Journal of Economic Dynamics and Control

ELSEVIER 20 (1996) 1763-1773

The business cycle model with stable limit cycle

Kazuyuki Sasakura

a unique

School qf Political Science and Economics, Waseda Univrrsiy. Shinjuku-ku. Tokyo 169-50, Japan

(Received May 1994; final version received July 1995)

Abstract

This paper provides the mathematical foundation to the long-standing academic he/i<f

that Goodwin’s 1951 nonlinear business cycle model has a unique stable limit cycle. In spite of the asymmetric nonlinearity of investment function, the model has certainly a unique stable limit cycle in an economically meaningful region. Once solution paths start from any initial point in the region, they all tend to the limit cycle without escaping from the region or hitting the ceiling or floor of investment during a transition period. The structural stability of the model prevents the limit cycle from vanishing in the face of small perturbations.

Key words: Goodwin’s 1951 model; Asymmetric Rayleigh-type equation; Unique stable limit cycle; Structural stability IEL c/assijication: E32

1. Introduction

Richard M. Goodwin is one of the pioneers in nonlinear business cycle theory. Among his works, the 19.51 paper ‘The Nonlinear Accelerator and the Persist- ence of Business Cycles’ can be regarded as a breakthrough in the field. Needless to say, as very long time has gone and the nonlinear business cycle theory has developed, it only represents a theory nowadays. Nevertheless we have still

I owe a great deal to Professors Tatsuji Owase (Waseda University), Hideyuki Adachi (Kobe University), and Akitaka Dohtani (Toyama University). Particularly I must thank the third profes-

sor for kindly informing me of the Luo-Chen theorem. Detailed comments of two anonymous

referees have improved this paper greatly. Of course, the usual caveat applies.

0165-1889/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved

SSDI 0165-1889(95)00897-S

1764 K. Sasakura / Journal of Economic D_vnamics and Control 20 (1996) 1763-I 773

much to learn from it. Whatsoever, he argues in it that nonlinear cycle theory is necessary because (i) it explains the maintenance of oscillation, (ii) does away with the necessity for initial conditions, and (iii) so treats the acceleration principle that we may make the depression as different from the boom as we wish. Mathematically (i) and (ii) correspond to the existence of a stable limit cycle and its uniqueness, respectively. (iii) is comparatively easy to accomplish once general conditions to satisfy (i) and (ii) can be found. The model construc- ted to establish his claims above is certainly based on the formal nonlinear theory. It has been taken up as an ideal or a mere example of an endogenous business cycle model by many people believing Goodwin (1951).

Here I pose a fundamental question: Does the model have really a unique stable limit cycle? To my surprise, there are only a limited number of works concerning this question. Yasui (1952) and Puu (1986) obtained a unique stable limit cycle using respectively the method of piecewise linearization and the singular perturbation method. Although both methods have the merit of de- scribing an explicit cycle, numerical specification of functions is required for the methods to be applicable. Therefore, their results are not warranted to hold in other cases. This also applies to Goodwin (1951) who used the Lienard method of graphical integration. Morishima (1953) and lchimura (1954) dealt with a general case. They applied the Levinson-Smith theorem to Goodwin-type model to prove the existence and uniqueness of a limit cycle. The famous theorem guarantees the existence and uniqueness at once if prescribed condi- tions are satisfied all at once. As Lorenz (1986) pointed out, however, such conditions are usually too restrictive from an economic point of view, i.e., the symmetry of functions is necessary. Thus, their results are ambiguous. The question posed above remains open after all. No one has succeeded in resolving

the question completely. In this paper I untie the ‘Goodwinian knot’ for the first time. In Section 2

Goodwin’s (1951) model is briefly reviewed. Section 3 is concerned with the existence of a closed orbit, while Section 4 is concerned with its uniqueness and stability. In Section 5 the relationship between Goodwin’s result and mine is made clear. The answer to the open question is stated in the final section.

2. Brief review of Goodwin’s 1951 model

Analysis starts from the following model:

y(t) = c(t) + R(t) - t+(t),

c(t) = ay(t) + B, (1)

k(t + 0) = cp[j(t)] + 1.

K. Sasakura / Journal of Economic Dynamics and Control 20 (I 996) 1763- I773 1765

Fig. I. The nonlinear accelerator cp( j) and related functions.

q(j)=(p2forj.5a4, = Kj for a, < y <: h,, = CD, for j 2 h,,

q’(j) 3> 0 for a4 i j i b,.

q”(9) > 0 for a4 < j < a,, < 0 for h, <j < h,,

q7’(az) = E + 0(1 - c() = cp’(hz),

q’(a3) = i: = cp’(k),

Here y, c, and k are income, consumption, and capital stock, respectively. I:, x ( < l), 8, 0, and I are positive constants. The first equation of (1) is the dynamical multiplier. The second is a linear consumption function, while the third represents the nonlinear accelerator. Net investment & consists of an autonomous part 1 and an induced part cp( j) , the details of which are stated in Fig. 1.

By a Taylor expansion (1) can be transformed to

EH” + [E + O(l - cL)]j - q(y) + (1 - x)y = fi + 1.

Substituting y = z + (/I + I) /(l - x) into (2) gives us

(3

(3) COY + [F: + fI(l - a)] i - cp(i) + (1 - a)z = 0.

1166 K. Sasakura / Journal of Economic Dynamics and Control 20 (1996) 1763-I 773

z is the deviation from the equilibrium income. As is shown in Fig. 1, it is

assumed that

dq(O)/di = q’(O) = K > E + d(l - a). (4)

Let

I&) = [E + 0(1 - CX)] i - cp(i),

tl = J(1 - LX)/&B&

x = J_ (ZJio),

where i,, is any convenient unit in which to measure velocity. Then (3) is reduced to the dimensionless form:

j;_ + x(i) + x = 0, (5)

where 1 = dx/dt,, .? = d’x/dt:, and x(i) = $(&a)/i,J~0(1 - a). The general shape of x(x) is shown in Fig. 2. It should be noted that the x function is

asymmetric with respect to the origin. The conclusion Goodwin (1951, pp. 13-14) drew about the solution paths of

(5) was justified mathematically by citing Andronow and Chaikin (1945, pp. 4, 302 ff.). However, it is not satisfactory because they work with the equation jl + x = ,u~ f(x, i), pL1 being a small positive parameter, and the van der Pol equation in the form .% + p2(x2 - 1) z? + x = 0, pz being an arbitrary positive

parameter, neither of which is equivalent to (5). In fact, Goodwin obtained a ‘single, stable limit cycle’ by the Litnard method.

3. The existence of a closed orbit

Simple calculation shows that (2) is equivalent to

j = (l/F){i - (1 - a)y + B},

; = (l/0) {q(j) + 1 - i}

= (l/@{qOC(l/&){i - (1 - 4y + P}] + l- i],

(6)

where i is investment, i.e., i = k, and the differentiation is with respect to t, not tl. For the purpose of this section, (6) is much easier to treat than (2), though they are the same mathematically.

K. Susakura 1 Journal of Economic Dynamics and Control 20 (I 996) 1763- 1773 1167

Fig. 2. The general shape of I(.<)

On the assumption (4), the configuration of the two loci, j = 0 and i = 0, is as shown in Fig. 3. It is found from it that the i = 0 focus is Z-shaped. The equilibrium point E(y*, i*) is unique and unstable. Since the induced invest- ment has a ‘ceiling’ and a ‘floor’, it is natural to restrict the value of y as well to an economically meaningful interval, e.g., 0 < y 5 j, where j corresponds to capacity output. Then put y1 = (/I + I + cp,)/(l - x) and y2 = (p + I + cpz)i (1 - r), and impose the following condition:

yl I j and y, > 0.

Lemma 1. (6) has at least one closed orbit in the economically meaningful region S, where S = {(y, i)lO -C y I j, (p2 + IS i I ‘pl + 1:.

Proof: To prove this lemma, according to the PoincarkBendixson theorem, it remains to detect a compact region including the equilibrium point E, on the boundary of which the solution paths ultimately enter the interior. Such a region S1 = ABCD is also shown in Fig. 3, where A(yz, cpl + I), B(y,, cp, + 0,

1768 K. Sasakura /Journal of Economic Dynamics and Control 20 (1996) 1763-1773

im

Fig. 3. The behavior of y and i in (6)

C(y,, qpz + 0, and WY,, (p2 + 0, i.e., S1 = {(y, 91yI I y 5 y2, (p2 + 1 I i s cpt + I>. Considering also A’( yi - sb,+/(l - a), ql + 1) and c’(y, - ea4/ (1 - CI), cpz + 1) it is easy to check that any solution path starting from the boundary of S1 ultimately enters the interior. Therefore at least one closed orbit is guaranteed to exist in the region Si c S by the Poincare-Bendixson theorem. Q.E.D.

It is sure that Goodwin tried to study the interaction between income and investment. But it is not clear only by seeing Fig. 9 of Goodwin (1951) since he examined his model in the form of (5) exclusively on the xi plane and derived income cycle only. On the other hand, I analyzed (6) exclusively on the yi plane and succeeded in observing income cycle and investment cycle at the same time. Thus, for example, I can answer to the question: Can investment hit its ceiling or floor? The answer is, of course, no, except when disturbances intervene.

4. The uniqueness and stability of a limit cycle

In the previous section, the existence of a closed orbit in (6) has been proved. Then, how many closed orbits exist? As was noticed in the introduction, sufficient conditions for uniqueness are usually too strict to economic dynamic models. Uniqueness proven is meaningless if it is based on unrealistic assump- tions. As far as (6) is concerned, however, there is the following nice theorem.

Uniqueness Theorem (Luo Ding-jun and Chen Xiang-yan). For the system

1 = y - F(x), ,’ = - x,

K. Sasakura /Journal of Economic Dynamics and Control 20 (1996) 1763-1773 1769

if

F’(x) - F(x)/x 2 0 (or IO)

,for all x # 0, and in the strip where the limit cycle exists the left side of the above ,$~rmula is not identically zero, then the system has at most one limit cycle.

Proof. See Ye et al. (1986, pp. 139-140).

It is intuitively clear from the proof of Lemma 1 that if (6) has a unique closed orbit it must be asymptotically stable and, as a matter of fact, it is true mathematically.’ If the closed orbit is unique, it naturally turns out to be a limit cycle. Furthermore, the stability of the limit cycle implies that (6) is structurally stable in the region S and it maintains a ‘similar behavior’ in the face of small perturbations.2 Thus, the proof of the uniqueness of a closed orbit warrants the existence of a stable limit cycle and the structural stability of (6) at once.

Before proving the uniqueness (and also stability), let’s show the following lemma.

Lemma 2. For any closed orbit L in the region S, = {(y, i)l y, I J’ < y2, ‘pz + 1 I i I ‘pl + l}, consider the set SL = {(y,, iL) ( A point (yL, iJ is on L) and put jL( yL, iL) = (l/s) {iL - (1 - x) yL + p). Then there exists a point (PI,, G,) E SI, such that jL(jL, 7’) I a2 or j&, FL) 2 bz.

Proof. Consider the simply connected region S2 = i(y, i) E Sr 1 a2 < (l/c) x {i - (1 - X) y + fi} < b,} and evaluate the trace of (6) in S2. Then it is positive for all (y, i) E S2. It is found from the Bendixson theorem that (6) does not possess any closed orbit which lies entirely in S2. From Lemma 1, however, there exists a closed orbit which lies entirely in Sr ( 3 S,). Therefore, any closed orbit

L in the region Sr, passes outside S2. Take a point (FL, TL) E SL which lies outside S2. Then the above lemma holds. Q.E.D.

Return from (6) to (5) and note that (5) is equivalent to the following equations:

ti = 1: - x(u), tj= -_u

where u = 1 and v = - x.

‘See Hale (1969, p. 57).

‘This statement is derived from the DeBaggis’ converse theorem. For the necessary and sufficient

conditions for structural stability, see DeBaggis (1952).

1770 K. Sasakura / Journal of Economic Dynamics and Control 20 (1996) 1763-I 773

Theorem. (6) has a unique stable limit cycle in the economically meaningful region S, and the solution path starting from any initial point in that region is the limit cycle or tends to the limit cycle without escaping from it during a transition period.

Proof Since the proof of Lemma 1 and the first part of this theorem imply the second, we have only to prove the first, in which only the uniqueness remains unproved. In order to prove the theorem, therefore, it suffices to show that x(u) satisfies two conditions in the above-mentioned uniqueness theorem. The formula in this case is as follows:

for all i # 0. Moreover, from Lemma 2, in the strip where the limit cycle exists the value of i& ( = i = j) can be less than or equal to a2, or greater than or equal to b2. Therefore the left side of the above formula is not identically zero, i.e., it can be positive in that strip. Since x(u) satisfies the two conditions, a closed orbit in (6) is unique. Q.E.D.

cp’(i&) and cp(&l) /i& which appear in the above proof, can be interpreted respectively as the marginal and average propensities to invest with respect to the increase in income, and the uniqueness conditions can be understood in terms of the two economic concepts, i.e., the former must not exceed the latter in order to guarantee the uniqueness. Here it should be noted that even the asymmetric function x(i) or q(j) generates a unique cycle. If it is symmetric, uniqueness is fairly easy to prove, e.g., using the Levinson - Smith theorem. In economic literature, as far as I know, there has been no relevant discussion in the case of asymmetric function.3

5. The relationship between Goodwin’s result and mine

The remaining task to do is to connect my result (a unique limit cycle in (6)) with Goodwin’s (a unique limit cycle obtained by the Liinard method). For this

30ne of the referees suggested another easier method for proving at least the uniqueness and

stability parts as follows: ‘This is to compute the derivative of the Poincark map and show that (2) is hyperbolic and orbitally asymptotically stable. Then uniqueness drops out easily too.’

K. Sasakura 1 Journal of Economic D.vnamics and Control 20 (1996) 1763-l 773 1771

it is important to note the fact that (6), (2), and (5) are all mathematically equivalent and so all Goodwin’s model. (2) can be written as

j = w, (2’)

bti = (l/E@ [Q(W) - { & + O(l - CC,} w - (1 - LX) J’ + j + I],

while (5) can be written as

i = u, (5’)

d = - X(Ll) - u.

(2’) differs from (5’) only in the scale of time and income, i.e.,

l-x t,= ~

J-

I - x y - y*

i-:0 t and x = __ ___

$^ Fbl io

The movements of solution paths in the yj and xi planes are essentially the same. Thus, it is sufficient to relate (6) and (2’).

Since a unique limit cycle of y exists in the yi plane, the corresponding limit cycle must be observed in the yj plane, too. In fact, the yj cycle is connected with the yi cycle through the plane .sj + (1 - CC) y - i = /3. The details of the relation- ship are as depicted in Fig. 4.4 It is found from the comparison of this figure and Goodwin’s (1951) Fig. 9 that Goodwin did not recognize in his yj analysis an economically meaningful region, i.e.,

6. Conclusion

In Goodwin’s (195 1) model there has been something mysterious to business cycle theorists, since, in spite of the simple structure, the question: ‘Does the

4 The yj limit cycle in Fig. 4 is so described that it is a two-stroke oscillator, where the point ():*, as)

and the equilibrium point (y*, 0) are inside the limit cycle but the point (y*, h,) is outside it. This

corresponds to the xl limit cycle in Fig. 9 of Goodwin (1951). The two-stroke oscillator breathes

only once within each period, i.e.. the total energy increases for j > al, and decreases for j > h5.

Thus, it is the floor of investment that is essential to the persistence of business cycles. Goodwin (1982, pp. viii-ix) attaches importance to the ceiling, but his model works as an endogenous

business cycle model without it (the Goodwin characteristic !). For two-stroke oscillators, see Le

Corbeiller (1960) and also Velupillai (1991).

1772 K. Sasakura / Journal of Economic Dynamics and Control 20 (1996) I763-I 773

i*

0

‘p? + 1

4 !

/

: ; /

I

:

/

: ,

;

Fig. 4. The relationship between (6) and (2’)

K. Sasakura / Journal of Economic qvnamics and Control 20 (1996) 1763-l 773 1773

model have really a unique stable limit cycle?’ could not be solved in general

circumstances. In this paper I gave a correct answer to the question: ‘Yes, as was expected.’ The model has a unique stable limit cycle in an economically mean- ingful region. Such a fact is very ‘unique’ in terms of business cycle models in general. Once solution paths start from any initial point in the economically meaningful region, they all tend to the limit cycle without escaping from the region or hitting the ceiling or floor of investment during a transition period. Since the model turned out structurally stable, the limit cycle does not vanish in the face of small perturbations.

References

Andronow, A.A. and C.E. Chaikin, 1949. Theory of oscillations (Princeton University Press.

Princeton, NJ).

DeBaggis, H.F., 1952, Dynamical systems with stable structure, in: S. Lefschetz, ed., Contributions to

the theory of non-linear oscillations, Vol. II (Princeton University Press, Princeton, NJ) 37759.

Goodwin, R.M., 1951, The nonlinear accelerator and the persistence of business cycles.

Econometrica 19, 1-17; reprinted in Goodwin (1982).

Goodwin, R.M., 1982, Essays in economic dynamics (Macmillan, London).

Hale, J.K., 1969, Ordinary differential equation (Wiley, New York, NY).

Ichimura, S., 1954, Towards a general nonlinear macrodynamic theory of economic fluctuations, in:

K.K. Kurihara, ed., Post-Keynesian economics (Rutgers University Press, New Brunswick, NJ)

192 -226.

Le Corbeiller, P., 1960, Two-stroke oscillators, Institute of Radio Engineers Transactions of the

Professional Group on Circuit Theory 7, 387-397; reprinted in Goodwin (1982).

Lorenz, H.-W., 1986, On the uniqueness of limit cycles in business cycle theory, Metroeconomica 38,

281-293.

Morishima, M., 1953, Three non-linear models of the trade cycle (in Japanese). Economic Studies

Quarterly 4, 2133219.

Putt. T.. 1986. Multiplier-accelerator models revisited, Regional Science and Urban Economics 16.

81-95.

Velupillai, K., 1991, Theories of the trade cycle: Analytical and conceptual perspectives and

perplexities, in: N. Thygesen, K. Velupillai, and S. Zambelli, eds., Business cycles: Theories.

evidence and analysis (Macmillan, London) 3338.

Yasui, T., 1952, Self-excited oscillations and the business cycle (in Japanese). Economic Studies

Quarterly 3. 1699185.

Ye, Y. et al., 1986, Theory of limit cycles (American Mathematical Society, Providence. RI).