Measuring Consumer Welfare with Mean Demands

Economics Department of the University of PennsylvaniaInstitute of Social and Economic Research -- Osaka University

Measuring Consumer Welfare with Mean DemandsAuthor(s): Edward E. SchleeSource: International Economic Review, Vol. 48, No. 3 (Aug., 2007), pp. 869-899Published by: Wiley for the Economics Department of the University of Pennsylvania and Instituteof Social and Economic Research -- Osaka UniversityStable URL: http://www.jstor.org/stable/4541994 .

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INTERNATIONAL ECONOMIC REVIEW Vol. 48, No. 3, August 2007

MEASURING CONSUMER WELFARE WITH MEAN DEMANDS*

BY EDWARD E. SCHLEE1

Arizona State University, U.S.A.

The welfare change from a price increase-for example, the compensating variation (cv)-is often calculated using the expenditure function from an estimated demand. If there is unobserved preference heterogeneity, then the estimated demand is an average over households with different preferences. And the cv from the mean demand does not generally equal the mean cv. We give conditions en- suring that the cv from the mean demand equals the mean cv, is less than the mean cv, and approximates the mean cv better than the change in consumers' surplus. A necessary condition is that demands become more dispersed as income rises.

1. INTRODUCTION

Suppose that the price of a good rises because of a tax and we have an estimate of the good's demand. Moreover, we agree that either the aggregate compensating or aggregate equivalent variation is an acceptable measure of the welfare loss.2 How should we use the estimated demand to calculate either measure? As is well understood, the change in consumers' surplus does not equal either of the two welfare measures if there are income effects on demand. Willig's (1976) apology for consumers' surplus is that the percentage error from using it to approximate either measure is small if the price change is small, the good's budget share is small, or its income elasticity of demand is small. Hausman (1981) counters that there is no need to settle for an approximation: Often the estimated demand can be "integrated back" to find the expenditure function, from which both the compensating and equivalent variation can be calculated directly.

Hausman's (1981) approach requires a parametric demand function. Since welfare calculations are sensitive to functional form choices, Hausman and Newey (1995) nonparametrically estimate the demand for gasoline using household level data. To estimate the welfare loss of a tax to households, they numerically integrate the estimated demand to calculate the equivalent variation. How we interpret the welfare estimates in either approach depends on how we interpret the regression

* Manuscript received September 2005; revised June 2006. 1 I thank three referees for detailed comments. Part of this work was done while I was visiting Stan-

ford University on Sabbatical leave from Arizona State. I thank both institutions for support. An earlier version circulated under the title "Measuring Consumer Welfare Using Statistical Demands." Please address correspondence to: Edward E. Schlee, Department of Economics, Arizona State University, Tempe, AZ 85287, U.S.A. E-mail: [email protected].

2 Boadway (1974) shows that the aggregate willingness to pay as measured by the compensating variation can be positive for a change, yet the winners cannot compensate the losers, so this agreement should not be taken lightly. Of course, the aggregate willingness to pay is Pareto Consistent: If it is positive, then at least one household must be better off.

869



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errors (Foster and Hahn, 2000; Blundell et al., 2003, Section 3.4). If they are en- tirely due to measurement error (of the dependent variable), then we may safely ignore them. If, however, they are due in part to unobserved preference heterogeneity, then the demand being estimated is really an average demand, in which the mean is taken over households with different preferences. And in general, the equivalent variation calculated by integrating the mean demand does not equal the mean equivalent variation, so ignoring the regression errors can lead to wrong inferences about the mean welfare loss.3 Hausman and Newey (1995, pp. 1453-54) offer two justifications for ignoring the regression errors: They are due solely to mismeasurement (of demand) or the welfare estimates simply apply to a (hypothetical) household whose demand equals the estimated demand. The second justification amounts to using the preferences of a "representative household" to make welfare judgments about the underlying group.

It is unlikely that regression errors are due solely to mismeasurement. A natural question is how welfare measures calculated using only the mean demand compare with the mean welfare measures. Indeed, can we even be sure that the equivalent variation calculated from the mean demand approximates the mean equivalent variation better than the change in consumers' surplus? For concreteness, we consider the compensating variation. (Analogous results hold for the equivalent variation.) Suppose that we observe prices, household incomes, some demographic variables, and household consumption choices, but that the demographic variables do not capture all of the preference heterogeneity. Suppose that the price of a single good increases and consider a group of of households with the same income and demographic characteristics. We give conditions under which the following three assertions about the compensating variation calculated from the expenditure function for the mean demand are true:

(i) it always equals the mean compensating variation; (ii) it is a lower bound for the mean compensating variation; and (iii) it approximates the mean compensating variation better than the change

in consumers' surplus.4

A necessary and sufficient condition for (i) is that preferences are Gorman: Engel curves for the single good are straight lines with common slope across households (Proposition 1). A necessary condition for (ii) is that demands satisfy increasing dispersion: On average, households with higher consumption of a good have a higher marginal propensity to consume it. If the demands satisfy a strict version of increasing dispersion and the mean demand increases with income, then (ii) and (iii) hold for all small enough price increases. In the two-good case, increasing dispersion is equivalent to a plausible form of heteroskedascity:

3 In particular, even if one had data on the entire population of households, this procedure does not give the true mean welfare loss: The welfare loss estimate is inconsistent.

4 If the mean demand is increasing in income then (iii) follows from (ii) since, for a normal good, the change in consumer's surplus is less than the compensating variation for a price increase. The mean change in consumers' surplus equals the change in consumers' surplus calculated using the mean demand. See Equations (5) and (6).



CONSUMER WELFARE WITH MEAN DEMANDS 871

the variance of the error increases with income.5 These results thus give some support for using the compensating variation from the mean demand to bound the welfare loss from taxes. Unfortunately, increasing dispersion does not imply either (ii) or (iii) (Example 1 and Proposition 3). If, however, we impose an additional restriction-that the indirect utility functions are additively separable in preference type and income-then increasing dispersion is equivalent to (ii), and therefore implies (iii) under normality (Proposition 4). We characterize the family of demands that admit a family of additive indirect utilities (Proposition 5).

Clearly, ours is a special kind of aggregation question, in which only households with the same income and observed demographic characteristics are aggregated. The closest paper in the aggregation literature to ours is Jerison (1994). He shows that, with a fixed income distribution, increasing dispersion together with symmetry of the Slutsky matrix for aggregate demand imply the existence of a (positive) representative household-that is, aggregate demand is the demand of a hypothetical household who owns all the wealth. (The qualifier "positive" is used to emphasize that the representative household's utility function need not be a valid measure of social welfare: The "positive" representative household might be nor- matively unrepresentative.) More importantly, he shows that increasing dispersion is necessary for a representative household to satisfy a version of Kaldor consistency (Proposition 4)-roughly if the aggregate willingness to pay for a change in prices is positive, then the representative household must prefer the change. He also poses as an open question whether a strict version of increasing dispersion implies that the representative household is Kaldor consistent. We give examples showing that the answer is "no." We spell out the relationship between our results and Jerison's in Section 3.5.

In another closely related paper, Blundell et al. (2003) allow for unobserved preference heterogeneity in their nonparametric treatment of Engel curves for household-level data (see especially their Section 3.4). They assume that the error in the demand equation is multiplicatively separable in the preference type and the price-income pair (equation (28) on page 221). Under this assumption, they show that, for households with the same income and observed demographic characteristics, the compensated variation calculated from the mean demand is a first-order approximation to the mean compensating variation, and it is a second-order approximation if the marginal propensity to consume is the same across households (p. 223). We extend these findings to more general functional forms (Proposition 2). We show, however, that identical marginal propensities to consume does not ensure that the bias is zero (Proposition 3).

A natural question is why we should use just the mean demand if we have household-level data. Besides unobserved preference heterogeneity, the regression errors in practice also include measurement error.6 And mean demands at

5 Prais and Houthakker (1955, pp. 55-63) is a classic reference on heteroskedasticity in demand estimation. Hirdle et al. (1991), Hildenbrand and Kneip (1993), and Hildenbrand (1994, chapter 3) find that increasing dispersion holds across data from several countries.

6 Blundell et al. (2003, p. 223). Household-level demand studies rely on surveys instead of actual consumption choices, so measurement error is apt to be important.



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least eliminate (zero mean) measurement error. Our results on bounding the true welfare loss using mean demands are useful for studies that average over households because of measurement error-or because the entire distribution of preferences cannot be recovered.

The results just described all assume that the households face no uncertainty: The uncertainty, embodied in the preference heterogeneity, is in the mind of the statistician. If households are uncertain of future prices, incomes, or states of the world that affect their preferences, then risk preferences matter for welfare. By Lewbel (2001, Theorem 2), if the indirect von Neumann-Morgenstern (vN-M) utility is additively separable in income and any variable that is random (including the preference type), then the household's expected indirect vN-M utility function generates the mean demand. In this case, the ex ante compensating variation for a price change equals the compensating variation from the mean demand. As Lewbel points out, this additive separability assumption can justify disregarding the error term (if we interpret the welfare calculation to be ex ante). We show that if the compensating variation from the mean demand is merely a first-order approximation to the ex ante compensating variation, then the indirect vN-M utility must be additively separable in income and any variable that is random (Proposition 6). If this separability fails, then the change in consumers' surplus can easily be a better approximation (Proposition 7): It is, for example, if preferences are Cobb-Douglas, the price of the taxed good is not too high, and each household's relative risk aversion for income gambles exceeds one.

We assume that household demands are continuously differentiable. This assumption ensures that the Slutsky equation holds for each household, a fact we repeatedly exploit in our proofs. Unfortunately, even continuity of demands rules out discrete goods. We cannot therefore directly apply our results to probabilis- tic discrete choice models-a potentially important omission, since these models emphasize unobserved preference heterogeneity. Early studies typically assumed that the marginal utility of income across all discrete alternatives was constant and equal-the Gorman condition (see, e.g., Domencich and McFadden, 1975; McFadden, 1981)-so the representative household assumption held. More re- cently, this assumption has been relaxed (e.g., Petrin, 2002). And studies that do relax it routinely estimate the mean compensating variation.' Since so few studies that estimate discrete choice models approximate the mean compensating variation by the compensating variation of a representative household, our exclusion of discrete goods is perhaps not a serious flaw.

2. PRELIMINARIES

We follow the random utility models of McElroy (1987), Brown and Walker (1989), and especially Lewbel (2001). We assume that prices, household income, and household consumption choices are observable. Preference heterogeneity is

7 Herriges and Kling (1999) survey approaches to estimating the mean compensating variation in discrete choice models. And Dagsvik and Karlstr6m (2005) write down a formula for the mean

compensating variation from a discrete choice model with nonlinear income effects.




usually modeled by including demographic variables. We assume that the observed demographic variables do not fully explain preferences. Since household income is observed, in what follows we only aggregate households with the same income and demographic characteristics.

Let p denote the price of good 1 and m household income. We assume thatp e P and m e M, where P and M are bounded, strictly positive open intervals. We assume that the prices of the other goods are fixed, so that we can aggregate all other goods into a composite commodity with price equal to 1, as do most studies that estimate the welfare loss of taxes. Preferences of a household are determined by a pair (r, z) e T x Z, where T and Z are each nonsingleton subsets of a (for simplicity) Euclidean space. z is a vector of observed demographic variables and r is the unobserved type of the household. (Note that we allow r to be multidimensional.) Let d(p, m, r, z) denote the demand for good 1 at price-income pair (p, m) for a type-r household with demographic variables equal to z. We assume throughout that d(., r, z) is single-valued and arises from utility maximization for every (r, z) e T x Z. Let (p, m) ý-+ v(p, m, r, z) be a representation of a (r, z) indirect utility function, (p, u) + e(p, u, r, z) the associated expenditure function, and (p, u) H h(p, u, r, z) the compensated demand, where u is utility.8 To ensure that expectations are well defined, we assume that d(p, m, ., z) is Borel measurable for every (p, m, z) E P x M x Z. The distribution of r is given by a cumulative distribution function (c.d.f.), F(.; z), for each z e Z. Let F be the set of all c.d.f.'s with bounded support in T.

ASSUMPTION 1. For every z e Z, F(.; z) e F.

ASSUMPTION 2. For every (r, z) e T x Z, the functions v, e, d, and h are twice continuously differentiable in (p, m, u) with vi < O and v2 > O on P x M x Range(v(.;

The mean demand, D(p, m, F, z), equals

(1) D(p, m, F, z) = d(p, m, r, z) dF(r; z).

Assumptions 1 and 2 ensure that the mean demand is bounded and is differentiable in p and m and that, for i = 1, 2, we have Di(p, m, F, z)= fdi(p, m, r, z) dF(r).

Define the error to be the difference between the mean demand and the true demand for a (r, z)-household with income m:

(2) E(p, m, r, F, z) = d(p, m, r, z)- D(p, m, F, z).

By construction, the error has mean zero for every (p, m, z) E P x M x Z.

8 For a review of the definitions and properties of these functions, see Mas-Colell et al. (1995) or Deaton and Muellbauer (1980).

9 Range(v(.; r, z)) = {v(p, m; r, z) I (p, m) E P x M}. Subscripts denote partial derivatives; e.g., vl = av/ap. The condition vl < 0 implies that demand is positive for all price-income pairs in P x M and all types in T.



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Of course, an empirical study would start by specifying D and then (possibly) impose assumptions on the distribution of the error. The empirical error is just the difference between the estimated demand D and the choice of a household with characteristics z at price-income pair (p, m). (In a sample, there may well be only a single observation at any particular (p, m, z), since it is unlikely that two households will report exactly the same expenditure level m. Thus, the aggregation is really over the population of potential households, not just those included in the sample.) We start with d, then define D to be the mean of d; the error is just the difference between d and D.

The next assumption ensures that D is generated by utility maximization.10

AssUMPTION 3. For every (p, m, F, z) E P x M x Fx Z,

Di (p, m, F, z) + D(p, m, F, z)D2(p, m, F, z) < 0.

Although not all empirical studies assume that aggregate demand is generated by a utility function, our question presupposes it: We need to calculate the compensating variation from the mean demand, which requires a well-defined expenditure function. Assumption 3 ensures that the mean demand is generated by some utility function, U(.; F, z). This utility function represents the preferences of a (positive) representative household for the group. Let (p, m) + V(p, m, F, z) denote the associated indirect utility function: V(p, m, F, z) = maxx[o,m/p] U(x, m - px; F, z). Let (p, U) + E(p, U, F, z) denote the associated expenditure function-the inverse of V(p, ., F, z)-and (p, U) ý- H(p, U, F, z) the compensated function for good 1.

ASsUMPTION 4. For every (F, z) E T x Z, the functions V, E, and H are twice continuously differentiable in (p, m, U) on P x M x Range(V(.; F, z)).

For two results (Propositions 1 and 6), we require that the demands for any two distinct types be unequal almost everywhere.

AssuMPTION 5. For each (m, z) e M x Z and any two distinct types rl and r2 in

T,7 d(., m, r 1, z) d(., m, T2, z), exceptpossibly on a set P* C P of Lebesgue measure zero.

3. EX POST WELFARE: ERRORS AS UNOBSERVED

PREFERENCE HETEROGENEITY

From now on, we aggregate over households with the same income and demographic characteristics (a restriction, to repeat, we justify by the existence of household-level expenditure data and unobserved heterogeneity). So, from now

10 Since there are only two goods and d(., -, r, z) is continuously differentiable and is generated by utility maximization, the two-good demand system (D, m - pD) with good 2 as numeraire satisfies

adding up, homogeneity, and Slutsky symmetry (Lewbel, 2001, Corollary 2 or Mas-Colell et al., 1995,

pp. 34-36). It is easy to show that the additional requirement of Assumption 3 implies that the Slutsky matrix is negative semidefinite, so that the mean demand is generated by utility maximization.




on, we suppress the vector of demographic variables, z, for all functions. Although we aggregate over households with the same income m, we keep it as an argument, since the relationship between demand and income across types turns out to be crucial.

Suppose that the price rises from Pi to P2. Define the compensating variation (cv) to be the extra income needed to compensate a type r household for the change:

(3) cv(pl, P2, m, ) e(p2, v(pl, m, r), r) - m = h(p(p, v(pl, m, r), r) dp.

(Many authors define the cv to be minus the expression in (3).) The mean compensating variation is

(4) fTcv(pl, p2, m, r) dF(r) = fe(p2,2

v(p, m, r), r)dF(r) -m

= f h(p, v(pi, m, r), r) dp dF(r).

One common way to approximate the mean compensating variation is to use (minus) the change in consumer's surplus:

(5) -ACS(pl, P2, m, F)= D(p, m, F) dp.

Since D is the mean demand, (5) also equals (minus) the change in expected consumers' surplus,

(6) [ d(p, m, r)dp dF(r).

Another way to approximate the mean compensating variation is to use the mean demand D to recover the expenditure function E for the representative household, in the spirit of Hausman's (1981) suggestion. This gives us the compensating variation for the representative household,

(7) CV(pi, p2, m, F) = E(p2, V(pl, m, F), F) - m = ] H(p, V(pl, m, F), F) dp.

It is evident that neither (5) nor (7) equals the mean compensating variation (4). In (4), we first calculate the compensating variation for each type r by integrating each demand, then average to get E[cv(r)] (where E[.] denotes the expectations operator); in (7) we first average the demands to get E[d(r)], then calculate the compensating variation by integrating the average demand. Loosely, (7) and (4)



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reverse the order of expectation and integration. We compare the bias from using (5) and (7). Define

(8) CSBias(p2;p1,m, 5F) = D(p, m, F)dp - J cv(pl, P2,5m, r)dF(r)

and

(9) CVBias(p2; pi, m, F) = E(p2, V(pi, m, F), F) -fe(P2, v(p, m, ),r),dF.

3.1. A Diagrammatic Overview. Before plunging into details, we give an overview using social indifference curves over price-income pairs. This overview also helps place our results in the large literature on welfare and aggregation initi- ated by Gorman (1953). Consider a group of households, each with price-income pair (pa,

mi), so that m, is also the mean income of the group. A social indifference curve through (pl, mi) (with income initially distributed equally) gives, for each P2 e P, the smallest mean income mi + Am that can be distributed among the households so that each one is indifferent between the initial price-income pair, (pi, mi), and (p2, m1 + Amr), where Am, is type r's allocation of the additional mean income, Am. Of course, for each type r, Am, is precisely its compensating variation for the price increase, and the change in mean income, Am, is the mean compensating variation for the group. Consider Figure 1, and suppose that the curve ab is a social indifference curve through (pi, mi) (assuming income is initially distributed

equally). If the price rises frompl to p2, then mi + Am is the smallest mean income that can be redistributed among the households to make each indifferent between

Income

b mi +Am

a

Pi P2 Price of Good 1

FIGURE 1

A PRICE-INCOME INDIFFERENCE CURVE




the new and initial situation. (Of course, at (P2, m1 + Am), income need not be equally distributed.) As noted, Am = E[cv(pi, P2 , , r)]. It follows easily that the slope of the social indifference curve through (pl, mi) is the mean demand at

Pl, E[d(pl, mi, r)].11 Now consider the indifference curve for the group's representative household

through (pl, mi). If the curve ab in Figure 1 depicts such an indifference curve, then the increment Am is CV(pl, P2, mi, F), and the slope at (pl, mi) is the mean demand, D(pi, mi, F). (Of course, the curve is a graph of the representative's expenditure function at utility V(pi, mi, F).) Thus CV is a first-order approximation of E[cv], a result we report in Proposition 2 (a).

Clearly, if the representative household's compensating variation always equals the mean compensating variation, then, for any (pl, F)E P x F, the social indifference curve through (pi, mi) (with income initially distributed equally) must coincide with the indifference curve of the representative household. Consider the social indifference curve through (pl, mi). Its slope at (P2, mi + E[cv(pl, P2, ml, r)]) is E[h(p2, v(pi, mi, r), r)] (see footnote 11), or equivalently,

E[d(p2, e(p2, v(pl, mi, r), r), r)].

Consider the representative household's indifference curve through (pi, mi). Its slope at (p2, m1 + CV(pl, p2, ml, F)) is H(p2, V(pl, mi, F), F), or equivalently,

D(p2, e(p2, V(pl, mi, F), F), F).

Equating the two slopes, we have

(10) D(p2, e(p2, V(pl, ml, F), F), F) f d(p2, e(p2, v(pl, mi, r), r), r) dF(r).

Proposition 1 says that if (10) holds for all prices, incomes, and type distributions, then the mean demand cannot depend on the distribution of income, implying that the Engel curves are straight lines with common slope-the Gorman condition. The standard aggregation literature arrives at this condition by allowing the distribution of income to vary arbitrarily (see, e.g., Mas-Colell et al., 1995, pp. 100-7). Note, however, that the income distribution on the right side of (10) is pinned down by the initial equal income mi and the compensating variation of each household; this restriction on the income distribution complicates the proof of Proposition 1. We show in that proof that the third derivatives of the social and representative agent's indifference curve are always equal if and only if each household's demand is both additively separable in income and type and affine in income.

We also give sufficient conditions for CV to be a lower bound for E[cv] (for a price increase). Figure 2 shows social and representative household indifference

11 The slope of the social indifference curve through (p1, ml) is just the mean derivative of cv. Using Equation (3), the derivative of cv with respect to p2 is the compensated demand for a type-r household, h(p2, v(pi, mi, r), r). Setting P2 = Pl gives the ordinary demand for a type-r household at (pl, mI).



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Income

Social IC

ml ••

epresentative Household's IC

Ap Price of Good 1

FIGURE 2

SOCIAL INDIFFERENCE CURVE (IC) AND REPRESENTATIVE HOUSEHOLD'S IC WITH CV < E[CV]

curves in this case. Obviously, a necessary condition is that the social indifference curve has less curvature than the representative household's indifference curve for every (pi, F) e P x F. Proposition 2(b) and Proposition 3 give the relationship between this curvature condition and increasing dispersion.

3.2. Zero Bias. If each household has preferences that are quasilinear in some good other than good 1, then the demand for good 1 does not depend on income and the change in consumers' surplus from a price increase equals the compensating variation. That strong requirement is a special case of the following well-known aggregation condition.

DEFINITION 1. Preferences for types in T are Gorman on P x M if the indirect utility functions can be written in the form v(p, m, r) = a(p, r) + mb(p) on P x M x T (with the function b(.) common to all types).

Equivalently, preferences are Gorman if the Engel curves for good 1 are straight lines with common slope (Gorman, 1961): The demands take the form d(p, m, r) = a(p, r) + 6(p)m. (All proofs are in the Appendix.)

PROPOSITION 1. Let Assumptions 1-5 hold. Then CVBias(p2; p1, m, F) = 0 for all (pl, p2, m, F) E P2 x M x F if and only if preferences for household types in Tare Gorman on P x M.

If the initial income distribution of the group were unrestricted, then the conclusion of Proposition 1 would follow immediately from Gorman (1953), and we could dispense with Assumption 5, which rules out the case of identical




preferences. We, however, only aggregate across households with the same income (a point also emphasized by Blundell et al., 2003, p. 222 in their treatment of unobserved heterogenity). Obviously if preferences are also the same, then CVBias is zero, no matter what the common preferences are. Put another way, we may ask: "If we restrict households to have the same income, can we substantially weaken the conditions found in Gorman (1953) for which social indifference curves will be nonintersecting (in the price-income space)?"12 Proposition 1 shows that the answer is "no": Unless preferences are identical, we are back to the same conclusion we would get if the income distribution were allowed to be arbitrary.

Another difference with the standard aggregation literature is that we quantify over all type distributions (i.e., all F e F) to prove necessity of the Gorman form, whereas in Gorman (1953) the distribution of tastes is fixed. Proposition 1 thus leaves open the possibility that CVBias might be zero for some restriction on the type distribution. The proof of Proposition 1 reveals, however, that if CVbias is zero for a given type distribution F, then the marginal propensity to consume must be uncorrelated with demand (see Equation (A.8), which is equivalent to E[e2e] = 0). Since zero correlation contradicts (strictly) increasing dispersion, CVBias is surely not zero.13 Note that Proposition 1 does not imply the economy is Gorman, since we only aggregate across households with the same observed demographics and income.

3.3. Small Price Changes: Approximations and Bounds. The next best thing to unbiasedness is for CVbias to be small; failing that we want to know the direction of the bias and whether it is smaller than CSbias.

Let o(.) denote a real-valued function on the real line equal to 0 at 0 with limto o(t)/t = 0.

PROPOSITION 2. Suppose that Assumptions 1-4 hold. Fix (pl, m, F) E P x M x

Y. For all p2 E P with p2 > p1 we have

(a) CSBias(p2; P1, m, F) = o(Ipl - p21) and CVBias(p2; p1, m, F) =

o(Ipl - P2 );

(b) CVBias(p2; ,pi m, F)= - (pl - p2)2f8(p1, m, r, F)e2(Pl, m, r, F) dF(r) + o(Ipl - p212);

(c) CSBias(p2; P1, m, F) = -(pi - p2)2D(pl, m, F)D2(p1, m, F) - (P1 - p2)2 f E(pl ,

, r, F)82(p1, m, r, F) dF(r) + o(Ipl - p212).

Proposition 2(a) says that both the change in consumers' surplus and the representative household's compensating variation are first-order approximations to

12 Gorman (1953) deals with social indifference curves over aggregate consumption bundles, not price-income pairs. Gorman (1961) gives equivalent conditions on the demands and indirect utility function.

13 If the type distribution is fixed, then zero correlation between e and e2 for all prices and income does not imply that the group of households in T is Gorman. Let the mean demand be given by some function 0(p, m) that is generated by utility maximization. Suppose that the error is given by e =

a(M l(r) + ( /1

-oa(m)r2(r), where 0 < a(m) < 1 for all m e M, a(.) is continuously differentiable

with a' > 0 on M, and rl (), r2('),

and F are such that q1 and 02 are identically and independently distributed random variables with zero mean and positive variance. It is easy to confirm that E[e2e] = 0, but the group of households is not Gorman.



880 SCHLEE

the mean compensating variation; parts (b) and (c) give the sign and magnitude of the second-order bias for small price increases. The covariance between e and E2 figures prominently.

We say that the demands indexed by r E T exhibit increasing dispersion at (p, m, F) if f e(p, m, r, F)E2(p, m, r, F) dF(r) > 0; they exhibit strictly increasing dispersion at (p, m, F) if f E(p, m, r, F)e2(p, m, r, F) dF(r) > 0. The demands

satisfy increasing dispersion at F if they satisfy increasing dispersion at (p, m, F) for every (p, m) E P x M. Intuitively, increasing dispersion says that, on average, households with a higher demand have a higher marginal propensity to consume the good.14 Let a be the variance of . Since E[2] = - , increasing dispersion is equivalent to a plausible form of heteroskedasticity: The variance of the error increases with income. The following corollary follows from Proposition 2(b).

COROLLARY 1. Suppose that Assumptions 1-4 hold, fix (pl, m, F) E P x M x F and suppose that D2(p1, m, F) > 0 (the last implying that CSBias(p2; p1, m, F) <

CVBias(p2; p1, m, F) for all p2 > p1 in a neighborhood of pl).

(a) If CVBias(p2; pi, m, F) < 0 for all P2 > pl, then the demands exhibit

increasing dispersion at (pi, m, F). (b) If the demands exhibit strictly increasing dispersion at

(pl, m, F), then

CVBias(p2; pi, m, F) < 0 for all p2 > p1 on a neighborhood of pi. (c) CVBias(p2; p1, m, F) = o(0p2 - pl12) for all p2 E P only if

f E(pi, m, r,

F)E2(Pl, m, r, F)dF = 0.

Corollary 1(c) formally records a fact mentioned at the end of Section 3.2: CVBias = 0 at a given type distribution F only if income effects are uncorrelated with demand at F, contradicting (strictly) increasing dispersion. Since increasing dispersion has strong empirical support (see note 5), Corollary 1(c) implies that CVBias is surely not zero. Put more positively, CV + (P2 - pl)2E[882] is a second-order approximation to E[cv].

Corollary 1 (a) gives a necessary and (b) a local sufficient condition for I CVBiasI to be smaller than ICSBiasl. In Figure 2, strict increasing dispersion implies that, locally, the representative household's indifference curve through (pt, mo) has more curvature than the social indifference curve through (pi, mo).

Since increasing dispersion has broad empirical support, it is worth knowing whether imposing it globally implies that ICVBiasJ < ICSBiasl. The following example shows that it does not.

EXAMPLE 1. Let the demands be d(p, m, r) = (a(t) + b(r)m - -cm2)/p,

where a, b, c > 0. The demand d(., r) is generated by the indirect utility function

2 1 (cm - b(r) r(p, m, r) =-lnp+ 2a(t)c +bTr)2

k 2a(r)c+b(r)2

/

14 For discussions of increasing dispersion (for n > 2 goods and under much more general conditions on the population of households) see Hildenbrand (1994, chapter 3) and Jerison (1994).




The compensating variation for an increase in price from pi = 1 to P2 > 1 for a type-r household is

S

tanh ln(p2) + tanh-1 cm - b() - m + c 2 \ Ar))) c

where A(r) = 2a(r)c + b(r)2. Since d is affine in a(r) and b(r), we can calculate V and CV by substituting the expected values of a and b into the formulas for v and cv. Suppose first that r only affects a, so that dispersion is constant. Let c = 1/2, m = 1 = b, a E {1, 3} with equal probability, pl = 1, and p2 = 2.7. For this price increase fcv dF

. 2.7097, CV 2.7675, and -ACS

. 2.7314. By continuity, the

inequalities between each of these magnitudes are preserved if b is random with small enough support. If b is positively correlated with a, then dispersion is strictly increasing.

Intuitively, increasing dispersion is a second-order property (Proposition 2(b)). For large price changes, it is not enough to ensure that I CVBiasl is smaller than I CSBiasI or that CV < f cv dF. In Example 1, the marginal propensity to consume, d2, is the same across types and is strictly concave in income. The next result shows that these two conditions ensure that CV > f cv dF-the compensating variation from the mean demand is not a lower bound for the mean compensating variation.

PROPOSITION 3. Let Assumptions 1-4 hold and fix (pl, m, F) E P x M x F. If D22(Pl, m, F) < 0, f(e(pl, m, r, F))2 dF # 0, and 82(p, m, r, F) = 0 for all (p, r) e P x T, then CVbias(p2; pi, m, F) > 0 for all p2 > P1 on a neighborhood ofpl.

3.4. Additive Separability and Bounds for Large Price Changes. Summing up the argument so far, CV is almost surely biased, since unbiasedness contradicts strictly increasing dispersion. If dispersion is strictly increasing, then CV bounds the mean compensating variation from below for small enough price increases. Example 1 and Proposition 3 show, however, that increasing dispersion does not imply that CV approximates the mean compensating variation better than the change in CS, or that CV bounds the mean compensating variation from below.15 For these implications, we must restrict the preference profile further. In particular, suppose that the indirect utility functions can be written in the additively separable form v(p, m, r) = f(p, r) + g(p, m) for some functions f and g (where g is the same for all r). This condition obviously generalizes the Gorman condition (Definition 1). Unlike the Gorman condition, however, it does not restrict any individual household's demand in isolation; it only restricts the relationship between demands. In particular, if all households with the same income and demographics have the same preferences, then additive separability holds trivially. By Roy's identity, additive separability implies that

f (p, r) gl (p, m) (11) d(p, m, r) = .

g2(p, m) g2(p, m)" 15 Hausman and Newey consider "large" tax increases in the gasoline tax to show that their estimates

for the equivalent variation are far from the change in consumers' surplus.



882 SCHLEE

Differentiate both sides of Equation (11) with respect to m and rearrange to find that

g22(p, m) g2(p,_m) (12) d2(p, m, r) - d(p, m, ) - 2(, m)

g2(p, m) g2(p, m)

Taking expectations of both sides with respect to r and rearranging yields

(13) d2(p, m, r) - D2(p, m, F) g22(P, m) d(p, m, r) - D(p, m, F) g2(p, m)

The left side of Equation (13) gives the ratio between two magnitudes: the difference between type-r's marginal income effect and the mean marginal income effect; and the difference between type-r's demand and the mean demand. Equation (13) says that this ratio does not depend on (r, F). Strict concavity of g implies that this ratio is positive: The marginal propensity to consume is higher for higher demand types (at each (p, m)). This form of additive separability with g strictly concave in income thus implies a strong form of increasing dispersion.

PROPOSITION 4. Let Assumptions 1-4 hold and suppose that the indirect utility functions indexed by r Tcan be written in the form v(p, m, r) = f(p, r) + g(p, m) (where g is common to all types r). If the corresponding demands exhibit increasing dispersion at Fe F, then CVBias(p2; P1, m, F) < 0 for all (pl, P2, m) e p2 X M with p2 > pi. If in addition D2(p, m, F) > O for all (p, m) E P x M, then CSBias(p2;

pl, m, F) < CVBias(p2; Pl, m, F) < 0 for all (pl, pI) e P2 with p2 > pl.

In words, under additive separability of the indirect utility in type and income, increasing dispersion implies that the CV bounds the mean compensating variation from below. Since additive separability has this important implication, it is worth spelling out what exactly it imposes on the relationship between demands. Equation (12) obviously implies that

(14) d2(p, m, r) = a (p, m) + 8(p, m) d(p, m, r)

for some functions a and 6.16 Since the demands are twice continuously differentiable in (p, m), Equation (12) also implies the "integrability" condition that

a2(P, m) = l(p, m) for all (p, m) e P x M. It turns out that (14) and this integrability condition characterize additive separability of the indirect utilities in type and income.

PROPOSITION 5. The following two conditions are equivalent:

(a) The indirect utilities indexed by r e T can be written in the additively separable form v(p, m, r) = f(p, r) + g(p, m) for all (p, m) e P x M.

16 In particular, alng2(p, m)/am = -p(p, m) and alng2(p, m)/Ip = -U(p, m).




(b) For all r e T, d2(p, m, r) = a(p, m) + 8(p, m) d(p, m, r) for some continuously differentiable functions a and P with a2(p, m) = (1(p, m) for all (p, m) e P x M.

We now illustrate condition (b) for some examples satisfying strictly increasing dispersion (implying that P(p, m) > 0 in (14)).

EXAMPLE 2 (Linear Demands). Consider the linear demands integrated by Hausman (1981): d(p, m, r) = a(r) + b(r)p + c(r)m. If these demands satisfy strictly increasing dispersion, then the function c(.) is not constant. But if c(.) is not constant and Equation (14) holds, then f(p, m) = 1/m and a(p, m) = -(a + bp)/m, so that P1(p, m) = 0 : (a + bp)/m2 = o2(p, m), violating (b) in Proposition 5.

EXAMPLE 3 (PIGLOG Demands, Muellbauer, 1976). Let the demands be given by

d(p, m, r) = m(a(p, r) + b(p, r) In m).17

Here Equation (14) implies that P(p, m) = 1/m and a(p, m) = b(p, r) so that

/ (p, m) = 0 = a2(p, m) = 0. Thus condition (b) holds precisely when the function b(.) does not depend on r. Since / is positive, dispersion is increasing. Assuming that b does not depend on r, these demands are generated by the indirect utility functions

In m - In A(p, r) v(p, m, r ) = In B(p)

where A(., r) and B are positive, strictly increasing, and concave functions.

EXAMPLE 4 (Nonparametric Engel Curves, Blundell et al., 2003). In their discussion of unobserved heterogeneity, Blundell et al. write budget shares as

a(r)q(p, m) + b(p, m),

so that

d(p, m, r) = ma(r)4(p, m)/p + mb(p, m)/p

Here Equation (14) implies that f(p, m) = 1/m and a(p, m) = m(b2 + /2 a)/p. Since a cannot depend on r, additive separability requires that 0 cannot depend on income, implying increasing dispersion. Moreover, since 61 (p, m) = 0, we have

az(p, m) = 0, which implies that b is linear in In m: budget shares for each good must be affine in log income, and we are back to the PIGLOG case of Example 3.

These examples suggest that additive separability of the indirect utilities in type and income is a strong restriction. One way to assess it empirically is to model heteroskedasticity explicitly, since additive separability restricts the functional form of the errors. By Equation (13), E82/e is independent of (r, F); in particular,

17 PIGLOG demands include the log-linear demands that Foster and Hahn (2000) use in their nonparametric estimation of consumers' surplus.



884 SCHLEE

by Equations (11), (14), and (13), we have that

(15) a82(p, m, r, F) e(p, m, r, F)

where f(p, m) is the function in Equation (13). Equation (15) implies that the error must be multiplicatively separable in m and (r, F).18 This functional form restriction on the heteroskedasticity could in principle be tested. To develop one implication of multiplicative separability, multiply both sides of (15) by E2 and take expectations to find that

(16)f e2(p, m, r, F)(p, m, r, F) dF(r) = /(p, m) f (p, m, r, F)2 dF(r).

Recall that or2 = f e2 dF and acZ/8m = f e2e dF. These facts and Equation (16) imply that

(17) aor/m = =(p, m).

Additive separability implies that the left side of (17) does not depend on F. In words, the growth rate of the error variance (as income rises) does not depend on the distribution of types.

To illustrate the condition in (17), consider Prais and Houthakker's (1955, p. 56) finding that the variance of the error in the demand for tea is proportional to expenditure on tea. In our notation, they find that a2 (p, m, F) = k(F)(pD(p, m, F))2, for some real-valued, positive function k(.) on F. One can easily verify that Equation (17) holds for this specification if and only if D2/D does not depend on F. Under Assumption 5, D must depend on F, so, by (14), we have a(p, m) = 0 for all (p, m) E P x M. Setting a(p, m) = 0 for all (p, m) E P x M in (13) has two immediate and strong implications: The income elasticity of demand does not depend on the type and the price elasticity of demand does not depend on income. Since one clear stylized fact about food demand is that price elasticities do depend on income (Blundell et al., 1993), additive separability is implausible if the error variance is proportional to expenditure on a food item.19

Another, more informal, way to evaluate additive separability would be to look at how observed demographic variables enter estimated demands, on the (perhaps heroic) assumption that variables that account for unobserved heterogeneity enter demand the same way that observed demographic variables do. As an example, Schmalensee and Stoker (1999) extend Hausman and Newey's (1995) gasoline demand estimates by including demographic variables. In particular, they use

18 That is, e(p, m, r, F) = 4(p, m)*(p, r, F) for some real-valued functions 4 and *r. 19 If the error variance is simply proportional to the square of total expenditure, i.e., o2(p, m, F) =

k(F)m2, then additive separability could hold (it does, for example, if each household in the group has a common and constant income elasticity).




(1999, eq. (3.1)) the partial linear specification

LogGallons = G(LogIncome, LogAge) + H(Demographics)

where H is a linear function of nonage demographic variables, and G is an arbitrary function of household income and the age of its oldest member. Since G is arbitrary, this specification does not imply that the indirect utility function is additive separable in income and the vector of demographic variables. Their nonparametric estimate of the function G suggests that it is additively separable (pp. 649-50). In our notation, their estimated demand function is20

(18) In d(p, m, r) = f(p, r) + b In m,

where b > 0 and "r" is the vector of all demographic variables. One can easily check that condition (b) of Proposition 5 holds.21

To sum up, additive separability of the indirect utility in type and income is a strong assumption. Unless we test for it, we cannot be sure that the representative household's compensating variation for a price increase bounds the mean compensating variation from below (unless the price increase is small).

3.5. Relationship to Jerison (1994). Jerison (1994, Section 5) determines when a positive representative household's preferences over price vectors are norma- tively representative of households as a whole. In particular, he considers a version of Kaldor consistency: If the aggregate willingness to pay for a change in prices is positive, then the representative household prefers the change, and if the aggregate willingness to pay is nonnegative, then the representative household is not worse off with the change. He finds that, when the income distribution is fixed, Kaldor consistency implies (an n-commodity version of) increasing dispersion, and if dispersion is strictly increasing in a neighborhood of a price vector, then Kaldor consistency holds for some neighborhood of that price vector (1994, p. 755). Jerison allows more than one price to change, while normalizing aggregate income to 1. Of course, if we consider a single price change in our two-good world, then a representative household is necessarily Kaldor consistent since all households dislike price increases. Nonetheless our Corollary 1 can be deduced from his Propositions 3 and 4. We prove this assertion in Section A.7.

In the last sentence of his Section 5, Jerison poses as an open question whether strictly increasing dispersion implies Kaldor consistency. Our Proposition 3 con- structs a class of examples that satisfy constant dispersion, yet fails his version of Kaldor consistency (which can be shown using an argument similar to that in Section A.7); as in Example 1, one can perturb the demands to yield strictly increasing dispersion, yet preserve the inconsistency. Hence, this strengthening of

20 Schmalensee and Stoker do not have price data. 21 Yatchew and No (2001, eqs. 3.1 and 3.2) begin with specifications already of the form (18), so we

cannot use their results to evaluate additive separability.



886 SCHLEE

increasing dispersion does not imply Kaldor consistency. If, however, the indirect utilities are additively separable in type and income, then by Proposition 4 increasing dispersion is equivalent to this version of Kaldor consistency.

4. EX ANTE WELFARE: ERRORS AS HOUSEHOLD UNCERTAINTY

So far we have assumed that the only uncertainty is that the statistician does not know households' preferences; households themselves face no uncertainty. One reason why we estimate welfare changes is to evaluate policies to be taken in the future. Consider, for example, a proposed increase in the gasoline tax. At the time the tax increase is being considered, a household may well be uncertain of several things: what the future price of gasoline will be (with or without the tax increase), what its income will be, and what its preferences for gasoline and other goods will be (for example, due to weather shocks). If the tax increase must be decided on before this uncertainty is resolved, then we should calculate welfare ex ante, not ex post. Since, on this interpretation, each household faces uncertainty, risk preferences matter for welfare.22

Define cvante(Pl, P2, m, F) to be the solution to

f v(pl, m, r) dF =

f v(p2, m + Cuante(pl, P2, m, F), r) dF.

Here the function v is a von Neumann-Morgenstern (indirect) utility. The number

CVan,,te gives the ex ante welfare loss from a price increase for a household facing uncertainty. Note that by allowing the vector r to enter the price and income arguments of v, we can accommodate uncertainty over the future price of good 1 and income, as well as preference uncertainty. For example, we could have v(p, m, r) = f(p + tl, m + r2, r3), where r = (rl, T2, r3). In this case, we might know that the tax will raise the average price by P2 - P1, but not precisely what the post- tax price will be. The ex ante compensating variation takes this uncertainty into account. The question here is how well CV(pl, P2, m, F) approximates CVante(Pi, P2, m, F). By Theorem 2 in Lewbel (2001), if the marginal utility of income, v2(p,

m, r), is independent of r, then f v(pl, m, r) dF generates the mean demand. It then follows that cvante(Pi, P2, m, F) = CV(pl, P2, m, F). As Lewbel points out, this result can justify Hausman and Newey's disregard of the error term-provided that we interpret their welfare calculation to be an ex ante one.

PROPOSITION 6. Let Assumptions 1-5 hold. The following four conditions are

equivalent.

(a) CV(pl, p2,m, F) = cvante(pl, p2,m, F) +o( IP2 - p lI) for every (pl, m, F) e P x Mx F.

(b) CV(pl, p2, m, F) = CVante(Pl, P2, m, F) for every (pl, p2, m, F) E P2 x M x FT.

22 The question of how to evaluate nonrandom price changes under taste uncertainty was posed by Schmalensee (1972). Lewbel (2001) considers ex ante welfare for demand systems with unobserved

heterogeneity.




(c) -ACS(pi, P2, m, F) = CVante(Pi, P2, m, F) + o(|p2 - 1ll) for every (p, m, F) E P x M x F.

(d) The indirect utility function v is additively separable in m and r: For every (p, m) E P x M, v2(p, m, r) does not depend on r.

The main contribution of Proposition 6 is that (d) is necessary even for (a), which just says that CV (derived from the mean demand) is a first-order approximation to CVante. The condition in (d) is stronger than the additive separability condition of the last section since here v is a von Neumann-Morgenstern utility: (d) implies that all household types have the same risk preferences over income gambles, whereas the additive separability of the last section says nothing at all about risk preferences. If the household is uncertain of its future income (so that some coordinate of r enters the income argument of v), then Proposition 6 implies that v must be linear in m: The household is risk neutral for income gambles and has quasilinear preferences.

If the marginal utility of income is negatively correlated with demand (when both are viewed as random variables with states T), and the mean demand is increasing in income, then consumer's surplus from the mean demand approximates the ex ante compensating variation better than the compensating variation from the mean demand.

PROPOSITION 7. If the covariance of d(pl, m, -) and v2(pl, m, .) is negative, then

CV(pl, p2, m, F) > cvante(pi, P2, m, F) for all p2 in some neighborhood of pl. If in addition D2(pl, m, F) > 0, then

-ACS(pl, P2, m, F) approximates cvante(Pi, P2,

m, F) better than CV(pl, p2, m, F) at (pl, m, F).

To illustrate the covariance condition in Proposition 7, suppose that the ordinal version of additive separability from the last section holds: v(p, m, r) = ¢( f(p, r) g(p, m)), where 0 is a strictly increasing function. If v is more concave in income than g and higher demand states are preferred, then the covariance condition of Proposition 7 holds. For example, if preferences are homothetic, then g(p, m) = In m; if in addition - fi(p, -') > - fl(p, r) if and only if f(p, r') > f(p, r) and v is more concave in income than In m, then the covariance condition holds-a plausible condition on risk aversion, if not demand.23

We pointed out in Section 4 that additive separability is a strong condition, even absent household uncertainty. Propositions 6 and 7 suggest that, if households face uncertainty, then the compensating variation calculated using the mean demand is apt to be a poor approximation to the true welfare loss from a price increase.

5. CONCLUSION

If errors-the difference between actual and mean demands-arise from unobserved preference heterogeneity, then using just the mean demand for welfare

23 If preferences are Cobb-Douglas and we write f in the "natural" form f(p, r) = a(r) In p, then the co-monotonicity condition on f and -fl automatically holds if p < 1. We can always add a function f(r) involving only r to f without changing the demand. If 4 is an increasing transformation of a(r), then this cutoff price will exceed one.



888 SCHLEE

amounts to using the preferences of a representative household. And it is unlikely that the representative household's compensating variation equals the mean compensating variation for a group of households, since that equality contradicts the well-established property of increasing dispersion. Under (strictly) increasing dispersion, the representative household's compensating variation is a lower bound for the mean cv for two cases: the price change is small enough or the indirect utilities are additively separable in income and the preference type. An open question is how far additive separability can be relaxed for the second case, since it tightly constrains the relationship between demands.

If households face uncertainty, then matters are less clear. Here additive separability of the indirect vN-M utility in income and any variable that is random (including the preference type) is necessary if the compensating variation calculated by integrating the mean demand is merely a first-order approximation to the ex ante compensating variation; the change in expected consumer's surplus can easily do better if additive separability fails.

Clearly, one solution is to estimate the distribution of welfare measures instead of just using information from the mean demand. Brown and Matzkin (1998), Foster and Hahn (2000), Matzkin (2003), Berry et al. (1995), Petrin (2002), and Beckert and Blundell (2005) give approaches to the identification and estimation of random utility models with nonlinear income effects.

We have focused on the case of a price change for a single good largely for simplicity, allowing us to aggregate up to two goods. Many of the results extend to the case of more than two goods (after strengthening Assumptions 3 and 5 for the n-good case). In particular, our characterization of additively separable indirect utilities by demands in Proposition 5 and the equivalence noted in Section 3.5 between Kaldor consistency and increasing dispersion in this case hold for n goods whose prices all change simultaneously.

APPENDIX: PROOFS

A.1. Two Preliminary Lemmata

LEMMA A.1. Suppose that Assumptions 1-4 hold, and fix a (pl, m, F) E P x M x .

(a) CVante(Pi, P2, m, F) = (pl

- P2)f (pr)dF( (12 - P1)

(b) CV(pl, p2, m, F) = (pl - p2) f ( ) dF(r) + o(p2 - p i)

(c) -ACS(pl, P2, m, F) = (pi - P2) f v(P:')dF(r) + o(IP2 -

P1 ).

PROOF. Fix (P1, m, F) EP x M x . For any (P2, r) EP x T, we have by the Mean Value Theorem that there is a point 4r(p2, r) on the line segment between

(pl, m) and (p2, m + CVante(P1, P2, m, F)) such that

(A.1) v(p2, m + CVante(Pa,

P2, m, F), r) - v(pl, m, r)

=- v1(u(P2, t), t)(p2 - Pl) + v2(f(p2, t), T)CUvante(P1, P2, m, F).




Let

f vi(pl, m, r) dF(r)

f v2(Pl, m, r) dF(r)

Integrating both sides of (A.1) with respect to r and rearranging gives us

-CVante(Pi, P2, m, F)/(p2 - Pl) f vi(4(p2, r), r) dF(r)

f v2 *(P2, r), r)dF(r)

fA+ ( vl(*(P2, T), T) vi (pl, m, ) dF().

S+ f V2((P2, r), r) dF(r) f v2-Pl m, r) dF(rt)

Since the support of F is bounded by Assumption 1, and v is continuously differentiable on its bounded domain by Assumption 2, the integrand in the last integral is bounded and converges pointwise to zero as p2 --+ p1-, so the integral converges to zero by the Lebesgue Convergence Theorem, establishing part (a). The proofs for parts (b) and (c) are similar. 0

LEMMA A.2. Let X C R be a Borel measurable set, and let Yx be the set of cumulative distribution functions on X with bounded support. Let f and g be real- valued, bounded, Borel measurable functions on X, with f one-to-one. If

(A.2) f f(x) dF(x) J g(x) dF(x) = f f(x)g(x) dF(x)

for all Fe Fx, then g is constant on X.

PROOF. Let all the hypotheses hold, and suppose that (A.2) holds for all Fe .Fx. If X is a singleton, then the conclusion follows immediately, so suppose that X has more than one element, let xl and x2 be any two distinct points in X, and let F put equal probability on the two points. Thus Equation (20) implies (after rearranging and multiplying both sides by 2) that

f(Xl)g(x2) + f(x2)g(x1)= f(xl)g(x1)+ f(x2)g(x2),

or

(f(x2) - f(xl))(g(x2)

- g(x1))

= 0.

Since f is one-to-one, this last equality implies that g(xl) = g(x2). Since xl and x2 were arbitrary, g is constant on X. M

A.2. ProofofProposition 1. Suppose that preferences are Gorman for T on P x M, so that a type-r household has an indirect utility function of the form v(p,



890 SCHLEE

m, r) = a(p, r) + mb(p). The expenditure function for a type-r household is

u - a(p, r) e(p, u, r) = b(p)

so that

e(p2, v(pi,m, r), r) dF(r) f(a(pi, r) - a(p2, r))dF(r) + mb(p1) b(p2)

The mean demand

f i(p, c) b'(p) D(p, m, F)= al(p, r dF(r) + m

b(p) b(p)

is generated by the indirect utility function

V(p,m, F) = f a(p, r)dF(r) + mb(p) = v(p, m, r)dF(r).

The associated expenditure function is

U - f a(p, r) dF(r) E(p, U, F) = b(p)

so that

V(pi, m, F) - f a(p2, r) dF(r) E(p2, V(pl, m, F), F) - b(p2) b(p2)

f a(pi, r)dF(r) + mb(pi) - f a(p2, r)dF(r) b(p2)

f(a(pi, r) - a(p2, r)) dF(r) + mb(pl) b(p2)

Hence E(p2, V(pl, m, F), F) = fe(p2, v(pi, m, r), r)dF(r) and CVBias(p2; P1, m, F) = 0 for all (pl, P2, m, F) E p2 x M x FT.

To prove the necessity of Gorman, let CVBias(p2; P1, m, F) = 0 for all (P1, P2, m, F) E p2 x Mx FT, so that

(A.3) E(p2, V(pi, m, F), F) = 2, e (p2 , m, r), r)dF

for any (pi, P2, m, F) E p2 x M x F. Differentiating both sides of (A.3) with respect to p2 yields24

E(p2, V(p, m, F), F) = f el (p2,

v(pl, m, r), )dF

24 Since F has bounded support by Assumption 1 and since e and v are continuously differentiable on their bounded domains by Assumption 2, the derivative of the integral on the right side of (A.3) with respect to P2 equals the integral of the derivative of e(p2, v(pl, m, r), r) with respect to p2.




which, by Shepard's Lemma, is the same as

(A.4) H(p2, V(pi, m, F), F) = fh(p2, v(pl, m, r), r) dF,

or, equivalently,

(A.5) D(p2, m + CV(p, p2, m, F), F) = d(p2, m + c(p, 2,m, r), r)dF.

Differentiating both sides of (A.5) with respect to p2 yields

(A.6) D1(p2, m + CV(pl, P2, m, F), F)

+ D2(P2, m + CV(pl,

P2, m, F), F)CV2(pl, P2, m, F)

= f d(p2, m + cv(p, P2, m, r), r) dF(r)

+ (d2(p2, m + cv(pl,

P2, , r), )cu2(pl,p2,

m, r))dF.

Setting p2 = P1 yields (using CV2(P1, P2, m, F) = H(p2, V(pi, m, F), F)) and

H(pl, V(pi, m, F), F)) = D(pl, m, F))

(A.7) D (pl, m, F) + Dz(pi, m, F)D(pl, m, F)

= f (di(pl,

m, r) + d2(pi, m, r)d(pl,

m, r)) dF(r).

Since D(pl,m, F) = f d(pl,m, r)dF(r), we have Di(pl,m, F) = f d(pl,m, r) dF(r), so Equation (A.7) implies that

(A.8) D2(pl, m,

F)D(p1, m, F) = f d2(p, m, r) d(p, m, r) dF(r).

Let r' and r" be two distinct points in T. By Assumption 5, we may take d(pl, m, r') = d(pi, m, r") (otherwise increase pl slightly and the inequality will be true). Since d(pl, m, -) is one-to-one on X= {r', r"}, if (A.8) holds for all F E Fx, then by Lemma A.2, we have that d2(P1, m, r) does not depend on r on the set X. Since (pi, m) was arbitrary and d2(., m, r) is continuous, d2(., r) is constant on X. And since r', r" were arbitrary, d2(p, m, .) is constant on T for every (p, m) eP x M.

It remains to show that d is affine in income. Return to Equation (A.6). Differ- entiate this equation with respect to P2 and evaluate at p2 = P1 to find that

(A.9) D + 2D12D + D22D2 + D2D1 + DzD

= (dl

+ 2d12d

+ d22d2 + d2di + did)dF,



892 SCHLEE

where all price arguments have been set equal to pl, and for the left side of (A.9) we have used the following facts: CV2(Pl, P2, m, F) = H(p2, V(pl, m, F), F);

H(pa, V(pa, m, F), F)) = D(pi, m, F); CV22(p1, P2, m, F) = Hi(p2, V(pi, m, F),

F); and the Slutsky equation,

Hi(pl, V(pl, m, F), F) = DI(pl, m, F) + D2(pi, m, F)D(pi, m, F).

For the right side of (A.9), we have used the analogous facts for the individual demands. Of course D11 = f dl1 dF. And since d2 does not depend on r, we have that 2D12D + D2D1 + D2D

= f(2d12d + d2dl

+ did) dF, so that (A.9) reduces to

(A.10) D22(pl, m, F)(D(pl, m, F))2 = d22(pl, m, r') f(d(pl, m, r))2 dF.

Since d2 does not depend on r, neither does d22, SO

D22(P1, m, F) = d22(P1, m, r)

for all (pl, m, r, F) e P x M x Tx . If d22(pl1, m', r) : 0 for some (p', m') E P x M, we have from (A.10) that

(A.11) (D(p', m', F))2 = f(d(p', m', r))2 dF,

for all Fe F. Let F have support on two distinct points in T. If (A.11) holds, then by Assumption 5 there is a p* E P such that (A.11) fails at p' = p* and d22(P*, m', r) 0 (since d22(', m', r) is continuous by Assumption 2). Thus, if Equation (A.10) holds for all (pl, m, F) e P x M x F, then d22(p, m, r) = 0 for all (p, m, r) E P x M x T, so demand is affine in income and additively separable in type and income: the household types in T are Gorman on P x M. N

A.3. Proof of Proposition 2. Since F is fixed throughout, we often omit it as an argument. Consider part (a). By the Mean Value Theorem,

CSBias(p2; Pi, m, F)

= (P2- Pl)[D(k(p2),

m) -

h(P(p2), v(pI,

im,gr), r)dF(r),

where P(p2) lies between pland P2. Rewrite this expression as

(P2 - Pl)

[f(d(p(p2), m, r) - h((p2), v(p,

m, r),

r))dF(r).

The integrand converges pointwise to zero as p2 - 1 P and the integrand is bounded (by Assumption 1 and the continuity of d, h, and v from Assumption 2).




Hence, the Lebesgue Convergence Theorem implies that

lim CSBias(p2; p1, m, F) =0.

P2--P P1 - P2

Now consider

CV(pi, p2, m, F) - cv(pi, p2, m, r) dF(r) + f D(p, m) dp

f H(p, V(pl, m)) dp - D(p, m, F) dp + CSBias(p2;

p1, m, F).

The Mean Value Theorem implies that the first expression is of higher order than IP1 - P21, so that CVBias (p2; p1, m, F) = o(Ipl - p2 ).

To prove parts (b) and (c), note that di(p, m, r) = DI(p, m) + el(p,

m, r) and

d2(p, m, r) = D2(p, m) + 82(p, m, r). By the Slutsky equation,

(A.12) hi(pa,

v(pl, m, r), r) = dl(pl, m, r)+ d2(p1, m, r)d(pl, m, r)

= Di(pi, m) + el(p1, m, r) + D2(p1, m)D(pl, m)

+ E(PI, m, t)D(pl, m) + E(p1, m, r)e2(p1, m, r) (A.13) = Hi (p, V(pl, m)) + e(pl, m, r)D2(pl, m)

+ E1(P1, m, r) + E(pI, m, r)82(P1, m, r).

Since fT, (p, m, r) dF(r) = 0, and e is continuously differentiable by Assump- tion 2, we have fT e1 (p, m, tr)dF(r) = 0 and fT e2(p, m, r) dF(r) = 0. By Assump- tions 1, 2, and 4, all the demand, expenditure, and indirect utility functions are twice continuously differentiable on their bounded domains. Hence, each Bias function is C3 the bounded domain P2 x M. Now differentiate (9) twice with respect to p2 and use (A.13) to get

(A.14) a2CVBias (P2; P1, m, F) (A.14) = H(p2, V(pl, m)) aP2

- f h(p2, v(p,, m, r), r)dF(r),

so that by (A.13) we have

a2CVBias (p2; Pl, m, F) =

(A.15) - - pl, m, OS2(pl,

m, r) dF(r). 8p2 P2= p

Similarly from (8) and we have

a2CSBias (p2;

Pl, m, F) -

(A.16) Op2

= DI(p2, m) -l hl(p2, v(pm, m, r), r) dF(r),



894 SCHLEE

so that by (A.12) we have

82CSBias (P2; Pl, m, F) (A.17) = - D(pl, m)D2(pl, m) ap22 P2=Pl

- f(pl,

m, r)82(pl,

m, r) dF(r).

Parts (b) and (c) now follow from Taylor's Theorem and by (A.17) and (A.15). I

A.4. Proof of Proposition 3. We often omit F as an argument since it is fixed. Since 82(p, m, r) = 0 for all (p, r) E P x T, the right side of (A.15) in the proof of Proposition 2 is zero. Differentiating the right side of (A.14) again with respect to

p2 gives

(A.18) 3 CVBias(p2; pi, m, F) (A.18) = H1 (P2, V(pl, m))

aP2

- hJ1(p2, v(pl,m, r), r) dF(r).

By the Slutsky equation, we have

(A.19) hal(pi,

v(pi, m, r), r) = dil + 2d12d + d22d2 + d2dz

+ d d,

where all the functions on the right side of the equality are evaluated at (pl, m, r). Similarly,

(A.20) Hi1(pi, V(pa,

m, F), F) = D11 + 2D12D + D22 D2 + D D1 + D2D,

where the functions on the right side are evaluated at (pl, m, F). Since 82(p, m,

r) = 0 for all (p, r) E P x T, it follows from (A.19) and (A.20) that25

(A.21) hi l (p, v(p, m, r), r) dF(r)

= Hu(pi, V(pi, m, F), F) + D22(pi, m, F) f((pi, m, r, F))2 dF(r).

The result follows from Taylor's theorem, (A.18), and (A.21). U

25 Since e2(p, m, r) = 0 for all (p, r) E P x T, we have d2(p, m, r) = D2(p, m, F) for all (p, r, F) e

P x T. From this equality, we get that d2(p, m, r) dl(p, m, r) = D2(p, m, F)D1(p, m, F), d12(p, m, r) d(p, m, r) = D12(P, m, F)D(p, m, F), and d2(p, m, T)2d(p, m, r) = D2(p, m, F)2 D(p, m, F) for all (p, r) E P x T.




A.5. Proof of Proposition 4. Suppose that the demands exhibit increasing dispersion at F. By Equation (13), g is concave in m. Letting g-1 (p, -) denote the inverse of g with respect to income, we have

e(p, u, r) = g-'(p, u - f(p, r)).

Thus

e(p2, v(pl, m, r), r) = g-'(p2, f(Pl, r) 1 g(pl, m) - f(p2, r)).

The mean demand

D(pm, F)

= f fi(P, r) dF + gi(p, m)

g2(p, m)

is generated by the indirect utility function

V(p, m, F) = f (p, r) dF + g(p, m),

with associated expenditure function

E(p, U, F) = g-1 (p U- f f(p r)dF).

Thus

E(p2, V(p, m, F)) g-1 (2, (f(, )- (p2, r))dF+ g(pl, m)) .

Since g is concave (and strictly increasing) in income,

f e(p2, v(pi, m, r), r)dF = f g(p2f( , )- f(P2, r)+ g(p, m))dF

> g'l (P2, (f(pi, r) -

f(p2, r)) dF + g(pl, m))

= E(p2, V(pl, m)),

and CVBias(p2; p1, m, F) < 0. If D2 > 0, then CSBias(p2; pl, m, F) < CVBias(p2; p, m, F) < O. I

A.6. Proof of Proposition 5. The argument that (a) implies (b) is given in the text. Suppose (b) holds. We will find functions f and g so that the indirect utilities can be written in the form in (a). Since a(.) and ~(.) are continuously differentiable, by Froebinius's Theorem there is a function G(p, m) such that GI(p, m) = a(p, m)



896 SCHLEE

and G2(p, m) = m8(p, m). For any mo e M define

g(p, m) = Jexp(-G(p, )) d4.

Next define

H(p, m, r) = g2(p, m) d(p, m, r) + gi(p, m).

Differentiating both sides with respect to m gives us

H2(p, m, r) = g22(P, m) d(p, m, r) + g2(p, m) d2(p, m, r) + g12(P, m)

= -g2(p, m)(a(p, m) + P(p, m) d(p, m, r) - d2(p, m, r))

=0,

so that H is independent of m. Set (for any po e P) f(p, r) = - jP H(4, m, r) Applying Roy's Identity to the function f(p, r) + g(p, m) shows that it generates the demands. I

A.7. Proofofthe Assertion in Section 3.5. We show that our Corollary 1 can be deduced from Jerison's (1994, p. 755) Propositions 3 and 4. For part (a), it suffices to show that our requirement that CV < f cv dF for all changes in the price of good 1 implies his version of Kaldor consistency when we restrict attention to the two-commodity case. Let (pI/m, qi/m, 1) denote the initial price-income pair (qi is the initial nominal price of good 2 and the third component is income, which has been normalized to one) and (p2/m, q2/m, 1) the second price-income pair. Suppose that the representative household does not strictly prefer (p2/m, q2/m, 1) to (pl/m, ql/m, 1), or equivalently, does not strictly prefer (P2/q2, 1, m/q2) to (pi/qi, 1, m/q1). We want to show that the mean willingness to pay for the change from (pl, ql, m) to (P2, q2, m) is nonpositive. Let CV denote the representative household's compensating variation for a change in the price of good 1 from pl/ql to P2/q2 at income ml/ql. By hypothesis, CV + m/qi > m/q2. Now let cv*(r) denote a type-r household's compensating variation for the change between (pilql, 1, mlql) and (P2/q2, 1, m/q2).26 It follows that a type-r household's compensating variation for the change in price of good 1 from pllql to p2/q2 at income mlql is cv(r) = (m/q2) - (m/ql) + cv*(r), so that f cv(r) dF < CV + f cv*(r) dF. If CV < f cv(r) dF, then f cv*(r) dF > O0, implying that households' aggregate willingness to pay for the change is nonpositive. A similar argument shows that if the representative agent prefers the first situation, then the aggregate willingness to pay is negative, so the condition CV

< f cv dF for all price changes implies Kaldor consistency, and our Corollary 1(a) now follows from Jerison's (1994)

26 That is, v(p2/q2, cv*(r) + m/q2, r) = v(pa/ql, mlql, r), so that cv*(r) is nonnegative if and only

if the household does not prefer the change.




Propositions 3 and 4(a). (A similar argument shows that our Corollary 1(b) follows from his Propositions 3 and 4(b).)

A.8. Proof of Proposition 6. Obviously (b) =i (a). The implication (d) =ý follows from Theorem 2 in Lewbel (2001, p. 614).

Consider the implication (a) =4 (d). By Lemma 1, we have

(A.22) cvante(P, P2, m, F) - = fv(, m, r)dF (P2 - Pl) + o(p2 - P I), f v2(p, m, r) dF(r)

and

(A.23) CV(pi, P2, m, F) = - I vl(pl,

m, r) dF(r)(p2 -

Pl) + o(0p2 - Pll).

VZ1pM, m, r)

Clearly, (a), (A.22), and (A.23) imply that

f (A.24) (pl, m, r) fvi(pl, m, r) dF(r)

Sv2(P, m, t) f 2(P1, m, r) dF(tr)

or, equivalently,

(A.25) vi(p, m, r)dF(r) v2Pl,m,

r)dF(r) (pl, m, )dF(r), v2(Pl, m, )

for any F e F. Now apply Lemma 2 with X= {r', r"} C T, f = vl/v2(= -d), and

g = v2 (viewed as functions of r for fixed (pa,

m)). Assumption 2 (and the continuity of demands in price) implies that v2(pl, m, .) is constant on X. Since (pi, m) and X are arbitrary, (a) implies (d). Hence we have that (a), (b), and (d) are equivalent.

By Lemma 1,

(A.26) -ACS(pl, p2, m, F) = - l(pl m, dF(r)(p2 - Pl) + o(1p2 - pAl). v2(pl, m, r)

The argument that (c) =, (d) parallels the argument that (a) = (d) and (d) =, (c) follows from Lemma A.1 (since additive separability implies that f(vI/v2) dF = f vi dF/f v2 dF). U

A.9. Proof of Proposition 7. From the proof of Proposition 6,

lim CV(pL, p2, m, F) - cvante(pl, p2, m, F)

P2•p• P2 - Pl

equals the left side of (A.24), which is the same as

IA27 ', v2(pli, m, r)dF )



898 SCHLEE

If the covariance between d and v2 is negative, then the left side of (A.27) is positive, so that CV(pl, P2, m, F) > Cv,,ante(p, 2, m, F) for all P2 > Pi in some

neighborhood of Pl. By (A.25) and (A.23),

lim CV(pl, P2, m, F) + ACS(pl, p2, m, F) = 0, P2 P- P2 - P1

and if D2(p1, m, F) > 0, then CV(pa,

P2, m, F) > -ACS(pl, P2, m, F) for all P2 >

pl in some neighborhood of pl. U

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