Ex-Post Efficiency and Resource Allocation under Uncertainty

The Review of Economic Studies, Ltd.

Ex-Post Efficiency and Resource Allocation under UncertaintyAuthor(s): Richard HarrisSource: The Review of Economic Studies, Vol. 45, No. 3 (Oct., 1978), pp. 427-436Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2297245 .

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Ex-Post Efficiency and Resource

Allocation Under Uncertainty RICHARD HARRIS

Queen's University, Kingston

1. INTRODUCTION

Ross Starr (1973) has recently pointed out that under uncertainty about the future an ex-ante optimal allocation of resources over time or an " optimal allocation of risk bearing ", following the terminology of Arrow (1964), does not necessarily imply an ex-post optimal allocation of resources over time. Starr's general result is that an ex-ante optimum will not be an ex-post optimum unless individuals' subjective probability distributions over states of nature are identical.'

The conflict between ex-post and ex-ante Pareto efficiency of intertemporal resource allocation under uncertainty is an example of the problems caused by changing tastes. The problem has serious implications for making welfare judgements, as there may well be a divergence between ex-ante choice and ex-post preference. This casts doubt on the validity of the principle of consumer sovereignty as a means of evaluating resource allocations.

Suppose it has been decided by " government " or " society " that the desirability of alternative resource allocations, when there is uncertainty about the future, is to be judged in the Pareto sense relative to the ex-post preferences of the consumers in the economy. Note that this is an explicit value judgement on the part of " society " in that they respect individual preferences, but only ex-post preferences. Some economists have expressed the opinion that it is only these preferences which should " count ". Starr states that " the achievement of an Arrow (ex-ante) optimum is a normative dead end. After all, we are not so much interested in expectations as in results ".'2 We do not propose to discuss whether ex-ante or ex-post optimality is to be ethically preferred, but rather to examine the logical implications of accepting the ex-post view. Even if welfare judgements are made from an ex-post viewpoint, any actual allocation of resources will continue to be made ex ante. Furthermore, individuals may make ex-ante choices with respect to ex-ante preferences. We will assume in particular that these choices are made in accordance with the Von Neumann-Morgenstern-Savage axioms of behaviour under uncertainty, and that individuals' subjective probabilities are not necessarily the same. Two questions then arise. Can an ex-ante decentralized resource allocation mechanism be found, such that when ex-ante choices are made within the context of this mechanism, all equilibrium allocations are ex-post Pareto efficient? Secondly, given an ex-post efficient allocation, can an ex-ante decentralized resource allocation mechanism be found which attains that allocation as an equilibrium? It should be evident that these are natural analogues to the usual welfare questions which use ex-ante Pareto efficiency as the welfare criterion.

In this paper we propose a resource allocation mechanism which is " market like" as consumers use prices to guide decision making. Arrow-Debreu markets have the feature that ex-ante marginal rates of substitution are equated as all consumers face the same set of prices. When a state of the world is realized, however, the actual intertemporal marginal rates of substitution will differ across individuals unless all individuals have the same ex-ante beliefs. The resource allocation mechanism proposed in this paper corrects for these ex-post divergences among marginal rates of substitution by quoting to each

427



428 REVIEW OF ECONOMIC STUDIES

individual ex ante a different set of prices. It turns out that each individual's personal prices for delivery of contingent commodities in any one state are proportional to a common price system, and the factor of proportionality is precisely equal to the subjective probability that the individual attaches to that state.

When evaluating resource allocations on an ex-post basis there is more than one ex- post efficiency criterion which one might adopt. In this paper we examine three alternative ex-post efficiency criteria and evaluate the proposed resource allocation mechanism in the light of these criteria. The resource allocation mechanism suggested does not do equally well for all three ex-post efficiency criteria and some examples are given to illustrate the problems which occur.

The rest of the paper proceeds as follows. In Section 2, we present the notation used, a description of the economy and various definitions of efficiency. Section 3 presents a discussion of the proposed resource allocation mechanism and a statement and proof of a theorem on the optimality of equilibrium of this mechanism for exchange economies. In Section 4, a basic decentralization theorem on ex-post optimal allocations is presented. Finally, Section 5 provides a discussion of production and the limitations of ex-post efficiency criteria.

2. THE ECONOMY AND CONCEPTS OF EX-POST EFFICIENCY

The basic issues of ex-post efficiency can be illustrated quite readily in the context of a two- period state preference model. All of the following results could be extended to a many- period model as in Debreu (1959). In period 1 there are 1 physical commodities, where x E RI denotes the period 1 commodity vector. In period 2 there are n possible states of nature which may be realized from the state space Q = {S1, .S*, sn}. In each state sk of Q, k E N _ {1, ..., n}, which we shall refer to henceforth as state k, there are 1 physical commodities, where Yk e RI denotes the state k vector of 1 commodities. The economy consists of I consumers indexed i = 1, ..., L We shall also use I to denote the index set {1, ..., I}.

An allocation in the economy is a list of consumption vectors c _ (c%)c61, one for each consumer, such that for each i, c' e Rm, where m = l(n +1), and each consumer's consumption set is taken to be the non-negative orthant of m-dimensional Euclidean space. An allocation (C')iei is feasible provided that

zieI C < C,

where c is a strictly positive aggregate endowment vector. Consumers have two sets of preferences. The ex-ante preferences of an agent over

consumption bundles c' = (x', yi, ..., yn) E R+, are given by

(C)= Ek 6NkV (x Yk). ....(2.1)

Hence, we assume each agent's ex-ante preferences can be scaled in terms of subjective probabilities, {ik}, and state-dependent Von Neumann-Morgenstern utilities { Vik( . Furthermore we assume that each Von Neumann-Morgenstern utility function is inter- temporally additively separable.3 Hence

Vik(xi, yi) Ui(xi) + Uik(yi), i E I, k E N. ...(2.2)

We assume all Ui and Uik are concave, continuous and strictly monotone increasing. Thus all consumers are risk averse. Furthermore, we assume that the ex-ante beliefs of agents, 7r= ( 7i, ..., it") are such that all states of nature are given positive probability by all agents; together with strict monotonicity of the Uik this ensures all agents are non-satiated in all states of nature, ex ante.



HARRIS ALLOCATION UNDER UNCERTAINTY 429

For a state of nature k, an agent's ex-post k-preferences are represented by the utility function

Ui(xi) + Uik(y), (2.3)

where (2.3) is the appropriate kth utility function from the ex-ante utility function (2.1). This assumption implies that if state k occurs, the agent has the same tastes he had ex ante with respect to period 1 and state k consumption, but clearly by experience his beliefs (ex post) have been altered. We turn now to consider various efficiency concepts.4

An allocation, c*, is ex-ante efficient, if c* is feasible, and there exists no other allocation c, such that

(a) c is feasible

(b) V'(c') > VL(cL*) for all i E I,

(c) Vi(ci) > Vi(ci*) for some i E L . . .(2.4)

An allocation c* is ex-post efficient for state k, or ex-post k-efficient, if c* is feasible, and there exists no other allocation c, such that

(a) c is feasible,

(b) ui(xi) + Uik(yD) > ui(x) + Uik(yi*) for all i E I,

(c) Ui(xi) + uik(yi) > Ui(xi*) + Uik(yi) for some i E I,

where (x, yi) and (xi*, yi*) are the appropriate components of cl and ci* respectively. In what follows we shall require a stronger notion of ex-post efficiency-one which is

independent of which particular state occurs. An allocation c* is universally ex-post efficient if c* is ex-post k-efficient for all states

k = 1, ..., n. It is straightforward to show using the intertemporal additivity of the ex-post utility functions and the absence of production that the set of universally ex-post efficient allocation is non-empty. (The proof is an adaptation of Theorem 6.2.1. of Debreu (1959) and is omitted for the sake of brevity.)

Universal ex-post efficiency is not the only possible type of ex-post efficiency criterion one might be interested in. Two other ex-post efficiency criteria we shall be concerned with are as follows.

A feasible allocation c* is ex-post efficient if there is no alternative feasible allocation c which, in terms of all ex-post utility functions in all states of the world is Pareto superior, i.e.:

Ui(x) + Uik(y) > Ui(xi*) + Uik(yi*) for all i E I, all k E N, and

Uj(xi) + Uis(yi) > Uj(xj*) + Uis(yj*) for some j E I, s E N.

Clearly any universally ex-post efficient allocation is ex-post efficient but the converse is not true. A simple example with one physical good, two consumers and two states will readily illustrate this (see example 4.2 below).

Starr proved that if all agents have identical positive probability beliefs then an ex-ante efficient allocation is universally ex-post efficient. This suggests the following ex-post criterion.

A feasible allocation c* is expected ex-post efficient with respect to the set of beliefs 1[ = (ar1, .., r) if and only if it is ex-ante efficient in terms of the expected utility functions

kNXkV(X, yk) all i E I.

Starr's theorem can thus be re-phrased as saying that any allocation which is expected ex-post efficient is universally ex-post efficient.




3. PRICES AND EX-POST EFFICIENCY

In this section we consider a resource allocation mechanism which operates ex ante, and has the property that the equilibrium allocations are ex-post efficient, universally ex-post efficient and expected ex-post efficient. The mechanism is " competitive-like " in the sense that prices are used to guide individual decision making. We shall refer to this mechanism as the Personalized Price Mechanism or PPM for short.

The economy consists of (n +1)l ex-ante markets operated at date 1. There are I markets for commodities currently deliverable and nl contingent-commodity markets for I commodities deliverable contingent on the occurrence of a state of nature k, k = 1, ..., n. All contracts made ex ante are binding, no ex-post revision is admissible nor will markets be re-opened in the second period. We shall assume all agents make choices in these markets with respect to their ex-ante preferences given by (2.1) and (2.2).

The key factor in the operation of these markets is that all prices are individual specific. In this sense the mechanism is similar to the Lindahl mechanism for allocating public goods. A price system is a vector p e Rm\{0}, one price for each market. A personal price system for the ith individual is a vector p' = (q, ai p1 ., nnpn), where p (q, Pi, ..., Pn) is the market price system. Thus for all individuals the contingent commodity prices, (Pi, ..., p), are modified by their subjective probabilities, (ir), i E I, in a multiplicative fashion. Compared to Lindahl prices, these " personal prices " are very special, since the relative prices of two goods to be delivered in the same state of the world are the same for all persons.

If consumer i has an income m' his demands at market prices p are given as the solution to

max V'(ci) subject to p'c' < mi.

For an equilibrium of the PPM it must be the case that income is distributed such that consumers can purchase their equilibrium demands. The following definition is a straightforward generalization of the concept of a competitive allocation.5

An allocation (ci*)ieI is competitive relative to the personalized price system (pi) i,1 if

(a) zi e -* = C

(b) Vi(ci*) > V'(c') for all c' e R' such that p'c' < pici*

(c) pi = (q, ni pi, ..., inpn) for all i e I, where p e Rm\{Q_.

At this stage it will be convenient to state a lemma the proof of which is provided in an appendix.

Lemma 3.1. Suppose that i, (9k) is a solution to the problem

max U(x) + keN akUk(Yk) subject to qx + rkeN fNkPkYk < m.

Then, provided that ak>O (all k), fik 2 0 (all k) and Yk > 0 (all k) it follows that , (9Pk) is also a solution to the problem

max U(x) + Sk e N Yk Uk(Yk) subject to qx + Sk e N 'kPkYk _ # + 2k e- N 'kPkPk

where Ak -lkYkl/ak

An immediate implication of the lemma is the following.

Corollary 3.1. If X, (9k) is the solution to

max U'(x') + Sk NrkU(yk) subject to qx' +J k e N7kPkYk ? m (all i)




then x, 9, is the solution to

max Ui(xi) + Uik(yk) subject to qx +Pkyk < q50 + PkA

for all k e N and all i - L

Proof Take cxj = j-= 7i all j E N in Lemma 3.1 and yj = 1 if j = k, yj = 0 otherwise. Then Aj = 1 if j= k and )j = 0 otherwise.

We now show

Theorem 3.1. An allocation (ci*)ife which is competitive relative to a personalized price system (p')ije is universally ex-post efficient.

Proof By Corollary 3.1 (xi*, yk*) is competitive at prices (q, Pk) for all i E I. Thus by the first theorem of welfare economics6 the allocation (xi*, yi*)i I is Pareto efficient with respect to the utility functions Vik( ), i E L As this is true for all k E N the allocation (c'*)ici is universally ex-post efficient. 11

Thus a personalized price mechanism yields universally ex-post efficient allocations, and hence ex-post efficient allocations.

Theorem 3.2. An allocation (cl )ieI which is competitive relative to a personalized price system (p')icI is expected ex-post efficient with respect to any set of probabilities z = (ir1, gn., rca).

Proof Using Lemma 3.1 we take Ock = 13k = 7iri (all k) and Yk = 7rk. Thus fJkykI/k = 7rk

which implies the allocation (c'*)ici is competitive at prices (q, rplP, ..., ir,pn) for the utility functions Ui(xi)+SkeN lkU(yk). The theorem follows by the first theorem of welfare economic. 11

4. EX-POST EFFICIENCY AND SUPPORTING PRICES

In the last section we presented a resource allocation mechanism for an exchange economy whose equilibrium states were universally ex-post efficient, ex-post efficient and expected ex-post efficient. In this section we pose the traditional converse of this question. Can an ex-post efficient allocation be sustained as an equilibrium of an ex-ante resource allocation mechanism?' In particular can it be sustained as an equilibrium of the personalized price mechanism? As it turns out the answer depends upon the efficiency criterion adopted.

The following lemma will be useful in what follows.

Lemma 4.1. Suppose (c'*)fic is a universally ex-post efficient allocation and,

(a) U'(x') is a continuously differentiable function on the interior of R' for all i E I,

(b) xi*Q>>i for all i EI L

Then (ci*)icI is a weakly expected ex-post efficient allocation with respect to any probability distribution rt = ..., isn) satisfying 2tk > 0 for all k E N.

Proof. Let k 0 denote the first-period and Ak the utility possibility set for state k=0,...,n. Thatis

Ak {(u1, ..., UI) e RI J u < Uik(yik) (all i E I) and (ya)i,I feasible}.

By the properties of the utility functions Ak are closed convex sets in RI. As (ci*)iei is universally ex-post efficient the corresponding utility vectors (ui*+u*)ieI, where ui*-Ui(xi*) and ui* Uik(y), are efficient points in the convex sets AO +Ak, all k e N. Thus by the supporting hyperplane theorem there exists a non-zero, non-negative vector wk E RI such that

Wk (UO +uk) < wk (u* +u*) for all u0 e Ao, uk E Ak (4.1)




where u =- (ui*)j and u* = (uki)i 1. As this is true for all k E N, (4.1) implies that

Wk uo<wk u* for all uo EAo,allkcEN. ...(4.2) The hypotheses (a) and (b) of the lemma imply that at u*, Ao has a differentiable boundary and consequently from (4.2) the vectors wk can all be taken to be equal to a common vector w.

Consider now the utility possibility set for utility functions U'(xi) + XkENikU (yk)

where 7r- = (71, ..., 7Tr) is any set of positive probability beliefs. Clearly this is given by the set

B = Ao +YkeNitkAk-

From (4.1) it follows that w -v w v* for all v E B,

where v *= Uo+ 2keN7kUk* is the expected utility allocation corresponding to the probability beliefs it. As the vector w is non-negative and non-zero the utility allocation v* must be weakly Pareto efficient8 in B. 11

Before proceeding we need one further corollary of Lemma 3.1.

Corollary 4.1. If ii, (9k) is a solution to

max U'(x') +keNitkUk(A) subject to qx $Ske NPkks c m'

where 7Ck>O, all k E N, then ii, (9k) is a solution to

max Ui(x')+2keN7kUk(Yk) subject to qx' +1ke N(7tkitk)PkYk ? q5i +YXk.N(itkitk)PkSki

Proof. In Lemma 3.1 take Gxk = itk, fJk =1 and Yk = Xit. Then fkykIak = zikltk 1

We now introduce the standard minimum wealth or " cheaper point" assumption needed to exclude the exceptional " Arrow" case.

(A. 1) Given a price vector p # 0 and a consumption vector t' there exists ci E Rm+ such that pc <pY'.

Theorem 4.1. If (ci*)icI is a universally ex-post efficient allocation such that conditions (a) and (b) of Lemma 4.1 hold, and each ci* satisfies (A. 1), then there exists a price system p --A 0 such that the allocation is competitive relative to the personalized price system (P')i e

Proof. By Lemma 4.1 (ci*)i61 is weakly expected ex-post Pareto efficient with respect to any positive probability distribution it. Using assumption (A.1) by the second theorem of welfare economics there exist prices q, (Pk)keN, not all zero, such that (Ci*)i6N is competitive at these prices for the utility functions Ui(Xi)+1keNiNkUk(Yk). But from Corollary 4.1 it then follows that (ci*)iel is competitive relative to the personalized price system (p)i)., wherep' = (q, it1p1/i1, ..., 7trp./n.) and the price systemp=-(q,p1/l,.p ./n/,).

Theorem 4.2. If (ci*)ie-I is an expected ex-post efficient allocation with respect to some probability beliefs it, itk > 0, all k E N, and assumption (A. 1) holds, then there exists a price system p # Q such that the allocation is competitive relative to the personalized price system (p')i-'.

Proof. Follows immediately from the second theorem of welfare economics and Corollary 4.1. 11

It is worth while considering whether the regularity conditions (a) and (b) of Lemma 4.1 could be dispensed with. We shall argue that some such condition must be invoked, for otherwise it is possible to have an allocation which is universally ex-post efficient but not weakly expected ex-post efficient for any positive probability beliefs, as the following example demonstrates.




Example 4.1. Consider an economy with two consumers, a and b, and two states in the second period, I and 2. In the first period there are two goods, and each consumer desires only one of the goods which is not desired by the other consumer. The resulting utility possibility set is illustrated in Figure 1. In the second period there is only one good in each state, with a fixed endowment of 1 unit. Men a and b have utility functions in each state given by

ua(c') = 2c' ub(c') = Cb

ua(ca) = Ca ub(cb) = 2cb

The resulting utility possibility sets are illustrated in Figure 1.

ua Ul a Ua

ug~~~~~ LJ

ub ub U

PERIOD I STATE I STATE 2

FIGURE 1

Now consider the utility allocation corresponding to points A, B and C in Figure 1. Clearly this allocation is universally ex-post efficient; but note that it cannot be weakly expected ex-post efficient for any set of positive probabilities on both states. Standard comparative advantage arguments imply that a necessary condition for an allocation to be expected ex-post efficient is that at least one state specialize in providing utility to one man. For the allocations B and C this is not the case. Thus A, B and C cannot correspond to weakly expected ex-post efficient allocations.

If there are allocations which are universally ex-post efficient but not expected ex-post efficient for positive probabilities then they cannot be sustained as personalized price equilibrium. The argument is simple. Suppose they could. Then such allocations would be expected ex-post efficient by Theorem 3.2, providing a contradiction. Thus while all expected ex-post efficient allocations with positive probabilities can, subject to the usual qualifications of (A.1), be sustained as personalized price equilibrium with the appropriate redistribution the same is not true of universally ex-post efficient allocations.

Do there exist supporting prices for ex-post efficient allocations? Again, problems similar to those with universal ex-post efficiency arise. If an allocation is ex-post efficient but not expected ex-post efficient then by the same argument used above it cannot be sustained as a personalized price equilibrium. In this case however the problem is pervasive. Even for the most well-behaved economies there is a continuum of ex-post efficient allocations which are not expected ex-post efficient nor universally ex-post efficient, as the following example demonstrates.

Example 4.2. The example has two consumers with two states in the second period and one good in each state. The economy can be represented in terms of two parallel Edgeworth boxes illustrated in Figure 2.

Any feasible allocation must have the same vertical co-ordinate as first period consumption must be the same independent of which state occurs. The indifference curves in each box represent each man's ex-post preferences in that state. Clearly the allocation corresponding to AB is universally ex-post efficient and expected ex-post efficient for any




man b man b

E A _ A F _

PERIOD I C

man a STATE I man a STATE2

FIGURE 2

set of positive probabilities. The allocation CB is neither universally ex-post efficient nor expected ex-post efficient, but it is ex-post efficient; any reallocation with respect to state 1 utilities which yields Pareto improvements necessarily implies that man a in state 2 is made worse off. In this example any allocation on the line EF in state 1, together with B in state 2 is ex-post efficient, but of these only one, AB, is universally ex-post efficient and expected ex-post efficient.

5. PRODUCTION

The extension of the results of the previous sections to an economy with production presents some difficulties.9 If production allows trade offs between time periods or between states then universally ex-post efficient allocations will not exist. This is because increases in utilities in one state can be had at the expense of a decrease in utilities in another state. Equilibrium of the PPM with production, where production decisions are made with respect to the market prices p, can be shown to be ex-post efficient. As shown in the last section, however, there are many ex-post efficient allocations which cannot be sustained as personal price equilibria and the introduction of production does not change this.

Expected ex-post efficiency presents no difficulties however. The generalizations of Theorems 3.2 and 4.2 to include convex production possibilities are straightforward and omitted for the sake of brevity. The procedure, given a price system (q, Pl, ..., pn), is to have producers maximize profits,

qZO+k NPkZk subject to (z0, z1, .., zn) eZ

where Z is a convex production set. The main result of interest is the relationship between producer and consumer price systems. If expected ex-post efficiency is with respect to a set of positive probability beliefs 7t, Jk >0, k e N, and producer prices are given by (q, Pm' ..., Pn) then consumers' personalized prices are given bypi = (q, ni I1P 1 ..., * jitnln1Pn) for all i E L If one thinks of n as being a set of social beliefs, then the ith consumer's prices in state k are computed by multiplying producer prices by the ratio of the personal probability individual i attaches to state k to the social probability of state k occurring. Note that as in the Arrow-Debreu model producers are neither required to take attitudes either towards risk nor to form probability judgements as to the likelihood of various states of nature occurring.

APPENDIX

In this appendix we provide a proof of Lemma 3.1. I am grateful to Peter Hammond who suggested this Lemma as a means of proving Corollaries 3.1 and 4.1.

Lemma 3.1. Suppose that i, (Pk) keN is a solution to the problem

max U(x) + Sk E Nck Uk(yk) subject to qx + Sk E NfkPkyk ? m




Thlen, provided Xkk>O (all k), /Jk > 0 (all k) and Yk _ (all k) it follows that i, (Pk)kkeN is also a solution to the problem

max U(x) + keNYk Uk(Yk) subject to qx + Yk e NAkPkYk _ qX + Ek G NIkPk&k

where 4 -lkVk/ak

Proof. Suppose x', (yk) satisfies the second budget constraint, but

U(X') + Ek e NYk Uk(Yk) > U(M) + Xk e NYk Uk(&k)

Choose x = (x'/2X) + (&i(2AZ- 1)/2A) and Yk = Pk + ((yk/ck)(Yk'-k)/22A) for all k e N, where I ? 1 is chosen such that 7k/2Axck ? 1 for all k e N. Then

qx + -k e NPkPkYk = qX + -k . N kPkYk + (q(x' -x)/2A) + (Yk e N4kPk(Yk -k)/2') < m But, using the concavity of the utility functions

U(X) + Ik e NLkUk(Yk) - U(*) - 6NYkUk(Yk) ? ({ U(X') k-U(Y)}/2A) + (Sk e NYk{ Uk(Yk)

which yields a contradiction. - Uk(Pk)}2A)>0.

First version received July 1976; final version accepted November 1977 (Eds.). The author is indebted to Richard Arnott, Frank Flatters, James MacKinnon and Nancy Olewiler

for discussions on the problem. In particular the author wishes to acknowledge the comments of Peter Hammond and an anonymous referee which lead to a greatly improved version of the paper. All remaining errors are the responsibility of the author.

NOTES 1. Actually Starr shows that if individual utility functions are not differentiable then it is possible to have

ex-post efficiency without identical probability beliefs-he requires instead that individuals' probabilities be what he terms universally similar. See Starr (1973, p. 85).

2. Starr (1973, p. 82). In addition to Starr, Dr6ze (1970) and Guesnerie and de Montbrial (1974) have expressed doubts about the appropriate efficiency concept for resource allocation under uncertainty.

3. The intertemporal additive separability appears to be crucial to most of the results in Sections 3 and 4. The efficiency concepts of this section, however, are well defined for any Von Neumann-Morgenstern utility function.

4. Hammond (1976) introduces the concept of expected ex-post social welfare optima. While different from the efficiency concepts introduced here it is closely related. Expected ex-post social welfare optima correspond to the maxima of a social welfare function

W = Yke N7Tk WI(Vlk, . . . VI)

where Wk(.) is the ex-post social welfare function for state k and is a function solely of individual ex-post utilities in state k. Note that Wdoes not depend on individuals' probability beliefs and that social attitudes towards risk bearing will be affected by the form of the Wk functions.

5. A competitive allocation is analogous to Debreu's term of an " equilibrium relative to a price system ". See Debreu (1959, p. 93).

6. The first and second theorems of welfare economics correspond to Theorems 6.3 and 6.4 respectively of Debreu (1959, pp. 94-96).

7. Hammond (1976) demonstrates that expected ex-post social welfare optima can be attained through a spot market structure, one market for each state in each period, with appropriate lump-sum transfers, provided consumers are not allowed to make any trades either between periods or across states. This last qualification is quite important as these trades will in general reflect individuals' beliefs and attitudes towards risk. Since we are concerned with ex-post efficiency the allocation mechanism need only negate the effect of individuals' beliefs on their trades.

8. An allocation is weakly Pareto efficient if there exists no alternative feasible allocation which makes everyone better off.

9. Starr (1973) proves that if production allowed no trade-offs among either states or time periods then provided individuals' subjective probabilities are alike an ex-ante optimum is universally ex-post efficient.

REFERENCES ARROW, K. J. (1964), " The Role of Securities in the Optimal Allocation of Risk-Bearing ", Review of

Economic Studies, 31, 91-96. DEBREU, G. (1959) The Theory of Value (New York: Wiley).




DRtZE, J. (1970), " Market Allocation Under Uncertainty ", European Economic Review, 2, 133-165. GUESNERIE, R. and DE MONTBRIAL, T. (1974) ," Allocation Under Uncertainty: A Survey ", in

J. Dr6ze (ed.) Allocation Under Uncertainty (New York: MacMillan). HAMMOND, P. J. (1976), " Ex-Ante and Ex-Post Welfare Optimality Under Uncertainty " (Essex Dis-

cussion Paper No. 83, revised). STARR, R. (1973), " Optimal Production and Allocation Under Uncertainty ", Quarterly Journal of

Economics, 87, 81-95.



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Ex-Post Efficiency and Resource Allocation under Uncertainty

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