Consumer surplus analysis under uncertainty: A general equilibrium perspective

Journal of Mathematical Economics (

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Consumer surplus analysis under uncertainty: A general equilibrium perspective Takashi Hayashi University of Glasgow, United Kingdom

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Article history: Received 23 August 2013 Received in revised form 7 February 2014 Accepted 8 February 2014 Available online xxxx Keywords: Partial equilibrium analysis General equilibrium analysis Incomplete asset markets Hicksian aggregation Expected consumer surplus

abstract This paper derives an exact form of partial equilibrium efficiency measure under uncertainty which is consistent with expected utility maximization in a general equilibrium situation with ex-post spot markets for many goods and asset markets which are in general incomplete. We consider that the good under consideration tends to be negligibly small compared to the entire set of commodity characteristics which is assumed to be a continuum, and look into the limit property of preferences over state-contingent consumption of the good and state-contingent income transfer associated to it. We show that the limit preference exhibits risk neutrality, not only that it exhibits no income effect, meaning that the two conditions are tied together. We also show that the marginal rate of substitution between extra income transfers at different states of the world converges to the ratio between the Lagrange multipliers associated to those states. When the asset markets are complete such ratios are equalized between consumers, but it is not the case in general when the asset markets are incomplete. This means that using the aggregate expected consumer surplus as the welfare measure will be in general inconsistent with individuals’ expected utility maximization in the general equilibrium environment or with ex-ante Pareto efficiency. © 2014 Elsevier B.V. All rights reserved.

1. Introduction 1.1. Motivation The role of (aggregate) expected consumer surplus as an efficiency measure is prominent in many fields which adopt the partial equilibrium framework under uncertainty, such as mechanism design, industrial organization, environment, health, agriculture and others. To recall the definition, suppose there are S states of the world, let (x1 , . . . , xS ) denote the vector of an individual’s statecontingent consumptions of the commodity under consideration, where xs denotes his consumption of the good at state s = 1, . . . , S. Let (a1 , . . . , aS ) denote the vector of state-contingent income transfers to him, where as denotes ex-post transfer at state s = 1, . . . , S. Then the expected consumer surplus for the given individual takes the form S 

(v(xs ) + as ) πs

s=1

or its arbitrary monotone transformation, where (π1 , . . . , πS ) denotes the probability vector.

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmateco.2014.02.001 0304-4068/© 2014 Elsevier B.V. All rights reserved.

The usual textbook/classroom remark for this is that it relies on two assumptions: (1) no income effect in the sense that the marginal rate of substitution of income transfer at state s and consumption of the good contingent on state s is independent of as ; (2) risk neutrality in the sense that evaluation of uncertain prospects in the above form depends only on the expectation of S consumer surplus s=1 (v(xs ) + as ) πs without any further adjustment, or in other words the marginal rates of substitution between income transfers at different states are constant. We would point out one more assumption which is critical in aggregation: (3) values of income transfers are equal across states of the world, and also equal across individuals; or at least, the way how the values of income transfers differ across states is the same across individuals. It remains unclear, however, if and how the three assumptions can be said precisely and consistently in the words of the general equilibrium theory. In complete market settings, the assumption of no income effect has been given a general equilibrium theoretic characterization. Vives (1987) considers an increasing sequence of sets of commodities, and shows that income effect on each single commodity vanishes as the number of commodity and income tend to infinity at the same rate (see also Hayashi, 2008 for some follow-up to it). Hayashi (2013) instead starts with presenting the whole set of

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T. Hayashi / Journal of Mathematical Economics (

commodities as a continuum and subdivide it into many pieces so that each piece tends to be arbitrarily small, and shows that willingness to pay for a commodity is established in the limit as a density notion and exhibits no income effect. The key here is to make a distinction between the pool of income held by the consumer beforehand (let us call it background income to emphasize the distinction) and relative change from it by means of transfer. The small income effect result therefore states that if the background income is sufficiently large compared to the given commodity or the commodity is sufficiently small compared to the background income then the effect of income transfer on the demand for it is negligible. For the case of uncertainty, the above arguments are readily extended to Arrow–Debreu markets (Arrow, 0000; Debreu, 1959) in an essentially deterministic manner, in which commodities are differentiated by date-events at which they are delivered as well as by their material characteristics. Let T be the set of material characteristics of commodities and S be the set of states of the world. Then we can consider an extended set of commodity characteristics T × S and take let us say (t , s) to be the object of partial equilibrium analysis, in which we look at how consumers are willing to substitute between commodity (t , s) and income transfer to be spent over the rest of commodities (T × S ) \ {(t , s)}. However, this way of extension presumes that value of income is uniform across all date-events, and it does not answer our question about if and why values of income transfers are equalized across states of the world and also equalized across individuals. This necessitates to look into a Hicksian-type aggregation problem in a general equilibrium setting under uncertainty, such that: 1. there are ‘‘many’’ commodities in the spot markets to be opened at each state of the world, and the commodity under consideration is a ‘‘negligibly small’’ one; 2. there are asset markets in which individuals allocate their incomes across states, either completely or incompletely. The first element captures the above-noted issue on the noincome-effect assumption. It reconfirms the classic ‘‘excuse’’ since Marshall (1920), saying that when the commodity is negligibly small compared to the entire set of commodities one can ignore income effect on it. The second element captures the fact that consumers’ risk attitudes in partial equilibrium and the values of income transfers at different states are determined endogenously by how they take positions in the asset markets. This fact is captured better by adopting the asset market model due to Radner (1968), rather than the Arrow–Debreu model (Arrow, 0000; Debreu, 1959), which is the case particularly when the asset markets are incomplete. 1.2. Outline of results In order to deal with the above issues, we work on the problem of Hicksian aggregation under uncertainty in the setting due to Radner (1968). We derive consumer’s indirect preference over state-contingent consumptions of a given material good and statecontingent transfers of income that is to be spent on the other goods in the spot markets at each state. We take the set of material characteristics of commodities as a continuum, and take the given good as an element of its finite partition. We consider a limit in the sense that the partition becomes arbitrarily finer and the good tends to be arbitrarily small, while the magnitude of income transfers is adjusted to the smallness of the good and tends to be small as well. Given a finite partition, the argument falls in the standard demand theory in the literature of general equilibrium with incomplete markets (GEI) such as Geanakoplos and Polemarchakis (1986) and Magill and Quinzii (2002). The current work may be viewed as a contribution to the demand theory in the GEI setting with infinitely many commodities in the spot markets, as we establish the existence and uniqueness of the limit.

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We look into the limit property of preference over statecontingent consumption of the good and state-contingent income transfer associated to it. We show that the limit preference is riskneutral, not only that it exhibits no income effect, meaning that the two conditions are tied together. We also show that the marginal rate of substitution between extra income transfers at different states of the world converges to the ratio between the Lagrange multipliers associated to those states. When the asset markets are complete such ratios are equalized between consumers, but it is not the case in general when the asset markets are incomplete. This means that using the aggregate expected consumer surplus as the welfare measure will be in general inconsistent with individuals’ expected utility maximization in the general equilibrium environment or with ex-ante Pareto efficiency. 1.3. Related literature Let us conclude the introduction by discussing the relation between the present paper and papers on evaluating uncertain price–income pairs or uncertain incomes. It is known that the expected consumer surplus criterion concludes price instability is good (see for example Waugh, 1944 and Massell, 1969). Consider that inverse demand curve is linear or that it is locally approximated linearly. Say it is p(x) = 1 − x. Then consumer surplus given price p is (1 − p)2 /2, which is convex in p and implies p being more risky is good. This point has motivated careful examination of preference over price uncertainty. Rogerson (1980) considers preference over probability distribution of price–income pairs in ex-post spot markets, basically repre sented in the expected utility form U (F ) = V (p, m)dF (p, m), in which the von-Neumann/Morgenstern index V defined over price vector p and income m is supposed to play the role of indirect utility function in the ex-post spot markets as well as to describe the consumer’s risk attitude toward price–income uncertainty.1 , 2 Rogerson shows that expected consumer surplus from a good represents the consumer’s preference over distributions of its price given the same income if and only if V is additively separable between the price and income. The result is sophisticated by Schlee (2008), who shows a stronger result that the equivalence is true even for approximate representation, and also that the aggregate expected consumer surplus is consistent with Kaldor criterion if and essentially only if V is linear in income. In a more recent paper, Schlee (2012) provides a more robust sufficient condition under which maximization of expected consumer surplus leads to Pareto efficiency. In the above approach the asset markets and how consumers take positions there are taken to be an implicit fixed factor. The current paper is taken to be endogenously deriving consumers’ risk attitude over prices and incomes from their decisions in the asset market. 2. Hicksian aggregation under uncertainty 2.1. Consumption and price spaces First we describe the consumption space and the associated price space in a deterministic setting, before introducing uncertainty, and introduce some relevant mathematical concepts. Let T = [0, 1] be the set of commodity characteristics, Σ be the family of Lebesgue measurable sets, and µ be the Lebesgue measure.

1 In basically the same setting, Turnovsky et al. (1980) provide a sufficient condition under which price stabilization is good. 2 Grant et al. (1992) provide characterizations of when preference over lotteries over consumption of many goods, which is taken to be the primitive, can be described as a preference over money lotteries.

T. Hayashi / Journal of Mathematical Economics (

Let L∞ (T ) be the space of essentially bounded measurable functions from T to R, which is endowed with the sup norm. Let ∞ L∞ +++ (T ) = {f ∈ L (T ) : ∃ l > 0, a.e. t ∈ T , f (t ) ≥ l}

be the set of essentially bounded measurable functions which are positively bounded away from zero. We take L∞ +++ (T ) to be the set of deterministic consumptions. Let L1 (T ) be the space of Lebesgue integrable functions from T to R, which is endowed with the integral norm. Let L1+++ (T ) = {f ∈ L1 (T ) : ∃ l > 0, a.e. t ∈ T , f (t ) ≥ l}

3

Subdivisions of the set of commodity characteristics We describe the process of subdividing the set of commodity characteristics by a sequence of increasing partitions, denoted by {J n }, which is for simplicity assumed to be generated by the binary expansion, as

Jn =

    n  1 2 2 − 2 2n − 1 , , , . . . , , , 2n 2n 2n 2n 2n  n  2 −1 , 1 n



0,

1

for each n. Notice that for every t ∈ T , there exists a unique sequence {J n (t )} such that J n (t ) ∈ J n and t ∈ J n (t ) for all n and lim inf J n (t ) = limnsup J n (t ) = t. Given n, let RJ be the corresponding finite-dimensional subspace of L∞ (T ), consisting of simple functions measurable with reJn

spect to J n , and let R++ be the finite-dimensional consumption space as a subset of L∞ +++ (T ). The lemma below states that the union of finite-dimensional consumption subspaces, in other words the set of simple functions, is dense in the weak-∗ topology. Lemma 3.

ν

⟨f , p ⟩ → ⟨f , p⟩ as ν → ∞ for all f ∈ L∞ (T ). Say that a sequence in L∞ (T ), denoted by {f ν }, weak-∗ converges to f if

⟨f ν , p⟩ → ⟨f , p⟩ as ν → ∞ for all p ∈ L1 (T ). We introduce two more convergence notions. Let C ⊂ Rm be a compact set and consider a sequence of functions from C to L1 (T ), denoted by {pν }. Say that {pν } weakly converges to p, a function from C to L1 (T ), uniformly on C if as ν → ∞

c ∈C

for all f ∈ L∞ (T ). Also, consider a sequence of functions from C to L∞ (T ), denoted by {f ν }. Then say that {f ν } weak-∗ converges to f , a function from C to L∞ (T ), uniformly on C if sup |⟨f ν (c ) − f (c ), p⟩| → 0

–

2

be the set of integrable functions which are positively bounded away from zero. We assume that any price system for the spot markets to be opened at each possible state is in L1+++ (T ). Given a  price system p ∈ L1+++ (T ) and a set K ∈ Σ , let pK = K p(t )dµ(t ) denote the price of consumption bundles characterized by K . It is known that L1 (T )∗ = L∞ (T ), with the dual operation given by ⟨f , p⟩ = T f (t )p(t )dµ(t ) where f ∈ L∞ (T ) and p ∈ L1 (T ). Hence one can consider weak convergence in L1 (T ) and weak-∗ convergence in L∞ (T ). Say that a sequence in L1 (T ), denoted by {pν }, weakly converges to p if

sup |⟨f , pν (c ) − p(c )⟩| → 0

)

as ν → ∞

c ∈C

for all p ∈ L1 (T ). We will use some mathematical claims. First one follows from the sequential Banach–Alaoglu theorem, since L1 (T ) is separable. Lemma 1. Fix z , z ∈ R. Then the set [z1, z1] ≡ {f ∈ L (T ) : z1 ≤ f ≤ z1} is weak-∗ sequentially compact, where 1 denotes the constant function of one. ∞

Second is a generalization of the Ascoli–Arzela theorem, which has been shown in Hayashi (2013, Lemma 3). Lemma 2. Let C be a compact metric space. Let {f ν } be a sequence of functions from C to L∞ (T ). Suppose (i) there exists a weak-∗ sequentially compact subset G ⊂ L∞ (T ) such that f ν (c ) ∈ G for all ν and c ∈ C ; (ii) for all q ∈ L1 (T ), for all ε > 0, there is δ > 0 such that for all ν and c , c ′ ∈ C , d(c , c ′ ) < δ H⇒ |⟨f ν (c ), q⟩ − ⟨f ν (c ′ ), q⟩| < ε. Then there exist a subsequence {f k(ν) } and a f function from C to L∞ (T ) such that for all q ∈ L1 (T ), sup |⟨f k(ν) (c ), q⟩ − ⟨f (c ), q⟩| → 0 as ν → ∞, c ∈C

where f is continuous in the sense that f (c l ) weak-∗ converges to f (c ) as c l → c.



n ≥1

Jn

R++ is weak-∗ dense in L∞ +++ (T ).

2.2. State-contingent consumptions, spot markets and asset markets S Let S be a finite set of states of the world. We take L∞ +++ (T ) to ∞ be the set of state-contingent consumptions. Given f ∈ L+++ (T )S , fs (t ) denotes the consumption of commodity t ∈ T to be received at the spot markets to be opened at state s ∈ S. We take L1+++ (T )S to be the set of state-contingent price systems in the spot markets. Given p ∈ L1+++ (T )S , ps (t ) denotes the price of commodity t ∈ T at the spot markets to be opened at state s ∈ S. Let w ∈ RS++ be the vector of state-contingent deliveries of background income, where ws denotes the income to be received at state s ∈ S. Let H be a finite set of securities available in the asset markets and let R be an |S | × |H | matrix describing the payoff structure, which says that security h pays Rsh units of income at state s which can be used in the spot markets there. Let q ∈ RH ++ be the vector of security prices. We make the following assumptions on wealth and prices. Wealth and prices:

(i) w ∈ RS++ ; (ii) p ∈ L1+++ (T )S and there exist p, p with 0 < p < p such that ps (t ) ∈ [p, p] for almost all t ∈ T and all s ∈ S. (iii) q ∈ RH ++ . (iv) (q, R) admits no arbitrage. (v) rankR = |H |. Condition (v) is imposed without loss of generality, because otherwise there may be many portfolio choices which yield the same state-contingent consumption plans. Given a measurable set of commodity characteristics K ⊂ T and  a state of the world s ∈ S, let psK = K ps (t )dµ(t ) be the price of K in the spot markets at state s. 2.3. Basic assumptions on preference We assume that the preference over state-contingent consumptions follows the expected utility theory, and it is represented in the form U (f ) =



u(fs )πs ,

s∈S

where u : L∞ +++ (T ) → R is the von-Neumann/Morgenstern index and πs denotes the probability of each state s.

4

T. Hayashi / Journal of Mathematical Economics (

First we impose an assumption on preference regularity. Regular preference: (i) u : L∞ +++ (T ) → R is norm-continuous and Frechet differentiable. Moreover, Du(f ) ∈ L1+++ (T ) for all f ∈ L∞ +++ (T ), and 1 the mapping Du(·) : L∞ ( T ) → L ( T ) is continuous in the +++ following sense: given any compact set C ⊂ Rm , for any seν quence of functions from C to L∞ +++ (T ), denoted by {f }, and a ∞ ν function from C to L+++ (T ) denoted by f , if {f } weak-∗ converges to f uniformly on C then Du(f ν ) weakly converges to Du(f ) uniformly on C . (ii) u : L∞ +++ (T ) → R is strictly concave. We also make an assumption on the preference induced on the finite dimensional subspaces generated by {J n }. Regular preference on finite dimensions: (i) For all n, the restriction of u onto the finite-dimensional Jn

n

Jn

subspace R , denoted by u : R++ → R, which is defined by



 

un (z ) = u

zK 1K

Jn

Dun (z ) ∈ R++ Jn

for all z ∈ R++ . (iii) Denote the second derivative of un by D2 un . Then, for all z ∈ Jn

R++ , the |J n | × |J n | matrix D2 un is negative definite. We assume that the Inada-type condition holds in the uniform manner across n, which is parallel to what Vives (1987) assumes for increasing numbers of commodities. Uniform Inada property: There exist non-increasing functions φ, φ from R++ to R++ such that (i) φ(y) ≤ φ(y) for all y ∈ R++ ; (ii) φ(y) → ∞ as y → 0 and φ(y) → 0 as y → ∞; Jn

(iii) for all n, z = (zK )K ∈Jn ∈ R++ and K ∈ J n ,

∂ un (z ) /µ(K ) ≤ φ(zK ). ∂ zK

2.4. Hicksian aggregation Given n, pick J ∈ J n to be the object of partial equilibrium analysis in the approximate sense. Now consider the preference induced over pairs of state-contingent consumptions of commodity piece J and associated state-contingent income transfers. n Given n and J ∈ J n and for each s ∈ S, let (xs , zs,−J ) ∈ RJ denote the vector such that xs refers to its J-th entry and zs,−J ∈ n n RJ \{J } refers to the rest of the entries. Also, let (x, z−J ) ∈ RJ ×S be (J n \{J })×S given by (x, z−J ) = (xs , zs,−J )s∈S and z−J ∈ R be given by z−J = (zs,−J )s∈S . We consider vectors of state-contingent income transfers associated with the consumption of given commodity piece J ∈ J n , which is to be spent on the rest of commodity pieces J n \{J }. When a ∈

− µ(wJ ) , ∞



Definition 1. Given n, J ∈ J n and (x, a), (y, b) ∈ RS++ × − µ(wJ ) ,

S ∞ , the relation (x, a) %n,J (y, b) holds if V n,J (x, a) ≥ V n,J (y, b), where V n,J (x, a) =

max



(J n \{J })×S

ζ ∈RH ,z−J ∈R++

un xs , zs,−J πs





s∈S

qh ζh = 0

h∈H

Jn

S

At a social level, such income transfers must satisfy certain budget-balance condition ex-post at each state. Since we are looking into individual preferences over any potentially possible transfers they might receive or pay, however, we assume that any transfer can be considered at an individual level, unless one’s incomes after transfer are negative.



for z = (zK )K ∈Jn ∈ R++ , is twice continuously differentiable, where 1K denotes the characteristic function over K ∈ J n . (ii) Denote the first derivative of un by Dun . Then,



–

subject to

K ∈J n

φ(zK ) ≤

)

is given, it means that the consumer receives

as µ(J ) units of income transfer at state s ∈ S, where it is taken to be a payment when it is negative. Note that this income transfer is adjusted to the size of commodity piece J, because as it tends to be small associated income transfers tend to be small as well.



psK zsK = ws +

K ∈J n \{J }



Rsh ζh + as µ(J ) for each s ∈ S .

h∈H

Under the current assumption the finite dimensional problem above for each fixed n has a unique interior solution, since it falls in the standard demand analysis in the literature of general equilibrium with incomplete markets (GEI), such as Geanakoplos and Polemarchakis (1986), Magill and Quinzii (2002), with the modification that there is no consumption in the ex-ante stage and the ex-ante income is fixed to be zero. Since ws is positive for all s ∈ S and the asset markets are assumed to admit no arbitrage, the solution is now guaranteed to exist. Because of the regularity of the preference the solution is unique and in the interior. n ,J Let zsK (x, a) denote the demand for commodity piece K ∈ n ,J

J n \ {J } in the spot markets st state s ∈ S, let zs,−J (x, a) =

(x, a)



 n,J n ,J and let z−J (x, a) = zs,−J (x, a)

s∈S

. Let ζh (x, a) den ,J

note the demand for security h ∈ H and let ζh (x, a) =

 (x, a)

n,J

zsK

n ,J



K ∈J n \{J }





ζhn,J

n ,J

h∈H

. Let λ0 (x, a) > 0 be the Lagrange multiplier on the conn ,J

straint h∈ > 0 be the multiplier on the H qh ζh = 0 and λs (x, a) constraint K ∈Jn \{J } psK zsK = ws + h∈H Rsh ζh + as µ(J ) for each s ∈ S. Then we have the first order condition



λn0,J (x, a)qh =



λns ,J (x, a)Rsh for each h ∈ H

s∈S

πs

∂

 u xs , zs,−J ∂ zsK = λns ,J (x, a)psK for each K ∈ J n \ {J }, s ∈ S ,  n

where the derivatives are evaluated at optimal consumption and security demand when no confusion arises. Similar comments apply to subsequent writings. By direct calculations we obtain

 ∂ V n,J (x, a) ∂ n = πs u xs , zs,−J ∂ xs ∂ xs ∂ V n,J (x, a) = λns ,J (x, a)µ(J ). ∂ as Let MRSnxs,,J as (x, a) denote the marginal rate of substitution of ex-

post income transfer at state s for the good under consideration to

T. Hayashi / Journal of Mathematical Economics (

be delivered at state s. Then from the above formula we obtain MRSxns,,J as

(x, a) =

πs λsn,J (x, a)

·

∂ n u ∂ xs

xs , zs,−J



µ(J )

)

–

5

From the uniform Inada condition again, we have

 ∂ n n u xsn , zsn ,−J n /µ(K ) ≥ n n K ∈J \{J } ∂ zK

 .

=φ

Likewise, we obtain n,J s

λ (x, a)

MRSans,J,a ′ (x, a) =



n ,J n

max φ zsn ,K (xn , an ) K ∈J n \{J n }

max



n ,J n min zsn ,K K ∈J n \{J n }



 (x , a ) . n

n

Jn

From the assumed property of φ , we have minK ∈Jn \{J n } zsn ,K (xn , an ) → ∞ as n → ∞. However, since

λsn′,J (x, a)   πs ∂∂xs un xs , zs,−J n ,J   MRSxs ,x ′ (x, a) = s πs′ ∂ x∂ ′ un xs′ , zs′ ,−J s   ∂ n u xs , zs,−J π s ∂ xs n ,J . · MRSxs ,a ′ (x, a) = n,J s µ(J ) λ ′ (x, a) s

wsn +



Rsn h ζhn + ans µ(J n ) =

n,J n

 K ∈J n \{J n }

h∈H

pK zsn ,K (xn , an )

n ,J zsn ,K (xn , an ), ≥ pµ(T ) min n n n

K ∈J \{J }

s

we have h∈H Rsn h ζhn → ∞. This is impossible under the current assumptions on the asset markets and the positive consumption assumption on the ex-post spot markets at all states. Next we prove uniform boundedness away from zero. Suppose not. Then without loss of generality there is a sequence {(xn , an ), J n , K n , sn } such that



2.5. Behavior of the conditional demand First we show that the conditional demand choice is uniformly bounded from above and away from zero.

n,J n

Lemma 4. There exist ζ , ζ with ζ < ζ , z , z with 0 < z < z and

zsn K n (xn , an ) → 0

λ, λ with 0 < λ < λ, such that

Note that we have

∂

n,J h

ζ (x, a) ∈ [ζ , ζ ]

∂ zsn K n

n ,J

zsK (x, a) ∈ [z , z ]

for all n, (x, a) ∈ C and h ∈ H , J ∈ J n , K ∈ J n \ {J } and s ∈ S. Proof. First we show the uniform boundedness of state-contingent consumptions from above. Suppose not. Then without loss of generality there is a sequence {(xn , an ), J n , K n , sn } such that





πsn

n,J n



→ 0 as

n → ∞, which implies

min

K ∈J n \{J n }

 ∂ n n u xsn , zsn ,−J n /µ(K ) ∂ zK

n

K ∈J \{J }

≥ λnsn,J (xn , an )p, n

Since u

πsn

= λnsn,J (xn , an ) min n n

  ∂ un xnsn , zsn ,−J n /µ(K n ) → 0 as n → ∞. ∂ zsn K n

∂ zsn K n

  ∂ psn K n n ,J n un xnsn , zsn ,−J n /µ(K n ) = λsn (xn , an ) ∂ zsn K n µ(K n ) n ≤ λnsn,J (xn , an )p, n ,J n

 /µ(K n ) ≤ ψ zsn K n (xn , an ) . 

πsn



we have λsn (xn , an ) → ∞ as n → ∞. On the other hand, also from the first order condition, we have

n,J n

By the assumed property of ψ , we have ψ zsn K n (xn , an )

n ,J n

xnsn , zsn ,−J n /µ(K n ) = λsn (xn , an )

 n

n ,J n

from the first-order condition and πsn is bounded away from above,

Note that we have

∂



Since

n ,J n

∂ zsn K n

n ,J n

  ∂ un xnsn , zsn ,−J n /µ(K n ) → ∞ as n → ∞. ∂ zsn K n

zsn K n (xn , an ) → ∞ as n → ∞.

xnsn , zsn ,−J n



n → ∞, which implies

λ (x, a) ∈ [λ, λ]

u





λ (x, a) ∈ [λ, λ]

 n



un xnsn , zsn ,−J n /µ(K n ) ≥ ψ zsn K n (xn , an ) .

By the assumed property of ψ , we have ψ zsn K n (xn , an ) → ∞ as

n ,J 0 n ,J s

∂

as n → ∞.



n ,J n sn

≥λ

psn K n

µ(K n )

(xn , an )p, n ,J n

from the first-order condition, we have λsn (xn , an ) → 0 as n → ∞. On the other hand, also from the first order condition, we have

 ∂ n n πsn max u xsn , zsn ,−J n /µ(K ) n n K ∈J \{J } ∂ zK = λsnn,J (xn , an ) max n n n

K ∈J \{J }

pK

µ(K )

≤ λnsn,J (xn , an )p, n

hence the left hand side converges to zero. Note that πsn is bounded away from zero.

pK

µ(K )

hence the left hand side diverges to infinity. Note again that πsn is bounded away from above. From the uniform Inada condition again, we have

  n  ∂ n n n,J u xsn , zsn ,−J n /µ(K ) ≤ min φ zsn ,K (xn , an ) n n K ∈J \{J } ∂ zK   n n,J =φ max zsn ,K (xn , an ) . n n

min

K ∈J n \{J n }

K ∈J \{J }

Jn

From the assumed property of φ , we have maxK ∈Jn \{J n } zsn ,K (xn , an ) → 0 as n → ∞. However, since

wsn +



Rsn h ζhn + ans µ(J n )

h∈H

=

 K ∈J n \{J n }

n ,J n

n,J n

pK zsn ,K (xn , an ) ≤ pµ(T ) max zsn ,K (xn , an ), K ∈J n \{J n }

we have wsn +



h∈H

Rsn h ζhn + ans µ(J n ) → 0 as n → ∞.

6

T. Hayashi / Journal of Mathematical Economics (

n ,J

v n ,J ( x s , w s ) =

max

J n \{J } zs,−J ∈R++

u

xs , zs,−J

 n

n,J

obtained by replacing the column of H (z ) for commodity piece sK by the column of Gn,J (z ) for commodity piece J at s′ . For each n ,J s ∈ S , s′ ∈ S and K ∈ J n \ {J }, let  HsK ,s′ (z ) be the matrix obtained



subject to K ∈J n \{J }

–

For each s, s′ ∈ S and K , ∈ J n \ {J }, let HsK ,s′ (z ) be the matrix

Let



)

by replacing the column of H n,J (z ) for commodity piece sK by the column of Gn,J (z ) for income transfer at state s′ . Then, by Cramer’s rule we have

psK zsK = w s .

Then it holds V n,J (x, a) = max

w ∈W (a)



dζh

v n (xs , w s )πs

dxs′

s∈S

where

dζh

 W ( a) =

w ∈ RS++ : ∃ζ ∈ RH ,



das′

qh ζh = 0,

h∈H

dzsK

 ws +



Rsh ζh + as µ(J ) for each s ∈ S

dxs′

.

h∈H

∂v n,J (x , w)

n ,J

n ,J

s s Now direct calculation shows = λs , where λs is ∂ ws the corresponding Lagrange multiplier. Under the present assumption this diverges to infinity when it is evaluated along {(xn , an ), J n , K n , sn }. However, since w s = ws + as µ(J ) is always available by taking ζ to be the zero vector, the choice of corresponding w  is not optimal for sufficiently large n. Since consumption is uniformly bounded from above and below, the corresponding Lagrangian multipliers are also bounded from above and away from zero. Because of the budget constraints and the no-arbitrage condition the asset holdings are uniformly bounded as well. 

dzsK das′

   n ,J  Hh,s′ (x, z n,J (x, a))  , h ∈ H , s′ ∈ S =  n,J H (x, z n,J (x, a))   n,J  Hh,s′ (x, z n,J (x, a))  , h ∈ H , s′ ∈ S =  n ,J H (x, z n,J (x, a))    n ,J  HsK ,s′ (x, z n,J (x, a))   = , s, s′ ∈ S , K ∈ In \ {J } H n,J (x, z n,J (x, a))   n,J  HsK ,s′ (x, z n,J (x, a))   = , s, s′ ∈ S , K ∈ In \ {J }. H n,J (x, z n,J (x, a))

Condition below guarantees that the sensitivity terms given above are uniformly bounded as the consumption vectors are uniformly bounded. Uniform boundedness of sensitivity: For any fixed z , z > 0, there n exist α, α and β, β such that for all n, z ∈ [z , z ]J , h ∈ H , J , K ∈ J n ′ and s, s ∈ S, it holds

      n ,J   n ,J Hh,s′ (z ) HsK ,s′ (z ) α 5  n ,J  ,  n ,J  5 α H (z ) H (z )

Next we derive comparative statics properties of the conditional demand. From the second-order argument, we have

and

 dζ    dx1  dz1,−J   .   ..   ..   .        dz dx|S |  |S |,−J  n,J n ,J H n,J (x, z n,J (x, a))    dλ0  = G (x, z (x, a))   da1     .   dλ1   .    .  ..  da|S | . dλ|S |

2.6. The limit theorem

J n ×S

where H n,J (z ) for arbitrary z ∈ R++ is an (|H | + 2n |S | + 1) × (|H | + 2n |S | + 1) matrix given by H n,J (z ) in Box I and Gn,J (z ) is an (|H | + 2n |S | + 1) × 2|S | matrix given by Gn,J (z ) in Box II and pns ,J = (psK )K ∈Jn \{J }

D−J u (zs ) = n



∂ 2 un (zs ) ∂ zsL ∂ zsK



DJ D−J u (zs ) =



∂ 2 un (zs ) ∂ zsJ ∂ zsK

n, let {f n,J (τ ) } be the sequence of functions from C to [z1, z1] ⊂ S L∞ +++ (T ) given by n

n fsn,J (τ ) (x, a) = xs 1J n (τ ) +

n,J n (τ )



zsK

(x, a)1K

K ∈J n \{J n (τ )}

for each n, s ∈ S and (x, a) ∈ C .

d((x, a), (y, b)) < δ

K ,L∈J n \{J }

H⇒ |⟨fsn,J

is a (2n − 1) × (2n − 1) matrix for each s ∈ S, and n

Hereafter, fix τ ∈ T arbitrarily and let J = J n (τ ) for each n, and fix a compact set C ⊂ RS++ × RS . Also, for all sufficiently large

Lemma 5. For all q ∈ L1 (T ), for all ε > 0, there is δ > 0 such that for all n and (x, a), (y, b) ∈ C ,

is a row vector with 2n − 1 entries for each s ∈ S, 2

    n,J  n,J  Hh,s′ (z ) HsK ,s′ (z ) β 5  n,J  ,  n,J  5 β. H (z ) H (z )

n (τ )

(x, a), q⟩ − ⟨fsn,J

n (τ )

(y, b), q⟩| < ε

for all s ∈ S.



Proof. Note that for any (x, a), (y, b) ∈ C ,

K ∈J n \{J }

is a column vector with 2n − 1 entries for each s ∈ S. Also, for given sets X and Y , let 0X denote the row zero vector with |X | entries and let 0X ×Y denote the |X | × |Y | zero matrix. n,J For each h ∈ H and s′ ∈ S, let Hh,s′ (z ) be the matrix obtained by

replacing the column of H n,J (z ) for asset h by the column of Gn,J (z ) n,J for commodity piece J at s′ . For each h ∈ H and s′ ∈ S, let  Hh,s′ (z ) n,J

be the matrix obtained by replacing the column of H (z ) for asset h by the column of Gn,J (z ) for income transfer at state s′ .

⟨fsn,J

n (τ )

(x, a), q⟩ − ⟨fsn,J

= (x − y)qJ n (τ ) +

n (τ )

(y, b), q⟩   n,J n (τ ) zsK

n,J (x, a) − zsK

n (τ )

 (y, b) qK

K ∈J n \{J n (τ )}

for all s ∈ S. Hence the proof is done if it is shown that for all  ε > 0 there n,J n (τ )

is δ > 0 such that d((x, a), (y, b)) < δ implies zsK



n,J n (τ ) zsK

  (y, b) < ε for all n, K ∈ J n \ {J n (τ )} and s ∈ S.

(x, a)−

T. Hayashi / Journal of Mathematical Economics (

0Jn \{J }

q



    −R    H n ,J ( z ) =  OH 2    O(Jn \{J })×H   ..  .

.. .

0Jn \{J }

··· ··· ··· .. . ···

OH × S

π1 D−J u (z1 ) .. . 2

O(Jn \{J })×H

0Jn \{J }

··· ··· .. .

n ,J p1

n

O(Jn \{J })2

)

–

7

0S

0



0Jn \{J }

.. .

0tS

n ,J p|S |

OS 2

qt

OH × S

(

t

O(Jn \{J })2

0Jn \{J }

.. .

π|S | D2−J un (z|S | )

             

.. .

0tJn \{J }

−R t ··· .. . ···

n ,J p1 t

.. .

)

0tJn \{J }

0tJn \{J }

(

.. .

n ,J p|S | t

)

Box I.

0S

       n ,J G (z ) =    −π1 DJ D−J un (z1 )   ..  .

0S

µ(J ) .. .

OS 2

··· .. . ···

0 OH × S

0

.. . µ(J )

OH ×S 0tJn \{J }

··· .. . ···

0tJn \{J }



.. .

O(Jn \{J })×S

−π|S | DJ D−J un (z|S | )

           

Box II.

Moreover, (ζ , f ) is the unique solution to the problem (we call it unconditional problem)

By the mean value theorem we have n,J n (τ ) zsK

=



(x, a) −

n,J n (τ ) zsK

n,J n (τ ) Dv zsK

(y, b)

(v, e),

n,J n (τ ) De zsK

(v, e)



x−y a−b

ζ ∈RH ,f ∈L∞

+++

for some (v, e) between (x, a) and (y, b). By the Uniform Boundn,J n (τ ) Dv zsK

n,J n (τ ) De zsK

edness of Sensitivity assumption, (v, e) and (v, e) are uniformly bounded. This delivers the equi-continuity property.  Lemma 6. For all ε > 0, there is δ > 0 such that for all n and (x, a), (y, b) ∈ C , d((x, a), (y, b)) < δ H⇒ ∥ζ

n,J n (τ )

(x, a) − ζ

n,J n (τ )

(y, b)∥ < ε.

Proof. The proof is done if it is shown that for all  ε n> 0 there is δ > 0 such that d((x, a), (y, b)) < δ implies ζ n,J (τ ) (x, a)−

ζ n ,J

n (τ )

 (y, b) < ε for all n.

By the mean value theorem,

ζ

n,J n (τ )

(x, a) − ζ

n,J n (τ )



k(n)

,f

n,J n (τ )



sup ∥ζ

(x,a)∈C

, −ζ ∥ → 0 as n → ∞

and sup |⟨f n,J

(x,a)∈C

ps (t )fs (t )dµ(t ) = ws + T



Rsh ζh

for each s ∈ S .

h∈H

n n Proof. From the equi-continuity property, {ζ n,J (τ ) , f n,J (τ ) } has k(n)

k(n)

a subsequence {ζ k(n),J (τ ) , f k(n),J (τ ) } which uniformly and weak-∗ converges uniformly on C . Denote its limit by ζ τ , f τ , then for all (x, a) ∈ C , we have ζ τ (x, a) ∈ [ζ , ζ ]H and f τ (x, a) ∈ [z1,

S z1]S ⊂ L∞ +++ (T ) . Since n,J n (τ )

qh ζh

=0



n,J n (τ ) zsK



 (x, a)1K , ps

K ∈J n \{J n (τ )}

=

n,J n (τ )



psK zsK

(x, a)1K

K ∈J n \{J n (τ )}

= ws +

} has a convergent subse-

quence {ζ n,J (τ ) , f n,J (τ ) } with the limit ζ ∈ [ζ , ζ ]H and f ∈ [z1, z1], which are constant over (x, a), in the sense that n,J k(n) (τ )

qh ζh = 0

h∈H

for some (v, e) between (x, a) and (y, b). nBy the Uniform Boundn edness of Sensitivity assumption, Dv ζ n,J (τ ) (v, e) and De ζ n,J (τ ) (v, e) are uniformly bounded. This delivers the equi-continuity property.  k(n)

u(fs )πs

s∈S

h∈H

   n n x−y = Dv ζ n,J (τ ) (v, e), De ζ n,J (τ ) (v, e) a−b

Lemma 7. The sequence {ζ

(T )S

subject to



(y, b)

n,J n (τ )



max





n,J n (τ )

Rsh ζh

+ as µ(J n (τ ))

h∈H

for all n and s ∈ S, the uniform convergence implies



qh ζhτ (x, a) = 0

h∈H



ps (t )f τ (t , s)(x, a)dµ(t ) T

k(n) (τ )

(x, a), q⟩ − ⟨f , q⟩| → 0 as n → ∞

for all s ∈ S and q ∈ L1 (T ).

= ws +



Rsh ζhτ (x, a) for each s ∈ S

h∈H

for all (x, a) ∈ C .

8

T. Hayashi / Journal of Mathematical Economics (

Fix any (x, a) ∈ C . We show that (ζ τ (x, a), f τ (x, a)) is a solution to the unconditional problem. Suppose not. Then there exist ξ ∈ S RH and g ∈ L∞ +++ (T ) with qξ = 0 and ⟨ps , gs ⟩ = ws + Rs ξ for each s ∈ S such that U (g ) > U (f τ (x, a)). Since the uniform weak-∗ convergence implies pointwise weak-∗ convergence,  n one can  find  U with U (g ) >  U > U (f ) such that  U > U f n,J (τ ) (x, a) for all sufficiently large n. Since the subspace of J n -measurable simple functions is  weak-∗ dense, one can find sufficiently large n and xs 1J n (τ ) +





zsK 1K s∈S , as well as some portfolio vector, so that it satisfies the corresponding budget constraint and its value is larger than  U. However, it contradicts to the optimality given n. From strict quasi-concavity of preference and the full-rank property of R, the unconditional problem has at most one solution. Therefore, (ζ τ (x, a), f τ (x, a)) is constant over (x, a) and τ , hence rewrite it by (ζ , f ).  K ∈J n \{J n (τ )}

Lemma 8. The corresponding subsequence of {λn,J (τ ) (x, a)} con{0}∪S verges to λ ∈ R++ uniformly on C , which is the vector of Lagrange multipliers associated with the solution (ζ , f ) in the unconditional problem. n

Proof. Pick any K ∈ J r \ {J r (τ )} for some fixed r. From the firstorder condition we have

λns ,J

n (τ )

n,J n (τ )

 (x, a) =

T

Du(fs

(x, a))(t )1K dµ(t )  p (t )dµ(t ) K s

for all s ∈ S and n = r.

k(n)

As the subsequence {f k(n),J (τ ) } uniformly weak-∗ converges to f , from the Regular Preference assumption the sequence k(n)

{Du(fsk(n),J (τ ) (x, a))} uniformly weakly converges to Du(fs ) for all  s ∈ S. Therefore the right-hand-side uniformly converges to (f )(t )1K dµ(t ) T Du  s K ps (t )dµ(t )

. Since the limit of the right-hand-side is indepen-

dent of (x, a) and τ , so is the limit of the left-hand-side. Thus, for k(n),J k(n) (τ )

each s ∈ S let λs be the uniform limit of λs , which is constant over (x, a) and τ . Note that this does not depend on the choice of K ∈ J r \ {J r (τ )} and r. From the first-order condition

λk0(n),J

k(n) (τ )

(x, a)qh =



λks (n),J

k(n) (τ )

(x, a)Rsh

s∈S

holds for all n, we obtain the uniform convergence of λ (x, a) to λ0 .  Finally we assume the following continuity property.

Continuous marginal utility density: For almost every τ ∈ T , for any compact set D ⊂ R++ and f ∈ L∞ +++ (T ), there exists a function 1u(·, τ ; f ) : D → R such that

   ∂    sup  u x1J + f 1T \J − 1u(x, τ ; f )µ(J ) = o(µ(J )), ∂ x x∈D where J is any interval containing τ with µ(J ) > 0, and o(µ(J )) is a function which vanishes faster than µ(J ). Moreover, 1u(x, τ ; f ) is continuous in f in the following sense: Given any compact set C ⊂ Rm , if a sequence of functions from C to L+++ (T ), denoted by {f ν }, weak-∗ converges to f uniformly on C , then sup sup |1u(x, τ ; f ν (c )) − 1u(x, τ ; f (c ))| → 0

as ν → ∞.

c ∈C x∈D

Note that when preference is separable over deterministic consumptions allowing representation U (f ) =

   φ v(fs (t ), t )dµ(t ) πs , s∈S

T

–

the above condition follows from the fundamental theorem of calculus and we have

1u(xs , τ ; fs ) = φ

′



 ∂ v(fs (t ), t )dµ(t ) v(xs , τ ) ∂ xs T

for each s ∈ S. Here we present the main result. Theorem 1. Given almost every τ ∈ T and any compact set C ⊂ R++ × R, there exists a subsequence of {n}, denoted by {k(n)}, such that as n → ∞ it holds

 πs 1u(xs , τ , fs )  →0 λs (x,a)∈C S    λs  k(n),J k(n) (τ )  sup MRSas ,a ′ (x, a) − →0 s λs′  (x,a)∈C S    πs 1u(xs , τ , fs )  k(n) sup MRSkxs(,nx),′J (τ ) (x, a) − →0 s πs′ 1u(xs′ , τ , fs′ )  (x,a)∈C S    πs 1u(xs , τ , fs )  k(n) sup MRSkxs(,na),′J (τ ) (x, a) − →0 s λs′ (x,a)∈C S  

k(n) sup MRSkxs(,na),s J (τ ) (x, a) −

for all s, s′ ∈ S, where λ0 , {λs }s∈S are the Lagrange multipliers to the unconditional problem. Proof. We establish the first one. The rests follow similarly. From the assumption of Continuous Marginal Utility Density, we have     ∂    u xs 1 k(n) + fs 1 k(n) J (τ ) T \J (τ )  ∂x sup  s µ(J k(n) (τ )) (x,a)∈C             ∂ u x 1 z 1 k(n) (τ ) +   s sK K J ∂ xs   k(n) (τ ) K ∈J k(n) \{J k(n) (τ )} k ( n ), J z =z (x,a)  −  k ( n )  µ(J (τ ))         ∂   ∂ xs u xs 1J k(n) (τ ) + fs 1T \J k(n) (τ ) − 1u(xs ; τ , fs )µ(J k(n) (τ )) 5 sup

k(n),J k(n) (τ ) 0

)

(x,a)∈C

+ sup

µ(J k(n) (τ ))   k(n)   1u(xs ; τ , fs )µ(J k(n) (τ )) − 1u(xs ; τ , f k(n),Js (τ ) (x, a))µ(J k(n) (τ )) µ(J k(n) (τ ))

(x,a)∈C

      + sup  (x,a)∈C    

 ∂ ∂ xs

u xs 1J k(n) (τ ) +

 K ∈J k(n) \{J k(n) (τ )}

   

zsK 1K 

z =z k(n),J

k(n) (τ )

(x,a)

µ(J k(n) (τ ))

    k ( n ) k(n),J (τ ) k(n) 1u(xs ; τ , fs (x, a))µ(J (τ ))  −   µ(J k(n) (τ ))      o(µ(J k(n) (τ ))) k(n)   = sup 1u(xs ; τ , fs ) − 1u(xs ; τ , fsk(n),J (τ ) (x, a)) + µ(J k(n) (τ )) (x,a)∈C

for each s ∈ S. k(n) Since f k(n),J (τ ) (x, a) uniformly weak-∗ converges to f , the right-hand-side uniformly converges to zero, from the assumption of Continuous Marginal Utility Density. k(n)

Combining this with the fact that {λk(n),J (τ ) (x, a)} converges to λ uniformly on C , we obtain the desired result. 

T. Hayashi / Journal of Mathematical Economics (

2.7. Limit preference Let %τ denote the preference over state-contingent allocations of commodity τ ∈ T and associated income transfers, which is defined over a sufficiently large compact set C ⊂ RS++ × RS as in the above theorem, such that the marginal rates of substitutions it induces are the limits of the marginal rate of substitutions: πs 1u(xs , τ , fs ) MRSτxs ,as (x, a) =

λs λs MRSas ,a ′ (x, a) = s λs′ πs 1u(xs , τ , fs ) MRSτxs ,x ′ (x, a) = s πs′ 1u(xs′ , τ , fs′ ) πs 1u(xs , τ , fs ) . MRSτxs ,a ′ (x, a) = s λs′ τ

By integrating back the MRS, we see that the limit preference is represented in the form

   1  xs λs 1u(zs , τ , fs )dzs + U (x, a) = as πs λ0 0 λ0 πs s∈S τ

or its arbitrary monotone transformation. Here the limit preference describes willingness to pay for a ‘‘negligible commodity’’, which is in our argument given as a ‘‘density’’ notion. In the limit, x describes state-contingent delivery of a ‘‘negligible’’ commodity, and a describes income transfer to be spent on the other commodities, which is adjusted to the size of the commodity at the general equilibrium level and hence ‘‘negligible’’. Nevertheless, in the sense of ‘‘density’’ the marginal rate of substitution between them is well-defined. Two notable observations follow. 1. The limit preference is risk neutral, not only the preference over ex-post allocations exhibits no income effect. The no income effect property follows from the marginal rate of substitution of income transfer for consumption being independent of the amount of income at every state. The risk neutrality property follows from the marginal rates of substitution between income transfers across states are constant. This is because when income effect of a commodity is small because of the smallness of commodity it also means that income transfer associated to it is small as well and we already know from Arrow’s result that smooth expected utility preference exhibits risk neutrality toward small risks (Arrow, 1971). Thus the two conditions are tied together. Note also that the risk attitude as a primitive matters of course at a general equilibrium level, and here it enters the above willingness-to-pay formula as a fixed constant. 2. The marginal rate of substitution in the limit between extra income transfers at different states of the world equals to the ratio between the Lagrange multipliers associated to those states. Therefore, the values of state-contingent income transfers may differ between states and the way how they differ between states may in general differ between consumers. 3. Effectiveness as an efficiency measure Here we examine the effectiveness of the aggregate of expected adjusted consumer surplus as an efficiency measure. Definition 2. Say that (x, a) = (xi , ai )i∈I yields larger aggregate expected adjusted consumer surplus than (y, b) = (yi , bi )i∈I if

      1  xis λis 1u(zis , τ , fs )dzis + ais πis λi0 0 λi0 πis i∈I s∈S       1  yis λis > bis πis . 1u(zis , τ , fs )dzis + λi0 0 λi0 πis i∈I s∈S

)

–

9

The result below states that the aggregate expected adjusted consumer surplus criterion is consistent with Kaldor criterion when the asset markets are complete. Proposition 1. Suppose the asset markets are complete. Then if (x, a) yields larger aggregate expected  adjusted consumer surplus than (y, b), there exists ( aiS )i∈I with i∈I  ais = i∈I ais for all s ∈ S such that (xi , ai0 , aiS ) ≻τi (yi , bi ) for all i ∈ I. It says that an option generating larger aggregate expected adjusted consumer surplus can be made Pareto superior by making suitable redistribution of ex-post income transfers. Note that the result holds independently of the distribution of beliefs, this is because in complete markets each individual’s marginal rate of substitution between extra income transfers at different states fully takes her subjective belief into account through asset allocation, and the rates are equalized across individuals. Proof. Notice that the above condition is equivalent to

   πis  i∈I

>

λi0

s∈S

 i∈I

s∈S

xis

1u(zis , τ , fs )dzis +

0

πis λi0

yis



λis ais λi0

1u(zis , τ , fs )dzis +

0



 λis bis . λi0

Without loss of generality, assume that R is an |S | × |S | matrix and invertible. Then from the first order condition we have



λi|S | λi1 ,..., λi0 λi0



= qR−1 ,

for every i ∈ I, which is common across all consumers.



When the asset markets are incomplete, the aggregate expected adjusted consumer surplus is in general inconsistent with the exante Kaldor criterion, however. Competitive equilibrium allocation is constrained efficient at partial equilibrium level in the following sense: if (fi )i∈I is a competitive equilibrium allocation given equilibrium prices p and q and asset trade ζ , then for almost all τ ∈ T , the partial equilibrium allocation (fis (τ ))i∈I solves the maximization problem

    1  xis λis 1u(zis , τ , fs )dzis − ps (τ )xis πis . max (xi )i∈I λi0 0 λi0 πis i∈I s∈S However, just like competitive equilibrium allocation is only constrained efficient at a general equilibrium level it is only constrained efficient at the partial equilibrium level as well and can be improved upon by state-contingent income transfers with ex-post budget balance. To illustrate, consider a two-state example with no production, in which the two states are equally likely and there is only one asset which yields one unit of income per one unit regardless of states. Also let T = [0, 1]. Let ωA = (ωA1 , ωA2 ) and ωB = (ωB1 , ωB2 ) denote statecontingent endowments for A and B, respectively. They are given by

ωA1 = (δ + ε)1, ωA2 = ε1 ωB1 = ε1, ωB2 = (δ + ε)1. For simplicity, both A and B have the same preference which additively separable and uniform over commodity characteristics. Thus it is represented by U (f ) =

1 2

φ



   1 v(fi1 (t ), t )dµ(t ) + φ v(fi2 (t ), t )dµ(t ) ,

T

for both i = A, B.

2

T

10

T. Hayashi / Journal of Mathematical Economics (

Without loss of generality, normalize the spot price system by



for all s = 1, 2 and set q = 1 because there is just one asset. Then, in competitive general equilibrium we have

which is seemingly consistent with the Pareto criterion because the marginal rates of substitution between state-contingent income transfers are assumed to be equalized across individuals. However, such assumption of equality cannot come from underlying preference maximization of the individuals in general when the asset markets are incomplete, even when their beliefs are identical. On the other hand, the criterion based on aggregate adjusted expected consumer surplus

p1 = p2 = 1 fA2 = ε 1

fB1 = ε 1, fB2 = (δ + ε)1 1 ′ λA1 = φ (v(δ + ε))v ′ (δ + ε), 2 1 ′ ′ 2

φ (v(ε))v (ε),

λA0 = λB0 =

1 2

λB2 =

λA2 =

1 ′ φ (v(ε))v ′ (ε) 2

1 ′ φ (v(δ + ε))v ′ (δ + ε) 2

      1  xis λis 1u(zis , τ , fs )dzis + ais πis λi0 0 λi0 πis i∈I s∈S

φ ′ (v(δ + ε))v ′ (δ + ε) + φ ′ (v(ε))v ′ (ε) ≡ λ0 . 

According to our result, consumer surplus in the statecontingent allocation problem for a single commodity and associated income transfers is given by UA (xA , aA )

 φ ′ (v(δ + ε)) φ ′ (v(δ + ε))v ′ (δ + ε) v(xA1 ) + aA1 2 λ0 λ0   1 φ ′ (v(ε)) φ ′ (v(ε))v ′ (ε) + v(xA2 ) + aA2 + aA0 2 λ0 λ0   φ ′ (v(ε))v ′ (ε) 1 φ ′ (v(ε)) v(xB1 ) + aB1 UB (xB , aB ) = 2 λ0 λ0  ′  φ ′ (v(δ + ε))v ′ (δ + ε) 1 φ (v(δ + ε)) v(xB2 ) + aB2 × 2 λ0 λ0 + aB0 . =

1



UA ((δ + ε, ε), (0, −(δ + ε), −ε))

= UB ((ε, δ + ε), (0, −ε, −(δ + ε))) =

u+u 2

,

where

 φ ′ (v(δ + ε))  v(δ + ε) − v ′ (δ + ε)(δ + ε) λ0  φ ′ (v(ε))  u= v(ε) − v ′ (ε)ε . λ0

u=

Now consider the following reallocation of state-contingent income transfers: move γ1 units of income from A to B, and move γ2 units of income from B to A, where

φ ′ (v(δ

+

λ0 (u − u) . + ε) + φ ′ (v(ε))v ′ (ε)

ε))v ′ (δ

Then the reallocation improves both parties’ expected adjusted consumer surplus, since UA ((δ + ε, ε), (0, −(δ + ε) − γ1 , −ε + γ2 ))

= UB ((ε, δ + ε), (0, −ε + γ1 , −(δ + ε) − γ2 )) u+u (u − u)(φ ′ (v(ε))v ′ (ε) − φ ′ (v(δ + ε))v ′ (δ + ε)) = + 2 φ ′ (v(δ + ε))v ′ (δ + ε) + φ ′ (v(ε))v ′ (ε) u+u > . 2

Now how should we interpret the criterion of aggregate expected consumer surplus as used in practice? It isinterpreted asasλ

λ

suming that (πi1 , . . . , πi|S | ) = (π1 , . . . , π|S | ) and λi1 , . . . , λi|S | i0 i0

is consistent with underlying preference maximization of the individuals whether the asset markets are complete or incomplete. However, when the asset markets are incomplete it is not in general consistent with the ex-ante Pareto criterion. 4. Conclusion

Then competitive partial equilibrium in the market for a single commodity delivers

γ1 = γ2 =

(π1 , . . . , π|S | ) for all i for some common (π1 , . . . , π|S | ) and pre      1  xis 1u(zis , τ , fs )dzis + ais πs , λi0 0 i∈I s∈S

T

λB1 =

–

scribing the maximization of

ps (t )dµ(t ) = 1

fA1 = (δ + ε)1,

)

=

In the current paper we have worked on the problem of Hicksian aggregation under uncertainty in an environment with asset markets and ex-post spot markets due to Radner (1968). We took the set of material characteristics of commodities as a continuum, and take the given good under consideration as an element of its partition. We have derived consumer’s indirect preference over statecontingent consumptions of the given good and state-contingent transfers of income that is to be spent on the other goods in the spot markets at each state. We considered a limit in the sense that the partition becomes arbitrarily finer and the good tends to be arbitrarily small, whereas the magnitude of income transfers is adjusted to the smallness of the good and tends to be small as well. We have shown that the limit preference exhibits risk neutrality, not only that it exhibits no income effect. This is because when income effect of a commodity is small due to the smallness in the set of commodity characteristics, it also means that uncertain income transfer associated to it is small as well and we already know that smooth expected utility preference exhibits risk neutrality toward small risks. The two conditions are therefore tied together. We have shown that the marginal rate of substitution in the limit between extra income transfers at different states of the world equals to the ratio between the Lagrange multipliers associated to those states. Therefore, the values of state-contingent income transfers may differ between states and the way how they differ between states may in general differ between consumers. When the asset markets are complete such ratios are equalized between consumers, but it is not the case in general when the asset markets are incomplete. This means that using the aggregate expected consumer surplus as used in practice will be in general inconsistent with individuals’ expected utility maximization in the general equilibrium environment. This is a negative result somehow, in the sense that when the underlying asset markets are incomplete willingness to pay is not an accurate description of how much the object is valuable for the given individual at a social level. This gap will be critical particularly when the patterns of insurability of endowments and earnings are severely different across individuals. A future research will be to find a positive result, in the sense that the aggregate expected consumer surplus is nevertheless a reasonable efficiency measure in certain sense. Also, because the uncertainty treated in this paper is about exogenous and verifiable

T. Hayashi / Journal of Mathematical Economics (

states and not quite about types which are to be private information, a natural way to think further is to look for the partial equilibrium efficiency measure which is consistent with expected utility maximization in a general equilibrium environment with rational expectation. Acknowledgments I thank Atsushi Kajii for helpful comments and suggestions. I thank seminar participants at Bath and conference participants at WEAI Pacific Rim Conference 2013, JEA Spring Meeting 2013 and AMES 2013 for helpful comments. References Arrow, K.J., 1971. Essays in the Theory of Risk Bearing. Markham Publishing Company, Chicago, IL. Arrow, K.J., Le role des valeurs boursieres pour la repartition la meilleure des risques. Econometrie, CNRS, Paris, pp. 41–47. Translated as The role of securities in the optimal allocation of risk-bearing, Rev. Econ. Stud. 31 (1964) 91–96. Debreu, G., 1959. Theory of Value: An Axiomatic Analysis of Economic Equilibrium. Yale University Press, New Haven, London.

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Geanakoplos, John, Polemarchakis, Heraklis M., 1986. Existence, regularity and constrained suboptimality of competitive allocations when the asset market is incomplete. In: Uncertainty, Information and Communication: Essays in Honor of KJ Arrow, Vol. 3. pp. 65–96. Grant, Simon, Kajii, Atsushi, Polak, Ben, 1992. Many good choice axioms: when can many-good lotteries be treated as money lotteries? J. Econom. Theory 56 (2), 313–337. Hayashi, T., 2008. A note on small income effects. J. Econom. Theory 139, 360–379. Hayashi, T., 2013. Smallness of a commodity and partial equilibrium analysis. J. Econom. Theory 148, 279–305. Magill, Michael J.P., Quinzii, Martine, 2002. Theory of Incomplete Markets: Vol. 1. The MIT Press. Marshall, A., 1920. Principle of Economics. Macmillan, London. Massell, B.F., 1969. Price stabilization and welfare. Quart. J. Econ. 83 (2), 284–298. Radner, R., 1968. Competitive equilibrium under uncertainty. Econometrica 36 (1), 31–58. Rogerson, W.P., 1980. Aggregate expected consumer surplus as a welfare index with an application to price stabilization. Econometrica 48 (2), 423–436. Schlee, E., 2008. Expected consumer’s surplus as an approximate welfare measure. Econom. Theory 34 (1), 127–155. Schlee, E., 2012. Surplus maximization and optimality. Working Paper. Turnovsky, S.J., Shalit, H., Schmitz, A., 1980. Consumer’s surplus, price instability, and consumer welfare. Econometrica 48 (1), 135–152. Vives, X., 1987. Small income effects: a Marshallian theory of consumer surplus and downward sloping demand. Rev. Econom. Stud. 54 (1), 87–103. Waugh, F.V., 1944. Does the consumer benefit from price instability? Quart. J. Econ. 58, 602–614.