The Flawed Foundations of General Equilibrium Theory

Critical Essays on Economic Theory

Frank Ackerman, Alejandro Nadal, Kevin P. Gallagher

• Publisher: Taylor & Francis
• Year: 2004
• Print ISBN: 9780415654869
• eBook ISBN: 9781135997373
• Edition: 1
• Topic: Business
• Subtopic: Business General
• Index: Business

This book, as the title suggests, explains how General equilibrium, the dominant conceptual framework in mainstream economics, describes a perfectly impossible world. Even with its counterfactual assumptions taken for granted, it fails on many levels. Under the impressive editorship of Ackerman and Nadal, this book will appeal to students and resea

1 Still dead after all these years

Interpreting the failure of general equilibrium theory

Frank Ackerman

For years after the Spanish dictator actually died, the mock television newscast on Saturday Night Live was periodically interrupted with a “news flash” informing viewers that “General Franco is still dead!” This served both to satirize the breathlessly urgent style of television news reporting, and to suggest that after many decades of taking an absolute ruler for granted, the world needed more than one reminder that he was no longer alive and well.

Much the same is true for general equilibrium theory. In the course of its long decades of rule over the discipline of economics, general equilibrium became established as the fundamental framework for theoretical discourse. Its influence continues to spread in policy applications, with the growing use of computable general equilibrium models. At its peak it even colonized much of macroeconomics, with the insistence on the derivation of rigorous microfoundations for macro models and theories. General equilibrium theory is widely cited in a normative context, often in textbooks or semitechnical discussion, as providing the rigorous theoretical version of Adam Smith’s invisible hand and demonstrating the desirable properties of a competitive economy.

Yet those who follow the news about microeconomic theory have known for some time that general equilibrium is not exactly alive and well anymore. The equilibrium in a general equilibrium model is not necessarily either unique or stable, and there are apparently no grounds for dismissing such ill-behaved outcomes as implausible special cases. This conclusion is clearly at odds with established modes of thought about economics; several more “news flashes” will be required to assimilate and interpret the failure of earlier hopes for general equilibrium models, and to formulate new directions for economic theory.

The first section of this chapter, “The limits of general equilibrium,” presents one such news flash, summarizing and explaining the evidence of fundamental flaws in general equilibrium theory. But simply hearing the news one more time is not enough. The goal of this chapter is to develop a basic, intuitively comprehensible understanding of why it happened, as a guide to future theorizing. What features of the general equilibrium model led to its failure? What changes in economic theory are needed to avoid the problem in the future?

The second section, “Explanations of the fall,” examines contemporary interpretations of the findings of instability. Some attempts have been made to avoid the issue, without success. Despite occasional claims to the contrary, general equilibrium remains fundamental to the theory and practice of economics. Analysts who have faced the problem have identified two underlying causes: the inherent difficulties of the aggregation process, and the unpredictable nature of individual preferences.

The third section, “The limits of analogy,” pursues the roots of the problem in the early history of general equilibrium theory: a mathematical framework transplanted from nineteenth-century physics was far less fruitful in economics, due to fundamental differences between the two fields. The provocative treatment of this topic by Philip Mirowski asks the right questions, but falls short of adequately answering them.

The final section, “Alternatives for the future,” briefly describes alternative approaches that might remedy the earlier flaws in neoclassical theory. Post-general equilibrium economics will need a new model of consumer behavior, new mathematical models of social interaction, and an analysis of the exogenous institutional sources of stability.

The limits of general equilibrium

Now the reason for this sterility of the Walrasian system is largely, I believe, that he did not go on to work out the laws of change for his system of General Equilibrium.

(John Hicks 1939: 61)

The best-known results of general equilibrium theory are the two theorems proved by Kenneth Arrow and Gerard Debreu in the 1950s. First, under familiar assumptions defining an idealized competitive market economy, any market equilibrium is a Pareto optimum. Second, under somewhat more restrictive assumptions, any Pareto optimum is a market equilibrium for some set of initial conditions.

There is a long-standing debate about the interpretation of the Arrow–Debreu results, in light of the obvious lack of realism of some of their assumptions. For example, nonconvexities, such as increasing returns to scale in production, are common in reality. If they are allowed into the theory, then the existence of an equilibrium is no longer certain, and a Pareto optimum need not be a market equilibrium (i.e., the second theorem no longer holds).

Yet despite awareness of this and other qualifications, economists frequently talk as if deductions from general equilibrium theory are applicable to reality. The most common and most important example involves the relationship between efficiency and equity. (For a critical review of the standard approach to the subject, see Putterman et al. (1998).) The second fundamental theorem is often interpreted to mean that any efficient allocation of resources – for instance, one based on a preferred distribution of income – could be achieved by market competition, after an appropriate lump-sum redistribution of initial endowments.

This interpretation is a mistaken one. Even if the conditions assumed in the proofs applied in real life (which they clearly do not), meaningful application of the Arrow–Debreu theorems would require dynamic stability. Consider the process of redistributing initial resources and then letting the market achieve a new equilibrium. Implicitly, this image assumes that the desired new equilibrium is both unique and stable. If the equilibrium is not unique, one of the possible equilibrium points might be more socially desirable than another, and the market might converge toward the wrong one. If the equilibrium is unstable, the market might never reach it, or might not stay there when shaken by small, random events.

Beyond stability

In the 1970s, theorists reached quite strong, and almost entirely negative, conclusions about both the uniqueness and the stability of general equilibrium. There is no hope of proving uniqueness in general, since examples can be constructed of economies with multiple equilibria. The fundamental result about uniqueness, achieved by Debreu in 1970, is that the number of equilibria is virtually always finite (the set of parameters for which there are an infinite number of equilibria has measure zero). There are certain restrictions on the nature of aggregate demand that ensure uniqueness of equilibrium, but no compelling case has been made for the economic realism of these restrictions.

For stability, the results are, if anything, even worse. There are examples of three-person, three-commodity economies with permanently unstable price dynamics (Scarf 1960), showing that there is no hope of proving stability of general equilibrium in all cases. The basic finding about instability, presented in a limited form by Sonnenschein (1972) and generalized by Mantel (1974) and Debreu (1974), is that almost any continuous pattern of price movements can occur in a general equilibrium model, so long as the number of consumers is at least as great as the number of commodities.¹ Cycles of any length, chaos, or anything else you can describe will arise in a general equilibrium model for some set of consumer preferences and initial endowments. Not only does general equilibrium fail to be reliably stable; its dynamics can be as bad as you want them to be.

A common reaction to this Sonnenschein–Mantel–Debreu (SMD) theorem is to guess that instability is an artifact of the model, perhaps caused by uncommon or unrealistic initial conditions, or by the nature of the assumed market mechanisms. Investigations along these lines have failed to revive general equilibrium, but instead have driven more nails into its coffin.

The SMD result cannot be attributed to a specific, rigid choice of individuals’ preferences, nor to a particular distribution of income. In a sweeping generalization of the SMD theorem, Kirman and Koch (1986) proved that the full range of instability can result – this is, virtually any continuous price dynamics can occur – even if all consumers have identical preferences, and any arbitrarily chosen income distribution is used, as long as the number of different income levels is at least as great as the number of commodities. This means that the SMD theorem can be established even for a population of nearly identical consumers – with identical preferences and almost, but not quite, equal incomes (Kirman 1992).

Another important generalization shows that “SMD instability” may be a property of an economy as a whole even if it is not present in any part, or subset, of the economy (Saari 1992). Suppose that there are n commodities; even if every subset of the economy with n – 1 or fewer commodities satisfies conditions that guarantee stability of equilibrium, it is still possible to have “arbitrarily bad” dynamics in the full n-commodity economy. This means, among other things, that the addition of one more commodity could be sufficient to destabilize a formerly stable general equilibrium model. More generally speaking, dynamic results that are proven for small general equilibrium models need not apply to bigger ones.

Might instability be just a result of the unrealistic method of price adjustment assumed in general equilibrium models? Again, the answer is no. In Walrasian general equilibrium, prices are adjusted through a tâtonnement (“groping”) process: the rate of change for any commodity’s price is proportional to the excess demand for the commodity, and no trades take place until equilibrium prices have been reached. This may not be realistic, but it is mathematically tractable: it makes price movements for each commodity depend only on information about that commodity.² Unfortunately, as the SMD theorem shows, tâtonnement does not reliably lead to convergence to equilibrium. An early response to the problem of instability was the exploration of alternative mechanisms of price adjustment; but several economically plausible mechanisms failed to ensure stability except under narrow special conditions (Fisher 1989).

On the other hand, any price adjustment process that does reliably converge to equilibrium must be even less realistic, and far more complex, than tâtonnement. There is an iterative procedure that always leads to a market equilibrium, starting from any set of initial conditions (Smale 1976). However, there is no apparent economic justification for this procedure, and it requires overwhelming amounts of information about the effects of prices of some goods on the demand for other goods.

A final negative result has been achieved on this question, showing that any price adjustment process that always converges to an equilibrium has essentially infinite information requirements (Saari 1985). Consider any iterative price adjustment mechanism, in which current prices are a smooth function of past excess demand and its partial derivatives. If there is an upper bound on the amount of information used in the adjustment process – that is, if it relies solely on information about any fixed number of past periods and any fixed number of derivatives of the excess demand function – then there are cases in which the process fails to converge. These cases of nonconvergence are mathematically robust; that is, they occur on open sets of initial conditions, not just at isolated points.

Safety in numbers?

Not much is left, therefore, of the original hopes for general equilibrium. One direction in which theoretical work has continued is the attempt to deduce regularities in aggregate economic behavior from the dispersion of individual characteristics. This approach abandons efforts to prove that market economies are generically stable, and instead suggests that conditions that lead to stability are statistically very likely to occur, even if not quite guaranteed.

In particular, Hildenbrand (1994) and Grandmont (1992) have explored the hypothesis that the dispersion of individual preferences is a source of aggregate stability. That is, predictable, smoothly distributed differences in individuals’ demand functions and consumption patterns, of the sort that are observed in reality, could lead to a definite structure of aggregate demand that might imply stability of equilibrium. (For reviews of this line of work, see Kirman (1998), Lewbel (1994), and Rizvi (1997).)

There are two problems with the statistical approach to economic stability. First, it has not yet succeeded. The assumption of a smooth distribution of consumer characteristics seems to help, but has not entirely freed the proof of market stability from arbitrary restrictions on individual preferences or aggregate demand functions.

Second, even if the statistical approach were to succeed in explaining past and present market stability, it would remain vulnerable to future changes in preferences. Suppose that it is eventually demonstrated that the empirically observed dispersion of consumer preferences is sufficient to ensure stability in a general equilibrium model. This finding might not be reliable for the future, since, in the real world, fads and fashions episodically reorganize and homogenize individual preferences. That is, coordinated preference changes involving the media, fashions, celebrities, brand names, and advertising could, in the future, reduce the dispersion of consumer preferences to a level that no longer guaranteed stability.

Explanations of the fall

In the aggregate, the hypothesis of rational behavior has in general no implications.

(Kenneth Arrow 1986)

The mathematical failure of general equilibrium is such a shock to established theory that it is hard for many economists to absorb its full impact. Useful interpretations of its causes and significance have been slow to appear. This section begins with a presentation and critique of three views that suggest that the SMD theorem is not as important as it looks. It then turns to other interpretations offered by two of the theorists whose work was referred to in the previous section.

Three styles of denial

Is the SMD result only a mathematical curiosity, of limited importance for economics? At least three major arguments make that claim, on the basis of disinterest in dynamics, disinterest in abstraction, and disinterest in the particular theories of the past. As we will see, none of the three is persuasive.

First, some essentially say that we’re just not a dynamic profession. A recent graduate text in microeconomic theory presents a detailed explanation and proof of the SMD theorem and then, a few pages later, tells students that

A characteristic feature that distinguishes economics from other scientific fields is that, for us, the equations of equilibrium constitute the center of our discipline. Other sciences, such as physics or even ecology, put comparatively more emphasis on the determination of dynamic laws of change.

Mas-Colell et al. 1995: 620)

Second, perhaps it was always silly to care so much about empty abstractions. According to Deirdre McCloskey, the whole category of general equilibrium theorizing is merely “blackboard economics,” exhibiting the “rhetoric of mathematical formalism”:

None of the theorems and countertheorems of general equilibrium theory has been surprising in a qualitative sense . . . But the qualitative sense is the only sense they have. . . . The problem is that the general theorem of Arrow and Debreu or any of the other qualitative theorems do not, strictly speaking, relate to anything an economist would actually want to know.

(McCloskey 1994: 135; emphasis in original)

Things an economist would actually want to know, for McCloskey, necessarily involve information about how big something really is compared to something else.

Finally, it could be that those on the inside track already have learned to avoid the theoretical dead ends of the past. This view is common in conversation with economists, if not in writing. “No one,” it is alleged, believes in general equilibrium theory anymore; the profession has moved on to game theory, complexity theory, evolutionary frameworks, and other techniques, allowing the creation of sophisticated new models that do not fit into the old Arrow–Debreu mold.

Each of these claims is narrowly true and broadly false. In a narrow sense, they describe the behavior of numerous economists: many do focus on static rather than dynamic theoretical problems; many others are predominantly engaged in empirical work; and there are theorists who no longer use a general equilibrium framework. Yet in a broad sense, each of these observations misses the point.

The first claim, the dismissal of dynamics, fails because all significant applications of theo...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contributors
• Acknowledgments
• Introduction
• 1 Still Dead after All These Years
• 2 Behind the Building Blocks
• 3 Money and Prices
• 4 The Law of Supply and Demand in the Proof of Existence of General Competitive Equilibrium
• 5 Consumed in Theory
• 6 Choice of Technique Revisited
• 7 Existence Values and Priceless Externalities
• 8 Contradictions of the Open Economy Model as Applied in Mexico
• 9 An Offer You Can’t Refuse
• 10 Computable Abstraction
• 11 Freedom and Submission
• References