The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Optimality and Efficiency

  • Peter Newman
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_1594

Abstract

An exchange economy consists of a group of people, each of whom has preferences concerning what commodities he or she likes, and initial holdings of the various commodities available. Operating under whatever institutional rules permit freedom of contract, the society redistributes the initial holdings among itself so as to achieve a distribution that is in some sense a solution to the exchange problem.

Keywords

Allais, M. Competitive equilibrium Completness Convexity Core Compensated core Compensated equilibrium Efficiency Optimality and efficiency Pareto efficiency Quasi-equilibrium Transitivity 

JEL Classifications

D0 

An exchange economy consists of a group of people, each of whom has preferences concerning what commodities he or she likes, and initial holdings of the various commodities available. Operating under whatever institutional rules permit freedom of contract, the society redistributes the initial holdings among itself so as to achieve a distribution that is in some sense a solution to the exchange problem.

But in what sense? Over the years three common meanings of solution have emerged, each with ever greater clarity. In order of increasing structural content rather than historical origin they are: (a) optimality in the sense of Edgeworth (1881) and Pareto (1909), or for brevity EP-optimality; (b) core solutions, which originated wholly with Edgeworth (1881) but had to wait until the advent of game theory before they were properly understood; and (c) competitive equilibria, which owe most to Walras (1874). Diverse as they are these three concepts are linked by a common thread, that each agent’s objective is to seek the greatest satisfaction possible within the constraints that bind him.

If the roles of objectives and constraints are interchanged in (a), (b) and (c) we obtain three new concepts of solution, which are in effect mirror images of the earlier ideas. Thus corresponding to EP-optimality there is (a)′ efficiency in the sense of Allais (1943, pp. 610–16, 637–44) and Scitovsky (1942), or in brief AS-efficiency. Corresponding to the core is the idea (b)′ of a compensated core (for which see below), and corresponding to competitive equilibria is the concept (c)′ of compensated equilibria due to Arrow and Hahn (1971, p. 108), although the closely related quasi-equilibria were defined earlier by Debreu (1962).

(Curiously enough, the passage in Scitovsky (1941–2) that gives his definition of (a)′ was omitted from the reprinted version in his collected essays (1964). Very clear accounts of his approach may however be found in Samuelson (1956) and Graaff (1957), while Allais has published many further elaborations of his ideas, for example in Allais (1978). Those ideas were clearly at work in the pioneering paper by Debreu (1951), where he used them to overcome the problem ‘that no meaningful metrics exists in the satisfaction space, [that is, that utility is not cardinally measurable]’ (1951, p. 273). Later, Debreu’s proof of the Second Fundamental Theorem of Welfare Economics (the ‘pricing-out’ of EP-optima) in Chap. 6 of (1959) also depended quite explicitly on the use of AS-efficiency.)

The interrelations between competitive and compensated equilibria are well recognized (see for example cost minimization and utility maximization) and concern such matters as the existence of locally cheaper points, which in turn necessarily imply market valuations of commodity bundles. The analogous interrelations between EP-optimality and AS-efficiency, and between cores and compensated cores, do not involve market phenomena and perhaps as a consequence are not so well known.

Preliminaries

We need appropriate language and notation, and some general assumptions. The exchange economy consists of m agents, indexed by h, and n goods, indexed by i. Each agent has a preference relation ≾k that is defined over some subset of the non-negative orthant of Rn and whose meaning is ‘at least as good as’. It is assumed to be composed of two disjoint sub-relations, strict preferences ≻k and indifference ~k. Completeness and convexity of preferences are never assumed, and only partial transitivity is required, in the sense that the two cases \( \left({z}_k^1 \succ_k{z}_k^2 \mathrm{and} {z}_k^2 \succ_k{z}_k^3\right) \) and \( \left({z}_k^1 \succ_k{z}_k^2\ \mathrm{and}\ {z}_k^2 \sim_k{z}_k^3\right) \) are each assumed to lead to the conclusion \( {z}_k^1 \succ_k{z}_k^3 \). In particular, ~k need not be transitive.

A distribution (or allocation) Z is any m × n matrix of the individual holdings zhi, so that Z0 is the distribution of initial holdings \( {z}_{hi}^0 \) or the endowment. If two distributions Z1 and Z2 are such that \( {z}_k^1 \succ_k{z}_k^2 \) for every agent h, then we write Z1 ≻≻ Z2 and say that Zi is better than Z2. Similarly, Z1 ≿ Z2 means that \( {z}_k^1 \succ_k{z}_k^2 \) for every h; we say that Z1meets Z2. If Z1 meets Z2 and the number of agents k for whom \( {z}_k^1 \succ_k{z}_k^2 \) is at least 1 and not m, then we write Z1 ≻ Z2.

An agent’s holdings are written zh = (zh1, zh2,…, zhn). The symbolic expression ΣZ means the commodity vector z = Σzh (all summations here are over the index h for agents). In particular, ΣZ0 = z0, the vector of total endowments of each good; O is the vector with zero amounts of every good. Z1 ≺≺ Z2 means (z2z1) ≺≺ O, that is, ΣZ1 is less in every component (good) than ΣZ2; Z1 is then less than Z2. Similarly, Z1Z2 means that ΣZ1 is not greater than ΣZ2 in any component. If ΣZ1 ≤ ΣZ2 and the number of goods j for which \( \sum {z}_{hj}^1=\sum {z}_{hj}^2 \) is at least 1 and not n, we write Z1 < Z2.

The notation Z1 ≻≻ Z2 means (z1z2) ≺≺ O. In an exchange economy it is natural to assume z0 ≻≻ O. A distribution Z is feasible if ΣZ = z0, the use of = here rather than ≤ implying that free disposal is not assumed; quantities have to be conserved during exchange.

The assumptions made in this section will be maintained throughout.

EP-Optimality and AS-Efficiency

Purely for simplicity the definition of EP-optimality given by Arrow–Hahn (1971, p. 91) is used here, generalized to allow for incompleteness of preferences but specialized to an exchange economy. For compactness, that exchange economy will always be denoted E(≿k, Z0).

Definition 1 (D1)

A distribution Z1 is EP-optimal for E(≿k, Z0) if: (a) it is feasible; and (b) there is no other feasible distribution Z that is better than Z1.

Notice that D1 depends only on the totals z0 and not on their distribution Z0. Applying a weaker meaning of being better, namely: that ZZ1, produces a smaller set of allocations that can withstand such tests, the strongly EP-optimal allocations. Proofs of the interrelations between this more usual type of EP-optimality and the strong AS-efficiency defined below are similar to those given here, but with more complication, much as the theory of non-negative matrices is basically similar to but more complicated than the theory of positive matrices.

The following definition of AS-efficiency is implicit in the original works of Allais and Scitovsky.

Definition 2 (D2)

A distribution Z2 is AS-efficient for E(≿k, Z0) if: (c) it is feasible; and (d) there is no other distribution Z which meets Z2 and is less than Z0.

Again, D2 depends only on z0 and not on Z0, since ‘less than Z0’ actually involves only z0. As before, a weaker meaning of being less, namely: that ZZ2, produces a smaller set of allocations that can withstand such tests, the strongly AS-efficient allocations.

The first special assumption asserts a kind of monotonicity of preference for the society considered as a whole.

Assumption 1 (A1)

For any Z and any commodity vector s ≻≻ O there exists Zs such that ΣZs = ΣZ + s and Zs ≻≻ Z.

Theorem 1

Assume A1. If Z1 is EP-optimal then it is AS-efficient.

Proof

This and all other proofs are by contraposition. If Z1 is not AS-efficient, there exists Z such that ZZ1 and ΣZ ≺≺ Z0. So there is a vector of surpluses in every commodity, i.e., s =(z0 − ΣZ) ≻≻ O. Hence from A1 there is Zs such that ΣZs = ΣZ + s = z0 and Zs ≻≻ Z. But then Zs ≻≻ ZZ1 implies Zs ≻≻ Z1, and Zs is feasible. So Z1 is not EP-optimal.

The second special assumption does not involve the topology of Rn but nevertheless plays the role of a continuity condition on preferences.

Assumption 2 (A2)

For any agent h, \( {z}_h^1\;{\succ}_n{z}_h^2 \) implies the existence of μh ∈ (0, 1) such that \( \lambda {z}_k^1 \pm_k{z}_k^2 \) for all λ ∈ [μh, 1).

Theorem 2

Assume A2. If Z2 is AS-efficient then it is EP-optimal

Proof

If not, there exists Z such that ΣZ = z0 and Z ≻≻ Z2. From A2 there exists for each zh in Z some μh ∈ (0, 1) such that \( \lambda {z}_k \succcurlyeq_k{z}_k^2 \) for all λ ∈ [μh, 1). Put μ equal to the maximum of these μh, so that μ < 1 and write μZ for the m × n matrix of the μzh. Then μZZ2. But by construction and the fact that z0 ≻≻ O, ΣμZ = μΣZ ≺≺ ΣZ = z0. So Z2 is not AS-efficient.

Cores and Compensated Cores

The language and notation of section “Preliminaries” need modification to cope with cores. A coalition C is any non-empty subset of the m agents in the economy, and |C| denotes its cardinality, so that 1 ≤ |C| ≤ m. A distribution over C is the |C| × n matrix Zc whose rows are the n-vectors zk for kC. The notation ΣZc means the sum over the |C| rows of Zc, and for any Z0 and any C we write \( {z}_c^0=\Sigma {Z}_c^0 \), the total endowments available to the coalition C. Given any distribution Z for the whole economy and any coalition C, the C-section of Z is the distribution Zc over C.

The notion and language of section “Preliminaries” for preferential and quantitative relations between distributions will be applied freely to C-sections. But rather than writing \( {Z}_c^1\succ \succ_c{Z}_c^3 \) and \( {Z}_0^2\le {\;}_c{Z}_c^4 \) etc., the simpler notation \( {Z}_c^1\succ \succ {Z}_c^3 \) and \( {Z}_0^2\le {Z}_c^4 \) will be used.

Just as in section “EP-Optimality and AS-Efficiency,” stronger concepts of core and compensated core could be defined and corresponding results proved for them; but that is not done here.

Definition 3 (D3)

A distribution Z1 is in the core of E(≿k, Z0) if: (i) it is feasible; and (ii) there is no coalition C and no distribution Zc over C such that (a) \( \sum {\mathrm{Z}}_{\mathrm{z}}={z}_c^0 \)and (b) Zc is better than the C-section \( {Z}_c^1 \) of Z′.

D1 is the special case of D3 in which the only coalition allowed is the whole society, and similarly for D2 and D4, which is given next.

Definition 4 (D4)

A distribution Z2 is in the compensated core of Zc if: (iii) it is feasible; and (iv) there is no coalition C and no distribution Zc over C such that (c) Zc meets \( {Z}_c^2 \) and (d) \( \sum {Z}_c\prec \prec {z}_c^0 \).

The rationale for D4 is clearly similar to that for D2. Equally clearly, it is the appropriate ‘mirrored’ version of the core, in which objectives and constraints are interchanged. For example, it is easy to show that any compensated equilibrium of E(≿k, Z0) is in its compensated core. A much deeper result, for an exchange economy with a continuum of agents, is a ‘ compensated’ version of the core equivalence theorem of Aumann (1964), namely: the set of the compensated equilibria is precisely the compensated core (Newman 1982). Moreover, the assumptions and proof needed for this result are significantly simpler than in the classic paper of Aumann; in particular, only a non-topological separating hyperplane theorem is needed.

Since a coalition can be of any size, from one agent to every agent, the monotonicity of preference for the society as a whole asserted by A1 is quite inadequate to prove interrelations between cores and compensated cores. Instead, we use a more standard monotonicity assumption:

Assumption 3 (A3)

For any agent h, \( {z}_h^1\succ \succ {z}_h^2 \) implies \( {z}_h^1\;{\succ}_h{z}_h^2 \).

Theorem 3

Assume A3. If Z1 is in the core of E(≿k, Z0) then it is in its compensated core.

Proof

If not there is a coalition C and a distribution Zc over C such that \( {\mathrm{Z}}_{\mathrm{c}}\succsim {Z}_c^1 \) and\( \sum {Z}_c\prec \prec {z}_c^0 \). So for C there is a vector sc of surpluses in every commodity, that is, \( {s}_c=\left({z}_c^0-\sum {Z}_c^3\right)\succ \succ \)O.

Now form a new distribution \( {Z}_c^s \) over C by adding the vector (|C|−1) s ≻≻ O to each zk for kC, and denote the result by \( {z}_k^s \). Since \( {z}_k^s\succ \succ {z}_k \), from A3 \( {z}_k^s \succ_k{Z}_k \). Then \( {Z}_c^s\succ \succ {Z}_c\succsim {Z}_c^1 \) so that \( {Z}_c^s\succ \succ {Z}_c^1 \). Moreover, by construction, \( \sum {Z}_c^{\mathrm{s}}={z}_c^0. \) Hence Z1 is not in the core.

Theorem 4

Assume A2. If Z2 is in the compensated core of E(≿k, Z0) then it is in its core.

Proof

If not, there exists a coalition C and a distribution Zc over C such that \( \sum {Z}_c={z}_c^0 \) and \( {Z}_c\succ \succ {Z}_c^2 \) The proof then proceeds as in Theorem 2.

Conclusion

There is remarkable symmetry between the solution concepts (a) and (b) on the one hand, and (a)′ and (b)′ on the other. However, there is a major asymmetry. The concepts (a) and (b) implicitly give each member of the society a positive weight, that is, each person ‘counts’ for something. Hence, as Edgeworth first observed (1881, p. 23), it is easy to show that a distribution is (strongly) EP-optimal if and only if it maximizes the satisfaction of any agent picked at random, given both the total endowments and the levels of satisfaction of the remaining (m − 1) agents.

The corresponding statement for strong AS-efficiency is not so obvious. Suppose (and this is Scitovsky’s original argument) that we fix the levels of satisfaction of everyone in the society and the total amounts of all but one commodity chosen at random, say zi. Then it is tempting to say that a distribution is AS-efficient if and only if it minimizes the usage of zi. The trouble with this is that, in the situation prevailing zi just might be a commodity that nobody wants. So it is not scarce, its shadow price is zero, and there is no point in trying to economize on its use. Unlike the case with persons, we cannot be sure that a commodity chosen at random will carry positive weight.

The obvious way of dealing with this point is to put sufficient structure on the problem to make zi always desired. But then it ceases to be an arbitrary commodity, unless all commodities are always so desired; and that is a strong assumption indeed.

Exactly the same difficulty arises of course with efficient production programmes, if they are defined as allocations that maximize the output of an arbitrary product yj given the supplies of all the factors and the quantities of all products other than yj. This is really not surprising, since such ‘Pareto-efficiency’ is the analogue in a production economy of AS-efficiency in an exchange economy.

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© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Peter Newman
    • 1
  1. 1.