Using conditional utility models for measuring welfare

6 Pages·1994·0.34 MB·English

by  Mendelsohn, Robert O., 1952-

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outhwest Plains
Research Note RM-527
February 1994
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Forest Service
Rocky Mountain Forest and
Range Experiment Station
Using Conditional Utility Models
for Measuring Welfare
Robert Mendelsohn1
Rosa Matzkin1
George Peterson2
Donald Rosenthal3
This paper argues that the functional form of conditional indirect
utility functions is more restrictive than practitioners realize. The
conditional indirect utility must be a function of the difference be-
tween income and price. Many empirical examples of discrete choice
violate this constraint, implicitly assuming that the marginal utility of
income in the conditional indirect utility function is constant across
choices. When applied to conditional utility functions, this is equiva-
lent to assuming that people of different incomes would make the
same choices, which is frequently not the case. Empirical discrete
choice studies should be interpreted using unconditional utility.
Keywords: Welfare measurement, conditional utility, discrete choice,
marginal utility of income, unconditional utility
Introduction is jeopardized. Given the widespread adoption of
discrete choice models for modeling contingent valu-
In recent years, logit and probit models have grown ation responses (Bishop and Heberlein 1979; Bishop
in popularity as a tool for analyzing discrete choices et al. 1983; Bowker and Stoll 1981; Boyle and Bishop
because of their analytic simplicity and powerful 1985; Loehman and De 1982; Sellar et al. 1985, 1986)
empirical properties. In order to interpret these mod- and recreation decisions (Caulkins et al. 1986, Kling
els, McFadden (1981], followed by Small and Rosen 1987, Morey and Rowe 1985, Smith and Kaoru 1986)
(1981) and Hanemann (1984), have proposed the use it is important that researchers conducting future
of random conditional utility models. Although this empirical work understand the restrictions theory
analytical development has been flawless, its adop- places on the form of conditional utility functions.
tion by the empirical literature has been more prob- This paper begins by clarifying the limitations of
lematic. Empirical analysts appear not to understand using conditional utility models for predicting wel-
the restrictions that must be placed on the form of fare when consumers must make discrete (1,0)
conditional utility functions to make them consistent choices. Following McFadden, conditional indirect
with theory. By using functional forms that have no utility functions are deduced from an underlying
utility basis or that require strong underlying as- direct utility function. The correct specification of
sumptions, the welfare implications of this research the conditional utility model for discrete choices is
emphasized. Although the architects of this method-
' Mendelsohn and Matzkin are Professors, Yale University, New ology may argue that there is no need to clarify the
Haven, Connecticut. theory, the applied literature appears not to under-
2 General Research Supervisory Engineer, Rocky Mountain For- stand the limitations of the model. Unless these
est and Range Experiment Station, Fort Collins, Colorado. Head-
methodological problems are openly reviewed, there
quarters is in Fort Collins in cooperation with Colorado State Uni-
versity. is every reason to suspect empirical studies will con-
3 Senior Economist, U.S. Department ofEnergy, Washington, D.C. tinue to be flawed by suspect specifications.
In light of this theoretical discussion, several ap- where P, is the price of the good that is selected.4
X
plications of discrete choice from the resource valu- Substituting the equation for N+1 back into [1]
ation literature are reviewed. The specifications used yields:
in many of these studies are analyzed for their theo- MAX
retical consistency. It is shown that a few studies are U[l,0,Q...0,Y-P1,Z1,e)
correctly specified, but most are either correct only or U{0,%0,...0,Y-P2,Z2,e) [2]
under strict assumptions or are simply misspecified. or U[0,0,...0XY-PN,ZN,e).
Each element above is a conditional indirect utility
Specifying Conditional Utility Differences function. The conditional indirect utility function for
a discrete choice consequently has the following
The general model we are analyzing is a discrete form:
cahsoiincgelebypucrocnhassuemefrrsomeiotnheerotrommoarkeemourtunaoltlytoemxcalkue- V[Y,Pi,Zi,ei) = V[Y-Pi,Zi,ei). [3]
sive options. This choice is assumed to be motivated
by an underlying conditional indirect utility func- The essential aspect of [3] that we wish to draw at-
tion, VjiXPjjZ^ei), associated with choosing good i, tention to is that neither income nor price enters the
where Y is income, P, is the price of good i, Z, is a conditional indirect utility function alone. The price-
vector of characteristics associated with choice and income variable in the conditional utility function is
i,
e, is an error term. The error term could be viewed the difference between income and price.
as known to the decision-maker or a random vari- For example, suppose the indirect utility function
able. Since this distinction is not relevant to our ar- [3] is linear:
gument, we assume for simplicity of exposition that
the error term is known to the decision-maker. Each V^Ai +BiUr-PJ+CiZf+e,. [4]
choice is presumed to have a conditional utility func-
tion associated with it. The model has the general In this case, the income and price coefficient must
be the same magnitude and opposite sign for the dis-
form:
crete choice model to be consistent with the under-
If V(Y, ePiq,uaZlhteo,)i,>thV(eYn,gPoko,dZki,iesk)c,hofosrena;ll k not laysisnugmdeirtehcattu[t3i]lihtyasfuancsteimoin-.loAlgtefrunnacttiivoenlayl, foonrem:could
otherwise good i is not chosen.
V,- = At +B, log {Y-Pi)+C, log Z, +e,-. [5]
In the single good example Vk[Y, Pk, Z^, ek) = V {Y Regardless of the functional form, the difference be-
0, 0, e„).
We derive, following McFadden (1981), the condi- tween income and price is the appropriate argument
tional indirect utility function from a general utility (variable) in all conditional indirect utility functions.
function in order to clarify the restrictions on its As discussed above, the decision calculus is based
functional form. We focus entirely upon a pure (1,0) upon the difference between the conditional indirect
discrete choice. We begin with a continuous direct utility functions of each choice. For example, in the
utility function U[X,Z,e) defined over a set of goods linear case, where the choice is to purchase the good
X where e is a vector of individual characteristics at price P or not, the utility difference is:
known to the consumer but not to the econometri-
cian. Thus, we assume that the consumer makes AV = V[Y,0)-V[Y-P,Z) = A -A +
choices with certainty. The consumer faces the prob- BqY-B, [Y-P)+C^Z+u l [6]
lem of choosing a bundle of goods to maximize his/
her utility function subject to a budget constraint:
where u = e -ei. Similarly, with the semi-log indi-
MAX U{X1,X2,...XN,XN+1,e) rect utilitv function [5], the utilitv difference is:
^*P [1] SV = V[Y.0)-V[Y-P,Z) = A -A +
Y = X +X o l
s.t. t i N+1 B log y-B, log (T-PJ-C, log Z+u. [7]
i=i
where prices have been normalized so that the price
ocfhotihceecmomopdoeslitteakgeosoad XsnNa+pXsihsoetquoafltthois1.deTchiesidoinscprreot-e Oecfoanlolmtihce elimtpeirraitucrael, poanpleyrsBdinwktehre aapnpdliSetdolrles(o1u9r81c)e
cess during which the consumeNr chooses only one 4 Many applications of discrete choice in environmentalresource
unit of one good from a set of goods. Given this economics involve preferences forpublicgoods for which there may
X
constraint, one can solve for N+1 in the budget con- be no market price. In the case of outdoor recreation, the effective
straint: price, however, may be transportation costs to the site. Further, in
= Y-P contingent valuation exercises, the price is the payment being re-
quested of the individual on behalf of the public good.
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and Smith and Kaoru (1986) choose a functional form often reveal that income elasticities are nonzero.
that is clearly consistent with [6] or [7]. Empirically, income does affect public recreation and
In contrast, many empirical studies have introduced environmental choices. In order to reflect the impact
the following simplification of these two models: of income in a conditional utility function frame-
work, one must assume that Bj * Bk even for relatively
AV= V{Y,0)-V{Y-P,Z)=A -A +B P+C Z+u [8] inexpensive goods. In which case, the utility difference
1 1 1
model must be specified as [6] or [7], not [8] or [9].
and One final specification in the literature is provided
by Bowker and Stoll (1981) and Loehman and De
AV = V(Y,0)-V(Y-P,Z) = A - + (1982), who estimate a log model:
A-y
By log (P)+C, log Z+u. [9] AV = V -V = Aj + B logF+C, log Pj. [10]
i 1
For example, Bowker and Stoll (1981), Caulkins et al. This model follows from an underlying indirect util-
(1986), Kling (1987), Morey and Rowe (1985), and ity function of
Sellar et al. (1985) all adopt [8]. Bishop and Heberlein
(1979), Bishop et al. (1983), Boyle and Bishop (1985), Vj = Aj + Bj log [YlPil [11]
Hanemann (1984), and Sellar et al. (1986) all adopt
[9]. It is not possible to derive [9] from [7] unless Y However, [11] is not consistent with [3], and [10] is
approaches 2P. a poor approximation of [7].
With [8], the literature has either explicitly or im- In this paper, we argue against interpreting the
plicitly assumed that B = Bx. Is this a reasonable as- results of models such as [8], [9], and [10] using con-
sumption? Since Bj is the marginal utility of income ditional utility models. Because the specifications are
and it is unlikely that the marginal utility of income inconsistent with the theory, the interpretations will
will change depending upon the decision to consume be biased. For example, Bowker and Stoll (1981) re-
one good, several authors argue that, in most circum- veal that the misspecified equations (relative to the
stances, it is a reasonable assumption. correctly specified equation) can lead to biases in
In this paper, we argue that Bj * Bk in most circum- predicted welfare of between 25% to 100% of the
stances. Bj is the marginal utility of income for a true value.
conditional utility function, not an unconditional However, we are not arguing that models such as
utility function. If Bj was the marginal utility of in- [8], [9], and [10] are unattractive econometric mod-
come for an unconditional utility function, the argu- els. In fact, these models perform admirably in em-
ment for equality would be reasonable. However, the pirical applications. Bowker and Stoll (1981), Boyle
fact that Bj is the marginal utility of income for a and Bishop (1985), and Sellar, Chavas, and Stoll
conditional utility function is very important. As- (1986) all find that theoretically incorrect functional
suming that the conditional marginal utility of in- forms fit the data more closely than tested theoreti-
come is the same for all choices is effectively assum- cally correct specifications. Given that [8], [9], and
ing that income will not affect the choice between the [10] can all be justified using a standard uncondi-
available alternatives. That is, people with high in- tional utility framework, the empirical results suggest
come and people with low income will choose alter- they are preferable functional forms. The results sug-
native i with the same probability.5 In the context of gest that conditional utility functions are not useful
a traditional demand function, assuming income will frameworks for analyzing discrete choices because they
not affect choice is equivalent to assuming that the impose restrictions on the data that are not justified.
income elasticity of the good is zero. Alternatively,
assuming that B, = Bk for a class of goods is the same
as assuming that the income elasticity of all goods k Conclusion
and i is the same.
Although there are special cases where income will This paper examines the derivation of discrete
not affect choice, the frequency by which people choice (1,0) conditional indirect utility functions.
choose any environmental or recreation good i is The derivation implies that the conditional utility
often affected by their incomes. Empirical evidence function is a function of the difference between in-
from multiple site analyses reveals that Bj is not the come and price. Simplified functional forms that just
same across sites. For example, in the three sites in depend upon price can only be used if income does
Smith and Kaoru (1986), Bj varies from -.37 to -3.6, not affect choice. The empirical valuation literature,
a statistically significant difference. Multiple site, however, reveals that income frequently does affect
travel cost analyses such as Burt and Brewer (1971) choice. Further, valuation exercises that compare
and Cichetti et al. (1976) also reveal that Bj will vary correctly specified and incorrectly specified equa-
across sites. Individual travel cost analyses of sites tions reveal that they imply quite different welfare
values. Logit and probit models with broad specifi-
su5mpSteieo,nfoerxpleixcaimtpilne,hisHa[6n].emAasnhne(d1i9s8c4u)sswehserien hhies mfoaoktenostethi6,s atsh-e cations may be perfectly reasonable methods of esti-
more general formulation is the one adopted in this paper. mating empirical relationships. Conditional utility
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models can be used to measure welfare from these Hanemann, W.M. 1984. Welfare evaluation in contin-
estimates, but only with very rigid specifications for gent valuation experiments with discrete re-
these models. In general, for welfare analysis of dis- sponses. American Journal ofAgricultural Econom-
crete choice, the unconditional utility framework is ics. 66: 332-41.
far more promising than conditional utility models. Kling, C. 1987. A simulation approach to comparing
multiple site recreation demand models using
Chesapeake Bay survey data. Journal of Marine
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