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Weak Complementarity, Path Independence, and the Intuition of the Willig Condition by Raymond B. Palmquist Department of Economics North Carolina State University Raleigh, NC 27695-8110 USA [email protected] January 2004 The author is grateful to Dan Phaneuf, Kerry Smith, Wally Thurman, Spencer Banzhaf, and Roger von Haefen, as well as the editor and two referees, for comments on an earlier version. Weak Complementarity, Path Independence, and the Intuition of the Willig Condition Valuing public goods is important, but typically markets for those goods do not exist. Using information revealed in the markets for private goods to infer information about the demand for a public good is often the best option. However, this is not possible unless certain conditions are met. One of the most popular of these maintained hypotheses is weak complementarity, but even with this condition it is complex to derive useful results. The purpose of this paper is to provide a new interpretation of the conditions under which information on the value of an improvement in a public good can be derived from estimates of the uncompensated demand for a related private good and to consider the implications for welfare measurement. The concept of weak complementarity was introduced by Karl-Göran Mäler in 1974. Since that time it has been used widely in valuing non-market goods from observed behavior, particularly in recreation studies. The assumption of weak complementarity requires that there is a private good that is consumed with the non-market environmental good. The environmental good can be thought of as a public good or equivalently as the unpriced quality of the private good. It is assumed that the private good is nonessential, which implies that there is a choke price sufficiently high that the consumption of the private good falls to zero. When consumption of the weakly complementary private good falls to zero, changes in the environmental good will be of no value to the consumer. Under these circumstances, it is possible to estimate the willingness to pay for the improvement in the environmental good by estimating the change in the compensating variation the consumer receives from the consumption of the private good. Mäler emphasized that this would necessitate the use of Hicksian demands or the expenditure function, neither of which is 1 directly observable. Thus, there are two main options. Mäler (1974) and Larson (1991) describe methods for integrating back from Marshallian demands to obtain the expenditure function.1 The more straight-forward alternative is to estimate Marshallian demands and to use them to approximate the true welfare measures. As Bockstael and McConnell (1993) show, this is more complex than was originally thought. In the case where the demand for quality is known, Randall and Stoll (1980) and Hanemann (1991) have established that the compensating and equivalent measures bound the uncompensated welfare measure. However, the uncompensated measure in terms of the weakly complementary private good will not necessarily equal the uncompensated measure in terms of the environmental good. This is where the Willig condition is relevant. In 1978 in an article on dealing with quality change in price indexes, Willig established three equivalent conditions under which quality changes could be converted into price changes. However, the link between the Willig condition and more traditional welfare measurement techniques was unclear until an important series of papers by Bockstael and McConnell (see, for example, Bockstael and McConnell, 1993) established the importance of the Willig condition for valuing quality changes.2 However, the source of the importance of the Willig condition can still be clarified. It should be emphasized that the two conditions, weak complementarity and the Willig condition are completely independent. Either can hold with or without the other. This paper shows that the Willig condition can be derived easily from the condition for path independence of a line integral. The importance of path independence in measuring the welfare effects of multiple price changes was established by Silberberg (1972).3 Path independence with quality changes has received considerably less attention. When there is path independence with an environmental good and its weakly complementary private good, the 2 uncompensated welfare measure in the environmental good space is equivalent to the uncompensated welfare measure in the private good space. The uncompensated measure derived in private good space can provide a bound for the compensated measure of the value of a quality improvement. In addition to deriving the three versions of the Willig condition from the condition for path independence, this note derives a fourth version that is equivalent and links the Willig condition in quality space to the Silberberg conditions in price space. Examples of the application of these conditions to representative empirical specifications are given, and the implications for implementation are discussed Path Independence and Weak Complementarity In a typical application of weak complementarity there is a change in one environmental good or environmental quality. If the uncompensated demand for environmental quality, q, were observable, the path of integration would not be an issue. If the demand were continuous, the welfare measure would simply be a Riemann integral, since the direct path of integration moves between the two levels of quality without allowing the price of the private good to vary. However, such demands for environmental quality usually are not revealed directly. As soon as we use changes in the private good, x, whose price is p, to reveal information about the environmental good, we must consider changes in both the quality of the environmental good and the price of the private good. This indirect path first holds q at its initial value, increases p to its choke price with quality at q0, and then increases the price further to its choke price with quality at q1. Then quality can be increased from q0 to q1 (without affecting the level of satisfaction because of weak complementarity), and then p can be returned to its initial equilibrium level. We now have a line integral, and the value of that integral may vary with the path of integration that is chosen. We are really only interested in two paths between (p0,q0) and 3 (p0,q1), this indirect path and the direct path with only q changing. Path independence would insure that the latter measure (which is want we want) would equal the former measure (which is what we can observe). 4 The intuition of this integration can be seen using two diagrams similar to those in Bockstael and McConnell (1993). Figure 1 shows the quality change directly in quality space using inverse demands and virtual prices. The welfare measures when quality changes from q0 to q1 can be obtained by integrating under the various inverse demands between q0 and q1. The true uncompensated welfare measure would be the area q 0 abq1 , while the compensated measures would be the areas q 0 acq1 or q 0 dbq1 depending on whether utility was at its initial level or its new level. Figure 2 is the typical weak complementarity diagram showing the demands for the weakly complementary private good. The compensated demands are only shown for the initial level of utility. H and M superscripts on the demand functions represent Hicksian (compensated) and Marshallian (uncompensated) demands. The Hicksian choke prices are labeled p̂ j and the Marshallian choke prices are labeled ~ p j with j being 0 or 1 depending on the level of q. In the diagram, the market price is p*. These various prices are also used as labels for points on the price axis. The uncompensated welfare measure in this diagram would be the area p * ~ p1 g − p * ~ p 0 e , whereas the compensated measure would be the area p * pˆ 1 f − p * pˆ 0 e . Because of path independence, the compensated welfare measures in the two diagrams are identically equal. The uncompensated measures will differ unless a further path independence condition holds. For simplicity assume that in addition to the weakly complementary private good and environmental quality, there is a composite numeraire good, z. The two goods and quality provide utility, u = U(x,q,z) with x, q and z all greater than or equal to zero. Since q is a non- 4 market good, the budget constraint is M = px + z. Solving this constrained maximization problem yields conditional Marshallian demands, x = xm(p,q,M) and z = z(p,q,M), which can be substituted back into the direct utility function to yield the conditional indirect utility function, u = V(p,q,M). Changes in the parameters p, q, and/or M will result in changes in utility. The change in utility can be expressed in terms of the conditional indirect utility function, ∆u = V(p1,q1,M1) - V(p0,q0,M0) = ∫ dV L = ∫ [V p dp + Vq dq + VM dM ] L = Vp ∫ λ ( p, q, M )V L dp + M dq + dM . VM Vq (1) where L is the path of integration, and λ is the Lagrange multiplier, which, by the Envelope Theorem, is the marginal utility of income, VM. As in the case of pure price changes, if λ were constant, it could be taken outside the line integral and used to convert the change in utility to an equivalent change in income. However, λ cannot be constant with respect to all three of the parameters5. Fortunately, the uncompensated weak complementarity argument does not require income to change, so λ may be constant with respect to p and q. Even without assuming λ to be constant with respect to p and q, it is possible to define an uncompensated surplus measure as Vq Vp S = ∫ dp + dq V VM L M (2) (see Silberberg, 1972). This is a line integral that is no longer an exact differential and thus not necessarily path independent. Thus, integrating along the two paths described above need not yield the same answers. Path independence requires 5 Vp ∂ VM ∂q ≡ Vq ∂ VM ∂p . (3) Taking the partial derivatives V pqVM − VMqV p (VM ) 2 = VqpVM − VMpVq (VM )2 . Using Young's Theorem6, the first terms in the numerators on each side of the equation are equal. This implies that VqM V p = V pM Vq if and only if the line integral is path independent. Since Vq ∂ V p ∂M V V −V V pM q = qM p , 2 (V p ) Vq ∂ V p ∂M =0 then (4) if and only if the line integral is path independent. This is one version of the Willig condition. When income changes, the relative change in Vq must be equal to the relative change in Vp. This is the counterpart of quasi-homotheticity of the utility function when there are only private goods.7 Additional insights are available by further consideration of the conditional indirect utility function, V(p,q,M). The familiar Envelope Theorem results, Vp = -λxm(p,q,M) and VM = λ imply that 6 Vq Vp = Vq − λx ( p, q, M ) m =− Vq 1 x ( p, q, M ) VM m . (5) The term in parentheses is the marginal willingness to pay for environmental quality. With path independence, the identity in equation (3) holds. Integrating (3) over p from p0 to ∞, Vq ∂ ∞ dp = VM ∫ ∂p p0 Vp ∂ VM ∫p ∂q 0 ∞ dp (6) The right-hand integral in (6) can be evaluated as Vq VM − p =∞ Vq VM =− p = p0 Vq VM p = p0 because weak complementarity and nonessentiality implies that Vq = 0 at p = ∞. The left-hand integral in (6) is ∞ ∂x ∫ − ∂q dp p0 by Roy’s Identity. Thus, Vq VM ∞ = ∫ x q dp . (7) p0 Combining (5) and (7) yields Vq ∞ 1 − = m x q dp . V p x ( p, q, M ) p∫0 (8) This is the second version of the Willig condition.8 The fact that Vq/Vp is independent of income, as implied by equation (4), implies that 7 ∞ 1 x q dp m x ( p, q, M ) p∫0 (9) is independent of income,9 which is the third version of the Willig condition. The marginal willingness to pay for environmental quality per unit of the weakly complementary private good must be independent of income. Virtual Prices Introducing the virtual price of quality, as done by Bockstael and McConnell (1993), provides another view of the path independence condition. Let πM(p,q,M) be the Marshallian virtual price of quality. It can be derived by solving the hypothetical problem ~ max U ( x, q, z ) subject to M ≡ M + π M q = px + π M q + z . x,q , z The first order conditions include Uq=λπM, so πM = Uq/λ=Vq/VM. Substituting these results and Roy's Identity in equation (2) gives [ ] S = ∫ − x M ( p, q, M ) dp + π M ( p, q, M ) dq . L In this case, the path independence condition is ∂x M ∂π M 10 − = . ∂q ∂p (10) Let compensated (Hicksian) and uncompensated (Marshallian) functions be denoted with H and M superscripts respectively and let E(p,q,u) be the conditional expenditure function. The identity π H ( p, q, u ) ≡ π M ( p, q, E ( p, q, u )) implies that ∂π M ∂π H ∂π M , +x = ∂M ∂p ∂p (11) 8 while the identity x H ( p, q, u ) ≡ x M ( p, q, E ( p, q, u )) implies that ∂x H ∂x M ∂x M , = −π ∂q ∂q ∂M (12) using the fact that Eq = -πH.11 Since ∂x H ∂π H , = − E qp = − E pq = − ∂q ∂p (13) combining equations (10), (11), (12), and (13) and rearranging yields M ∂π M π ∂M M ∂x = . x ∂M (14) The income flexibility of quality must be equal to the income elasticity of the private good in order for the uncompensated welfare measure to be path independent. This is analogous to the Silberberg result that the income elasticities must be equal for private goods whose prices change. Specific Preferences The implications of these results can be illustrated by considering some common specifications of preferences and demand functions. In these examples with known preferences, the true welfare measures in quantity space can be calculated and compared with those derived in price space. This will highlight the importance of the path independence condition. The linear expenditure system (LES) can be modified to incorporate quality and impose weak complementarity (see, for example, Larson, 1991). The LES is generated by a StoneGeary utility function, which is a Cobb-Douglas utility function that has been translated by the 9 subsistence levels of the goods. Quality affects the exponent of x. With this utility function weak complementarity is guaranteed only if the subsistence level of x is -1.12 To keep the example simple, the subsistence level of z is taken to be zero and the exponents other than q are set to one.13 Finally, the log transform of the utility function is used. Maximizing u = q ln(x+1) + ln z subject to the budget constraint, M = px +z yields the Marshallian demand x= qM − p . (1 + q ) p (15) Substituting (15) and the Marshallian demand for z in the direct utility function yields the indirect utility function V ( p, q, M ) = q ln(q ) + (1 + q ) ln( M + p ) − (1 + q ) ln(1 + q ) − q ln( p ) . (16) and by inverting (16), the expenditure function is (u − q ln(q ) + q ln( p ) M = − p + (1 + q ) exp 1+ q (17) Vq = ln(q ) + ln(M + p ) − ln(1 + q ) − ln( p ) (18) Since and Vp = (1 + q ) q − , ( M + p) p (19) it is apparent that the Willig condition (equation 4) cannot hold. How much of a problem does this pose? For example, assume M =10, p = 1, and quality increases from an initial level of 5 to 6. At the initial level of quality, equation (15) can be solved for the Marshallian choke price equal to 50, whereas at q = 6, the choke price is 60. The two Marshallian consumer surpluses are 10 50 1 ∫1 6 p − 6 dp = 24.4335 and 50 60 60 1 ∫ 7 p − 7 dp = 26.6658 , so the change in consumer surplus from 1 x because of the change in q is 2.2323. Using equation (16), one can calculate that the initial utility, u0, is 11.684 and after the quality improvement u1 is 13.914. These values for u and the parameter values can be used in the expenditure function (equation 17) to calculate the expenditure necessary to get to u0 with q=6 and u1 with q=5. The expenditures necessary to get to u0 with q=5 and u1 with q=6 are both 10, the actual income. Thus, the compensating surplus or quantity compensating variation, QCV, is 3.0015, while the equivalent surplus or quantity equivalent variation, QEV, is 4.9528. The Marshallian consumer surplus measure, CS, should be between QCV and QEV, yet estimating it from the uncompensated demands for x comes up with a value that is significantly lower than either value. The absence of path independence can lead to significant miscalculations. The correct uncompensated welfare measure can be calculated by integrating the Marshallian virtual price, π M = Vq VM , as q changes from 5 to 6. ( M + p )[ln(q ) + ln( M + p ) − ln(1 + q ) − ln( p )] dq ∫ + 1 q 5 6 (20) While this integral cannot be solved analytically, numerical integration at the parameters above yields 3.7815. This true uncompensated measure is between QCV and QEV, but it is roughly 70 percent larger than the incorrect measure that is often invoked.14 This example also highlights another point. While x and q are weak complements, this does not necessarily imply anything about their complementarity at the actual price. When q increases from 5 to 6, the uncompensated quantity of x demanded does increase from 8.167 to 11 8.429, but the compensated quantity demanded is reduced to 5.856. The compensated quantity goes down at the market price, but the integral of the Hicksian demand increases as expected because the two Hicksian demands cross above the equilibrium price. While the demands with this utility function are nonlinear, this type of shift can be visualized using Figure 2. Suppose DH(p,q1,u0) were steeper. Point h would be further to the left, and, more surprisingly, point f could be to the left of point e. This is the case in this example. The typical weak complementarity demand diagram is a useful expository device, but actual cases can be more complex. A second example, one where the Willig condition holds, is a variant on the simple repackaging model introduced by Fisher and Shell (1971) and generalized by Willig (1978). Suppose the indirect utility function is q V ( p, q, M ) = ln( M ) + ln + 1) . p The partial derivatives of V are V p = − (21) 1 q , and VM = 1 / M . Roy’s Identity , Vq = (q + p) p q+ p gives x= qM , (q + p ) p (22) and inversion of equation (23) gives the expenditure function M = exp(u ) . q + 1 p (23) The Marshallian and Hicksian choke prices are now infinity, but these are valid choke prices because the integrals to the left of the demands are finite. The utility function is weakly 12 complementary because as p → ∞ , changes in q have no effect on utility. Since neither Vq nor Vp depends on M, neither does their ratio. The Willig condition holds. Using the same parameter values as in the first example, one can calculate the initial and final utility levels as u0 = 4.0943 and u1 = 4.2485. Substituting these values in the expenditure function yields the welfare values QCV = 1.4286 and QEV = 1.6667. Integrating the Marshallian demands for x over p from the actual price to the choke price at the two quality levels yields ∞ ∞ p ∫ 5 p + p 2 dp = 10 ln p + 5 = 17.9176 1 1 ∞ 50 ∞ p ∫ 6 p + p 2 dp = 10 ln p + 6 = 19.4591 1 1 60 so the change in consumer surplus is CS = 1.5415. Now the uncompensated measure is between the two compensated measures as expected. This can be confirmed using the uncompensated virtual price πM = M/(q+p) and integrating to get the same value.15 Also, the income flexibility of this uncompensated inverse demand for quality is unitary and therefore equal to the income elasticity of x. A final example is the common linear demand for x, x = α + βp + γq + δM . Larson (1991) shows that weak complementarity requires that the indirect utility function be β 1 V ( p, q, M ) = M + α + βp + γq + δ δ This means that 13 δ exp − (γδ + βp ) . β (24) − δγ β γ 1 M + α + βp + γq + + Vq δ δ β δ = , Vp β β 1 − δ M + α + βp + γq + + δ δ δ (25) which obviously varies with M unless δ = 0 so that equation (25) equals γ/β. If income does not affect x, then the Marshallian and Hicksian demands are identical and path independence holds by default. This special case is not very interesting. Otherwise, the Willig condition does not hold, and the consumer surplus cannot be based on the Marshallian demands for x. Implications The estimation of the willingness to pay for environmental quality using related markets often invokes weak complementarity. If compensated demands are used, weak complementarity guarantees that the welfare measure in the private good space is identical to the welfare measure in the public good space. However, when uncompensated demands are used, the two measures need not be equivalent. This note shows that the conditions under which the uncompensated measures are equivalent correspond to the condition for path independence of a line integral generating the welfare measure. This path independence provides intuition for the three expressions that have become known as the Willig condition. The implications of path independence for the virtual price of environmental quality also imply the equivalence of the income elasticity of the private good and the income flexibility of the public good. This is a strong assumption that is unlikely to be satisfied in most cases, just as most private goods do not have identical income elasticities. If it is not satisfied, there can be major errors in the uncompensated welfare measure in terms of the weakly complementary private good. The Willig condition has played a major role in the evolution of our understanding of ways to recover welfare measures for environmental changes using related market goods (e.g., 14 Bockstael and McConnell, 1993, and Smith and Banzhaf, forthcoming) and therefore has substantial pedagogical value. However, it is only possible to confidently derive welfare measures for quality changes directly from estimates of the demand for the weakly complementary private good if the path independence conditions hold. This requires conditions on the indirect utility function or the uncompensated virtual price function. Yet if one knows either of those functions, it is unnecessary to derive the welfare measures from the estimated uncompensated demands for the private good. It is preferable to specify an indirect utility function that satisfies nonessentiality and weak complementarity but not the Willig condition and derive compensated and uncompensated welfare measures directly as environmental quality changes. By not requiring the Willig condition, greater flexibility in specifying functional forms is possible. If a Willig form were nested within the estimated form, one could test the implied restrictions. 15 1 Larson (1991) analytically solves the differential equation implied by the Marshallian demand. Numerical solutions as in Vartia (1983) might seem an attractive alternative. However, as Bockstael and McConnell (1993) show, determining the bounds for the numerical integration requires additional information such as that provided by the Willig condition discussed here. 2 Smith and Banzhaf (forthcoming) present a graphical interpretation of the Willig conditions. 3 See Just, Hueth, and Schmitz (1982) for a good summary of this literature. 4 The issue of path independence does not arise when Hicksian demands are used because the expression being integrated is an exact differential of the conditional expenditure function introduced below. Line integrals of exact differentials are path independent. 5 See, for example, Just, et al. (1982). 6 Young’s Theorem proves that as long as V is twice continuously differentiable, Vpq=Vqp. 7 Silberberg (1972) showed that when there were only private goods and income did not change, consumer surplus was path independent only when the direct utility function was homothetic. Homotheticity implies that the marginal rate of substitution between two goods is invariant to income and all income elasticities are equal to 1. Lau (1970) showed that if the direct utility function was homothetic, the indirect utility function was also homothetic. If not all prices change, the income elasticities of those goods must be equal but not necessarily equal to 1. This is quasihomotheticity. With the conditional system here, Vq>0 while VP<0, and these concepts of homotheticity have not been used. Nonetheless, the condition for path independence here is in the same spirit. When virtual prices are used later, this link will be further clarified. 8 The minus sign was inadvertently omitted in Bockstael and McConnell (1993). 9 The change of sign does not influence the independence. 10 Bockstael and McConnell (1993) derive a similar equation from Willig rather than path independence. However, a minus sign is omitted. They start with their equation (12), which, in the notation here, is Vq VM ∞ ( p0 ) = ∫ x q dp , p0 which is equivalent to equation (7) here. The left-hand side is the marginal willingness to pay for q when p=p0, so it is equal to the uncompensated virtual price of quality when p=p0. They suggest differentiating both sides by p, perhaps using the Fundamental Theorem of Calculus (integration and differentiation are inverse operations), to get ∂π M ∂x M . = ∂p ∂q However, this is a definite integral, and differentiating with respect to the equilibrium price is differentiating with respect to the lower limit of integration. This means that (e.g., see Taylor and Mann (1972)) ∞ ∂ ∫ x q dp p0 ∂p 0 ∂x M ( p 0 ) =− , ∂q which yields the equation (10) here. 16 11 See, for example, Hanemann (1991). 12 Herriges, et al. (2004) introduce a modified form of this utility function where the ‘subsistence” level can take on any values while maintaining weak complementarity. This generalization is empirically important, but for the current example the simpler form is used. 13 In empirical work, these parameters would be estimated. 14 The Marshallian virtual price also shows that the Willig condition does not hold. The income elasticity of demand for x is equal to 1, while the income flexibility of the inverse demand for quality is equal to M/(M+p). 15 Even though the Willig condition holds, the compensated changes in x at the equilibrium price show that an increase in quality leads to a reduction in the quantity demanded of x. The Hicksian demands again cross when q changes. 17 References Bockstael, N.E., and K.E. McConnell (1993). "Public Goods as Characteristics of Non-Market Commodities." Economic Journal 103: 1244-1257. Fisher, F.M., and K. Shell (1971). “Taste and Quality Change in the Pure Theory of the True Cost-of-Living Index.” in Z. Griliches, ed., Price Indexes and Quality Change, Cambridge MA: Harvard University Press. Hanemann, W. Michael (1991). "Willingness To Pay and Willingness To Accept: How Much Can They Differ?" American Economic Review 81:635-647. Herriges, J.A., C.L.Kling, and D.J. Phaneuf (2004). “What’s the Use? Welfare Estimates from Revealed Preference Models when Weak Complementarity Does Not Hold.” Journal of Environmental Economics and Management 47: 55-70. Just, Richard E., Darrell L. Hueth, and Andrew Schmitz (1982). Applied Welfare Economics and Public Policy. Englewood Cliffs: Prentice-Hall. Larson, Douglas M. (1991). "Recovering Weakly Complementary Preferences." Journal of Environmental Economics and Management 21: 97-108. Lau, Lawrence J. (1970). "Duality and the Structure of Utility Functions." Journal of Economic Theory 1: 374-396. Mäler, Karl-Göran (1974). Environmental Economics: A Theoretical Inquiry. Baltimore: Resources for the Future. Randall, Alan, and John R. Stoll (1980). "Consumer's Surplus in Commodity Space." American Economic Review 71: 449-457. Silberberg, Eugene (1972). "Duality and the Many Consumer's Surpluses." American Economic Review 62: 942-952. Smith, V. Kerry, and H. Spencer Banzhaf (forthcoming). "A Diagrammatic Exposition of Weak Complementarity and the Willig Condition." American Journal or Agricultural Economics. Taylor, Angus E., and W. Robert Mann (1972). Advanced Calculus, 2nd edition. New York: John Wiley and Sons. Vartia, Y.O. (1983). “Efficient Methods of Measuring Welfare Change and Compensated Income in Terms of Ordinary Demand Functions.” Econometrica 51:79-98. 18 Willig, Robert D. (1978). "Incremental Consumer's Surplus and Hedonic Price Adjustment." Journal of Economic Theory 17: 227-253. 19 p d p H (P*,q,u1) a p b c q0 p q1 Figure 1 H M (P*,q,M) (P*,q,u0) q P P1 P1 P0 P0 P* h DM(p,q0,M) g f e DH(p,q0,u0) DM(p,q1,M) DH(p,q1,u1) x Figure 2