The Moving Finger Writes...

...and, having writ,
Moves on: nor all your piety nor wit
Shall lure it back to cancel half a line,
nor all your tears wash out a word of it.

1.0 Introduction
Neoclassical economics emphasizes equilibria, for example in General Equilibrium models. In equilibria, all agents are optimizing and their plans are all pre-coordinated. But no reason exists for economists to expect actually existing more-or-less capitalist economies to ever be in such equilibria.

This post demonstrates that economies need not be near equilibria by means of an example. This example has been available for almost a half century (Scarf 1960) and is often referenced (e.g., Ackerman 1999, Hahn 1961 and 1970, McCauley 2004, Saari 1995, Sonnenschein 1972). The example is of a pure exchange economy. Since no production occurs in the example, it cannot be considered an example of a Sraffian model. Furthermore, the example is of brain-dead tâtonnement dynamics. No trading occurs at any prices other than equilibrium prices. Since the example has one locally unstable equilibrium, equilibrium prices are never achieved.

Neoclassical economic theory imposes almost no restriction on excess demand functions. The most substantial restriction is the unfounded conservation law expressed by the symmetry of the Slutsky matrix. This lack of any empirical implications of neoclassical theory for market behavior is an implication of the Sonnenschein-Mantel-Debreu results. Another implication is that any price dynamics are possible in a tâtonnement process, including chaos. So this example does not even represent the most general or complex dynamics possible in neoclassical models.

2.0 Data
This example economy consists of three individuals, each endowed with one unit of a different commodity (Table 1). The individuals also differ in tastes, as expressed by the utility functions in Table 2. Our problem is to determine equibrium prices for this simple economy and the price dynamics.

Table 1: Agents' Endowments

Agent	Endowment
	Apples	Bananas	Cantalopes
Mary	1	0	0
Nancy	0	1	0
Olivia	0	0	1

Table 2: Agents' Preferences

Agent	Utility Function
Mary	UM = min( xA, xB )
Nancy	UN = min( xB, xC )
Olivia	UO = min( xA, xC )

3.0 Demand Functions
Each agent maximizes their utility, subject to their budget constraint. Consider a single agent, for example, Mary. Mary chooses non-negative x_A, x_B, x_C to maximize

U_M = min( x_A, x_B )          (1)

such that

p_A x_A + p_B x_B + p_C x_C ≤ p_A          (2)

Since Mary derives no utility from cantalopes, she will not consume any of them. Thus, Mary's problem can be graphed in a two-dimensional space (Figure 1). The graph also shows the quantities Mary demands of apples and bananas. These quantities are on the intersection of the budget constraint with a particular isoquant of the utility function. Symbolically:

x_A^* = x_B^* = p_A/(p_A + p_B)          (3)

x_C^* = 0          (4)

Figure 1: Mary's Utility Maximizing Problem

One can find Nancy and Olivia's demand functions by symmetrical arguments. Aggregate excess demand functions are the difference between aggregate demands and aggregate supplies. Aggregate demands are individual demand functions summed across the individuals. Aggregate supplies, in this pure exchange economy, are endowments summed across individuals. In fact, the aggregate supply of each commodity is one unit here. A bit of algebra yields:

z_A = p_C/(p_A + p_C) - p_B/(p_A + p_B)          (5)

z_B = p_A/(p_A + p_B) - p_C/(p_B + p_C)          (6)

z_C = p_B/(p_B + p_C) - p_A/(p_A + p_C)          (7)

where z_B, z_B, and z_C are the aggregate excess demand functions for apples, bananas, and cantelopes, respectively.

The numeraire is arbitrary. One can confine prices to lie on the unit sphere:

p_A² + p_B² + p_C² = 1          (8)

4.0 Equilibrium

In equilibrium, aggregate excess demand functions are zero. The only equilibrium is one in which all prices are equal:

p_A^* = p_B^* = p_C^* = (1/3)^1/2          (9)

5.0 Tâtonnement Dynamics
I postulate that when the aggregate excess demand for a particular commodity is positive, the price of that commodity rises. Likewise, when aggregate excess demand is negative, the price falls. The simplest dynamical system with these properties is one in which the rate of change of prices with respect to time is equal to the aggregate excess demands:

dp_A/dt = p_C/(p_A + p_C) - p_B/(p_A + p_B)          (10)

dp_B/dt = p_A/(p_A + p_B) - p_C/(p_B + p_C)          (11)

dp_C/dt = p_B/(p_B + p_C) - p_A/(p_A + p_C)          (12)

Under these dynamics, the equilibrium is unstable. Solutions around the equilibrium spiral out on the unit sphere to a limit cycle. Figure 2 shows a two-dimensional projection of that limit cycle.

Figure 2: Two-Dimensional Projection of Dynamics

6.0 Conclusion
The failure of General Equilibrium Theory to limit dynamics, I gather, is intrinsic to methodological individualism, in which independent agents can have arbitrary preferences and endowments. Attempts to explain economies seem to need to postulate influences on tastes and income above the level of the individual, for example, by others in one's social class or through some sort of structuralist theory. In other words, there is too such a thing as society. I take Kirman (1989) to point in this direction.

I might as well mention that the arbitrary dynamics implied by orthodox economic theory undermines a certain political outlook. I refer to the idea that we ought to loosen restrictions on trade, but ensure some sort of redistribution so as to ensure that everybody participates in the supposedly enlarged pie. I take the second welfare theorem to be the basis for this view. But that redistribution doesn't necessarily lead to the economy converging to the original equilibrium, as altered by free trade.

References

• Frank Ackerman (1999) "Still Dead After All These Years: Interpreting the Failure of General Equilibrium Theory", Global Development and Environment Institute, Working Paper No. 00-01
• Frank H. Hahn (1961) "A Stable Adjustment Process for a Competitive Economy", The Review of Economic Studies, V. 29, N. 1 (October): pp 62-65.
• Frank H. Hahn (1970) "Some Adjustment Problems", Econometrica, V. 38, N. 1 (January): 1-17
• Alan Kirman (1989) "The Intrinsic Limits of Modern Economic Theory: The Emperor Has No Clothes", Economic Journal, V. 99, N. 395: 126-139
• Joseph L. McCauley (2004) Dynamics of Markets: Econophysics and Finance, Cambridge University Press
• Donald Saari (1995) "Mathematical Complexity of Simple Economics", Notices of the AMS, V. 42, N. 2 (February): 222-230
• Herbert Scarf (1960) "Some Examples of Global Instability of the Competitive Equilibrium", International Economic Review, V. 1, N. 3 (September): 157-172
• Hugo Sonnenschein (1972) "Market Excess Demand Functions", Econometrica, V. 40, N. 3 (May): 549-563