Sunday, June 15, 2008

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
AgentEndowment
ApplesBananasCantalopes
Mary100
Nancy010
Olivia001

Table 2: Agents' Preferences
AgentUtility Function
MaryUM = min( xA, xB )
NancyUN = min( xB, xC )
OliviaUO = 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 xA, xB, xC to maximize
UM = min( xA, xB )          (1)
such that

pA xA + pB xB + pC xCpA          (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:

xA* = xB* = pA/(pA + pB)          (3)

xC* = 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:

zA = pC/(pA + pC) - pB/(pA + pB)          (5)

zB = pA/(pA + pB) - pC/(pB + pC)          (6)

zC = pB/(pB + pC) - pA/(pA + pC)          (7)
where zB, zB, and zC 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:

pA2 + pB2 + pC2 = 1          (8)

4.0 Equilibrium

In equilibrium, aggregate excess demand functions are zero. The only equilibrium is one in which all prices are equal:
pA* = pB* = pC* = (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:
dpA/dt = pC/(pA + pC) - pB/(pA + pB)          (10)

dpB/dt = pA/(pA + pB) - pC/(pB + pC)          (11)

dpC/dt = pB/(pB + pC) - pA/(pA + pC)          (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

13 comments:

Anonymous said...

"Furthermore, the example is of brain-dead tâtonnement dynamics."

So why is it relevant for your purpose --to show that economies need not be near equilibria-- given that tatonnement dynamics is not the best way to analyze the problem (despite what economists used to think at one time).


"This lack of any empirical implications of neoclassical theory for market behavior is an implication of the Sonnenschein-Mantel-Debreu results."

This is nonsense. It is just as nonsensical as saying that classical mechanics lacks any empirical content beyond some conservation laws and thus it is empirically useless. Of course, once you take into account the initial conditions, the behavior of the system becomes quite well specified.


"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 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."

Let's assume, for the purpose of this discussion, that your first sentence is true (although it is not justified at all, if you just note that tatonnement is not the only way to model dynamics).

How does the desire of economists to have simple and pretty models have any implications about what the world is really like?


It is remarkable how many bad criticisms of GE are out there, given that GE should not be that hard to criticize.

Alex

Robert Vienneau said...

I must have been unclear on my thesis. My thesis is not "that [real-world] economies need not be near equilibria." It is that orthodox economists have no reason to expect actually existing more-or-less capitalist economies to ever be in such equilibria.

In other words, my post presents an internal critique of neoclassical economics. Even model economies, conforming to the assumptions of orthodox economics, need not have any tendencies to converge to equilibrium.

Anyways, I don't think the issues raised by Scarf's example are limited to tatonnement processes. Other mainstream dynamic processes I've seen seem either ad hoc or have similar issues (e.g., in Franklin Fisher's book).

I don't see how this question is raised by any of the preceding discussion:

"How does the desire of economists to have simple and pretty models have any implications about what the world is really like?"

I don't see how any discussion confined to neoclassical General Equilibrium theory can have anything to say about what the world is really like.

I hope that a concrete example might help lurkers, if any, with the prior discussion.

Anonymous said...

"My thesis is not "that [real-world] economies need not be near equilibria." It is that orthodox economists have no reason to expect actually existing more-or-less capitalist economies to ever be in such equilibria."

OK, I do actually agree that economists in general have provided little realistic justification for treating equilibria as the normal state of the economy. So it remains an open question to what extent this is so.

But I do think that by treating only the tatonnement dynamics, your example does not go that far into the direction of proving your clarified thesis.

I'm also curious as to why you limit your claim to orthodox economists. Do you know of some convincing dynamics for justifying Sraffian equilibria for instance?

"I don't see how this question is raised by any of the preceding discussion"

I meant that the jump from the claim that 'methodological individualism can not put sufficient restriction on dynamics' to the claim that 'therefore we should use structuralist models' is not justified. The assumed failure of methodological individualism to provide nice models does not imply that structuralist models are better connected to reality.

Alex

Anonymous said...

I'll continue on with Alex's line of thought here. This looks like an interesting exposition of why price adjustment processes themselves are important to study--especially in small or very frictional economies--which is something that neoclassical macroeconomists in particular are obsessed with, perhaps to an unhealthy degree.

What does NOT follow are the entire contents of the conclusion. Not A does not imply B when an unknown C might be the truth. The conclusion makes exactly the same argument that creationists make--since biology does not explain everything then we must have divine intervention. Since tatonnement does not explain everything, a Marxian or institutionalist theory must.

To blatantly crib something notsneaky said a while back, the idea that the aggregate of individual behavior cannot be modeled as an individual representing the aggregate is a huge problem for Marxists, since it destroys the idea of using classes as the unit of analysis. It is much less a problem for neoclassical economists who prefer to look at individual behavior, even macroeconomists who adopt highly restricted models as a means of storytelling and rough quantitative guesstimation. The very existence of macro (and labor and finance and public, etc) as fields indicates economists' understanding that most interesting questions are quantitative and not qualitative. The existence of 'labor' or 'capital' or 'society' as coherent entities seems to matter much less than the old Marshallian questions like "How do people adjust their behavior when gasoline becomes more expensive? And what happens when governments impose price controls?" And lo and behold, neoclassical economists got that one right using old Marshallian theory as a heuristic.

-Chris

Michael Greinecker said...

Robert,I think you will like this paper (JSTOR) by William Bryant. He basically makes the same point about instability and the second welfare theorem.

Michael Greinecker said...

"the idea that the aggregate of individual behavior cannot be modeled as an individual representing the aggregate is a huge problem for Marxists, since it destroys the idea of using classes as the unit of analysis."

Class is not the unit of analysis in Marxism.

YouNotSneaky! said...

"Even model economies, conforming to the assumptions of orthodox economics, need not have any tendencies to converge to equilibrium."

Robert,
If these three people found themselves on a deserted island, and they had exactly these preferences, and exactly these endowments, what trading ratios would you expect to observe?
(I'd go with the 1/3's, unstable or not). In other words, so what? Sometimes the real world is unstable.

YouNotSneaky! said...

"Class is not the unit of analysis in Marxism"

This needs elaboration. Class is the unit of analysis in a lot of Marxism. In a lot of classical economics for that matter. Workers eat their wages, capitalist reinvest their profits, that kind of thing.

YouNotSneaky! said...

Scarf's paper available here:
http://cowles.econ.yale.edu/P/cd/d00b/d0079.pdf

Debreu's paper on regular economies here, though you need JSTOR.

http://ideas.repec.org/a/ecm/emetrp/v38y1970i3p387-92.html

Michael Greinecker said...

"This needs elaboration. Class is the unit of analysis in a lot of Marxism. In a lot of classical economics for that matter. Workers eat their wages, capitalist reinvest their profits, that kind of thing."

In Marxism, class is constituted by the production relationships of individual, it is not the basic unit.

Anonymous said...

"In Marxism, class is constituted by the production relationships of individual, it is not the basic unit."

I'm far from being well read in Marxist literature, but I know that Jon Elster style analysis of Marxism from the standpoint of methodological individualism is not a universally shared methodological assumption among Marxists.

Indeed "Marxism" is claimed by so many grouplets --amusingly many, given the small number of Marxists-- that it is best not to say much at all about "Marxism" in general.

Alex

Michael Greinecker said...

"Indeed "Marxism" is claimed by so many grouplets --amusingly many, given the small number of Marxists-- that it is best not to say much at all about "Marxism" in general."

In that case there should be little general problems for Marxism.

Apart from bad pubclic relations, shared by all of them.... ;-)

Anonymous said...

"In that case there should be little general problems for Marxism."

They can be all generally wrong, while having different specific mistaken claims.

Who says you can't have unity within diversity?

Alex