Abstract
If the world is not linear (and there is no qualitative reason to assume the contrary), it should be natural to model dynamical economic phenomena nonlinearily. However, there will not always exist an advantage in such a modelling. It depends crucially on the kind of nonlinearity in a model and sometimes on the subject of the investigation whether techniques appropriate to nonlinear systems provide new insights into the dynamical behavior of an economic system. Nonlinearities may be so weak that linear approximations do not constitute an essential error in answering qualitative questions about the system, e.g., whether or not the system converges to an equilibrium state. While this is certainly true for low-dimensional systems, the effects of nonlinearities in higher-dimensional systems cannot always be anticipated with preciseness, implying that linear approximations should be treated with scepticism especially when the nonlinearities obviously diverge from linear structures.
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As will be obvious throughout this section, the dependence of the variables on t is taken for granted and will be omitted for notational convenience.
Note that this definition of a limit set is identical with the notion of an w-limit set, which is frequently used in the literature. The ce-limit set of x is the set of points from which trajectories ending in x originate.
Actually, an attractor must be distinguished from an attracting set. There exist examples showing that the attractor may be only a subset of an attracting set. For a graphical example, cf. Eckmann/Ruelle (1985), p. 623.
Cf. Guckenheimer/Holmes (1983), p. 236.
The following list of stability concepts is not complete but, instead, mentions only those notions which will be used in the present book. For other stability concepts compare, e.g., Hahn (1984), pp. 748ff. and Takayama (1974), p. 356.
The Taylor expansion of a Cm function f: R — R at a point x* is defined as with J1x=X* as the Jacobian matrix of partial derivatives evaluated at x*. When x* is an equilibrium value, f(x*) is, of course, equal to zero.
Cf. Hirsch/Smale (1974), pp. 192ff. and Guckenheimer/Holmes (1983), pp. 4f. Extensive treatments of the usage of Lyapunov functions can be found in Hahn (1967) and Lasalle/Lefschetz (1961).
Cf. Chapter 6 for the role of potentials in catastrophe theory.
Cf. Guckenheimer/Holmes (1983), p. 46.
Cf. Guckenheimer/Holmes (1983), p. 45.
A rigorous. textbook presentation of the Poincaré-Bendixson theorem can be found in Hirsch/Smale (1974), Chapter 11, to which the interested reader is strongly referred. Further presentations can be found in Boyce/DiPrima (1977), Chapter 9, Coddington/Levinson (1955), Chapter 16, and Arrowsmith/Place (1982), pp. 109ff. A concise overview is Varian (1981).
Cf. Boyce/DiPrima (1977), p. 445, and Hirsch/Smale (1974), p.252.
A “simply connected” set is a set that consists of one piece and does not contain any holes in it. For example, the basin of attraction in Figure 2.3.a is simply connected while the basin in Figure 2.3.b is connected but not simply connected (cf. Debreu (1959), p. 15 and Arrowsmith/Place (1982), p. 111.
Cf. Andronov/Chaikin (1949), p. 227 and Boyce/DiPrima (1977), p. 446.
Cf. Section 2.3 for sufficient conditions for the uniqueness of limit cycles.
A more intensive discussion of the Kaldor model and its formal reconsideration by Chang/Smyth (1971) can be found in Gabisch/Lorenz (1989), pp. 122ff.
Cf. Gabisch/Lorenz (1989), pp. 122–129 for economic justifications of these assumptions.
Kaldor himself assumed a sigmoid shape of S(Y, •). The linearity assumption does
This assumption is not very convincing. Chang/Smyth (1971) therefore assumed that SK 0, i.e., a standard wealth effect prevails. However, the different signs do not essentially effect the results.
In other examples the search for this set D can be difficult. Cf. Gabisch/Lorenz (1989), pp. 143ff for a discussion of a non-Walrasian business cycle model by Benassy (1984) with a complicated compact set D. Cf. also Mas-Colell (1986).
Cf. Hirsch/Smale (1974), p.215 and Boyce/DiPrima (1977), pp.447 ff.
Cf. Levinson/Smith (1942), p.397 f.
A function is even if f(x) = f (—x), e.g. a parabolic function with the origin as the center. A function is odd if —g(x) = g(—x), e.g., a cubic equation.
For example, it can easily be shown that the van der Pol equation (2.3.3) fulfills the requirements of the Levinson/Smith theorem.
Cf. Guckenheimer/ Holmes (1983), pp. 166ff. Chiarella (1986) discusses several endogenous business cycle models with the help of averaging methods.
An inspection of the Liénard conditions in the Kaldor model, outlined in Section 2.2.2., can be found in Gabisch/Lorenz (1989), pp. 158ff. Compare also Schinasi (1981).
Cf. Lorenz (1988a) for the following model.
Cf., e.g., the original Liénard equation in Section 2.3., where f’(x)± represents a damping term.
The term stems from considerations of physical systems with a permanent input of energy which dissipates through the system. If the energy input is interupted, the system collapses to its equilibrium state.
The obvious physical example of a conservative dynamical system is, of course, the perfect pendulum where no friction is involved. Note that the harmonic oscillator, shortly mentioned in Section 1.2.1., is an example of a conservative system.
Cf. Clark (1976) for a survey of economic approaches to biological phenomena.
Cf. Andronon/Chaikin (1949), pp. 99ff. and Arrowsmith/Place (1982), pp. 101pp. and 144pp. for the following ideas.
Cf. Gandolfo (1983), pp. 450ff.
Cf. Hirsch/Sma1e (1974), p. 262.
When the dynamical system is linear (or linearized) the Lie derivative is thus identical with the trace of the Jacobi an matrix.
Cf. Arnold (1973), pp. 198f. and Arrowsmith/Place (1982), pp.103f for identical
Goodwin (1967) investigated the solution to (2.4.10) by means of graphical integration. Cf. Gabisch/Lorenz (1989), pp. 153ff. for a presentation of Goodwin’s method.
Further developments of Goodwin’s model can be found in Desai (1973), Velupillai (1979), Pohjola (1981), Ploeg (1983, 1985 ), Flaschel (1984), and Glombowski/ Krüger (1987).
Economically more reasonable modifications can be found, e.g., in Wolfstetter (1982) in an investigation of the influence of stabilization policies in the Goodwin model and in an elaborate discussion of Wolfstetter’s results in Flaschel (1987). However, the effects of these modification are not as easily to trace as the simple perturbation given here.
Samuelson did not refer specifically to the Goodwin model but to the biologically oriented Lotka-Volterra framework. In the Goodwin model, diminishing or increasing returns to scale can be taken into account by assuming that the capital-output ratio a changes with Y.
The case of an initial point identical with a fixed point attractor is, of course, trivial.
The claim for modelling time-irreversibilities often expressed by authors in the field of the so-called evolutionary economics is therefore superfluous because it will be difficult to construct economically reasonable time-reversible models.
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© 1989 Springer-Verlag Berlin Heidelberg
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Lorenz, HW. (1989). Nonlinearities in Dynamical Economics. In: Nonlinear Dynamical Economics and Chaotic Motion. Lecture Notes in Economics and Mathematical Systems, vol 334. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-22233-1_3
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DOI: https://doi.org/10.1007/978-3-662-22233-1_3
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