A NOTE ON GLOBAL INSTABILITY O F COMPETITIVE EQUILIBRIUM*

David Gale

Department of Mathematics Brown U n i v e r s i t y

P r o v i d e n c e 1 2 , Rhode Island

INTRODUCTION

librium which is globally unstable. The examples involve three consumers and three goods; the consumers' utility functions a re obtained one from another by permuting in a cyclic manner.

It is our purpose here to give what appears to be an even simpler class of examples involving only two individuals and three goods, one of which behaves a s numeraire, so that prices of other goods a r e measured in terms of it. Further, these examples tie in with the elementary ideas of consumption theory as taught in undergraduate courses in economics, and global instability turns out to be a consequence of the familiar "Giffen paradox," The whole situa- tion can be conveniently illustrated by a two-dimensional diagram, as we shall see.

In the final section we give a very simple example of a symmetrical two-person, two- good economy whose only symmetrical price equilibrium turns out to be completely unstable.

Mr. H. Scarf' has given some examples of an economy with a unique competitive equi-

THE MODEL We consider a model consisting of two individuals, o r perhaps preferably, two sectors

which a re themselves aggregates of individuals. The first we call the consuming sector o r simply "Consumption," which is denoted by C. C is equipped with a single good G o , in the amount go which behaves as numeraire so that its price po is always unity (we may occasion- ally break down and simply refer to Go a s Money). The second sector of the economy is called "Production," is denoted by P, and produces annually two goods G1 and G2 in amounts gl and g2.

We next assume, a s is usual, that C consumes so as to maximize his satisfaction, which is described by a nondecreasing, continuous, concave utility function q5 defined for all non-negative pairs (xl,x2) of amounts of goods G1 and G2. C places no value at all on Go, while P has a strictly increasing utility for Go, but zero utility for G1 and G2. Under these conditions, as any educated sophomore will willingly explain, one can find an equilibrium price vector 5 = (il, p2) by looking at C's indifference curve, which passes through (gl, g2), and choosing in the direction of the normal to this curve while choosing the length of satisfy the "budget constraint" plgl + p2g2 = go. Further, the vector 3 will be unique if, say, $ is differentiable at the point (gl, g2).

so a s to

*Research was supported by the Office of Naval Research under Contract Nonr-562(15). lHerber t Scarf, "Some Examples of Global Instability of Competitive Equilibrium," Interna- tional Economic Review, Vol. l , No. 3 (September, 1960).

81

82 DAVID GALE

Suppose now that prices a re disturbed slightly from equilibrium, say, p1 is increased. Then under %ormal” circumstances one would expect a decrease in the demand for G1, and by the usual dynamic assumptions this decrease should produce a drop in the price pl, tending to bring it back toward the equilibrium value. However, under suitable circumstances the Giffen effect may operate so that increasing p1 produces an increased, rather than a decreased, demand for gl, which according to our dynamics will tend to increase p1 still further. This suggests that when the Giffen effect is present in the neighborhood of equilibrium, global stability will not obtain. This turns out to be the case, but the above rough argument is not itself conclusive, for it is conceivable that when prices are disturbed in a certain way, they do move away from equilibrium for a time, but eventually they turn around and sneak back to the equilibrium values from some non-Giffenian direction. In order to prove instability, we shall therefore have to use other methods which amount, in essence, to a variant on the famous Lyapunov method. For those unfamiliar with this method, we present the following brief, self- contained treatment of our special case.

THE INSTABILITY THEOREM A little more notation will be needed in order to state our result. Given p = (p1,p2) we denote the excess demand functions for G1 and G2 by f and g,

respectively. Thus, i f x1 and x2 a r e the amounts of G1 and G2 demanded at price p, then f(p) = x1 - gl and g(p) = x2 - g2. If we denote the partial derivatives of f and g with respect to p1 and p2 by f l , f a and gl, g2 , respectively, then the Giffen effect obtains at equilibrium with respect to G1, precisely when fl@) is positive.

Now, following Arrow and Hurwicz,2 we have for the dynamic equations of price adjustment

where hl and $ are positive numbers.

the previously mentioned system approach p, a s t approaches infinity. We shall be interested in the opposite situation described a s follows:

The unique equilibrium price vector is called globally stable if all solutions p(t) of

DEFINITION: The point is the constant solution p(t) = p.

Our result now is

is anti-stable i f the only solution of (1) which approaches

THEOREM:

For those familiar with stability theory of differential equations the reason for this

If I+ is twice differentiable a t p and the Giffen effect obtains there, then for a range of values of the constants hl and h2 the point Tj will be anti-stable.

result is easily given. The starting point for stability theory is the fact that the system (1) is locally stable at p i f , and only i f , the Jacobian matrix

2K. J. Arrow and L. Hurwicz , “On the Stability of Competitive Equilibrium, I,” Econornetr ica , 26 (1958).

GLOBAL INSTABILITY O F C O M P E T I T I V E EQUILIBRIUM 83

has eigen-values with negative real parts. Analogously, Eq. (1) is anti-stable if its eigen-values all have positive real parts. In fact, in this case by the Lyapunov theory, one can find a family of small ellipses about p such that all solution paths of Eq. (1) a re directed outward across them, hence any solution which starts outside of such an ellipse remains outside ever after.

Therefore, toprove anti-stabilityit is only necessary to show that hl and h2 can be chosen so that the matrix A has eigen-values with positive real parts. But this will be the case provided the trace, y = h l f l + h 2 f 2 , of A is positive, since y is the sum of the real parts of the roots of A, while the product of the roots is the determinant of A, which is always positive for a demand function obtained from a concave utility function in two dimensions. Fi- nally, i f the Giffen effect obtains so that, say, f l is positive, then X l f l + h2f2 can be made positive for hl large enough.

For the sake of keeping things elementary, we present a short self-contained proof of the anti-stability of Eq. (1) without using results of differential equations. From the previous paragraph we may suppose that the trace and determinant of A a re positive. To simplify nota- tion, let us rewrite A as

where trace (A) = y = a + d is positive, and det(A) = 6 = ad - bc is positive.

(1) in vector matrix form Now, denoting p - j5 by Ap and using our differentiability assumptions, we may rewrite

Ap = AAp + R(Ap),

2 where R(Ap) goes to zero like IIApII by Taylor's Theorem. We now define a quadratic function @J by the rule

where

B = t+d 2

- bd -bd ), and b + e d 2

where 6, b, and d have been defined and E is a positive number sufficiently small for all future purposes. Now by the standard test, B is positive definite, for 6 + d2 is positive since 6 is positive by assumption, and

det(B) = b26 + Ed(6 + d2)

84 DAVID G A L E

is also positive for the same reason for E small enough. Now, let q(t) be any non-zero solu- tion of Eq. (2) and define

Then

and combining with Eq. (2) gives

but evaluating BA, we get

Now, C is also positive definite, i.e., xCx 2 0 for x f 0 as one verifies by examining its symmetric part

which also has a positive diagonal and positive determinant for E small. Note that we here use the positivity of both 6 and y .

small, then the first term on the right-hand side dominates, hence x i s positive. Hence x i s increasing for small q(t), so i f q(t) approached zero, x would have to approach zero from below, which is impossible, a s x is positive definite.

We now asser t that q(t) cannot approach zero, for from Eq. (3) we see that i f q(t) is

AN EXAMPLE There could remain one doubt about the above argument. Can one actually find a utility

function @ and initial endowments go, gl, g2 which make all the things we have postulated actually happen, i.e., differentiability, concavity, Giffen effect, etc. ? The example illustrated in Figure 1 shows that one can. We have plotted a set of indifference curves for the function @

given by

2 2 $(x1, x2) = 28x1 + 28x2 - 2x1 - 3x1 ~2 - 2x2 ,

which i s valid in the region OAPB. Outside of this region the indifference curves straighten out, indicating saturation of one of the goods, and complete saturation occurs at the point P = (4 ,4) so that additional goods beyond that point have no additional value.

GLOBAL INSTABILITY O F COMPETITIVE EQUILIBRIUM 85

F i g u r e 1

Initially, the consumer has 12 units of go and the producer has 6 units of G1 and 1 unit of G2. The price line CD passes through the equilibrium point Q = ( 6 , l ) and is, of course, tangent to the indifference curve through that point. One easily calculates that the unique equi- librium prices prices a re p1 = 1, p2 = 6.

The line CD' is the price line for a slightly increased value of p1 and it is tangent to an indifference curve at the point Q' which is to the right of Q so that the demand for x1 has increased in accordance with the Giffen effect.

By repeated calculations the writer has convinced himself that the Jacobian matrix for the excess demand function is given by

which has a positive trace and determinant, so that the dynamic adjustment process with price derivatives equal to excess demand is anti-stable.

A TWO-GOOD EXAMPLE Arrow and Hurwicz have shown that for the case of two goods, one always has global

stability, that is, starting from any set of prices, the price path will in time approach some equilibrium price vector. Nevertheless, some queer things can happen even in this case. We shall here give a very simple symmetric two-person two-good economy in which there i s an obvious equilibrium which is, however, anti- stable.

A consumer C1 holds one unit of a good X. A second consumer C2 holds one unit of good Y. Their utility functions a re of the 'fixed-proportions type," specifically, C1 wishes to hold goods in the ratio two parts X to one part Y, any excess of X or Y beyond this ratio being of no value to him. Analytically, his utility G1 for the goods vector (x, y) is given by

86 DAVID GALE

Symmetrically, C2 wants goods in the ratio one to two so that his utility function i s

Now, the fair and natural equilibrium solution is completely obvious. The prices of the two goods should be the same, and each consumer should sell the other one-third unit of the good he holds. But suppose one s ta r t s out with prices p and q. Let u s compute the demand of C1. If he demands cy units of Y, then he wil l demand 2cu units of X, and the budget con- straint requires

p (1 - 2cY) = qa

so that

P a=- 2 P + q '

and C1 demands the bundle

By symmetry, C2 demands

and adding gives a total demand for X of

2 2 where 6 = 2p + 5pq + 2q and substracting 1 gives the excess demand for X as

Symmetrically, the excess demand for Y is

But note that if , say, p > q, then Ax i s positive while Ay i s negative, so that the more expensive good i s the one which is over-demanded and hence, according to our dynamics, its price will go still higher, while that of Y will do just the reverse. Thus, prices will move further from the symmetric equilibrium until, in fact, q becomes zero, at which point Y is a free good, and we arr ive at the "unfair" equilibrium, i n whichC gives away at least half of his good Y and gets nothing in return.

2

GLOBAL INSTABILITY OF COMPETITIVE EQUILIBRIUM 87

A s a final remark we suggest the reader amuse himself by analyzing stability for the same model except that this time C1 holds the good Y and C2 the good X. He will find that in this case the symmetric equilibrium is stable while the one-sided ones are anti stable. Thus, stability occurs when each consumer places a higher value on the other's good than on his own, while instability obtains when he is more interested in his own good than h i s competitor's, which is perhaps the opposite of what one might have guessed.

* * *