Journal of Mathematical Economics 1 (1974) 63-66. 0 North-Holland Publishing Company

EXCHANGE EQUILIBRIUM AND COALITIONS An example

David GALE

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

Received 30 June 1973

1. Introduction

The Core Theorem for exchange economies asserts that if a given set (coalition) of agents receives a certain distribution of goods under a price equilibrium then there is no redistribution of the initial endowments of these agents among them- selves which will make all of them better off than they were under this equili- brium distribution. This fact is sometimes thought of as implying a sort of stability property of equilibrium. Namely, even if some agents are dissatisfied with what they are receiving at equilibrium there is nothing they can do solely among themselves to improve their situation. Thus, since no group can improve its lot, there is no sense in struggling and hence the equilibrium once established will tend to persist.

Now there is a result which has rather the opposite flavor from the above and that is Samuelson’s (1939) theorem on gains from trade which deals with the following situation: A certain set of agents (the Home Country) which has been operating ‘autarchically’, producing and distributing goods among its own agents, is presented with the opportunity of trading with the Outside World. Under these conditions it will generally no longer be Pareto optimal for the country to continue its autarchic program. On the other hand, it is quite possible, in fact likely, that in a World Exchange Equilibrium some of

the agents in the Home Country will find themselves worse off than they were under the autarchic regime (agents could, for example, be put out of business by foreign competition). Samuelson’s important theorem asserts, however, that it is always possible for the Home Country to make suitable lump sum transfers among its own agents in such a way that in the World Equilibrium aI/ agents of the Home Country will be at least as well off and in general better off than they were under the autarchic regime. [For the case of pure exchange economies Samuelson’s result is rather obvious, but for models with production the result appears to be mathematically nonelementary requiring the use of fixed point theorems. A correct proof of the result has been given only recently by Grandmont and McFadden (1972).]

64 D. Gale, Exchange equilibrium and coalitions

We will here consider a situation combining the features of the two described above. Imagine a World Equilibrium involving the Home Country. By the Core Theorem there is no autarchic exchange program making everyone in the Home Country better off. However, Samuelson’s theorem suggests another course of action which is open to the Home Country, namely that of making a re- distribution of wealth among its own agents. The question is then whether there is some program of lump sum transfers among agents of the Home Country which will lead to a new World Equilibrium in which all members of the Home Country are better off than they were under the old equilibrium. Observe that, since the old equilibrium was Pareto optimal, any new equilibrium which benefits everyone in the Home Country must do so at the expense of the Outside World. To put it somewhat over-dramatically, the question is whether a group of agents can conspire together to plunder the rest of the world by a purely domestic redistribution of wealth.

The answer to the above question turns out to be affirmative as will be shown by a simple 2-good, 3-agent example. The result is perhaps not very surprising. It is well known, for example, that an agent can improve his equilibrium position by falsifying his utility function. By pretending he doesn’t like a good as much as he really does he can force it to have a lower price and thus get more of it. We will see here that the same effect can be achieved by a coalition of agents without resorting to falsification. An agent of the Home Country who has a high utility for a foreign good can bring down its World Price simply by giving away some of his own wealth to another Home Country agent who has a lower utility for the good in question. If the magnitudes are right this drop in the price of the desired good will more than compensate the agent for his loss of wealth. I should point out however, that my example is one with nonsmooth preferences, and several attempts to construct examples involving smooth preferences have been unsuccessful, so that for the present the question of existence of smooth examples of this pnenomenon remain open. ’

2. The example

There are three agents, A,, AZ and B. A, and A, own a, and a2 units of

apples where a 1 + a2 = 2 and B owns three units of bananas. All agents have

preference pre-orderings given by utility functions of the fixed proportions type. Specifically, the respective utility functions U,, Uz and V are given by

U,(X, Y) = min[% ~1, U,(x, Y) = mW, 4~1,

V(x, y) = min[x, y] .

ID. McFadden has also constructed an example showing the possibility of the kind of behavior described here but it too involves nonsmooth preferences. Also it has been brought to my attention that DrBze and Gabszewicz (1971) present an example with a similar flavor to the one given here, again involving nonsmooth preferences.

D. Gale, Exchange equilibrium and coalitions 65

Thus A, likes apples and bananas in the ratio of 1 to 4, A, in the ratio 4 to 1 and B in equal amounts.

We seek prices p and q of apples and bananas giving a competitive equili- brium. For this purpose we make the further assumption that a, > 3/4. It now follows that neither p nor q is zero, for if apples were free then B will demand at least 3 units of apples to go with his 3 units of bananas, but A, and AZ have only 2 units of apples between them. On the other hand, if bananas are free then A, will demand at least 4~2, units to go with his al units of apples, but by assumption 4a, > 3 which exceeds the supply of bananas owned by B. We may therefore normalize by choosingp = 1, so that apples serve as numer- air-e. If the price of bananas is q then A, will demand x units of apples and 4x units of bananas subject to his budget equation,

x+4qx = a,, (1)

and AZ will choose y units of bananas and 4y units of apples subject to his budget equation

4y+qy = a2 = 2-a,, (2)

and B will choose z units of apples and z units of bananas subject to his budget equation

z+qz = 3q. (3)

Note that we are making use of the positivity of q in writing these equations, for if q were zero we could have inequalities in (1) and (2).

Finally, we require balance of supply and demand for apples,

x+4y+z = 2, (4)

and bananas,

4x+y+z = 3. (5)

This gives a system of 5 equations for the 4 quantities x, y, z and q. Of course the equations are dependent. In fact adding (I), (2), (3) and subtracting (4) gives

q(4x+y+z) = 3q,

which gives (5) on dividing by q, since q is positive. To solve these equations eliminate z between (4) and (5) giving

3x-3y = 1,

and eliminate q between (1) and (2) giving

15xy+a,y+4(u,-2)x = 0.

(6)

(7)

66 D. Gale, Exchange equilibrium and coalitions

Finally eliminating x between (6) and (7) gives

15y2+(5a,-3)y-4/3(2-a,) = 0.

For the special case a 1 = 1 we get a unique equilibrium

X = (1/21+4)/15 w 0.57

y = (1/21- 1)/15 w 0.24

z = (6-d/21)/3 w 0.47

q = (31/21-13)/4 E 0.185.

(8)

By the implicit function theorem there will also be a unique equilibrium for all a, sufficiently close to 1. Without making any more calculations we can see that by suitably changing aI we can increase the equilibrium values of both x and y. Namely, eq. (6) shows that, in fact, any change in y must be accompanied by an identical change in x. But if x and y both increase so do the utilities of AI and A,. To see that the implicit function theorem is applicable we simply observe that the partial derivation of the left side of eq. (8) with respect to a, is positive at a, = 1, y = (d(21) - 1)/15. Finally implicit differentiation shows that y is a decreasing function of a, at a, = 1, so a small decrease in a, will produce an increase in the welfare of both A, and A, at the expense of B.

The economic interpretation of the above is straightforward. A, is the agent with a large utility for bananas. By transferring some of his initial wealth (apples) to A, who has a lower utility for bananas one brings down the equilibrium price of bananas by so much that even though AI now has less initial wealth in price units, his ‘real wealth’ has increased because of the even greater drop in banana prices.

References

Grandmont, J.M. and D. McFadden, 1972, A technical note on classical gains from trade, Journal of International Economics 2,109-125.

Jaskold-Gabszewicz, J. and J. Dreze, 1971, Syndicates of traders in an exchange economy, in: H.W. Kuhn and G.P. SzegB, eds., Differential games and related topics (North-Holland, Amsterdam).

Samuelson, P.A., 1939, The gains from international trade, Canadian Journal of Economics and Political Science 5,195-205.