The Mathematics of Nonlinearity
In this chapter we consider in a more detailed way the role of mathematical techniques in shaping theories of business cycles, which have been characterized by an increasing use of nonlinearities, with particular reference and applications to the specification of the labor market.
KeywordsHopf Bifurcation Real Wage Periodic Point Chaotic Behavior Chaotic Attractor
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- 1.See Adelman-Adelman (1959). This philosophy has been confirmed by Gordon (1986).Google Scholar
- 2.According to Goodwin (1987), the trouble is that discrete time models occur in the context of economic activity which is substantially continuous, so that one should formulate mixed difference-differential systems, a procedure whose complications are well known.Google Scholar
- 3.On these points, see Devaney (1986), Guckenheimer-Holmes (1985) and Lauwerier (1986).Google Scholar
- 4.Other examples can be found in Benhabib-Miyao (1981) and Reichlin, (1986).Google Scholar
- 5.The conditions refer to the question concerning complex eigenvalues and the transversal condition. See Iooss (1979).Google Scholar
- 6.For a criticism of the concept of the detectability of chaos, see Melese-Transue (1986). For a discussion on the applicability of these concepts in economic models, see Frank and Stengos (1988).Google Scholar
- 7.In this perspective, a limit cycle will ordinarily have an unstable (repellor) fixed point but a global attractor, so that between the two must lie at least one closed boundary, which constitutes a stable fixed motion, or limit cycle.Google Scholar
- 8.See also Beltrami(1987) and Thompson-Steward (1986).Google Scholar
- 9.In this excursus, we did not analyze catastrophe theory in detail. It seems that the initial debate about the relevance of this theory has settled down and that agreement exists that it is at least a very useful method for studying dynamical systems heuristically. Moreover, catastrophe theory is able to elucidate what the structural requirements for a model are if certain dynamical phenomena should be explained. (See Gabisch-Lorenz, 1987, p. 188.) Catastrophe theory mainly deals with sudden jumps in the variables, classified as fast or slow. (See also Medio, 1984, Haken, 1983 and Saunders, 1980) For an economic application, see Varian (1977).Google Scholar