Charles Blackorby, David Donaldson, and John A. Weymark

Transcription

1 HICKSIAN SURPLUS MEASURES OF INDIVIDUAL WELFARE CHANGE WHEN THERE IS PRICE AND INCOME UNCERTAINTY by Charles Blackorby, David Donaldson, and John A. Weyark Working Paper No. 06-W18R August 2006 Revised March 2007 DEPARTMENT OF ECONOMICS VANDERBILT UNIVERSITY NASHVILLE, TN

2 Hicksian Surplus Measures of Individual Welfare Change When There is Price and Incoe Uncertainty Charles Blackorby 1, David Donaldson 2, and John A. Weyark 3 1 Departent of Econoics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UNITED KINGDOM and GREQAM, Centre de la Vieille Charité, 2 rue de la Charité, Marseille, FRANCE. c.blackorby@warwick.ac.uk 2 Departent of Econoics, University of British Colubia, East Mall, Vancouver, B.C, V6T 1Z1, CANADA. dvdd@telus.net 3 Departent of Econoics, Vanderbilt University, VU Station B #35189, 2301 Vanderbilt Place, Nashville, TN , U.S.A. john.weyark@vanderbilt.edu 1 Introduction When there is no uncertainty, it is well known that the Hicksian copensating and equivalent variations are exact easures of individual welfare change. That is, the sign of either of these easures of Hicksian consuer s surplus correctly identifies whether a change in prices and incoe akes an individual consuer better or worse off. 4 It is also well known that Marshallian consuer s surplus is not an exact easure of individual welfare change except under restrictive assuptions. 5 The use of the expected value of a Hicksian or Marshallian easure of consuer s surplus to evaluate the welfare consequences of price changes in uncertain environents can be traced back to the seinal analysis of Waugh (1944), who showed that under standard assuptions about individual deand, expected Marshallian consuer s surplus and expected copensating variation are both negative if a stochastic price is stabilized at its arithetic ean. For an individual whose preferences satisfy the expected utility hypothesis, the use of an expected surplus easure, whether it be Hicksian or Marshallian, is a valid easure of individual welfare change under uncer- 4 Although both the copensating and equivalent variations are exact easures of individual welfare change, they are defined using different reference prices and, hence, do not, in general, assign the sae nuerical value to a given change in prices and incoe. 5 See, for exaple, Boadway and Bruce (1984, Chapter 7) and Chipan and Moore (1976, 1980).

3 2 Charles Blackorby, David Donaldson, and John A. Weyark tainty if and only if its sign correctly deterines whether his expected utility increases or decreases as a result of a change in the distribution of prices and incoes across states. Anderson and Riley (1976) have argued that these expected surplus easures do not not correctly track individual preferences when a stochastic price is stabilized unless the arginal utility of incoe (in the utility representation of preferences used to copute expected utilities) is independent of both the level of incoe and the value of this price. Rogerson (1980) and Turnovsky, Shalit, and Schitz (1980) have identified restrictions on preferences for which expected Marshallian surplus is a valid indicator of individual welfare change when prices and, in the case of Rogerson, incoes are stochastic. For the case in which only one price is uncertain, Hels (1984, 1985) has characterized the restrictions on preferences for which expected copensating variation is a valid easure of individual welfare change both when the aount of price variability after the change in the distribution of this price is unrestricted and when the stochastic price is stabilized at its ean value. In each case, these restrictions are quite stringent. In the odels considered by Hels, the consuer allocates a certain incoe over one or ore coodities whose prices are certain and one coodity whose price is uncertain. However, whether uncertainty is generated by, for exaple, trade shocks (Anderson and Riley, 1976) or by factors that affect the volatility of coodity prices (Newbery and Stiglitz, 1981), it is often the case that the incoes of consuers are also uncertain and one or ore prices are state dependent. Furtherore, incoes ay be directly affected by rando events such as health and the tiing of a worker s entry into the labour arket, in which case the design of social insurance prograes needs to be evaluated. See, for exaple, Varian (1980). In this article, we extend Hels s analyses by identifying the circustances under which a consuer s surplus criterion based on a Hicksian surplus easure in each state is a valid easure of individual welfare change when incoe and soe or all of the prices vary across states. For concreteness, we use copensating variations in our analysis, but our theores are also valid for equivalent variations. Although the echanis that generates a change in the state distribution of prices and incoes can take any fors, for concreteness, we suppose that it is a governent project. In order to evaluate the welfare consequences of a project for an individual consuer whose preferences satisfy the expected utility hypothesis, we eploy a surplus evaluation function that aggregates the ex post copensating variations in each state into an overall surplus easure. Such a surplus evaluation function is a consistent easure of individual welfare change if it is positive valued whenever the project akes the consuer better off ex ante. For the kinds of state-dependent prices and incoes that we consider, we show that a consistent easure of individual welfare change based on the ex post copensating variations ust regard a project as being welfare iproving if and only if its expected copensating variation is positive. Furtherore, the indirect utility function that the consuer uses to evaluate prices and

4 Uncertainty and Hicksian Surplus Measures 3 incoe in each state and that is used to copute expected utilities ust be affine in incoe with the origin ter a constant and the weight on incoe independent of those prices that are uncertain. These restrictions iply that preferences are hoothetic. If all prices are uncertain, these conditions are inconsistent with the hoogeneity properties of an indirect utility function and, hence, we obtain an ipossibility result. In Section 2, we describe our state-contingent alternatives odel of uncertainty and forally define the copensating variations obtained in each state. We introduce our consistency criterion and the doains that we consider in Section 3. In Section 4, we adapt a result due to Blackorby and Donaldson (1985) in order to provide a partial characterization of the restrictions iplied by consistency. A coplete characterization of the restrictions iplied by consistency on our doains is established in Section 5. In Section 6, we discuss our theores and relate the to results on consistent easures of welfare change that have been obtained in a variety of different contexts. We provide soe concluding rearks in Section 7. 2 Copensating Variations for the State-Contingent Alternatives Model of Uncertainty We eploy the state-contingent alternatives odel of uncertainty with a finite nuber of states due to Arrow (1953, 1964). For discussions of the expected utility theore for this odel, see Arrow (1965), Blackorby, Davidson, and Donaldson (1977), and Diewert (1993). We assue that there are M states (M 2) and let M = {1,...,M} denote the set of states. In each state, there are N coodities (N 2). The set of coodities is N = {1,...,N}. In state, the prices of the coodities are p =(p 1,...,p N ) RN ++ and the consuer has incoe y R +. 6 Ex ante, the consuer faces the state-contingent price-incoe vector (p, y) R MN ++ R M +, where p =(p 1,...,p M ) and y =(y 1,...,y M ). Ex post consuption in state is c =(c 1,...,c N ) RN +. The consuer s ex ante state-contingent consuption vector is c =(c 1,...,c M ) R MN +. The probability that state occurs is π > 0, where π =1. These probabilities can be either subjective or objective, but are fixed throughout our analysis. We assue that the consuer s preferences over state-contingent coodity vectors in R MN + are continuous, strictly onotonic, convex, and satisfy the expected utility hypothesis. Hence, these preferences can be represented by a utility function U : R MN + R for which U(c) = π u(c ) (1) 6 R + and R ++ denote the set of nonnegative and positive nubers, respectively.

5 4 Charles Blackorby, David Donaldson, and John A. Weyark for all c R MN +, where the function u: R N + R is continuous, increasing in each of is arguents, concave, and state independent. Following Arrow (1965), u is called a Bernoulli utility function. 7 Note that u represents preferences over ex post consuption bundles. Any increasing transfor of u also represents these preferences. However, only increasing affine transfors of u are Bernoulli utility functions; i.e., only increasing affine transfors of u can be used to represent the ex ante preferences in the expected utility for given in (1). The Bernoulli indirect utility function v : R N ++ R + R is defined by setting v(p,y ) = ax {u(c ) p c y } (2) c R N + for all (p,y ) R N ++ R +. Hence, the consuer preferences for statecontingent price-incoe vectors can be represented by the indirect expected utility function V : R MN ++ R M + R defined by setting V (p, y) = π v(p,y ) (3) for all (p, y) R MN ++ R M +. It follows fro our assuptions on u that the function v is continuous, decreasing, and convex in prices, increasing in incoe, and hoogeneous of degree zero in prices and incoe. 8 When there is no price or incoe uncertainty, with p =(p 0,...,p 0 ) and y =(y 0,...,y 0 ) say, then V (p, y) =v(p 0,y 0 ). (4) Thus, v represents preferences over certain price-incoe vectors. As is the case with u, any increasing transfor of v represents these preferences over certain outcoes, but only increasing affine transfors of v can be used to copute expected utilities as in (3). Suppose that the price-incoe pair in state is initially ( p, ȳ ) and, therefore, the consuer has utility ū = v( p, ȳ ) in state. Now consider changing this price-incoe pair to (ˆp, ŷ ). The consuer then has utility û = v(ˆp, ŷ ) in this state. The copensating variation associated with this change, s = S ( p, ȳ, ˆp, ŷ ), (5) is the axiu aount the consuer would pay for the change. It is defined iplicitly by v(ˆp, ŷ s )=v( p, ȳ )=ū. (6) 7 A Bernoulli utility function in the state-contingent alternatives odel of uncertainty is the analogue of a von Neuann and Morgenstern (1944) utility function in the lottery odel of uncertainty. 8 The function v is decreasing in prices if the value of v decreases when the price of every good is increased.

6 Uncertainty and Hicksian Surplus Measures 5 Note that u = v(p,y ) y = e(p,u ), (7) where e is the expenditure function dual to u. Thus, the copensating variation in state can be written as s = e(ˆp, û ) e(ˆp, ū ) =ŷ e(ˆp, ū ) (8) =[ŷ ȳ ]+[e( p, ū ) e(ˆp, ū )]. Because the expenditure function is increasing in its last arguent, s 0 û ū for all M. (9) Therefore, this state-specific easure of willingness-to-pay is nonnegative if and only if the consuer is no worse off in state as a result of the change fro ( p, ȳ )to(ˆp, ŷ ). Hence, the copensating variation correctly identifies whether a change in prices and incoe in a given state akes the consuer better off or not. This observation is siply a reflection of the well-known fact that the copensating variation is a valid indicator of individual welfare change when there is no uncertainty. 3 Consistency A project affects the consuer by changing the vector of state-contingent prices and incoes. Let ( p, ȳ) (resp. (ˆp, ŷ)) denote the pre-project (resp. postproject) prices and incoes. This project changes the consuer s indirect expected utility fro V ( p, ȳ) tov (ˆp, ŷ). We assue that the sae set D of vectors of state-contingent prices and incoes are possible both before and after the ipleentation of a project. The question is whether the vector of state-contingent copensating variations s =(s 1,...,s M ) defined in (8) can be used to easure the change in the well-being of the consuer for a project that changes ( p, ȳ) to(ˆp, ŷ). More precisely, for the doain D, we ask if there exists soe real-valued function of the state-contingent copensating variations that is positive valued for projects that iprove the well-being of the consuer and that is nonpositive for those that do not. This surplus evaluation function is a function Γ : S(D) R, where S(D) R M is the set of vectors of state-contingent copensating variations that are achievable when the doain is D. We assue that Γ is continuous and increasing. Consistency of the surplus evaluation function with consuer well-being on the doain D is defined as follows. Consistency. (Γ, V ) is consistent on D R MN ++ R M + if and only if for all ( p, ȳ), (ˆp, ŷ) D. Γ (s 1,...,s M ) 0 V (ˆp, ŷ) V ( p, ȳ) (10)

7 6 Charles Blackorby, David Donaldson, and John A. Weyark Let and, for all K N, let D y = R M + (11) D K p = {p R MN ++ j K,, M,p j = p j } (12) and D K = D K p D y. (13) The sets D K, K N, are the doains that we consider for our project evaluations. For the doain D K, the pair (Γ, V ) has to be consistent for all nonnegative incoes and for all positive prices for which the prices of the goods with indices in the set K are the sae in each state. In this doain, the prices with indices in N\K are peritted to differ across states. Clearly, the ore prices that are peritted to differ across states, the ore restrictions that Γ and V ust satisfy. 4 A Useful Lea By interpreting M as a set of individuals, instead of a set of states, Blackorby and Donaldson (1985) have defined an indirect Bergson Sauelson social welfare function V BS : R MN ++ R M + R by setting V BS (p, y) =W (v 1 (p 1,y 1 ),...,v M (p M,y M )) (14) for all (p, y) R MN ++ R M +, where W : R M R is a continuous, increasing Bergson Sauelson social welfare function and, for all M, p are the prices person faces, y is his incoe, and v : R N ++ R + R is his indirect utility function. Note that individuals ay face different prices in (14). As above, we can copute a copensating variation for each individual M(using the function v instead of v) and define consistency (using V BS instead of V ) as in (10). Blackorby and Donaldson (1985) have shown that for the doains D (all prices can be person specific) and D N (no price can be person specific), consistency iplies that the vector of individual incoes ust be separable fro the prices in the indirect Bergson Sauelson social welfare function and that for every vector of copensating variations in the doain of Γ, the sign of the surplus evaluation function Γ ust be the sae as the sign of a linear function of the individual surplus easures. 9 Their proofs apply equally well to any doain D K with K N. 9 The separability result for the doain D N was first established by Roberts (1980, Proposition 1).

8 Uncertainty and Hicksian Surplus Measures 7 Because our indirect expected utility function is forally a special case of Blackorby and Donaldson s indirect Bergson Sauelson social welfare function (with v = π v and and V BS = V ), their results also hold for our odel. Hence, consistency iplies that the state-contingent incoes ust be separable fro the state-contingent prices in the indirect expected utility function of the consuer and that for every vector of state-contingent copensating variations in S(D), the surplus evaluation function Γ ust have the sae sign as a linear function of these copensating variations. Lea 1. For all K N,if(Γ, V ) is consistent on D K, then (i) for every (p, y) D K, V can be written as V (p, y) = V (p, φ(y)), (15) where V is continuous, increasing in φ(y), and hoogeneous of degree zero in (p, y) and (ii) there exist a > 0 for all Msuch that φ can be written as φ(y) = a y (16) for all y R N +. Furtherore, Γ (s 1,...,s M ) 0 a s 0 (17) for all (s 1,...,s M ) S(D K ). For a foral proof of Lea 1, see Blackorby and Donaldson (1985, Lea 1, Theore 1, and Corollary 1.2). The proof strategy is as follows. By considering a project in which only the state-contingent incoes change, (8) iplies that the copensating variation in any state is siply the difference between the new and the old incoe. Because the left side of (10) is independent of prices for such a project, so is the right side, fro which the separability result in (15) follows. The hoogeneity of degree zero of V iplies that φ can be chosen to be hoogeneous of degree one. Using this hoogeneity property, it can be shown that φ satisfies an additive Cauchy equation, whose solution is given by (16). 10 Because prices have not been changed, the sign of the change in indirect expected utility is the sae as the sign of the change in the value of φ, fro which (17) follows. As we have seen, when a project only changes incoes but not prices, the copensating variation in a state is equal to the difference between the preand post-project incoes in that state. Thus, the surplus evaluation function ust ignore inforation about incoe levels. What Lea 1 deonstrates is that in order for this function to be sensitive only to incoe differences and at the sae tie respect the hoogeneity properties of the indirect expected utility function, it ust assign each state a weight and then use these weights to copute a weighted su of copensating variations. 10 See Aczél (1969, Chapter 2) or Eichhorn (1978, Chapter 1) for an introduction to Cauchy equations.

9 8 Charles Blackorby, David Donaldson, and John A. Weyark 5 The Theores Lea 1 has identified soe restrictions that ust be satisfied by the indirect expected utility function V and by the surplus evaluation function Γ if they are to be consistent with each other. However, we have yet to identify the restrictions iplied by consistency on the Bernoulli indirect utility function v or on the choice of the weights that are used to aggregate the state-contingent copensating variations other than that these weights are positive. In this section, we characterize these restrictions. The proof of Lea 1 does not exploit the assuption that V (p, y) isthe expected value of the Bernoulli utilities v(p,y ) obtained in each state. The conclusions in this lea are also valid if the ex ante utility V (p, y) isany continuous, increasing function of the ex post utilities v(p,y ). We now show that Lea 1 and the assuption that the consuer s preferences satisfy the expected utility hypothesis iply (i) that the Bernoulli utility function v ust be affine in incoe with the origin ter a constant and the weight on incoe possibly price dependent and (ii) for all vectors of state-contingent copensating variatons in S(D), the sign of the surplus evaluation function ust be the sae as the sign of the expected value of these copensating variations. The first of these conditions iplies that the Bernoulli direct utility function u is hoothetic. Theore 1. For all K N,if(Γ, V ) is consistent on D K, then there exists a function α: R N ++ R ++ and a scalar β for which v(p 0,y 0 )=α(p 0 )y 0 + β (18) for all (p 0,y 0 ) R N ++ R +, where α is continuous, decreasing, convex, and hoogeneous of degree inus one. Furtherore, Γ (s 1,...,s M ) 0 π s 0 (19) for all (s 1,...,s M ) S(D K ). Proof. Fro (3), (15), and (16), we obtain ( V p, ) a y = π v(p,y ). (20) Consider any p 0 R N ++ and let p =(p 0,...,p 0 ). That is, there is no price uncertainty. Define z := a y for all M, (21) ˆv (z ):=π v(p 0,y ), (22)

10 Uncertainty and Hicksian Surplus Measures 9 and ( ) ( ˆV z := V p, ) z. (23) Substituting (21), (22), and (23) into (20) yields ( ) ˆV z = ˆv (z ). (24) Equation (24) is a Pexider equation whose solution is ˆv (z )=ᾱ(p 0 )z + β (p 0 ) for all M, (25) where ᾱ(p 0 ) > 0 because ˆv is increasing in z. 11 Note that β (p 0 )=ˆv (0) = π v(p 0, 0) for all M. (26) Define β(p 0 ):=v(p 0, 0). (27) Fro (26) and (27), we obtain β (p 0 )=π β(p 0 ). (28) Substituting (25) and (28) into (22) and using (21) yields π v(p 0,y )=ᾱ(p 0 )a y + π β(p 0 ) (29) or, equivalently, v(p 0,y )=ᾱ(p 0 ) [ a π ] y + β(p 0 ). (30) Because v is state independent and D y = R M +, (30) iplies that a = κπ for all M, (31) where κ>0 because v is increasing in y. Defining α(p 0 ):=κᾱ(p 0 ) (32) yields v(p 0,y 0 )=α(p 0 )y + β(p 0 ). (33) In order for v to be hoogenous of degree zero in prices and incoe, α ust be hoogenous of degree inus one and β ust be hoogenous of degree zero. 11 See Aczél (1969, Chapter 3) or Eichhorn (1978, Section 3.1).

11 10 Charles Blackorby, David Donaldson, and John A. Weyark An indirect utility function with the functional for given in (33) is called quasi-hoothetic. That is, it exhibits the Goran (1961) polar for. In order for the deands generated by these preferences to be nonnegative for all prices and incoes, β ust be independent of prices. See Blackorby, Boyce, and Russell (1978). 12 Thus, v ust satisfy (18). The other properties of α follow straightforwardly fro the corresponding properties of v. Using (31), (17) yields (19). As indicated in the proof of Theore 1, if v satisfies (30), then it cannot be an indirect utility function unless the function α satisfies the restrictions stated in Theore 1 and β is a constant. However, if we choose a utility level ū for ex post utility such that the consuption of any good is positive if this utility level is achieved, then it is not necessary for β to be a constant if (Γ, V ) is only required to be consistent on the subset of D K for which ex post utilities are at least ū. Instead, β only needs to be continuous, nonincreasing, convex, and hoogeneous of degree zero in prices. Soe intuition for Theore 1 can be obtained by considering the special case in which the Bernoulli indirect utility function v is differentiable. Suppose that there is no price uncertainty and that a project only changes the statecontingent incoes arginally. Let p 0 be the price vector in each state both before and after the project is ipleented, y be the intial state-contingent incoe vector, and dy = (dy 1,...,dy M ) be the vector of incoe changes that result fro this project. Note that dy is also the vector of copensating variations associated with this project. By Lea 1, consistency requires that a dy 0 π v y (p 0,y )dy 0. (34) Because the left side of (34) does not depend on the level of y, in order for this equivalence to hold for all y R M + and all dy in a neighbourhood of the origin for which y + dy R M +, the arginal utility of incoe function v y ust be positive and cannot depend on incoe. Hence, v is an increasing affine function of incoe. That is, v satisfies (33). Using (33), (34) siplifies to a dy 0 α(p 0 ) π dy 0, (35) which can only hold for all dy in a neighbourhood of the origin if the weights a =(a 1,...,a M ) are proportional to the probabilities π =(π 1,...,π M ). A striking feature of Theore 1 is that consistency requires that projects be evaluated in ters of expected copensating variation. That is, a project 12 This result can be shown quite easily for the case in which v is differentiable. By Euler s Theore, n i=1 p0 i β(p 0 )/ p 0 i = 0. Hence, if β is not independent of prices, there ust exist soe price vector p 0 and good j for which β( p 0 )/ p 0 j > 0. Using Roy s Identity, the deand for good j at ( p 0,y ) is c j( p 0,y )= [y α( p 0 )/ p 0 j + β( p 0 )/ p 0 j]/α( p 0 ), which is negative when y is sufficiently close to 0.

12 Uncertainty and Hicksian Surplus Measures 11 is welfare iproving for an individual if and only if the expected copensating variation of the project is positive. As we have noted, previous studies of cost-benefit analysis under uncertainty for a single individual siply assue that the surplus evaluation function is the expected value of soe easure of consuer s surplus. For the doains we are considering, we have shown that this ust be the case, at least when surplus is easured using the copensating variation. In particular, it is not possible to require the surplus evaluation function to exhibit inequality aversion in the distribution of copensating variations across states. In Theore 1, K is the set of goods for which prices are certain across states. If K = N, then there is no price uncertainty, whereas if K =, then all prices can vary across states. Note that any p Dp K can be written as p = ( p 0 K,p 1 K,...,p 0 K,p M K), (36) where p 0 K are the prices of the goods that are certain across states. Theore 2 shows that the weight on incoe in the Bernoulli indirect utility function (18) can only depend on the prices of the goods that are certain. Furtherore, requiring the Bernoulli indirect utility function v to satisfy this restriction and requiring the surplus evaluation function to identify a project as being welfare iproving if and only if the expected copensating variation is positive are jointly necessary and sufficient for (Γ, V ) to be consistent on any of the doains we are considering except for the doain in which all prices are uncertain. Theore 2. For all K N\, (Γ, V ) is consistent on D K if and only if (18) and (19) hold and there exists a function α K : R K ++ R ++ for which α(p 0 K,p 0 K) =α K (p 0 K) (37) for all (p 0 K,p0 K ) RN ++, where α K hoogeneous of degree inus one. is continuous, decreasing, convex, and Proof. Suppose that (Γ, V ) is consistent on D K. Fro (20) and Theore 1, we have ( V p, ) π y = π [α(p )y + β] (38) for all (p, y) D K. Consider any j K and, contrary to the theore, suppose that there exist distinct p,p R N ++ for which p i = p i for all i j and α(p ) α(p ). Consider any p Dp K for which p = p and p = p. Next, consider any distinct ȳ, ŷ D y for which ȳ =ŷ for all, and π ȳ + π ȳ = π ŷ + π ŷ. (39)

13 12 Charles Blackorby, David Donaldson, and John A. Weyark By construction, the value of the left side of (38) is the sae when evaluated at ( p, ȳ) and ( p, ŷ). Thus, (38) iplies that π α(p )ȳ + π α(p )ȳ = π α(p )ŷ + π α(p )ŷ. (40) Because (40) ust hold for any nonnegative ȳ, ȳ, ŷ, and ŷ that satisfy (39), it follows that α(p )=α(p ), a contradiction. Thus, (37) is satisfied. The necessity part of the arguent is copleted by noting that the properties of α K in the theore stateent follow iediately fro the properties of α in Theore 1. Now, suppose that (18), (19), and (37) are satisfied. Consider any ( p, ȳ), (ˆp, ŷ) D K, where ( p, ȳ) = ( p 0 K, p1 K,..., p0 K, pm K, ȳ1,...,ȳ ) M and (ˆp, ŷ) = 0 K, ˆp 1 K,...,ˆp0 K, ˆpM K, ŷ1,...,ŷ ) M. Then, V (ˆp, ŷ) V ( p, ȳ) = = π v(ˆp 0 K, ˆp K, ŷ ) π [ αk (ˆp 0 K)ŷ α K ( p 0 K)ȳ ]. π v( p 0 K, p K, ȳ ) (41) Hence, V (ˆp, ŷ) V ( p, ȳ) 0 π [ αk (ˆp 0 K)ŷ α K ( p 0 K)ȳ ] 0. (42) Fro (6), the copensating variation s in state is defined iplicitly by α K (ˆp 0 K)[ŷ s ]+β = α K ( p 0 K)ȳ + β. (43) Thus, s 1 [ = α K (ˆp 0 K ) αk (ˆp 0 K)ŷ α K ( p 0 K)ȳ ]. (44) It follows fro (42) and (44) that V (ˆp, ŷ) V ( p, ȳ) 0 π s 0, (45) which copletes the sufficiency arguent. In Theore 2, we have assued that there is at least one price that is certain. If all prices and incoes can be stochastic, then there is no surplus evaluation function Γ that can provide a consistent cost-benefit test on D for an individual whose preferences satisfy the expected utility hypothesis. Theore 3. There is no function Γ : S(D ) R such that (Γ, V ) is consistent on D.

14 Uncertainty and Hicksian Surplus Measures 13 Proof. The necessity part of the proof of Theore 2 applies equally well when K =. As a consequence, α ust be independent of all prices. That is, for all (p 0,y 0 ) R N ++ R +, v(p 0,y 0 )=ξy 0 + β for soe ξ>0. (46) However, if v has this functional for, then it cannot be hoogeneous of degree zero and, hence, consistency is ipossible. It is also possible to use Theore 1 to prove Theore 3 without relying on Theore 2. Suppose that (Γ, V ) is consistent on D. Project 1 changes the prices and incoes fro ( p, ȳ) to(ˆp, ŷ). These prices and incoes can be chosen so that the expected copensating variation π s of this project is negative and the copensating variation s M in state M is positive for the individual under consideration. Because (Γ, V ) is consistent, Theore 1 iplies that the change in expected utility is negative and, hence, this individual is worse off as a result of Project 1. Now let ( p, ỹ) be the vector of prices and incoes for which ( p, ỹ )=(ˆp, ŷ ) for all M and ( p M, ỹ M )=λ(ˆp M, ŷ M ), where 0 <λ 1. In Project 2, prices and incoes are changed fro ( p, ȳ) to( p, ỹ). Because the Bernoulli indirect utility function v is hoogenous of degree zero in prices and incoe, the indirect expected utility is the sae with ( p, ỹ) as it is with (ˆp, ŷ). Therefore, Project 2 akes the individual worse off. Let s M be the copensating variation in state M for this project. Because the expenditure function is hoogeneous of degree one in prices, it follows fro (8) that s M = λs M. The copensating variations for the other states are the sae with both projects. Because π M > 0, by choosing λ to be sufficiently large, the expected copensating variation for Project 2 is positive, violating consistency. Because the Bernoulli indirect utility function v is hoogeneous of degree zero in prices and incoe, expected utility is unaffected if the prices and incoe in each state are divided by the price of good one. With this price noralization, the price of good one is certain and always equal to one. It ight see then that Theore 3 contradicts the special case of Theore 2 in which K = {1}. However, this is not the case. While such a price noralization does not change the expected utility either before or after a project is ipleented, it does change the value of the copensating variation in any state for which the post-project price of good one is not initially equal to one. In other words, noralizing by setting the price of good one so that it is always equal to one is innocuous fro the perspective of calculating expected utility, but it is not innocuous fro the perspective of calculating expected copensated variation. As we have seen, it is for precisely this reason that an ipossibility result is obtained when all prices and incoe can vary across states because we can scale the prices and incoe in any state without changing the prices

15 14 Charles Blackorby, David Donaldson, and John A. Weyark and incoe in any other state. This independent scaling is not possible if any price ust have the sae value in every state Discussion When the Bernoulli indirect utility function v is differentiable, we can easure the consuer s risk aversion with respect to incoe with the Arrow (1965) Pratt (1964) coefficient of relative risk aversion: ρ y (p 0,y 0 )= v yy(p 0,y 0 ) v y (p 0,y 0 ) y0 (47) for all (p 0,y 0 ) R N ++ R +. By Theore 1, this coefficient ust be identically zero. In other words, the consuer ust be risk neutral towards incoe uncertainty. Analogous to the Arrow Pratt coefficient of relative risk aversion ρ y for incoe, Turnovsky, Shalit, and Schitz (1980) have defined a coefficient of relative risk aversion ρ pi for the price of good i by taking derivatives with respect to p i instead of with respect to y in (47) and then ultiplying the resulting fraction by p 0 i instead of by y0. If it is assued that the consuer s indirect utility function has the expected utility for given in (3), then the consuer s attitudes towards incoe and price uncertainty can be easured using the coefficients ρ y and ρ pi, i N. These easures are only invariant to increasing affine transfors of the Bernoulli indirect utility function v, which are also the class of transfors that do not affect the consuer s ex ante preferences over state-contingent prices and incoes. However, when coputing the copensating variation in each state, only the ordinal properties of v are used. As a consequence, if v is any increasing transfor of v, then the copensating variation associated with a project in any state is the sae with v as it is with v, even if the risk attitudes associated with v differ fro those associated with v. Therefore, when the surplus evaluation function in (19) is used to deterine whether a project is worthwhile or not, it akes the sae recoendations for a consuer whose preferences are characterized by the function v as it does for a consuer whose preferences are characterized by v. For this reason, as we have seen, restrictions ust be placed on v in order for this cost-benefit test to be consistent Also note that the doain obtained fro D by noralizing the price of good one is not the sae as D {1}. For the doain D {1}, the price of good one is certain in any price vector p D p {1}, but it need not be the sae as the price of good one in soe other price vector q D p {1}. However, with the price noralization, not only is the price of good one constant across states in a given state-contingent price vector, it has the sae value in every state-contingent price vector. 14 See Hels (1985, p. 609) for siilar observations about the use of expected copensating variation as a test for whether stablizing a single stochastic price is beneficial for an individual.

16 Uncertainty and Hicksian Surplus Measures 15 The restrictions on the Bernoulli indirect utility function v that we have identified for consistency iply that the consuer is risk neutral towards incoe and that the arginal utility of incoe does not depend on any price that can vary across states. If there is no price uncertainty and a project only changes incoes, then our cost-benefit test declares a project to be worthwhile if it increases the expected value of incoe. For a consuer who is risk neutral towards incoe, this is all that he cares about. However, if the consuer is not risk neutral, then he cares about the distribution of incoes, not just its expected value, and consistency would be lost. If a project also changes prices, by using the copensating variation to easure the surplus in each state, price changes are converted into an equivalent incoe change using the post-project prices. In order for expected copensating variation to provide a consistent cost-benefit test when soe of the prices are stochastic, the arginal utility of incoe ust be constant across states. 15 Because any distribution of incoes across states is possible, this requires that the arginal utility of incoe be independent of any price that can be state dependent. In a odel with a continuu of states, Hels (1984) has investigated when expected copensating variation is a consistent easure of individual welfare change when the only source of uncertainty is in the price of one good and there are no restrictions on the stochastic variability that this price ight exhibit. Hels has shown that risk neutrality towards incoe and independence of the arginal utility of incoe with respect to this price are necessary and sufficient for consistency provided that the deand for this good is positive. That is, the Bernoulli indirect utility function ust satisfy (33) with α independent of the price that is stochastic. 16 Our theores are closely related to results about the consistency of costbenefit tests based on copensating or equivalent variations established by Blackorby, Donaldson, and Moloney (1984) and Blackorby and Donaldson (1985, 1986) in a variety of contexts. Blackorby and Donaldson (1985) have shown (i) that no continuous, increasing surplus evaluation function defined on individual copensating or equivalent variations can be consistent with an indirect Bergson Sauelson social welfare function when all prices and incoes can be person specific and (ii) that when everyone faces the sae prices, consistency requires individual preferences to be quasi-hoothetic with everyone having the sae price-dependent weight on incoe. 17 The latter condition is the necessary and 15 In (44), α K(ˆp 0 K) is the arginal utility of incoe at the post-project prices. If this value depends on any price that is not certain, then it could not be factored out in going fro (44) to (45). 16 Neither Hels (1984) nor the other articles considered in the rest of this section take account of the restrictions required to ensure that deands are nonnegative for all adissible prices and incoes. For this reason, they only show that preferences ust be quasi-hoothetic, rather than being fully hoothetic. 17 Related results ay be found in Haond (1977, 1980) and Roberts (1980).

17 16 Charles Blackorby, David Donaldson, and John A. Weyark sufficient condition identified by Goran (1953) for the existence of counity indifference curves. 18 In Blackorby, Donaldson, and Moloney (1984), a single consuer, whose utility is a continuous, increasing function of the instantaneous utilities obtained fro his consuption in each of a finite nuber of periods, chooses these consuptions to axiize lifetie utility given the prices of the goods and his wealth in a perfect capital arket. They have shown (i) that no discounted su of the copensating or equivalent variations in each period can serve as a consistent easure of welfare change for such a consuer if prices are free to vary across periods and (ii) that the instantaneous preferences ust be quasi-hoothetic if all prices are constant across periods. 19 In Blackorby and Donaldson (1986), there is a single period and each individual consues an aount of a single coodity should he live, which occurs with positive probability. For the case in which each person s preferences satisfy the expected utility hypothesis and everyone has soe level of consuption that akes life just worth living, they have shown (i) that the su of the individual copensating or equivalent variations is not consistent with the ranking of alternative distributions of survival probabilities and of consuptions obtained with any continuous, increasing Bergson Sauelson social welfare function if both the probabilities of survival and the consuptions can be person specific and (ii) that when everyone has the sae survival probability, then consistency requires that the individual preferences to be quasi-hoothetic with utility functions that are affine in consuption with an incoe weight that can only depend on the coon survival probability. As Blackorby and Donaldson (1986, Section III) have noted, the probabilities in this odel correspond to prices in the riskless ulti-good odel. There is clearly a close faily reseblance between these results and those obtained here. This is not surprising. Although our odel and those described above differ in soe iportant respects, the overall easure of individual or social welfare in each case is a continuous, increasing function of the utility functions that are used to copute the individual or state-contingent or period-contingent copensating or equivalent varitions. Furtherore, the surplus evaluation function is, in each case, a continuous, increasing function of these surpluses. It is these coon structural features of these odels that accounts for the siilarity of the results about the consistency of welfare evaluations based on Hicksian easures of consuer s surplus that are obtained with the. 18 Blackorby and Donaldson (1999) have established siilar results about the consistency of the su of individual Marshallian consuers surpluses with an indirect Bergson Sauelson indirect social welfare function. For discussions of the use of expected Marshallian consuer s surplus as a easure of individual welfare change, see Rogerson (1980), Turnovsky, Shalit, and Schitz (1980), and Stennek (1999). 19 The analogue of a perfect capital arket in our odel is a perfect insurance arket that perits an individual to transfer wealth across states.

18 7 Concluding Rearks Uncertainty and Hicksian Surplus Measures 17 The restrictions on preferences that Hels (1984, 1985) has shown are required for expected copensated variation to be a consistent easure of individual welfare change are uch less restrictive when a single stochastic price is stabilized at its ean value copared with the case in which all distributions can be stochastic. However, they are still quite stringent and are unlikely to be satisfied in practice. For the prices that are allowed to vary across states and for incoe, we have placed no restrictions on the pre- and post-project distributions. We could instead restrict our doains by, for exaple, considering projects that stabilize incoe or soe of the prices. On such doains, the conditions required for consistency would be weaker than those obtained here. However, Hels s theores suggest that they will nevertheless be quite restrictive, so considering ore specialized doains does not appear to be a proising direction in which to seek ore positive results. In view of the rather stringent conditions required for a surplus evaluation function based on the ex post copensating variations to be a consistent easure of individual welfare change, it is natural to ask if there is any easure of consuer s surplus that applies ore generally when prices or incoes are uncertain. An affirative answer is provided by the ex ante copensating and equivalent variations introduced by Schalensee (1972). 20 The ex ante copensating variation s c for a project that changes the statecontingent prices and incoe fro ( p, ȳ) to(ˆp, ŷ) is defined iplicitly by π v(ˆp, ŷ s c )=V( p, ȳ). (48) That is, s c is the aount by which an individual s incoe can be reduced in each state in order for the post-project situation to give hi the sae ex ante expected utility as is achieved before the project is ipleented. Because v is increasing in incoe, s c is positive if and only if the project akes the consuer better off ex ante. Thus, s c can serve as an exact easure of welfare change for any individual whose preferences satisfy the expected utility hypothesis. Siilarly, the ex ante equivalent variation s e is the aount of incoe that needs to be provided to an individual in each state in the pre-project situation in order to give hi the sae ex ante expected utility as is achieved after the project is ipleented. It too is an exact easure of individual welfare change. Schalensee (1972) did not advocate the use of these ex ante easures because he thought that they are non-operational. However, assuing that the appropriate coefficients of risk aversion can be deterined fro analyzing 20 Schalensee refers to these easures as copensating and equivalent option prices. An alternative proposal for easuring individual welfare change under uncertainty ay be found in Boadway and Bruce (1984, Chapter 7).

19 18 Charles Blackorby, David Donaldson, and John A. Weyark behaviour under uncertainty, Anderson (1979) has argued that these easures are operational, and so has endorsed their use, as has Hels (1985). 21 Given that inforation about risk attitudes is needed in order to deterine if expected copensating (or equivalent) variation is a consistent easure of individual welfare change, it therefore sees that there is little reason to use the expected value of soe consuer s surplus easure to evaluate projects that involve price and incoe uncertainty instead of the ex ante copensating or equivalent variation. Acknowlegeents This article was presented at the International Conference on Rational Choice, Individual Rights and Non-Welfaristic Norative Econoics in Honour of Kotaro Suzuura at Hitotsubashi University, the Workshop on Advances in Collective Choice at the University of Maastricht, the Workshop on Econoic Decisions at the Public University of the Navarre, the Workshop on Advances in Microeconoics and Social Choice Theory at the University of Verona, the Eighth International Meeting of the Society for Social Choice and Welfare in Istanbul, and at a seinar at GREQAM. We are grateful for the coents that we have received on these occasions. We are particularly grateful to Peter Haond for his coents at the Hitotsubashi conference. References Aczél, J. (1969). Lectures on Functional Equations and Their Applications. Acadeic Press, New York. Anderson, J. E. (1979). On the easureent of welfare cost under uncertainty. Southern Econoic Journal, 45, Anderson, J. E. and Riley, J. G. (1976). International trade with fluctuating prices. International Econoic Review, 17, Arrow, K. J. (1953). Le rôle des valeurs boursières pour la répartition la eillure des risques. In Econoétrie, pages Centre National de la Recherche Scientifique, Paris. Translated as Arrow (1964). Arrow, K. J. (1964). The role of securities in the optial allocation of risk-bearing. Review of Econoic Studies, 31, Arrow, K. J. (1965). Aspects of the Theory of Risk-Bearing. Yrjö Jahnssonin säätiö, Helsinki. Blackorby, C. and Donaldson, D. (1985). Consuers surpluses and consistent costbenefit tests. Social Choice and Welfare, 1, The relationship between these ex ante easures and the expected value of a Hicksian or Marshallian easure of ex post consuer s surplus has been explored in Choi and Johnson (1987) and Stennek (1999).

20 Uncertainty and Hicksian Surplus Measures 19 Blackorby, C. and Donaldson, D. (1986). Can risk-benefit analysis provide consistent policy evaluations of projects involving loss of life? Econoic Journal, 96, Blackorby, C. and Donaldson, D. (1999). Market deand curves and Dupuit Marshall consuers surpluses: A general equilibriu analysis. Matheatical Social Sciences, 37, Blackorby, C., Davidson, R., and Donaldson, D. (1977). A hoiletic exposition of the expected utility hypothesis. Econoica, 44, Blackorby, C., Boyce, R., and Russell, R. R. (1978). Estiation of deand systes generated by the Goran polar for: A generalization of the S-branch utility tree. Econoetrica, 46, Blackorby, C., Donaldson, D., and Moloney, D. (1984). Consuer s surplus and welfare change in a siple dynaic odel. Review of Econoic Studies, 51, Boadway, R. W. and Bruce, N. (1984). Welfare Econoics. Basil Blackwell, Oxford. Chipan, J. S. and Moore, J. C. (1976). The scope of consuer s surplus arguents. In A. M. Tang, F. M. Westfield, and J. S. Worley, editors, Evolution, Welfare, and Tie in Econoics: Essays in Honor of Nicholas Georgescu-Roegen, pages Lexington Books, Lexington, MA. Chipan, J. S. and Moore, J. C. (1980). Copensating variation, consuer s surplus, and welfare. Aerican Econoic Review, 70, Choi, E. K. and Johnson, S. R. (1987). Consuer s surplus and price uncertainty. International Econoic Review, 28, Diewert, W. E. (1993). Syetric eans and choice under uncertainty. In W. E. Diewert and A. O. Nakaura, editors, Essays in Index Nuber Theory, volue 1, pages North-Holland, Asterda. Eichhorn, W. (1978). Functional Equations in Econoics. Addison-Wesley, Reading, MA. Goran, W. M. (1953). Counity preference fields. Econoetrica, 21, Goran, W. M. (1961). On a class of preference fields. Metroeconoica, 13, Haond, P. J. (1977). Dual interpersonal coparisons of utility and the welfare econoics of incoe distribution. Journal of Public Econoics, 7, Haond, P. J. (1980). Dual interpersonal coparisons of utility and the welfare econoics of incoe distribution: A corrigendu. Journal of Public Econoics, 14, Hels, L. J. (1984). Coparing stochastic price regies: The liitations of expected surplus easures. Econoics Letters, 14, Hels, L. J. (1985). Expected consuer s surplus and the welfare effects of price stabilization. International Econoic Review, 26, Newbery, D. M. and Stiglitz, J. E. (1981). The Theory of Coodity Price Stabilization: A Study in the Econoics of Risk. Clarendon Press, Oxford. Pratt, J. W. (1964). Risk aversion in the sall and in the large. Econoetrica, 32, Roberts, K. (1980). Price-independent welfare prescriptions. Journal of Public Econoics, 13, Rogerson, W. P. (1980). Aggregate expected consuer surplus as a welfare index with an application to price stabilization. Econoetrica, 48, Schalensee, R. (1972). Option deand and consuer s surplus: Valuing price changes under uncertainty. Aerican Econoic Review, 62,

21 20 Charles Blackorby, David Donaldson, and John A. Weyark Stennek, J. (1999). The expected consuer s surplus as a welfare easure. Journal of Public Econoics, 73, Turnovsky, S. J., Shalit, H., and Schitz, A. (1980). Consuer s surplus, price instability, and consuer welfare. Econoetrica, 48, Varian, H. R. (1980). Redistributive taxation as social insurance. Journal of Public Econoics, 14, von Neuann, J. and Morgenstern, O. (1944). Theory of Gaes and Econoic Behavior. Princeton University Press, Princeton. Waugh, F. V. (1944). Does the consuer benefit fro price instability? Quarterly Journal of Econoics, 58,