DEBATING THE ROLE OF ECONOPHYSICS
J. Barkley Rosser, Jr.
Department of Economics
James Madison University
Harrisonburg, VA, 22807 USA
rosserjb@jmu.edu
January, 2008
Acknowledgements: The author thanks Mauro Gallegati, Steve Keen, Rosario Mantegna,
Joseph McCauley, and two referees for useful materials provided. The usual caveat
applies.
1
Outbreak of the Debate
Self-conscious research in econophysics has been going on for more than a
decade. Numerous conferences, journal articles, and books have appeared under its
rubric along with considerable publicity in some of the leading general science journals
such as Science and Nature. While most of this research has appeared in physics
journals, some has appeared in economics journals as well, more often when at least one
author is an economist. Strong claims have been made by some advocates regarding its
reputed superiority to economics (McCauley, 2004), with arguments that in fact the
teaching of microeconomics and macroeconomics as they are currently constituted should
cease and be replaced by appropriate courses in mathematics, physics, and some other
harder sciences. The lack of invariance principles in economics and the failure of
economists to deal properly with certain empirical regularities are held against it in this
line of argument, although most econophysicists do not go as far as McCauley in
proposing the complete replacement of economics as such by econophysics.
Unsurprisingly such strong claims have brought forth responding arguments. A
particularly striking set of such arguments have been made by Gallegati, Keen, Lux, &
Ormerod (2006). It must be noted before proceeding further that all of these economist
authors have been involved to some degree in research in econophysics and that all have
been critics of conventional economic theory, some quite vigorously so to the point of
also calling for its demise and replacement with something superior (Ormerod, 1994;
Keen, 2001).1 However, their arguments take the form less of defending economics as
1
This author has even heard Keen in particular advocate econophysics as a superior approach to dealing
with economics problems as compared with conventional economic theory at a 2002 conference in Tokyo.
2
such rather than mounting a criticism of much of what is done in econophysics and how it
is done. Economics may have lots of problems, but the would-be replacers do not clearly
have the solution in their argument.
This group argues four points. The first is that many econophysicists lack
awareness of what has been done in economics and thus sometimes claim a greater
degree of originality and innovativeness in their work than is deserved. The second is
that econophysicists do not use as sufficiently rigorous or sophisticated statistical
methodology as econometricians, an argument that potentially stings given the usual
stance of physicists relative to economists in the area of mathematics.2 The third is that
econophysicists naively believe and search for universal empirical regularities in
economics that probably do not exist. Finally, that the theoretical models they adduce to
explain empirical phenomena have many difficulties and limits.
This paper will consider their arguments in more detail.3 While there is
considerable substance to all of them, it can be argued that in places they have overstated
their case. An obvious solution is for there to be a greater degree of collaboration
between economists and econophysicists to resolve and to avoid these problems,
something that these critics probably agree with. However, before getting into the details
of these arguments, we shall review aspects of the development of econophysics and its
accomplishments up to now in comparison with related ideas in economics. We also note
2
We need to distinguish here between the mathematical rigor of statistical methods used in empirical
research and the mathematical rigor of theory. Thus, at the first Santa Fe conference (Anderson, Arrow,
and Pines, 1988) between physicists and economists, which some see as a major stimulus to the rise of
econophysics, the economists were mostly conventional general equilibrium theorists who proved rigorous
theorems, which exercise the attending physicists found to be absurdly empty and a cover for the lack of
real science that the economics were doing.
3
McCauley (2006) has responded to Gallegati, Keen, Lux, & Ormerod (2006), agreeing with many of their
critiques, but repeating his general argument that economics is so flawed that econophysics should replace
it. Ball (2006) has commented on this debate as well.
3
at this point that the overwhelming majority of these models involve nonlinear dynamics
of one sort or another, with dynamic discontinuities, chaotic dynamics, fractality,
criticality, self-organization, and other such phenomena well-known to students of
nonlinearity commonly appearing in them.
The Development and Accomplishments of Econophysics
Neologizers of the term econophysics,4 Rosario Mantegna and H. Eugene Stanley
proposed the following to define “the multidisciplinary field of econophysics …[as] a
neologism that denotes the activities of physicists who are working on economics
problems to test a variety of new conceptual approaches deriving from the physical
sciences” (Mantegna & Stanley, 2000, pp. viii-ix). Curiously this definition is not an
intellectual one, but a sociological one, based on who is doing the working on economics
problems: in this case, physicists. The more usual way to define a “multidisciplinary
discipline” (or interdisciplinary or transdisciplinary one)5 is to do so in terms of the ideas
or methods that it deals with, as for example political economy or biophysics. This
sociologically oriented definition resembles more the contrast between socio-economics
and social economics, which deal with nearly identical ideas and subject matter, differing
4
See (Chakrabarti, 2005, p. 225) for an account of its neologization, with H. Eugene Stanley being
specifically credited for this innovation at the second Statphys-Kolkata conference in 1995.
5
It can be argued that “transdisciplinary” might be a better label for econophysics. “Multidisciplinary”
suggests distinct disciplines discussing as with an economist and a physicist talking to each other.
“Interdisciplinary” suggests a narrow specialty created out of elements of each separate discipline, such as
a “water economist” who knows some hydrology and some economics. However, “transdisciplinary”
suggests a deeper synthesis of approaches and ideas from the disciplines involved. This is the term favored
by the ecological economics for what they are trying to develop.
4
largely in that the former is largely done by sociologists while the latter is largely done by
economists.6
Given that econophysicists work on economic problems with new conceptual
approaches from the physical sciences, the issue arises as to what these economic
problems are. Perhaps the most intensively studied have been the distributions of returns
in financial markets, (Mantegna, 1991; Gopakrishnan, Plerou, Amaral, Meyer, & Stanley,
1999; Bouchaud & Cont, 2002; Farmer & Joshi, 2002; Sornette, 2003). This has
probably been due to the availability of large quantities of high frequency data in many of
these markets compared to other areas of economics. Also studied have been the
distribution of income and wealth (Levy & Solomon, 1997; Drăgulescu & Yakovenko,
2001; Chatterjee, Yarlagadda, & Chakrabarti, 2005), the distribution of economic shocks
and growth rate variations (Bak, Chen, Scheinkman, & Woodford, 1993; Canning,
Amaral, Lee, Meyer, & Stanley, 1998), the distribution of firm sizes and growth rates
(Stanley, Amaral, Lee, Buldyrev, Havlin, Leschhorn, Maass, Salinger, & Stanley, 1996;
Takayasu & Okuyama, 1998), the distribution of city sizes (Gabaix, 1999), and the
distribution of scientific discoveries (Plerou, Amaral, Gopakrishnan, Meyer, & Stanley,
1999)..
In terms of approaches within physics, widely used models have come from
statistical mechanics (Spitzer, 1971), geophysical earthquake models (Sornette, 2003),
and self-organized criticality models of sandpile avalanche dynamics (Bak, 1996), among
6
This is symbolized by two competing professional associations. The Society for the Advancement of
Socio-Economics (SASE) is dominated by sociologists, although there are economists associated with it,
while the Association for Social Economics (ASE) is essentially the reverse. The ASE holds meetings in
conjunction with the American Economic Association annual meetings, while the SASE does not.
5
others, with all of these involving nonlinear dynamics. Probably most widespread has
been the statistical mechanics approach and variations on it.
Arguably the clearest contrast between the approach of econophysicists and more
regular economists has been in the conviction of the former that many of these
phenomena can be better described using scaling laws that imply non-Gaussian
distributions exhibiting skewness and leptokurtosis in contrast to Gaussian distributions.
A major area of contention regarding this has been in the area of financial returns
distributions. The Gaussian theory dates to Bachelier in 1900, who developed the theory
of Brownian motion five years before Einstein did in order to model financial markets.
This approach would develop through the mean-variance approach to risk analysis and
culminate in such conceptual outcomes as the Black-Scholes formula (Black & Scholes,
1973), widely used for the pricing of options and derivatives in financial markets.
The contrasting approach using scaling laws derives from initially from Pareto in
1897, who used it to study income distribution, and was first applied to financial markets
by Mandelbrot in 1963. The Paretian distribution was also used by Lotka (1926) to
study the pattern of scientific discoveries and by Zipf (1941) to study city size
distributions, as well as by Ijiri and Simon (1977) to study firm size distributions.
A rather curious point is that it is not clear who really preceded whom
disciplinarily in these developments. Models of physics in some cases came from
economics, and the much-derided standard models of economics largely came from
physics. “Econophysics” may well be a new name for an old reality.
Thus, Vilfredo Pareto was an economist and sociologist, even as he preceded
physicists in studying distributions exhibiting scaling laws. Bachelier was a
6
mathematician who studied financial markets, but developed the idea of Brownian
motion before the physicists got to it. On the other hand, it was a physicist, Osborne
(1959), who played a major role in ensuring that standard financial economics would use
the Gaussian Brownian motion model, with the lognormal distribution assumed for the
crucial Black-Scholes model.
The role of physics models as the foundations for the standard neoclassical model
that current econophysicists seek to displace has been much studied (Mirowski, 1989).
Canard in 1801 first proposed that supply and demand were ontologically like
contradicting physical forces. The founder of general equilibrium theory in economics,
Léon Walras (1874), was deeply influenced by the physicist Louis Poinsot (1803) in his
formulation of this central concept.7 The father of American mathematical economics in
its neoclassical form, Irving Fisher (1930), was a student of the father of statistical
mechanics, J. Willard Gibbs (1902). The culmination of this transfer of essentially
nineteenth century physical concepts into standard neoclassical economics came with
Paul Samuelson’s Foundations of Economic Analysis in 1947, Samuelson originally an
undergraduate physics major at the University of Chicago. Furthermore, the more direct
introduction of statistical mechanics applications into economics originally came from
economists themselves (Föllmer, 1974).
Controversies over Empirical Analysis
In discussing the points raised by the critics of econophysics, let us dispense
quickly with the initial charge by agreeing that it has been a problem. There is little point
7
See Walker (2006) for further discussion of this point.
7
in noting specific claims made in particular econophysics articles, but many have made
exaggerated claims of originality that do not hold up on closer examination and where the
problem seems to be simply one of ignorance of the relevant economics literature. An
obvious example has been repeated articles that seem to think that economists have never
been aware that some economic data may exhibit power law distributions. This is clearly
the sort of problem that should disappear in time as the relevant literatures are pointed out
to the relevant researchers, which hopefully will happen sooner rather than later.
However, it is already the case that the more influential econophysicists have largely
remedied this lacunae and are becoming fully cognizant of the relevant literatures,
sometimes aided by working with economists.
This brings us to the second and third charges that have been leveled against
econophysicists, which appear to be linked as they both involve matters regarding
empirical analysis. These are the claims that on the one hand econophysicists resist the
use of rigorous and more advanced statistical methods while on the other hand arguing
for the universality of certain perceived empirical regularities. These are really
intertwined charges as the most serious claim that is not seen as being verified by fully
rigorous statistical methods is precisely that of the universality of various distributions
and parameter values.
Gallegati, Keen, Lux, & Ormerod (2006) specifically cite work in the area of
income and wealth distribution, notably by Clementi & Gallegati (2005), that supports
the idea that while the upper end of the distribution may be Pareto (and thus subject to
power law scaling), the lower 95% or so is more likely lognormal, although some argue
for the exponential distribution. More importantly they argue that the distribution has
8
changed over time, citing the Kuznets curve hypothesis (Kuznets, 1955), which
hypothesizes that over the course of economic development countries tend to move from
equality to inequality and back toward equality. They also note that other distributions
have been studied as well, such as the beta and quadratic elasticity probability density
functions (Bordley, McDonald, Mantrala, 1996). Furthermore, there are serious
problems with data for this case because so much of it has been “binned,” thereby
introducing distortions and making it even more likely that no way of distinguishing
definitively between one series and another can be done. Evidence for universality
simply is very weak for this example.
In the more central area of financial market studies the same problem appears
with more persistent and dramatic claims being made. The failed market crash prediction
based on log-periodic studies by Johansen and Sornette (2001) was oversold. While
Avnir, Biham, Lidar, & Malcai (1998) in a review of papers find scaling exponents
clustering around 1 and 3, fractality would imply an infinite order of magnitudes. While
disagreeing with Gallegati, Keen, Lux, & Ormerod in other areas, McCauley (2006) in a
comment on their work supports them in this area, noting findings of fat tail exponents of
2 to 7 (Dacorogna, Gençay, Müller, Olsen, & Pictet, 2001)..
Gallegati, Keen, Lux, & Ormerod find that the biggest problem is a tendency to
rely on visual inspection of log-log regressions rather than using a variety of available
more rigorous techniques, a point also made in Alfarano, Lux, & Wagner (2006). This
tendency has also been observed on numerous occasions by this author as well. Given
that one of the strongest arguments by econophysicists against economists has been the
widespread refusal of economists to recognize the empirical ubiquity of scaling
9
phenomena in economic data, it is curious indeed that econophysicists have been so slow
in general to use the most sophisticated statistical tests available to examine their
hypotheses. In this area, the critics clearly have made very strong arguments, even if not
all econophysicists are subject to them.
The Problem of Theory
Which brings us to the fourth argument of the critique, the reputed failure of
econophysicists to provide adequate theories to explain the phenomena that they explain.8
While Gallegati, Keen, Lux, & Ormerod are critical of established economic theory on a
variety of grounds, with its tendency to assume equilibrium at the top of the list of their
complaints, they nevertheless defend economic theory as not being “all an empty box.”9
While this observer agrees that economic theory is not all an empty box, the criticisms
against the econophysicists will be seen to go too far.
Two main problems are identified regarding the translation of statistical physics
into economics: 1) the models are only of exchange and do not allow for production, and
2) they often confuse basic concepts such as transactions and income, this latter
especially for models of income and wealth distribution. While the latter has been a
problem in some cases, it is with the first of these that we shall concern ourselves as it is
8
While the critics may find the theories proposed insufficient, some econophysicists have tried to respond
to this issue by attempting to develop theories. Thus Stanley, Gabaix, Gopakrishnan, & Plerou (2006)
propose an argument based on earlier work of Stanley (1971, Appendix B; 1999) on Ising spin models
wherein universal power law distributions arise because of the canceling out of exponential functions on a
lattice as exponential decay occurs with rising temperature while exponential growth of the number of links
between spins with distance. But this approach does not overcome all the critiques of Gallegati et al [2].
9
McCauley (2006) responds in effect that it is a completely empty box, citing a critique made by Osborne
(1977) in doing so. Osborne essentially argues that economists have never succeeded in econometrically
identifying supply and demand curves in any market. Rosser (2006) responds to this, showing serious
flaws in Osborne’s argument. McCauley’s argument effectively amounts to his complaint that the usual
view of economics that equilibria shift with changes in exogenous variables violates the need for invariance
principles and therefore destroys its scientific nature.
10
clearly a deep charge, especially for models of income, which has grown dramatically
over time. The argument here will be that while indeed the first charge holds for most
econophysics models, it need not ultimately for theories based on ideas of statistical
mechanics and physics as applied in economics.
It is certainly the case that while their developers have not made much of the fact,
the overwhelming majority of econophysics models based on statistical mechanics have
been pure exchange models. This has been true of the financial markets models where a
set of assets are assumed to exist along with a set of traders who trade them. While
prices may not be limited and may go up and down nonergodically, there is no production
and no change in the number of objects being traded. Nothing is produced. Empirically
it is not entirely unreasonable to view financial markets as pure exchange phenomena.
However, it is also the case that even in financial markets new assets appear and old ones
disappear. The system is not necessarily conservative in any definite sense over time.
However, that using conservative pure exchange models is a tempting approach is seen in
that many of the economists who have introduced statistical mechanics approaches into
economics have made similar assumptions (Föllmer, 1974).
Can Entropy Provide a Solution?
Which brings us back to the problem of can a model allowing production be based
upon a model of physics, particularly the so-widely used statistical mechanics approach?
There does seem to be a possibility. It is a foundational concept of the original Gibbs
system (1902), namely the maximum entropy principle. Now, we must be very cautious
here. The history of efforts to apply the entropy concept in economics, both by physicists
11
and economists, has been fraught with difficulties and even embarrassment. That great
applier of physics models in economics, Paul Samuelson (1990, p. 263), recipient of the
second Nobel Prize awarded in economics, has observed as follows.
“I have come over the years to have some impatience and boredom with those
who try to find an analogue of the entropy of Clausius or Boltzman or Shannon to put
into economic theory. It is the mathematical structure of classical (phenomenological,
macroscopic, nonstochastic) thermodynamics that has isomorphisms with theoretical
economics.”
Physicists who argued for the application of the law of entropy date back to Helm
in 1887 and Winiarski in 1900, who argued that gold was socio-biological energy. This
was followed by the econometrician Harold Davis in 1941, who suggested that the utility
of money might be “economic entropy.” Aside from the fact that the utility of money
itself is not an easily observed or even well-defined concept, critics such as Lisman
(1949) noted that money simply does not perform an equivalent function in economics to
what entropy does in physics.
A slightly different, and arguably more reasonable tack was taken by Ostwald
(1908), who argued that the conversion factor from energy to value depends on the
availability of specific forms of energy, an idea that inspired the economist Julius
Davidson (1919) to argue that law of diminishing returns ultimately depends on the law
of entropy. Such ideas were criticized by Lotka (1925, p. 355) by noting that there is no
general rule regarding the ratio of energy applied to energy set free, as for example with a
finger pressing a trigger that sets off a pile of dynamite.
12
Georgescu-Roegen (1971) would later argue that entropy constitutes the ultimate
limit and driving force of the economy through ecological relations and processes, but
nevertheless dismissed more simplistic applications of the law of entropy to the
determination of value in economics. Thus, the high entropy beaten egg may be viewed
as more valuable than the low entropy raw egg. Just as conversion factors of energy may
ultimately have arbitrary relations to supply aspects of the economy, so may entropic
elements have arbitrary relations to the utility side of the economy.
While these have been focused on social science areas extending beyond
economics, other efforts have been made to use the entropy concept for such purposes by
researchers coming out of physics backgrounds. One such was Alan Wilson (1970) who
moved from physics to geography and used entropy formulations to model stochastic
models of interregional migration. This topic would also concern researchers associated
with the Stuttgart Institute of Theoretical Physics, notably Weidlich and Haag (1983),
who would eventually favor the master equation approach to such stochastic modeling.
Weidlich’s work (2000) has served as an inspiration to others who have gone on to more
explicitly develop sociophysics since (Chakrabarti, Chakraborti, & Chatterjee, 2006).
So, where then might there be such an application? Returning to the Gibbsian
statistical mechanics and a concern with the nature of statistical distributions may be a
way to proceed, considering the mathematical structure as such, to follow the advice of
Samuelson. Now there have been some efforts made in this direction, such as by Stutzer
(1994) in his analysis of how the Black-Scholes model of asset prices can be derived
from an Arrow-Debreu contingent claims pricing framework (Arrow, 1974, Chap. 1)
13
using entropic formulations. But his analysis also depends on a conservative system in a
pure exchange model framework.
A possible opening is offered in work by Duncan Foley (1994). While most of
the specific examples he presents involve pure exchange models, his framework allows
for both production and exchange. The model does involve some strong assumptions,
most notably that all possible transactions within the economy have an equal probability.
But it does generate outcomes with “entropy prices” that reflect a statistical distribution
of behaviors within the economy. In particular, the number of agents achieving a
particular transaction is inversely proportional to the exponential of its equilibrium
entropy price. He follows the Gibbsian method of assuming that entropy is maximized,
and the solution gives the canonical form of Gibbs from which the entropy shadow prices
are derived and the proportions of each kind of transaction are given. This outcome is
more general than a Walrasian general equilibrium and includes that as a special limiting
case (one in which “temperature” is zero). Unsurprisingly then, this more general
solution does not possess the welfare implications that one obtains with the Walrasian
general equilibrium. Some will gain while some may lose.
Assuming there are m commodities, n agents of r types k with offer set Ak, he
defines hk[x] as the proportion of agents of type k who achieve a particular transaction x
of which there mn. The multiplicity of an assignment is the number of ways n agents can
be assigned to S actions, with ns being the number of agents assigned to action s and is
given by
W[{ns}] = n!/(n1! . . . ns! . . . nS!).
(1)
14
A measure of this multiplicity is the informational entropy of the distribution and is given
by
H{hk[x]} = - Σk=1r Wk Σx=1mn hk[x] ln hk[x].
(2)
This is then maximized subject to feasibility constraints
ΣxεAk hk[x] = 1,
(3)
Σk=1r Wk ΣxεAk hk[x]x = 0,
(4)
and has a unique solution if the feasible set is non-empty of the canonical Gibbs form
hk[x] = exp[-пx]/Σx exp[-пx],
(5)
where the vectors п є Rm are the entropy shadow prices.10
While this formulation poses a possible nonlinear alternative foundation that
might be able to accommodate production and growth, it must be noted that it can be
argued that it is not strictly a physics foundation. This is because this is based on an
informational entropy formulation rather than a strictly thermodynamic one. The
conservative nature of the latter may constrain analysis to the sort of pure exchange
models that the critics find inadequate to explain economic phenomena suitably.
Conclusions
We have considered the debate regarding the use of physics models based on
statistical mechanics to explain economic phenomena, so-called econophysics. Critics
from within economics have charged that some of this work does not fully recognize
relevant work done by economists, uses non-rigorous statistical techniques, claims
10
A curious outcome of this solution is that these prices may be negative. While this has been traditionally
viewed as something not acceptable in economic theory, such prices are sometimes observed in the real
world, e.g. negative prices for water when it is overabundant in a flood, or people paying to get rid of a
piece of property that is encumbered with lawsuits to clean up costly environmental problems. Probably
physicists are more open to such ideas than economists.
15
universalities that do not exist, and does not provide a reasonable theoretical foundation
for understanding economic phenomena. While the first three of these are granted to
have some foundation, the latter is argued to be possibly resolved by the use of the theory
of informational entropy, following the Gibbsian statistical mechanics formulation.
However, it is recognized that some might argue that this is not a true physics model.
More generally, probably the best way forward that will avoid the kinds of
problems raised by the critics of econophysics will be to involve both economists and
physicists in the research jointly. This will best achieve the goals enunciated by the
econophysicists and their predecessors such as the late Ettore Majorana (1942), who
advocated integrated statistical approaches to the physical and social sciences.
Inevitably, nonlinear dynamics will be central to this integrated approach.
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