## Multiple Equilibria

## Multiple Equilibria

# Multiple Equilibria

DEFINITION OF MULTIPLE EQUILIBRIA

ORIGIN AND STATUS OF THIS CONCEPT

THE SIGNIFICANCE OF MULTIPLE EQUILIBRIA

Derived from the Latin, the word *equilibrium* means “equal weight” or “balance” as illustrated by an equal-armed scale. Physically, a body acted upon by two or more forces in a state of equilibrium maintains a stationary position. This state is known as *static equilibrium*, as distinct from *dynamic equilibrium*, in which the body’s position may change over time but maintains a state of static equilibrium at any given instant. The orbital motion of planets exemplifies dynamic equilibrium. Along with the notion of equilibrium comes that of *stability,* the tendency of a system to return to an equilibrium position after experiencing small perturbations of parameters influencing its prior equilibrium state. In the late nineteenth century, these concepts were adapted to economics and later game theory. In economics, the aim was, in part, to address the central question of prices that coordinate supply and demand.

## DEFINITION OF MULTIPLE EQUILIBRIA

For the most part, economic equilibria are studied through the development, analysis, and application of mathematical models. The values of interest are solutions of systems of equations and inequalities. Economists distinguish between *general* and *partial* equilibrium theory. Partial equilibrium theory differs from general equilibrium theory by having a specific set of variables held constant for the analysis. The French economist Léon Walras (1834–1910) is credited with being the father of general equilibrium theory. His great seminal work, *Elements of Pure Economics* (1874), sets forth his conception of the subject in increasing levels of completeness and detail. In dealing with a pure exchange economy with multiple markets, he developed a mathematical model in the form of a system of simultaneous equations having exactly as many unknowns as equations; a solution of the system would presumably yield an equilibrium. As it happens, neither the existence nor the uniqueness of a solution to the formulated system is guaranteed on the grounds of having as many equations as variables. Moreover, even if a solution exists, there is no guarantee it will be nonnegative (or even real).

Walras’s system was first rigorously addressed in the mid-1930s by Abraham Wald (1902–1950), a doctoral student (and participant in the Mathematical Colloquium) of Karl Menger (1902–1985). Between the work of Walras and Wald there appeared a series of significant contributions. Gustav Cassel (1866–1945) simplified some of Walras’s writings. Independently, Frederik Zeuthen (1888–1959), Hans Neisser (1895–1975), and Heinrich von Stackelberg (1905–1946) emphasized the importance of modeling some or all constraints with inequalities rather than equations. Around this time, Karl Schlesinger (1902–1985) introduced the notion of *complementary slackness*, which says that if a resource is not fully utilized, then its price must be zero, and if a price is positive, then the corresponding inequality constraint must hold as an equation. These advances culminated in Wald’s existence proof of competitive equilibrium. This, however, was to be superseded almost twenty years later by the work of Kenneth Arrow and Gerard Debreu (1921–2004).

## ORIGIN AND STATUS OF THIS CONCEPT

It is difficult to pinpoint the discovery of multiple equilibria. On the history of general equilibrium analysis, Arrow and F. H. Hahn remark, “in general, there is no need that equilibrium be unique, and examples of non-uniqueness have been known since Marshall” (1971, p.15). In fact, in section 64 of his *Elements*, Walras points out that in a two-commodity exchange problem there could be no solution, and in section 65 he notes that there could be multiple equilibria. Irving Fisher’s (1867–1947) 1891 PhD thesis contains an equilibrium model for an exchange economy and an ingenious hydraulic physical model for computing equilibrium prices and the resulting distribution of endowments. More recently, William Brainard and Herbert Scarf have elaborated this work and simulated the capability of Fisher’s device to compute equilibrium prices and even to find multiple equilibria.

If an economy has multiple equilibria, their number may be finite or infinite. Debreu argues that “such economies still seem to provide a satisfactory explanation of equilibrium as well as a satisfactory foundation for the study of stability provided that all the equilibria of the economy are locally unique. But if the set of equilibria is compact (a common situation), local uniqueness is equivalent to finiteness” (1970, p. 387). He gives sufficient mathematical conditions for the existence of finitely many equilibria. This line of investigation employs the concept of a *regular economy*, a detailed exposition of which is beyond the scope of this entry. The key concept involves the rank of the Jacobian matrix of the excess supply mapping. An economy is *regular* if this Jacobian is of full rank at every equilibrium point. Taking this a step further, if the determinant of the Jacobian at an equilibrium point *p* is positive (negative), define *i* (*p* ) = 1 (*i* (*p* ) = –1). The sum of *i* (*p* ) over all the finitely many equilibria is 1. This implies that the number of equilibria of a regular economy is finite odd. Hence if *i* (*p* ) = 1 for every equilibrium point *p*, then there can be only one of them. Conditions for local uniqueness have received considerable attention in literature. Michael Allingham (1989) gives a readable account of the subject.

## THE SIGNIFICANCE OF MULTIPLE EQUILIBRIA

The possibility that a general equilibrium model can have multiple equilibria presents challenges on a variety of fronts. Timothy Kehoe (1998) gives necessary and sufficient conditions for uniqueness of equilibrium. But, as he concedes, “useful conditions that guarantee uniqueness of equilibrium are very restrictive.… The problem is that translating these mathematical conditions into easy-to-check and interpretable economic conditions, they lose their necessity” (1998, p. 38). Kehoe identifies the crux of the matter saying “it may be the case that most applied models have unique equilibria. Unfortunately, however, these models seldom satisfy analytical conditions that are known to guarantee uniqueness, and are often too large and complex to allow exhaustive searches to numerically verify uniqueness” (1998, p. 39).

The size of models can require aggregation and therein lies another problem. A type of result separately discovered by Hugo Sonnenschein, Rolf Mantel, and Debreu implies that aggregate excess-demand functions do not inherit all the properties known to be sufficient for proving uniqueness of equilibrium. In short, there could be more than one price vector at which excess demand is zero. This finding is sometimes called the *anything goes theorem*.

Franklin Fisher (1983), among many others, has emphasized the importance of *disequilibrium analysis*, the study of the process by which prices change when the economy is not in equilibrium. This is made all the more complicated by the presence of multiple equilibria.

In addition to dynamics, critiques of the Arrow-Debreu general equilibrium theory have signaled the need to consider features such as uncertainty, (asymmetric) information, money, and taxes. Some of this is addressed in Stephen Morris and Hyun Song Shin (2001) and Fabio Petri (2004). It has been said that there is a role for public policy in the reconciliation of cases where there are multiple equilibria in applied problems.

**SEE ALSO** *Equilibrium in Economics; General Equilibrium; Jacobian Matrix; Nash Equilibrium; Partial Equilibrium; Prisoner’s Dilemma (Economics); Walras, Léon*

## BIBLIOGRAPHY

Allingham, Michael. 1989. Uniqueness of Equilibrium. In *General Equilibrium: The New Palgrave*. eds. John Eatwell, Murray Milgate, and Peter Newman. 324–327. New York: Norton.

Arrow, Kenneth J., and Gerard Debreu. 1954. Existence of Equilibrium for a Competitive Economy. *Econometrica* 22: 265–290.

Arrow, Kenneth J., and F. H. Hahn. 1971. *General Competitive Analysis*. San Francisco: Holden-Day.

Debreu, Gerard. 1970. Economies with a Finite Set of Equilibria. *Econometrica* 38: 387–392.

Eatwell, John, Murray Milgate, and Peter Newman, eds. 1989. *General Equilibrium: The New Palgrave*. New York: Norton.

Fisher, Franklin M. 1983. *Disequilibrium Foundations of Equilibrium Economics*. Cambridge, U.K.: Cambridge University Press.

Kehoe, Timothy J. 1998. Uniqueness and Stability. In *Elements of General Equilibrium Analysis,* ed. Alan Kirman, 38–87. Malden, MA: Blackwell.

Morris, Stephen, and Hyun Song Shin. 2001. Rethinking Multiple Equilibria in Macroeconomic Modeling. In *NBER Macroeconomics Annual 2000*, 139–161. Cambridge, MA: MIT Press.

Petri, Fabio. 2004. *General Equilibrium, Capital, and Macroeconomics: A Key to Recent Controversies in Equilibrium Theory*. Cheltenham, U.K.: Edward Elgar.

*Richard W. Cottle*