Journal of Mathematical Economics 3 (1976) 205-209. 0 North-Holland Publishing Company

THE LINEAR EXCHANGE MODEL

David GALE

University of California, Berkeley, CA 94720, U.S.A.

Received February 1976

1. Introduction

In the preceding article B.C. Eaves presents a finite algorithm for finding a competitive equilibrium for a pure exchange economy in which all traders have linear utility functions. The method either finds an equilibrium if such exists or else it obtains a ‘reduction’ of the model. This provides a constructive proof of a result obtained by the author in Gale (1957) where it was shown that any ‘irreducible’ model has an equilibrium. The purpose of this note is to round out the theory of these models by (1) giving the simple necessary and sufficient condition for the existence of an equilibrium, and (2) showing that equilibria when they exist are essentially unique in the sense that all equilibria provide the same utility to all traders so that traders are indifferent as to which equilibrium prevails.

It is easy to give examples for which there are no equilibria. Suppose I own apples and oranges but I have positive utility only for apples while you have positive utility for both, but only own oranges. Then, it is clear that there can be no equilibrium price for oranges since at any price I will want to exchange my oranges for apples, but I already own all the apples. Our Existence Theorem asserts that equilibrium will fail to exist if and only if the above situation,

suitably generalized to m traders and IZ goods, prevails. It is also easy to give examples in which there is more than one equilibrium

price vector. Suppose I own all the applies and you all the oranges and we both prefer our own fruit. Then it is clear that there can be a range of relative prices which give equilibrium in which we each consume our own endowment. Of course, in this case we each obtain the same utility regardless of the prices. Again, it turns out that the above situation, suitably generalized, is, in fact, the only one in which there can be more than one equilibrium price vector. The proof of this fact, however, turns out to be quite involved.

I should remark that the uniqueness result is contained in Gale (1957), but has not hitherto appeared in print. With the present interest in computing equilibria it seemed worthwhile presenting it now.

206 D. Gale, The linear exchange model

2. Model

There is a finite set T of traders and a finite set G of goods. A G-vector v is a function from G to R’. We will denote the value of v on the good g of G by vg. To specify a linear model one assigns to each trader t in T his endowment G-vector w’ and his utility G-vector u’. Given any G-vector x the value of x to trader t is then the product u’x.

A feasible allocation (x*) is an assignment of G-vectors x’ to each t in T, such that

An equilibrium consists of a price G-vector p and a feasible allocation (x’) such that, for all t in T,

x’ maximizes z/x’, (2)

subject to

px’ $ pw’. (3)

It will be assumed from here on that for every g in G there exists at least one t in T such that w: > 0, and at least one t in T such that U: > 0. In words, every good must be desired by at least one trader and owned by at least one trader. The assumption is clearly harmless for if it were violated one would simply cut down the set G appropriately. It is an obvious consequence of this assumption that any equilibrium price vector must be (strictly) positive. We further choose units in such a way that

that is, there is exactly one unit of each good in the model.

3. Existence

If S is any subset of T we denote by s’ the complement of S in T.

Definition. A subset S c T is called self-sufficient (s-s) if, for any s in S,

ui > 0 then w”#’ = 0, for all s’ E S’. (4)

In words, this means that the members of S place no value on goods owned by the members of 5”.

D. Gale, The linear exchange model 207

The set S is called super self-sufficient (ss-s) if in addition there exists s in S such that, for some g in G,

n$, > 0 but U; = 0, for all t in S. (5)

In words, this means that someone in S owns a good which is not valued by any member of S.

Existence Theorem. The model has an equilibrium if and only if no subset of T is SM.

Suppose there is an equilibrium and suppose S is a s-s subset of T. Then members of S trade only with each other so the equilibrium allocation xt must satisfy

(remember, there are no free goods) but if S were ss-s then for some g and some s,-, in S we would have w? > 0, ‘& xi = 0 contradicting (6).

Conversely, suppose there is no equilibrium. From the preceding article by Eaves and from Gale (1957) it follows that there is some proper s-s subset of T. We must show that there must, in fact, be an ss-s subset. We proceed by induction on ITI (the cardinality of T). For ITI = 2 suppose both traders t and t’ are s-s. Then one easily verifies that p = Au’+@‘, x’ = w’, x” = wt’ gives an equili- brium for any positive 1 and p. Therefore suppose only {t] is s-s, but not ss-s. Then define 2’ to be ut’ for goods held by t’, but u”:’ = 0 for goods held by t. Now choose p = u’+Ez?‘. It is claimed that for E sufficiently small, this will give an equilibrium, for t will be indifferent among the goods he holds and may therefore be assigned wt as his allocation. Further, if E is small enough, all goods held by t and valued positively by t’ will be too expensive for t’. It follows then that (t} must be ss-s if there is no equilibrium.

For the case of many traders the argument is similar. We know there must be some proper s-s set S. Now, consider the equilibrium problem for the set S alone. It will exist by induction hypothesis unless S has some proper subset 5 which is ss-s in J, but then clearly s” is also ss-s in T as claimed. Next, consider the equilibrium problem for S’ where we assume the only goods in the model are those held by members of S’. If an equilibrium fails to exist, then there must be a proper subset ss-s, subset 9 of S’. But now one sees from the definition of ss-s that S u s”l is ss-s in T as claimed.

The only remaining case is that in which equilibrium prices p and p’ exist for the separate problems for S and S’. In this case, as above, a price vector p = p+&p’ will give an equilibrium for the original model provided E is suffici- ently small. Thus equilibrium fails to exist only under the asserted conditions. n

208 D. Gale, The linear exchange model

4. Uniqueness

As noted in the introduction equilibrium prices need not be unique. However, we have the following :

Uniqueness Theorem. If(p,(x’)) and @, 2’)) are equilibria for the same model, then uxt = u? for all t in T.

There seems to be no easy direct proof of this result. The following somewhat roundabout proof requires :

Definition. Let S be a subset of T and let Hs be the subset of all goods held by at least one member of S. Then S is called independent under the allocation

(x9 w;r’ = 0, for s’ES’, hEHs (7)

(in words, members of S own all the goods in Hs), and

-g xs = 7 ws

(in words, members of S trade only among themselves).

(8)

Lemma. If (p, (2)) and @, (2’)) are equilibria and p and j are not proportional then there is a proper subset of S of T which is independent under both (2) and (?).

Let 0 = max (p,/p,) and H = (gl&,/pg = 0).

G

Sincep and p are not proportional it follows that H is a proper subset of G. Let S = (tl2; > 0 for some h E H). We first show that if ,x; > 0 then g E H. That is, members of S only receive goods of H at prices p (from this it will follow that S is a proper subset of T since there must be other traders to receive the goods of H’). Suppose then that s E S. Then by definition of S there is some h in H such that XS, > 0. Suppose g E H’, then fi,/p, < 8. It follows easily from (2) and (3) that 2: > 0 only if

41~~ 2 u;ifi, > u:/ep,.

Multiplying the above by 0 gives

4IPh ’ $lP,>

so that xi = 0. Thus, xi > 0 only for h in Has asserted. It follows that

(9)

D. Gale, The linear exchange model 209

since at prices p the entire wealth of the set S is spent on goods in H. (We are here using the fact that ‘& w: = 1.) On the other hand, from the definition of S, its members must be able to afford all of H at prices p so

Combining (9) and (10) gives

(pep> c ws >= 0, s

(11)

but by definition of 8, p” - tip S 0, and for g E H’, j, - f3p, < 0, so from (11) we have Es wi = 0 for g E H’, so wi = 0 for all s E S, g E H’. Thus, members of S own only goods of H. But at prices 8 they buy all the goods in H. It follows that they must own all of H, otherwise they couldn’t afford to buy all of H. This is condition (7). But now (8) also follows because at prices p, members of S spend their entire wealth on goods of H which they alone hold, and at prices p” they buy only the goods of H since that is all they can afford. n

The proof of the Uniqueness Theorem is an easy consequence of the Lemma. Ifp and p are proportional but (x’) and (2’) are different, then from (2) and (3) it is immediate that u’x’ = ~‘2’. If p and p” are not proportional we argue by induction. For ITI = 2, say {t} is independent under both (x’) and (2’). Then clearly x’ = X’ = w’, and therefore x” = If’ = w”. For the general case suppose S is the independent subset. All we need is condition (8) which assures that p and p are not only equilibrium vectors for the original model but also for the submodels consisting only of S, and also the submodel consisting only of S’ [(8) for S implies (8) for S’, clearly]. But since ISI, IS’1 < ITI, the uniqueness property follows by the induction hypothesis. H

References

Eaves, B.C., 1976, A finite algorithm for the linear exchange model, Journal of Mathematical Economics, this issue.

Gale, D., 1957, Price equilibrium for linear models of exchange, Rand Corporation Technical Report P-M56, Aug. 15.