# Optimality and Efficiency

**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 TransitivityAn 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 *R*^{n} 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 *z*_{hi}, so that Z^{0} is the distribution of initial holdings \( {z}_{hi}^0 \) or the *endowment.* If two distributions Z^{1} and Z^{2} are such that \( {z}_k^1 \succ_k{z}_k^2 \) for every agent h, then we write *Z*^{1} ≻≻ *Z*^{2} and say that *Z*^{i} is *better* than *Z*^{2}. Similarly, *Z*^{1} ≿ Z^{2} means that \( {z}_k^1 \succ_k{z}_k^2 \) for every *h*; we say that *Z*^{1}*meets Z*^{2}. If *Z*^{1} meets *Z*^{2} 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 *Z*^{1} ≻ Z^{2}.

An agent’s holdings are written *z*_{h} = (*z*_{h1}, *z*_{h2},…, z_{hn}). The symbolic expression ΣZ means the commodity vector *z* = Σ*z*_{h} (*all* summations here are over the index *h* for agents). In particular, Σ*Z*^{0} = *z*^{0}, the vector of total endowments of each good; **O** is the vector with zero amounts of every good. *Z*^{1} ≺≺ *Z*^{2} means (*z*^{2} − *z*^{1}) ≺≺ **O**, that is, Σ*Z*^{1} is less in every component (good) than Σ*Z*^{2}; *Z*^{1} is then *less* than *Z*^{2}. Similarly, *Z*^{1} ≤ *Z*^{2} means that Σ*Z*^{1} is not greater than Σ*Z*^{2} in any component. If Σ*Z*^{1} ≤ Σ*Z*^{2} 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 *Z*^{1} < *Z*^{2}.

The notation *Z*^{1} ≻≻ *Z*^{2} means (*z*^{1} − *z*^{2}) ≺≺ **O**. In an exchange economy it is natural to assume *z*^{0} ≻≻ **O**. A distribution Z is *feasible* if ΣZ = *z*^{0}, 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}, *Z*^{0}).

### Definition 1 (D1)

A distribution *Z*^{1} is *EP*-*optimal* for *E*(≿_{k}, *Z*^{0}) if: (*a*) it is feasible; and (*b*) there is no other feasible distribution *Z* that is better than *Z*^{1}.

Notice that D1 depends only on the totals *z*^{0} and not on their distribution *Z*^{0}. Applying a weaker meaning of being better, namely: that *Z* ≻ *Z*^{1}, 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 *Z*^{2} is *AS*-*efficient* for *E*(≿_{k}, *Z*^{0}) if: (*c*) it is feasible; and (*d*) there is no other distribution *Z* which meets *Z*^{2} and is less than *Z*^{0}.

Again, D2 depends only on *z*^{0} and not on *Z*^{0}, since ‘less than *Z*^{0}’ actually involves only *z*^{0}. As before, a weaker meaning of being less, namely: that *Z* ≺ *Z*^{2}, 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 *Z*^{s} such that Σ*Z*^{s} = Σ*Z* + *s* and *Z*^{s} ≻≻ *Z*.

### Theorem 1

Assume A1. If *Z*^{1} is EP-optimal then it is AS-efficient.

### Proof

This and all other proofs are by contraposition. If *Z*^{1} is not AS-efficient, there exists *Z* such that *Z* ≿ *Z*^{1} and ΣZ ≺≺ Z^{0}. So there is a vector of surpluses in every commodity, i.e., *s* =(*z*^{0} − ΣZ) ≻≻ **O**. Hence from A1 there is *Z*^{s} such that Σ*Z*^{s} = ΣZ + *s* = *z*^{0} and *Z*^{s} ≻≻ *Z*. But then *Z*^{s} ≻≻ *Z* ≿ *Z*^{1} implies *Z*^{s} ≻≻ *Z*^{1}, and *Z*^{s} is feasible. So *Z*^{1} is not EP-optimal.

The second special assumption does not involve the topology of *R*^{n} 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 *μ*_{h} ∈ (0, 1) such that \( \lambda {z}_k^1 \pm_k{z}_k^2 \) for all λ ∈ [*μ*_{h}, 1).

### Theorem 2

Assume A2. If *Z*^{2} is AS-efficient then it is EP-optimal

### Proof

If not, there exists Z such that Σ*Z* = *z*^{0} and *Z* ≻≻ *Z*^{2}. From A2 there exists for each *z*_{h} in *Z* some *μ*_{h} ∈ (0, 1) such that \( \lambda {z}_k \succcurlyeq_k{z}_k^2 \) for all λ ∈ [*μ*_{h}, 1). Put *μ* equal to the maximum of these *μ*_{h}, so that *μ* < 1 and write *μZ* for the *m* × *n* matrix of the *μ**z*_{h}. Then *μZ* ≿ *Z*^{2}. But by construction and the fact that *z*^{0} ≻≻ **O**, Σ*μZ* = *μ*ΣZ ≺≺ Σ*Z* = *z*^{0}. So *Z*^{2} 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 *Z*_{c} whose rows are the *n*-vectors *z*_{k} for *k* ∈ *C.* The notation Σ*Z*_{c} means the sum over the |*C*| rows of *Z*_{c}, and for any *Z*^{0} 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 *Z*_{c} 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 *Z*^{1} is in the *core* of *E*(≿_{k}, *Z*^{0}) if: (i) it is feasible; and (ii) there is no coalition *C* and no distribution *Z*_{c} over *C* such that (*a*) \( \sum {\mathrm{Z}}_{\mathrm{z}}={z}_c^0 \)and (*b*) *Z*_{c} 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 *Z*^{2} is in the *compensated core* of *Z*_{c} if: (iii) it is feasible; and (iv) there is no coalition *C* and no distribution *Z*_{c} over *C* such that (c) *Z*_{c} 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}, *Z*^{0}) 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 *Z*^{1} is in the core of *E*(≿_{k}, *Z*^{0}) then it is in its compensated core.

### Proof

If not there is a coalition *C* and a distribution *Z*_{c} 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 *s*_{c} 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 *z*_{k} for *k* ∈ *C*, 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 *Z*^{1} is not in the core.

### Theorem 4

Assume A2. If *Z*^{2} is in the compensated core of *E*(≿_{k}, *Z*^{0}) then it is in its core.

### Proof

If not, there exists a coalition *C* and a distribution *Z*_{c} 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 *z*_{i}. Then it is tempting to say that a distribution is AS-efficient if and only if it minimizes the usage of *z*_{i}. The trouble with this is that, in the situation prevailing *z*_{i} 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 *z*_{i} 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 *y*_{j} given the supplies of all the factors and the quantities of all products other than *y*_{j}. 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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