Journal of Economic Dynamics & Control 31 (2007) 3255–3280
The rise and fall of catastrophe theory
applications in economics:
Was the baby thrown out with the bathwater?
J. Barkley Rosser Jr.
Department of Economics, MSC 0204 James Madison University, Harrisonburg, VA 22807, USA
Received 20 September 2004; accepted 15 September 2006
Available online 29 January 2007
Abstract
This paper discusses the mathematical origins of catastrophe theory, the various
applications of it in economics, the controversy over its use, and the criticism of it as a fad,
with the subsequent general disappearance of its use in economics. It presents a criticism of the
criticism of the most famous application and a discussion of its current relevance and available
alternatives. It concludes that indeed the baby was largely thrown out with the bathwater, and
that catastrophe theory should be openly and properly used again in economics.
r 2006 Elsevier B.V. All rights reserved.
JEL classification: B23; C02
Keywords: Catastrophe theory; Multiple equilibria; Structural stability; Intellectual fads
1. Introduction
The science writer, Horgan (1995, 1997), has ridiculed what he labels
‘chaoplexology,’ a combination of chaos theory and complexity theory. A central
charge against this alleged monstrosity is that it, or more precisely its two component
parts separately, are (or were) fads, intellectual bubbles of little consequence. They
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www.elsevier.com/locate/jedc
0165-1889/$ - see front matter r 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.jedc.2006.09.013
E-mail address: rosserjb@jmu.edu.
would soon disappear and deservedly so, once scholars and intellects realized what
worthless dross they truly were (or are). As the culminating centerpiece of his
argument, Horgan introduced the label, ‘the four C’s,’ which consist of cybernetics,
catastrophe theory, chaos theory, and complexity theory.1
More particularly,
Horgan singled out catastrophe theory as the supreme example of an intellectual fad
to which he compared chaos and complexity theory. This comparison was supposed
to constitute the definitive proof of his argument, its pie` ce de re´ sistance, the point
that would send any right-minded or sensible individual running for their intellectual
respectability while screaming in horror at the very idea of taking either chaos theory
or complexity theory even remotely seriously. Clearly, Horgan considered
catastrophe theory to be such an utterly worthless intellectual stock that the very
mention of it in conjunction with another idea would trigger an immediate and
relentless crash of the other idea’s value in the intellectual bourse once and for all.
Such is the current status of catastrophe theory as perceived by many observers of
the intellectual scene.
That this is supposedly the status of catastrophe theory more generally means that
it is also largely its status in economics as well. It is a discounted idea or approach or
method or theory that no ambitious junior scholar would dare to refer to in a paper
except either in ridicule or in a remote footnote with little further discussion. Within
the last decade hardly any paper in a leading journal in economics has appeared that
had any reference to ‘catastrophe theory.’ Recent papers that apparently use some
version it often avoid mentioning that they are doing so (Wagener, 2003), although
there seem to be fewer inhibitions in physics (Leahy, 2001; Kuznetsova et al., 2004)
and biology (Li et al., 2004). Whether catastrophe theory was a loveable baby or a
bucket of worthless bathwater, it has been largely thrown out by economists. The
case would seem to be closed, with the widespread nature of its rejection seeming
the final proof that it was really just bathwater after all and that Horgan was
fully justified to hold it up as the prime example of a ridiculous and worthless
intellectual fad.
This paper suggests that this viewpoint needs reconsideration. The reference to
‘the baby being thrown out with the bathwater’ was first applied to the question of
catastrophe theory during a debate over its use by Oliva and Capdeville (1980) in
Behavioral Science. This suggests that there was some bathwater that needed to be
thrown out, but that catastrophe theory itself was not that bathwater, that it was in
fact a not totally unloveable baby that deserved to be preserved and raised in a
proper household. Sins of intellectual hype and exaggeration were committed as
were inappropriate applications of the theory. It is not as widespread in application
as its original proponents claimed and is not a general intellectual panacea. There
was a fad and an intellectual bubble, and it was perfectly reasonable that there
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1
Rosser (1999) agrees with Horgan that the four C’s are linked through their common use of nonlinear
dynamical systems. But he argues that this is something to be celebrated and appreciated rather than
denigrated or dismissed. Just as the term ‘impressionism’ in art was originally bestowed by a critic, so the
‘chaoplexologists’ should accept the label originally provided in derision and wear it in pride. Ironically at
the time that catastrophe theory was first criticized in 1977, its main founder, Thom (1969, 1972) worried
that it could ‘have the same fate as cybernetics.’ (Aubin, 2001, p. 274).
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–32803256
should have been a discounting and a downgrading. However, this has been
overdone; the intellectual marketplace has inefficiently overshot on the downside. A
sign of this is that the field in which the attitude towards catastrophe theory did not
fall so low is mathematics, probably because it did not rise so high in the first place.
Economists should reevaluate the former fad and move it to a more proper
valuation.
2. The emergence of catastrophe theory out of general bifurcation theory
Catastrophe theory studies structurally stable singularities of dynamical systems,
an important aspect of bifurcation theory. Bifurcation theory was principally
invented/discovered by Poincare´ (1880–1890) as part of his qualitative analysis of
systems of nonlinear differential equations. This arose from his study of celestial
mechanics and the famous three-body problem in particular. Would the orbits of the
planets in the solar system escape to infinity, remain within certain bounds, or would
the planets crash into each other or the sun? Beyond this question he investigated the
structural stability of the system, studying if small perturbations to it would leave it
relatively unchanged in its behavior or cause it to move in a very different manner,
the central focus of bifurcation theory.
Although Poincare´ was the first to formally analyze bifurcation theory, Arnol’d
(1992, Appendix) has provided a list of precursors to Poincare´ . According to
Arnol’d, although an observer can find hints of it in some work of Leonardo da
Vinci, the first clear presentation of the structural stability of a cusp point came in
the study of light caustics and wave fronts was by Huygens in 1654. Critical points in
geometrical optics were studied by Hamilton in 1837–1938, and by the late 19th
century algebraic geometers such as Cayley, Kronecker, and Bertini were examining
the singularities of curves and smooth surfaces, even in textbooks on algebraic
geometry. Nevertheless, it was Poincare´ who brought structure to this discussion.
Consider a general family of differential equations whose behavior is determined
by a k-dimensional control parameter, m
dx=dt ¼ f mðxÞ; x in Rn
; m in Rk
. (1)
Equilibrium solutions are given by f mðxÞ ¼ 0. This set of equilibria will bifurcate into
separate branches at a singularity, or a degenerate critical point. More precisely, a
singularity occurs where the Jacobian matrix Dxf mðxÞ has zero eigenvalues.
Intuitively a single stable curve of equilibrium points may split into several curves
at such a point, with some stable and others unstable locally. At such points the first
derivative may be zero but the function may not be at an extremum. There are many
different kinds of bifurcations, with Guckenheimer and Holmes (1983) and
Kuznetsov (1998) summarizing various types, with this analysis also being viewed
as the study of non-hyperbolic equilibria of vector fields.
The distinction between critical points of functions that are non-degenerate
(associated with extrema) and degenerate ones (singular, non-invertible Hessian) was
further studied by Morse (1931) who showed how a function with a degenerate
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J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–3280 3257
singularity could be slightly perturbed to a new function that would now exhibit two
distinct non-degenerate critical points instead of the singularity. This was a
bifurcation of the degenerate equilibrium and indicates the close connection between
the singularity of a mapping and its structural stability (see Fig. 1).
Whitney (1955) followed Morse by discovering/inventing the two singularities
associated with the two most commonly studied kinds of elementary catastrophes,
the fold and the cusp (see Fig. 2), showing that these were the only two kinds of
structurally stable singularities for differentiable mappings between two planar
surfaces. Thus Whitney can be viewed as the real founder of catastrophe theory.
Following his discovery of transversality Thom (1956), developed further the
classification of singularities, or of elementary catastrophes, although a more
complete categorization would eventually be carried out by Arnol’d et al. (1985),
who showed that for systems beyond a dimensionality of 11, the categories of
catastrophes become infinite and thus difficult to categorize. Thom (1972,
pp. 103–108) would label such catastrophes as ‘generalized’ or ‘non-elementary.’
More particularly, Thom (1972) studied the seven elementary catastrophes going up
through six dimensions in control and state variables. Within the context of
dynamical systems the control variables are usually conceived as moving slowly,
perhaps exogenously to some degree, while the state variables are viewed as moving
more rapidly, adjusting endogenously to changes in the control variables and moving
to the equilibrium manifold. This became standard (or ‘elementary’) catastrophe
theory.
Consider a dynamical system given by n functions on r control variables, ci. The n
equations determine n state variables, xj
xj ¼ f jðc1 . . . crÞ. (2)
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Fig. 1. Bifurcation at a singularity.
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–32803258
Let V be a potential function on the set of control and state variables
V ¼ Vðci; xjÞ (3)
such that for all xj
qV=qxj ¼ 0. (4)
Under regularity conditions this set of points constitutes the equilibrium manifold,
M, and an example is seen in the cusp catastrophe seen in Fig. 2, which is
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Fig. 2. (a) Fold catastrophe, (b) cusp catastrophe.
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–3280 3259
characterized by two control variables and one state variable. In much discussion the
control variables are characterized as being ‘slow,’ whereas the state variables are
characterized as being ‘fast.’ The usual presumption has been that the state variables
adjust quickly to be on the equilibrium manifold while the control variables move
the system around on the manifold. The catastrophe function is the projection of the
equilibrium manifold into the r-dimensional control variable space, with its
singularities the main focus of catastrophe theory.
Thom’s Theorem, proven by Malgrange (1966) and Mather (1968), states that if the
underlying functions f j are generic (qualitatively stable under slight perturbations), if
ro6, and if n is finite with all but two state variables being represented by linear and
non-degenerate quadratic terms, then any singularity of a catastrophe function will be
structurally stable (generic) under slight perturbations and can be classified into 11
different types. There are seven such types for ro5, and Thom (1972) provided colorful
names for each of these, along with detailed discussions of their various characteristics,
with further discussion carried out by Trotman and Zeeman (1976). For r45 and more
than two control parameters, the set of possible catastrophes is infinite, although as
noted above Arnol’d et al. (1985) have extended the finite set somewhat.
Overviews of the theory and some applications across disciplines can be found in
Poston and Stewart (1978), Woodcock and Davis (1978), Gilmore (1981), Thompson
(1982), Arnol’d (1992), and Castrigiano and Hayes (1993). Portions of the
mathematical presentation given here as well as the discussion of the controversy
about catastrophe theory can be found in Rosser (2000a, Chapter 2).
Although some applications of higher dimensional catastrophes have appeared in
urban and regional economics, most applications in economics have involved the
two simplest forms known to Whitney in 1955, the fold and the cusp depicted in
Fig. 2. In order to analyze a particular model using one of these one must make
assumptions regarding how the system moves between equilibria in situations of
multiple equilibria, which is a more general problem in bifurcation theory (Wiggins,
1990, p. 384). This is often dealt with via conventions. Under the Maxwell
convention a system will immediately jump to a new equilibrium zone, whereas
under the delay convention a system will remain in the old equilibrium zone until the
last possible point before it vanishes (Gilmore, 1981, Chapter 5). In his original
discussion Thom (1972, p. 44) explicitly simplified his analysis by assuming the
Maxwell convention. For real systems all kinds of intermediate possibilities abound
and must be determined empirically. More generally for the fold catastrophe four
kinds of behavior can occur: hysteresis, bimodality, inaccessibility, and sudden jumps,
with divergence also happening for the cusp catastrophe.
The question of Thom’s assumption of the Maxwell convention is associated with
a deeper problem that we have avoided so far, that the definition of catastrophe
theory itself is somewhat fuzzy (although we effectively presented one in the opening
sentence of this section). His use of the Maxwell assumption rules out a variety of
cases such as the cusp case studied by Whitney originally.2
Thus the neologizer of the
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2
An anonymous referee provides a summary definition of the Thom version of a metabolic catastrophe
point that exhibits this problem. ‘Let M and L be the manifolds of ‘internal parameters’ and ‘control
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–32803260
term, Zeeman (1974, p. 623) complains that ‘there is strictly speaking no ‘catastrophe
theory,’ but then this is more or less true for any non-axiomatic theory in
mathematics that attempts to describe nature.’
Disagreements about the definition are relevant to the discussion of the
controversy of catastrophe theory made below. One issue is exactly the degree to
which time necessarily enters into the definition. Thus, Thom assumes that
catastrophe theory is necessarily about dynamical systems, and this is the approach
used above. However, it can be argued that the broader singularity theory of
Whitney is about mappings more generally that may not involve time at all. Arnol’d
(1992, p. 6) especially emphasizes that ‘the combination of singularity theory and its
applications should be called catastrophe theory,’ although he attributes this
formulation to Zeeman.3
This split between the original Thom formulation and the more generalized
Arnol’d formulation also shows up in regard to the question of whether or not the
systems must have a potential function, for which there is a necessary symmetry
condition that all cross-partial derivatives must be equal. Again, the broader
singularity theory does not require this, and forms that resemble the elementary
catastrophes can appear within this theory even while not fulfilling the stricter
assumption about the existence of a potential function. This would become a central
issue in the later controversy over catastrophe theory.
3. Some applications in economics
Critics of catastrophe theory have argued that many applications of it in many
fields violated necessary assumptions or were carried out questionably for one reason
or another. Let us list a few examples of applications in economics, most of which
were done in a reasonable manner.4
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(footnote continued)
parameters.’ For every l 2 L, there is a vector field Xl given on M, and an attracting equilibrium aðlÞ 2 M
of the vector field Xl. A metabolic catastrophe point is any value of l for which aðlÞ, or one of its
derivatives, is discontinuous.’
3
Arnol’d (1992, p. 2): ‘Singularity theory is a far-reaching generalization of the study of functions at
maximum and minimum points. In Whitney’s theory functions are replaced by mappingsyCatastrophes
are abrupt changes arising as a sudden response to a smooth change in external conditions.’ Arnol’d (1992,
p. 7): ‘Since smooth mappings are found everywhere, their singularities must be everywhere also, and since
Whitney’s theory gives significant information on singularities of generic mappings, we can try to use this
information to study large numbers of diverse phenomena and processes in all areas of science. This simple
idea is the whole essence of catastrophe theory.’ I thank two referees for bringing to my attention the
importance of this issue. I also note the irony that there are competing definitions of chaos theory as well
(Rosser, 2000a, Chapter 2).
4
For more extensive discussion of the urban and regional examples see Rosser (1991, Chapters 9–11),
the ecological and environmental examples see Rosser (1991, Chapters 12–14), and for the international
finance examples see Rosser (1991, Chapters 15–16). For more extensive discussion of the microeconomic
theory examples see Rosser (2000a, Chapter 3), macroeconomic models see Rosser (2000a, Chapter 6),
capital theory examples see Rosser (2000a, Chapter 8), and for the financial market examples see (Rosser,
2000a, Chapter 5).
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–3280 3261
The earliest published application was due to Zeeman (1974) and was an effort to
model bubbles and crashes in stock markets. This example has been criticized
(Sussman and Zahler, 1978a; Weintraub, 1983), using arguments we shall discuss
later.
Debreu (1970) set the stage for applying catastrophe theory to general equilibrium
theory in distinguishing regular from critical economies, the latter containing
equilibria that are singularities. Discontinuous structural transformations of general
equilibria in response to slow and continuous variation of control variables can
occur at such equilibria. Analysis of this possible phenomenon was carried out using
catastrophe theory by Rand (1976)5
and Balasko (1978).6
Rand in particular derives
such a case when at least one trader in a pure exchange economy has non-convex
preferences, as depicted in Fig. 3, which shows the Edgeworth–Bowley box for such
a case.
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Fig. 3. Pareto set with catastrophe thresholds.
5
Two years later, Rand (1978) would publish the first self-conscious model of chaotic dynamics in
economics, although others had provided such examples earlier without realizing what they were, e.g.
Strotz et al. (1953). This latter paper drew on the nonlinear accelerator model of Goodwin (1951), which
could also potentially be analyzed using catastrophe theory in the same manner as that of Kaldor (1940),
but which this author is unaware of anyone actually doing so.
Although not specifically using catastrophe theory, Debreu’s colleague in mathematics at Berkeley,
Smale (1974) also studied structural stability of general equilibria, drawing on his earlier work on
genericity that also provided a foundation for chaos theory (Smale, 1967). Smale was in close contact with
Thom and Zeeman during the early 1970s (Aubin, 2001), and would later encourage Hal Varian to study
catastrophe theory, leading him to develop his business cycle model based on the Kaldor model, discussed
below (I thank Weintraub, 1983, for this information, who interviewed Varian).
6
Balasko ultimately sided with the critics of applications of catastrophe theory in economics by noting
that some of the necessary mathematical conditions are rarely fulfilled, especially that of a potential
function.
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–32803262
Bonanno (1987) studied a model of monopoly in which there were non-monotonic
marginal revenue curves due to market segmentation. Multiple equilibria can
arise with smoothly shifting cost curves, which he analyzed using catastrophe
theory.
Perhaps the most influential application of catastrophe theory in economics was to
the analysis of business cycles in a paper by Varian (1979). He adopted a nonlinear
investment function of Kaldor (1940) as modified by Chang and Smyth (1971) to
construct the following model:
dy=dt ¼ sðCðyÞÞ þ Iðy; kÞ À y, (5)
dk=dt ¼ Iðy; kÞ À I0, (6)
CðyÞ ¼ cy þ D, (7)
with y as national income, k as capital stock measured against a long-run trend, CðyÞ
as the consumption function, Iðy; kÞ as the gross investment function with I0 an
autonomous level of replacement investment, and s the speed of adjustment
parameter assumed to be rapid relative to the movements of the capital stock. The
nonlinear investment function was assumed to have a sigmoid shape and would shift
with the capital stock as depicted in Fig. 4, with S ¼ I being the equilibrium
condition.
Within this model a hysteresis cycle with discontinuities can arise as the investment
function shifts back and forth during the course of a business cycle, as depicted in
Fig. 5.
Varian then extended this model by allowing the consumption function to include
wealth, w, as a control variable as follows:
Cðy; wÞ ¼ cðwÞy þ DðwÞ, (8)
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Fig. 4. Nonlinear investment function.
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–3280 3263
with c0
ðwÞ40 and D0
ðwÞ40. This formulation allows for a tilting of the savings
function such that there are no longer any multiple equilibria. This allowed Varian to
distinguish between simple recessions and longer term depressions. This was depicted
by a cusp catastrophe in which wealth is the splitting factor, as depicted in Fig. 6.
One of the few efforts to empirically estimate a catastrophe theory model in
economics was of a model of inflationary hysteresis involving a presumably shifting
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Fig. 5. Business cycle as fold catastrophe.
Fig. 6. Business cycle as cusp catastrophe.
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–32803264
Phillips Curve. This was due to Fischer and Jammernegg (1986). The method they
used was a multi-modal density function due to Cobb (1978, 1981), with further
discussion by Cobb et al. (1983) and Cobb and Zacks (1985, 1988). For U.S. data for
the period of June 1957 to June 1983, they found a cusp point in the space of the
unemployment rate and inflationary expectations of about 7% for each variable.
This drew on an ad hoc model suggested by Woodcock and Davis (1978), and in
effect argued that this system could be viewed as a cusp catastrophe, with the
economy jumping to the ‘higher’ sheet of the equilibrium manifold during 1973–1974
and then back down again, but then jumping up again at the end of 1977 only to
gradually come back down by going around the cusp point after 1980. It is curious
that this multi-modal approach developed by Cobb is mentioned by few of the critics
of practical applicability of catastrophe theory.
Drawing on models due to Bruno (1967) and Magill (1977), Rosser (1983)
analyzed dynamic discontinuities in an optimal control theoretic growth theory
model that contained capital theoretic paradoxes. Ho and Saunders (1980)
developed a catastrophe theory model of bank failure when risk factors go beyond
critical levels.
The areas of urban and regional economics saw especially large numbers of
applications of catastrophe theory, including the use of catastrophe theory models of
higher dimensionality than the three-dimensional cusp catastrophe seen above,
although some of this work was done by geographers rather than economists and
much of it is open to questions regarding ad hocracy. Amson (1975) initiated the
formal use of it with a cusp catastrophe model of urban density as a function of rent
and ‘opulence.’ Mees (1975) modeled the revival of cities in medieval Europe using
the five-dimensional ‘butterfly’ catastrophe. Wilson (1976) studied modal transportation
choice as a fold catastrophe. Dendrinos (1979) modeled the formation of
urban slums using the six-dimensional parabolic umbilic or ‘mushroom’ catastrophe.
Andersson (1986) modeled ‘logistical revolutions’ in interurban transportation and
communications relations and patterns as a function of long run technological
change using a fold catastrophe. Following a fully mathematically satisfying
framework, structural change in regional trading systems was analyzed using
the five-dimensional hyperbolic and elliptic umbilic catastrophes by Puu (1979,
1981a, b)7
and by Beckmann and Puu (1985).
Within ecologic–economic systems considerable focus has been paid to systems in
which there are discontinuous changes in biological populations, including collapses
to extinction as a result of interaction with human activities. The multiple equilibria
model of fishery dynamics in the case of backward-bending supply curves was
initially studied by Copes (1970), and Clark (1976) examined it in the context of
catastrophe theory. The basic pattern is depicted in Fig. 7 in which outward shifts of
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7
Puu (1989) was probably the first to analyze an economic model using both catastrophic and chaotic
dynamics in a model of business cycles in which an economy experiences temporary periods of chaotic
dynamics immediately after catastrophic jumps occur. Rosser (1991) has labeled such a phenomenon as
being ‘chaotic hysteresis.’ Puu was specifically inspired by the Goodwin (1951) model. Almost certainly the
first to combine catastrophe theory with a cellular automata model were Albin and Hormozi (1983) in
their model of technological change limited by information and institutions.
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–3280 3265
the demand curve due to rising incomes or preferences for fish can lead to
discontinuous changes in equilibria. A somewhat similar model with improved
fishing technology as a control variable was due to Jones and Walters (1976).
Another vein of argumentation drew on models of predator–prey dynamics, such
as the spruce budworm dynamics in forests modeled by Ludwig et al. (1978). Walters
(1986) examined a fold catastrophe model of Great Lakes trout dynamics using such
a predator–prey model to study how yields could be maximized while avoiding a
catastrophic collapse by using a so-called ‘surfing’ strategy, refuting the widespread
argument that catastrophe theory has no practical application. Unsurprisingly there
has continued to be more interest in catastrophe theory models, or variations on
them, in ecologic–economic modeling than in other areas of economics, although
often with multiple equilibria in fold or cusp patterns that are not identified with
catastrophe theory explicitly (Wagener, 2003).
In international finance, George (1981) studied foreign currency speculation in a
model with non-convex risk preferences, using a cusp catastrophe that essentially
followed Rand’s (1976) general equilibrium model. Although he did not put it
formally into a catastrophe theory framework, Krugman’s (1984) model of multiple
equilibria in the demand for foreign currencies could rather easily be put into such a
framework following along the lines of the Varian (1979) approach.8
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Fig. 7. Overfishing catastrophe with backward-bending supply.
8
Krugman’s (1991) core-periphery model of regional economic structure could also be easily put into a
catastrophe theory framework, although he has not done so. However, Baldwin et al. (2001) have used the
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–32803266
4. The debate and downfall of catastrophe theory
The major controversy and debate surrounding catastrophe theory erupted quite
early in the process during the late 1970s. The outcome of this debate would be a
residue that gradually corroded the support for using catastrophe theory and
culminated in the current widespread disdain. Again, among mathematicians the
view is widely held that although there was an overhyping of catastrophe theory in
the first place, the current disdain is overdone and that catastrophe theory is a proper
method if used correctly.
The most important criticisms of catastrophe theory applications in general were
by Zahler and Sussman (1977), Sussman and Zahler (1978a, b),9
and Kolata (1977).
Responses appeared in Science and Nature in 1977, with a more vigorous and
extended set of defenses appearing in Behavioral Science (Oliva and Capdeville, 1980;
Guastello, 1981),10
with the first of these being the source of the line that ‘the baby
was thrown out with the bathwater.’ More balanced overviews came from
mathematicians (Guckenheimer, 1978; Arnol’d, 1992).
The critics succeeded in pointing out some dirty bathwater.11
The most salient
points include: (1) excessive reliance on qualitative methods, (2) inappropriate
quantization in some applications, and (3) the use of excessively restrictive or narrow
mathematical assumptions. The third point in turn has at least three sub-points: (a)
the necessity for a potential function to exist for it to be properly used, (b) that the
necessary use of gradient dynamics does not allow the use of time as a control
variable as was often done in many applications, and (c) that the set of elementary
catastrophes is only a limited subset of the possible range of bifurcations and
catastrophes. These arguments relate to applications of catastrophe theory in general
rather than to economics specifically.
Regarding excessive reliance on qualitative methods, it is true that the majority of
catastrophe theory models have had that character. Indeed, this criticism can be
leveled at most of bifurcation theory as it was developed by Poincare´ and his various
followers, especially those in Russia (Andronov et al., 1966). Nevertheless, this does
not rule out specific quantitative models under the right circumstances. Even critics
Sussman and Zahler agree that there are possible applications, especially in physics
and engineering such as with structural mechanics where specific quantifiable models
can be derived from underlying physical laws. This is harder in economics, but not as
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(footnote continued)
language of ‘catastrophic agglomeration’ in connection with a closely related model, a continuing sign that
in urban and regional economics there has remained more openness to such approaches as there has also
within ecological economics. Another paper to mention possibly using catastrophe theory for the model
studied was Gennotte and Leland’s (1990) model of financial market dynamics. Krugman (1996) once
wisecracked that he had ‘forgotten more catastrophe theory than most people ever knew in the first place.’
9
A satire of Sussman and Zahler appeared under the alleged authorship of ‘‘Fussbudget and Snarler’
(1979).
10
Psychology is a field that has remained somewhat more open to using catastrophe theory than some
others (Guastello, 1995).
11
They also made arguments that looked serious at the time but petty in retrospect, such as that some
crucial papers in catastrophe theory initially appeared in unrefereed Conference Proceedings.
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–3280 3267
difficult as in some of the ‘softer’ social sciences. Most examples in economics have
been ultimately qualitative in nature.
Closely related has been the second criticism regarding spurious quantization.
Putting these first two together, some critics argued that the only viable applications
of catastrophe theory were the qualitative ones, which were ultimately useless, and
the ones that attempted to be useful and quantitative were improperly done, at least
in the softer social sciences. Some of Zeeman’s work in particular was among the
most fiercely criticized in this regard, particularly his study of prison riots (Zeeman,
1977, Chapters 13, 14) in Gartree Prison in 1972 using a cusp catastrophe, with
‘alienation’ measured by ‘punishment plus segregation’ and ‘tension’ measured by
‘sickness plus governor’s applications plus welfare visits’ as the control variables.
Data points drawn from these were imputed to exhibit two cusp surfaces. This study
was validly criticized as using arbitrary and spurious variables as well as improper
statistical methodology.
Sussman and Zahler (1978a, b) went further to argue that any surface can be fit to
a set of points and thus one can never verify that a global form is correct from a local
estimate. One should be cautious about extrapolating a particular mathematical
function beyond a narrow range of observation, but this argument smacks indeed of
‘throwing the baby out with the bathwater,’ seeming to deny the possibility of any
kind of confidence testing for nonlinear econometric models. There are critics who
hold such positions, but they are generally held for all econometric models, not
merely the nonlinear ones.
As noted above in the discussion of Fischer and Jammernegg (1986), it is possible
to use multi-modal methods developed by Cobb (1978, 1981) and others. Crucial to
these techniques are data adjustments for location, often using deviations from the
sample mean, and for scale that use some variability from a mode rather than the
mean. These methods have problems and limits, such as the assumption of a perfect
Markov process in dynamic situations. An alternative proposed specifically for
estimating the cusp catastrophe model is the GEMCAT method due to Oliva et al.
(1987), although Guastello (1995, p. 70) has criticized this technique as subject to
Type I errors due to an excessive number of parameters. In any case, the general
issue here is that empirical studies of quantitative models should conform to
accepted statistical and econometric methodologies, and they are not in principle
more difficult to apply to the estimation of catastrophe theory models than they are
to any other kind of nonlinear model.
An outcome from this debate over qualitative methods and spurious quantization
was a split between the two main promulgators of catastrophe theory, Thom and
Zeeman. Whereas Zeeman was the main author of the quantitative studies that came
under criticism, Thom had always been more the abstract theoretician and
philosopher of catastrophe theory.12
He eventually came to agree with Zeeman’s
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12
It is not accidental that there remains a more favorable attitude towards catastrophe theory in Thom’s
homeland of France, favorable to highly abstract thought, with Lordon (1997) providing a recent
application in economics. This may also have to do with the less dramatic meaning that the word
‘catastrophe’ has in the French language than it does in English, with minor social faux pas regularly
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–32803268
critics (Thom, 1983), arguing that ‘There is little doubt that the main criticism of the
pragmatic inadequacy of C.T. [catastrophe theory] models has been in essence well
founded’ (1983, Chapter 7). More broadly he defended the strictly qualitative
approach, criticizing what he labeled as ‘neo-positive epistemology.’ Catastrophe
theory was to be used for ‘understanding reality’ and for the ‘classification of
analogous situations.’ Even before the controversy broke he had declared (Thom,
1975, p. 382):
On the plane of philosophy properly speaking, of metaphysics, catastrophe theory
cannot, to be sure, supply any answer to the great problems which torment
mankind. But it favors a dialectical, Heraclitean view of the universe, of a world
which is the continual theatre of the battle between ‘logoi,’ between archetypes.
Such remarks led Arnol’d (1992) to refer to the ‘mysticism’ of catastrophe theory.
More generally Thom would argue that catastrophe theory showed how qualitative
changes could arise from quantitative changes as in Hegel’s dialectical formulation
(see Rosser, 2000b for further discussion).
Regarding the first of the arguments about mathematical assumptions, the need
for a potential function to exist is one that is a serious problem for many economics
applications. Some clearly satisfy this assumption, with the general equilibrium ones
by Rand (1976) and Balasko (1978) fulfilling this, as well as the regional models of
Puu (1979, 1981a, b) and Beckmann and Puu (1985). This requirement that led
Balasko to argue that proper applications of catastrophe theory to economics would
necessarily be limited. One response due to Lorenz (1989) is that the existence of a
stable Lyapunov function may be a sufficient alternative, which will hold for many
models, although such cannot in general be demonstrated for purely qualitative
models. Another response is that implied by the above discussion of definitions, that
this is really a problem for the more narrowly defined original version of catastrophe
theory, but is not so much of one for more broadly defined singularity theory
versions.
Varian (1981, p. 108) suggested that the narrower version of catastrophe theory ‘is
well developed only for studying local catastrophes of gradient systems’ (italics in
original), citing Golubitsky (1978). However, Golubitsky and Schaeffer (1985,
p. 167) argue that the complaining about the non-existence of a potential function is
‘a red herring’ for systems that can be reduced to n ¼ 1, in which case using contact
equivalence will generate the ‘same set of pictures’ and there will be a
‘correspondence between catastrophe theory and singularity theory that can be
made in either direction, through differentiation or integration, as appropriate. In
other words, for n ¼ 1, potential functions always may be constructed.’13
In this
regard, Leahy (2001) notes that the quartic cusp catastrophe ðf ðxÞ ¼ Àx4
=4 þ ax2
=2Þ
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(footnote continued)
described as ‘catastrophes.’ Weintraub (2002, p. 182) argues that Thom was a ‘Bourbakist.’ Although
Thom was initially trained by French Bourbakist mathematicians, the form of intellectual abstraction he
pursued in this later period was very anti-Bourbakist in spirit and abjured formal, axiomatic approaches.
13
I thank Brock for bringing these references to my attention.
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–3280 3269
and its negative are symmetric and form the universal family of one-parameter
catastrophe functions.
The second mathematical limitation involves the fact that gradient dynamics do
not allow for time to be an independent (or control) variable, a point especially
emphasized by Guckenheimer (1973). Thom (1983, pp. 107–108) responded to this
by arguing that an elementary catastrophe form may be embedded in a larger system
with a time variable. Indeed in his original discussion Thom (1972, pp. 38–40) makes
clear that what is being discussed are dynamical systems in a B Â t space, where the
structural characteristics are in the B part of it. If the larger system is transversal to
the catastrophe set in the enlarged space, then there will be no problem. Of course, it
will be very difficult to determine this in practice. However, while most reject the idea
of time as an explicit variable in catastrophe theory models, we note the discussion
above regarding the definition of catastrophe theory, and that broader views of it
may not find this so objectionable.
Finally there is the argument that the elementary catastrophes are only a limited
subset of the possible range of bifurcations and discontinuities. As Arnol’d (1992)
shows that there are infinite such sets and even infinite families of such sets as the
number of dimensions exceeds 11, this is clearly true, but is only a criticism of the
idea that catastrophe theory is some kind of general answer to all questions and
problems. It is not a valid criticism of using catastrophe theory in situations for
which it is appropriate.
5. Criticism of an economic application
Regarding economic applications of catastrophe theory, there was relatively little
specific discussion during the debates in the late 1970s. The main economic
application discussed was probably the first one ever made, Zeeman’s (1974) model
of stock market crashes. However, much of the criticism directed at this model was
misguided. That these criticisms fed into the current negative attitude towards
applying catastrophe theory in economics therefore calls for correction.
Zeeman models stock market dynamics as reflecting the interactions of two
different kinds of agents, fundamentalists who know what the true value of an asset is
and who buy when the asset is below that true value and sell when it is above that
value, and chartists who chase trends, who buy as price rises and sell as price falls.
The formulation is somewhat different from most economic models in that what is
modeled is the rate of change of price rather than the level of price. This rate of
change of price is J, the state variable. It is modeled as determined by the excess
demands of the two groups, F for the excess demand of the fundamentalists and C
for the excess demand of the chartists. These two are the control variables then for a
cusp catastrophe in which F is the normal factor and C is the splitting factor, as
shown in Fig. 8. If all agents are fundamentalists, then the market will be wellbehaved
and stable, with a unique equilibrium that is actually a rate of change of
price, although if the equilibrium for the midpoint equals zero then that will coincide
with the random walk model. As C increases and the cusp point is passed, possible
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J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–32803270
instability and discontinuous changes in J appears. Zeeman’s original story involved
C rising as the price accelerates until there is a crash; then C declines as chastened
investors revert to more cautious fundamentalist behavior.
In their most sustained critique of applied catastrophe theory, Sussman and
Zahler (1978a) criticized Zeeman’s model on three grounds. First (pp. 133–134), they
argue that the model is essentially tautological, arguing that it does not explain
crashes in a ‘nontrivial’ way, but ‘at best restate[s] the fact that there are crashes, not
account[s] for ityThere are jumps because there are jumps.’ It is true that for
Zeeman’s model to generate discontinuities, he must assume his accompanying story
regarding the respective behavior of his traders. However, this story is quite
reasonable and is consistent with many more recent models of heterogeneous agents
in financial markets in which the balance between fundamentalists and chartist types
centrally determines the dynamics of bubbles and crashes (Arthur et al., 1997; Lux
and Marchesi, 1999; De Grauwe and Grimaldi, 2006). In addition, Brock and
Hommes (1998) provide a detailed bifurcation analysis of such a model, and broader
overviews can be found in Hommes (2006) and LeBaron (2006).
Second (pp. 138–139), they argue that not all crossings of the bifurcation value will
be identified as crashes if they are ‘too small.’ This may be true, but it is also silly to
deny Zeeman’s observation that if it is ‘very large, causing a steep descent, then we
are liable to call the recession a crash.’ Third (p. 198) they argue that ‘Zeeman’s stock
market model predicts that a purely speculative market cannot crash,’ which they
claim is ‘simply wrong.’ However, it is Sussman and Zahler who are simply wrong
about this, as it is the market in which there are no chartist speculators that cannot
crash.14
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Fig. 8. Bubble and crash as cusp catastrophe.
14
Of course one can observe a discontinuous decline, or ‘crash,’ in a rational expectations model of
financial dynamics without speculators if a large, negative information shock arrives suddenly. However,
this is not the usual sense in which the term crash is used and is simply an instantaneous and efficient
adjustment as modeled in the jump-diffusion literature (Merton, 1976). While a crash may be triggered by
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–3280 3271
Another criticism was made by Weintraub (1983), although he did not argue
against ever using catastrophe theory in economics. Coming from a discussion of the
taˆ tonnement process in general equilibrium theory and the stability conditions for
that process, he concluded that Zeeman’s model implied that chartist traders must
have upward-sloping demand curves. He then cited Stigler (1948) who argued that
there never has been a true Giffen good with an upward-sloping demand curve.
Regarding the idea that a stock market participant might believe that tomorrow’s
price will depend on what the price has been, Weintraub (1983, p. 80) declared,
‘There is no evidence whatsoever to support such an hypothesis,’ citing Malkiel’s
(1975) random walk model.
As regards the claim that a chartist speculator must have an upward-sloping
demand curve, which cannot exist because Stigler said so in 1948, what is involved
is not a static demand curve, but a situation where the demand curve shifts outwards
as the price rises (or accelerates). Weintraub might well respond that in
this case it is meaningless to describe the surface in Fig. 8 as an equilibrium manifold.
But, he himself notes it is not a proper general equilibrium manifold anyway
because J is a rate of change of price rather than price itself. Thus it is a different
kind of equilibrium, one about a pattern of shifts in demand curves rather
than about movements along fixed demand curves. Stigler’s argument is simply
irrelevant.
Weintraub’s argument that somehow chartist traders cannot to be allowed in
economic models because of their apparent irrationality reflects its era. This was
before Black’s (1986) speech to the American Finance Association on noise traders
and probably more importantly before the stock market crash of 1987.15
These
events, and the continued appearance of more bubbles and crashes since, have made
models using heterogeneous agents, not all of them perfectly rational, much more
acceptable.
The Zeeman model has its oddities, but many criticisms of it were fallacious. The
idea that these criticisms constituted a case for not using catastrophe theory in
economics was absurd. Although current models of financial market dynamics do
not generally use catastrophe theory per se, the Zeeman model has regained its
respectability and is now recognized as a source of useful insights, and with
Rheinlander and Steinkamp (2004) specifically studying it anew.
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(footnote continued)
the arrival of negative information, it is thought to involve a larger change in price than is justified by that
new information, reflecting the reversal of the speculative dynamics.
15
An immediate response to the 1987 crash was to use chaos theory to model it, then near its intellectual
bubble peak, although in retrospect Zeeman’s catastrophe model looks more relevant for explaining such
large discontinuities. Guastello (1995, pp. 292–297) later did this. Rosser (1991, Chapter 5; 1997) extended
the model to a five-dimensional butterfly catastrophe to explain the phenomenon of the ‘period of distress’
observed in historical bubbles by Kindelberger (2000, Appendix B), in which there is a gradual decline for
a period after the peak but prior to the crash. This occurred in 1987 with the peak in mid-August and the
crash on October 19. An alternative approach to modeling the period of distress has been done using a
heterogeneous interacting agents model with financial constraints (Gallegati et al., 2005).
J.B. Rosser Jr. / Journal of Economic Dynamics & Control 31 (2007) 3255–32803272
6. What are the alternatives today?
If one wishes to examine the structural stability of a particular pattern of
bifurcation or to compare the topological characteristics of two distinct patterns of
discontinuities in economics, then catastrophe theory is the most appropriate method
to use for sufficiently low dimensional systems with gradient dynamics derived from a
potential function. If, however, one is simply modeling dynamic discontinuities within
economic processes, other alternatives exist, most not relying on specific mathematical
assumptions that frequently do not hold. Alternative approaches are indicated
especially if one of the control variables in the process is time.
Among modern complexity theorists many methods have emerged that can
produce phase transitions or dynamic discontinuities in models with heterogeneous
interacting agents. Interacting particle models from statistical mechanics in physics
have been a source (Fo¨ llmer, 1974), with the mean field method one that provides
distinct bifurcations describing phase transitions between different forms of system
organization (Brock, 1993). Another arises with multiple equilibria when the basins
of attraction boundaries are fractally interwoven with each other as in Lorenz
(1992). Yet another involves self-organizing criticality wherein small exogenous
shocks can trigger much larger endogenous reactions (Bak et al., 1993). Still another
uses synergetics, especially the use of the master equation approach (Weidlich and
Braun, 1992). In its emphasis upon distinguishing between slowly changing control
variables and more rapidly changing slaved variables, the synergetics approach
resembles catastrophe theory.
Finally we note the increasing use in economics of models positing multiple
equilibria. Often these models generate equilibrium surfaces similar to the
equilibrium manifolds of catastrophe theory, although they may fail to fulfill all
of the mathematical characteristics of true catastrophe theory. Nevertheless, these
models can produce dynamic discontinuities as control parameters are varied in
ways that cause the system to cross bifurcation points that separate one equilibrium
zone from a discretely different equilibrium zone, thus effectively being close siblings
of catastrophe theory. While they have long been known as possibilities (von
Mangoldt, 1863; Walras, 1874; Marshall, 1890), it is only recently that their use has
become widespread.
An example of such an approach uses Skiba points (or regions or surfaces),
originally studied for convex–concave production functions in optimal growth
models (Skiba, 1978; Dechert and Nishimura, 1983). The Skiba point separates the
basins of attraction of the distinct equilibria and for this model was used to explain
dualistic growth outcomes like endogenous growth models. More recently this has
been applied more widely (Deissenberg et al., 2001).
A striking example is due to Wagener (2003) of multiple equilibria in an
ecologic–economic model of pollutants in a lake system (Brock et al., 1999; Ma¨ ler
et al., 2003; Brock and Starrett, 2003; Dechert and O’Donnell, 2006). Catastrophe
theory is not mentioned per se, but Wagener finds a sufficient condition for a Skiba
point to exist for this lake system is for a Hamiltonian cusp bifurcation as described
in Thom’s book (1972, p. 62) to exist.
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Brock has argued that catastrophe theory may be limited in its applicability to
this case because of symmetry condition requirements (Colander et al., 2004, p. 173)
and notes that the lake game may not fulfill these conditions, even as it may well
fit into the broader view of catastrophe theory. Symmetry conditions are required
to prove the Morse Lemma (Castrigiano and Hayes, 1993, Chapter 1), but are
not required for the Malgrange–Mather preparation theorem, key to proving
the Thom theorem (Castrigiano and Hayes, Chapters 10 and 11). However, as
noted above, as long as catastrophe theory is limited to gradient systems with
potential functions, the symmetry condition of equal cross-partial derivatives is
necessary, but not sufficient (Apostol, 1969, pp. 340–341). Symmetry also guarantees
integrability, which has important implications for welfare analysis of dynamic
games, such as the lake model, which fails to fulfill these conditions, although the
special category of ‘potential games’ does fulfill these conditions (Sandholm, 2001,
2006). This question links back to the somewhat unresolved question regarding the
definition of catastrophe theory, with more generalized views tied to generalized
singularity theory less dependent on such symmetry conditions, whereas the
narrowest of definitions closely tied to the original Thom formulation assumed
their presence.
7. Conclusion
Catastrophe theory experienced one of the most dramatic intellectual bubbles ever
seen. After a gradual development over many decades, it burst onto the intellectual
scene in the early and mid-1970s following the publicizing of the work of Thom and
Zeeman. Part of the reason for its faddishness at that time was the socio-cultural
environment. Radical political movements abounded, and dramatic changes in the
world economy were happening such as the extreme shocks to food and oil prices in
the early 1970s. The idea that huge, sudden, and revolutionary changes might
happen had considerable widespread appeal, especially among dissident intellectuals,
but inappropriate applications of the theory undermined its credibility. A counterattack
came in the late 1970s, and as the 1980s wore on fewer applications of
catastrophe theory were seen, especially in economics, although catastrophe theory
always retained more respectability among mathematicians as a special case of
bifurcation theory. Nevertheless, there were many proper applications of catastrophe
theory in economics before the counterattack’s influence was fully felt.
Criticisms of applications of catastrophe theory included that it involved excessive
reliance on qualitative methods, that many applications involved spurious
quantization or improper statistical methods, and that many models failed to fulfill
mathematical conditions such as possessing a true potential function. Also,
responding to the claimed universal applicability of catastrophe theory it was noted
that the elementary catastrophes are only a subset of the more general set of
bifurcations and singularities. Nevertheless, empirical methods such as multi-modal
models can be used for estimating catastrophe theory models, which have been little
used in economics.
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The general critics of catastrophe theory also subjected Zeeman’s (1974) of
financial market dynamics to harsh criticism. However, reevaluation suggests many
of these criticisms were misguided. To the extent that economists have avoided using
catastrophe theory because of those critiques, they should no longer do so. With the
decline of catastrophe theory, other methods of modeling dynamic discontinuities in
economic models have appeared, some of them around longer than catastrophe
theory and some closely connected to catastrophe theory such as analyzing Skiba
points in multiple equilibria dynamical systems.
In sum, it would appear that the baby of catastrophe theory was largely thrown
out with the bathwater of its inappropriate applications. Although there are serious
limits to its proper application in economics, there are many potential such
applications, especially when one considers its broader version in generalized
singularity theory rather than the narrower version originally formulated by Thom.
Economists should no longer shy away from its use and should include it in the
family of methods for studying dynamic discontinuity. It should be revalued from its
currently low state on the intellectual bourse, righting the wrong of its excessive
devaluation, while avoiding any return to the hype and overvaluation that occurred
in the 1970s. A reasonable middle ground can and should be found.
Acknowledgments
The author is especially grateful to Roy Weintraub for advice and encouragement,
this paper resulting initially from his invitation to discuss this topic in a Duke
University seminar. Others who have provided useful advice are William A. Brock,
Cars H. Hommes, Erik Mosekilde, William H. Sandholm, and four anonymous
referees. I also wish to thank participants in a History of Economics conference in
Washington, a workshop on Optimal Control Theory, Nonlinear Dynamics, and
Adaptive Games in Vienna, and a conference on Chaos Theory in Psychology and
the Life Sciences in Milwaukee, as well as in seminars in various other locations.
None are responsible for any errors or misinterpretations still in this paper.
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