Abstract
The insight of Adam Smith (1776) that the “invisible hand” can efficiently allocate resources in many markets simultaneously inspired Léon Walras (1874) to define and establish the existence of competitive general equilibrium1 in algebraic systems of linear supply and demand equations. Although Walras’ approach of counting equations and unknowns was insufficient for truly establishing such existence, it was carried forward by Pareto (1909) who defined optimality and used calculus to establish the coexistence of optimality and Walrasian general equilibrium.
“At the still point of the turning world. Neither flesh, nor fleshness; Neither from nor towards; At the still point, there the dance is, But neither arrest nor movement. And do not call it fixity, Where past and future are gathered. Neither movement from nor towards, Neither ascent nor decline. Except for the point, the still point, There would be no dance, and there is only dance.”
T.S. Eliot, 1943 “Burnt Norton”, The Four Quartets
“Things fall apart; the centre cannot hold”
William Butler Yeats, 1921 “The Second Coming”
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Notes
According to Schumpeter (1954), Boisguillebert in the early 1700s was the first to envision something like a general equilibrium.
See Rosser (1999a) for further discussion of this issue. Day and Pianigiani (1992) and Day (1994) examine chaotic dynamics in Walras’ original model when there are multiple equilibria, as well as in single equilibrium cases.
Following Debreu, much of this literature focuses on pure exchange economies where endowments are the only exogenous variables. Mandler (1995, 1999) argues that Debreu’s genericity result breaks down in sequential production economies, with indeterminacy of equilibria obtaining for positive measure sets of initial endowments. He links this to an argument that Sraffa’s (1960) criticism of neoclassical marginal productivity theory is really about indeterminacy of general equilibria (Mandler, 1997, 1999 ).
Balasko (1978) suggests that the application of catastrophe theory will be limited in general equilibrium theory because of the lack of a potential function and because generally dimensionality will be too great.
Somewhat more formally, the mapping of the continuous utility functions cross the prices into the set of continuous tangent vector fields on the “epsilon-trimmed” (p>0) n-1 price simplex is surjective if and only if there are at least as many agents as commodities (Saari, 1995, p. 224).
As discussed in Chapter 2, Bala, Majumdar, and Mitra (1998) have examined a model of controlling chaotic Walrasian adjustment processes.
George recognizes the argument of Rothenberg (1960) that the discontinuous behavior of individuals due to nonconvexities may be swamped by aggregation in the total market to yield overall smoothness.
Although in intense conversions there is usually a particular “moment of truth,” Nock (1933) showed that sometimes that moment is preceded by a long gradual buildup, as in the case of Saint Augustine of Hippo, suggesting a slow underlying dynamic that reaches a bifurcation point and suddenly becomes a fast dynamic.
A competing theory of addiction involves competing “Jekyll-Hyde” personalities (or sets of preferences) within a single individual which alternate back and forth (Elster, 1979; Winston, 1980; Schelling, 1984). These observers object to the Stigler-Becker-Murphy view of addicts as “rational utility maximizers.”
For analysis of how economic motives encouraged conversion to Islam, see Ensminger (1992, 1994), and Rosser and Rosser (1998a).
A special case is discussed for religious and political movements by Kuran (1997) in which many people are concealing their true private beliefs. Then, the public affirmation of this suppressed belief by a single individual (or critical mass of people) can trigger a sudden wave of open statements and a revolutionary shift of public opinion, as with the fall of Communism in the former Soviet bloc.
The term “cobweb” was originally due to Kaldor (1934) based on the appearance of graphs of the dynamics in supply-demand space. Other early work on cobweb models includes Schultz (1930), Ricci (1930), and Tinbergen (1930).
Although these are the three possible outcomes of Ezekiel’s (1938) formal model, he remarked in his paper on the possibility of irregular dynamic patterns endogenously arising in real world cobweb systems, arguably a premonition of chaos.
It is less clear that such an argument applies in the labor market when we are dealing with the major field decisions of college students who are essentially first time decision makers in the labor market.
Copes (1970) proposed the backward-bending supply curve for fisheries. The open access case is the same as the myopic optimum (Gordon, 1954; Clark, 1990) which maximizes the backward bend of the supply curve. Conklin and Kolberg (1994) have shown chaotic dynamics for the halibut fishery with a backward-bending supply curve. Rosser (1999b) and Hommes and Rosser (1999) consider chaotic dynamics in fisheries with backward-bending supply curves with consistent expectations equilibria or self-fulfilling mistakes.
Finkenstädt and Kuhbier (1992) carry out a similar analysis for an agricultural market cobweb with a linear supply curve but a nonlinear monotonic demand curve somewhat resembling Chiarella’s (1988) supply curve. See Chiarella (1990) and Finkenstadt (1995) for more detailed discussion.
Standard cobweb models involve discrete time periods. However Mackey (1989) has shown the possibility of cyclical dynamics in continuous time models with production and storage lags. Invernizzi and Medio (1991) show the possibility of chaotic dynamics in models with continuous lags.
In a model based on that of Hommes (1994), Gallas and Nusse (1996) show that with two control parameters there can exist fractal structures in the space of the control parameters.
Rosen, Murphy, and Scheinkman (1994) show how endogenous cycles in cattle production can arise even with rational expectations for “time-to-build” reasons in a world of lags due to investment in breeding stock.
Chavas and Holt (1991) also claim to have empirically observed evidence for chaotic dynamics in the pork cycle, or at least nonlinear dynamics not fully explained by GARCH analysis. This paper does not present a theoretical model, but the corn-hog cycle can be expected to more closely resemble the predator-prey model rather than the cobweb, although these have considerable mathematical similarities and many early developers of the cobweb model had the hog cycle in mind, such as Ezekiel (1938). Holzer and Precht (1993) also claim to observe chaotic dynamics in the pork cycle.
Another variation has focused on diffusion of competing techniques for producing a good with the techniques exhibiting increasing returns to scale in a two-good market. Besides the usual arguments regarding path dependence and lock-in (Arthur, 1994), Greiner and Kugler (1994) have shown chaotic dynamics for a discrete time version of a model due to Amable (1992).
On the other hand, Sterman (1989) has shown chaos arising from simplistic decision rules in 40% of the parameter space of an experimental economic setting.
Sargent (1993) has come to accept that agents are unlikely to possess rational expectations in the face of computational complexity. Thus, he accepts adaptive learning with bounded rationality, although he remains optimistic about the possibility of converging on rational expectations equilibria. He (1999) has developed the concept of approximate rational expectations equilibrium in which agents operate with optimally misspecified forecasts.
In a laboratory experiment of learning from feedback an unknown cobweb with a unique steady-state rational expectations equilibrium solution, Hommes, Sonnemans, and van de Velden (1998) found that only about one third of the subjects were able to learn it, with another third following some kind of naive prediction strategy with persistent and systematic forecasting errors. Baak (1999) uses a Kalman filter to test for the fraction of agents in the US cattle market following boundedly rational strategies and found it to be about one third.
Sorger’s (1998) example is of a macroeconomic model with overlapping generations (OLG). Others finding learning to believe in chaos in OLG macroeconomic models include Bullard and Duffy (1998a) and Schönhofer (1999a,b).
The concept of learning equilibrium is due to Bullard (1994), although arguably implicit in Grandmont (1998) which was first presented in lecture form at the beginning of the 1990s.
Another source of discontinuous behavior in oligopoly models arises when the marginal revenue curve is discontinuous which happens when the demand curve is kinked as in the case due to Sweezy (1939) where firms react asymmetrically to output increases or decreases by their competitors. The most notable implication of this model is price stickiness.
Walters (1980) proposed as possible examples ports, such as Singapore, where transhipment is possible, and electric utilities with segmented markets.
This is somehow appropriate given that Cournot’s model was the first clear-cut example of an economic model using calculus ever published.
Strotz, McAnulty, and Naines (1953) demonstrated chaotic dynamics in a nonlinear accelerator model of the Goodwin (1951) type, but did not fully understand the mathematics of what they had shown.
Puu (1997) also studies global bifurcations of these systems. This system is actually a special case of coupled oscillators that Puu (1997) examines in the context of interrelated regional models as well.
These results are not especially dependent on the nature of the demand function and Koppel shows similar systems arising with the iso-elastic reciprocal demand function studied by Puu (1991, 1996, 1997, 1998).
Similar issues arise in the modeling of firms themselves where there may be critical levels of agents beyond which a firm divides into two firms (Axtell, 1999). Arthur and Ruszczynski (1992) show in a model of strategic pricing with increasing returns that the crucial variable determining whether or not there is a stable or unstable market solution could be the discount rate.
Probably the most common way to deal with the question of cooperation versus noncooperation involves the study of the prisoner’s dilemma where it is well known that evolutionary strategies are very different from one-shot ones (Axelrod, 1984). For modeling of complex dynamics in repeated prisoner’s dilemma game settings see Lindgren (1997), Albin with Foley (1998), and Young (1998).
Van Dijk and Nijkamp (1980) also model energy prices with a cusp catastrophe. But they focus on inertias of supply and demand in time rather than market structure changes, although this is not inconsistent with a contrast between short-term and long-term elasticity effects.
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Rosser, J.B. (2000). Discontinuities in Microeconomic Systems. In: From Catastrophe to Chaos: A General Theory of Economic Discontinuities. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1613-0_3
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