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Recovering weakly complementary preferences
,O”RNAL
OF ENVIRONMENTAL
ECONOMICS
AND
MANAGEMENT
21, 97-108 (1991)
Recovering Weakly Complementary Preferences DOUGLAS M . LARSON* Department of Agricultural Economics, University of California, Davis, California 95616
Received September 12, 1990; revised January 4, 1991 This paper demonstrates a method for recovering weakly complementary expenditure and indirect utility functions while integrating back from Marshallian demand specifications that do not intrinsically embody weak complementarity. Weak complementarity of a good with quality provides information about how the constant of integration over price must behave with respect to quality, and this information can be incorporated into the process of recovering preferences. The weakly complementary expenditure and indirect utility functions are recovered for the linear demand specification and for the Stone-Geary utility function with any value of the subsistence parameter. o 1991 Academic press, hc.
INTRODUCTION
Interest in valuing environmental quality improvements has been around for several decades, but is especially keen at present, due perhaps to the large publicity attendant to recent oil spills in Alaska and elsewhere, the treks of garbage barges, and the spate of hypodermic needles washing ashore on Northeastern beaches. In addition to being of considerable practical interest, the issue of measuring willingness to pay for quality is important in theoretical terms, as the increasingly common approach of determining “exact” measures of welfare change (i.e., the compensating or equivalent variations or surpluses) from e m p irical specifications has shown. As is by now well known, it is possible for a number of commonly used e m p irical specifications to integrate back and obtain the quasiexpenditure function implied by the estimated demands (see, e.g., Hausman [6], Bockstael et al. [l], LaFrance and Hanemann [7, 81). W h ile this approach is quite useful for evaluating the welfare effects of price changes, things are not so straightforward for measuring the values of changes in non-price parameters such as quality (Willig [lo], Bockstael and McConnell [3, 41, Hanemann [5]). The basic problem is that by integrating back over prices to recover facsimiles of the expenditure and preference function corresponding to particular specifications of Marshallian demand a constant of integration is obtained. W h ile the constant of integration clearly does not depend on prices (since these are the variables integrated over), it may depend on quality or other non-price parameters in arbitrary ways. W ithout some additional information or structure, it proves virtually impossible to measure the values of quality changes. Fortunately, one avenue of analysis has proven highly fruitful in analyzing the values of quality changes. Weak complementarity, introduced by M a ler [9] and a m p lified upon by W illig, is an intuitively appealing and often plausible condition *I acknowledge the encouragement of Ted McConnell in the development of this paper, and the helpful comments of Michael Hanemann and an anonymous Journal referee. This is Giannini Foundation Research Paper 971. 97 00950696/91 $3.00 Copyright 6 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
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on preferences that permits the measurement of willingness to pay for quality changes. If quality is weakly complementary to a set of market goods, when consumption of the complementary market goods is zero the individual’s welfare is unaffected by changes in quality. That is, a person has to be consuming at least one good complementary to quality in order to care about the change in quality. For preferences that satisfy this condition, Mller showed that the total value of a discrete change in quality can be obtained as the difference between two integrals over price, one corresponding to (Hicksian) demand conditioned on initial quality level and the second conditioned on subsequent quality. The conventional approach to integrating back from demands to recover preferences has been to treat the constant of integration as independent of quality, assuming implicitly that all the relevant information about how quality affects preferences is embodied in the demand system. Using this approach, it is straightforward to show that certain demand functions integrate back to quasi-expenditure functions which exhibit weak complementarity intrinsically, while others do not. Since it is plausible that preferences for many non-unique environmental facilities exhibit weak complementarity, empirical demand specifications which embody the condition (such as the semilog and Cobb-Douglas forms) have been used frequently. Other common functional forms which have often been used for measuring the welfare effects of price changes, such as the linear, have not been used for valuing quality changes because they have not generally been thought consistent with weak complementarily. Researchers have often been forced to choose the functional form for an analysis based on whether it exhibits weak complementarity under the “conventional” approach to integrating back, without much apparent opportunity to influence this key attribute. The purpose of this paper is to introduce and demonstrate a method for recovering weakly complementary preferences when integrating back from any Marshallian demands to recover the quasi-expenditure function.’ Requiring preferences to be weakly complementary amounts to providing more structure for the behavior of the constant of integration (that arises from integrating over price) with respect to the quality argument. This additional structure provides for an explicit representation of how the expenditure function must behave with changes in quality, which is obtained by an integration over quality, and the remaining constant of integration is independent of price and quality. The method permits researchers to conveniently apply the logic of Maler’s argument for calculating the value of quality changes to any empirical demand specification. Section I describes the theoretical background involved with integrating back from Marshallian demand specifications to recover preferences and the additional structure which imposing weak complementarity provides for this process. Section II illustrates the approach by recovering the weakly complementary expenditure function and indirect utility function implied by the linear demand specification, which is often thought to be incompatible with weak complementarity. Section III ‘The approach developed here is similar in concept to a method described by MIler [9, pp. 186-1871, but may be somewhat easier to apply in practice. Using the approach of this paper, a single constant of integration arises from integrating back, and the weak complementarity condition places additional structure (i.e., a differential equation) on the curvature of this constant of integration. With the Mller approach, two constants of integration result from integrating back and the assumption of weak complementarity results in two differential equations which must be solved jointly. I thank an anonymous referee for bringing this similarity to my attention.
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analyzes the Stone-Geary utility function, which is well known to be weakly complementary for quality changes when the “subsistence” parameter is - 1. The advantage of working forward from the utility function to derive demands and the expenditure function is that one knows exactly how expenditure changes with quality. This provides a check for how well the approach of integrating back from Marshallian demands to obtain the behavior of expenditure with quality works. The analysis shows that it is possible to recover weakly complementary preferences when integrating back from demands for all admissible values of the subsistence parameter and that, for the particular value - 1, the same expenditure function is obtained. Section IV concludes.
I. SOME THEORETICAL BACKGROUND The consumer is presumed to maximize the utility function u(x, z, q), where x is a vector of consumption goods with corresponding price vector p, q is a quality variable, and z is a composite commodity such that z = m - px, where m is income. The solution to this problem is the set of Marshallian demands x = x(p, q, m) and z = z(p, q, m) = m - px. Substituting these demands into the utility function yields the indirect utility function v(p, q, ml = u(x(p, q, m), z(p, q, m), q) + h[m - px - z], where h is a Lagrange multiplier. Applying the envelope theorem to the indirect utility function, one can write -uJu, = hxJh, or Roy’s identity, where ui and u, are derivatives of indirect utility with respect to price of good i and income, respectively. The inverse of indirect utility with respect to the income argument is the m inimum expenditure function e(p, q, u) = m in,, ,{px + z: u(x, z, q) = u}, which is dual to the utility function. Using the envelope theorem, it can be seen that as expenditure varies to hold utility constant and compensate for changes in any price pi, du = ui dpi + u, de( p, q, u) = 0,
which can be rewritten as the differential equation de(p,q,u)
-: hi
=
=Xi(P,4Te(P7q7u))7 m
(1)
with the latter equality resulting from Roy’s Identity. Equation (1) provides the linkage between Marshallian demands and the price slope of the expenditure function, and given a parametric representation of x(p, q, ml one can in principle integrate back from Eq. (1) sequentially for all i’s to obtain the quasi-expenditure function kX(p, q, Nq, UN (see, e.g., Hausman [61, LaFrance and Hanemann [SD. This quasi-expenditure is related to the true expenditure function by e(p,q,u)
= ~(P,q,fJ(q,U)),
(2)
where e’(. ) is a known function that represents the part of the expenditure function which is identified parametrically from (11, and f3(-1 is the unknown constant of integration.
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The welfare measure which is often of interest in assessing the effects of a quality change is the compensating variation. 2 For a change in quality from 4’ to ql, it is
c(4O,d)
=e(p,4O,u)
-e(p,q’,u)
= [TEd4,
(3)
and it can be seen immediately from (2) that without knowing 8(4, U) one cannot calculate the compensating variation in (31, since
ae/aq = ae'/aq + (aa/au)(ae/a4>. The way that weak complementarity resolves the problem of not knowing the constant of integration is as follows. Suppose a good xi and quality are weakly complementary.3 By Shephard’s lemma, ae( p, 4,~) aPi
(4)
+Q7,4,u);
that is, the ith price slope of the expenditure function is the compensated demand XT exists and is one to one for XT 2 0. Denoting the Hicksian choke price for good i as j$ = pi(0,4, U) and substituting into Eq. (41, one way of defining weak complementarily in terms of the expenditure function is the identity
(5) Now by the fundamental theorem of calculus and Eq. (51, under weak complementarity the change in expenditure with a change in quality, as consumption is held at zero, is
where S,! = fii(4’) and fij = $i(41). Since the left side of (6) is zero, it can be added and subtracted to (3) without changing that expression, so that c(4’,4’)
=e(lj!,4’,u)
- e(Pi,4’,u)
= lBJXi( Py qly U) dp - ~‘~i( Pi Pi
- e(fiP,4O,u)
+ e(Pi,4O,u)
p, 4’, U) dp
20ther Hicksian welfare measures such as equivalent variation or surplus measures can be obtained from this analysis with only minor modifications. 3This discussion focuses on weak complementarity of a single good with quality; for a treatment of the case where several market goods and quality are weakly complementary, see Bockstael and Khng [2]. More extensive and very useful discussions of welfare measurement with quality changes can be found in Bockstael and McConnell [3, 41. 4That is, negative definiteness of the Slut&y matrix; see, e.g., LaFrance and Hanemann [S].
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which expresses the compensating variation of the quality change as the difference between two integrals over price. Under weak complementarity, then, the desired measure of welfare change due to the quality change can be expressed as the difference between the area under the Hicksian demand conditioned on subsequent quality and the area under Hicksian demand conditioned on initial quality. Weak complementarity can be used as a restriction on preferences when integrating back from observed demands to recover the expenditure function e(p, 4, u). This is accomplished by using both (1) and (5) in the process of integrating back, instead of just (1) as is usually done. What this adds to the process of integrating back is a second integration over quality that yields a parametric representation of how the constant of integration tl(q, u), and therefore the expenditure function, changes with quality. Using (2) in (5), and letting fi = (j$(q, u>, p) be the price vector when consumption of good i is held at zero, the weak complementarity condition requires that
~qBdL~(w)) a4
+ qB,q,e(w)) ae
aqq,u) = o aq
-'
and since Z(a) is known, Eq. (7) can in principle be integrated to obtain 8(q, u) = &q, #J(U)>,where e’<.> is a known function and 4(e) is an unknown constant of integration independent of p and q.5 II. ILLUSTRATING THE APPROACH: THE CASE OF LINEAR DEMANDS
If the Marshallian demand function for a good x of interest is of the form6 x=a+pp+yq+&n,
(8)
then using Eq. (1) it is possible to integrate back (e.g., Hausman 161,Bockstael et al. [l] to obtain a( P, 4, qq,
4)
= qq,
u)e sp - (W>b
+ PP + Y4 +
P/S],
(9)
where B(q, u) is the constant of integration that in general depends on q. This model is well defined for 0 I x < -p/S; i.e., epp < 0 and x 2 0 over this range. In Hausman’s procedure, which was not concerned with quality changes, B(q, u) was used as the utility index to define reference utility levels for welfare purposes. This cannot be done when quality or other nonprice parameters change because 8 will also change; a utility index invariant to changes in the variable of interest must be found. 51n (7), there is another term [&?(b,. )/3piXd$JJq) that reflects the fact that as quality changes, the choke price of good i adjusts to maintain XT = 0. However, this term is zero since from the envelope theorem &$j, I/Q = XT 5 0. 6The method of integrating back is illustrated here and in the next section for the case where a single market good is believed weakly complementary to quality. The logic of the approach extends in a straightforward way to recovering expenditure functions that exhibit weak complementarity with respect to a set of goods and quality, though cross-equation parameter restrictions arise from integrating over price. See, for example, LaFrance and Hanemann [8] and Bockstael and Khng [2] for discussions of some issues that arise in this context.
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Now the Hicksian choke price fi is found by setting Hicksian consumption equal to zero; from Eq. (9), this means @I( q, up
- p/s
= 0,
from which it can be seen that fi = (1/6)ln
{S’GLU)
1.
Using (9) and (10) in (5) results in the weak complementarity condition for this model,
-&/a2
- (W[ CY+ (p/6)ln{P/t-j2e(q, u)] + 74 + P/S]) = 0.
Performing the indicated differentiation and gathering terms yield the differential equation de(q, u) = ?dq, cL4
P
which can be solved for B(q, u) = +e(ys/B)4,
where 4 is a constant of integration that depends on u but not on q nor on p. Substituting f?(q, u) into (9) yields the weakly complementary expenditure function corresponding to (8), e(p,q,u)
= 4e (s’@xyq+@p)- (1/6)[ (Y + /3p + yq + p/s-J,
where 4 has been chosen as the utility index. The corresponding indirect utility function is u(p,q,m)
= [m +
(l/S)(cr
+/3p + yq + /3/S)]e-(S/BXyq+BP),
and Roy’s identity, weak complementarity, and the other properties of v( .) and e(a) can be seen to hold for 0 5 x < -p/S. It is worth mentioning that quasi-concavity of preferences in this model implies that the constant of integration e(q, U) I 0. Also, for q > (I) 0, it can be shown that du/aq > ( 5) 0 and ae/aq < (2) 0 when the second-order condition holds (i.e., when C#I 0). III. CORROl3ORATlNG THE APPROACH: THE LINEAR EXPENDITURE SYSTEM
The previous section showed how the weakly complementary expenditure and indirect utility functions implied by linear Marshallian demands can be recovered while integrating back. While the approach may appear to have promise, it is difficult to know whether all the relevant information about quality in the expendi-
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ture function is recovered by integrating back, since there is not an expenditure function based on maximizing a weakly complementary utility function available for comparison. To evaluate this question, this section analyzes the Stone-Geary utility function. By choosing a parametric form for utility, one can derive the demand, indirect utility, and expenditure functions that correspond to that utility function, so that these functions are known in advance and can be compared with the functions recovered by the approach suggested in this paper. The Stone-Geary utility function can be written u = b(q)ln(x
- c) + In(z -c*),
(11)
where b(q) > 0 is an arbitrary function of quality, x is a market good and z is a composite good, and c and c* are parameters that are often referred to as subsistence quantities in demand studies. This model is well defined for x > max{O, c) and z > max{O, c*}; it will be convenient algebraically to restrict consideration to values of c < 0. It is well known that when the subsistence parameter c = - 1, the utility function exhibits weak complementarity with respect to q and x, since when x = 0, u = b(q)ln(O + 1) + ln(z - c*) = In(z - c*), so utility is unaffected by quality if x is not consumed. What is perhaps not generally recognized is that it is possible, when integrating back from demands, to recover weakly complementary expenditure and indirect utility functions for any admissible value of c. That is, while c = - 1 in the direct utility function (11) implies weak complementarity, c # - 1 in the demand’function does not imply that preferences are not weakly complementary. Depending on the behavior of the constant of integration fI(q, u), preferences may or may not exhibit this property when c # - 1. Deriving
the Expenditure
Function
Implied
by the Direct
Utility Function
Maximizing (11) subject to the budget constraint m - px - z = 0 gives the Marshallian demands C
+ 1 + b(q)
(12)
and z=m-px
1 = 1 + b(q)
(m + b(q)c*
-PC).
Substituting (12) and (13) into (11) yields the indirect utility function m - c* - cp + [l + b(q)]ln
1 + b(q)
(13)
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and expenditure function u -
cp + [l + b(q)lw
bWn(b(q)/p)
1 + b(q)
i
cp + [ 1 + b( q>] eU(l+b(q))[ p/b(q)]
b(q)/(l +b(q)).
(14)
The expenditure function in (14) provides a basis for evaluating how the suggested approach of this paper performs. If the method of imposing weak complementarity while integrating back is to recover all relevant information about how expenditure changes with p and q, (14) must be recovered up to the constant u that indexes utility when the direct utility function also exhibits weak complementarity. It is easy to verify that for the expenditure function (141, weak complementarity implies c = - 1. The Hicksian demand for x based on (14) is q
p, q, u) = c + eU(l+W))[ p/b( q>] -(l/(l+b(q)))e
(15)
The Hicksian choke price j3 is defined implicitly by x*@, q, U) = 0; using this with (15) and solving explicitly for $ give b(q)e” ’
=
(+l+b(d
and when (Hicksian) consumption of x is held at zero, the expenditure function is e( B, q, u) = ( -~)l+~(~)e’
+ c*.
The only way for weak complementarity to hold (&z($, q, u)/aq b(q) not equal to a constant, is for -c = 1, or c = - 1. Recovering
the Expenditure
Function
by Integrating
= 01, for arbitrary
Back
Suppose instead that one begins with the Marshallian demand in (12); Eq. (1) used for integrating back over price can be written as the nonhomogeneous first-order different equation de( p, 4, u) B(q) - -e(p,q,u) P dr,
= --
where to simplify notation B(q) = b(q)/(l factor p -E(q), (16) can be solved for q P, q,e(q,
u))
BWc* P
+ b(q)).
+
c 1 + b(q)
’
(16)
Employing the integrating
= c* + cp + e( q, u)pE(@,
(17)
where 6%q, u) is the unknown constant of integration. Now in order to employ the weak complementarity condition (5) it is necessary to obtain the Hicksian choke price @. Differentiating (17) with respect to p yields the identity which implicitly defines fi, cG(cf, q, u)/ap
= c + B(q, u)B(q)j+-’
= 0,
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from which it can be seen that
B(4YU)
(18)
=
since l/(B(q) - 1) = -Cl + b(q)). Now, using (18) in (5) and simplifying, if preferences are to exhibit weak complementarity between x and q, the constant of integration 8(q, u) must behave in the following way,
-de/& = db/& 8
(19)
l+b
where the dependence of b and B on q has been suppressed. Since 13= tI(s, u) and b = b(q), and utility is being held constant, Eq. (19) can also be expressed as a differential equation relating the constant of integration 8 to the quality function b, de -=8
db
(20)
l+b
and with a change of variables 8 = In 8, (20) can be written as the nonhomogeneous first-order differential equation
d5
5
z+-=
l+b
ln( -c/B} l+b
’
which, using the integrating factor 1 + b, has solution
5= -&
ln( -c)
4
+ In(1 + b) - &
ln( b) + l+b’
where C#Iis a constant of integration that does not depend on 4, but does depend on the utility level u. Using the inverse transformation 8 = exp([), this means that in order to satisfy the weak complementarity condition (5), the constant of integration Nq, u) in the quasi-expenditure function (2) must be of the form q4,
u) = [l + b(q)]e4/(l+b(q))(
-c/b)b(q)/(l+b(q)).
Using this in the quasi-expenditure function (17), the weakly complementary expenditure function implied by the demand function (12) is e(p,q,u)
= c* + cp + [l + b(q)]e4/(1+b(q)) x(-c>
b(q)/(l+b(q))[
p,b(q)]
b(q)/(l+b(q)). (21)
Note that if the constant of integration 4 is scaled to be the utility index, which is appropriate for analyzing quality changes since C#J does not depend on q, (14) and
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DOUGLAS
M. LARSON
(21) are identical with the exception of the term involving ( - ~)~(q)/(i+~(~))in (21). When c = - 1, as noted earlier the direct utility function imposes weak complementarity between x and q regardless of the quality function b(q). In this case, the expenditure function in (141, obtained from the direct utility function, and the expenditure function in (211, obtained by integrating back, are identical aside from scaling of the utility index. But it can be easily verified that the expenditure function in (21) exhibits weak complementarity for all values of c < 0. The difference between (14) and (21) is that (14) was derived by treating the utility index u as simply a constant, whereas in integrating back the utility index (constant of integration) was treated as a function of quality on which additional structure (the weak complementarity condition) was placed. This illustrates the following point. Any demand function (or system) can be generated by a weakly complementary utility function (as well as an infinitude of other utility functions which do not exhibit this property). If weak complementarity is judged to be an appropriate relationship between quality and a market good (or goods), the preferences which exhibit this property and generate the observed demand function can be obtained by imposing the weak complementarity condition while integrating back. In the context of the example of this section, the Stone-Geary direct utility function in (11) exhibits weak complementarity only if c = - 1, but the demand functions in (12) derived from this direct utility function integrate back to a broader class of expenditure and utility functions of which the Stone-Geary is one member. For values of the subsistence parameter c # - 1, the Stone-Geary utility function (11) does not display weak complementarity, but other members of this class, whose expenditure functions are given in (211, do.
IV. CONCLUSION
The evaluation of willingness to pay for changes in environmental quality is an important topic both for its application to a wide variety of contemporary settings and for the theoretical issues it raises. In many cases, analysts have found that weak complementarity of a set of market goods and quality is a reasonable assumption to make in the determination of the value of quality changes, but may have felt restricted in the choice of functional forms to select for demand analysis because some functions exhibit weak complementarity intrinsically, while others apparently do not. This paper has demonstrated an approach to the recovery of weakly complementary preferences from any empirical demand specification. When one integrates back over price(s) from a Marshallian demand function or functions to recover the implied quasi-expenditure function, a constant of integration that does not depend on prices but may depend on quality is also obtained. As LaFrance and Hanemann [8, pp. 266-2671 have noted, for incomplete demand systems there is an infinite number of expenditure functions that generate the same demand functions for any subset of the goods consumed, because information about the other goods is not being fully used. The assumption of weak complementarity of a good or goods with quality provides additional information about the way the constant of integration in the
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quasi-expenditure function must behave with respect to quality changes. If the prices of the set of goods weakly complementary to quality are at the choke levels, the change in expenditure as quality changes is zero. This is in effect a differential equation that relates the change in the constant of integration with quality to other known parts of the quasi-expenditure function and can in principle be solved for the constant of integration as it depends on quality, leaving a second constant of integration independent of both price and quality. This means there is still a class of expenditure functions corresponding to an incomplete demand system which all yield the same demands and which exhibit weak complementarity between quality and market goods, but these differ from each other only with regard to arguments other than quality and prices. The method of recovering weakly complementary preferences should broaden the set of functional forms used in empirical analyses of willingness to pay for quality changes. It was demonstrated on the linear Marshallian demand specification, and the weakly complementary expenditure and indirect utility functions implied by this demand function were found. It was applied also to the Stone-Geary utility function, for which it is well known that when the value of the subsistence parameter for a good x is - 1, preferences are weakly complementary with respect to that good and quality. This provides a check for whether the approach is picking up all the relevant information about quality in the preference structure when integrating back. It was seen that the weakly complementary expenditure function can be recovered for all admissible values of the subsistence parameter and that when the parameter is - 1 the same expenditure function with respect to price and quality is recovered as is obtained by maximizing forward. There remains the important question of whether weak complementarity should be imposed on empirical demand systems. The intent here is not to argue that weak complementarity should be imposed on principle, but that it can be imposed if judged appropriate. Clearly, this will not be reasonable for well-known, unique, and highly valued environmental amentities for which there are likely to be substantial nonuse values. Unfortunately, for cases where the structure provided by weak complementarily is inappropriate, there is not much guidance about alternative structure that would enable the measurement of welfare changes due to quality changes. Judgment by the researcher is also important in the selection of which goods are weakly complementary to quality, as this choice will affect the form of the expenditure function recovered and the resulting estimates of the value of quality changes. REFERENCES 1. N. E. Bockstael, W. M. Hanemann, and I. E. Strand, “Measuring the Benefits of Water Quality Improvements Using Indirect or Imputed Market Methods,” Report to Environmental Protection Agency (19861. 2. N. E. Bockstael and C. L. Kling, Valuing environmental quality changes when quality is a weak complement to a set of goods, Amer. J. Agr. Econom. 70, 654-662 (1988). 3. N. E. Bockstael and K. E. McConnell (19871, “Welfare Effects of Changes in Quality: A Synthesis,” University of Maryland (19871. 4. N. E. Bockstael and K. E. McConnell (19871, “Some Comparative Statics of Quality-Differentiated Goods Models,” University of Maryland (1987). 5. W. M. Hanemann, Quality and Demand Analysis, in “New Directions in Econometric Modeling and Forecasting in US Agriculture” (G. C. Rausser, Ed.), North-Holland, New York (1982).
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6. J. A. Hausman, Exact consumer’s surplus and deadweight loss, Amer. Econom. Rec. 71, 662-676 (1981). 7. J. T. LaFrance and W. M. Hanemann, “On the Integration of Some Common Demand Systems,” Montana State University, Bozeman, MT (1983). 8. J. T. LaFrance and W. M. Hanemann, The dual structure of incomplete demand systems,” Amer. J. Agr. Econom. 71, 262-274 (1989). 9. K.-G. MLler, “Environmental Economics: A Theoretical Inquiry,” Johns Hopkins Univ. Press, Baltimore (1974). 10. R. Willig, Incremental consumer’s surplus and hedonic price adjustment, J. Econom. 7heory 17, 227-253 (1978).
OF ENVIRONMENTAL
ECONOMICS
AND
MANAGEMENT
21, 97-108 (1991)
Recovering Weakly Complementary Preferences DOUGLAS M . LARSON* Department of Agricultural Economics, University of California, Davis, California 95616
Received September 12, 1990; revised January 4, 1991 This paper demonstrates a method for recovering weakly complementary expenditure and indirect utility functions while integrating back from Marshallian demand specifications that do not intrinsically embody weak complementarity. Weak complementarity of a good with quality provides information about how the constant of integration over price must behave with respect to quality, and this information can be incorporated into the process of recovering preferences. The weakly complementary expenditure and indirect utility functions are recovered for the linear demand specification and for the Stone-Geary utility function with any value of the subsistence parameter. o 1991 Academic press, hc.
INTRODUCTION
Interest in valuing environmental quality improvements has been around for several decades, but is especially keen at present, due perhaps to the large publicity attendant to recent oil spills in Alaska and elsewhere, the treks of garbage barges, and the spate of hypodermic needles washing ashore on Northeastern beaches. In addition to being of considerable practical interest, the issue of measuring willingness to pay for quality is important in theoretical terms, as the increasingly common approach of determining “exact” measures of welfare change (i.e., the compensating or equivalent variations or surpluses) from e m p irical specifications has shown. As is by now well known, it is possible for a number of commonly used e m p irical specifications to integrate back and obtain the quasiexpenditure function implied by the estimated demands (see, e.g., Hausman [6], Bockstael et al. [l], LaFrance and Hanemann [7, 81). W h ile this approach is quite useful for evaluating the welfare effects of price changes, things are not so straightforward for measuring the values of changes in non-price parameters such as quality (Willig [lo], Bockstael and McConnell [3, 41, Hanemann [5]). The basic problem is that by integrating back over prices to recover facsimiles of the expenditure and preference function corresponding to particular specifications of Marshallian demand a constant of integration is obtained. W h ile the constant of integration clearly does not depend on prices (since these are the variables integrated over), it may depend on quality or other non-price parameters in arbitrary ways. W ithout some additional information or structure, it proves virtually impossible to measure the values of quality changes. Fortunately, one avenue of analysis has proven highly fruitful in analyzing the values of quality changes. Weak complementarity, introduced by M a ler [9] and a m p lified upon by W illig, is an intuitively appealing and often plausible condition *I acknowledge the encouragement of Ted McConnell in the development of this paper, and the helpful comments of Michael Hanemann and an anonymous Journal referee. This is Giannini Foundation Research Paper 971. 97 00950696/91 $3.00 Copyright 6 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
98
DOUGLAS
M. LARSON
on preferences that permits the measurement of willingness to pay for quality changes. If quality is weakly complementary to a set of market goods, when consumption of the complementary market goods is zero the individual’s welfare is unaffected by changes in quality. That is, a person has to be consuming at least one good complementary to quality in order to care about the change in quality. For preferences that satisfy this condition, Mller showed that the total value of a discrete change in quality can be obtained as the difference between two integrals over price, one corresponding to (Hicksian) demand conditioned on initial quality level and the second conditioned on subsequent quality. The conventional approach to integrating back from demands to recover preferences has been to treat the constant of integration as independent of quality, assuming implicitly that all the relevant information about how quality affects preferences is embodied in the demand system. Using this approach, it is straightforward to show that certain demand functions integrate back to quasi-expenditure functions which exhibit weak complementarity intrinsically, while others do not. Since it is plausible that preferences for many non-unique environmental facilities exhibit weak complementarity, empirical demand specifications which embody the condition (such as the semilog and Cobb-Douglas forms) have been used frequently. Other common functional forms which have often been used for measuring the welfare effects of price changes, such as the linear, have not been used for valuing quality changes because they have not generally been thought consistent with weak complementarily. Researchers have often been forced to choose the functional form for an analysis based on whether it exhibits weak complementarity under the “conventional” approach to integrating back, without much apparent opportunity to influence this key attribute. The purpose of this paper is to introduce and demonstrate a method for recovering weakly complementary preferences when integrating back from any Marshallian demands to recover the quasi-expenditure function.’ Requiring preferences to be weakly complementary amounts to providing more structure for the behavior of the constant of integration (that arises from integrating over price) with respect to the quality argument. This additional structure provides for an explicit representation of how the expenditure function must behave with changes in quality, which is obtained by an integration over quality, and the remaining constant of integration is independent of price and quality. The method permits researchers to conveniently apply the logic of Maler’s argument for calculating the value of quality changes to any empirical demand specification. Section I describes the theoretical background involved with integrating back from Marshallian demand specifications to recover preferences and the additional structure which imposing weak complementarity provides for this process. Section II illustrates the approach by recovering the weakly complementary expenditure function and indirect utility function implied by the linear demand specification, which is often thought to be incompatible with weak complementarity. Section III ‘The approach developed here is similar in concept to a method described by MIler [9, pp. 186-1871, but may be somewhat easier to apply in practice. Using the approach of this paper, a single constant of integration arises from integrating back, and the weak complementarity condition places additional structure (i.e., a differential equation) on the curvature of this constant of integration. With the Mller approach, two constants of integration result from integrating back and the assumption of weak complementarity results in two differential equations which must be solved jointly. I thank an anonymous referee for bringing this similarity to my attention.
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analyzes the Stone-Geary utility function, which is well known to be weakly complementary for quality changes when the “subsistence” parameter is - 1. The advantage of working forward from the utility function to derive demands and the expenditure function is that one knows exactly how expenditure changes with quality. This provides a check for how well the approach of integrating back from Marshallian demands to obtain the behavior of expenditure with quality works. The analysis shows that it is possible to recover weakly complementary preferences when integrating back from demands for all admissible values of the subsistence parameter and that, for the particular value - 1, the same expenditure function is obtained. Section IV concludes.
I. SOME THEORETICAL BACKGROUND The consumer is presumed to maximize the utility function u(x, z, q), where x is a vector of consumption goods with corresponding price vector p, q is a quality variable, and z is a composite commodity such that z = m - px, where m is income. The solution to this problem is the set of Marshallian demands x = x(p, q, m) and z = z(p, q, m) = m - px. Substituting these demands into the utility function yields the indirect utility function v(p, q, ml = u(x(p, q, m), z(p, q, m), q) + h[m - px - z], where h is a Lagrange multiplier. Applying the envelope theorem to the indirect utility function, one can write -uJu, = hxJh, or Roy’s identity, where ui and u, are derivatives of indirect utility with respect to price of good i and income, respectively. The inverse of indirect utility with respect to the income argument is the m inimum expenditure function e(p, q, u) = m in,, ,{px + z: u(x, z, q) = u}, which is dual to the utility function. Using the envelope theorem, it can be seen that as expenditure varies to hold utility constant and compensate for changes in any price pi, du = ui dpi + u, de( p, q, u) = 0,
which can be rewritten as the differential equation de(p,q,u)
-: hi
=
=Xi(P,4Te(P7q7u))7 m
(1)
with the latter equality resulting from Roy’s Identity. Equation (1) provides the linkage between Marshallian demands and the price slope of the expenditure function, and given a parametric representation of x(p, q, ml one can in principle integrate back from Eq. (1) sequentially for all i’s to obtain the quasi-expenditure function kX(p, q, Nq, UN (see, e.g., Hausman [61, LaFrance and Hanemann [SD. This quasi-expenditure is related to the true expenditure function by e(p,q,u)
= ~(P,q,fJ(q,U)),
(2)
where e’(. ) is a known function that represents the part of the expenditure function which is identified parametrically from (11, and f3(-1 is the unknown constant of integration.
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The welfare measure which is often of interest in assessing the effects of a quality change is the compensating variation. 2 For a change in quality from 4’ to ql, it is
c(4O,d)
=e(p,4O,u)
-e(p,q’,u)
= [TEd4,
(3)
and it can be seen immediately from (2) that without knowing 8(4, U) one cannot calculate the compensating variation in (31, since
ae/aq = ae'/aq + (aa/au)(ae/a4>. The way that weak complementarity resolves the problem of not knowing the constant of integration is as follows. Suppose a good xi and quality are weakly complementary.3 By Shephard’s lemma, ae( p, 4,~) aPi
(4)
+Q7,4,u);
that is, the ith price slope of the expenditure function is the compensated demand XT
(5) Now by the fundamental theorem of calculus and Eq. (51, under weak complementarity the change in expenditure with a change in quality, as consumption is held at zero, is
where S,! = fii(4’) and fij = $i(41). Since the left side of (6) is zero, it can be added and subtracted to (3) without changing that expression, so that c(4’,4’)
=e(lj!,4’,u)
- e(Pi,4’,u)
= lBJXi( Py qly U) dp - ~‘~i( Pi Pi
- e(fiP,4O,u)
+ e(Pi,4O,u)
p, 4’, U) dp
20ther Hicksian welfare measures such as equivalent variation or surplus measures can be obtained from this analysis with only minor modifications. 3This discussion focuses on weak complementarity of a single good with quality; for a treatment of the case where several market goods and quality are weakly complementary, see Bockstael and Khng [2]. More extensive and very useful discussions of welfare measurement with quality changes can be found in Bockstael and McConnell [3, 41. 4That is, negative definiteness of the Slut&y matrix; see, e.g., LaFrance and Hanemann [S].
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which expresses the compensating variation of the quality change as the difference between two integrals over price. Under weak complementarity, then, the desired measure of welfare change due to the quality change can be expressed as the difference between the area under the Hicksian demand conditioned on subsequent quality and the area under Hicksian demand conditioned on initial quality. Weak complementarity can be used as a restriction on preferences when integrating back from observed demands to recover the expenditure function e(p, 4, u). This is accomplished by using both (1) and (5) in the process of integrating back, instead of just (1) as is usually done. What this adds to the process of integrating back is a second integration over quality that yields a parametric representation of how the constant of integration tl(q, u), and therefore the expenditure function, changes with quality. Using (2) in (5), and letting fi = (j$(q, u>, p) be the price vector when consumption of good i is held at zero, the weak complementarity condition requires that
~qBdL~(w)) a4
+ qB,q,e(w)) ae
aqq,u) = o aq
-'
and since Z(a) is known, Eq. (7) can in principle be integrated to obtain 8(q, u) = &q, #J(U)>,where e’<.> is a known function and 4(e) is an unknown constant of integration independent of p and q.5 II. ILLUSTRATING THE APPROACH: THE CASE OF LINEAR DEMANDS
If the Marshallian demand function for a good x of interest is of the form6 x=a+pp+yq+&n,
(8)
then using Eq. (1) it is possible to integrate back (e.g., Hausman 161,Bockstael et al. [l] to obtain a( P, 4, qq,
4)
= qq,
u)e sp - (W>b
+ PP + Y4 +
P/S],
(9)
where B(q, u) is the constant of integration that in general depends on q. This model is well defined for 0 I x < -p/S; i.e., epp < 0 and x 2 0 over this range. In Hausman’s procedure, which was not concerned with quality changes, B(q, u) was used as the utility index to define reference utility levels for welfare purposes. This cannot be done when quality or other nonprice parameters change because 8 will also change; a utility index invariant to changes in the variable of interest must be found. 51n (7), there is another term [&?(b,. )/3piXd$JJq) that reflects the fact that as quality changes, the choke price of good i adjusts to maintain XT = 0. However, this term is zero since from the envelope theorem &$j, I/Q = XT 5 0. 6The method of integrating back is illustrated here and in the next section for the case where a single market good is believed weakly complementary to quality. The logic of the approach extends in a straightforward way to recovering expenditure functions that exhibit weak complementarity with respect to a set of goods and quality, though cross-equation parameter restrictions arise from integrating over price. See, for example, LaFrance and Hanemann [8] and Bockstael and Khng [2] for discussions of some issues that arise in this context.
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Now the Hicksian choke price fi is found by setting Hicksian consumption equal to zero; from Eq. (9), this means @I( q, up
- p/s
= 0,
from which it can be seen that fi = (1/6)ln
{S’GLU)
1.
Using (9) and (10) in (5) results in the weak complementarity condition for this model,
-&/a2
- (W[ CY+ (p/6)ln{P/t-j2e(q, u)] + 74 + P/S]) = 0.
Performing the indicated differentiation and gathering terms yield the differential equation de(q, u) = ?dq, cL4
P
which can be solved for B(q, u) = +e(ys/B)4,
where 4 is a constant of integration that depends on u but not on q nor on p. Substituting f?(q, u) into (9) yields the weakly complementary expenditure function corresponding to (8), e(p,q,u)
= 4e (s’@xyq+@p)- (1/6)[ (Y + /3p + yq + p/s-J,
where 4 has been chosen as the utility index. The corresponding indirect utility function is u(p,q,m)
= [m +
(l/S)(cr
+/3p + yq + /3/S)]e-(S/BXyq+BP),
and Roy’s identity, weak complementarity, and the other properties of v( .) and e(a) can be seen to hold for 0 5 x < -p/S. It is worth mentioning that quasi-concavity of preferences in this model implies that the constant of integration e(q, U) I 0. Also, for q > (I) 0, it can be shown that du/aq > ( 5) 0 and ae/aq < (2) 0 when the second-order condition holds (i.e., when C#I 0). III. CORROl3ORATlNG THE APPROACH: THE LINEAR EXPENDITURE SYSTEM
The previous section showed how the weakly complementary expenditure and indirect utility functions implied by linear Marshallian demands can be recovered while integrating back. While the approach may appear to have promise, it is difficult to know whether all the relevant information about quality in the expendi-
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ture function is recovered by integrating back, since there is not an expenditure function based on maximizing a weakly complementary utility function available for comparison. To evaluate this question, this section analyzes the Stone-Geary utility function. By choosing a parametric form for utility, one can derive the demand, indirect utility, and expenditure functions that correspond to that utility function, so that these functions are known in advance and can be compared with the functions recovered by the approach suggested in this paper. The Stone-Geary utility function can be written u = b(q)ln(x
- c) + In(z -c*),
(11)
where b(q) > 0 is an arbitrary function of quality, x is a market good and z is a composite good, and c and c* are parameters that are often referred to as subsistence quantities in demand studies. This model is well defined for x > max{O, c) and z > max{O, c*}; it will be convenient algebraically to restrict consideration to values of c < 0. It is well known that when the subsistence parameter c = - 1, the utility function exhibits weak complementarity with respect to q and x, since when x = 0, u = b(q)ln(O + 1) + ln(z - c*) = In(z - c*), so utility is unaffected by quality if x is not consumed. What is perhaps not generally recognized is that it is possible, when integrating back from demands, to recover weakly complementary expenditure and indirect utility functions for any admissible value of c. That is, while c = - 1 in the direct utility function (11) implies weak complementarity, c # - 1 in the demand’function does not imply that preferences are not weakly complementary. Depending on the behavior of the constant of integration fI(q, u), preferences may or may not exhibit this property when c # - 1. Deriving
the Expenditure
Function
Implied
by the Direct
Utility Function
Maximizing (11) subject to the budget constraint m - px - z = 0 gives the Marshallian demands C
+ 1 + b(q)
(12)
and z=m-px
1 = 1 + b(q)
(m + b(q)c*
-PC).
Substituting (12) and (13) into (11) yields the indirect utility function m - c* - cp + [l + b(q)]ln
1 + b(q)
(13)
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and expenditure function u -
cp + [l + b(q)lw
bWn(b(q)/p)
1 + b(q)
i
cp + [ 1 + b( q>] eU(l+b(q))[ p/b(q)]
b(q)/(l +b(q)).
(14)
The expenditure function in (14) provides a basis for evaluating how the suggested approach of this paper performs. If the method of imposing weak complementarity while integrating back is to recover all relevant information about how expenditure changes with p and q, (14) must be recovered up to the constant u that indexes utility when the direct utility function also exhibits weak complementarity. It is easy to verify that for the expenditure function (141, weak complementarity implies c = - 1. The Hicksian demand for x based on (14) is q
p, q, u) = c + eU(l+W))[ p/b( q>] -(l/(l+b(q)))e
(15)
The Hicksian choke price j3 is defined implicitly by x*@, q, U) = 0; using this with (15) and solving explicitly for $ give b(q)e” ’
=
(+l+b(d
and when (Hicksian) consumption of x is held at zero, the expenditure function is e( B, q, u) = ( -~)l+~(~)e’
+ c*.
The only way for weak complementarity to hold (&z($, q, u)/aq b(q) not equal to a constant, is for -c = 1, or c = - 1. Recovering
the Expenditure
Function
by Integrating
= 01, for arbitrary
Back
Suppose instead that one begins with the Marshallian demand in (12); Eq. (1) used for integrating back over price can be written as the nonhomogeneous first-order different equation de( p, 4, u) B(q) - -e(p,q,u) P dr,
= --
where to simplify notation B(q) = b(q)/(l factor p -E(q), (16) can be solved for q P, q,e(q,
u))
BWc* P
+ b(q)).
+
c 1 + b(q)
’
(16)
Employing the integrating
= c* + cp + e( q, u)pE(@,
(17)
where 6%q, u) is the unknown constant of integration. Now in order to employ the weak complementarity condition (5) it is necessary to obtain the Hicksian choke price @. Differentiating (17) with respect to p yields the identity which implicitly defines fi, cG(cf, q, u)/ap
= c + B(q, u)B(q)j+-’
= 0,
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from which it can be seen that
B(4YU)
(18)
=
since l/(B(q) - 1) = -Cl + b(q)). Now, using (18) in (5) and simplifying, if preferences are to exhibit weak complementarity between x and q, the constant of integration 8(q, u) must behave in the following way,
-de/& = db/& 8
(19)
l+b
where the dependence of b and B on q has been suppressed. Since 13= tI(s, u) and b = b(q), and utility is being held constant, Eq. (19) can also be expressed as a differential equation relating the constant of integration 8 to the quality function b, de -=8
db
(20)
l+b
and with a change of variables 8 = In 8, (20) can be written as the nonhomogeneous first-order differential equation
d5
5
z+-=
l+b
ln( -c/B} l+b
’
which, using the integrating factor 1 + b, has solution
5= -&
ln( -c)
4
+ In(1 + b) - &
ln( b) + l+b’
where C#Iis a constant of integration that does not depend on 4, but does depend on the utility level u. Using the inverse transformation 8 = exp([), this means that in order to satisfy the weak complementarity condition (5), the constant of integration Nq, u) in the quasi-expenditure function (2) must be of the form q4,
u) = [l + b(q)]e4/(l+b(q))(
-c/b)b(q)/(l+b(q)).
Using this in the quasi-expenditure function (17), the weakly complementary expenditure function implied by the demand function (12) is e(p,q,u)
= c* + cp + [l + b(q)]e4/(1+b(q)) x(-c>
b(q)/(l+b(q))[
p,b(q)]
b(q)/(l+b(q)). (21)
Note that if the constant of integration 4 is scaled to be the utility index, which is appropriate for analyzing quality changes since C#J does not depend on q, (14) and
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(21) are identical with the exception of the term involving ( - ~)~(q)/(i+~(~))in (21). When c = - 1, as noted earlier the direct utility function imposes weak complementarity between x and q regardless of the quality function b(q). In this case, the expenditure function in (141, obtained from the direct utility function, and the expenditure function in (211, obtained by integrating back, are identical aside from scaling of the utility index. But it can be easily verified that the expenditure function in (21) exhibits weak complementarity for all values of c < 0. The difference between (14) and (21) is that (14) was derived by treating the utility index u as simply a constant, whereas in integrating back the utility index (constant of integration) was treated as a function of quality on which additional structure (the weak complementarity condition) was placed. This illustrates the following point. Any demand function (or system) can be generated by a weakly complementary utility function (as well as an infinitude of other utility functions which do not exhibit this property). If weak complementarity is judged to be an appropriate relationship between quality and a market good (or goods), the preferences which exhibit this property and generate the observed demand function can be obtained by imposing the weak complementarity condition while integrating back. In the context of the example of this section, the Stone-Geary direct utility function in (11) exhibits weak complementarity only if c = - 1, but the demand functions in (12) derived from this direct utility function integrate back to a broader class of expenditure and utility functions of which the Stone-Geary is one member. For values of the subsistence parameter c # - 1, the Stone-Geary utility function (11) does not display weak complementarity, but other members of this class, whose expenditure functions are given in (211, do.
IV. CONCLUSION
The evaluation of willingness to pay for changes in environmental quality is an important topic both for its application to a wide variety of contemporary settings and for the theoretical issues it raises. In many cases, analysts have found that weak complementarity of a set of market goods and quality is a reasonable assumption to make in the determination of the value of quality changes, but may have felt restricted in the choice of functional forms to select for demand analysis because some functions exhibit weak complementarity intrinsically, while others apparently do not. This paper has demonstrated an approach to the recovery of weakly complementary preferences from any empirical demand specification. When one integrates back over price(s) from a Marshallian demand function or functions to recover the implied quasi-expenditure function, a constant of integration that does not depend on prices but may depend on quality is also obtained. As LaFrance and Hanemann [8, pp. 266-2671 have noted, for incomplete demand systems there is an infinite number of expenditure functions that generate the same demand functions for any subset of the goods consumed, because information about the other goods is not being fully used. The assumption of weak complementarity of a good or goods with quality provides additional information about the way the constant of integration in the
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quasi-expenditure function must behave with respect to quality changes. If the prices of the set of goods weakly complementary to quality are at the choke levels, the change in expenditure as quality changes is zero. This is in effect a differential equation that relates the change in the constant of integration with quality to other known parts of the quasi-expenditure function and can in principle be solved for the constant of integration as it depends on quality, leaving a second constant of integration independent of both price and quality. This means there is still a class of expenditure functions corresponding to an incomplete demand system which all yield the same demands and which exhibit weak complementarity between quality and market goods, but these differ from each other only with regard to arguments other than quality and prices. The method of recovering weakly complementary preferences should broaden the set of functional forms used in empirical analyses of willingness to pay for quality changes. It was demonstrated on the linear Marshallian demand specification, and the weakly complementary expenditure and indirect utility functions implied by this demand function were found. It was applied also to the Stone-Geary utility function, for which it is well known that when the value of the subsistence parameter for a good x is - 1, preferences are weakly complementary with respect to that good and quality. This provides a check for whether the approach is picking up all the relevant information about quality in the preference structure when integrating back. It was seen that the weakly complementary expenditure function can be recovered for all admissible values of the subsistence parameter and that when the parameter is - 1 the same expenditure function with respect to price and quality is recovered as is obtained by maximizing forward. There remains the important question of whether weak complementarity should be imposed on empirical demand systems. The intent here is not to argue that weak complementarity should be imposed on principle, but that it can be imposed if judged appropriate. Clearly, this will not be reasonable for well-known, unique, and highly valued environmental amentities for which there are likely to be substantial nonuse values. Unfortunately, for cases where the structure provided by weak complementarily is inappropriate, there is not much guidance about alternative structure that would enable the measurement of welfare changes due to quality changes. Judgment by the researcher is also important in the selection of which goods are weakly complementary to quality, as this choice will affect the form of the expenditure function recovered and the resulting estimates of the value of quality changes. REFERENCES 1. N. E. Bockstael, W. M. Hanemann, and I. E. Strand, “Measuring the Benefits of Water Quality Improvements Using Indirect or Imputed Market Methods,” Report to Environmental Protection Agency (19861. 2. N. E. Bockstael and C. L. Kling, Valuing environmental quality changes when quality is a weak complement to a set of goods, Amer. J. Agr. Econom. 70, 654-662 (1988). 3. N. E. Bockstael and K. E. McConnell (19871, “Welfare Effects of Changes in Quality: A Synthesis,” University of Maryland (19871. 4. N. E. Bockstael and K. E. McConnell (19871, “Some Comparative Statics of Quality-Differentiated Goods Models,” University of Maryland (1987). 5. W. M. Hanemann, Quality and Demand Analysis, in “New Directions in Econometric Modeling and Forecasting in US Agriculture” (G. C. Rausser, Ed.), North-Holland, New York (1982).
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6. J. A. Hausman, Exact consumer’s surplus and deadweight loss, Amer. Econom. Rec. 71, 662-676 (1981). 7. J. T. LaFrance and W. M. Hanemann, “On the Integration of Some Common Demand Systems,” Montana State University, Bozeman, MT (1983). 8. J. T. LaFrance and W. M. Hanemann, The dual structure of incomplete demand systems,” Amer. J. Agr. Econom. 71, 262-274 (1989). 9. K.-G. MLler, “Environmental Economics: A Theoretical Inquiry,” Johns Hopkins Univ. Press, Baltimore (1974). 10. R. Willig, Incremental consumer’s surplus and hedonic price adjustment, J. Econom. 7heory 17, 227-253 (1978).