium [p,(x ),z], the
i
government must decide on some discrete increment q in the public good that
it will propose for production. Once the consumers are informed of the pro-
posal q they can form their own valuations of q in one of several ways whose
merits will be discussed below.
Since good 1 is used for input into the production of any changes in z
we will state all valuation in units of good 1.
C.V. Measure of Valuation
One of the most common ways of measuring person i's valuation of the
proposed increment of the public good is to determine the maximum amount of
good 1 he would be willing to give up in order to have the increment q pro-
duced. This measure is called (in certain contexts) the compensating varia-
tion (CV) associated with increment q. However, CV is usually defined in
terms of a fixed nominal income and known prices, therefore it does not lend
itself well to use in a general equilibrium context [see K-G. Maler, p. 126].
Under two different assumptions we consider the following CV measures.
Fixed Price Assumption
vP = [Ax |h.(x -Ax ,x ,z+q,p ,p ) = h.(x ,x ,z,p ,p )] (2.9)
i il i il il i2 12 i il i2 12
-------
where h is the maximum utility function:
i
. i .
h. (oo > 0) >z,p ,PO) = max u (x..»x ,z)
s.t. p x. + p x. = p u + p 0) . (2.10)
v measures the maximum amount of good 1 that person i would be willing
i
to give up if he knew that after the increment q were produced he would be
able to trade in the private goods market at the current prices p and p .
The problem with this measure is that the prices at which he will be able to
trade after q is produced (if indeed it is produced) are not known at the
time when y. is needed. By using current prices as the ones he will be able
to trade at, he may overstate his valuation and end up at a utility level
that is lower than his present level. This would destroy the Pareto-impro-
ving property of the allocation procedure. One way of avoiding this is to
use the following conservative approach.
Fixed Utility Assumption
v = [Ax u (x -Ax ,x ,z+q) = u (x..,x ,z)] (2.11)
i il il il i2 il iz
This measure assumes that the consumer will not be allowed to trade af-
ter he is taxed and the project is produced. Of course, if later he is able
to trade then he xjill only do so if he is able to move to a preferred posi-
tion. Therefore this method can never overstate the person's valuation of q,
but it car understate the true valuation. An allocation procedure that is
based on this measure will move only to Pareto superior points, but it may
fail to move to some points that are Pareto superior.
E.V. Measure of Valuation
A frequently discussed measure of public good valuation is the minimum
amount that a consumer would have to be given to make him as happy as he
would be if he had the increment in the public good. The two EV measures
that correspond to the CV measures given above are:
PP = [Ax., h.(x. +Ax ,x ,z,p ,p ) = h.(x ,x ,z+q,p p )] (2.12)
i il i il il i2 12 i il i2 rl 2
u. = [Ax u1(x. +Ax ,x ,z) = u (x...,x ,z+q)] (2.13)
i il il il i2 il i2
Although the EV measures may have some theoretical interest in a partial
equilibrium framework, it is clear from the expressions (2.12) and (2.13) a-
bove that they are not relevant to the type of public good allocation deci-
sion under consideration here. In order for the government to know whether
to produce the increment q, it needs to know if the required resources for
that production can be obtained x^ithout making someone worse-off. The dif-
ficulty x>/ith the EV measures is that they ask the consumers to compare txco
allocations that are technologically infeasible. The two allocations, as
seen in (2.12) and (2.13) are [(x +Ax. ,x. J,z] and [(x ,x ),z+q]. It is
il il i2 il i2
-------
clear that if the competitive allocation [(x ,x ),z] is both feasible and
il i2
efficient, then the two allocations compared in the EV measure are either
infeasible or inefficient except when Ax. =0, for all i, and when q=0. This
fact renders the EV measures useless for decisionmaking in a general equili-
brium context. Therefore we will use only CV measures in the following pro-
cedures .
Using one of the CV measures of valuation of the proposed increment q
in the public good, the government decision procedure in the discrete frame-
work is described below.
The marginal social valuation of the public good in the discrete case is:
V(z,q) = 2X (2.14)
i
The marginal social cost associated with a change from z to z+q of the pub-
lic good is:
AC = C(z+q) - C(z) (2.15)
Therefore the net marginal social valuation is [V(z,q) - AC], and the
government's decision rule will be to produce the increment; q if [V(z,q) -
AC] > 0, and to not produce it otherwise. If it is to be produced then the
necessary resources AC of good 1 are collected from the consumers according
to the following formula:
As. = v. - y. [V(z,q) - AC] (2.16)
111
where As. denotes the discrete change in person i's total tax and y. is per-
son i's share of the net surplus, where Ey. = 1 and -y. > 0, i = 1, . . . ,1.
i
Summing the tax changes over all consumers we see that:
X As = AC (2.17)
i
i
which is the needed amount of good 1 for input to produce the increment q.
Features of the Discrete Decision Process
Once the government has chosen which valuation measure to use, the pro-
cess just described can be applied, and it is clearly more applicable than
the previous continuous procedure since it will need only a finite amount of
information for each proposed incremental ciiange in the public good. The
method works equally well for proposals where q < 0, therefore it can also
be used to consider reductions in the public good level. Unfortunately the
method has several weaknesses that detract somewhat from its greater degree
of applicability. They are:
a. The procedure will, in general, stop before reaching a Pareto opti-
mum, for any given q.
b. The procedure may cause reallocations that will make some consumers
p
worse-off if the valuation measure v is used. Therefore the pro-
cedure would not be Pareto-improving.
9
-------
Both of these weaknesses can be eliminated through modification of the
procedure, however, the modifications reduce the applicability by increasing
the informational demands.
Problem (a) can be resolved by changing the size or the sign of q when-
ever a stop is encountered. As q becomes smaller the procedure requires
more information per unit change in the public good, however, the government
could make some judgment about how close is "close enough" to a Pareto opti-
mum, in view of the cost of information for each decision.
Problem (b) can be eliminated by using v rather than v as the valua-
tion measure. The difficulty with using v 5 as mentioned earlier, is that
it systematically understates the person's true valuation of the public good,
given that there will be some trading possibilities in private goods if the
project is approved. The valuation measure v is based on the assumption
that the consumers will not engage in private goods trade after the public
good decision. To guarantee that the understatement is not preventing the
detection of a possible Pareto improving move, the size of q must be reduced
whenever a stop is encountered in order to see if there remain any possible
Pareto improvements. The reduction in q increases the information require-
ments of the procedure.
A separate approach to this problem is to attempt to get accurate esti-
mates of what the equilibrium prices will be if the size q proposal is ap-
proved. This is a difficult task since the prices will depend on market in-
teractions that cannot be theoretically calculated without knowing all con-
sumers' utility functions. Such information is equal in order of magnitude
to that required in the continuous procedure. However, if rather than doing
theoretical calculations of prices we allow a contingent claims market to
operate then each consumer not only gets an accurate estimate of the future
prices if the project is approved but he is able to hedge completely against
possible loss due to price changes. The claims would be on private goods
and they would be contingent on the approval of the increment q. Each per-
son would have (x -v >x ) units of contingent goods 1 and 2 to trade with,
il i ±2
and would alter their valuations v as the contingent goods market moved to-
i
ward equilibrium. Once the contingent go'ods market reached an equilibrium
the government could use the already described decision criteria to make the
project approval and taxation decisions. The procedure would be guaranteed
to move only to a Pareto superior allocation. If the project were not ap-
proved then the contingent claims would not be binding. Although this meth-
od requires the functioning of a competitive market for contingent claims,
it uses an essentially decentralized procedure to determine accurate price
estimates. It will be seen later that this type of contingent market can be
very useful in applied procedures where the public good project is relatively
large.
So far we have assumed that the government is able to get the consumers
to reveal their correct valuations of public good changes. Unfortunately,
whenever the consumers understand how their individual valuations are to be
used for taxation purposes they have incentive to misrepresent their true
10
-------
valuations. This problem is widely referred to as the "free-rider" problem,
and until recently it was thought to be unavoidable even in a purely theore-
tical model of an economy with public goods. Recent research has shown that
it is possible to provide the proper incentives for individuals to submit
accurate messages to the government concerning their true valuation func-
tions. J3/ This work is extremely important for theoretical development in
this area, however, it is very far from a form that is applicable to actual
public goods decision problems.
A different approach that also pays close attention to the individuals'
incentives is one developed by Vernon Smith and tested by him and others in
many experimental situations involving collective decisions.4_/ This approach
is not so fully developed theoretically, but it currently offers more promise
in terms of application to public goods allocation problems in both a partial
and a general economy framework. The; method uses a system of bidding to over-
come some of the distortionary effects of the free-rider problem.
In the following section we develop an extension of Vernon Smith's bid-
ding mechanism that can be used to make Pareto improving decisions concerning
public goods production in a general economy framework. The important thing
about this method is that it does not require that the government know the
consumers' preferences.
A Bidding Mechanism for Public Goods Decisions
In this section we develop an extension of Vernon Smith's Auction Mech-
anism for public good decisions to a general economy framework where private
goods are traded in competitive markets, and the public good is produced by
the government using private good inputs.
The bidding procedure developed here incorporates a market for contin-
gent claims on private goods in order to avoid the type of unanticipated
price changes that are associated with movements from one equilibrium to an-
other. The claims are contingent on the approval of the public good project.
Gambling on the. outcome of the bidding procedure (by trading current goods
for contingent claims) is prohibited since that would tend to bias people's
bids and possibly cause some people to be worse off after the project deci-
sion. By trading in the contingent claims market each individual is able to
determine the full value of his maximum willingness to pay for the public
goods, and he can then form his bids in the same manner as in the partial
equilibrium auction mechanism of Vernon Smith.
In Section 2.1 we examine the individual incentives in a partial equili-
brium bidding procedure used to approve and finance a public good project.
This procedure modifies Vernon Smith's Auction Mechanism_5/ by: (1) adding an
initial non-binding round of bidding used to determine if bidding should con-
tinue and to provide the group with an estimate of the net project surplus;
and (2) including a positive and increasing stop-probability to induce the
members to avoid a stalling strategy. Without analyzing all of the possible
strategies that individuals could use we look at the type and the strength of
the incentives that pull the group toward (or away from) a cooperative solu-
tion that is Pareto superior to the initial position. Section 2.2 develops
the bidding procedure for an economy with two private goods and one public
11
-------
good. The public good is produced by the government using private good in-
puts obtained from consumers. The nature of the price uncertainty problem
and its adverse effect on bidding decisions is explained. A market for con-
tingent claims is designed to clear simultaneously x^ith the bidding rounds
in order to overcome the problems caused by price uncertainty. Section 2.3
gives the summary and concluding remarks.
2.1 Partial Equilibrium Procedure
The purpose of the bidding procedure described in this section is tc
provide a framework within which a group can decide whether to approve the
production of a given amount of a public good. The framework is based on the
Auction Mechanism used in Smith for experiments in public good decisions.
The bidding procedure should enable the group to jointly approve and
finance the production of public good projects that have a positive net sur-
plus and to reject projects that do not. The procedure should not lead any-
one into the position of being worse off after the decision, and it should
provide the incentive and guidelines for quickly arriving at a cooperative
Pareto superior solution when one exists. Although we will deal here with
only a single discrete decision, it is clear that by using a sequence of
such decisions the group could move toward a Pareto optimum.
Individual group members indicate their support for (opposition to) a
project by submitting anonymous positive (negative) bids which establish the
maximum amounts they can be assessed if the project is approved. Project
approval occurs when the sum of the bids is at least as great as the project
cost.
The total project cost is known to all, and after each round of bidding
the sum of the bids is announced. As long as an individual's own project
valuation is greater than his bid, he favors approval of the project. There
are a finite number of bidding rounds, and if the project is not approved by
the last round then it is judged infeasible and is abandoned. All potential
gains from the project are lost if it is not approved by the last round. Mem-
bers are not allowed individually to purchase small amounts of the public
good.
If each person never bids higher than his true valuation then the method
will never approve a project that makes anyone worse off, and in particular
will not approve a project with a negative net social valuation. The proce-
dure should then be considered successful if it is able to arrive at cooper-
ative approval of projects having positive net valuations more frequently
than other methods of unanimous social choice. Such a comparison can be made
using experimental methods,6/ but cannot be done theoretically.
The fact that there is incentive for each member to keep his bid low in
the hope that others will fill in the gap and cause the project to be ap-
proved may make it appear that this procedure has not really avoided the
classic "free-rider" problem, and of course it hasn't entirely. However, it
is important to recognize that the problem is greatly changed and is dimin-
ished in strength in this framework. In a contingent bidding procedure (one
where bids are contingent on prcject acceptance) each person knows the amount
12
-------
of public good to be produced if his bid is accepted. Therefore he knows
exactly what it is that he is valuing when he forms his bid. The same thing
is not true in the case of private uncoordinated purchases of a public good
or under systems of uncontingent donations toward production of a public
good. As long as the sum of bids is less than the project cost, the incen-
tive to free ride is offset by the incentive to increase the sum toward pro-
ject approval. The strength of this incentive is diminished as one's bid
gets close to his own project valuation. In the bidding procedure each per-
son knows that he can signal a willingness to support the project without
the fear that he will be left "holding the bag" if others don't cooperate
sufficiently. Also the addition of bids for the same project corresponds to
the way in which valuations must be added to determine the group value of a
public good.
These features all tend to diminish the strength of the "free-rider"
effect within this context. The results that Vernon Smith has obtained in
experimental studies of his Auction Mechanism for public good decisions indi-
cate that the free-rider effect is indeed diminished in such a context. The
following modified auction mechanism was designed after observing the results
of experiments conducted by Smith.
Project Apprcval
Consider a group of N individuals, indexed i = 1, . . .,N, who will all
be affected by the; production of a public good project costing C. Person i
has true valuation V for the proposed project. The following bidding pro-
cedure will be followed to determine if the project will be constructed and
how much each person must pay toward the total cost C. There will be two
stages of bidding composed of a total of T+l rounds of bids. There will be
only one round of bidding in Stage I. The purpose of this round of bidding
is to determine whether or not tht: project will be considered further and to
give everyone an estimate of the net project surplus, therefore the bids will
be non-binding in terms of tax purposes.7_/
Stage I (The Non-Binding Bids)
Each person anonymously submits his initial bid b . The decision rule
for Stage I is: If zb < C, then stop bidding and abandon the project. If
If Zb > C, then proceed to Stage II.
i
The purpose of Stage II is to decide on individual payments that will
cover the total cost of the project. Each person determines his own bid of
offered support for the project knowing that if the total of the bids is not
high enough then the project may fail.
Stage II (The Binding Bids)
There will be at least one and at most T rounds of bidding in this
stage. After each round in which the total bids fall short of cost there is
a known probability that the procedure will be stopped and the project
13
-------
abandoned. The probability of this type of stop is t/T, where t = 1, . . .,T
is the number of the round. The purpose of this increasing "stop" probabili-
ty is.to provide the incentive to the group to move quickly toward a solu-
tion.8/ At round t = 1, . . .,T the decision procedure will be:
If Zb > c, then stop bidding, tax each member b - l/N(£b - C), and pro-
i duce the public good. i
If Zb <_ C and 6 =1, then post the value Zb and proceed to the next round.
i i t~
If Zb < C and 6 =0, then stop bidding and do not produce the public good.
. t t
The distribution of 6 is: P(0 = 0) = t/T, t = 1, . . .,T and
P(0t =!)=!- P(0t = 0)
The complete bidding procedure is explained to each member before round
0 of bidding.
There is no attempt made here to model completely the behavior or stra-
tegy of each individual. However, by looking at the situation from the
point-of-view of a single agent we can get some idea of the incentive struc-
ture facing him. I will argue here that each person references his behavior
to a commonly held notion of "fairness" which in this situation is defined
as an equal sharing of the apparent gains. A person does not always feel
obliged to abide by exact "fairness," and will at times attempt to get more
than his "fair" share, and at other times be willing to accept less than his
"fair" share in order to prevent the failure of the project.
Person i's true valuation of the public -good is V . During Stage I of
the bidding process he can bid any arbitrary value since he knows that he is
not accountable for his bid in terms of future taxes, and no one else will
ever know the value of his initial bid. However, he has incentive to make
his initial bid close to his true valuation V . The reason for this is that
if he overbids (i.e., bids b > V ) in an attempt to help carry the project
into Stage II then he is contributing to the overstatement of the apparent
consumer surplus (Zb - C) associated with the project. An overstated ap-
i
parent surplus will make it difficult to obtain joint approval in Stage II
even if there is a large real surplus since unless he makes his Stage II bids
greater than V (which would be foolish) then the other members must absorb
his initial overbid be]ieving that they are .getting less than their fair
share. On the other hand, if person i bids b < V in an attempt to under-
state Che apparent surplus so that he can get a larger share of the true sur-
plus when the project is approved he increases the likelihood that the pro-
ject will fail in round 0. Now it is certainly true that there may be some
overbidding in Stage I for various possible reasons, however, if there are
strong tendencies in one direction then this will result in a high proportion
14
-------
of failures in either Stage I or Scage II of the process. This high failure
rate would presumably provide the incentive to correct this type of misbici-
ding.
In Stage II person i is aware of the total apparent surplus (Eb' - C)
i
established in Stage I. If he takes this number as being the true surplus
then his fair share is l/N(F.b - C) and his corresponding fair bid is bft = b
- l/N(Ib - C). He knows that if everyone bids his fair bid that the pro-
i
ject will he exactly approved on the first round and each will obtain an
equal share of the apparent surplus. However, he may bid higher or lower
than his fair bid depending on how urgently he wants the project approved
and on what he believes that others will do. In general if he bids higher
then he is contributing to rapid project approval, and if he bids lower he is
attempting to get a larger share of the surplus while some socially benefi-
cial projects will fail.
It was mentioned earlier that the procedure is designed to enlist every-
one's support by giving each person a vested interest in the approval of the
project. There is, of course, the possibility that one of the members de-
rives his pleasure from foiling the plans of the others. There is no way
that the procedure can offset this type of behavior if the person is deter-
mined to foil every project. Whether or not this type of behavior is fre-
quent enough to cause problems for the method would most likely be brought
out; in experimental studies.
Project Size and Approval Determination
The two-stage bidding procedure can be extended to a procedure that de-
termines both the-size and approval of the public good project. This proce-
dure takes advantage of the incentives present during the first stage to ob-
tain information about the group valuation function of the public good.
Suppose that each of I members has the individual valuation function
V (Z), where Z ^ 0 is the level of the public good. Suppose that C(Z) is the
total cost of Z units of the public good. For convenience we assr.me that V
is concave with V (0) = 0, for all i, and that C is convex and increasing
with C(0) = 0.
Stage I
Each member anonymously submits a bid function b (Z), knowing that the
aggregate function £b (Z) - C(Z) will be used to determine the project size
i
to be considered for approval in Stage II. The project size Z is selected
to maximize Zbj!.(Z) - C(Z), and Z, Zb^(Z) and C(Z) are announced to all mem-
i i
bers.
15
-------
Stage II
This stage is handled exactly as in the previous procedure where Z =
the project size, ib (Z) = Zl/, and C(Z) = C.
i i
The interesting question here is whether there is incentive for the in-
dividual members to misrepresent their valuation functions V (z) in their
Stage I bid functions b (Z) . The incentive for making one's initial bid
function very close to one's true valuation function is the same as before,
however in this case since the person cannot know what project size will be
selected he is induced to bid "honest].y" over the whole range. He wants the
project to succeed in Stage I (i.e., to have the selected project to be Z ^
0), but does not want the apparent surplus to be inflated so that approval is
more difficult in Stage II.
2.2- Bidding Procedure for a General Economy
All of the previous sections rested on the assumption that people's val-
uations of a public good do not change as a result of the production of the
public good. We assumed that the valuations were in units of money that the
person is willing to give up to obtain the public good and that only money is
required for the production of the public good. Of course, in reality, the
production of a public good requires real resources which when demanded as
inputs into public good production may affect the prices of all other goods.
These price changes will alter both the money valuation and the real valua-
tion of the public good, therefore raising some serious doubts about decision
criteria that assume no changes take place. The difficulties are caused by
the fact that changes in the level of the public good are associated with a
movement from one genera] equilibrium to another, but at the time that agents
are expected to make bids on such a change they do not know the prices that
will prevail in the new equilibrium. Therefore, they are unable to know
their own maximum willingness to pay for the proposed public good, and conse-
quently they have inadequate basis for bidding. The following bidding proce-
dure incorporates a market for claims that are contingent on project approval
to provide the type of information needed by each agent. This contingent
claims market allows the group to get close to the full valuation of the pro-
posed public good and it protects each agent from ending up worse off after
project approval due to unanticipated price changes. Therefore, by using
this method the group will be more likely to find a Pareto superior solution
if one exists since the element of price uncertainty will be removed, and we
can be assured that projects will only be approved if they lead to Pareto su-
perior allocations. The method uses the incentive structure of the previous
section to induce members toward a cooperative decision. We will consider
only the problem of project approval.
General Equilibrium Method
Consider an economy with two private goods and one public good. The
public good is produced by the government using inputs of private good 1 ob-
tained from the consumers. There are N consumers, indexed i = 1, . . .,N,
16
-------
who each have a utility function u (x ,z), where x is the consumer's vector
of private goods and z is the amount of public good. The economy's initial
resources of private goods is co = ( ), and there is initially no public
0)2
good. The public good production function is z = f(y), where y is the in-
put of private good 1. There is no production of private goods, so the econ-
i y
omy resource constraint is given by Ex 4- ( ) < w.
0
i
The public choice problem faced by this economy is whether to produce
z units of the public good and if so how to distribute the taxes among the
consumers to obtain the needed input. The total input of good 1 that is
needed to produce z is denoted C = f (z). The society wants to approve
this public good project if and only if it can do so in a Pareto improving
way. The economy is assumed initially to be at the competitive equilibrium
[(x ),Q,p], where (x ) is the allocation of private goods among the consu-
mers, 0 is the current amount of public good, and p = ( ) is the equilibrium
P2
price vector. As before there will be T+l rounds of bidding indexed t = 0,
1, . . . ,T. There will be two stages of bidding consisting of the non-bind-
ing bids in Stage II. At each round of bidding a contingent claims market
will be conducted, and the bids for that round become official when the mar-
ket clears. No trading of uncontingent claims (i.e., contributing to pos-
sible non-approval of the project. He is never tempted to bid higher than V
during Stage II since if the project is approved then he will suffer a net
loss.
As t gets larger and closer to T (increasing the probability of a stop)
the persons whose bids are much lower than their valuations have strong in-
centive to raise their bids in order to increase their bids since their gains
would be small even if approval is accomplished. In this way the bidding
procedure tends to put the greatest individual pressure for bid increases on
those who are attempting to get the largest gains. It is they who have the
largest vested interests in the project's success.
Ignoring the costs associated with conducting the bidding, the process
will move only to Pareto superior points. This is true because no one will
make a Stage II bid that is higher than his true valuation. Therefore, we
know that the process will not move if there are no longer projects having a
positive net surplus. So, in this partial equilibrium sense, the process
will only move toward Pareto superior points and will not move from a Pareto
optimum. However, there is the possibility that even though there is posi-
tive net surplus associated with a project that it will not be approved since
the procedure may stop before approval is reached. It may seem wasteful that
some projects having positive consumer surplus will fail due to a stop occur-
ring before the cooperative solution is reached. However, if we imagine a
procedure where, whenever there is a positive apparent surplus in Stage I,
the Stage II bidding will continue until the group arrives at a cooperative
solution, then we see that there is almost no incentive for the individuals
to raise their bids up toward their valuations. By using a system that may
cause a loss due to non-cooperative behavior at each round we provide some
17
-------
disincentive for holding out for a "free ride." The cost is that claims
contingent on the failure of the project is allowed during the entire bid-
ding procedure. This rule is used to prevent speculation on the success or
failure of the project which might cause some members to end up worse off .
than originally. At the beginning of each round of bidding person i has x
as his initial endowment of contingent claims. His choice of contingent
i
i
claims at the end of round t is denoted u
The current contingent
claims prices are denoted p and p . Person i's bid in round t is denoted
i
b , and it represents the maximum amount of good 1 that he is willing to de-
liver to the government upon the approval of the project.
Stage I (The; Non-Binding Bid)
Stage I will consist of one round of bids used only to determine if the
project should be considered further. Since the contingent claims market in
this round (and in other rounds) is competitive we will first look at the de-
cision faced by the price taking agents. Given z, p and p , person i chooses
i l i
person i chooses a bid b and a contingent claims vector u such that:
U1(vjj,z) > u^.O) and (2.18)
U
p maximizes u (u ,z) (2.19)
subject to PI^O + P240 1 P/** - bj) + p2x*
Let b denote the bid when (2.18) is an equality. Then b is the person's
true maximum willingness to pay for the public good. In general, b is
greater than the standard measure known as the compensating variation (CV),
since the calculation of CV ignores price and trading considerations. Let
q denote the compensating variation, in units of good 1, for z units of the
public good. Mathematically, q satisfies the equation:
u1(qQ,z) = u^x1,^ (2.20)
Clearly q £ b , and except for a unique price ratio q < b . This relation-
ship is illustrated in the indifference curve diagram of Figure 2.1, where
U = u (x ,0) denotes the.-, indifference curve when there is zero public good,
and U' denotes the indifference curve at the same utility level when there
i *i
are z units of public good, q is the distance BA on the diagram, and b
is the distance CA. The slope of the line CD indicates the price ratio for
18
-------
Good 2
Figure 2 .1
The Non-Binding Bid
0
U1
0
Good 1
19
-------
contingent claims. Therefore, we see that the contingent claims market al-
lows the society to determine its full social valuation of the proposed pub-
lic good, whereas CV measure does not because it doesn't allow for possible
private goods trading. The Stage I bids become effective when the folloxjing
market clearing condition holds:
' i\
-o
0
(2.21)
The decision rule for Stage I is:
If Eb _< C, then abandon the project.
i
If Ib1 > C, then post the values C and £b , and proceed to Stage II.
i i
As in the partial equilibrium procedure each person here has some incen-
tive to give an honest bid on round 0 since he knows that his bid will not be
used to assign his tax and he has a vested interest in Stage I approval, but
he realizes that an overstated apparent surplus will cause difficulty in
Stage II approval.
Stage II (The Binding Bids)
Each person knows the value of the apparent consumer surplus established
during round 0, therefore they each have some idea of their own fair bid b u =
b1 - l/N(£bX - C) . Also, each person is aware that the "stop" probability
i
after round t is given by t/T. During round t with given values p and p
i "i
person i chooses b and p such that:
1 X
u(Mt,z) >. u (x,0), and (2.22)
"i . . i i
p maximizes u (p )Z) (2.23)
subject to p u + p y _< p (x - b ) + p x
-L -*- Z.Z. ,i,X I- tL £.
The bids are effective once the prices p and p are such that the contingent
claims market clears:
1 + (i l) = w (2.24)
t 0
Each person will bid in such a way that (2.22) is a strict inequality. The
social decision rule in round t is:
If £bt > C, then stop bidding, tax each member and produce the public good.
i
i
20
-------
If Eb _^ C and 9 = 1, then post the value Eb and proceed to the next round.
i i
If Eb _< C and 6 =0, then stop bidding and do not produce the public good.
i
The-: distribution of 0 is:
t
P(8 = 0) = t/T, t = 1, . . .,T and
t
P(8 =!)=!- P(0 = 0).
t t
This rule is exactly the same as in the partial equilibrium procedure except
that here bids and the tax are in units of good 1 rather than money. If the
project is approved in round t, then person j's holdings of the two goods
after taxes is:
- C)
This means that the contingent claims become real claims and if the sum of
the bids is greater than the cost of producing T. units of public good, then
the households share the excess. Once the project has been approved, then
the trading of private goods can resume.
It is clear from the description of the procedure that a project will
only be approved if it leads to a Pareto superior allocation. Therefore, the
procedure does guarantee that no one will be hurt as a result of unanticipa-
ted price changes.
Even though the general economy procedure was explained using a simple
3-good economy, it should be clear that there would be no theoretical pro-
blems involved in going to economies having n private goods, m public goods,
and more general production sets for the public goods. The main feature that
was introduced in order to use the partial equilibrium technique in a general
economy was the market for contingent claims.
It is important to recognize the way that the contingent claims market
is being used in this procedure to avoid a rather difficult problem concern-
ing price expectations. The contingent claims market artificially creates a
close approximation to the real market that will exist once the taxes are
collected and the public good produced. With this market the agents are able
to have accurate price expectations and therefore to accurately calculate
their valuations of the public good. By prohibiting trades involving current
(uncontingent) goods we avoid all of the problems caused by mixing people's
preferences with their subjective probabilities that the project will be ap-
proved. Allowing only trade of contingent commodities once the project has
been proposed separates the two types of markets so that gambling on the out-
come of the project approval decision through trade is avoided. If this were
allowed then the nature of the process would be altered considerably.
21
-------
The use of contingent claims markets tends to conceal a severe problem
in the applicability of the general economy procedure. We have assumed that
the contingent claims market will clear simultaneously with each round of
bidding without recognizing the substantial difficulty in finding the market
clearing equilibrium in practice. Economists usually do not dwell on the
difficulties involved in attaining the competitive equilibrium, so I will
not do so here. However, in any application of this technique the problem
would have to be dealt with.
2.3 Conclusions
By framing the public good decision within a general equilibrium model
we are able to see clearly some of the problems associated with the use of
the standard partial equilibrium techniques. Some of the features that are
brought out in this framework are the following:
1. It emphasizes the fact that public good production is a realloca-
tion process that moves the economy from one competitive equili-
brium to another. This is especially important when dealing with
projects that are not infinitesimal in size, since the discrete
reallocation will lead to price changes that cannot automatically
be anticipated. On the other hand, the partial equilibrium method
views the government as a type of Marshallian firm whose actions
will not have any effect on the rest of the economy.
2. The framework allows us to see clearly why the application of par-
tial equilibrium methods of cost-benefit will not lead to alloca-
tions that are Pareto superior if the project is of discrete size.
3. The approach emphasizes the logical impossibility of separating
costs from benefits and valuation from taxation and trade.
4. The inappropriateness of the EV measure for use in public goods
decisions is made obvious by the technical infeasibility of the
allocations it compares.
5. Changing the size of the project proposals brings out the tradeoff
between information and a3locative efficiency within this framework.
2.A Recommenda t ions
Based on the models developed in this report, there are several recom-
mendations that can be made for avoiding the types of distortions caused by
either unanticipated price changes or "the free-rider effect." They are:
1. Although it may not be practical to hold contingent markets for all
commodities, it is conceivable that the government could organize
markets for those goods that are highly likely to undergo substan-
tial price changes. In the air pollution example, it would be use-
ful to have a contingent market for real estate. Another likely
candidate for contingent trading is any major input into the public
good production. Thus, if the proposed project is to reduce air
pollution by requiring (or prohibiting) the use of certain types of
22
-------
fuels, then the government could organize contingent markets for
various sources of energy among which there may be substantial
substitution. The sponsorship of such markets would improve the
valuation estimates of the public good project and it would allow
consumers and producers to hedge against possible losses due to
price uncertainty caused by the project. Furthermore, their exis-
tence would provide the means and the incentive for the public to
stay informed about proposed public goods projects. The reason
that the government should sponsor such markets rather than let
them simply evolve due to normal market forces is to prevent the
substantial danger of moral hazard that is present when people are
allowed to gamble on the outcome of a decision they can influence.
The government could insure that the contracts are only binding if
the project is approved. The legal machinery required to enforce
a contract that is contingent on a government decision would have
to be developed very carefully since it is not now in existence
and is not likely to develop on its own.
Another, less radical, suggestion for reducing the distortion
caused by unanticipated price changes resultin£ from the public
good decision is to have the government attempt to estimate the
nature of important market interactions in supply and demand in
order to calculate adjustments to the valuation and cost figures
that are based on current prices. Econometric models for this
type of estimation require more information than those used to es-
timate single supply or demand functions, however such techniques
are currently in wide use and could be easily applied to this type
of scheme.
The difficulty involved in applying the bidding mechanism to a real
public good proposal depends on the exact nature of the public good.
It is important in any application of this technique that the par-
ticipant bidders realize the exact nature of the proposal, the cur-
rent total of bids, and the fact that their own bid will be a
bi.nding obligation. If it is simply a number which they know will
have no relationship to their tax, then it cannot provide a measure
of their true valuation.
23
-------
FOOTNOTES: CHAPTER II
See Milleron for a survey to this literature.
2/
See Champsaur and Malinvaud for procedures for allocating public
goods in a planned economy.
3/
See Groves and Ledyard for this result in a general equilibrium
framework, and see Clarke, Groves and Loeb, and Tideman and Tullock for
the result in partial equilibrium models.
4 /
See Bohm, Ferejohn and Noll, Scherr and Babb, and Smith for
descriptions and results of these experiments.
Reported in Smith.
It is clear that as the positive net surplus becomes smaller
that there is less incentive for the members to cooperate. In
experiments we could measure the approval rate as a function of the net
surplus in order to determine hov? effective the method is.
The usefulness of an initial round of non-binding bids is shown
clearly by the experimental results reported in Smith. He designed this
trial as a "practice trial" used to provide familiarity with the
procedure but noted that it also provided the subjects with valuable
information about the potential surplus available. I have made the
continuation of the bidding contingent on obtaining a positive net
surplus in the initial trial in order to provide disincentive to
underbidding here.
8/
It is apparent in some of the experimental results reported in
Smith that the bidding didn't get serious until the process got close
to the last trial. Incorporating an increasing random stop probability
makes each of the stage II rounds a potential last round. This should
increase the seriousness of the bidding very early in the procedure.
24
-------
REFERENCES
Bohm, P., "Estimating Demand for Public Goods: An Experiment,"
European Economic Review 3 (1.972), 111-130.
Champsaur, P., "Neutrality of Planning Procedures in an Economy
with Public Goods," Review of Economic Studies 43 (June 1976),
293-299.
Clarke, E.H., "Multipart Pricing of Public Goods," Public Choice
11 (1971), 17, 33.
Ferejohn, J. and R. Noll, "An Experimental Market for Public Goods:
The PBS Station Program Cooperative," American Economic Review,
Papers and Proceedings (May 1976), 267-273.
Groves, T. and J. Ledyard, "Optimal Allocation of Public Goods: A
Solution to the Free Rider Problem," Econometrica 45 (1977),
783-809. ~~~
Groves, T. and M. Loeb, "Incentives and Public Inputs," Journal of Public
Economics 4 (1975), 211-226.
MMler, K-G., Environmental Economics: A Theoretical Inquiry, Baltimore:
Johns Hopkins University Press, 1974.
Malinvaud, E., "Prices for Individual Consumption, Quantity Indicators
for Collective Consumption," Review of Economic Studies 120
(October 1972), 385-406.
Milleron, J-C., "Theory of Value with Public Goods: A Survey Article,"
Journal of Economic Theory 5 (1972), 419-477.
Scherr, B. and E. Babb, "Pricing Public Goods: An Experiment with Two
Proposed Pricing Systems," Public Choice (Fall 1975), 35-48.
Smith, V.L., "Incentive Compatible Experimental Processes for the
Provision of Public Goods," NBER Conference on Decentralization,
April 23-25, 1976. Forthcoming in Research in Experimental Economics,
edited by V.L. Smith, Greenwich, Conn., JAI Press.
Smith, V.L., "The Principle of Unanimity and Voluntary Consent in Social
Choice," Journal of Political Economy 85 (1977).
Tideman, T. and G. Tullock, "A New and Superior Process for Making
Social Choices," Journal of Political Economy 84 (1976), 1145-1159.
25
-------
CHAPTER III
THE VALUE OF LEARNING ABOUT CONSUMPTION
HAZARDS
by
Robert A. Jones
This report examines the implications of reducing uncertainty about the
hazards associated with various forms of consumption. Section 3.1 focuses
on the determinants of the dollar valuation of such a reduction in uncertain-
ty, measured as the willingness to pay. The chapter begins with the simplest
'Marshallian' case and then successively generalizes the results at the cost
of making Taylor's series approximations. It is shown that the value of re-
ducing uncertainty is readily determined once estimates have been made of the
ex-post shifts in demand associated with the information.
A major simplifying feature of the models in Section 3.1 is that all
p'rices are exogenous. While this is perhaps a reasonable first approximation
for many applications, it is surely inappropriate for non-produced commodities
of uncertain quality. One important case is the adjustment of land prices to
reflect differences in air quality in an urban environment. This case is the
primary focus of Section 3.6. First the equilibrium location of a population
with different incomes is described. It is shown that there is only a mild
presumption in favor of location in the less hazardous areas by the more
wealth. Optimal location of an identical population is then examined. Fin-
ally, it is shown that the expected value of research which reduces uncertain-
ty about an environmental hazard may be fully reflected in land values.
Section 3.11 introduces time into the analysis, taking account of the
fact that the prospect of future information will affect consumption decisions
made prior to the receipt of the information. The central result is that if
the possibly harmful effects of consuming a particular good depend on its
accumulated consumption over the lifetime, then the prospect of receiving in-
formation abcut the maximum safe level of consumption reduces current consump-
tion of that good.
3.1 The Value of Information
If a consumer is uncertain about the va]ue of some parameter, for exam-
ple the 'quality' of a particular product or the probability it will result
in early death, he will in general be willing to pay to obtain a better esti-
mate of the unknown parameter. In the following section we ask how much a
consumer would be willing to pay for perfect information.
Formally, suppose uncertainty is captured by a parameter s and the util-
ity of the consumer in state s is:
26
-------
u = u(x(s);s) (3.1)
where x(s) = (x,(s),...,x (s)) is consumption in state s.
To focus upon uncertainty about the quality of a product we assume that
neither the price vector p nor income M are state dependent. Then with
perfect information about the state provided at a cost of V, the consumer
chooses x(s) to maximize u subject to his budget constraint. That is x(s)
yields the solution of:
u(s) = Max{u(x;s) p'x ^ M - V}. (3.2)
x
Since the cost of obtaining the information is incurred prior to knowing
the true state, anticipated benefit is a random variable u(s). Assuming
that the consumer's preferences satisfy the von Neumann-Morgenstern axioms
we can express the benefit as the expectation of this random variable, that
is :
U*(V) = Eu(s)
s
= /u(s)dF(s)
seS (3.3)
where F(s) is the consumer's subjective probability distribution over the
set of feasible states S.
Without the information, the consumer simply chooses x to maximize his
expected utility. That is x yields the solution of:
U° = Max{Eu(x;s) |p'x <_ M} (3.4)
Since x is a feasible solution to problem (3.3)when V = 0, U*(V) >_ U at
V = 0. Moreover U*(V) is a non-increasing function of V. Therefore for
some V*- the expected utility associated with being perfectly informed at the
time of purchase is equal to the expected utility in the absence of this
information. V* is therefore the most the consumer would be willing to pay
to be perfectly informed. That is, V* is the reservation price or value
of perfect information.
In the following sections we derive expressions for V* under alternative
assumptions about the utility function u(x;s). Section 3.2 considers the
simple Marshallian case in which the marginal utility of expenditure on
other goods is constant and independent of the state. This generates a
particularly simple expression for the value of information. Section 3.3
introduces the more plausible situation in which marginal utility varies.
After obtaining an expression for V* using the logarithmic utility function,
a first order approximation is derived. The accuracy of this approximation
is then discussed.
In Section 3.4 a first order approximation of the value of being
perfectly informed is obtained for a general utility function u(x;s).
The results are related to those of the previous two sections and several
other special cases are then considered.
Finally, in Section 3.5 we turn to the value of becomming better
27
-------
informed rather than perfectly informed. A general definition of better
information is provided and the first order approximation developed in
section 1.3 is then extended.
3.2 Marshallian Analysis
Beginning with the simplest possible case suppost the utility associated
with the consumption bundle x can be expressed as:
n u(x1,x?, . . . ,xn>s) = UI(X;L;S) + y (3.5)
where y = Ep.x. is expenditure on other goods. Suppose further that
S = {1,2}, that is, s takes on two possible values with probabilities ir, and
IT.-,. Then expected utility:
U = ir1u1(x1;l) + ir^Cx ;2) (3.6)
The consumer faces a budget constraint:
P1x1 + y = I
Since we are only dealing with uncertainty about the value of a single
commodity we drop subscripts on x,, P1, and u (x,;s). Substituting for y in
(3.5)we have:
U = {iT1u(x;l) + TT?u(x,2)} - px + M (3.7)
Then the consumer chooses x°(p) to maximize (3.7).
At an interior option we therefore have:
,ij(x;2) = p (3.8)
Interpreting this in Marshallian terms, the function p (x) defined by(3.8)is
the price that would generate a demand of x.
Compare this with decisionmaking when the state of the world is known
prior to trading:
u = u(x;s) - px+M- V
At an interior option
ir(x,s) = p (3.9)
dx
Therefore the function ps (x) = (x,s) is the perfect information
Marshallian demand curves. These are depicted in Figure 1 for s = 1 and
s = 2. Note that the incomplete information demand curve:
O / \ v- S s \
p (x) = ZTTSP (s)
is simply a probability weighted average of the perfect information demand
curves. With full information the consumer chooses either x-"- or x^ at the
prive p. With imperfect information the consumer chooses x° where from' (3.8)
0,0, s , o>.
p (x ) = £TTSP (x ) = p
In the latter case expected utility is, from (3.7).
28
-------
u°
ITT (u(x°;s) - px°)-f M
, r 3u o
= Z7TS(J g-(q?s)dq - px )+ M
s 0
_ p)dq + M
If the true state is known to be s utility is:
x
s s
J(p (q) - p)dq + M - V
0
Thus the expected utility with perfect information prior to trading is:
- p)dq + M - V
s 0
Choosing V* so that U and U* are equal we have finally
s
V* = ZTT J (ps(q) - p)dq
s S o
x
(3.10)
For the two state case depicted in Figure 3.3, this can be rewritten as:
x1
V* = -nj (p'(q) - p)dq 4- TT J (p - p2(q))dq
o 2
* x
= T^CAREA ABC) + TT^AREA ADE)
The value of perfect information is then equal to the expected net increase
in consumer surplus.
Returning to the S state case, suppose we approximate the demand curves
Q
p (x) by parallel linear demand curves of shape
dp (x )
dx
Substituting into (3.]D)we then have
1 ,-. r s
s , o
(x )
s dx
, CK 0 , O. ,2
(X ) - P (X )]
V* =
o
(3.11)
d{
dx
29
dP0(x°)
dx
-------
Figure 3.1
The Value of Information
(x)
1
i '
1
i 1
1 0
X X
' 2
P 00
i
i
i
.... ^
X
X
'
demands for
coin:noditv 1
30
-------
var{ps(x )} is the variance of full information demand prices for the
quality of x pruchased with imperfect information. dp0(x°) is the steepness
dx
of the incomplete information, inverse demand curve. The value of
information is therefore an increasing function of the dispersion of demand
prices and of the price sensitivity of demand.
3.3 Logarithmic Utility Functions
We noxvr begin the process of relaxing the strong assumption of constant
marginal utility. First we consider the issues for the special case in
which the utility function takes on the simple form:
n
U = Z Oiln6ixi, 9±, Si 1 0
where 0 = 0(s) and g = g(s)
In the absence of further information about the true state the consumer
chooses a consumption bundle x° yielding the solution of:
Max{Eu(x;s) p'x <_ M}
s
Note first that we can rewrite U as
U = Z0 Ing. + I0.1nx.
.1 i .1 i
i i
Therefore x is the solution of
Max{EI0 Inx |p'x < M]
s ± ±
It follows that information leading to a change in beliefs about the vector
6 but not a has no effect upon the optimal consumption bundle. In particular
suppose the only uncertain parameter is ^. For example a consumer might be
uncertain about the quality per unit of a particular commodity. Then for
the logarithmic case information about the true value of (3 has no effect
upon the optimal consumption bundle x°. Moreover the knowledge that S-, will
be known prior to the time of purchase has no effect upon the ex ante
utility level. That is, the value of perfect information about g is zero.
To generate a model in which information changes actions we therefore
focus upon cases in which the vector 0 = (0^,...,0 ) is uncertain. Without
further loss of generality we may set 3 - (1,!,...?!)-
Consider the case in which
0J = S
0± = (1 - S)YI i = 2,...,n
n
where I y- = 1
2
Such a consumer is uncertain about his marginal valuation of commodity
1 relative to all other commodities but always spends his income on
commodities 2,...,n in the same proportion. Given constant prices we may
31
-------
apply Hick's aggregation theorem and write the objective as
Max{E(slnx1 + (} - s)lny) p^ + y = M} (3.12)
In the absence of further information about the true state this
problem reduces to the certainty equivalent problem;
Max {slnx, + (1 - s)lny|px + y < M} (3.13)
1 JL I
x]_.y
Solving we have:
U° = sins + (1 - s)ln(l - s) - slnp- + InM (3.14)
Having paid V for perfect information about the true state the consumer
chooses x(s) to yield the solution of:
Maxfslnx, + (1 - s)lny P-.X, + y < M - V}
vy
Since this problem has exactly the form of problem (3. 13)the solution u(s)
takes the form of (3.14). We have
u(s) = sins + (1 - s)ln(l - s) - slnp-j^ + ln(M - V)
Then the expected utility with full information prior to purchase is:
U* = E{slns + (1 - s)} - slnp1 + ln(M - V) (3.15)
s
The value of information V* is then the level of V such that U and U* are
equal. Equating (3.l4)md (3.15^nd rearranging we have:
V* - - - -
-ln(l - ^-) = Efslns + (1 - s)ln(l - s)] - fslns + (1 - s)ln(l - s)]
M s
(3.16)
The first bracketed term is a strictly concave function and the second term
is the value of this function at s, the mean level of s. Then by Jensen's
inequality this expression is necessarily positive. Expanding both sides
using Taylor's approximation we also have,
) (3.17)
= var(s)/2(l - s)s
It is interesting to compare this with the 'consumer surplus' estimate of
the previous section. For the logarithmic utility function:
o
p (x) = sM/x
Substituting into (3.13) the Marshallian approximation can be written as
Mvar(s)2s
Comparing this with(3.17)it follows that the Marshallian estimate of the
value of perfect information is biassed downwards by a factor of (1 - s) .
The two estimates differ because in the logarithmic case a change in s
changes not only the demand curves for x, but also the damand for other
goods y. When the triangles corresponding to those in Figure 1 are computed
for both x1 and y and the average areas are added together the resulting
32
-------
estimate of V* is indeed (3.17). All this suggests that the average area
calculation is capable of further generalization. In Section 3.4 we shall
see that this is indeed the case.
We conclude this section with a comparison of the exact value of
information given by equation (3. 16), with the approximation given by equation
(3. 17). Suppose s takes on two values s + e and s - e with equal probability.
Let
V*
-ln(l - -) = A
Then
V* =
where from (3.1$,).
A = [
Also from (3. 17)the approximation to the value of information can be
expressed as:
V, M2I _L
a 2s 1 - s'
Computational results are summarized in the following tables.
Note that V*(s) = V*(1-I) and V* = V*(l-s). Therefore the value ol
tion for i = .7, .9 .99 can also be obtained from the two tables.
Comparison of these tables indicates that the approximation is
remarkably good over the whole range of feasible values of s. For example
the mean difference between the ten computed values of V* and V* expressed
as a percentage of V*, is less than 6.5%. This is reason for having some
confidence that the results developed in the next sections tield reasonably
good approximations of V*.
3.4 General Utility Functions
We now consider the value of perfect information for any utility
function u(x;s) which is twice differentiable in x and s and strictly quasi-
concave in x. In contrast to the above discussion we allow not only x but
also s to be a vector.
Suppose first that perfect information is provided at no cost. Then
the consumer chooses x(p;s) yielding the solution of;
u(s) = Max{u(x,s) jp'x <_ M} (3.18)
The expected utility thereby achieved is:
u*(M) = Eu(x(p,s);s)
S
Without the information the consumer chooses x° to achieve an expected
utility of:
u°(M) = Max{Eu(x,s)|p'x <_ M}
x
33
-------
Table 3.1
The Value of Perfect Information as a Percentage of Income
\^
.01
.10
.30
.50
.01
.696
.10
.056
7.215
.30
.02 4
2.387
23.994
.50
.020
1.994
17.532
50.000
Table 3.2
Approximation of the Value of Perfect Information
as a Percentage of Income
^x. s
£ \.
.01
.10
.30
.50
.01
.505
.10
.056
5.556
.30
.024
2.417
21.750
.50
.020
2.000
18.000
50.000
34
-------
Let s be that value of s so that:
o , CK
x = x(p,s )
Then the increase in utility associated with having perfect information
is:
u*(M) - u°(M) = E[u(x(p,s);s) - u(x(p,s°);s)]
Expanding the right hand side according to Taylor's approximation we have
u*(M) - u°(M) = Ef(x-x°)'u +l(x-x°)'u (x-x°) + (x-x°)'u (s-s°)+...]
s X L XS (3.19)
Since x(p,s) is the solution of (18) it must satisfy the Kuhn-Tucker
necessary conditions for the following Lagrangian:
L(x;A;s)= u(x;s) + A(m-p'x)
Assuming that x(p,s) is an interior solution we have:
L = u (x(p,s);s) - AP=0 (3.20)
.X ?v
Then the first term inside the bracket of expression (3.^g^reduces to:
(x-x°)'Xp = Xp'(x-x°) = A(p'x-p'x°) = 0
Moreover, differentiating(3.20)with respect to both s and p we have:
u x = Al + PA' (3.21)
xx p p
and
u x + u = pA! (3.22)
xx s xs ' s ^ '
Linearizing the demand curves x(p;s) x^e have:
(x-x°) * xs(s-s°) (3.23)
Prior- to the receipt of information x(p,s) is a random variable. Then
actual demand x, can be thought of as a random drawing from the set
X = {x|x = x(p,s);s e S}. The Marshallian demand price vector associated with
consumption vector x° is therefore:
p = (p:x(p,s) = x°}
Then
x - x
°
(3.24)
Utilizing (3. 22)we can rewrite the third term in the bracket of (3.19)as
follows :
(x-x°)'u (s-s°) = (x-x°)'(PA' -u x )(s-s°)
XS s XX 5
= (x-K°)IPAI (s-s°) - (x-x°)'u x (s-s°)
S XX s
The first term on the right hand side is zero since p'x = p'x . Then
using the linear approximation (3 . 23)we have:
(x-x°)'uxs(s-s°) ~ _(x-x°)Uxx(x-x0)
35
-------
The increase in utility associated with having perfect information can
therefore be approximated as follows:
u*(M) - u°(M) : 7E(x-x°)'u
s
(3.25)
Substituting for u and x-x from (21) and (24) we have:
xx
u*(M) - u°(M) : -
From the first order conditions we have:
I?00 - !»<>
Therefore, ignoring the impact of variation across states in the marginal
utility of income we have:
u*(M) - u°(M) = - E{(p-p)ix (P-P)J (3.26)
For the final step we note that the value of information is that level V*
such that:
u*(M-V*) = u
Taking first order approximation about V* = 0 we have:
u*(M - V*) : u*(M) - |H V* (3.27)
Comparing(3.26)and(3.27)it follows that:
V* :-jE(p-p)'xp(p-p) (3.28)
Suppose only the demand price of commodity 1 varies with s. Then:
3P var(PrPl}
Comparing this with expression (11) it follows that our approximation does
correspond to that obtained in Section 3.2.
Similarly, for the logarithmic utility functions it is a straight-
forward exercise to show the approximation given(3.28)reduces to the
expression obtained in Section 3.3.
3.5 The Value of Imperfect Information
The preceding sections were concerned with valuing information which
eliminated all uncertainty about the effects of consuming various goods.
V* represented what the consumer would pay for perfect information about s.
But it is seldom feasible for research to eliminate all uncertainty about
the characteristics of goods. Realistically, investigation only narrows
the range in which the true characteristics lie, decreasing but not
36
-------
eliminating the dispersion of the consumer's probability distribution over
s. In this section we ask how much a consumer would be willing to pay for
such imperfect information.
The outcome of the research the consumer commissions, or message he
receives, will be denoted by a e A where A is the set of possible results.
Before the research is conducted a is a random variable in the mind of the
consumer. Its relation to the uncertain state of the world is embodied in
a subjective joint probability distribution function F(a,s) over A x S;
F(s), F(a) , F(s|a) denote the associated marginal and conditional probability
distributions. This pair [A, F(a,s)] is the information structure whose
value we wish to determine.
If the information is provided at no cost, and if only s not the
message itself affects his ultimate welfare, then upon receiving o. the
consumer chooses x(p,a) E x to obtain conditional level of expected utility
E u(x;s) = Max{E u(x;s)| p' x <_ M} (3.29)
s
X
Prior to the receipt of a, x is a random variable, given a it is no
longer random even though s may still be unknown. The anticipated level
of expected utility prior to receipt of the message, depending both on the
information structure and income, is:
u*(M;A) = E . u(x;s) = E E u(x;s). (3.30)
S 9 Ct tJt o Q(
As before, the consumer chooses x° without the information to achieve an
expected utility of:
u°(M) = Max {E u(x;s)| p'x <_ M}
S
x
and the increase in expected utility associated with having the information
structure is:
u"(M;A) - u°(M) = E Eo n [u(x;s) - u(x°;s)j. (3.31)
a s
a
Expanding the inner expectation of the right hand side in a Taylor series in
x around x yields:
E [u(x;s) - u(x°;s)] = -E [u(x°;s) - u(x;s)]
s
g
a
- -v r (v -C-Vii -f i/? Cv -v^'n fv v\ 4- i
- Es a I U x) ux + 1/2 l.x x) uxx U -x) + ...j
(3.32)
Recalling that x was the solution to (29), and forming the Langrangian
L(x,A;a) = ES au(x;s) + A(M-p'x),
x must satisfy the first order condition:
Lx = Es a Ux(^;s) - Xp = 0. (3.33)
The scalar X denotes the expected marginal utility of income conditional on
research outcome a being received. Differentiating
with respect to p provides the additional relation (3.33)
37
-------
r p ** "~ *** i
s|a Uxx]xp = AI + pAp. (
Note that A, x , x, Ap are non-random once a is revealed.
Substituting (3.33) into the first component of the right hand side of (3. 32)
tells us that:
Es(a t*°-xr\ = (x° -x)'Es[aux = (x° -x)'PA = 0
i
since x° p - x'p = M from the budget constraints.
Hence(3. 3i)is approximated by:
u*(M;A) - u°(M) = EaEg|a [-1/2 (x0-x)'uxx(x°-x)] (3.35)
=-1/2 Ea (x°-x)' [Es|auxx](x°-x)
Now define the Marshallian demand price vector p associated with the
consumption vector xo conditional on message a being received as:
p = {p: x(p,ct) = x°}.
Linearly approximating the demand function for given a around p gives
(x°-x) = x (p-p). (3.36)
Substituting(3.36)into the right hand side of (3. 35)yields:
-1/2 E (x°-x)' [E i u li (P-P)
Q S | (X XX p
which can be written utilizing relation (34) as:
-1/2 Z (x°-x)' [XI + PA '] (p-p).
a. p
The (x -x)'pA'(p-p) portion of their expression vanishes
o ~ o " r
since (x -x)'p = p'x - p'x = M-M =0 from the budget constraints.
Using (3. 36)again on the remaining portion of the expression results in:
u*(M;A)-u°(M) = -1/2 EwX(p-p)'x (p-p). (3.37)
a p
Prior to receipt of the message the expected marginal utility of income is
u _
~9M~ \ V-
If the effect of messages on the slopes xp of the demand curves is
negligible, and if we ignore any correlation between A and the remaining
quadratic form in (3. 37) , then the expected gain in utility may be written
almost precisely as in (3.26): ^
u*(M;A)-u°(M) = -1/2 |g E^ (p-p) 'xp(p-p) ] . (3.38)
*
Analagously defining the value of the information structure as V .
for which:
u (M-V .;A) = u°(M),
*
one obtains a first order approximation to V of
38
-------
V" = -1/2 Ea(p-p)'xp(p-p). (3.39)
Although it is an approximation,(3.39)provides a consistent estimate
of the value of improving a consumer's estimate of s over a wide range of
information structures. For example, if the research will provide perfect
information, as when A coincides with S and a = s, then(3.39)is identical
to (3.28). If the research outcome in fact sheds no light on s, so that
x(p,a) = x° for all outcomes, then p = p for all a and(3.39)indicates
V*^ = 0. More importantly, (3.39)inakes it clear that research x^hose
results would not change consumers' behaviour is valueless, even though it
may significantly improve estimates of s in a purely statistical sense.
One final check on the plausibility of(3.39)as an approximate
indicator of the value of imperfect information about the consequences of
consuming various goods is to verify that information never has a negative
value. Such a result must follow if the outcome of the research itself,
as opposed to the true characteristics of goods s, has no direct effect
on the consumer's utility. That(3.39)has this property can be demonstrated
as follows. Assuming as we have that the slopes of the uncompensated
demand curves as indicated by x = [3x./3p.] are unaffected by the outcome
of the research a, these slopes'will be identical to those of the demand
curves if no information was to be received. Using the Slutsky relation
of conventional demand theory
3x./3p. = 3xc/3p - x.3x./3M
1 J i J J i
c
in which x./3p. is the slope of the income-compensated demand curve for
good i with 'respect to the price of good j, we can express x as xc - x x
in which x = x is the consumption point at which the derivatives are
evaluated. Inserting this expression for xp into(3.39)gives us the
alternate form
V* = -1/2 Eot(p-p)'[Xp - x^0'] (p-p).
But since p'x = p'x = M from the budget constraints and definition of p,
the second component of the inner bracketed expression becomes 0 when
multiplied by (p-p). Thus(3.39)can be alternately written as
V* = -1/2 Ea(p-p)'x^(p-p). (
The Stutsky matrix xc is known to be symmetric and negative semidefinite.
Hence the expectation of the quadratic form in(3.40)is non-positive and
V* must be non-negative for all information structures.
3.6 Information and Price Adjustment
As analyzed in Section 3.1 of this report, information is valuable to
the extent that consumption plans change with the message received.
Loosely, the greater the optimal adjustment to the different messages the
more an individual is willing to pay ex-ante for the provision of the
information. Ignored, however, is the possibility that the receipt of
information will have significant price effects.
Implicity in such a formulation is the assumption that prices are
39
-------
largely determined by cost conditions rather than the intersection of
supply and demand curves. While this is a natural first approximation
for a variety of applications it is particularly inappropriate for non-
produced commodities of uncertain quality. One important case is the
adjustment of land prices to reflect differences in air quality in an
urban environment. It is this case that we shall focus on in the follow-
ing sections.
We begin in Section 3.7 by illustrating the implications of price
adjustment on the value of information for a simple exchange economy.
It is shown that all agents in an economy may be made worse off by the
announcement that the true quality of a product will be made known prior
to trading. Essentially the anticipation of information introduces an
additional distributive risk which reduces each individual's expected
utility. It is shown that each agent would prefer to engage in a round
of trading prior to the revelation of product quality, thereby insuring
himself against an undesirable outcome.
The in Section 3.8 a simple urban model is developed in which a
fixed number of individuals must be located in two regions. The equilib-
rium allocation of individuals is first examined. Simple sufficient
conditions for higher income groups to locate in the preferred environ-
ment are established.
Surprisingly, it is shown that under non implausible alternative
conditions both tails of the income distribution may locate in the
preferred environment.
Section 3.9 asks what allocation of land and goods maximize a
symmetric social welfare function. Starting with income equally distribut-
ed it is shown that optimization in general requires an income transfer
from those living in one zone to those in the other. Under the conditions
which imply that in equilibrium the rich will locate in the better
environment, it is optimal to transfer income to those in the better
environment from the remainder of the population! The intuition behind
this paradoxical conclusion is then developed.
Finally, Section 3.10 focusses on the implications of conducting
research to resolve uncertainty about the nature of the environmental
hazard.
3.7 Information About Product Quality with Negative Social Value
Consider a two person economy in which aggregate endowments of two
commodities, X and Y, are fixed and equal to unity. Both individuals
have utility functions of the form:
1/0 1/9
u(xi,yi;0) = (0xi) 7 + y± ' i = 1,2
where 0 is a parameter reflecting the 'quality' of the product. Prior to
trading 0 is unknown but both individuals believe that with equal
probability 0 takes on the values 0 and 1.
40
-------
Then the expected utility of agent i is:
u°(x ,y.) - EU(X ,y ;0) = -L. 2 + y 1/2 (3.41)
0
Without loss of generality we may set the price of y equal to unity. Then
each agent chooses (x.,y.) to maximize U° subject to a budget constraint
px. + Y. < PX + y.
1
where (x.,y.) is the agent's endowment.
Since U° is strictly concave the following first order condition yields
the global maximum.
,/y.\l/2
Then :
Y , 2
^P i = 1,2 (3.42)
X
1
It follows that:
Thus the equilibrium price of_x is 1/2 and from (3.42) y. = x., i = 1,2.
Suppose (x^,y ) = (1,0) and (x2,y ) = (0,1). Then from'Szhe Budget
constraint it is a straightforward matter to show that:
(x1,y1) = (1/3,1/3) and (x2>y2) = (2/3,2/3)
From (3.41) theexpected utility of the agents is given by:
i1/2 /^/2
U° = J U° =/ I
Next suppose that research is to be conducted which will reveal the
true state prior to any trading. If 0 = 0 the endowment of agent 1 is
valueless hence there can be no trade ex post. Then:
u (0=0) = 0 and u2(0=0) = 1
If 0 = 1 each agent has an ex-post utility function:
1/2 1/2
ui = xi + >'i
Applying an almost identical argument to that made above, it can be shown
that for such preferences the equilibrium price of x is unity and both
agents consume half the aggregate endowment. Then:
1 /7
//-\__-is o -*- / * f r\~~\ \
u (0=1) =2 = u0(0=l)
1 2
Prior to the revelation of the information both agents place an equal
probability on the two possible states. Thus expected utility levels
41
-------
xj-ith the information are: ,
and i /2
U2 = "^Z^0^- + Iu2<-0= ^ = " 1 -
Then (U*)2 - (U°)2 = 1/2 - 3/4 < Q
and CV - (U°) = -f <0.
The prospect of information prior to trading therefore creates a distribu-
tive risk which reduces the expected utility of every agent!
Each agent would therefore like to insure himself against such risk.
It follows that there are potential gains to opening the commodity
market prior to the announcement of the true state. Since the future
spot price of X relative to Y, p, is independent of individual endowments
it follows from the above analysis that p = 0 if 0 = Q and p = 1 if 0 - 1,
that is:
p(0) = 0; G = 0,1
If the spot price of X is p, agent i can select bundles (x.,y.) satisfying
px± + y± = P^ + y± (3.43)
When the state is announced the agent then makes a second round of
exchanges subject to the contraint:
p(0)xi(.0) + yi(0) = p(.0)x_.. + y_L 0 = 1,2. (3.44)
But if 0 = 0 the future spot price p(0) = 0. It follows that there
will be no trading after the announcement, that is:
if 0 = 1 the future spot price, p(0) = 1. Given the symmetry of the
indifference curves each agent will trade in such a way as to equalize
his spending on the two commodities.
x. + y \l/2
t J T \
Then
/x. + y. x. + y \]
(x.UO,y.(l» 4 x 9 , 1 9 1)
1 L V l J
Expected utility of agent i is therefore
/x. + yAl/2
With a spot price of p, agent i chooses x. and y. to maximize U(x.,y.)
subject to his budget constraint(3.43).The first order condition Jor1
expected utility maximization is therefore:
42
-------
3U , x, -x -1/2
<*i = V. 71 + 1 /
_9U ^-1/2 , , -1/2 (3-45)
' x. \
+ 1v
y f i
x.. £x.
It follows that is the same for both agents, hence equal to - = 1.
Then from(3.45)p = 1/2. From the budget constraint(3,43)ic follows that
(xi,y1) = (1/3,1/3) and (x2,y2) = (2/3,2/3)
But this is exactly the consumption achieved by each agent in the
absence of the information. Therefore the prior trading just eliminates
the undesired utility risk, and the expected value of the information is
zero.
A central feature of this and the earlier results is that agents
correctly anticipate the price implications of the state revealing
message. If consumers are unaware of these implications the analysis of
section 1 applies. Each will therefore place a positive value on the
information.
Of course it is a long leap frcm this simple example to a general
proposition. However it does seem reasonable that there will, in general,
be a tendency for price adjustments to offset the anticipated gains
associated with better information. Thus except in cases where there
are solid ground for arguing that prices are cost determined, the
expressions for the value of information developed in Section 3.1 seem
likely to overstate true value.
3.8 Urban Location and Land Values with Environmental Hazards
. One very important case in which price adjustments to changes in
information are central, is that of urban location. To illustrate the
issues we shall consider a city which consists of two zones.
The utility of any individual living in the second zone is a concave
function U(x,y) of the area of his residence x and expenditure on other
commodities y. If provided the same bundle of commodities in the
environmentally affected first zone his utility drops to U(x,y)-s. That
is, s is the loss in utility associated with living in the "smoggy"
first zone.
Suppose each individual .purchases land from some outside landowner and
all have identical incomes. Let P. be the price of a unit of land in
zone i. For those locating in the second zone the utility level achieved
is:
V(p ,1) = Max{U(x,y) P2* + y = I) (3.46)
x,y
Similarly for those locating in the first zone the utility level achieved
is:
43
-------
V(p ,1) - s = Max{U(x,y)|p1x + y = 1} -s. (3.47)
In the absence of constraints on land purchases, the value of land in the
"smoggy" zone must fall until utility is equated in the two zones. This
is depicted in Figure 3.2.
At the level of an individual consumer, one measure of the cost of the
smog is the extra income H that a person living in the second zone would
have to be tgiven in order to make him willing to move at constant prices.
In formal terms this is the Hicksian compensation required to maintain the
utility level of an individual in the smoggy zone at the higher land value
P2, that is:
V(P2, I + H) = V(P1, I) = V(P2, I) -f- s (3.48)
This is also depicted in Figure 3.2.
With this background we can now ask which individuals live where, if
incomes are not equally distributed. For expositional ease we shall
restrict our attention to utility functions that are homothetic. Suppose
that income is distributed continuously. Then for some income level 1°
individuals will be indifferent between living in the two zones. We
therefore have:
V(P2, 1°) = V(P1, 1°) - S
An individual with income I > 1° locates in the smog free zone if and only
if:
V(P2, I) > V(P]_, I) - s
Consider Figure 3.2. Those with incomes of 1° are indifferent between C
and C2 and hence between C| and C2. Then:
V(P2, 1°) = V(P2, 1° + H°) - s. (3.49)
Moreover given our assumption that those with incomes of I locate in the
smog free zone, they must prefer D2 to D^ and hence prefer D~ to D'. Then:
V(P2, I) > V(P2, I + H) - s (3.50)
Combining (3.49) and (3.50) the higher income group prefer zone 2 if and
only if:
V(P2, I + H) - V(P2, I) < V(P2, 1° + H°) - V(P2, 1°) (3.51)
For the special case of homothetic preferences depicted in Figure 3.3 we
also have:
Moreover,
oc,
1
OD-L
°C1
OD
1
j
I. an
d°Di 14
d oc[- i.
H
+ H,
44
-------
Figure 3. 2
Urban Location and Land Values
e:-:pen011 ure on
other goods
V
PIX + y = I
45
-------
expenditure on
other ^;oods
Figure 3.3
tiomothetic Preference Ca;je
x + y = I
X
plot
size
-------
It follows immediately that:
We may therefore rewrite the necessary and sufficient condition (3.51) as
\>(P2,(.I°I+ H°)I) - V(Pr I) < V(P7,I0 + H0) - V(P I0) (3.51)'
~~ O "
Note that the left and right hand sides of (3.51)' are equal for I = I0.
Then a sufficient condition for all those with higher incomes to prefer zone
2 is that the left hand side of (3.51)' be decreasing in I, that is:
7 [(Io + HJV (P I0 + H0) - I0VT(P9,I0] < 0 (3.52)
-L o -L X ^
In turn a sufficient condition for inequality (3.52) to hold for the required
H0 is that it should hold for any H0. But this is the case if:
|-[IV (P I)] < 0
0 i 1 '-
that is:
-IV
> 1 (3.53)
VI
Thus with homothetic preferences a sufficient condition for the higher
income groups to prefer the smog free zone is that the income elasticity
of the marginal utility of income be greater than unity. Conversely, if each
of the above inequalities is reversed, it follows that with homothetic
preferences a sufficient condition for the higher income groups to prefer
the smoggy region is that the elasticity of marginal utility be less than
unity.
'We now note that this elasticity is also the coefficient of relative
aversion to income uncertainty. Arrow (1971) has argued that the latter
must be in the neighborhood of unity and increasing in income. Accepting
this conclusion it follows that there is no clear presumption that income
and environmental quality will be positively correlated. Indeed if relative
risk aversion is less than unity for low incomes, and rises above unity as
income increases it is possible for an equilibrium configuration with high
and low income groups sharing the smog-free region and middle income groups
in the smoggy region.
Of course this conclusion is very much dependent upon the underlying
assumptions. Suppose that instead of entering additively, the environmental
affects are multiplicative. That is, with the environment affected by an
amount s, utility is:
u (x,y)u2(s)
where u2(0) = 1 and ^(s) < 0.
Each consumer chooses x, y and his location to maximize the utility or,
equivalently, the logarithm of this utility, that is:
47
-------
(x,y) - Inu (s).
JL 2
Setting U(x,y) = Inu (x,y) the problem becomes equivalent to the one
already analysed. Therefore higher income groups will live in the smog
free areas if the relative risk aversion of an individual with a utility
function lnlL(x,y) exceeds unity. Since In () is a strictly concave
function, this individual's relative risk aversion exceeds that of an
individual with a utility function U,(x,y). Therefore the sufficient
condition is weakened and the presumption that higher income individuals
.will live in the less environmentally affected area is strengthened.
3.9 Optimal Urban Location
In the previous section we considered some of the positive implications
of intra urban environmental differences. It turns out that there are
also rather puzzling normative implications, at least if one adopts the
usual approach of maximizing a symmetric social welfare function. Suppose
that initially all individuals have the same income. Some locate in the
smog-free zone and the rest in the smoggy zone. A naive view might be
that those living in the smog should be compensated by an income transfer
from those in the smog free zone. Not so, an economist would almost
certainly respond. If individuals are free to move from one zone to the
other, land values will adjust to equalize utilities.
While the response is correct as far as it goes, it does not necessarily
follow that the sum of all the utilities, or indeed any symmetric function
of each utility, is maximized as a result. For expositional ease we shall
consider only the Benthamite welfare function. Let a^ be the are of zone i,
n. the number assigned to this zone, n the total population and y the total
income. We seek to maximize the utility sum:
2 a±
W = I n . [U( , y;. ) - s . ]
i=i ni
subject to the constraints:
n + n = n; n y + n?y? = y
J. £_ J. _L ~~
To solve we form a Lagrangian
L = W + A(n - r^ - n,,) + u(y - n y - n2Y2)
Necessary conditions for a maximum are therefore,
ly = ni(Uy- ~ u) = °' (3>54)
i i
and
3n.
where x. - a /n
i i i
Suppose that the optimal distribution of land and individuals is
48
(x(si),y(s )) i = 1,2
-------
Differentiating the two first order conditions with respect to s we
have:
and
ds
= 0
Uvy'(s) - 1 - x'(s) Ux - - , = 0
Substituting for v from (3.54) this reduces to:
0. It
follows that the optimal plot size is larger for those located in the smoggy
zone.
Furthermore, substituting from (3.58) we also have:
=x'
-------
Introducing the Lagrangian X (equal to the marginal utility of income) the
following first order conditions must be satisfied:
U = Ap.
x i
U = \
y
Suppose income I were increased. Differentiating the first order conditions
we have:
x! (i)
u u
xx xy
yy
Then applying Cramer's rule:
dx.
i
dl
Combining (3.59) and (3.60) we have:
dx
A (I)
U
1 dA
U U
x xy
U U
v yy
A dl |Hu|
(3.60)
dJJ
ds
-1
_ x
dl
1 dA
A dl
The expected utility of an individual residing in zone i is U(x.,y.) - s.
Therefore the change in expected utility as the smog level s increases i
E(x.,D
ds
(3.61)
Therefore if the right hand side is positive for any price P. and income
level I, it is optimal for those in the smoggy zone to have a higher
utility. Conversely, if the right hand side is always negative it is
optimal to transfer income to those in the less smoggy zone!
For the special case of homo thetic preferences examined in the previous
section E(x.,I)=l. Therefore in such cases it is optimal to transfer income
to those in the less smoggy zone if and only if the income elasticity of
marginal utility exceeds unity. Thus the condition obtained in section 2.2
ensuring that the higher income groups will locate in the less smoggy zone
also ensures that for-a population with equal incomes, the utility sum
is maximized with' a transfer of income to those in the less smoggy zone!
Such paradoxical results have already been noted in the urban literature
by Mirrlees (1972) Riley (1974) and others, although the usual emphasis has
been on the implications of differential transportation costs. Recently
Arnott and Riley (1977) have attempted to explain the origin of these
results as a production asymmetry. While their analysis does not carry
over directly, to this more complicated case the basic issues are the same.
50
-------
Suppose we begin with incomes equally distributed, as in Figure 3.2. Since
land is cheaper in the smoggy zone plot sizes are larger, unless land is a
Giffen good. That is, C^ lies to the right of 2- Moreover, if land is
a normal good c{ is above and to the right of C2- Arnott and Riley note
that for a normal good the marginal utility of income rises with a Hicks
compensated fall in the price of the good. That is, the marginal utility
of income rises around the curve from Cj to GI- With diminishing marginal
utility of income marginal utility falls in moving from 2 to C{. If the
latter effect outweighs the former (and this will be the case with a
sufficiently high income elasticity of marginal utility) marginal utility
is lower at C^ than at C2. Maximization of any dif f erentiable symmetric
social welfare function therefore requires a transfer of income from those
in the low marginal utility, smoggy zone to those in the less smoggy zone.
3.10 Uncertain Environmental Quality and the Prospect of Better Information
In the previous two sections we analysed the implications of environmental
quality differences for property values and locational choice. Given the
simple formulation of the model, none of the results are changed if s is
reinterpreted as the expected utility loss associated with a polluted
environment. We now consider the implications for property values of
conducting research which would resolve the uncertainty about the hazards
of the pollution. For expositional ease we consider the case in which
the polluted region is small relative to the unpolluted region. Then to a
first approximation land value and hence utility in the latter is un-
affected by such information. Continuing with our assumption of a
perfectly elastic response to any utility differential, it follows that
expected utility in the two regions will be fixed at some level U. Then
prior to any consideration of research resolving uncertainty about the
environmental hazard, the consumption bundle in the "rest of the world"
CQ and in the affected region C^ yield the same expected utility level.
This is depicted in Figure 3.4. Now suppose it is announced that research
will reveal the true level of s. For simplicity suppose this takes one of
two values SQ (=0) and s^. If s = 0 the utility level of individuals
in regions 1 rises to U + E(s). This attracts individuals into the region
and the price of land is bid up. Eventually the price of land reaches PQ
and outsiders no longer gain from relocation. Similarly, if s = s^ the
utility of those in region 1 is TJ + E(s) - s^ < u. Individuals therefore
leave until the price of land falls to the point where the utility
differential is eliminated. Assuming individuals own their own homes,
those remaining in region 1 have ex-post budget constraints:
PI(S)X + y = PI(S)XI + y1
Final consumption is therefore dependent upon the true state s. This is
also depicted in Figure 3.4. Note that in both states we have:
In anticipation of the release of the information about s, expected
utility in region 1 is therefore:
E(U(C1(s)) - s) = EU(C1(s)) - E(s) > U(C]L) - E(s) = U
51
-------
Figure 3.4
Uncertainty and Better Information
= I
-U=U + E(s)
Figure 3.5
me uase or wo bpeculation
U(x,y) = U(x ,y
land
x + y = I
land
52
-------
Therefore all homeowners in region I are made strictly better off by the
announcement of the proposed research. As a result outsiders will wish
to relocate in region 1. The value of land is therefore bid up to some
level p" where the expected utility achieved by relocation once again
falls to U.
The budget constrainst of those initially in region 1 and those moving
into the region are depicted in Figure 3.5 under the assumption that the
price of land jumps too quickly for significant speculative activity.
A* Suggose the former group chooses a bundle (x ,Y ) and the latter
(x , y ). Each group of course anticipates retrading at a later point.
Since both face an expected loss due to the environmental hazard of E(s)
we can write the utility differential as:
;; * &* ;VA
U(x ,y ) - U(x ,y )
where U(x,y) -EV(p p x + y) is the derived utility function for both groups,
?!
Of course there is no simple relationship between the indifference curves
for the derived utility function U(x,y) and the underlying function U(x,y).
However it musj^ be the case that those entering the region have the expected
utility level U. That is: , ft* **x
IKx ,y ) = U.
* * " &* >';*
It follows that U(x ,y ) - U(x ,y ) is the gain in expected utility
for those located initially in region 1. Consider again Figure 3.5. In
order for those entering region 1 to achieve as high a utility level as
the initial land owners, it would be necessary to increase the income of
each from I to I+A. Thus A is a measure of the dollar valuation of the
information. Note that AD=p 'x and BD=p x . Therefore the value of
information to each individual initially located in region 1 is:
A = (Px* - PI)XI
Aggregating over the whole region, the total value of the information is
equal to the increase in the value of the land in the region.
Unfortunately it is difficult to visualize how one might make a
quantitative prediction of the extent of this revaluation without working
back to the underlying preferences. In a later draft we intend to
illustrate how this might be done for the Cobb-Douglas case.
3.11 Precautionary Response to the Prospect of Information
Section 3.1 explores the value to an individual of receiving either
perfect or partial information about product quality prior to making
any consumption decisions. Consumption decisions were binding once made and
could not be altered if subsequent information about s arrived. It is
generally the case, however, that once an individual (or society) does
choose to acquire additional information about some good it takes some
time to produce it through experimentation and research. In the meantime
current consumption decisions must still be made, although future consump-
tion plans may be appropriately revised upon receipt of the experimental
53
-------
outcomes.
This chapter examines the impact of the prospect of future informa-
tion on decisions made prior to the receipt of that information. The
basic result is that if the possibily harmful effects of consuming a
particular good depend on its accumulated.consumption over one's lifetime,
then the prospect of receiving information about the maximum safe level
of consumption reduces current consumption of that good. Moreover the
sooner the information is anticipated the larger is the reduction in the
optimal current consumption.
Sections 3.12 and 3.13 characterize an agent's response to the prospect
of learning in a two period context: the agent must rely on prior beliefs
when choosing first period consumption but may receive additional informa-
tion before choosing second period consumption. Section 3.14 examines the
agents' response to variations in the expected time of arrival of the
information in a continuous time framework.
3.1'2 When Learning Prospects Do Not Affect Current Actions
We first examine circumstances in which the prospect of future
information about a good has no effect on current consumption decisions.
Let Xn and x2 denote an individual's consumption in two successive
periods of the good with uncertain characteristics s. Expenditures on all
other goods in the two periods will be denoted by y^ and y£. Ignoring
rate of time preference considerations and adopting the constant marginal
utility of income assumption of section 1.1, let us assume the agent's
"lifetime" utility has the form:
U(x1,x2,y1,y2;s) = u(x1?x2;s) + y]_ + y2-
Without loss of generality we may choose the urtits of x so that its price is
1 and, ignoring interest rate considerations, write the agent's budget
constraint as x.. + x2 + J-> + Y2 £ M>
This section examines the case in which lifetime utility is additively
separable in the two periods' consumptions. The consumer's objective is
thus to maximize the expected value of:
U = u-^x-^s) + u2(x2;s) + V1 + y2 (3.62)
subject to x, + x0 + y, 4- y < M.
1 6. JL jL
If no further information is forthcoming the agent relies on his prior
probabilistic beliefs about s to choose x, and x? maximizing:
EgU = EgUl(x ;s) + Esu2(x2;s) + M - y.^ - x2 (3.63)
Assuming concavity and differentiability of u., and u^, the necessary first
order conditions for a maximum are:
u au?
= 1, Ejp = 1. (3.64)
s 3x2
54
-------
The maximizing levels of consumption are denoted x and x°, and the ex ante
level of expected utility is:
U°(M) = Max ES[UI(XI;S) 4- u^^s) + M - X]_ - x2J. (3.65)
X1'X2
If perfect information about s is forthcoming after x, is chosen but
before x,-, is chosen, then x~ can be adjusted according to s revealed. The
level of expected utility attainable becomes:
U*(M) = Max E [u (x ;s) + u (x,(s);s) + M - x - x (s)].
(vL3-LJ^ £. £. A~ £
S)
The maximum principle of dynamic programming permits this to be rewritten as:
U*(M) = Max ES Max[u1(x ;s) + u2(x2;s) + M - x - x?] (3.66)
xl X2
= Max E2{u1(x1;s) + M - X;L + Max [u?;s) - x?]}
Xl X2
= Max E [u (x ;s) + M - x,] + E Max[u,(x :s) - xj.
sll J. s / 2. i
Xl X2
The last two equalities follow from additive structure of the utility
function. The first order conditions for an interior maximum are:
3u 9U
E - = 1, -T^ = 1 for all s.
s9x ox,-,
Denoting the optimum consumption levels by x* and x*(s), it follows from a
comparison of (3.64) and (3.67) that x° = x*. The prospect of learning the
true value of s before x? is chosen leaves unaltered the optimal current
level of consumption of the risky good x .
3.13 Utility Affected by Accumulated Consumption
In the study of environmental hazards, what is usually uncertain is the
effect of consuming particular goods, ingesting contaminants or continuing
polluting activities over long periods of time. Individuals and policy-
makers are concerned about the potential effect of current activities on
welfare in the future. Such potential effects are ruled out at the start by
the additive separability of section 3.1. One way to capture such concerns
is to suppose that lifetime utility depends in part on the accumulated
consumption of x over time. Hence let us assume the agent's utility
function is of the form:
U = U1(x1) + U2(x2) + yi 4- y2 + V(x1 + X2;s). (3.68)
To focus on cumulative consumption effect and yet maintain notational
simplicity we have further assumed that the immediate effects of consuming
55
-------
x are known: i.e., u, and u are independent of s.
J_ £
How might this cumulative consumption term v(x. + x^JS.} he interpreted
and what properties, might it have? In environmental problems the concern
is often that continuing an activity at high, levels over long periods
may ultimately have a large negative impact on welfare, although lower
levels of activity may be tolerated without ill effects.. Examples
include the ingestion of cumulative toxins such as heavy metals, exposure
to carcinogenic substances, and continued pollution of water bodies
leading to eutrophication. The accumulated level of contamination which
may be tolerated without harmful effects, however, is generally not known
for certain. The essential structure, of these situations is captured by
a v(x1 + x~;s) of the form:
[ 0 if x + x? <_ s
v(.x1 + x2;s) = J (3.69)
| -a if x, + x? > s..
The potential loss a is assumed very large, but finite. It is interpreted
as the cost of clean-up, cure or compensation if cumulative consumption
exceeds the initially uncertain "safe level" s. The agnet's prior beliefs
about s are represented by a probability density function f(s).
The following analysis shows that the prospect of receiving perfect
information about s before x2 is chosen, compared with no information,
reduces the optimal current compensation x-, of the risky good.
First, suppose no further information is forthcoming. Neither x-, nor
x? may be chosen contingent on s. The maximum level of expected lifetime
utility attainable is:
11° (M) = Max E [u (x ) + u (x ) + v + y + v(x + x 'sll
X1'X2
subject to x + x0 + y + y < M.
1 ^ i 2 ~~
Substituting the budget constraint and form of v from (3.69) into (3.70)
yields: x +x
U°(M) = Max[u,(x,) + u7(x ) + M - x - x9 - a / f(s)ds]. (3.71)
X i £- iL _L £~
0
Denoting by x^ and x2 the maximizing values of x1 and y. , the first order
conditions for an interior maximum are:
af(x1
(3.72)
Next, suppose that s is revealed to the agent prior to x? being chosen. In
contrast with (3.70), the maximum expected utility attainable is:
56
-------
U*(M). E Max ES[UI(XI) + u?(x2) + y^ + y*
s
X1'X2
yi'y2
S S
subject to x + x? + y.^ + >'2 <_ M for all s.
Eliminating the budget constraint through substitution and utilizing the
maximum principle yields:
U*(M) = Max Es Max[u1(x1) + u?(x2) + M -
(3.74)
The inner maximum with respect to x takes both s and x, as given; hence its
maximizer x*(x ,s) is a function of both variables.
This second period reaction function x*(x ,s) must be determined
before the optimal injLtial consumption level x* can be characterized.
First, let us define x^ to be the level of x? that would be consumed if the
good had no harmful consumption effects (i.eT, v = 0 for all_ x ,x9)- If
such x^ere the case then consumption would rise to where ul(x ) = j_, and
the corisumer_surplus realized from second period consumption of x would
be u2(xo) - X2- Second, let us make more pre_c_ise the meaning of a being
"large." By large we mean that a. exceeds u (x~) - x?, implying that once
s is revealed the agent would reduce x2 to 6, if necessary, to avoid the
penalty of exceeding the safe cumulative consumption level. Of course if
it turns out that previous consumption x-, already exceeded s, and if
negative consumption is ruled out, then nothing more can be lost and x? =
Xn will be chosen. The structure of v thus leads to the second period
reaction function:
x? if s < x,
s - x., if x1 <_ s <_ x.j + x2 (3.75)
f 4- V < C
^ ") -Li-A.-] * O ^ o
Turning our attention to the outer maximum of (3.74), x is chosen to
attain expected utility:
TI'tf'M^ = MTV P FM (v } + 11 ^v'V'i 4- M v v* -4- w (v + v^-c-^l 1
U \l 1) iiaA i-i [U^V^A^/ ! UrtVA^^/ ~ 11 AT A^ T^ V^A-i ' Oj^/j
Xl
x, (3.76)
= Max{u1(x1) + M - x1 + Eg[us(x*) - a/ f(s)ds).
x-j^ 0
Note that the penalty ex is incurred only if s turns out to be less than x.. .
The first order condition for an interior maximum is:
57
-------
af(x*). (3.77)
From (3.75) 3x*/3x is 0, except when x-j^ <_ s £ x][ + x2 in which case it is
-1. Substituting x* from (3.75) into (3.77) gives:
u'(x*) = 1 + af(x*) + / ~ [ul(s-x*) - l]f(s)ds. (3.78)
1 1 x* A l
The last term is the expected increase in second period consumer surplus
obtainable if one unit less x had been consumed in the first period, and
is non-negative.
Finally we may compare first period consumptions with and without
the prospect of information. Solving both (3.72) and (3.78) for 1 and
equating provides:
uj(x°) - af(x° + x°) = uj_(x*) - of (xj) - / [u^(S)-l]f (x*+6)d6.
The integral in (3.78) has been rewritten using the change of variable
5 = s - x*. One further assumption is needed to obtain unambiguous
results: assume the density function f(s) is non-increasing. The
exponential and uniform distributions would have this property, for example.
Our objective is to demonstrate that x.. >^ x*. If x? = x* then the
right hand side of (3.79) would generally be smaller than the left hand
side both because f(x,) > f (x° + x°) and because the right hand side
integral is non-negative. But the derivative of the right hand side of
(3.79) with respect to x, is negative at xj that is the necessary
second order condition for the maximum in (3.76). Hence x* must be
less than x, for (3.79) to hold. The prospect of learning trie true value
of s reduces the optimal consumption level of the risky good until the
information arrives.
The previous section demonstrated that if consuming a good is risky
because the safe level of cumulative consumption might be exceeded
(resulting in large loss), then the prospect of learning the safe level
part way through one's life reduces initial consumption of that good.
But the comparison between receiving such information and not is, in
essence, a comparison between receiving information before and after
second period consumption is chosen. It was the prospect of receiving
information sooner which curbed initial consumption until the safe level
was revealed. This section casts the problem in a continuous time
framework, in which information may arrive at any point during the agent's
life, to show that the sooner is perfect information forthcoming the
lower is the optimal pre-information consumption level.
Let the units in which time is measured be such that the agent lives
58
-------
one period, X denote the time at which s is revealed, and x , y represent
the rate of consumption of x and other goods at time 0 <_ t <_ 1. Assuming
the agent's instantaneous utility function for consuming x is identical
at all points in time, the lifetime utility function analogous to (3.68) is
111
U = / u(xt)dt + / ytdt + v(/ xcdt;s). (3.80)
000
The agent's budget constraint is:
1
/ (x + y )dt < M. (3.81)
Q t t
The maximum level of expected utility attainable with wealth M if s is
revealed at time A is:
U*(M,A) = Max E [U] subject to /(xS + yS)dt < M (3.82)
s t r
s s 0
Vyt
and xs = x | for t < A
5 S' , . ., . ,
y = y I tor all s, s .
The second constraint expresses the fact that if s is unknown before time A
then consumption prior to that time cannot be contingent on s. From the
form of the utility function and absence of positive interest it can be
readily shown that x_ is constant before and after time A, although the
T~ o
levels in these two intervals may differ. Denoting by x, and x' the
constant consumption levels before and after time A, eliminating the
budget constraint through substitution and integrating over these intervals
[0,A], [A,l] yields the more tractable expression:
U*(M,A) = Max EstAu(x1) + (l-A)u(x^) + M - \KI - (l-A)x;;
s
X1'X2 + v(AX] -I- (l-A)x;;;s)]. (3.83)
The loss v(-,s) from cumulative consumption Ax, + (l-A)x exceeding s is a
as in section 3.2. The prior probability density function on s is f(s).
The optimal initial level of consumption xif may now be characterized
using the same analysis as employed in the previous section. Let x denote
the optimal rate of consumption of x if the good had no harmful cumulative
consumption effects: i.e., u'(x) =1. If a is sufficiently large then the
optimal level of x~ as a function of x. , s and A is given by the reaction
function:
59
-------
if s < Ax,
=<
s - Ax
1 .
, if Ax.. < s < Axn + (l-A)x
1 A i i
x if Xx-, + (l-X)x < s.
(3.84)
Applying the maximum principle, x, is chosen to attain expected
utility :
U*(M,X) = Max ES[XU(XJ) + (l-X)u(x*) + M - AX.J - (l-A)x*
+ (l-X)x*;s)] (3.85)
= Max{Au(x,) + M - Ax,+(l-X)E [u(x*) - x*
i i s ^ ^
xl
Ax
- a / f(s)ds).
0
Again, the penalty a is incurred only if cumulative consumption AX prior to
s being revealed exceeds s.
The first order condition of an interior maximum with respect to x, in
(3.85) is:
A[u'(x*) - 1] + (l-A)EQ[u'(x*) -
^ - aXf(Ax*) = 0
OX, i
(3.86)
From (3.84) it is clear that 9x*;/9x, = 0 except when Ax, <_ s <_ Ax.. + (1-X)-
x, in which case 3x*/3x, = -A/(I-A). Making these substitutions into
(3.86) gives us:
Ax" + (l-A)x
u'(x*) - I - f
(l-A)x
tu'(
Ax*
s - Ax*
- r-i
1 - A
- l]f(s)ds -af(Xx*)=
u'(x*) - 1 - / [u'(yrx) - I]f(a+Axj)d6 - af(AxJ) = 0
The change of variable 6 = (s-Axj) was used to rev/rite the integral in the
latter expression. The second order condition for (3.87) to indicate
a maximum (used later) is:
u"(x*) - A / [u'(__) - i]f(6+Xx*)d6 - aAf'(Ax*) < 0
0 (3-88)
60
-------
We are now in a position to answer how x* varies with A. If X = 1 , so
that no information about s arrives before all consumption decisions have
been made, then (3.87) reduces to:
u' (x*) = 1 + af (x*). (3.89)
That is x is consumed at a rate where its immediate marginal utility of
consumption just equals the marginal utility of the other goods foregone
plus the marginal expected utility loss from increasing the likelihood
that s is exceeded. Totally differentiating (3.87) with respect to
A yields the relation between x* and A
x (3.90)
0
U-A)x
= -x[u'(x) - l]f(O + / u"(~r)- 6 ^f(')d<5
l~" d-A)2
(l-A)x
+ / [u'(TTT) - l]x*f'(')cI6 + ax*f'(Ax*).
0 J-
The expression multiplying dx-^/dA on the left side of (3.90) is negative
from the second order condition (3.88). The first term on the right
side is zero from the definition of x, the second negative since u is
assumed concave, the third and fourth negative under the assumption that
f(s) is non-increasing introduced in section 3.2. Hence it follows
that dxy'/dA >^ Q: The sooner knowledge of s is anticipated (smaller is
A), the' smaller is the optimal initial consumption of the risky good.
61
-------
FOOTNOTES: CHAPTER III
It should be noted that this result is not a general one. If
individuals assign different probabilities to the two states or have
different preferences, at least one individual will have a higher
expected utility with trading before and after announcement of the
true state. Moreover, by an appropriate redistribution of income both
can be made better off in our ex-ante sense.
2/
Alternatively the city owns the land and reimburses rents in
excess of the agricultural value of the land in a lump sum manner.
62
-------
REFERENCES
Arnott, R. and J.G. Riley, "Asymmetrical Production Possibilities, the
Social Gains from Inequality and the Optimum Town," Scandinavian
Journal of Economics 79 (1977).
Arrow, K.J., Essays in the Theory of Risk-Bearing, Markham, 1971, chapters
3 and 12.
Blackwell, D., "Equivalent Comparisons of Experiments," Annals of Mathematics
and Statistics 24 (June 1953), 265-272.
Bradford, D. and H. Kelejian, "The Value of Information for Crop
Forecasting in a Market System," ECON INC. Discussion paper,
(August 1975).
DeGroot, M.H., "Uncertainty, Information and Sequential Experiments,"
Annals of Mathematical Statistics 32 (June 1962).
Hirshleifer, J., "The Private and Social Value of Information and the
Reward to Inventive Activity," American Economic Review 61
(September 19.71).
Jones, R.A. and J. Ostroy, "Flexibility and Uncertainty," UCLA Discussion
Paper //73 (September 1977).
Marschak, J., "Role of Liquidity Under Complete and Incomplete Information,"
American Economic Review (May 1949), 182-195.
Marschak, J. and K. Miyasawa, "Economic Comparability of Information
Systems," Int. Econ. Review 9 (June 1969), 137-174.
Mirrlees, J.A., "The Optimum Town," Swedish Journal of Economics 74,
1972.
Riley, J.G., "Optimal Residential Density and Road Transportation,"
Journal of Urban Economics 2 (April 1974).
63
-------
CHAPTER IV
THE VALUATION OF LOCATIONAL AMENITIES: AN ALTERNATIVE
TO THE HEDONIC PRICE APPROACH
by
Maureen L. Cropper
It is widely recognized that the process of urbanization creates both
positive and negative externalities,. The important question from the
viewpoint of welfare economics is what value consumers, place on these
externalities. If consumers regard large cities as yielding net disutility
then a regression of wages on population and population density will
indicate how much individuals must be compensated for living in urban
areas. This figure, as suggested by Tobin and Nordhaua, may be used to
adjust welfare measures for the trend toward urbanization. Alternatively,
this information may be used to determine optimal city size (Henderson,
Tolley). Even if cities on net yield positive utility the valuation of
particular disamenities is useful for public decisionmaking. This has
led to a large number of studies (Getz and Huang, Hoch and Drake, Mayer
and Leone, Rosen 1977) which have computed hedonic prices for locational
amenities such as crime, pollution, congestion, and local public goods.
The purpose of this paper is not simply to add to a growing empirical
literature, but to present an alternative method of valuing locational
amenities. In the studies cited above, marginal valuations of amenities are
obtained by regressing the wage rate in city i on the level of amenities
in that city. This equation is usually interpreted as an equilibrium
locus of wage-amenity combinations since, if workers are mobile, wage
rates should adjust to reflect differences in site-specific amenities.
According to the theory of hedonic prices (Rosen 1974, 1977) the gradient
of the wage-amenity locus represents consumers' marginal willingness to
pay for amenities evaluated at market equilibrium.
In this paper valuations of environmental goods, are obtained by
estimating labor supply functions for various occupations, under the
assumption that the supply of labor will be lower in cities where disamenit-
ies are high. The labor supply functions to be estimated are derived from
a model of locational choice in which workers select not only the city in
which they live but their housing site within the city. Conditions for
equilibrium in the land market in each city lead to an equation in which
the real acceptance wage for each occupation in city i is a function of
employment in that occupation and the level of amenities in the city.
By specifying explicitly the form of individuals' utility functions it is
possible to relate the coefficients of the labor supply function to the
coefficients of the utility function, which in turn may he used to compute
willingness to pay.
64
-------
The novelty of this approach is that it explicitly considers the
spatial character of individual cities. By ignoring the spatial dimension
of the problem, previous studies have been forced to assume that individuals
within each city are exposed to the same level of amenities, regardless of
where they live. In our model it is possible to find assumptions about the
geographic distribution of amenities, and about utility functions, which
allow the acceptance wage to be expressed as a function of the level of
amenities at a single location within the city; or, when this is not
possible, to assess the bias introduced by measuring amenities at a single
point.
The spatial model also allows us to determine precisely what is meant
by the "value of reducing crime" or the "value of improving air quality."
Under the assumptions below the labor supply function captures the value
which individuals place on amenities both at their residence and at their
work site. The coefficients may therefore be used to estimate the maximum
willingness to pay for an equal proportionate change in an amenity
throughout the city.
The theoretical model which underlies the valuation of amenities is
presented in section I below. In order to obtain reliable estimates of
willingness to pay one must take account of factors affecting the demand
for labor x\'hich allow firms to compensate workers for urban disamenities.
This is accomplished in section I by developing a model in which industries
expand in cities where locational amenities proximity to input and out-
put markets, low property tax rates are favorable. In section II the
empirical counterpart of this model is developed and labour supply
functions are estimated for nine one-digit occupations using data from the
1970 Census of Population. The labor supply functions indicate which
amenities are most important in consumer location decisions and whether
they are valued equally by all occupational groups. The regression
results are used in section III to illustrate how marginal valuations of
amenities may be inferred from the coefficients of the labor supply
function.
4.1 An Equilibrium Model of Urban Location
To keep the notation simple the model below is presented for the case
of a single occupation and two industries, one of which produces for home
consumption and the other for export. Generalization to the case of
several occupations and industries is considered in section I.C.
The model used to justify our valuation of amenities consists of a
large number of cities, each one of which contains a business district
surrounded by residential areas. Below, it is assumed that each city is
circular with the business district at the city center; however, our
results continue to hold as long as all industry is located in a single
area and residential districts are indexed by their distance from this
area.
Within each city live identical workers who can costlessly migrate
from one city to another, but who must work in the city in which they
65
-------
reside. Outside of cities live landowners who rent land within the city
boundaries to workers and firms, the capital owners who own the capital
equipment used by firms.
For simplicity it is supposed that the size of the CBD and the boundary
of the city are both fixed. Thus what is analyzed is a short-run
situation where the period of analysis is long enough to allow workers to
move freely from one city to another but not long enough to allow the size
of the city to adjust to this migration. This short-run equilibrium
persists until the city re-zones agricultural areas as residential districts
and provides them with various public services (sewers, water, electricity).
Since it is unlikely that real-world data reflect a long-run equilibrium
situation, the assumption that the city boundary is fixed does not seem
inappropriate for empirical work.
For the purposes of empirical work it is also convenient to assume that
the land in the city center is located at a single point in space so that
no distinctions need be made among locations in the CBD. This may be
defended on the grounds that land in the CBD of a city is usually small
relative to the total area of the city. All land in the center of city i
is thus assumed to rent at the same price. The spatial character of the
rest of the city is acknowledged by expressing the rent on land in
residential areas as a function r.(k) of k, the distance of the annulus
from the boundary of the CBD.
A. Assumptions Regarding Workers
We shall assume that workers in all cities are identical and work a
fixed number of hours in the CBD of the city in which they live at a wage
of w. per period. Each period the worker makes a fixed number of trips
from his home to the CBD. In urban location models it is customary to
assume that the cost of commuting from the residence to the CBD is an
increasing function of distance traveled but does not depend on the
worker's income. This assumption, however, is incompatible with the log-
linear utility function employed below, which implies that a constant
fraction of income is spent on transportation. To be consistent with that
utility function transportation is treated as another good which the
individual purchases, and commuting costs are not subtracted from income.
The disutility associated with commuting is instead captured by including
the term k~^ in the utility function.
It is assumed that each worker receives utility from the size of his
residential site, q, from the quality of local goods consumed, x, and
from y, the amount of imports consumed. Utility is also received from
site-specific amenities, which may vary from one location to another
within the city.
In general, the fact that individuals in the same city are exposed to
different levels of crime, pollution, and even temperature, leads to
problems of aggregation when cities are the units of observation in
empirical work. This poses no problem here as long as the value of each
amenity at location k can be expressed as the product of the value of the
66
-------
amenity measured in Che CBD and a dispersion function which describes
how the amenity varies with distance from the point of measurement.
In the case of industrial pollution, for example, emissions are generated
in the CBD and spread to other parts of the city. Pollution at location
k can therefore be written P^a^(k) where P^ is pollution measured in the
CBD and a^(k) is a function which is decreasing in k.
Following this approach we denote by Aj_a-j_(k) the level of amenities
which the individual experiences at his housing site, k. (For convenience,
only a single A^ai(k) is included in the utility function.) The level of
amenities in the CBD, A-, enters the utility function separately since
most amenities which are consumed at home are enjoyed at the work site
also.
Since the individual takes locational amenities as given, utility
U,. = Bq6xUlya2Ai;T[Aia.(k)]6k-\ a 1+1*2+ 8 = 1 ^'^
will vary, for constant q, x, and y, according to the city and neighborhood
in which the individual lives. For any location (i,k) the individual can
determine his maximum utility be choosing q, x and y to maximize (4.1)
subject to the constraint:
w. = r.OOq + P .x + P y, (4.2)
where the prices of land, local goods, and imports are all taken as given.
The utility maximization problem yields demand functions for residential
land and for x and y. These can, in turn be substituted into (4.1) to
yield the indirect utility function:
~u f i *\ P fi > D J-T> «-* 7 i /i \ i ^ r n n 1 /
1 i i li 2i 'i ' ' 2 '(4.3)
which gives the level of utility in each neighborhood of each city as a
function of site-specific amenities, income and prices.
The fact that individuals are free to choose their residence implies
that in equilibrium the level of utility V.(k) must be identical in all
locations. Furthermore, if city i is small relative to the size of the
country, Vj_(k) may be regarded as exogenously determined and hence
V.j_(k)=V* for all i and k. Worker mobility thus implies that rents,
wages and the prices of local goods must adjust to compensate for
differences in amenities across locations. The extent of this adjustment
depends on how much individuals value amenities, as reflected by the
coefficient n+6.
It might at first appear that n+6 could be inferred by solving the
locational equilibrium condition V^(k)=V* for w^ and estimating the
resulting equation using data across cities. Unfortunately this leads to
an equation involving land prices and amenities., which vary within, as well
as across, cities. This problem is solved, however, if (4.3) is used to
67
-------
derive the supply function for labor.
In order to obtain the labor supply function (4.3) may be solved
explicitly for r.(k) to give each individual ' s maximum willingness to pay
for land at location k,
r.(k) =
Since land will be sold to the highest bidder (4.4) also represents the
equilibrium rent function in city i. Now for the land market to be in
equilibrium the population (labor force) in city i must be such that the
demand for land at distance k from the CBD equals the supply. Equivalently ,
if 2irkdk is the fixed supply of land at distance k, then the number of
persons living in ring k, n(k), must satisfy:
2Trkdkr. (k)
(4.5)
Substituting for r , (k) from (4.4) and integrating from k=0 to k-k., the
fixed boundary of the city, yields the number of workers in the city as
a function of amenity levels and the wage,
N = /^(Mdk = Mw.(1-B)/ep -°1/BP ~°2/3A (ri+6)/!ff (k ), (4.6)
i x li 2i i i " i
where M = 2-nsT1 (C/V*)1/B and f . (k . ) = / V~C/Ba . (k) 6/ 6dk.
i i 0 i
Equation (4.6) is the supply function of labor in city i, which may be
used to estimate the coefficient of amenities in the utility function.
For purposes of estimation, however, it is convenient to write the labor
supply function in the form:
(w./P.)* = c + rr N.* - A.* - F.(k.), P. =(a +a )/(l-B), (4.7)
i i 1-6 i 1-6 i 11 i 1 ^
where asterisks denote logarithms of the variables. The variable on the
left-hand side of (4.7) is the real acceptance wage the money wage
in city i divided by a price index in which all commodities except
residential land are weighted by the fraction of the budget spent on each.
The acceptance wage is an increasing function of N^ since, if land is fixed,
an increase in population will raise rents and thus the income necessary
to maintain V* . Amenities such as sunshine and clean air enter equation
(4.7) with negative coefficients-, while disamenities, for which individials
must be compensated, increase the acceptance wage.
68
-------
Note that due to che multiplicative nature of utility only the value of
amenities in the CBD appears in the supply function. The dispersion
function a-j_(k) which captures the fact that individuals in each city are
exposed to different levels of amenities, is subsumed jin F^Ck^). Since
in the short run k. is exogenous, we shall regard _t_he k. as independent
drawings from a probability density function. F.j_(k.) can then be regarded
as an error term which is independently though not identically distributed
for all cities. If, however, the dispersion functions are identical in
all cities, then the error terms F(k.) will be independently and identically
distributed for all i.
The coefficient of amenities in the utility function can therefore be
estimated by regressing the real wage in city i on employment and on
amenities in city i. In order to obtain consistent estimates of n+<5 >
however, it is necessary to first identify factors which determine the
demand for labour in each city.
B. Assumptions Regarding Firms
Rather than develop a model which explicitly treats firm migration we
assume that there is a production function for industry X and for industry
Y in each city. Differences in natural resource endowments, transportation
costs and locational amenities lead to differences in production costs among
cities which, in turn, explain the growth of industry in each city.
written :
For city i the production function of the export industry may be
Y. = D N^.L^.K^.S^.E^., a+b+c < 1, (4.8)
i 2 ^i 2i 2i 2i ^i
where L~ . denotes land and other raw material inputs, N? . , labor inputs,
K~ . , capital goods, S? . , pollution generated by the industry and E .
environmental goods which affect the production process. The latter
might include climate or the level of air pollution in the city. Population,
N., may also enter the production function as a proxy for agglomeration
economies if these are relevant for industry Y.
We shall assume that industry Y behaves as a price-taker in all markets.
Thus given output price, input prices, and a tax on effluents, the industry
determines profit-maximizing levels of inputs L, N and K and a level of
emissions, S. Industry X behaves analogously.
Although each industry regards input and output prices as exogenous,
the wage, the price of land in the CBD, and the price of local goods are
determined by equilibrium conditions, in product and factor markets in
city i. Equating the aggregate demand for land in the CBD to the size of
the CBD, the aggregate demand for labor to the right-hand side of (4.6)
and the supply of X to the aggregate demand for X yields a system of three
equations which may be solved for the price of land, the wage, and the price
of X. The equilibrium level of employment (population) may be found by
substituting the equilibrium wage into (4.6) and the quantity of local
69
-------
goods produced obtained by substituting P . into the aggregate demand
function for X. 1
Environmental goods which depend on output or on population are also
determined by market equilibrium conditions. The level of pollution in the
CBD of city i may be expressed as a function of industrial emissions,
S +S and weather conditions in the CBD. Crime, xohich depends on
population and on the wage, must also be regarded as endogenous.
In the model outlined here the size of industry in city i, and hence
the demand for labor, depends on the parameters of the production function
and on input and output prices. For the purposes of empirical work,
however, it is the exogenous factors which determine the size of industry
that are important. These enter the model through the variable E. and by
affecting the prices of capital goods, natural resources, and thexprice of
exports.
As indicated above the output of industry Y is sold in national
markets at a price p which may be regarded as exogenous to each city.
The price received by firms in city i, however, will fall short of p
by the cost of shipping Y to market. Since shipping costs depend on the
distance of city i from the central market and on the intervening top-
ography, one would expect the demand for labor to be higher in cities close
to output markets which have access to cheap sources of transportation.
The prices of natural resources and capital goods, which are
assumed to be traded in centrally located markets, may also be regarded
as exogenous to firms in city i. The delivered cost of these inputs
(and hence the damand for labor) depends on the proximity of the city to
input markets and on the feasibility of using low-cost means of transporta-
tion, e.g., water v. air.
Finally, the demand for labor should be higher in areas where land
prices are low. Although the price of land in the CBD is endogenous to
city i, it is affected by the size of the CBD and by the property tax rate,
both of which are determined by the government in the short run and are
treated as exogenous in our model.
C. Generalization to Several Occupations
The model of sections A and B, although locically consistent, is based
on assumptions which are difficult to accept in empirical work. By
treating all workers as identical the model ignores variations in skill
levels and job experience which explain a large proportion of variation
in wages across cities. The model also imposes the stringent requirement
that all individuals have identical preferences. These assumptions may be
relaxed by estimating labor supply functions for separate occupational
groups; however, it mus.t first be demonstrated that the coefficients of the
disaggregated labor supply functions have the same interpretation as the
coefficients, of equation (4.7).
Suppose in the model above that there are several classes of workers,
70
-------
with each class possessing different skills or years of job experience.
This means that a distinction will have to be drawn among categories of
labor in the production functions for X and Y, with each type of labor
entering the production function with a different coefficient. There will
as a result be a separate demand function for each type of labor;
however, as long as factor markets are perfectly competitive, generalization
to several occupational groups is straightforward.
Deriving the supply functions for labor presents more difficulties.
Suppose for simplicity that members of each occupational group are
identical and work a fixed number of hours in the CBD at the wage paid
to their group. While workers within each group have the same tastes,
it seems reasonable to allow preferences for consumption goods and
amenitities to differ among groups. The indirect utility function for
each group will thus be of the form;
-S. -cc . -a2. n. + S. 6. -C.
V..OO = C.w .r (k) Jp.. JP0. JA J Ja(k) Jk J (4.9)
ij 3 13 i li 2i
where parameters are subscripted to allow for differences in tastes among
groups.
As in the case of a single category of labor, the labor supply
function for each occupational group is derived from that group's location
decision. In locational equilibrium all members of the occupational class
must experience the same utility regardless of the neighborhood or city in
which they live. Thus V^- must be constant for all i and k and equal to
V-'. (If each city is small and open, the Vv can be considered exogenous
to the city.) This equilibrium condition is used to determine where in
each city members of group j will live. The group's labor supply function
is then derived by summing the number of persons in each neighborhood,
n(k), across all neighborhoods k in which members of the group reside.
The crucial step in the above procedure is determining the spatial
distribution of occupational groups within each city. Equilibrium in the
land market requires that land at each location be sold to the highest
bidder. To determine the bid function for each occupation the locational
equilibrium condition [(4.9) with vi-j=V*] may be solved for r.j-(k), group
j's maximum willingness to pay for land at each location. Under certain
assumptions these bid functions, if plotted against k, will be downward-
sloping and will intersect any number of times. Each city will thus be
divided into neighborhoods which are segregated on the basis of occupation,
with neighborhood boundaries determined by the intersections of the r^-OO's.
Summing the number of persons per annulus, n(k), across all k at which group
j resides (K .). yields group j's supply function for labor,
(1-B )./B -a /31 -«2./3. (Hj+6 )/(3 , ,1-^/^1 YBi
N = Mw J Jp,. V JP0- J JA. J J 3 J k J Ja(k) 3 Jdk.
U J ij li 2i x k£K
1J (4.10)
71
-------
The trouble with this procedure is that the boundaries of the group j
neighborhoods, which are determined by the intersections of the r..(k-)'s
cannot be treated as exogenous but themselves depend on w... The Integral
on the left-hand side of (4.10.) cannot therefore be regarctid as a random
error term, and omitting it from the equation will bias the coefficients
of N and A in (4.11).
ij i
B.J n.,+6.
, .1. I 1 i j_ r_
+ "ij. (A.11)
How serious this problem is depends on the extent to which current
neighborhood boundaries depend on current wages and levels of amenities.
To the extent that they do not the limits of integration in the supply
function may be regarded as independent of w.. and A., and the integral
in (4.10) may be treated as a random error term.
4.2 Empirical Specification and Estimation of the Model
The model of section I implies that one may value urban amenities
by estimating labor supply functions of the form (4.11). To illustrate
this approach supply function were estimated for one-digit occupational
categories using data from the. 1970 Census of Population. The results
of these regressions are presented below following a description of our
empirical model.
A. Specification of the Labor Supply Function
To estimate equation (4.11) one must find empirical counterparts to
the amenities A. which influence consumer location decisions. One group
of variables foind to be important in previous studies are the amenities
and disamenities associated with urbanization. Most regressions, for
example, include air pollution, crime and congestion (population density)
as measures of the disamenities of urban life while using some index of
availability of goods and services (number of sports franchises, number
of TV stations) to capture the advantages offered by large cities. In
the context of our urban location model all amenities and disamenities
associated with urban scale should be treated as endogenous variables.
Our small sample size (n=28), however, makes it difficult to treat more than
one or two variables as endogenous. Scale amenities must therefore be
treated as exogenous, causing simultaneous equations bias, or must be
omitted from the equation altogether.
To resolve this problem air pollution, measured by the arithmetic
mean of sulfur dioxide, is included in the labor supply function as an
endogenous variable. Crime is also included but is treated as exogenous
on the grounds that crime, rates, are affected by law enforcement practices,
by the racial composition of the population, and even by climate (Hoch),
all of which are exogenous to the model of section I. The only measure
of urban amenities explicitly included in the regression equation is
availability of health facilities number of hospital beds per 100,000
and number of doctors per 100,000. Unlike other measures of availability
72
-------
of goods and services these variables are not very responsive to
variations in income and can more reasonably be regarded as exogenous.
Scale amenities which are omitted from the labor supply function will
be captured in part by the endogenous employment variable, N... In
equation (A.11) this variable represents the effect of land prices on
wages and is expected to have a positive coefficient. If, however, N-^-j
enters the utility function as a proxy for scale amenities then its
coefficient should be wirtten (8-y)/(l-j3) where y represents the net
effect of scale amenities. If the amenities of urban life outweigh the
disamenities then the sign of employment may actually be negative.
Other factors which are likely to affect location decisions are
climate and scenic beauty. Although these variables can truly be regarded
as exogenous, high correlation between individual amenity measures, together
with a small sample, makes it difficult to include all relevant variables
in the regression equation. Of the one dozen climate variables considered,
only the two most significant, average July temperature and wind velocity,
appear in the final equation. These variables should therefore be
regarded as proxies for the amenities of climate, and their individual
coefficients should be interpreted with caution.
A similar situation arises in the. case of scenic amenities. Scenic
amenities, which may be measured by proximity to the ocean or to the
mountains, are closely related to the availability of recreational
facilities (beaches, parks, skiing). Unfortunately the measure of
recreational facilities used in our empirical work, number of national
parks, state parks and national forests within 100 miles of each city,
was highly correlated with a dummy variable = 1 if the city was located
on the ocean and with a dummy variable indicating the availability of
beaches. To avoid collinearity problems only a single variable, the
coastal dummy, was retained in the final equation. Its coefficient should
therefore be interpreted as a proxy for both recreational and aesthetic
amenities.
An additional category of amenities to be considered is employment
opportunities within each city. In our theoretical model employment
opportunities are captured entirely by the wage rate w. In reality,
markets are imperfect and individuals must consider the" probability of
being unemployed. For married males the relevant variables are the un-
employment rate in the individual's own occupation as well as some indicator
of employment opportunities for women. If the ratio of females to males in
the labor force were identical, in all cities, then the ratio of females to
males actually employed would indicate the availability of jobs for women.
This variable, first suggested by Getz and Huang, appears in one set of
regressions reported below. An alternate measure of employment opportunities,
which is more in the spirit of our model, is the real median earnings of
women in ea,ch city. This is included in the labor supply functions of
blue collar males, as reported in Table II.
While both measures of employment opportunities for women are signific-
ant for some occupations, the unemployment rate for males is not and has
73
-------
been deleted from the labor supply function. The poor performance
of the unemployment rate is probably due to the fact that aggregate un-
employment is of little significance to members of specific occupations.
Unemployment rates for one-digit occupations are, unfortunately, un-
available for the year 1970.
B. Identification of the Labor Supply Function
The model of Section I implies that the labor supply function must be
estimated as part of a simultaneous equation system in which the real
wage, employment, and air pollution are endogenously determined.
Exogenous variables in the system which affect the location of industry
but not of workers may be used to identify the labor supply function.
The discussion in I.B suggests at least three such variables availability
of raw material inputs, proximity to output markets, and availability of
cheap transportation. The empirical counterparts of these are used as
excluded exogenous variables in the 2SLS estimation of (4.11).
Availability of raw materials is measured by the value of farm
products, the number of acres of commercial timberland and by value added
in mining, all measured for the state in which the SMSA is located.
Proximity to other cities is measured by the percent of goods (by weight)
shipped at least 500 miles from the SMSA and by the percent of goods shipped
within 100. miles of th SMSA boundary. High values of the former variable
should indicate that a city is isolated from output markets, whereas high
values of the latter should indicate the reverse. A dummy variable
equal to 1 if the city is a port is included to indicate availability of
cheap transportation.
Finally, as noted at the beginning of section I, the size of each
city is regarded as fixed in our model on the grounds that we are dealing
with a short-run equilibrium situation. Since -land prices will affect
the growth of industry, city size (in acres) and the effective property
tax rate are both included as excluded exogenous variables in the estimation
of the labor supply function.
C. Estimation of the Labour Supply Function
The labor supply functions presented in Tables I-III have been
estimated using 1970 Census of Population data for 28 of the 39 cities
for which BLS Cost of Living indexes are available. (A list of these
cities and a description of data sources appear in the Appendix). In
each of the regressions the dependant variable is the median earnings of
all males who worked 50-52 weeks in 1969. The wage variable in each case
is deflated by the BLS intermediate budget cost of living index, with the
price of housing removed from the index, as indicated in I.A.
By including only those individuals who worked for the entire year, and
by estimating labor supply functions for specific occupations one is able
to control for some of the factors other than amenities which account for
inter-city variation in wage rates. Median earnings, however, may vary due
to differences in union membership, in educational levels and in years of
74
-------
job experience. Since data on union membership and on the ratio of union
to non-union wages are available by region for one-digit occupations it is.
possible to adjust the earnings, variable using the formula:
w = (l-a)w . + aw , (4.12)
observed non-union union
where a represents the percentage of workers in unions. The non-union wage,
obtained by solving (12), is the dependent variable in the regressions for
blue-collar occupations.
To test the significance of human capital factors and racial dis-
crimination in explaining variation in wages, median earnings in each
occupations (undeflated by the cost of living index but adjusted for
union membership) were regressed on the average age of workers in the
occupation, on the percent of non-whites in the occupation and on the
average school years completed by all males in the SMSA. In all cases
the years of schooling variable, which is unavailable by occupation, was,
not surprisingly, insignificant. The average age of the x^orkforce,
however, was positively related to the money wage for all occupations
and was significant at the .05 level in all but two cases. Percent non-
white was highly significant, with the expected negative sign, for
laborers and service workers-, the only two occupations employing a high
percent of non-whites.
In the context of our model it seems most appropriate to treat
average age and percent non-white as exogenous variables which affect the
productivity of labor, as perceived by firms. Average age and percent
non-white are therefore included as. exogenous variables in estimating the
labor supply function, the former for all occupations except managers and
the latter for laborers and service workers only.
Finally, wage rates- may vary .across cities due to disequilibrium
movements in workers and firms not allowed for in the model of section I.
For example, an increase in the demand for labor in city i will put upward
pressure on the wage rate and should be accompanied by an inflow of workers
into the city. To allow for this possibility the net migration rate is
included as an explanatory variable in one set of regressions.
4.3 Empirical Results
An important question to be answered by our empirical model is which
groups of variables are most important in individuals' location decisions.
A related question is whether these variables are the same for all
occupational groups. To answer these questions Table 4.1 presents
regression results for nine occupations with the same set of variables
appearing in each equation.
In examining these results one must be careful to interpret individual
variables as proxies for groups of ameni.ties. Viewed in this way scenic
amenities (coastal dummy), scale amenities (employment), and the availability
of health facilities seem to be the most important factors- in location
75
-------
Table /«. 1
Ks t: J in,') I Oil Labor Supply Kunc I .i on:-;
(n = 28)
Constant
F.niployment
S02
July temperature
Wind velocity
Doctors/100 ,000
Hospital beds/3 00, 000
Crimes/100, 000
Feina le/Male Employment
Coastal Dummy
R2
All
Earners
6.0536*-*
(1.4295)
0.0273**
(0.0160)
0.0219*
(0.016.1)
-0.4397***
(0.1327)
-0.1087**
(0.0576)
-0.1381**
(0.0681)
-0.0651**
(0.0338)
+0.0743**
(0.0349)
-0.0613
(0.1206)
-0.06 39 *
(0.0249)
.7429
Profession;! I
Worker?;
4.8012***
(1.7002)
0.0257
(0.0196)
0.0231
(0.0193)
0.0392
(0.1584)
-0.1545**
(0.0674)
-0.1065
(0.0807)
-0.0376
(0.0399)
0.10 70 <
(0.0422)
0.0335
(0.1443)
-0.0192
(0.0303)
.5916
Non-Kami
M'liiof.crs
5.947?-"*1-
(1.6454)
0. 034 2 '-'
(0.0185)
0.0255*
(0. 01.80)
-0.0768
(0.1525)
- 0.1 507 --
(0.0658)
-0.1008
(0.0739)
-0.0228
(0.0392)
0.0783*-
(0.0403)
0.0956
(0.1394)
-0.0690**
(0.0286)
.6047
Sales
Workers
4.303?
(1.9612)
0.0365**
(0.0206)
0.0151
(0.0204)
0.0386
(0.1815)
-O.OS55
(0.0781)
-0.0031
(0.0941)
-0.0637---
(0.0469)
0.0503
(0.0472)
0.0094
(0.1651)
- 0.09 38- :--
(0.0338)
.5499
Cluricnl
Ivorkors
4.0B9"-"
(1.5067)
0.0233*
(0.0ld4)
0.0209
(0.0170)
-0.2247*
(0.1397)
-0.07] 7
(0.060S)
-0.1156*
(0.0721)
-0.0380
(O.OT>5)
0.0709"*
(0.0367)
-0.1900*
(0.1277)
-0.0462**
(0.0265)
.5637
(continued)
Note: All variables are in natural logarithms.
*-'* = Significant at .01 level, one-tailed test.
** = Sic.nific.-int at .05 love]., one-tailed tost.
* = Significant at .10 level, one-tailed test.
76
-------
(n - 23) Craftsmen
Constant 1, . 3 4 1 9 **
(1.7232)
Employment 0.0360"*
(0.0200)
SO 0.0.185
(0.0192)
July temperature. -0.4680***
(0.1583)
Wind velocity -0.0904
(0.0&95)
Doctors/100,000 -0.1241*
(0.0814)
Hospital beds/ -0.0439
100,000 (0.0407)
Crimes/100,000 0.0496
(0.0414)
Fern,-} 1 e /M.-s 1 c -0.334 9**
Employment (0.1459)
Coastal Dummy -0.0869***
(0.0299)
R2 .7508
Table 4. 1
(coiu iniuTl)
Opernt: ives
2.4042*
(1.5790)
0.0014
(0.0.163)
0.0242*
(0.0179)
-0.4 339*
(0.144.1)
-0.0352
(0.0635)
-0.0401
(0.0736)
-0.1021***
(0.0368)
0.0048
(0.037/0
-0.554S***
(0.1356)
-0.04 10*
(0.0262)
.7998
Non-Fa rni
Laborers
6.4412***
(1.4466)
0.0344**
(0.0.143)
0.0340**
(0.0.150)
-0.8984***
(0.1332)
0.03.1.4
(0.0575)
-0.1657**
(0.0680)
0.00.14
(0.0338)
0.0335
(0.03/.7)
-0. 2327**
(0.1217)
-0.0293
(0.0250)
.8873
S . r v i c c
Workers
7.8618*'.
(2.2571)
0.0386*
(0.0242)
0.04SS*;
(0.0250)
-0.8524*='
(0.2117)
0.0342
(0.0899)
-0. 232.1 *'
(0.1071)
-0.0267
(0.0529)
0.0832
(0.0554)
-0.0220
(0.1905)
-0.0054
(0.0398)
.7366
Note: A.11 varinlilcs are in natural logarithms.
**:'. = Significant nt .01 level, one-tin i leil tosl.
** = Significant at .05 level, one-tailed test.
* = Significant at .10 level, one-tailed test.
77
-------
decisions.: Each, of thes,e variables, consistently has the expected sign and
is. asymptotically significant at the 0..1Q level or better in six out of
nine regressions.
The behaviour of employment is of particular interest since it is
this variable which represents the effects of city size. In all occupations
the coefficient of employment is positive, which would seem to imply that
individuals must be compensated for living in large cities. One must,
however, be cautious in drawing this conclusion. The coefficient of
employment in the labor supply function fi , depends not only on y>
the coefficient of city size in the utility function, but on (3, the
proportion of income spent on the housing site. Specifically,
B! = (3-Y)/(l-3).. (4.13)
Given |3 and 3 , equation (4.13) may be solved for y5 which is clearly
increasing in both variables. The smallest^value of y implied by Table I
occurs when j^ = .0014. Note that even if 3 were only .03, Y would still
be positive (although small) indicating that cities yield net amenities to
consumers. This conclusion, however, must be qualified by the fact that
crime and air pollution, two disamenities partially associated with city
size, are included separately in the regression equation and are often
significant and positive.
One must also be cautious in interpreting the variable doctors/
100,000, which may represent amenities other than health facilities. The
coefficient of this variable is particularly large for laborers and
service workers, groups for whom scenic amenities do not appear to be
significant. Conversely, in cases where MD's is insignificant the
coastal dummy is significant. This, suggests that MD's/100,000 may act
as a proxy for scenic amenities, an hypothesis which is not unreasonable
if doctors take part of their income in the form of locational amenities.
This hypothesis is also strengthened by casual inspection: San Francisco,
Denver and New York are among the cities with the highest number of
doctors per capita, whereas Wichita, Kansas is the sample minimum.
Of the remaining variables, crime is significant in five equations
and is clearly more important for white-collar than for blue-collar workers.
Air pollution, measured here by sulfur dioxide, has the expected positive
sign for all occupations but seems to be more significant for blue-collar
occupations. If this result appears surprising, it should he remembered
that blue-collar workers are more mobile than highly-paid white-collar
workers, whose location decisions are likely to depend on job-related
amenities. Pollution and other locational amenities are therefore more
likely to appear significant in the labor supply functions for blue-collar
occupations.
This reasoning may explain why climate, variables do not appear to be
very significant for white-collar workers. (The two exceptions in the
case of wind velocity are most likely due to the effect of wind on air
quality.) For blue-collar workers average July temperature is highly
significant and appears as an amenity in all cases. The extremely large
78
-------
coefficients of temperature may be due to the variable acting as a proxy
for other climate variables or, since July temperature is higher in Southern
cities, as a proxy for the large supply of unskilled labor often used to
explain the lower level of wages in the South.
The remaining variable in the supply function, the ratio of female
to male employment, is the more significant of the two measures of
employment opportunities for x^omen. As indicated in Table I this variable
is not significant in the supply functions for highly-paid white-collar
workers but is significant for clerical workers and for most blue-collar
occupations, implying that the importance of employment opportunities
varies inversely with the husband's income. It is interesting to note
that these results are similar to those of Getz and Huang, who find female/
male employment to be highly significant in labor supply functions
estimated from the same set of data.
The res.ults of using median earnings for women in place of female/
male employment are reported in Table II. Female earnings is significant
for only two occupations (operatives and laborers) but has a market
effect on the coefficients, of other variables whenever it is included in
the equation. In general the coefficients of other amenities increase
in absolute value and in significance. This may be the result of high
pairwise correlations between female earnings and employment, crime, and
doctors per 100,000 which are not present when female/male employment
is used. For this reason the results presented in Table I should be viewed
as more reliable.
To test the possibility that wage data reflect disequilibrium movements
of workers, the equations in Table I were re-estimated with net migration
included in non-log form. The net migration variable was significant only
for while-collar occupations and these results are reported in Table 4.3.
In all cases- net migration has a positive sign, suggesting that wages for
white-collar workers are higher in some cities due to an increase in the
demand for labor to which workers have not fully adjusted. Adding net
migration to the equation does not drastically alter the conclusions of
Table 4.1, but does affect the relative importance of the pollution and
employment variables. Sulfur dioxide is now significant in three out of
four white-collar occupations., whereas employment is significant only in
the aggregate labor supply function. This result is probably due to the
positive correlation between employment and air pollution, which makes it
difficult to separate th.e effects of the two variables.
The Valuation of Environmental Amenities. An Illustration
We shall not illustrate, using the results of Table 4.1 - 4.3, how
valuations, of locatipnal amenities can be inferred from the coefficients
of the labor supply function. In the model of section I. a given percentage
change in A. in. th.e CB£> of city i implies an equal percentage change in the
amenity throughout the city. The. amount an individual ia willing to pay
for this change may be defined as the largest amount of income one can take
away from the individual without altering his utility. If th.e change in A.
is so small that it does not affect prices, in city i then willingness to
79.
-------
T.'ll.U: '. . 2
Labor Supply Kiiiu: r iuns of 151 uo.-l"!o I l.i r Wiirl.i !'.;
(n = 2S)
Constant
Employment
so2
July temperature
Wind velocity
Doctors/100, 000
Hospital beds/
100,000
Crimes/100, 000
Median E.irninp.s,
Females
Coastal Dummy
R2
Cr;i f tstiK-'M
8. 12 98 ***
(1.0665)
0.0527="*
(0.0219)
0.0231
(0.021/,)
-0.5682***
(0.1709)
-0.1234*
(0.0776)
-0.2128"-*
0.0780
-0.0319
(0.0453)
0.068/t
(0.0525)
-0.1659
(0.1743)
-0.0817**
(0.0333)
.6849
Oj)L't'at ivos
8.9434***
(1.0216)
0.0228
(0.0182)
0.0402**
(0.0212)
-0. 59 6 A* -:*
(0.1658)
-0.0890
(0.07/i6)
- 0.170. V"-"'
(0.075/0
-0.0921**
(0.0/435)
0.0639
(0.0505)
-O./i 29 9 "**
(0.1637)
-0.0290
(0.0308)
.7155
Nnn-l-'.'i nil
Laborers
9. 2 9 10 *'
(0.8/.01)
0. 0/t/i 3***
(0.0150)
0.0410***
(0.0155)
-0.9733***
(0.1339)
0.0095
(0.0600)
-0.222]*"*
(0.0611)
0.0038
(0.0353)
0.0642*
(0.0409)
-0.1989*
(0.1355)
-0.0265
(0.0259)
.8764
S i r v i < o
U'urluTs
S. 1 321***
(1.2702)
0.0453**
(0.0254)
O.O/, '
-------
To Me. 4.3
L.tbor Supply Finnic, ions of Hii ice-Col 1 :ir V.'orkrrs
All. !
(n - 27)
Constant
Employii'ip.n t
so2
July temperature
Wind velocity
Doct.ors/100,000
llospic.nl beds/
100,000
Criincr./lOO.OOO
Female/Male
Kmploviiient
Coastal Dummy
Net Migralion
7
K
F.ar
5.
(1.
0.
(0.
0.
(0.
-0.
(0.
-0.
(0.
-0.
(0.
-0.
(0.
0.
(0.
-0.
(0.
-0.
(0.
0.
(0.
ncrs
302S***
4216)
0277*
0188)
0263*
0188)
4879---"*
1305)
0704
0593)
1488**
06S9)
0513*
0363)
0702**
0343)
1831*
1298)
07 iS***
0243)
0022**
0012)
7873
'ro i et;;;ional
Ncm- i;,-'.-;«
Wo rkor s Man.n f,o r s
3.
(1.
0.
(0.
0.
(0.
-0.
(0.
-0.
(0.
-0.
(0.
-0.
(0.
0.
(0.
-0.'
(0.
-0.
(0.
0.
(0.
6120**
3818)
0118
0215)
0332*
0210)
0478
1474)
0832*
0659)
0831
0755)
0346
0393)
0887**
0385)
161S
1447)
0315***
0275)
0036***
001.4)
7037
5 .
(1.
0,
(0,
0.
(0.
-0,
(0.
-0.
(0.
-0.
(0.
-0.
(0.
0.
(0.
-0.
(0.
-0.
(0.
0.
(0.
,0071***
6746)
.0206
.0224)
0338**
,0216)
, .1421
1537)
0937*
0698)
0802
0825)
0253
043V)
0631'*
0406)
0539
1538)
079/,***
0286)
,0030**
001.4)
6599
S;
ilo.s
WorkLM-s
3.
(1.
0.
(0.
0.
(0.
-0.
Co.
-0.
(0.
-0.
(0.
-0.
(0.
0.
(0.
-0.
(0.
-0.
(0.
0.
(0.
4306*
9937)
0322
0244)
0232
0240)
0176
1325)
0382
0823)
0086
0989)
0523
0324)
0450
0482)
1300
1822)
10 1.4 "
0338)
0026*
0017)
6086
c: i c
V.' ( 1 !
3.
(3-.
0.
(0.
0.
(0.
- 0
(0.
- 0
(0.
-0.
(0.
-0.
(0.
0.
(0.
-0.
(0.
-o.
(0.
0.
(0.
- r i c a 1
kors
«
5794)
0030
0200)
03! 3*
0210)
2622**
1449)
0304
0661)
0790
0769)
0477
0398)
0370*
0382)
2756
.1454)
04 9.1* *
02/3)
0017
0013)
5947
*>'<* = Asymptot u::i!]y significant: at the .01 level.
** = Asymptotical ly p i f,ni f ic.nn t at the .05 level.
* = Asymptotically significant at the .10 level.
81
-------
pay, Aw., is defined implicitly by:
.. . .
11 li 2i i ^
6Pii"alp2i-a2(Ai+AA.)n+Sa(k) k~C. (4. 14)
This can be simplified to:
Aw. = w.[l-(l+k)~(n+6)], k = AA./A. (4.15)
11- 11
where k denotes the proportional change in A .
i
Willingness to pay can thus be computed solely from knowledge of income
and the exponent of amenities in the utility function. To estimate n+6
from the coefficient of A^ in the labor supply function, -(n+<5)/(l-3) ,
requires knowledge of 3, the proportion of income spent on the
residential housing site. If employment acts as. a proxy for scale amenities
3 cannot be inferred from the coefficient of Nj_, however, valuations of
A^ can be computed for alternate values of g.
To illustrate the use of (4.15), willingness to pay for one-, ten-,
and twenty-percent changes in selected amenities are shown in Table 4.4
for an individual whose yearly income is $9,000.. These figures are based
on results reported in Table 4.1, and, in view of the discussion above,
should be interpreted with caution.
Table 4.4 implies that an individual with the same preferences as
a manager would be willing to pay between 0.68% and 0.80% of his income
for a 10% reduction in the total crime rate. Since the cost on insuring
one's possessions against theft is already included in the cost of living
index, this valuation represents the phychic disutility attached to crime.
These figures correspond closely to valuations of crime obtained by Rosen
(1977), who estimates that individuals would be willing to pay between 0%
and 1.16% of their income for a comparable reduction in the crime rate.
The coefficient of violent crime in the labor supply functions estimated
by Getz and Huang, 0-.05, also suggests that our estimates of willingness
to pay are reasonable.
The value placed on a reduction in sulfur dioxide, although low by
comparison with crime, is higher than the figure obtained by Ridker and
Henning in their important study of air pollution in the St. Louis SMSA.
By regressing property value by census tract (1960) on site-specific
amenities, Ridkex and Henning estimate that a permanent decrease in SO,.,
by approximately 30% would raise the. value of an average home by $245.
Based on figures in Table 4.1 the present discounted value. of a 30% reduc-
tion in S02 , calculated for a person earning the median income in St. Louis
in 19.60, is between $418 and $489, or roughly twice the figure cited by
Ridker and Henning. One reason for this discrepancy is that under the
82
-------
Table 4.4
Valuations of Environmental Amenities
03
1
| Crime
I (Managers)
J
j -1%
i
!
.19 ! $5.86
-10%
$61.2
i
.10 1 6.51 j 68.0
i i
.05 I 6.87 | 71.8
! <
! i
i I
(
Sulfur Dioxide
J (Laborers)
i
-20% -1%
$129 ! $2.50
'143
2.77
151
2.92
-10% S
(
$26.1
29.0
30.6
\
i
July Temperature
(Operatives) ;
-20% +1%
1
i
j
$55.1
$31.1
61.2 ! 34.6
I 1
+10% | +20% i
I S
$294 j $554 |
326 613
| j
64.6 j 36.5 344 j 646 i
j I * £
L
i ! i
NOTE: All figures represent annual values of willingness to pay, computed for an individual
with an income of $9,000.
-------
assumptions of I.A. our figures capture willingness to pay for reductions
in pollution at the work site and at home, whereas the property value
approach measures willingness to pay at the residence only. Furthermore,
part of our estimate may represent willingness to pay for a reduction
in suspended particulates. Particulates, being highly correlated with
sulfur dioxide, are omitted from the labor supply function to avoid
problems of multicollinearity.
The least reliable estimates in Table 4.4 are those for summer
temperature. In Tables 4.1 - 4.3 July temperature appears as an amenity,
with individuals willing to give up income for above-average temperatures.
Since the coefficient of temperature for laborers and service workers
likely represents the effects of lower skill levels in the South, the
estimates in Table 4.4 are computed using the more moderate coefficient
for operatives. If evaluated at the sample geometric mean, 75°F, this
figure implies that an individual earning $9000 is willing to pay between
$294 and $344 per year for an increase in average temperature from 75°
to 82.5°F. While not unreasonable, this figure is higher than valuations
implied by hedonic price regressions (see Meyer and Leone) and should be
regarded as purely illustrative.
In the case of a dichotomous amenity, e.g., the coastal dummy,
equation (4.15) no longer applies and willingness to pay must be
calculated from
Aw. = w.(l-e?) (4.16)
where £ is the coefficient of the dichotomous amenity in the utility
function. Using (4.16) Table 4.1 implies that a manager will give up
between $660 and $770 if his income is $12,000. This figure, of course,
must be regarded as approximate since the coastal dummy reflects other
scenic amenities as well.
Finally, equation (4.15) may be used to infer how much of the
husband's earnings a family would be willing to give up in order to
increase the earning opportunities for the wife. Theory suggests that a
family should not give up an equal amount of the husband's earnings if the
shadowprice of the wife's time at home exceeds that of the husband. In
Table 4.2 the highest significant coefficient of female earnings is
-0.43, obtained for operatives. This implies that a male operative will
relinquish at most 4% of his earnings for a 10% increase in real female
earnings. If this figure should seem small, recall that it is based on the
behavior of all operatives, some of whom are not married or do not have
working wives.
4.4 Conclusion
This paper has presented a method of valuing environmental amenities
using a model which describes the location of workers within as well as
among cities. This allows us explicitly to deal with the fact that
individuals within the same city are exposed to different levels of
amenities. As long as individuals have log-linear utility functions the
84
-------
value of an amenicy to an individual located anywhere in the city
can be computed from the coefficients of an aggregate labor supply function
which includes the level of the amenity measured at a single point within
the city.
To illustrate the proposed method of valuing amenities labor supply
functions were estimated for nine occupations using data from the 1970
Census of Population. The results of these regressions are of interest
quite apart from the problem of valuing amenities since they indicate which
groups of variables are important in inter-urban location decisions.
Based on the signs and asymptotic significance levels of the regression
coefficients crime and scenic amenities, measured here by a coastal dummy
variable, seem to be the most important environmental goods in the location
decisions of white-collar workers. Pollution (S0?) is significant for
three out of four blue-collar occupations, and is~important for white-
collar workers if net migration is included in the equation. Employment
opportunities for females, whether measured by median real earnings of
females or by the ratio of female workers to male workers, seems to be
an important consideration in the location decisions of blue-collar
workers, as does the availability of health facilities (MD*s/100,000,
hospital beds/100,000). Surprisingly, climate variables do not seem
very important, especially for white-collar workers, although this
conclusion must be qualified by the fact that it is hard to separate the
effects of climate from other variables.
The original motive for this paper was to place a value on the
amenities and disamenities associated with urbanization. Subject to
certain qualifications, willingness to pay for reductions in crime and
air pollution are presented in section 3.3 above. While one would not
want to place too much confidence in the figures, it is clear that certain
groups of individuals must be compensated for these urban disamenities.
The same, however, cannot be said for the other effects of city size.
For all occupations the coefficient of the urban scale variable is positive,
which appears to indicate that urbanization yields not disutility. One
cannot, however, regard the coefficient of employment as the marginal value
of city- size. The latter, as shown sbove, is very likely positive,
indicating that the effects of urbanization not captured by other variables
yield positive utility.
85
-------
CHAPTER V
VALUATION REVEALING GUESSES: A REPORT ON THE
EXPERIMENTAL TESTING OF A NON-MARKET VALUATION PROCEDURE
by
William R. Porter and Berton J. Hansen
This paper describes a survey method that can be used to measure the
public's valuation of a public good. In its simplest form, the method
attempts to determine the aggregate valuation of a public good (or change
in a public good) by a group of consumers. It is designed to provide
each respondent with strong incentive to (a) consider the valuation
question seriously and (b) to disclose unbiased information about the
public good valuation.
The method consists of asking each surveyed respondent to guess as
close as possible to the "true average valuation" of the others in the
group. Before guessing each person is told that if his guess is within "a%
of the actual average of the other peoples' guesses that he will be paid
a large prize of 8 dollars. The change of winning the price provides
each respondent with the incentive to attempt seriously to guess the
average guesses of others, and since his most important information about
others' true valuations is his own valuation, his guess will, if properly
interpreted, reveal unbiased information about his own true valuation of
the public good.
The underlying hypothesis in such a technique is that people base
their guesses about the average of a characteristic in others on the level
of that characteristic in themselves plus a partial but unbiased belief
about their own relative position in the group.
Now since it is impossible to test such a hypothesis for a
characteristic like people's true valuation of a public good, we have
designed and conducted an experiment in guessing about the average of a
measurable, but not commonly known, characteristic of members in a well-
defined group. The results of this experiment were used in designing and
interpreting a survey method of public good valuation.
5.1 Description of the Experiment-in-Guessing
A random sample of students drawn from the population of students at
the University of California, Riverside (enrolled during the Winter Quarter
of 1978) were sent copies of the attached letter.
The students who responded to the letter were scheduled for individual
86
-------
appointments during weekday mornings where they were read the following
instructions and questions:
Procedure During Interview of "Experiment in Guessing"
[Establish identity of interviewee and close door for privacy].
[Record student control number ].
The questions I will ask you are related to the amount of money that
is usually carried by UCR students. Your answers will be strictly
confidential.
First, will you please count the amount of money (U.S. currency and
coins only) that you are now carrying. [Record amount (M)]
Second, what is the average amount of money that you carry in the
morning of a school day? [Record amount (A)]
In the following question you will have an opportunity to win $50.00.
Therefore, please pay close attention to what I will ask you to do, and do
not answer until you are sure that you understand the situation.
You are one member of a group of 20 UCR students who will answer
this question. Each of you will guess a number based upon a clue that I
will give to all of you. The one member of the group who guesses closest
to the average of the 20 guesses will win $50.00. Here is the clue: The
number guessed should be close to the amount of money that an average UCR
student carries in the morning of a school day. [If the student indicates
that he does not understand, then tell him: "You are to guess as close
as possible to the average guess of the others, realizing that all of you
have been given the same clue." Reread the clue].
What is your guess? [Record amount (G) ]
Thank you very much. That concludes the interview. As soon as we
calculate the averages for each group, we will notify the winners. That
will be in approximately 3 weeks. Thank you again for your help.
A total of 107 students were interviewed, and upon completion of the
interviews the averages were calculated, and the winners were notified and
paid their prizes in cash.
The objective of the experiment was to see if there was a systematic
relationship between the value of a person's guess G^ about others' average
behavior and his idea of his own average behavior N.J_. The idea being that
in the analogous public good method we would be attempting to measure the
unknown £Nj_ by using the known IG^. Therefore the fundamental question
is: What is the nature of the random distribution of ZG-j_ about the true
value IN^, and how does that distribution change as the sample size n gets
large?
Our purpose in asking the first question in the procedure concerning
87
-------
i was to focus each respondent's attention on the exact amount of money
he currently was carrying so that he could more accurately form a judgment
about the average amount he normally carries, N^. It also provided an
objectively measurable quantity IM^ as a check on the accuracy of beliefs
about one's average behavior.
The characteristic the average amount of money that one carries
was chosen for the experiment because it is something (like one's own
valuation of a public good) that is known by each about himself but is
very imperfectly known by each about others. Therefore when asked to
guess about the average of this characteristic in others, it is natural
to use one's own best knowledge (of oneself) plus some idea of one's
relative position.
The results of the experiment provide a strong indication that
people do base their guesses about others on knowledge about themselves
and that their aggregate guesses are very accurate estimates of the
average true value of the characteristic. The statistical results are
presented below.
(Student number 25 was removed from the sample because his money
carrying behavior was so extremely different than the other students
that we could not expect their guesses to take account of his behavior.
Student number 25 was carrying $423.87 at the time of the interview and
he said that he carries an average of $150.00 each day).
Mean value of "Average Amount Carried":
_ 106
N = -±- I N. = 5.6715
106 i=1 i
Mean value of "Average Guess":
_ 106
G = >: G. = 5.8075
106 i=1 x
Suppose we assume that the average amount carried by a student is a random
variable
(1) N. = u + n.> where p is the "true" average amount carried by the
entire population and n^ has a normal (0,o ) distribution.
Suppose each student's guess G. is a random variable defined by
(2) G. = N. + c., where e. has a normal (0,o2) distribution.
Then we can write
(3) G. = y + oj., where CD. ~ N(0,o2) and a2 = a2 + o2 + 2cov(n,e)
88
-------
In a procedure where we do not know the value N- (such as in the public
good case), then using equation (3) we can use the observations on the
G^'s and our knowledge of the distribution of w- to estimate the value of p.
Suppose we consider the measured N to be the true population mean u,
then the estimate of the variance of w. calculated from the data is:
62 = (3.812)2
w
Using this estimate we can calculate the sample sizes that are required
to achieve various levels of accuracy in the measurement of jj. _
Let R-o(a) denote the sample size required to be (100B)% certain that G is
within (100a)% of p. Therefore Rp(a) is the smallest integer n required to
. P
guarantee that
1 n
P[ - I G. - y < ecu] > 3
n i=l x
The following table shows selected values of R (a) .
P
Table 5.1
a
R (a)
.80
R (a)
.90
R (a)
.95
.10
.05
.01
75
297
7,425
123
489
12,225
174
695
17,355
By analogy, if the value of a2 is similar, then these numbers indicate
that using a guessing technique for public good valuation will allow us to
be 90% certain of obtaining a measure that is within 10% of the true
social value by interviewing as few as 123 randomly selected consumers in
the area. Of course, the value of a2 may not be the same, however, we will
9 CO
ob.tain an estimate of o as the interviews proceed, and it is possible to
use a sequential technique to determine when the sample size is sufficient
for a given level of accuracy.
A very important property of this procedure is that it provides a
measure of the accuracy of the estimate obtained, and it is impossible to
say the same thing of previously used methods. Further experimental studies
are needed to substantiate the unbiasedness property of this type of
procedure, however the results of our own experiment indicate that the
method is quite promising.
89
-------
Based on the results of the guessing experiment we propose the
following method of determining the public good valuation by a specific
group.
Public Good Guessing Procedure
For each of the selected respondents:
1. Describe the exact proposed change in the public good from level
A to level B in a \ then you will win a prize of (3 dollars. What
is your guess?
The potential bias in the two-stage procedure originates with the
possibility of strategic behavior in response to the first question.
Since the respondent is offered no incentive to answer truthfully to
question (1), [indeed, it is impossible here to use a prize as
90
-------
incentive for truthfulness since the respondent knows that there is no
method of verification] it is natural for him to consider the effect
his response will have on either a project approval or a project
financing decision. As soon as he forms a belief about this relationship
then he is rational to give a stated valuation that he believes will
influence the outcome in his favor. The fact that his subjective belief
about the relationship may be incorrect does not alter the fact that it is
costless for him to overstate or understate his valuation in the direction
of his own perceived interest, and therefore, he probably will. When he is
asked to guess the average stated valuations of the others, he will
immediately realize that they also had incentive to distort their
responses; hence, in order to win the prize, he must guess in the
direction of their distortions rather than toward what he believes is their
true average valuation. To argue that people are too unsophisticated to go
quickly through this complicated chain of reasoning when responding to such
seemingly hypothetical questions is to ignore the fact that even ordinarily
dull people become quite suspicious when their own self interest may be
involved. The result of this is that the average guess in the Two-Stage
Procedure is likely to be biased in an unpredictable direction.
In contrast to the Two-Stage Procedure, the proposed Public Good
Guessing Procedure offers no net incentive for strategic behavior. Each
person has incentive to guess a number that is as close as possible to the
average of the guesses by others. If each believes that the others are
trying (as the clue suggests) to guess close to the true average
valuation, then he will seriously attempt to guess near what they believe
is the true average valuation. Neither he nor they have any incentive
for over or under bidding; therefore, the average of the guesses is likely
to be close to the average of the true valuations. The results of the
guessing experiment suggest that this is indeed the case. Any incentive to
state a guess that will strategically affect the outcome of the public
good decision is offset by the incentive to win the cash prize, if the prize
is high enough.
5.2 Incentive Structure of the Proposed Public Good Guessing Procedure
In contrast to the Two-Stage Procedure, the proposed Public Good
Guessing Procedure does not reference people's guesses to previously stated
valuations or bids. Instead, it uses a simultaneous guessing method having
only the given clue, "the average true valuation of the public good by the
people in the described population," as a common reference point. Each
respondent knows that none of the respondents can exactly know the
"average true valuation;" however, each has incentive (in the form of the
prize) to attempt to guess what other people think this value is, since
the prize is won by guessing close to the average guess of others. The
respondents will use strategic behavior; however, in this case (if the
prize is large enough), the objective of the strategic behavior will be to
win the cash prize rather than to affect the outcome of the public good
decision. The Guessing Experiment conducted at the University of
California, Riverside, indicated that if the respondents do use strategic
behavior to win the prize, then their aggregate guesses will accurately
reveal their aggregate true valuation of the public good. Therefore, we
91
-------
see that rather than attempting to eliminate strategic behavior, the
proposed method redirects the respondent's strategy in a way the reveals
public good valuation.
92
-------
TECHNICAL REPORT DATA
(I'/t'asc rcatl Instructions on il:e reverse licibri:
1. REPORT NO.
. RECIPIENT S ACCESSION NO.
4. TITLE ANDSUBTITU:
Methods Development for Assessing Air
Pollution Control Benefits: Volume IV, Studies on
Partial Equilibrium Approaches to Valuation of
Environmental Amenities
7. AUTHORIS!
Maureen L. Cropper, William R. Porter, Berton J. Hansen
Robert A. Jones, and John G. Riley
REPORT DATE
February 1979
G. PERFORMING ORGANIZATION CODE
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
University of Wyoming
Laramie, Wyoming 82071
12. SPONSORING AGENCY NAME AND ADDRESS
Office of Health and Ecological Effects
Office of Research and Development
U.S. Environmental Protection Agency
Washington, DC 20460
10. PROGRAM ELLMF.NT NO.
1HA616 and 630
1 1 . CONTRACT/GRANT NO.
R805059-01
13. TYPE OF REPORT AND PERIOD COVERED
_ Interim Final, 10/76-10/78
14. SPONSORING AGENCY COOK
EPA-600/18
15. SUPPLEMENTARY NOTES
16. ABSTRACT
The research presented in this volume of a five volume study of the economic
benefits of air pollution control explores various facets of the two central project
objectives that have not been given adequate attention in the previous volumes. The
valuations developed in these volumes have all been based on a partial equilibrium
framework. W.R. Porter considers the adjustments and changes in underlying assump-
tions these values would require if they were to be derived in a general equilibrium
framework. In a second purely theoretical paper, Robert Jones and John Riley
examine the impact upon the aformentioned partial equilibrium valuations under varia-
tion in consumer uncertainty about the health hazards associated with various forms
of consumption.
Two empirical efforts conclude the volume. M.L. Cropper employs and empiri-
cally tests a new model of the variations in wages for assorted occupations across
cities in order to establish an estimate of willingness to pay for environmental
amenities. The valuation she obtains for a 30 percent reduction in air pollution
concentrations accords very closely with the valuations reported in earlier volumes.
The volume concludes with a report of a small experiment by W.R. Porter and
B.J. Hansen intended to test a particular way to remove any biases that bidding
game respondents have to distort their true valuations.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
Economic analysis
Air pollution
b. IDENTIFIERS/OPEN ENDED TERMS
Economic benefits of
pollution control
c. COSATI iMcUl/Ciroup
13B
18. DISTRIBUTION STATEMENT
Release unlimited
19. SECURITY CLASS (This Report)
Unclassified
21. NO. OF PAGES
101
20. SECURITY CLASS (This pagei
Unclassified
22. PRICE
EPA Form 2220-1 (9-73)
ftU.S. GOVERNMENT PRINTING OFFICE: I979O620-007/3763 REGION 3-1
-------
United States
Environmental Protection
Agency
RD-683
Official Business
Penalty for Private Use
$300
Special Fourth-Class Rate
Book
Postage and Fees Paid
EPA
Permit No. G 35
Washington DC 20460
------- |