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Abstract More than 25 years after the discovery that the equilibrium point of a

general equilibrium model is not necessarily either unique or stable, there is still a

need for an intuitively comprehens ible explanation of the reasons for th is discovery.

Recent accounts identify two causes of the ®nding of instability: the inherent

di culties of aggregation , and the individualistic model of consumer behaviour.

The mathematical dead end reached by general equilibrium analysis is not due to

obscure or esoteric aspects of the model, but rather arises from intentional design

features, present in neoclassical theory since its beginnings. Modi®cation of

economic theory to overcome these underlying problems will require a new

model of consumer choice, nonlinear analyses of social interactions, and recogni-

tion of the central role of institutional and social constraints.

Keywords: general equilibrium theory, instability, Sonnenschein-Mantel-Debreu

theorem, mathematical modelling, economics and physics, Mirowski

1 INTRODUCT ION

For years after the Spanish dictator actually died, the mock television

newscast on `Saturday Night Live’ was periodically interrupte d with a

`news ¯ash’ 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 establishe d as the fundamenta l framework for theoretical discourse.

Its in¯uence 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 semi-

technical discussion, as providing the rigorous theoretica l version of Adam

Journal of Economic Methodology 9:2, 119±139 2002

Still dead after all these years: interpreting

the failure of general equilibrium theory

Frank Ackerman

Journal of Economic Methodology ISSN 1350-178X print/ISSN 1469-942 7 online #2002 Taylor & Francis Ltd

http://www.tandf.co.uk/journals

DOI: 10.1080/1350178021013708 3

Smith’s invisible hand and demonstratin g 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 any more.

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 outcome s as implausible special cases. This conclusio n is clearly at

odds with established modes of thought about economics ; several more `news

¯ashes’ 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 second section of this paper presents one such news ¯ash, summarizin g

and explaining the evidence of fundamental ¯aws in general equilibrium

theory. But simply hearing the news one more time is not enough. The goal of

this paper 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?

Section 3 examines contemporary interpretation s of the ®ndings 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 identi®ed two underlying causes: the inherent

di culties of the aggregation process, and the unpredictable nature of

individual preferences.

Section 4 pursues the roots of the problem in the early history of general

equilibrium theory: a mathematical framework transplanted from nine-

teenth-century physics was far less fruitful in economics, due to fundamenta l

di erences between the two ®elds. The provocativ e treatment of this topic by

Philip Mirowski asks the right questions, but falls short of adequately

answering them.

The ®nal section brie¯y describes alternative approaches that might

remedy the earlier ¯aws in neoclassical theory. Post-general-equilibriu m

economics will need a new model of consume r behaviour , new mathematica l

models of social interaction, and an analysis of the exogenous institutional

sources of stability.

2 TH E L IMI TS OF GEN E RAL E QU ILI BRIU M

Now the reason for this sterility of the Walrasia n 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)

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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 assumption s de®ning an idealized competitiv e market economy, any

market equilibrium is a Pareto optimum. Second, under somewhat more

restrictive assumptions , any Pareto optimu m is a market equilibrium for some

set of initial conditions.

There is a longstanding 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 marke t equilibrium (i.e., the second theorem no longer

holds).

Yet despite awareness of this and other quali®cations, economists fre-

quently talk as if deductions from gen eral equilibrium theory are applicabl e to

reality. The most common and most important example involves the relation-

ship between e ciency and equity. (For a critical review of the standard

approach to the subject see Putterman et al. 1998.) The second fundamenta l

theorem is often interpreted to mean that any e cient allocation of resources

± for instance, one based on a preferred distributio n of income ± could be

achieved by market competition, after an appropriate lump-sum redistribu-

tion 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 theorem s would require dynamic stability. Consider the

process of redis tributin g initial resources and then letting the market achieve a

new equilibrium. Implicitly, this image assumes that the desired new equili-

brium 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.

2.1 Beyond stability

In the 1970s theorists reached quite strong, and almost entirely negative,

conclusions about both the uniqueness and the stability of general equili-

brium. There is no hope of proving uniqueness in general, since examples can

be constructed of economies with multiple equilibria. The fundamenta l result

about uniqueness, achieved by Debreu in 1970, is that the number of

equilibria is virtually always ®nite (the set of parameters for which there are

an in®nite number of equilibria has measure zero). There are certain

restrictions on the nature of aggregate demand that ensure uniqueness of

Interpreting the failure of general equilibrium theory 121

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-commodit y economies with permanentl y unstable price

dynamics (Scarf 1960), showing that there is no hope of proving stability of

general equilibrium in all cases. The basic ®nding about instability, presented

in a limited form by Sonnenschein (1972) and generalize d 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-Debre u (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 co n.

The SMD result cannot be attributed to a speci®c, rigid choice of

individuals’ preferences, nor to a particular distribution of income. In a

sweeping generalizatio n of the SMD theorem, Kirman and Koch (1986)

proved that the full range of instability can result ± i.e., 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 di erent 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 consumer s ± with identical preference s and

almost, but not quite, equal incomes (Kirman 1992).

Another important generalizatio n 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 ncommodities;

even if every subset of the economy with n-1 or fewer commodities satis®es

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

su cient to destabiliz e 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 unrealisti c method of price

adjustment assumed in general equilibrium models? Again, the answer is

no. In Walrasian general equilibrium, prices are adjusted through a

ta

Ãtonnement (`groping’) process: the rate of change for any commodity’s

price is proportional to the excess demand for the commodity, and no trades

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take place until equilibrium prices have been reached. This may not be

realistic, but it is mathematicall y tractable : it makes price movements for

each commodity depend only on information about that commodity.

2

Unfortunately, as the SMD theorem shows, ta

Ãtonnement 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 ta

Ãtonnement. 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 apparen t economic justi®cation for this procedure, and

it requires overwhelming amounts of informatio n about the e ects of prices of

some goods on the demand for other goods.

A ®nal, negative result has been achieved on this question, showing that

any price adjustment process that always converges to an equilibrium has

essentially in®nite 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 adjustmen t process, i.e., if it

relies solely on information about any ®xed number of past periods and any

®xed number of derivatives of the excess demand function, then there are

cases in which the process fails to converge. These cases of nonconvergenc e

are mathematicall y robust; that is, they occur on open sets of initial

conditions, not just at isolated points.

2.2 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 behaviour from the dispersion of indi-

vidual characteristics . This approach abandons e orts to prove that market

economies are generically stable, and instead suggest s 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, smoothl y distributed di erences in

individuals’ demand functions and consumption patterns, of the sort that are

observed in reality, could lead to a de®nite structur e of aggregate demand that

might imply stability of equilibrium. (For reviews of this line of work, see

Kirman 1998, Lewbel 1994; Rizvi 1997).

There are two problems with the statistical approach to economic stability.

Interpreting the failure of general equilibrium theory 123

First, it has not yet succeeded. The assumptio n of a smooth distributio n 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 vulnerabl e to future changes in

preferences. Suppose that it is eventually demonstrate d that the empirically

observed dispersion of consumer preferences is su cient to ensure stability in

a general equilibrium model. This ®nding 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 advertisin g

could, in the future, reduce the dispersion of consumer preferences to a

level that no longer guarantee d stability.

3 EXPLANATIONS OF TH E FA L L

In the aggregate, the hypothesis of rational behaviour 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 signi®cance 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 o ered by two of the theorists whose work appeared in

Section 2.

3.1 Three styles of denial

Is the SMD result only a mathematica l 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 microeconomi c theory presents a detailed explanation

and proof of the SMD theorem and then, a few pages later, tells students that

A characteristic featur e that distinguishe s economics from other scienti®c

®elds 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 determinatio n of dynamic laws of

change.

(Mas-Colell et al. 1995: 620)

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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 mathematica l formalism’:

None of the theorems and countertheorems of general equilibrium theory

has been surprisin g in a qualitativ e 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 qualitativ e theorem s do not, strictly

speaking, relate to anything an economist would actually want to know.

(McCloskey 1994: 135)

Things an economist would actually want to know, for McCloskey, neces-

sarily 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 any more; the profession has moved on to game

theory, complexit y theory, evolutionary frameworks, and other techniques,

allowing the creation of sophisticate d new models that do not ®t into the old

Arrow±Debreu mold.

Each of these claims is narrowly true and broadly false. In a narrow sense,

they describe the behaviour of numerous economists: many do focus on static

rather than dynamic theoretical problems; many others are predominantl y

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 ®rst claim, the dismissa l of dynamics, fails because all signi®cant

applications of theory are inherentl y dynamic. This idea is not unknown to

economists: some of the earliest theoretical responses to the instability of

general equilibrium involved the exploration of alternativ e dynamics

(although those explorations ultimately failed, as explained in Fisher 1989).

In an ever-changing world, or a model of that world, static properties of

equilibrium have no practical meaning unless they persist in the face of small

disturbances. Advocacy of a policy based on its static optimality in a general

equilibrium framewor k ± a common conclusion in applied economics ±

implicitly assumes some level of dynamic stability, since otherwise the

optimum might not last for long enough to matter. Yet the dynamic stability

of the general equilibrium framework is precisely what is called into question

by the SMD theorem.

In the face of the ongoing mathematica l escalation in economic theory, the

second claim, McCloskey’s call to turn away from empty formalism toward

real empirical work, has a certain refreshing charm. But even applied

researchers often present their work in terms of the abstractions of what

Interpreting the failure of general equilibrium theory 125

she calls `blackboard economics’. Facts do not spontaneousl y assemble

themselves into theories; some theory is present, explicitly or implicitly, at

the beginning of any empirical study.

In applied economics today, it is increasingly common to ®nd explicit

reliance on a general equilibrium framework, often in the form of computable

general equilibrium (CGE) models. A typical CGE study is an exercise in

comparative statics: the model is run twice, once to calculate the equilibrium

before a policy change or other innovation, and once to calculate the new

equilibrium afte r the change. Such an approac h rests on what Paul Samuelso n

called the `correspondenc e principle’; he argued that in the neighbourhood of

a stable equilibrium, comparative statics would yield reliable results while

avoiding the ne ed for more complex dynami c analysis of adjustment processe s

(Samuelson 1947). Unfortunately , Samuelson’ s strongest conclusions apply

only to models with just two or three variables. For bigger models, the

instability re¯ected in the SMD theorem undermines the correspondence

principle as well (Hands and Mirowski 1998, Kehoe 1989).

The third claim, the notion that `no one’ believes in general equilibrium any

more, is true only of small circles of avant-gard e theorists. Look anywhere

except at the most abstractl ytheoretical journals, and general equilibrium still

characterizes the actual practice of economics. General equilibrium models

have become ubiquitous in such important areas as trade theory and

environmental economics, and are continuin g to spread. Macroeconomics,

a ®eld that once developed its own very di erent theories, has been led into the

pursuit of rigorous microfoundations ± in e ect seeking to deduce aggregate

behaviour from general equilibrium theory. An on-line search for publica-

tions on `general equilibrium’ turns up more than a thousand citations per

year, with no evidence of declining interest in the subject.

3

General equilibrium is fundamenta l to economics on a more normative

level as well. A story about Adam Smith, the invisible hand, and the merits of

markets pervades introductor y textbooks , classroom teaching, and contem-

porary political discourse. The intellectual foundation of this story rests on

general equilibrium, not on the latest mathematical excursions. If the

foundation of everyone’s favourite economics story is now known to be

unsound ± and according to some, uninteresting as well ± then the professio n

owes the world a bit of an explanation.

3.2 Individualism and aggregation

There are some theorists who have recognized the importance of the failure of

general equilibrium theory. Two of the authors cited above, Alan Kirman and

Donald Saari, have published thoughtful re¯ections on the subject. Kirman,

in a dramatically titled article (`The intrinsic limits of modern economic

theory: the emperor has no clothes’), argues that:

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The problem seems to be embodied in what is an essential feature of a

centuries-long tradition in economics , that of treating individuals as acting

independently of each other . . . This independenc e of individuals’

behaviour plays an essential role in the constructio n of economies

generating arbitrary excess demand functions [the source of instability in

the SMD theorem].

(Kirman 1989: 137±38)

Saari considers the problem from a mathematician’s perspective. Examin-

ing the `Mathematical Complexity of Simple Economics’ (Saari’s title), he

explores the SMD theorem and related results, and concludes that:

[T]he source of the di culty ± which is common across the social sciences ±

is that the social sciences are based on aggregation procedures . . . One way

to envision the aggregation di culties is to recognize that even a simple

mapping can admit a complex image should its domain have a larger

dimension than its image space . . . [T]he complexity of the social sciences

derives from the unlimited variety in individual preferences; preferences

that de®ne a su ciently large dimensional domain that, when aggregated,

can generate all imaginable forms of pathological behaviour.

(Saari 1995: 228±29)

There are two separate points here: one involves the methodology of

aggregation, and the other concerns the behavioural model of the individual .

Both are basic causes of the instability of general equilibrium.

Instability arises in part because aggregate demand is not as well behaved

as individual demand. If the aggregate demand function looked like an

individual demand function ± that is, if the popular theoretical ®ction of a

`representative individual’ could be used to represent marke t behaviour ± then

there would be no problem. Unfortunately , though, the aggregation problem

is intrinsic and inescapable. There is no representative individual whose

demand function generates the instability found in the SMD theorem

(Kirman 1992). Groups of people display patterns and structure s of beha-

viour that are not present in the behaviour of the individual members; this is a

mathematical truth with obvious importance throughou t the social sciences.

For contemporary economics, this suggests that the pursuit of micro-

foundations for macroeconomics is futile. Even if individual behaviour were

perfectly understood , it would be impossi bl e to draw useful conclusion s about

macroeconomics directly from that understanding, due to the aggregation

problem (Rizvi 1994, Martel 1996). This fact is re¯ected in Arrow’s one-

sentence summary of the SMD result, quoted at the beginning of this section.

The microeconomic model of behaviour contributes to instability because

it says too little about what individuals want or do. From a mathematica l

standpoint, as Saari suggests, there are too many dimensions of possible

variation, too many degrees of freedom, to allow results at a useful level of

Interpreting the failure of general equilibrium theory 127

speci®city. The consumer is free to roam over the vast expanse of available

commodities, subject only to a budget constraint and the thinnest possible

conception of rationality: anythin g you can a ord is acceptable, so long as

you avoid blatant inconsistency in your preferences .

The assumed independence of individuals from each other, emphasized by

Kirman, is an important part but not the whole of the problem. A reasonabl e

model of social behaviour should recognize the manner in which individual s

are interdependent; the standard economic theory of consumptio n fails to

acknowledge any forms of interdependence, except throug h market transac-

tions. However, merely amending the theory to allow more varied social

interactions will not produce a simpler or more stable model. Indeed, if

individuals are modelled as following or conforming to the behaviour of

others, the interactions will create positive feedback loops in the model,

increasing the opportunity for unstable responses to small ¯uctuations (see

Section 5).

4 TH E L IMI TS OF ANA L OGY

There is a fairly close analogy between the earlier stages of economic

reasoning and the devices of physical statics. But is there an equally

serviceable analogy between the later stages of economic reasoning and

the methods of physical dynamics? I think not.

(Alfred Marshal l 1898)

How did economists come to spend so much time and e ort on general

equilibrium, only to arrive at a mathematical dead end? What was the source

of such longstanding devotion to an ultimately unworkable theory? The

problems identi ®ed in the last section ± the inherent di culties of aggregation ,

and the underspeci ®ed model of individual behaviour ± are not new, and

cannot be blamed on the latest mathematica l wrinkles in the formulation of

general equilibrium .In particular, the microeconomi c behavioural model was

an intentional feature of the theory, and has been present in something like its

current form ever since neoclassical economics was born. A look at the history

of economic thought may help to identify what went wrong in the beginning .

In their history of the idea of economic equilibrium , Ingrao and Israel

(1990) argue that mathematical modelling of economic systems was a

continuation of a major current in eighteenth- and nineteenth-century Euro-

pean social thought, seeking to identify lawlike regularitie s in social life and

organization. Once the idea of equilibrium was given a mathematical form,

though, the mathematics itself became the predominan t in¯uence on the

further developmen t of the theory.

The outlines of general equilibrium theory ®rst appeared in the work of

Leon Walras, as part of the `marginalist revolution’ of the 1870s. The sudden

interest in marginalism in economics in the 1870s is commonly attributed to

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the in¯uence of mid-nineteenth-centur y advances in mathematics and the

physical sciences. Thus the original structure of general equilibrium theory

re¯ects the manner in which economists applied the new mathematica l

techniques of the era.

4.1 Breaking the conservation law

The in¯uence of physics on the origins of neoclassical economics is analyse d in

depth in an important, controversial work by Philip Mirowski (1989). The

pioneers of marginalism in economics, including Walras, tried to develop

analogies to mechanics in some detail. According to Mirowski, several of the

early neoclassical economists adopted very similar mathematica l models, and

incorporated similar ¯aws.

The physics of the day, much admire d by economists , assigned a central

role to the conservatio n of energy. Potential energy could be represente d as a

vector ®eld, indicating the direction in which particles would move unless

constrained by other forces. The economic analog y treated individuals as

particles moving in commodity space, where the spatial coordinates are

quantities of di erent commodities.

4

Utility was the vector ®eld indicating

the direction in which individuals would move, to the extent allowed by

budget constraints.

The problem with this economic analogy, for Mirowski, was the failure to

take or to understand the logical next step. In physics, the model of potentia l

energy as a vector ®eld induces predictable movements of particles and leads

to a related concept of kinetic energy, measured in the same units as potentia l

energy. The law of energy conservation applies to the sum of potential plus

kinetic energy, not to either one alone. Much of the power of the physical

theory and the e ectiveness of its use of mathematic s derives from the energy

conservation law.

In the economic analogy, if utility as a potential ®eld induces predictable

movements of individuals in commodity space, the `kinetic energy’ of that

movement should be consumer expenditure. The exact analogue of the law of

energy conservation would thus be the conservatio n of the sum of utility plus

expenditure, an economically meaningles s concept. At this point the analogy

breaks down. The resulting economic theory remained fragmented, using bits

and pieces of the mathematical apparatus related to energy conservatio n but

unable to draw on the full strength and coherence of the original physical

theory. In particular , the economists adapte d some of the relationships of

static equilibrium but failed to incorporat e the more complicated dynamic

relationships from physics.

The debate surroundin g Mirowski’s argument (see, for example, Varian

1991, Walker 1991, Cohen 1992, Hands 1992, Carlson 1997 and de Marchi

1993) raises many other issues, as does his original work. On the central

point about the close relationship between physics and early neoclassica l

Interpreting the failure of general equilibrium theory 129

economics, Mirowski poses the right question, but his answer is at best

incomplete. Mirowski certainly demonstrates that Walras, Jevons, Pareto,

Fisher, and other neoclassical pioneers discussed analogies to physics in great

detail, without always understandin g the mathematic s that was involved.

Moreover, he is persuasive in suggestin g that this episode of intellectual

history had a formative impact on modern economic theory. Yet Mirowski’s

version of what went wrong with neoclassical theory is frustrating on two

accounts.

First, why should economics need an exact analogue to energy conserva-

tion? The failure to create a precisely analogous principle might be taken as a

recognition that economics and physics are not identical in structure . While

the particular mathematical methods that work in physics are therefor e not

available, others , more appropriate to economics, could be created . In a sense,

it is true that something must be conserved in any economic theory that allows

quanti®cation and causal analysis; otherwise, there would be no way to

compare magnitudes and events at di erent times (Mirowski 1990). However,

this does not imply that the same thing, or the analogous thing, must be

conserved in two di erent theories.

Second, while Mirowski makes a remarkably strong case for the idea that

the earliest neoclassicals were mediocre mathematicians , that early history

does not explain the persistence of mistakes through successive generations of

economists. In the thorough reworking of neoclassical theory in the 1930s and

1940s by Hicks, Samuelson, von Neumann, and others, it seems unlikely that

past mathematical errors and oversights would have survived unnoticed.

Nonetheless, the problem remains as Mirowski describes it: the original

formulation of general equilibrium by Walras and others relied heavily on

analogies to physics, often using the same mathematica l structures, i.e., the

same mathematical metaphor for reality. Why did this metaphor prove so

much more fruitful in physics than in economics? Answers must be sought in

features of the economic model that are intentionally di erent from the

physical analogue, and have therefore persisted through more than a century

of development of economic theory. Two such answers are suggested in the

following subsections, involving the number of dimensions in the model, and

the individual, asocial nature of preferences.

4.2 Lost in commodity space

The analogy between mechanics and economics makes the spatial coordinates

of a particle correspond to the quantities of commodities held by an

individual. Once this step is taken there is already a signi®cant di erence

between the two theories, involving the number of dimensions.

Physics is, in this respect, the more modest of the two ®elds. Physical

particles have three spatial coordinates; they travel in the familiar world of

three-dimensiona l space. A paradigm-changin g innovation, the theory of

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relativity, adds just one more dimension to create a four-dimensional space-

time continuum . Abstract higher-dimensiona l construction s are common in

physics (e.g., the phase spaces of elaborate theoretical systems), but the

resulting theories have observable, testable implications for events in the

low-dimensional space of our physica l experience.

The analogous space of our economic experience is a commodity space

that, in a modern industrial economy , may have hundreds of thousands of

dimensions. Even in Walras’ day, there must have been thousands of distinct

commodities, and hence thousands of dimensions in a general equilibrium

model of the economy as a whole. This is no phase space used to explain a

simpler experiential space; the thousands of independent dimensions (com-

modities) are fundamenta l to the characterizatio n of economic experience in

neoclassical theory.

Intuition is a poor guide to the vast commodity spaces of economics. The

ability to visualize shapes and motion drops o rapidly as the number of

dimensions increases beyond three. The dynamic possibilities for a model are

far more complex in three dimension than in two; how much greater

complexity is introduced by going to thousands of dimensions? No system

of actual equations in such high-dimensiona l spaces can be comprehended or

manipulated. All that can be done is to prove completely generic results, or to

talk about low-dimensional ± usually visualizable, two- or three-dimensiona l

± examples and special cases. Yet as noted in Section 2, these special cases may

be misleading: dynamic results that can be proved for smaller general

equilibrium models need not apply to bigger ones.

The huge number of dimensions of commodit y space is not only a

mathematical problem. It also a ects the plausibility of the economic

model of the consumer . The consumer must be able to reveal her preferences

about any of the commodities on the market; this may require knowledge of

hundreds of thousand s of di erent items. Whenever a new commodity

appears on the market, she must be able to revise her preference ordering at

once to re¯ect the change. Clearly, no real person can come close to ful®lling

this role. Response s to this problem are discussed in Section 5.

4.3 A di erent drummer

The original neoclassical analogy to physics made utility comparable to

energy. Yet there are crucial ways in which utility and energy di er. In

physics, the same potential energy ®elds a ect all particles in the same

manner, allowing structure and predictability in the movements of large

groups of particles. In the analogous economic theories, a di erent utility

function motivate s each individual , giving a group of people a structureless,

unpredictable pattern of change. No common forces move them all in

parallel; no interactions with each other, save through market exchange ,

coordinate their motions.

Interpreting the failure of general equilibrium theory 131

This is no mathematica l accident, but rather a result of the microeconomic

behavioural model, which concentrate s on one aspect of human activity and

assumes away everythin g else. In fact, human behaviour involves a complex

combination of relatively predictabl e responses to social forces on the one

hand, and unpredictable individual preferences and choices on the other

hand. The former aspect is where an analogy to physical laws of motion might

have proved most valuable ± but the latter is the focus of neoclassical

economics.

At ®rst glance, physics and economics both appear to rely on an

unobservable force: potential energy, like utility, is not directly observable.

But potential energy is indirectly deducible from , and commensurabl e with,

observable data. The potential energy of any one particle is readily compared

to any other. In economics, in contrast, the lack of an observational

measure of utility and the absence of interpersonal comparability meant

that Walrasian general equilibrium was devoid of empirical content.

5

Not all the early neoclassical economists saw utility as a completely

individual and unmeasurable matter. Alfred Marshall and Arthur Pigou

maintained that interpersonal compariso n of utility was at least sometimes

possible, for group averages if not for individuals. If this `material welfare’

school of economics (Cooter and Rappopor t 1984) had remained dominant, a

di erent analysis of utility and preferences, and hence a di erent model of

equilibrium, might have emerged. However, any hints of a distinct Marshal-

lian paradigm were swept away by the `ordinalist revolution’ of the 1930s,

which resolved the conceptual problems with utility by banishin g it altogether

in favour of revealed preference.

The revealed preference account of consumer choice does not escape all the

philosophical problems surrounding the subject (Sen 1973, Sago 1994). Nor

does it eliminate the asocial individualism of the model, the feature which

subverts structure and prediction of group behaviour. Each individual still

marches to a di erent drummer, even if the drums are now labeled `revealed

preference relation’ instead of `utility function’. Naturally, this leaves no way

of telling where the parade is headed.

5 AL TERNATIV ES FO R T HE FU T U RE

How disappointing are the fruits, now that we have them, of the bright idea

of reducing Economics to a mathematica l application of the hedonistic

calculus . . .

(John Maynard Keynes 1963 : 155n)

General equilibrium is still dead. Exactly 100 years after the 1874

publication of Walras’ most important work, the SMD theorem proved

that there was no hope of showing that stability is a generic property of

market systems. More than a quarter-centur y of additional research has

132 Articles

found no way to sneak around this result, no reason to declare instability an

improbable event. These negative ®ndings should challenge the foundation s

of economic theory. They contradict the common belief that there is a

rigorous mathematical basis for the `invisible hand’ metaphor; in the original

story, the hand did not wobble.

6

While the SMD theorem itself appears mathematically esoteric, we have

seen that its roots are traceable to simple, intentional features of the

neoclassical model, which have been present since the beginning. What

happens to economic theory when those features are changed? Where

should we look for new alternatives for the future? This concluding section

examines three areas where new theoretical approaches are needed in

response to the failure of general equilibrium: the commodity-base d model

of consumer choice; the analysis of social interactions; and the role of

institutional sources of stability.

5.1 So many commodities , so little time

The discussion, in Section 4, of the high-dimensiona l nature of the traditiona l

model of consume r choice led to criticism of the implausibly large information

processing requirements which the theory imposed on consumers. This

criticism of the neoclassical model has been raised before, perhaps initially

in Herbert Simon’s arguments for bounded (rather than global) rationality.

It is also reminiscent of a classic series of attempts to reconceptualize

consumer choice. In roughly simultaneous, independent work, Kelvin Lan-

caster (1966), Richard Muth (1966), and Gary Becker (1965) each proposed

that what consumers actually want is not goods per se, but rather character -

istics of the goods, or experiences produced by consuming or using the goods.

This is consistent with the manner in which psychologists , sociologists, and

anthropologists generally understand the process of consumption . Yet

surprisingly little has come of this approach in economics. Of the two

major versions of the theory, Lancaster’s more rigidly ± probably too rigidly

± structured model was never developed much beyond its provocative initial

presentation. Meanwhile, Becker’s more amorphous `househol d production

function’ model is, like the neoclassical theory of consumption, capable of

being stretched to ®t virtually all possible situations, and hence ends up

explaining very little (Ackerman 1997, Goodwin et al. 1997).

Whether an alternative is based on these foundation s or other approaches ,

it remains importan t to create a mathematically manageable behavioural

model with informatio n requirement s on a human scale. Such a model cannot

be expressed primarily in terms of ownership, knowledge of, and response to

individual commodities, simply because there are so many of them. Human

needs and behaviour must be described in terms of other categories, more

limited in number.

Among other changes, this makes it virtually impossible to demonstrate

Interpreting the failure of general equilibrium theory 133

the optimality of consumer choices and market outcomes. Optimization

would require global rationality with its unrealisticall y high information

requirements. Any realistic behavioura l theory will, in contrast, embody

some form of bounde d rationality, de®ned over a far smaller set of choices

± because that is all that real people have time for.

5.2 Blowing bubble s

The weaknesses in neoclassical theory that ultimately led to the SMD

theorem, as described in Section 4, include not only the high-dimensiona l

model of consumer choice, but also the asocial, individualistic nature of

preferences. The absence of social forces that in¯uence individual s (with the

sole exception of market exchange) makes the results of the theory under-

determined and unpredictable. In reality, of course, there are numerous

nonmarket social interactions which impart structure to group behaviour.

The importance of these social interactions has been recognized in some

recent work in economics , leading to models that add a touch of realism to the

theory of economic behaviour.

However, models of interaction do not necessarily contribute to an

explanation of market stability. On the contrary, social conformit y or

emulation ± such as wanting a consumer good because other people already

have it ± can create positive feedback in the market, with the potential for

destabilization. In the simplest terms, erratic and fragile market bubbles or

`cascades’ can occur if individuals consider the behaviour of others to be a

better source of information than their own knowledge or preferences

(Bikhchandani et al. 1998).

Similar problems arise in more elaborate models. Deterministic nonlinear

models can lead to chaotic dynamics , while agent-based models, simulating

the actions of individuals under hypothesize d behavioura l rules, often display

nearly chaotic outcomes that have been dubbe d `complexity’ . In such models,

positive-feedback interactions ± in which one person’s action makes it more

likely that another will act in the same way ± are the source of either chaos or

complexity. These interactions would be commonplace in a realistic, com-

prehensive theory of individua l economic activity. As Saari puts it, `Eco-

nomics so e ortlessly o ers the needed ingredients for chaos that, rather than

being surprised about exotic dynamics , we should be suspicious about models

which always are stable’ (Saari 1996: 2268).

Chaos and complexity models are characterized by sensitive depend-

ence on initial conditions, the antithesis of stability. Thus some promising

approaches to modelling social interaction s threaten to compound the

problem of instability in economic theory. While the underspeci®ed social

structure assumed by neoclassical economics contributes to the indeter-

minacy of its outcomes, the most obvious cures seem to worsen the

disease.

134 Articles

5.3 Macrofoundation s of microeconomic s

Where, then, doe s stabilit y come from? Theoretical analysi s to date, which has

been impressive in its depth and breadth, has show n that stability is simply not

an endogenous mathematical property of market economies under all initial

conditions. This provides an elegant theoretical justi®cation for a return to

traditional styles of macroeconomics, in which cyclical ¯uctuation s and

potential instability of aggregate incomes are central topics of concern. Yet

the demonstration of the robustness of `SMD instability’, combined with

recent research on nonlinear dynamics, chaos, and complexity , appears to

prove too much. Market economies are only episodicall y unstable or chaotic;

it is certainly the norm, not the exception , for markets to clear and for prices to

change smoothly and gradually.

7

In short, it is more obvious in practice than in theory that large,

complicated market economies are usually stable. If it is so di cult to

demonstrate that stability is endogenous to a market economy, perhaps it is

exogenous. That is, exogenous factors such as institutional contexts, cultural

habits, and political constraints may provide the basis for stability, usually

damping the erratic endogenous ¯uctuations that could otherwise arise in a

laissez-faire economy. Variations on this theme can be found in several

alternative schools of thought, such as Marxist, feminist, and institutionalis t

economics. Mirowski (1991) has even rooted a similar argument in the

discourse of postmodernism .

There are other approaches , closer to conventional theory, that also make

institutions central to economic analysis. The need for exogenous sources of

stability is one of the tenets of what David Colander calls `post-Walrasia n

macroeconomics’. Colander (1996) identi®es three distinguishing character-

istics of the post-Walrasia n perspective. First, the equations necessary to

describe the economy have multiple equilibria and complex dynamics.

Second, individuals act on the basis of local, bounded rationality, since

global rationality is beyond anyone’s information processing capabilities.

Finally, institutions and non-price coordinating mechanisms are the source of

systemic stability in a market economy. Colander refers to the last of these

characteristics as establishing the macrofoundation s of microeconomics. The

`post-Walrasian’ initiative is an encouraging one, but much more remains to

be done to create a comprehensive alternative theory, building on what is now

known about the limitations of established models .

Colander’s approach , like any of the alternative schools of thought, would

lead economists to take a humbler stance than they often do in public debate.

The guaranteed optimality of market outcomes and laissez-faire policies died

with general equilibrium . If economic stability rests on exogenous social and

political forces, then it is surely appropriat e to debate the desirable extent of

intervention in the market ± in part, in order to rescue the market from its own

instability.

Interpreting the failure of general equilibrium theory 135

To recapitulate the main points in closing: Section 2 explained the fact that

general equilibrium is, indeed, still dead after all these years. There are two

principal causes of the death, as seen in Section 3. The instability of the

neoclassical model can be attribute d to the inescapable di culties of the

aggregation process, and the highly individual, asocial nature of consume r

preferences. These are not recent innovations, but design ¯aws that have been

present since the origin s of the theory in the late nineteenth century. Section 4

argued that two intentional feature s of the theory, present in the original

neoclassical analogy to physics, led economics astray : the huge number

of dimensions and information requirements of the `commodity space’

framework, and the individualisti c behavioura l model.

Repairing these ¯aws, as described in Section 5, will require a model of

human needs and behaviour that is not de®ned in term s of individual

commodities. It will involve mathematicall y complex analyses of social

interaction. And it will have to recognize the central role of social and

institutional constraints . These new departures will make economics more

realistic, but will not demonstrate the inherent stability or optimality of

market outcomes, as general equilibrium theory once seemed to do.

The death of General Franco was not a panacea for the problem s of Spain.

Yet it did open many democratic, pluralist options, no longer requiring the

whole country to follow one authoritarian leader. Spain after Franco looks a

lot more like neighbouring countries in the freedom of expression that it o ers

its citizens, and the diversity of opinions that can be expressed in public

debate. The same might yet be true of economics after general equilibrium.

Frank Ackerman

Tufts University

Frank.Ackerman@tufts.edu

AC KNO W LEDGEMEN T S

Helpful comments on earlier drafts were provided by Herb Gintis, Neva

Goodwin, Deirdre McCloskey, Philip Mirowski, Irene Peters, Donald Saari,

Diana Strassman, participants in the University of Massachusetts/Amherst

political economy seminar, and the referees for this journal.

NO T ES

1 Recent work in general equilibrium theory has typically assumed a pure exchange

economy, without production. The obstacles to proving uniqueness or stability

seem to arise on the consumer side of the market; including production would

make the mathematics more complicated, but would not change the results

discussed her e. Re al-world applications of the th eory, of course, require mo delling

of production as well as consumption.

136 Articles

2 Other relatively simple adjustment mechanisms have been proposed, such as

quantity adjustment in a ®xed-price environment. Rizvi (1994) argues that

analyses of such mechanisms have often relied on speci®c, ad hoc forms for

aggregate excess demand, making them vulnerable to the SMD critique.

3 Publications with the subject `general equilibrium’ listed in the EconLit database

of the American Economics Association increased from about 100 per year in the

early 1980s to more than 1,000 per year throughout the 1990s. Although the

number of all EconLit citations grew rapidly in those years, the number of `general

equilibrium’ citations grew even faster.

4 In comments on an earlier draft of this paper, Mirowski has objected that the

original neoclassical model, as described in his work, makes points in physical

space analogous to commodity bundles, but says nothing about physical

analogues to individuals. The distinction seems to me a rather thin one:

commodity bundles are meaningful as bundles only because they are, or could

be, held by individuals; conversely, individuals are located, in an exchange

economy, solely by the commodity bundles they possess.

5 More recently it has been argued that, if people respond rationally to lotteries, it is

possible to deduce their utility functions, which are unique up to a linear

transformation (solving the problem of measurement for an individual, though

still not allowing interpersonal comparison). However, this view, introduced by

von Neumann and Morgenstern, emerged only after the `ordinalist revolution’,

and has always been a minority perspective among neoclassical economists. Thus

it has played very little part in the historical developments described in the text.

Moreover, emp irical eviden ce suggests that people often do not res pond rationally

to lotteries, undermining this approach to measurement of utility.

6 The weakness of the metaphorical hand has attracted other comments along the

same lines: if the market economy is not stable, `one would be forced to

acknowledge that . . . Smith’s `invisible hand’ wavers Sisyphus-like around the

actually existing equilibrium position without having the strength to push the

economic system into it’ (Ingrao and Israel 1990: 331).

7 For my own empirical work on one of the exceptions, see Ackerman and

Gallagher (2002).

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