Radner’s cost–benefit analysis in the small: An equivalence result

Economics Letters 120 (2013) 570–572

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Radner’s cost–benefit analysis in the small: An equivalence result Edward E. Schlee ∗,1 Department of Economics, Arizona State University, United States

highlights • • • •

I show that local changes in five welfare measures are equal. Two are a measure proposed by Radner and the coefficient of resource utilization. Two others are consumers’ surplus and the Slutsky change in real wealth. The last is the Divisia quantity index.

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Article history: Received 1 April 2013 Accepted 7 June 2013 Available online 21 June 2013

abstract I show that local changes in five welfare measures are equal: a measure proposed by Radner (1993); consumers’ surplus; the Slutsky change in real wealth; the Divisia quantity index, and Debreu’s (1951) coefficient of resource utilization (the last two rescaled in units of a numeraire good). © 2013 Elsevier B.V. All rights reserved.

JEL classification: D61 D8 Keywords: Local cost–benefit analysis Consumers’ surplus Divisia index Slutsky compensation Coefficient of resource utilization

How does a ‘‘small’’ change in an economy’s aggregate production plan affect welfare? I show that five answers to this question are the same. Scholars have from time to time pointed out the equivalence between pairs of answers, as I indicate briefly at the end of this note. My contribution is to point out the equivalence of them all. The least obvious is between a measure proposed by Radner (1993) and the coefficient of resource utilization, Debreu (1951), a measure of the welfare change expressed as a percentage of the economy’s endowment. I use a variant of Radner’s (1993) model for local cost–benefit analysis (a model descended from Debreu (1954) and Hotelling

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Tel.: +1 4809655745. E-mail address: [email protected].

1 The paper originated from a question posed by M. Ali Khan about Divisia Indexes and Consumers’ Surplus during a presentation of my work ‘‘Surplus Maximization and Optimality’’ (Schlee, forthcoming) at Johns Hopkins University, May 2012. Afterward he suggested that I write a short paper on the local equivalence of welfare measures. The list quickly grew to five. The paper would not have been written without Ali’s encouragement. I also thank the Johns Hopkins University Economics Department for welcoming me into its lively intellectual environment during my Sabbatical visit there, and V. Kerry Smith for useful discussions. 0165-1765/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econlet.2013.06.012

(1938)).2 The economy has I consumers and L goods (L ≥ 2). Good L is numeraire. The consumption set of every consumer is X = RL+ , and the aggregate endowment is a nonzero point ω ∈ X . Consumer i’s preferences on X are represented by a utility function ui which is smooth: it is C 2 with positive first derivatives; it is differentiably strictly quasiconcave3 ; and it satisfies the boundary condition that ui (0) = ui (x) whenever xℓ = 0 for some 1 ≤ ℓ ≤ L. Consumer i’s demand is Di (p, wi ), where p ∈ RL+ is a price vector and wi > 0 is consumer i’s wealth. The smooth utility assumption implies that each consumer’s demand is single-valued, satisfies budget balance, and is continuously differentiable on the set of positive price–wealth pairs.4 Consumer i’s preferences over price–wealth pairs are represented by the indirect utility function

2 Radner (1993) reports that the paper grew from his lecture notes in the 1960s and 1970s. Varian (1978) contains the substance of the published version (his Section 7.5) and cites a 1974 working paper by the same title. 3 That is, at each x ∈ RL , ∇ 2 ui (x) is negative definite on {v ∈ RL | ∇ ui (x) · v ++

= 0}. 4 On differentiability of demand, see Katzner (1968) and Debreu (1972).

E.E. Schlee / Economics Letters 120 (2013) 570–572

V i (p, wi ) = ui (Di (p, wi )). Since ui is smooth, V i is C 1 with positive wealth derivative. Subscripts on V i denote derivatives: Vθi is the derivative of V i with respect to θ ∈ {p, w}. I index aggregate production plans by a real number α in an open interval A containing 0. I refer to elements of A as policies (Radner (1993) calls them projects). I do not model producers but simply assume that the production plan at α is a point y(α) ∈ RL with y(α)+ω ≫ 0 for every α ∈ A. Regarding consumers, I follow Radner (1993) in taking the equilibrium concept to be a valuation equilibrium (Debreu, 1959, Ch. 6): for every α ∈ A there is a price  p(α) and for every consumer i = 1, . . . , I a point xi (α) ∈ RL++ such  i i i i that (i) x (α) = D ( p(α),  p(α)· x (α)); and (ii) i  x (α) = ω+ y(α). Consumer i’s equilibrium wealth is w i (α) =  p(α) ·  xi (α).5 It foli lows from strict monotonicity of each u that  p(α) ≫ 0; and since  xi (α) ≫0 for every α ∈ A, w i (α) > 0 for every α  ∈ A. Let i   x(α) = x (α) be aggregate consumption and w  (α) = i (α) i iw aggregate wealth at α . I assume that y(·) and each  xi (·) are differentiable.6 Since the equilibrium price is pinned down by each consumer’s marginal rate of substitution between goods i and L, both  p(·) and each w i (·) are C 1 . Consumer i’s preferences over policies in A are represented by α → V i ( p(α), w i (α)), and I denote this function by  Vi (α). I now define five measures of the welfare change from policy 0 to α ∈ A. Radner’s measure is R(α) =  p(0) · (y(α) − y(0)), the value of the change in production at the original prices.7 The Slutsky change in real wealth for consumer i is SCi (α) = −Di ( p(0), w i (0)) p(α) + w i (α) − w i (0); it equals minus the change in wealth which keeps the choice at policy 0 just affordable  i at α . The aggregate Slutsky change in real wealth is SC (α) = i SC (α). I define the change in consumer i’s surplus to be the line integral Si (α) = −

 [0,α]

Di ( p(v), w i (v)) · d p(v) + w i (α) − w i (0),

(1)

and the change in aggregate consumers’ surplus to be S (α) = i Si (α) (of definition (1) more later). The Divisia quantity index is   p(v) ·  x(v)] · d x(v). The first three measures are ex[0,α] p(v)/[

The coefficient of resource utilization (Debreu, 1951) is the smallest number ρ(α) ∈ [0, 1] such that ( V 1 (α), . . . ,  V I (α)) is maximal on U(ρ(α)). To convert this measure into numeraire units, I scale it by aggregate wealth at α = 0: the nominal measure of resource utilization is RU (α) = ρ(α)×w(0). The main result is that the first-order welfare effect of a policy change is the same for all five measures. Proposition. The derivatives of all five measures at α = 0 are equal: S ′ (0) = SC ′ (0) = DI ′ (0) = R′ (0) = RU ′ (0). Proof. Differentiate (1) and sum over all consumers to find S ′ (0) = −



Di ( p(0), w i (0)) · p′ (0) + w ′ (0) = SC ′ (0).

 U(ρ) =

(s1 , . . . , sI ) ∈ RI | ui (xi ) = si ,  x ∈ X, i



x = ρω + y(α) . i

5 I index scalar functions of α for consumer i with a subscript, so I can use primes to denote derivatives; I index functions of vectors with a superscript i, so I can use subscripts to denote partial derivatives. 6 A justification of the differentiability assumption would require an excursion into the ‘‘smooth economies’’ literature (for example, Mas-Colell, 1985; Balasko, 2011). One special case, however, clearly satisfies the assumption, an economy in which each consumer has an endowment ωi ≫ 0 of goods and the production plan y(α) is transferred directly to consumers; they then trade in the private ownership exchange economy with total endowment ω+ y(α). The Remark on p. 390 of Debreu L+I (1970) implies that for almost-all (ω1 , . . . , ωI ) ∈ R++ , the economy with supply ω + y(0) has a finite number of equilibria, and each of them is C 1 on a interval owning α = 0. Pick one of these and set A equal to the corresponding interval. It follows that each consumer’s equilibrium consumption is C 1 on A. 7 More precisely, R is the integral of Radner’s local measure, p(0) · y′ (α), which he evaluates at α = 0. Hirshleifer (1966, p. 271), and Arrow and Lind (1970, p. 368), also hint at such a local measure.

(2)

i

The difference in aggregate wealth between policy α ∈ A and policy 0 is

w (α) − w (0) =  p(α) · x(α) − p(0) · x(0)  α  α   = p(v) · d x(v) + x(v) · d p(v). 0

(3)

0

Rearrange to find that DI (α) = S (α) for every α ∈ A, so DI ′ (0) = S ′ (0). It follows immediately from the equilibrium condition  x(α) = ω+ y(α) and the definitions of DI (·) and R(·) that DI ′ (0) = R′ (0). I am done if I show that S ′ (0) = RU ′ (0). As in Schlee (forthcoming), differentiate consumer i’s equilibrium utility and use Roy’s Identity and Eq. (1) to find

 Vi′ (0) = Vpi · p ′ (0) + Vwi w i′ (0)   = Vwi −Di · p′ (0) + w i′ (0) = Vwi × Si′ (0),

(4)

where Vpi and Vwi are evaluated at ( p(0), w i (0)). Divide both sides by Vwi (>0) and sum over all consumers to find



pressed in units of the numeraire; the Divisia index is not. To compare the Divisia index with the other  three I use the Nominal p(v) · d x(v). To define Divisia Quantity Index, namely DI (α) = [0,α]  the last measure, consider the utility possibility set for an economy with aggregate endowment of ρω, ρ ∈ (0, 1] and total supply of ρω + y(α) for some α ∈ A:

571

S ′ (0) =



 V ′ (0)

i

Vwi ( p(0), w i (0))

i

.

(5)

Eq. (5) on p. 17 of Debreu (1954) implies that

ρ ′ (0) =

1



 V ′ (0)

w (0)

i

Vwi ( p(0), w i (0))

i

.

By the definition RU (α) = ρ(α)w(0) and Eq. (5), it follows that S ′ (0) = RU ′ (0).  I should emphasize that α = 0 need not be Pareto optimal. If it were, it could happen that all five derivatives are equal to 0. The force of Proposition is that when the derivatives are nonzero, they agree in sign and magnitude. Recall that αˆ ∈ A is a potential Pareto improvement over α = 0 if there is an allocation (ˆx1 , . . . , xˆ I ) of the supply y(α) ˆ + ω that strictly Pareto dominates ( x1 (0), . . . , xI (0)). Radner (1993) proves that if R′ (0) > 0 then a small-enough increase in α from 0 is a potential Pareto improvement.8 His result and the Proposition yield the following corollary. Corollary. If any of the five derivatives in the Proposition are positive, then a small-enough increase in α from 0 is a potential Pareto improvement.

8 Intuitively, since the allocation at α = 0 is interior, the first order condition implies that the associated equilibrium price equals the vector of marginal utilities divided by the marginal utility of wealth for each consumer. If each consumer’s share of the new supply for α > 0 is proportional to the reciprocal of that consumer’s marginal utility of wealth, then each consumer’s utility is strictly increasing in α on a neighborhood of 0.

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I conclude with three remarks. First not everyone agrees on the definition of consumer’s surplus when both prices and wealth change. Chipman and Moore (1976) are agnostic about what function should be integrated along the path of prices and wealth (the function ‘‘f ’’ in their notation). Blackorby (1999) and Hammond (1990) differ from (1) in what to insert into the wealth argument of the integrand; Blackorby (1999) uses final wealth wi (α) and Hammond (1990) uses initial wealth wi (0). The Proposition and Corollary are arguments for using (1) as the definition of surplus.9 Second, as I mentioned, scholars have from time to time pointed out the relationship between pairs of welfare measures. Bruce (1977) points out the equivalence between the nominal Divisia quantity index and aggregate consumer’s surplus (noting that Samuelson (1947) and Silberberg (1972) hint at this fact). Radner (1993) derives the equality S ′ (0) = R′ (0), but without formally defining surplus as a path integral. Hammond (1990) identifies S ′ (0) as the change in real wealth (his Eq. (1)), and I call it the local Slutsky change in real wealth in Schlee (forthcoming).10 Third, there are clearly other locally equivalent measures. For example, consumer’s surplus as defined in (1) is a first-order approximation to both the compensating and equivalent variations (Hicks (1939) and Hicks (1942)), so the Proposition immediately extends to these two measures.11 References Arrow, K., Lind, R., 1970. Uncertainty and the evaluation of public investment decisions. American Economic Review 60, 364–378. Balasko, Y., 2011. General Equilibrium Theory of Value. Princeton University Press, Princeton. Blackorby, C., 1999. Partial-equilibrium welfare analysis. Journal of Public Economic Theory 1, 359–374. Boiteux, M., 1951. Le ‘revenu distruable’ et les Pertes economiques. Econometrica 19, 112–133.

9 Under Blackorby’s (1999) definition, for example, the conclusion of the Proposition fails unless one adds strong assumptions on preferences. 10 Diewert (1981) calculates the second order approximation of three welfare measures as a function of distortionary taxes: the coefficient of resource utilization, and measures proposed by Boiteux (1951) and Hotelling (1938). He does so starting from a Pareto Optimum level of taxes, so the first derivative of each of these measures is 0. I do not assume that α = 0 is Pareto Optimal. 11 Mosak (1942) is credited as the first to point out the derivatives of the Slutsky and Hicks compensations are equal. This equivalence lies at the heart of the famous approximation result of Willig (1976).

Bruce, N., 1977. A note on consumer’s surplus, the divisia index, and the measurement of welfare changes. Econometrica 45, 1033–1038. Chipman, J., Moore, J.C., 1976. The scope of consumer’s surplus arguments. In: Tang, A., Westfield, F., Worley, J. (Eds.), Evolution, Welfare and Time in Economics. D.C. Heath and Company, pp. 69–124. Debreu, G., 1951. The coefficient of resource utilization. Econometrica 19, 273–292. Debreu, G., 1954. A classical tax-subsidy problem. Econometrica 22, 14–22. Debreu, G., 1959. The Theory of Value. Yale University Press, New Haven. Debreu, G., 1970. Economies with a finite set of equilibria. Econometrica 38, 387–392. Debreu, G., 1972. Smooth preferences. Econometrica 40, 603–615. Diewert, W., 1981. The measurement of the deadweight loss revisited. Econometrica 49, 1225–1244. Hammond, P., 1990. Theoretical Progress in Public Economics: A Provocative Assessment. Oxford Economic Papers 42, 6–33. Hicks, J., 1939. Value and Capital. Oxford University Press, Oxford, UK. Hicks, J., 1942. Consumers’ surplus and index-numbers. Review of Economic Studies 9, 126–137. Hirshleifer, J., 1966. Investment decision under uncertainty: applications of the state-preference approach. The Quarterly Journal of Economics 80, 252–277. Hotelling, H., 1938. The general welfare in relation to problems of taxation and of railway and utility rates. Econometrica 6, 242–269. Katzner, D., 1968. A note on the differentiability of consumer demand functions. Econometrica 36, 415–418. Mas-Colell, A., 1985. The Theory of General Economic Equilibrium: A Differentiable Approach. Cambridge University Press, Cambridge. Mosak, J., 1942. On the interpretation of the fundamental equation of value theory. In: Lange, O., McIntyre, F., Yntema, T. (Eds.), Studies in Mathematical Economics and Econometrics: In Memory of Henry Schultz. Blackwell, pp. 69–74. Radner, R., 1993. A note on the theory of cost–benefit analysis in the small. In: Basu, K., Majumdar, M., Mitra, T. (Eds.), Capital, Investment and Development. Blackwell, pp. 129–141. Samuelson, P., 1947. Foundations of Economic Analysis. Harvard University Press, Cambridge, MA. Schlee, E., 2013. Surplus maximization and optimality. American Economic Review (forthcoming). Available at https://sites.google.com/a/asu.edu/edward-schlees-page/. Silberberg, E., 1972. Duality and the many consumer’s surpluses. American Economic Review 62, 942–952. Varian, H., 1978. Microeconomic Analysis. Norton, New York, NY. Willig, R., 1976. Consumer’s surplus without apology. American Economic Review 66, 589–597.