on nonlinear theories of economic cycles and the persistence of business cycles

Mathematical Social Sciences 12 (1986) 47-76 North-Holland

ON NONLINEAR THEORIES OF ECONOMIC CYCLES AND THE PERSISTENCE OF BUSINESS CYCLES

Willi SEMMLER

Department oj Economics. Graduate Faculty. New School Jar Social Research. New York. NY 10003. U.S.A.

Communicated by P .S. Albin Received 17 January 1985 Revised 17 May 1985

This paper. explores the dynamical structure of various linear and nonlinear theories of economic cycles. It focuses, in particular, on the importance of limit cycle theories for macroeconomic dynamics. It develops and simulates a model of self-sustained cycles and demonstrates the existence of a limit cycle by utilizing the Poincare-Bendixson theorem.

Key words: Economic cycle; limit cycle models; macroeconomic dynamics; dynamical systems; catastrophe; Poincare-Bendix son Theorem.

All things come in seasons - Herakleitos.

1. Introduction

This paper' attempts to revive the theory of endogenously produced cycles. There exists a widespread belief that if market economies do not experience exogenous shocks, then they are essentially stable. This is partly due to the interpretation of classical analysis and especially Adam Smith by neoclassical economists.2

The 'invisible hand' is supposed to make perturbation transitory with an imminent

1 This paper grew out of the work that I did with P. Flaschel on the stability properties of the classical concept of production prices. I would like to thank P. Flaschel, H. Minsky, A. Shaikh, P~ Albin and L. Taylor for helpful discussions. For computational assistance I want to thank Raul Zambrano.

2 From the point of view of classical economics the properties of the dynamics of competition and production prices have been discussed in many papers (Nikaido, 1983; Dumenil and Levy, 1984; Boggio, 1984; Franke, 1984; Flaschel and Semmler, 1985). The general idea is that in the classics there exists a theory of a long-period position of the economy which can be described by the concept of 'center of gravitation'. However, the question has been raised as to whether or not there is a long period position of the economy concerning not only prices but also outputs and employment. There is quite a lot of research going on in this area, and the current paper discusses some formulations of economic cycles and fluctuations in R2 which can be considered as possible formalizations of a classical dynamics of accumulation, output and employment. In the classics there already exist two examples of economic cycles. One is the classical dynamics between population growth and subsistence wage and the other the Marxian theory of a cyclically generated industrial reserve army.

48 W. Semmler I Nonlinear theories oj cycles

return to equilibrium after a sufficient period of time. Modern microeconomic analysis has in fact shown that asymptotic stability of market systems including production is rare (Hahn, 1982). But despite this research, the macroeconomic view of the self-adjusting and self-correcting system endures. The rational expectations school view of business cycles, for example, does allow for economic fluctuations, but such economic events are seen as originating with an unexpected random shock to a stable system (Lucas, 1975, 1977).

However, there is another theoretical tradition which has survived the classics, one which attempts to demonstrate the existence of endogenously produced cycles. In earlier theories of business cycles (Mitchel, 1928; Haberler, 1960,1965) as well as in the work of writers such as Preiser (1924,1933), Lowe (1965). Kalecki (1971), Goodwin (1951,1967,1972) and Kaldor (1940,1971) some features of endogenously produced cycles are present and are worth a second consideration in the light of recent mathematical advances in the theory of nonlinear oscillations.3 On the other hand there are cycle models where cyclicity is only generated through an unstable dynamics that is bounded by external constraints such as supply of labor or other resources (Hicks, 1950).

These early theories of economic cycles, except Goodwin's (1951) and Kaldor's (1940), utilized linear difference equations with time lags (Kalecki, 1971; Hicks, 1950) with either explosive, damped or stable solutions. The latter type of fluctuations is typical for dynamical systems, which are neither totally unstable nor totally stable but only of marginal stability, generating closed curves or harmonic cycles. Although cycles with marginal stability seem rather realistic models of economic fluctuations, they are not easily applied, since the results depend strongly on the parameters and the initial conditions (see Kalecki 1971, ch. 11). The application of a theory of nonlinear oscillations to economics yields much better results. Although the dynamics are more complicated, they allow for stable closed cycles which are generated independently of time lags, initial conditions or random shocks. These 'limit cycles' are characterized by the fact that no matter how the system is initiated, it will tend to a certain type of cycle. Goodwin, for example, in an early article argues that this type of economic cycle is characterized by some kind of selfnegating mechanism, where the 'boom generates its own ruin by fulfilling its purpose, and the depression brings about its own cure by removing the source of its being' (Goodwin, 1951, p. 6).

This theory of economic oscillations has recently gained popularity among many economists, and its application will be reviewed later. Most of the linear or nonlinear models of economic cycles, however, work with dynamical systems consisting of two differential equations and refer to a stationary economy only. Generalizations are still rare because of the quite complex dynamics involved, and in fact it is still not possible to work out models of limit cycles with systems of differential

3 See Lefschetz (l957), Minorsky (1962), Hirsch and Smale (1974), Guckenheimer and Holmes (1983), Godbillon (1983).

W. Semmler / Nonlinear theories oj cycles 49

equations of more than two interacting variables (Hirsch and Smale, 1974, p. 239). Yet some generalizations to non-stationary economies already exist and should also be discussed. Other generalizations have been presented recently as regards the structural stability of economic cycles (Flaschel, 1984) and concerning catastrophic events as a result of small shifts in parameters. Thus, beside linear or nonlinear cycle theories, new approaches which allow for a more complex behavior of dynamical systems than cyclicity, have been developed. Numerous papers have been published applying catastrophe theory (Varian, 1975,1979), or chaotic dynamics (Day, 1982,1983) to economic problems. The analytical treatments of such dynamics is rather complex, progress has been made here by means of computer simulations (see Albin, 1984). Since the paper focuses mainly on cycles theories, we will not be concerned with the preceding approaches to economic dynamics.

In Section 2 of this article many of the earlier approaches to economic cycles are reviewed. In particular, wage share-employment models, profit-investments models, and income-investment models will be addressed. Secton 3 will present my own contribution inspired by classical analysis. Here a profit-accumulation model will be developed, and it will be shown that this model is capable of generating endogenous business cycles with fluctuations in profits, capital accumulation and employment. Section 4 introduces numerical simulation studies and extensions of the dynamical system developed in Section 3. Finally, a mathematical Appendix is added to treat the more rigorous issues.

2. Linear and nonlinear models of economic cycles

Most recent approaches to the study of economic cycles have been modeled by using one of the following methods: (1) linear difference equations with time lags involved (Kalecki, 1971; Hicks, 1950; Phillips, 1954; see also Medio, 1979, ch. 1); (2) nonlinear differential equation with constant parameters (Goodwin, 1967,1972; Desai, 1973; Velupillai, 1979; Glombowski and Krueger, 1983); and (3) other non-. linear differential equations (Rose, 1967; Chang and Smyth, 1971; Varian, 1979). While the second form of differential equations can produce cycles in the sense of closed orbits, the third approach can actually produce limit cycles independently of the initial conditions. These types of economic cycles have already been discussed for a stationary or nonstationary economy. Here we will briefly review the wage share-employment cycle originally developed by Goodwin, the profit-investment cycle initially worked out by Kalecki (1971, chs. 9-11), well summarized in Medio (1979) and the income-investment cycle originally presented by Kaldor (1940) recently discussed in terms of a theory of nonlinear oscillations (Chang and Smyth, 1971; Varian, 1979).

50 W. Semmler / Nonlinear theories of cycles

2.1. Wage share-employment dynamics (Goodwin Model)

By utilizing a dynamical form, originally developed by Lotka (1925) and Volterra (1931) for models of interacting populations (see also Gael et al., 1971), we can rewrite the Goodwin model of wage-employment dynamics as follows. (A more general discussion is given in Appendix A.)

x == P(x, y) = (0 - by)x,

y= Q(x,y) = (cx- d)y,

xlx=a-by,

j/y=cx-d,

where x represents the time rate of change of the ratio of the employed to the total labor force and y is the change of the wage share. Both variables depend on the level of x and the constants (a, b, c, d) > O. The coefficient a denotes the trend of employment if all income is reinvested (y = 0) and d is the fall in real wage if x = O. The symbol by denotes the influence of the wage share on the employment ratio, and ex the positive influence of the employment on the wage share. Due to this interaction of the variables the employment ratio is prevented from rising and the wage share from falling without limits. For a growth model with trends as represented by Goodwin, the coefficients can be interpreted as follows: 0 == b - (m + n) where b is the output/capital ratio (YIK), m is the growth rate of productivity and n is the growth rate of the labor force. Assuming a linearized wage function (for instance, wlw = -e + ex) and with m the growth rate of productivity as before, we obtain for the growth rate of the wage share the term j/y = wlw - m, with d == e + m. Thus the differential equation system (2) can be written as

xix = b(1- y) - (m + n),

J/y=cx-(e+m), (3)

which is indeed equivalent to the form (1) except that it is written in terms of growth rates. The core of the system (3) shows that the change of the employment ratio depends on the profit share (1 - y) and that the change of the wage share depends on the employment ratio. This form has been used to explain the fluctuation of the employment ratio and the fluctuation of the industrial reserve army in Marx (Marx, 1967, vol. I, ch. 23; Goodwin, 1972). The basic structure of this model represents the interacting variables of the employment ratio and wage share as dynamically connected. The system (1) has two equilibria: (0,0) and (die, alb). The linear approximation of the system (l) is with ~ 1, ~2 as small deviations from the equilibrium values

W. Semmler / Nonlinear theories of cycles 51

The calculation of the Jacobian for the first linear approximation gives for the equilibrium (d/e,a/b)

J;=;: [0 -bdIC]. calb 0

The real parts of the eigenvalues are zero and the linear approximation of the equilibrium point represents the dynamical structure of a center (Hirsch and Smale, 1974, p. 258). With real parts of the eigenvalues zero, the linear approximation (5) does not allow conclusions regarding the behavior of the dynamical system in the neighborhood of the equilibrium (Lefschetz, 1957, p. 177). Yet, as can be shown, by constructing a Liapunov function for the system (1), which is constant in motion and hence has time derivatives V = 0, the wage share-employment dynamics results in closed solution curves (Hirsch and Smale, 1974, p. 258). The closed trajectories of the system (1) are, however, only closed curves and the wage share-employment dynamics does not allow for limit cycles (Hirsch and Smale, 1974, p. 262; Velupillai, 1979; Flaschel, 1984). In addition (see Flaschel, 1984), the dynamical system (1) is structurally unstable, since small perturbations can lead to additional interaction of the variables (1" or J22 can be come nonzero). This leads to a qualitatively different dynamical behavior of the system, hence it can become totally stable or unstable. Under certain conditions the system (1) can also become globally asymptotically stable. This can occur if the conditions for Olech's theorem are fulfilled (see Flaschel, 1984). Equivalent results are obtained when in place of a linear wage function a nonlinear wage function is substituted in (1) (see Velupillai, 1979). The wage share-employment dynamics worked out originally by Goodwin for a model of cyclical growth and then applied by him to explain the Marxian theory of an endogenously created industrial reserve army of labor depict a growing economy, whereas most of the other models of nonlinear oscillation refer only to a stationary economy (see Sections 2.2 and 2.3). However, this model of economic cycles does not really model business cycles but rather long cycles. On the other hand for a theory of long cycles the dynamical interaction of other important variables (such as waves of innovations, changes of capital/output ratio, relative prices and interest rates) are unfortunately neglected.

2.2. Profit-investment dynamics (Kalecki Model)

Kalecki in his early writings has developed a theory of the business cycle (Kalecki, 1971, chs. 9-11; see also Medio, 1979, ch. 2). He utilizes linear difference equations with time lags to describe such cycles in a capitalist economy. He analyses the dynamics of capitalist development based on the profit-investment dynamics. Kalecki's theory is in many respects close to the classical dynamics of economic development. The latter is also based on the dynamic interaction of profits and accumulation of capital (Marx, 1967, vol. I, ch. 23; vol. III, chs. 13-15). A similar approach to industrial cycles as developed by Kalecki can be found in earlier times

52 W. Semmler I Nonlinear theories of cycles

in Veblen (1904), Lowe (1965), Preiser (1933), and Mitchel (1928, p. 42). The profitinvestment dynamics in Kalecki has a lag structure. It can be written in a more simplified form by using a model of accumulation-decumulation of capital as utilized in Kaldor (1940) (see also Media, 1979, ch. 2). The core of Kalecki's business cycle theory (Kalecki, 1971, ch. 11) can then be represented by a system of linear differential equations (with f3 = 0 as the simplest form of a Kalecki cycle; see Appendix B)4

K=all,

if= -f3II- yK, (6)

where K = (J - t5K) ~ 0 with I as gross investment and t5K the depreciation. Since Kalecki analyses a stationary economy (the relation of cycles and trends are discussed in Kalecki, 1971, ch. 15) the net investment K can be positive or negative. According to his theory of investment and profits, the change in profit (if) is a decreasing function of the level of the capital stock. 'Finally, the net increment of capital equipment per unit of time affects adversely the rate of investment decision, i.e. without the effect the rate of investment decision would be higher. Indeed, an increase in the volume of capital equipment, if profits, P, are constant, means a reduction in the rate of profit... an accumulation of capital tends to restrict the boundaries of investment plans' (Kalecki, 1971, p. 98). This justifies the negative sign in the system (6) for the variable K. The interaction of if and II means that profits are generated by past investment of profits, but the accumulation of capital leads to if < 0 at some point (see below). Accordingly, for net investments below zero (this is the case when 1< t5K) profits will increase as a result of investment (see Kalecki, 1971, p. 123). The coefficients a,f3 and yare constants representing reaction coefficients (see also Media, 1979, p. 66). The system (6) can be written in a more appropriate form as (Minorsky, 1962, p. 14)

4 Ralecki summarizes his profit-investment dynamics in a difference equation with time lag as follows:

where i represents the net investment, e and ware lags, ex and J.I denote constant reaction coefficients (see Ralecki, 1971, pp. 124-137). Using Taylor's expansion series and writing the difference equation above as differential equation one can write

The second order differential equation above has the roots

J.lw;.2+ (B-J.l»). + (1- ex) =0.

The stability properties of the differential equation can be discussed by utilizing the results of the three cases elaborated in Appendix B for the accumulation-decumulation model.

W. Semmler I Nonlinear theories oj cycles 53

iI == -2bIJ - w2K, (7)

Here the coefficients 2b and w2 can be interpreted again as a combination of reaction coefficients and time lags determined by the dynamic behavior of the system (see Medio, 1979, ch. 2). The linear differential equations (7) with constant coefficients, as in Kalecki's cycle model can be reformulated in terms of differential equations with certain restrictions to produce oscillating solution curves. Since k = iI, we can write system (7/) as

K+2bK+W2K=0. (8)

For b2 - w 2 > 0 there is no oscillation in the system, only an aperiodic damped

motion of the solution curve for b > 0 (and an unstable solution curve for b < 0) (see Appendix B). The equilibrium point is a stable node, since for b2

- w2 > 0 and b > 0 the eigenvalues of (7') are real and negative. For b < 0 the eigenvalues are positive. For b2

- w2 < 0, we obtain a focus which is stable if b > 0 or unstable when b < O. The representation of the flow of K(t) as damped (b> 0), explosive (b < 0) or oscillatory movement is given by Fig. 1. Only when b = 0 are we assured an harmonic oscillation and a stable cycle (see Kalecki, 1971, p. 121). In this case, however, the dynamic system (7') will be structurally unstable. Thus the explosive damped or stable oscillations depend on the reaction coefficients and time lags, represented by the coefficients 2b and w2

• Kalecki realized that the profit-investment dynamics as he formalized them produced stable industrial cycles only for certain coefficients. Therefore, he believed that in reality there only exist damped oscillations, perturbed continuously by normally distributed random shocks which create the appearance of regular business cycles (Kalecki, 1971, p. 127). This requires that b<O. However, in some respects the profit investment dynamics formalized by Kalecki (1971, ch.

Fig. 1.

9) seem to be more complex than the above, since he uses difference equations with time lags. The parameters in his system are taken as constants but the rate of change of II changes signs in different stages of the industrial cycle. In his formulation of business cycles the increase of profits in the past, when K> ° and K < K*, where K* is the eqUilibrium value, has a positive effect on the change of profit in the later periods and a negative effect if K>K* (see Kalecki, 1971, chs. 9 and 11). This is characteristic for the profit-investment dynamics in Kalecki's model, which allows the generation of turning points. In his formalized model of business cycles, however, Kalecki depicts only a stationary economy where the capital stock remains constant in the long run. This and the fact that linear differential (or difference) equations cannot be used to produce limit cycles needed to depict economic cycles are limitations of his early attempt to model the dynamic interaction of profits and capital accumulation.

2.3. Income-investment dynamics (Kaldor Model)

In his 1940 article, Kaldor proposed a quite sophisticated nonlinear model of business cycles which has been recently reformulated in the light of mathematical advances in the theory of nonlinear oscillations (Kaldor, 1940,1971; Chang and Smyth, 1971). Kaldor relies on a geometric presentation of a business cycle model which depends on a nonlinear relation between income changes and capital stock changes and which seems to generate self-sustained cycles without rigid specifications for the coefficients, time lags and initial shocks. The geometric presentation of his model of persistent business cycles due to the dynamic interaction between income changes and accumulation and decumulation of capital, indeed also includes the possibility of limit cycles, i.e. asymptotically stable cycles regardless of the initial shocks and time lags. His ideas are also formulated for a stationary economic system and can be represented by nonlinear differential equations in the following way (Chang and Smyth, 1971)

Y = a(I(Y, K) - S(Y, K»,

K=I(Y,K), (9)

where a is a reaction coefficient, Y the rate of change of income, K the rate of change of the capital stock, I=investment and S=saving as functions of the level of income and capital stock. According to the assumptions underlying the model, there is a unique singular point (Chang and Smyth, 1971, p. 40). The linear approximation of the system (9) in the neighborhood of the equilibrium can be formulated by utilizing the Jacobian matrix

(~J = [~:: ~::] GJ, (10)

where Jij are the elements of the Jacobian evaluated at the equilibrium point. The Jacobian is

W. Semmler I Nonlinear theories oj cycles 55

o(Y,K) = [a(ly-Sy) a(Ix-SK)] , a(y, K) Iy Ix

where a(Ix-SK)<O, since /x<SK<O and ly>O (Chang and Smyth, 1971, p. 41). The determinant is a(SKly-SyIK), which is positive because for the existence of a unique singular point it is assumed that IKSy<SKly. The element, J22 =Ix, is always negative. The linear approximation (10) with the Jacobian (11) represents at its core the investment-income dynamics according to which the change of income depends negatively on the level of the capital stock (JI2 ) and the change of capital stock depends positively on the level of income (J21 ), but there is a negative feedback effect from the level of capital stock to the change of capital stock (122) and an ambiguous feedback effect from the level of income to the change of income (Jl d. This will be explained subsequently. Analysing the singular point one can conclude that the equilibrium is a focus or a node and that the equilibrium is stable or unstable accordingly as a(I y - S y) + I K;: 0. This singular point also allows for a limit cycle, since the necessary condition for a limit cycle is that the dynamic system has an index of a closed orbit which is 1 (Minorsky, 1962, p. 79). This excludes a saddle point as a singular point (see Minorsky, 1962, p. 77). Moreover, the most interesting point in this dynamic system is the ambiguous element J ll . According to Kaldor's graphical presentation, it is assumed (see Kaldor, 1940, p. 184) that

(1) ly>Sy for a normal level of income; (2) ly< Sy for abnormal high or abnormal low levels of income; and (3) the stationary state equilibrium has a normal level of income. This might be illustrated by Fig. 2 with Y the level of output which shows that

the normal level of Y is unstable and the extreme values of Yare stable. Mathematically this means that the trace J11 + J22 changes signs during industrial cycles. This is the negative criterion of Bendixson (Minorsky, 1962, p. 82) for limit cycles, i.e. if the trace J ll + J22 does not change signs, limit cycles cannot exist (see also Guckenheimer and Holms, 1983, p. 44). As proven by Chang and Smyth (1971, section V) there indeed exists the possibility of limit cycles under the assumption proposed by Kaldor. However, the three conditions as formulated above and

Fig. 2.

originally formulated by Kaldor (1940, p. 1984) are not necessary for the existence of cycles. What is actually necessary for cycles is only that /y>Sy (i.e. that JIl

switches signs) at some level of Y. Moreover, the singular point at the normal level of Y does not have to be unstable as a necessary condition for a limit cycle. The critical point can be stable (see Minorsky, 1962, p. 75). In addition there also is the possibility that the system (9) is globally asymptotically stable. This is the case if: (1) a(/y-Sy)+h<O and (2) Syh<SK/Y everywhere. The global asymptotic stability under these conditions follows from Olech's theorem (see Ito, 1978, p. 312).5 Evaluating Kaldor's model of a persistence of business cycles one can say that the replacement of Kalecki's profit-investment dynamics by Kaldor's formulation of an income-investment dynamics brought some advances regarding a theory of endogenously produced business cycles, especially formulations of the theory of cycles in terms of a theory of nonlinear oscillations (see also Kaldor, 1971). The reformulation, however, also implied a certain loss of information concerning the dynamics in capitalist economies which was originally more visible in models that referred to the profit-investment dynamics

3. Profit-accumulation dynamics and economic cycles

As mentioned earlier, theories of business cycles often refer to the dynamic interaction of profits and capital accumulation (see Veblen, 1904; Preiser, 1924,1951; Lowe, 1965, chs. VI and VII; Kalecki, 1971; and Mitchel, 1928, p. 42). However, most of these models of industrial cycles or 'long waves' (see Mandel, 1980; Gordon et aI., 1983) are not rigorously formulated in their endogenous dynamics. As already shown, it is not an easy task to formalize endogenously produced economic cycles, since the dynamics between the interacting variables become very complex for nonlinear differential equations, especially for more than two variables. Even cycles proper for a stationary economy are difficult to construct since economic cycles in the proper sense should be limit cycles which are selfsustaining and independent of time lags, specification of parameters and randomly distributed shocks. Thus the limit cycle model presented in Section 2.3. would be a good candidate for a theory of economic cycles. In the following section we explore the possibility of such cycles with regard to the profit-capital accumulation dynamics for (1) a generalized Lotka-Volterra-Goodwin model and (2) a general

5 Olech's theorem can be stated in the following form: given a nonlinear system of differential equations

x=P(x,y),

y=Q(x,y),

with P and Q having continuous partial derivatives with respect to x and y, if (i) the trace '(x, y) =

'II + '22 < 0, (ii) det '(x, y) = 'II . '22 - '12' J21 > 0 and (iii) either 'II '22 * 0 or J I2 J21 * 0 everywhere, then the equilibrium point (x*, y*) possesses global asymptotic stability (see Ito, 1978),

nonlinear model of cyclical capital accumulation similar to the model of Chang and Smyth (1971).

3.1. A generalized Goodwin and Kalecki model

A more general form than proposed either by Ka1ecki or Goodwin can be used to depict economic dynamics that reveal also different properties because it includes additional interactions of the main variables. This generalization of such dynamical models is briefly discussed here, because it serves as a convenient introduction to a more general nonlinear model which will be introduced in Section 3.2. (See also Appendix A.) The following form of a differential equation system is a generalization of the Lotka-Volterra system (see Hirsch and Smale, 1974, p. 258) as applied to economics by Goodwin. The profit-capital accumulation dynamics, therefore, might be written in the following form:

iI = (a - bK - J...Il)ll,

K = (cIl- d - fJ.K)K,

iIIIl=a-bK -All,

KIK=cIl-d-fJ.K.

In the above system, a, b, c, d, A, fJ. are all positive constants. As can be seen from (12) there is an additional interaction of variables: The change of profit is not only negatively influenced by the level of capital stock but also by the level of profits. The change of capital stock is influenced positively by the level of profits as well as negatively by the level of the capital stock. It is evident from (13) that the growth rate gK is an increasing function of Il and gn is a decreasing function of K.

Therefore, at the core of this model are the typical cross-over dynamics which is characteristic for models of economic oscillations or fluctuations. The coefficient d can be interpreted as a rate of decline of capital stock when no investment occurs (depreciation without additional accumulation of capital) and the coefficient a is the growth rate of profits which would grow exponentially with this rate if there were no negative feedback effects from the level of capital stock and profits to the change of profits. Moreover, -fJ.K denotes the negative feedback effect from the level of K to the growth of K and - All the negative feedback effect from the level of Il to the growth of II. The dynamical form with additional nonlinear perturbation terms - AIl2, - fJ.K2 is a generalization of the Goodwin as well as the Kalecki model of economic cycles. This generalized version of the dynamical models, discussed in Section 2 exhibits, however, a dynamics which does not necessarily result in cycles. It should be intuitively clear that the solution curves K(t) and Il(t) for any arbitrary K(O) and Il(O) are bounded in absolute values since for a large enough K(t), the change of K, K, turns negative and for a large enough Il(t), the change of profits, iI, also becomes negative. This relationship is reversed for small

W. Semmler / Nonlinear theories of cycles

Fig. 3.

values of K and n. A rise (fall) in K(t) entails a downward (upward) movement of n(t) and a rise (fall) in n(t) must sooner or later have also an upward (or downward) influence on K(t). Therefore it is clear that the system (12) cannot grow or decline without limits. The existence of outer boundaries indicates that limit cycles may exist. A sketch of a proof for the possible existence of a limit cycle is given in Hirsch and Smale (1974, p. 264). There it is shown that the system (12) has three equilibria for if = 0 and K = O. These are (0,0), z::::: (II*, K*) and (II*,O). The dynamics of the flows 0"[ K(t) and n(t) and their boundedness can be seen from Fig. 3: where the line if = 0 is derived from a - bK - An = 0 and the line K = 0 is derived from c - d -11K = 0 and fJ and q are the boundaries of II(t) and K(t). The economically most interesting equilibrium is z = (n*, K*). It is asymptotically stable, since the linear approximation of the system

~1 = [Ill 112](t!1) e2 121 122 e2

evaluated at the equilibrium z = (II*, K*) is

[ - AII* -bII*].

"cK* -I1K* (15)

As demonstrated in Hirsch and Smale (1974, p. 264) by using the PoincareBendixson theorem, every trajectory of II(t) and K(t) is bounded, i.e., for any initial value of n(O) and K(O) there exists t > 0 such that ll(t) <p and K(t) < q if t ~ to and the trajectories approach either the economically interesting equilibrium z = (II*, K*) or they spiral down into a limit cycle. On the other hand if one applies Olech's theorem (see footnote 5) to the Jacobian (15) one can demonstrate that a limit cycle cannot exist. By means of transformation of the variables with In II = u, In K = u and u=iflll, v=KIK (see FlascheI, 1984) we get

l(ll, K) = II II 12 v (I e" leV) 121 e h2e

since the signs of the Jacobian (16) do not change and (i) III + 122 < 0, (ii) det 1> 0

and (iii) III 122 *-° it follows that the system is asymptotically stable. Closed orbits or limit cycles cannot exist if the profit-accumulation dynamics as described in Section 2.2 is perturbed by the additional terms as included in (12) or (13). Such extensions of a Kalecki model (or Goodwin model, see Flaschel, 1984) may destroy the cyclical dynamics obtained in simpler versions of the model. Moreover, the above proposed dynamics of profit and accumulation is also too specifically formalized and it might not be immediately applicable. A nonlinear version of the profitaccumulation dynamics as presented in Section 2.2 which generates indeed limit cycles, will be discussed in the next section. The problem of growth in limit cycle models will also be touched upon at the end of the section.

3.2. General nonlinear cycles

We can also write the profit-accumulation dynamics in a more general form as system of nonlinear differential equations. Following the formulations of Chang and Smyth (1971) for an income-investment dynamics, we can consider for OUf purpose the following system:

1. iI=a[I(Il,K)-S(ll,K»),

2. K=I(Il,K). (17)

The linear approximation is again

( ~I)::= [III 112](~I), ~2 121 122 ~2

where the Jacobian is

a(lI,K)::= [a(ln-Sn) a(h-SK)] a(ll, K) In IK'

a is a reaction coefficient, 112 has a negative sign since it is assumed that aSlaK < 0, allaK < ° and allaK < aSlaK. 121 has a positive sign, since investment in capital stock is an increasing function of profit, thus aI/aIl> 0. The existence of a closed orbit is possible since there is a crossover dynamic represented by 112 and J21 , with different signs. The capital accumulation out of profit n(t) increases the capital stock K(t) and the increase in capital stock has a negative effect on the growth of profits. Moreover, 122 has also a negative sign which signifies that the change in capital stock is a decreasing function of the level of capital stock, an assumption made by many writers in the above-mentioned tradition and also in Kalecki (1971, pp. 69 and 123) and Kaldor (1940, p. 184). We also do not necessarily assume Say's Law, since we allow for investment out of profits to be unequal to savings out of profit (see Foley, 1985). Saving out of profit may be interpreted as the accumulation of money capital as source of loanable funds for other firms. Moreover, we want to allow for I" =a(In-Sn) to change signs during the cycle. In general it can be assumed that:

(1) allaII> aSian, for profits in an interval such as III <II < Il2 (see Fig. 4). This may be due to a previous decrease in capital stock, production and employment which entail low construction cost for plants, low material and wage costs (relative to productivity), low interest rates and easy access to credit. These factors then may give rise to an expectation of rising profits on investments.

(2) allall < aSian. (a) for II> Il2 due to capacity limits, rising construction cost for plants and

rising material and wage cost (relative to productivity), rising interest rates and thus falling expected profits.

(b) for II < III in a recessionary or slow recovery period, where capitalist firms invest in money markets instead of in real capital (see Levine, 1985; Minsky, 1983) but due to the economic conditions in a recessionary period, the rate of change of saving in response to falling profits tend to drop faster than the rate of change of investment.

The change of sign for III during the economic cycle was verbally anticipated by many writers on capitalist dynamics (see Preiser, 1924,1951; Kalecki, 1971, p. 123; Kaldor, 1940, p. 184) and can be regarded as an essential for a theory of fluctuations in economic development. Mathematically III + 122 must change signs in order to generate self-sustained cycles. if III and 122 were zero, 112 and J21 alone would determine the profit-accumulation dynamics. There would only be structurally unstable harmonic oscillations. The negative signs of 112 and 122 exert a retarding influence on accumulation, and 121 represents an accelerating force on capital accumulation, whereas III exerts a retarding influence in the boom period and an

Fig. 4.

I I I I I I I I

11 } t:: I I I I I

accelerating impact on profit and accumulation in the later phase of the recession. Intuitively, the existence of self-sustained cycles can be seen from the fact that the trajectories of Il(t) and K(t) are bounded in absolute values and the profit-investment dynamics follow certain directions in the plane. Roughly speaking, for large enough Il(t), iJ turns negative and for large enough K(t), K turns negative and vice versa. Geometrically, this can be illustrated by Fig. 4. For iJ == 0 we get the slope

dK:=:: Sn-In~O dIl Ix-SK:> ,

and for K == 0 the slope is

dK In -=-->0. dIl Ix

Thus in the plane there are four quadrants. For reasons of simplicity we have assumed a linear investment function in (17)2.

The system (17) has a unique solution at Il* and K* since the curve K = 0 has a steeper slope than iJ:;;;; 0 when the latter is upward sloping in a certain region. This follows from the assumption in the model. 6 The determinant for (19) is a(SKIn-:SnIK»O. The singular point is a focus or a node and is stable or unstable accordingly as a(ln-Sn)+Ix~O. A saddle is excluded, and the singular point has index 1 as necessary condition for a self-sustained cycle (Minorsky, 1962, p. 176). (The singular point does not have to be unstable as Kaldor originally assumed, Kaldor, 1940, p. 182.) The existence of a self-sustained cycle follows intuitively from the analysis of the vector fields in the different regions which correspond roughly to stages of economic cycles. (A rigorous proof using the PoincareBendixson theorem is given in Appendix C.)

For region I, which expresses the dynamics of a recovery period, K(t) is below the K = 0 curve and IJ(t) is below the iJ == 0 curve; the decline in capital stock and its effect on profit (i.e. the effect of equation (17)2 on (17)1) as well as other changes in economic conditions in a recessionary period will generate a positive rate of change of profit (since In> Sn in region I, see also condition 1). Therefore, in region I we will find iJ> 0 and K> O.

The increase of profits and investments after a recessionary period will lead to rising K(t), but through the influence of equation (17)2 on (17)1 (i.e. the negative effect of growth of capital stock on profits) the growth rate of II will become negative. Thus in region II, indicatiang a boom period, we have K>O and fI<o. Hence the arrows in Fig. 4, indicating the direction of the vector field of II and K, will start bending inward (see condition (2)(a) which leads to In< Sn). With capital

6 The curve ll=O is downward or upward sloping when Sn>In (or Sn<In). By assuming that for a certain region III <Il*<Jl2, n=o is upward sloping and K=O also has a positive but steeper slope, it follows that there is only one unique equilibrium point. For similar assumptions concerning an income/ investment model, see Chang and Smyth (1971, p. 40).

stock rising and iI < 0 due to a magnitude of capital stock greater than its stationary value K*, the capital stock must eventually decline (i.e. through the effect of equation (17)2). We also have iI<o due to In<Sn at the beginning of a downswing period (capital may be accumulated more as money capital than as real capital).

In region III, indicating a downswing period, through the influence of iI < 0 on K(t), K(t) also starts declining; thus II < 0 and K < O. Hence for fl(t) <fl* and K(t)<K* the vector field is pointing inward. A decline of capital stock below K* in region IV the recessionary period, however, causes profits eventually to rise. The recessionary period may slowly then turn into a recovery period, indicated by region I. This, of course, assumes again that eventually ll> O. The investment of money capital turns into investment in real capital, thus investment out of profit tends to become greater than savings out of profit. The recessionary period (with wage increase below productivity, low material and capital cost, low interest rates and easy access to credit as well as a decline in capital stock and thus rising profit expectation7 must have its impact on ll, for otherwise the recessionary period will endure. (For a related argument, though in a different model, see Varian, 1975 and 1979.)

Therefore under the economic conditions stated in conditions (1), (2)(a) and (2)(b) the profit-accumulation dynamics creates its own cycles by which profit, investment and thus output and employment cannot exceed certain boundaries. The dynamic system is self-correcting and fluctuates within limits: for large enough K(t) is K < 0 and for large enough fl(t) is iI < O. A similar argument holds for small enough K(t) and fl(t). Thus, whereas the system (17) becomes stable at its outer boundaries (indicated by the negative sign of 111 +122), it cannot converge toward the equilibrium, since the equilibrium is unstable (indicated by the positive sign of III + Id. Therefore, the dynamics of the system will result in cycles (see Appendix C). These self-sustained cycles, resulting from the profit-accumulation dynamics, can be regarded as close to classical dynamics and conceptions and the original Kalecki model.

4. Numerical simulations and some extensions

In addition to the analytical treatment of the aforementioned nonlinear profitaccumulation dynamics, numerical studies by means of computer simulation were performed in order to study the complexity of the dynamics of system (17) in Section 3. To simulate numerically the dynamic behavior of our cycle model proposed above, a simplified model was developed in order to avoid unnecessary complications. The following nonlinear differential equation system was simulated:

7 A very important factor for the change of signs in 'II for a monetary economy seems to be the financial condition of firms and the banking system (see Minsky, 1983).

For coefficients a==0.2, /J=0.4, y= 1, 0=0.09, CI :::;4.133 and c2= -1.3 the stationary values are ll*:::; 4 and K* = 30, which give a realistic stationary rate of profit n*= 13.30/0. The results of the numerical simulations are shown in Figs. 6-9.

1. n = allen) - Sell) - fJK] + CI,

2. K==yn-oK+C2, (20)

with a, fJ, y and 0> 0 and CI and C2 as appropriate constants to allow for reasonable stationary state values of nand K. An appropriate nonlinear function len) - Sen) of polynomial class was utilized which exhibit the properties of a Jacobian as required before. The results of the dynamic behavior of a numerical version of such a system will be presented in the following. Moreover, some other results concerning the extension of the model for a nonstationary economy will be reported as well.

There are certain polynomial functions which are appropriate for depicting the qualitative behavior as sufficiently described in the theory of the business cycle in Section 3 (for nonlinear differential equations of polynomial class, see Davis, 1962, ch. 8). For a function of polynomial class to represent J(n) - Sell) as stated in (20)

above we choose the following:

J(n) - S(n):::; 7.Sn + 2n2-1/6n3.

Hence, for the numerical test of the dynamical properties of system (17) in Section 3 we can use for example the following analytical form which has the properties as required.

(1) n=a[ -7.Sn + 2n2-l/6n3 - (fJI + fJ2)K] + CI (21)

where fJI and fJ2 denote the elasticity of investment and the elasticity of savings with respect to the level of capital stock and c] and C2 are appropriate constants to allow for reasonable stationary state values. By utilizing the analytical form (21) above to simulate the dynamics of system (17) in Section 3, we obtain Fig. S for n:::; 0 and K:::; O. The exploration of the direction of the vector fields show that they' are pointing inward at the boundaries as can be checked by calculating nand K for large and small enough n(t) and K(t). The relevant equilibrium point ll* and K* is unstable since the Jacobian of (21) for n* and K* is:

-afJ) -0 . (22)

The equilibrium is unique with a det > O. For the equilibrium points n* and K* is JII > 0 and for appropriate coefficients a and y the trace is JII + J22 > O. For large or small enough values of net) and K(t) is JII < 0 and Jl1 + h2 < O. The analytical form (21) revealed indeed an eqUilibrium dynamics around the stationary state as proposed in the theory of cycles in Section 3.2, i.e., n(t) and K(t) close to the equilibrium are pushed outside and ll(t) and K(f) at the outer boundaries are pushed inside.

16 -3 -'1 3 5 9 11

Fig. 5. Graph for profits and capstock functions.

-0.6 2 3 4 5 6

Fig. 6. Rate of change of n versus n actual.

25 27 29 31 33 35 37

Fig. 7. Rate of change of K versus K actual.

3 4 5 6

Fig. 8. Profits versus capstock for I1(O) = 3.9 and K(O)=29.8.

66 W. Semmler / Nonlinear theories oj cycles

40 ~------------------------------------------------~

: -t--r-----r1 ,~---,----,----,-~~ I I

I I I I I I I o 20 40 ffJ 80 100 120 140 160

Fig. 9. Profit, profit rate and capstock.

For coefficients Ct'=0.2, /J=0.4, y==l, 6==0.09, cI=4.133 and c2=-1.3 the stationary values are Il* = 4 and K* = 30, which give a realistic stationary rate of profit n* = 13.3070. The results of the numerical simulations are shown in Figs. 6-9.

In Fig. 6 the rate of change of II is plotted against II, and in Fig. 7 the rate of change of K against K. As can be seen, due to the interaction of the two variables large (small) enough Il(t) generate a negative (positive) fl. Similar properties hold, as displayed in Fig. 7, for K(t) and K.

The limit cycle properties are depicted well in Fig. 8, where the initial values Il(O) and K(O) are chosen to be close to the equilibrium, yet the equilibrium is not approached by the time path of Il(t) and K(t). The time path of Il(t) and K(t) converges toward a limit cycle instead. Initial start values Il(O) and K(O) outside the limit cycle revealed additionally that the time path of Il(t) and K(t) converges toward the same limit cycle from outside.

When Il(f) and K(t) are plotted against time the cyclical behavior of Il(t) and K(t) is revealed. Fig. 9 also exhibits the fluctuation of the rate of profit n(t) = Il(t)/K(t). If we assume an actual annual growth rate of capital stock of approximately 5070 which has been a realistic magnitude for the U.S. economy in the last 30 years, then the cycles have a length of 7 to 8 years. Further studies of the behavior of our dynamical system and its analytical form revealed that the generated limit cycle is not sensitive to the change in coefficients a, fl, y and 6 as long as the qualitative properties as described for the system (17) in Section 3 is preserved. In addition, system (17) was also tested for nonlinear investment functions. It was demonstrated numerically that the limit cycle properties of (17) were preserved for certain

nonlinear investment functions (see also Klein and Preston, 1969). A related dynamical model was tested for cyclical fluctuations not around the

equilibrium values of a stationary economy but around the trend ll*(t) and K*(t) of a growing economy. For a numerical simulation study the following system was used:

(1) Ii = a(I(ll) - S(fi) - PK)

(2) K = yfi - JK, (23)

with ll(t)=llO(t)-ll*(t), K(t)=KO(t)-K*(t) and rJG(t),K°(t) the actual values and ll*(t), K*(t) the trend values. For this model it was also numerically shown that a limit cycle as equilibrium dynamics around the long run growth path of ll*(t) and K*(t) exists and thus exhibits steady cycles around the long run trend of the economy (for details and the further complexity involved in models with cycles and growth, see Semmler, 1985).

Beside the problem of cycles and growth trends, there are many further problems arising in this context from which we have abstracted in our model (17) in Section 3 as well as in our numerical simulation studies.

First, there are several different but important additional factors which we have not considered yet that complicate the described dynamics. Accumulation takes place out of gross savings of all firms but also may be financed from loans created by the banking system. Thus total investment is allowed to exceed the level of current savings of all firms. Therefore prices may rise. In addition, financial constraints are as important for the dynamics of accumulation as are the financial conditions of the banking sector (see Minsky, 1957, 1983, 1984). There also may be waves of innovation which may be endogenously related to certain stages of business cycles or are created through some long run trends (see Schumpeter, 1939). The implementation of innovations may be concentrated at the beginning of the upswing period or more randomly distributed throughout the cycle. In our cycle model above, the innovations were considered part of the regular investment not influencing the dynamics of the cycle (for a further elaboration on this problem, see Semmler, 1985). The rate of price change and the degree to which the change of the price level is anticipated in the bargaining power over wages is also neglected in our qualitative model as well as in our numerical simulation model (for further discussion on this topic, see Wolfstetter, 1977; Flaschel, 1984; and Sargent, 1979). Moreover, large reserve capacity may exist throughout a considerable part of the cycle, influencing the actual amount of investment in new capital stock. Thus unanticipated price changes as well as the changing utilization of capacity may have their impact on the profit-accumulation dynamics. The change of inventories throughout the cycle is also of considerable importance for the dynamics of investment.

Second, if the sign J11 +J22 =a(In -Sn )+h does not change throughout the cycle, there is a strong possibility that the dynamic system considered here will not create cycles but instead will become globally asymptotically stable. This is the case

if (1) a(In-Sn)+IK<O, (2) S[JIK<SKln everywhere and (3) 111 .122 ,*0 or 112 .121 *0. This follows from Olech's theorem. Under these conditions the dynamic system outlined above would approach a stationary state after exogenous perturbations or random shocks. Also, there might occur a shift in parameters, such that the dynamics becomes stabilized at a lower level with a delay of the recovery or the dynamics might turn into a severe depression (see Varian, 1975, 1979, and Tobin, 1975 for similar arguments in related models).8

Third, most models of cycles, like our own presented and simulated above, refer basically to a stationary economy or to deviations from long run trends created by slowly moving variables (see also Kalecki, 1971, ch. 15). The relation of trend and cycles needs considerably more research effort, since even slowly moving trends can generate a complexity and irregularity in nonlinear models which are far more complicated than in most currently discussed limit cycles models. The problem of growth with irregularities has been treated in seminal papers by Day (1982,1983). A similar result has been obtained for a growth model version of system (17), where the actual values were allowed to grow over time unboundedly. In this case the simulation results of system (17) revealed either a convergence toward the steady state or a chaotic dynamics (depending on the choice of coefficients). Only if version (23) of a growth model was used, the actual values of nand K could be allowed to grow in a simulation test, yet the limit cycle property of the dynamics around the trend was preserved.

5. Conclusions

Since this paper was concerned with nonlinear dynamics in the context of cycle theories more general models with irregularities or chaotic behavior were not examined (see Day 1982, 1983; Albin 1984). Similarly, models with catastrophic dynamics such as explored by Varian (1975,1979) could not specifically be con-

s Such a delay in the change of signs of a(In-Sn)+/K may also be a result of the financial structure and the financial fragility of firms and the banking system, which in turn depend on the profit flows from investments (see Minsky, 1983, 1984). In referring the 1930s depresssion Keynes for example writes: 'Finally the loans and advances which banks have made to their customers for the purpose of their customers' business. These are, in many cases, in the worst condition of all. The security in these cases is primarily the profit, actual and prospective, of the business which is being financed, and in present circumstances for many classes of producers of raw material, of farmers and manufacturers, there are no profits and every prospect of insolvencies, if matters do not soon take a turn for the better'. (Keynes, 1972, p. 155). If there is, as Minsky argues (Minsky, 1983, 1984) a shift in parameters in recessionary periods such that the financial markets become suddenly fragile and the indebtedness of firms leads, due to the financial commitments of firms, to a sudden decline in net profits, the system (17) may exhibit a slow disequilibrium dynamics at the floor or a depression may arise. If there are severe financial constraints for firms in the downturn the drop of investment may continue to be stronger than the drop in saving, asset prices may fall and interest rate rise due to a shift into money. The trace of the Jacobian (19) might become positive everywhere and a financial crisis may trigger a depression (see Semmler, 1986).

sidered. We were interested more in a subclass of models with fairly well behaved dynamics, in particular in cycle models.

As can be seen from the review of the literature as well as from our model of profit-accumulation dynamics of Sections 3 and 4, there are still many speculative elements remaining in the theory of nonlinear cycles. But the revival of the theory of nonlinear oscillations, in its current state mostly presented as systems with two dynamically interacting variables, is quite an important development in the theory of endogeneously produced economic cycles. Extensions to models with more than two variables, to models with technical change, to growing economies with trends,9 to models including the impact of monetary variables on cycles as well as extensions to n-sector models are still rare and theoretically very complex. The current ongoing research on the dynamic properties of such systems, however, can very well complement macroeconomic analysis, especially if future research were able to refine further the connections between the theory of limit cycles and macroeconomic instability problems.

Appendix A: Stability analysis of system 2.1

The stability properties of system (1) in Section 2.1 can be discussed by using a Liapunov function. The proof of the oscillatory behavior of system (1) in Section 2.1 can be discussed by using a Liapunov function. The proof of the oscillatory behavior of system (1) will refer to a more general version (see Luenberger, 1979, p. 370). As described in Section 2.1, the coefficients a and -d in the system (1) in Section 2.1 represent the positive or negative growth rates of the variables x and y and -b and c the interacting terms. Due to those interacting terms, the system cannot grow or decline without limits. This idea can be formulated in a more general form as follows:

with au = 0 and aU the interacting coefficients. The interacting variables i and j lead to an increase in one and a decrease in the other variable. Therefore aij and aji have opposite signs (aU = - aji), ki is a linear growth constant, either positive or negative, and the b/s are positive normalization factors. The matrix A is skewsymmetric. The system (A. I) above can be written by using matrix notations as

X= (x)[k+(b)-lAx], (A.I')

with (x) and (b)-l diagonal matrices and k a vector. The equilibrium with all variables Xi nonzero is given by

9 Concerning a theory of long waves or long cycles, currently there does not seem to be much hope of a rigorously formulated dynamics of long cycles as is possible for business cycles because there are too many interacting variables in a long time period, and their dynamics cannot be analysed any more as limit cycles or even as closed solution curves.

k(b)+Ax*::=O. (A.2)

For the analysis of the stability of the equilibrium point a Liapunov function such as

Vex) = b[x - x* log x]

can be applied. The time derivatives of (A.3) are

. d \' * V(x) = dt '-' bi(Xi-Xi log Xi)

with x as defined in (1). Therefore we can write (A.4') as

V(x) = b[I-(x*)(X)-1 ](x)[k+ (b)-lAx] or

Vex) = b[(x) -(x*)][k+ (b)-lAx].

(A. 4)

(A.4')

(A.4/1)

Near the equilibrium point (x*) we can set x=x*+z with z==x-x*. Using (A.2) for the equilibrium point we obtain

Vex) = b[(x)- (x*)](b)-IAz (A.4I11) or

Vex) = e[ (x) - (x*) ]Az,

e the unit vector which gives

Vex) =zAz. (A.5)

It follows that Vex) == zAz == 0 for all x, because (A.S) can be written as Vex) == zt(A + A ')z and A + A' == O. In general, if A + A' is negative semi-definite the equilibrium of the variable x is stable (see Arrow and Hurwicz, 1958). This can be considered as a generalization of the model with two variables in Section 2.1. But it is also clear that the system (A. I) is structurally unstable, since an additional interaction of Xi with Xi (au not zero any more) can lead to a different qualitative behavior (see Section 3.1).

Appendix B: Oscillatory behavior of system 2.2

The differential equation of section 2.2

K=aIl,

fJ = -fJIl- yK,

can be written as a second-order differential equation

K fJI< -+-+yK=O. a a .

W. Semmler / Nonlinear theories 0/ cycles

Equation system (B. 1) can also be written as

if == -2bJI - w2K, with

2b=[J,

From (B.2) and (B.3) we get the characteristic equation

eV,2+,8)..+y=0,

with ii = 1/a and /1 = [Jla which has the roots

- iJ + UP - 4iiy) 1/2 .1. 1,2 = 2ii

and )..

2 + 2b)" + w2 == 0,

which has the roots

)..1,2 = -b ± (b 2 - W 2 )1!2.

(B.I')

(B.3')

Taking into account that (B.3) can be written as second-order differential equation, we obtain:

(B.3/1)

Concerning the stability of the systems (B.l) and (B.3) we can discuss three relevant cases:

Case (1). ,82> 4iiy or b2 > w 2• The equivalence follows from

b2 = (,812ii? and w2 = ylii

therefore

(iJ 12ii? > y Iii th us ,82 > 4iiy .

The general solution of the system (1) and (3) is then

K(t)=c,eA,i+c2eA2i, cl,czER.

FOr P>4iiy or b2>W2 and y>O we get AI <..1.z<O, for /1>0 (or' b>O) and 0<:)..1<..1.2 , for ,8<0 (or b<O). For ,8>0 (b>O) there is an aperiodic damped motion which has the form of a stable node in the plane. For iJ < 0 (b < 0), the node is unstable.

Case (2). /]2<4iiy or (b2<W2). The general solution is

--\----\---+---+-----/'--\--------

Fig. 10.

K(f) == eO[Cj COS(wt) + C2 sin(wt)], with

_ (4tiy - iP)1I2 W== 0-= -jJ/2ti and

2ti or

('ij -= (w2 _ b2)1I2.

For jJ> 0, therefore 0 < 0, we get a damped oscillation which is in the plane a stable focus as shown in Fig. ·10.

For jJ < 0 therefore 0> 0 we get an unstable focus as shown in Fig. 11.

Case (3). For jJ=O (b=O) we get

k+w2K=0.

The general solution is

K(f) = Cj cos(wf) + C2 sin(wt).

From case (2) we see that ('ij = w = (y/ti)1I2. This represents the standard case of a harmonic oscillation such as shown in Fig. 12. Case (3) is structurally unstable since a perturbation jJ:;;: 0 can lead to a qualitatively different dynamic behavior of the system.

H'----,AI--+-+--I- K

Fig. 11.

W. Semmler / Nonline(1r theories of cycles 73

Fig. 12.

Appendix C: Limit cycle property of the system 3.2

A proof of the existence of a limit cycle for the system 3.2 can be given by utilizing a specification of Poincare-Bendixson theorem.

Definitions. An co-limit cycle is a closed orbit y such that y E Lw(x) for some x $ y, where Lw(x) is called a limit set. Examples for a limit set can be a critical point or limit cycles. YEW is a co-limit point of x E W; if there is a sequence tn ~ 00 such that limn~oo(Oll/(x):::y, where (OII/(X) is the flow of a dynamical system. The set of all OJ-limit points of y is the limit set Lw(Y). Replacing tn-> 00 with tn ~ - 00 we get a so-called a-limit set. Here we are only concerned with an co-limit set. A limit cycle shows certain properties which are different from other closed orbits. y can be called an co-limit set if there exists x $ y such that lim d«(OI(x), y) = 0 where d denotes the distance from the flow to y.

Theorem 1. (Poincare-Bendixson): A 110nempty compact limit set of a planar dynamical system, which contains no equilibrium points, is a closed orbit. For the proo/, see Hirsch and Smale (1974, p. 248).

Theorem 2 (specification of Theorem 1): If a set K eX is compact and positively invariant and there exists exactly one critical point Xo in K and if K ,*xo with Xo as a source (i. e., if Re..1. (Df(xo» > 0», then there exists at last one limit cycle y in K. Moreover, if there is only one noncritical closed orbit in K, then y is asymptotically stable, i.e., lim d«(Ot(x), y) = 0 holds. For the proof of this specification of the Poincare-Bendixson theorem, see Amann (1983, p. 252).

Fig. 13.

Remark. The theorem 2 assumes that the set K is compact and positively invariant. A positively invariant set means that any point XEX is attracted by a set MCX, when t+(x)=oo and rt>(,,(x)-+Mfor (-+00. The set A(M): ::::xEXlxattracted by M) is called the attracting neighborhood of M and M the attractor. A(M) is positively invariant and the flows 'P(n(x) eventually approach the region inside A(M) (see Amann, 1983, p. 248). The set K is said to be compact if it is closed and bounded. Theorem 2, referring to a compact and positively invariant set with one unstable critical point xo, is illustrated by Fig. 13.

Theorem 2 allows the proof of the existence of at least one limit cycle y for the dynamical system 3.2 since the required properties of the theorem 2 are fulfilled. As shown in 3.2, for the system 3.2. (1) there is a set K which is compact, and it is positively invariant (the vector field points inward at the boundaries), (2) it has only one critical point ll*, K*, (3) the critical point ll*, K* is unstable, since Re J..(Df(ll*, K*» > 0 and (ll*, K*) *K is not an attractor (the critical point is a source). Therefore at least one limit cycle exists; the w-limit set y is asymptotically stable and produces periodic cycles. If there is more than one limit cycle, then they are alternately stable or unstable (Minorsky, 1962, ch. 3). If, however, there exists only one limit cycle, then all paths will converge to periodic cycles independently of initial conditions. In the case of more than one limit cycle, the ultimately cyclical fluctuation of the economy also depends on the parameters in the system 3.2 and the initial conditions

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on nonlinear theories of economic cycles and the persistence of business cycles

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