Discussion Paper No.303
Monetary policy in the unique growth cycle ofpost Keynesian systems

Hiroki Murakami
Faculty of Economics, Chuo University
October 2018

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Monetary policy in the unique growth cycle of post Keynesian systems
Hiroki Murakami∗
Faculty of Economics, Chuo University†
October 12, 2018

Abstract
This paper deals with the following two issues in a post Keynesian system: the existence and uniquenessof a growth cycle and the effectiveness of a counter-cyclical monetary policy for stabilization. It is found thatthe unique growth cycle, represented by the unique limit cycle, can be observed if the rate of interest is set
to a constant level while that the long-run equilibrium can be globally asymptotically stable if an appropriate
counter-cyclical monetary policy is conducted.

Keywords: Economic stability; Growth cycle; Keynesian economics; Monetary policy
JEL classification: C62; E12; E32; E52

1

Introduction

In economic theory, persistent cyclical fluctuations such as business cycles and growth cycles are usually describedby periodic orbits including limit cycles,1 and the main theme of theory of business cycles is, in a lot of cases, toestablish the presence of a periodic orbit in dynamic models. Since the contributions of Kaldor (1940), Hicks (1950)
and Goodwin (1951), in economic theory, the concept of “nonlinearity” has been recognized as a key element in“detecting” a periodic orbit.2 Indeed, mathematical theory of nonlinear dynamical systems, such as the PoincaréBendixson theorem and the Hopf bifurcation theorem,3 has intensively been employed to verify the existence of a
∗ E-mail: hmura@tamacc.chuo-u.ac.jp† 742-1, Higashi-Nakano, Hachioji, Tokyo 192-0393, Japan.
1 As is often the case in economics, limit cycles are often confused with periodic orbits, but mathematically, they are different (thoughslightly) concepts. Strictly speaking, a limit cycle is a periodic orbit such that there is another solution path of the dynamical system
under consideration converges to it as t → ∞ or as t → −∞. (cf. Coddington and Levinson 1955, pp. 391-392; Hirsch and Smale
1974, p. 250). In two-dimensional autonomous differential equations, periodic orbits are generically limit cycles due to the concept ofstructural stability (cf. Peixoto 1962), but there is a case in which a periodic orbit is not a limit cycle. In Goodwin’s (1967) model ofgrowth cycles, which can be reduced to a Lotka-Volterra system, all solution paths except for the equilibrium are periodic orbits, but
none of them is a limit cycle (cf. Velupillai 1979; Flaschel 1984). In particular, if the number of periodic orbits is finite, they are all
limit cycles, either stable or unstable.
2 See Yasui (1953), Ichimura (1955) or Morishima (1959) for early development of “nonlinear” economic theory. For the contributionsof these Japanese economists, see, for instance, Velupillai (2008) or Asada (2014).
3 For the Poincaré-Bendixson theorem and the Hopf bifurcation theorem, see, for example, Coddington and Levinson (1955, chap.
16) and Marsden and McCracken (1976), respectively.

1periodic orbit including a limit cycle.4
Although the mechanism of persistent business cycles can be explained by the existence of a periodic orbit,
the existence does not necessarily imply the uniqueness. Unless the uniqueness of a periodic orbit is obtained, thecharacteristics of (persistent) business cycles such as periods and amplitudes of fluctuations can dramatically bechanged after large external shocks occur because the macroeconomic system under consideration can converge toa periodic orbit different from the original one. With the uniqueness of a periodic orbit, on the other hand, themacroeconomic system converges to the unique (original) business cycle even after huge shocks.5 It is thus mucheasier to predict future economic situations when a periodic orbit is unique than when it is not. In this respect, theuniqueness of a periodic orbit has a practical merit, especially in terms of predictions.
Unfortunately, however, the uniqueness of a periodic orbit has not much been examined in economic theory due
to technical difficulty. Indeed, there have been only a few papers which address the uniqueness. Ichimura (1955),
Kosobud and O’Neill (1972), Lorenz (1986, 1993) and Galeotti and Gori (1989) examined the possibility that aperiodic orbit is unique in Kaldor’s (1940) model of business cycles, but they failed to find a realistic sufficientcondition for the uniqueness.6 Sasakura (1996), on the other hand, established the uniqueness in Goodwin’s (1951)
model of business cycles without any stringent assumption, especially on the saving function, but the investment
function in Goodwin’s (1951) model is itself an unrealistic one based on the “acceleration principle” of investment.7
Thus, the uniqueness of a periodic orbit has not, until recently, been established theoretically in realistic situations.
Recently, in Murakami (2018a, 2018b), I tackled the issue of the unique periodic orbit and verified the uniqueness,
as well as existence, of a limit cycle, which can be interpreted as persistent growth cycles in terms of economics,
in post Keynesian systems with realistic features. In these works, I made use of a linear Keynesian consumption
or saving function which is empirically plausible and of two investment functions, one of which is consistent with
Keynes’ (1936) marginal efficiency theory of investment (cf. Murakami 2018a) and the other of which is consistent
with the profit principle and the “utilization principle” of investment (cf. Murakami 2018b).8 In these studies, I
drew the conclusion that the unique persistent business (or growth) cycle is observed, without policy interventions,in the (post) Keynesian system, which is characterized by fast quantity (or utilization) adjustment or by frequent
revisions of the long-term expectation. In them, however, I took no account of the role of policy implementations.
In this paper, I begin with reviewing the conclusion on the unique growth cycle drawn in Murakami (2018b) and
4 For classical applications of the Poincaré-Bendixson theorem to economic theory, see, for instance, Rose (1967), Stiglitz (1967) or

Chang and Smyth (1971), while, for those of the Hope bifurcation theorem, see, for instance, Torre (1977), Benhabib and Nishimura(1979) or Benhabib and Miyao (1981).
5 Precisely speaking, the unique business cycle has to be (periodically) stable for obtaining this conclusion.
6 Specifically, all of them had to assume that aggregate saving is a sigmoid (nonlinear) function of aggregate income (and of aggregate
capital stock), but as Lorenz (1993) admitted, this assumption cannot be supported from an empirical viewpoint.
7 The acceleration principle, adopted in Harrod (1936), Samuelson (1939), Hicks (1950) and Goodwin (1951), is often confused withthe “profit principle” of investment, which states that investment is dependent on the level or rate of profit and was established by
Kalecki (1939) and Kaldor (1940). As argued by Kaldor (1940, 1951), however, these two principles are different and the profit principle
is more plausible from theoretical and empirical viewpoints than the acceleration principle is. Furthermore, as clarified by Chenery(1952), the acceleration principle implicitly assumes that capital stock is operated at full (or nearly full) capacity, but this assumption
is neither realistic nor reasonable. Also, if variations in capacity utilization are taken into account and the so-called flexible accelerator
is introduced, the accelerator principle is reduced to a simple version of the profit principle.
8 The utilization principle states that investment is affected by the rate of utilization and was postulated by Steindl (1952, 1979) andRowthorn (1981).

2then discuss the possibility that the monetary authority can, with policy interventions, stabilize the macroeconomic
system exposed to persistent economic fluctuations along the unique growth cycle. This paper is organized asfollows. In Section 2, I set up a post Keynesian model based on Murakami (2018b). In Section 3, I analyze thecharacteristics of the post Keynesian model, separately, in the cases in which the rate of interest is kept constantand in which it is changed by the monetary authority’s feedback (counter-cyclical) policy. Specifically, I confirm theconclusion given by Murakami (2018b) that, when the rate of interest is constant, a periodic orbit or a limit cycle,
which can be viewed as persistent growth cycles, uniquely exists, if the speed of quantity adjustment is high enough(Section 3.1) and then demonstrate that the unique limit cycle (or growth cycle) is eliminated and the long-run
equilibrium gains the “global” asymptotic stability, if the monetary authority conducts a proper (or “moderate”)
counter-cyclical interest rate policy (Section 3.2). In Section 4, I summarize my analysis and conclude this paper.
In Appendix, I provide the mathematical theorem to be utilized in my analysis.

2

The model

In this section, I first formalize the consumption and investment functions from Keynesian perspectives and thenset up a dynamical system composed of differential equations of the output-capital ratio, which can be identified
with the rate of utilization,9 and of per capita stock of capital.

2.1

Consumption

To begin, I describe aggregate consumption in the following way:C = c0 N + c(Y − T ),(1)

where c is a positive constant less than unity and c0 is a nonnegative constant. In (1), Y, C, T and N stand for
aggregate income or output, aggregate consumption expenditure, government income tax and the size of population,
respectively; c and c0 represent the marginal propensity to consume and the base level of individual consumption,
respectively. Equation (1) implies that aggregate consumption C is affected positively by aggregate disposable
income Y − T and that aggregate base consumption is proportionate to population N.10 The consumption function(1) can be adapted to the context of economic growth (or of economic decline) because aggregate base consumption
c0 N varies at the same rate as the (natural) rate of economic growth.
9 By assuming that the level of potential output is proportional to that of capital stock, I may identify the output-capital ratio withthe rate of utilization because the latter is proportional to the former.
10 The consumption function (1) is conceptually identical with those given by Murakami (2018a, 2018b). This consumption function
can be related to the saving functions presented in recent studies on the Kaleckian system (cf. Allain 2015; Lavoie 2016).

3

2.2

Investment

Based on the Keynesian theory of investment, I assume that the rate of gross capital formation (i.e., the ratio of
aggregate gross investment to aggregate capital stock) is expressed by a function of the rate of utilization (or theratio of aggregate output to aggregate capital stock) and the rate of interest in the following way:I
= f (u, r).
K(2)

In (2), I, K, u and r stand for aggregate gross investment, aggregate capital stock, the rate of utilization (orthe output-capital ratio) Y /K and the rate of interest, respectively; f is the (gross) capital formation function.
Equation (2) states that the rate of gross capital formation I/K is related to the rate of utilization u and to thatof interest r, and it is consistent with the Keynesian theory of investment.11
Following the same argument as in Murakami (2018b), I give a specific form to the capital formation function
f. For this purpose, I introduce the hypothetical postulate that all firms can choose only two types of investment
behavior: “optimistic” and “pessimistic” plans. A firm is assumed to set its rate of gross capital formation to fo(resp. fp ) if it chooses the “optimistic” (resp. “pessimistic”) plan, where fo and fp are nonnegative constants withfp < fo .12 Under this hypothesis, I can describe the rate of gross capital formation, in the aggregate sense, asa consequence of the distribution of heterogeneous firms.13 Denoting the share of “optimistic” firms by p, I cancalculate the (aggregate) rate of gross capital formation f as follows:f = pfo + (1 − p)fp .

(3)

I turn to the relationship of the rate of utilization u = Y /K and the rate of interest r to the share of “optimistic”
firms p. Since u is positively related to the rate of gross profit (on capital),14 I may say that the larger u is, the more
profitable (fixed) investment is. I can thus suppose that the share of “optimistic” firms p is affected positively byu and negatively by r. In the same way as in logistic regression analysis, I may relate p to u and r in the following
form:

ln

 p 
= ηu u − ηr r − η0 ,
1−p

(4)

where ηu and ηr are positive constants and η0 is a real constant.
11 Equation (2) reflects Keynes’ (1936, chap. 11) theory of investment and the profit and utilization principles of investment (cf.
Kalecki 1939, 1971; Kaldor 1940; Steindl 1952, 1979; Rowthorn 1981). Note that when aggregate capital share is constant, the profit
and utilization principles are identical with each other. For a microeconomic foundation of the profit principle, see Murakami (2016).
12 It is also assumed that f < δ + ν < f , where δ and ν are defined later.
po
13 Based upon the concept of statistical physics, Aoki and Yoshikawa (2007) and Yoshikawa (2015) suggested that aggregate data, as
a consequence of macroeconomic or collective behavior of heterogeneous agents, should be interpreted in terms of “distributions.” Myhypothetical treatment is consistent with their suggestions.
14 Denoting aggregate capital share by π ∈ (0, 1), I can give the rate of (gross) profit by πu. In this paper, I suppose that aggregate
capital share π is constant because it has been known to be roughly constant in the long run (cf. Kaldor 1961; Jones 2016). Under thispostulate, the rate of utilization u can be viewed as an index of profitability.

4

By substituting (4) in (3), I can provide the following form of f :fo (ηu u − ηr r − η0 ) + fp
I
= f (u, r) =.
K
1 + exp(ηu u − ηr r − η0 )(5)

Since p is allowed to vary in [0, 1], any number in [fp , fo ] can be realized as the resultant aggregate rate of grosscapital formation f. If fo and fp are taken as the upper and lower limits of each firm’s rate of gross capitalformation, respectively, our “binary-choice” hypothesis can be seen to describe any feasible outcome of aggregate
capital formation.15 Note that this capital formation function, illustrated in figure 1, is consistent with Kaldor’s(1940) sigmoid investment function.
I/Kδ + ν + φo

δ + ν − φp
η uu − η r r − η 0OFigure 1: Capital formation function

2.3

Government expenditure and income tax

For simplicity, I assume that government expenditure G is proportionate to the size of population N as follows:

G = gN,(6)

where g is a positive constant. In (6), it is assumed that the government conducts no counter-cyclical fiscal policy.
In this paper, I simply suppose that government expenditure G changes with the size of population N .
I also assume that the government collects its income tax T by the following simple rule:T = τ Y − τ0 N,(7)

where τ is a positive constant less than unity and τ0 is a nonnegative constant. In (7), τ and τ0 represent themarginal rate of tax and the base level of individual tax, respectively.
15 For similar specifications of the investment function, see, for instance, Skott (1989, chap. 6), Franke (2014) or Murakami (2018b).

5

Therefore, I can, substituting (7) in (1), obtain the following consumption function:

C = (c0 + cτ0 )N + c(1 − τ )Y.

2.4

(8)

Quantity or utilization adjustment

Following the Keynesian or Kaleckian tradition, I introduce the following formulation for utilization adjustment,
one kind of quantity adjustment:u̇ = αC + I + G
K

−

Y 
,
K

(9)

where α is a positive constant. In (9), α represents the speed of quantity (utilization) adjustment. Equation (9)means that the rate of utilization u is adjusted in response to that difference between aggregate demand per unitof capital stock (C + I + G)/K and u = Y /K, which is equal to aggregate excess demand or supply per unit ofcapital stock.16
Putting (5), (6) and (8) in (9), I can obtain the following equation:u̇ = αh f exp(η u − η r − η ) + f
o

u

r

0

1 + exp(ηu u − ηr r − η0 )p

− σu +

ai
.
k

(10)

where σ = 1 − c(1 − τ ) > 0 and a = c0 + g + cτ0 > 0. In (10), k stands for per capita stock of capital K/N.

2.5

Capital formation

The following equation describes aggregate capital formation process:
K̇
I
=
− δ,
K
K

(11)

where δ is a positive constant. In (11), δ represents the rate of capital depreciation.
Substituting (5) in (11), I can obtain the following expression:K̇
fo exp(ηu u − ηr r − η0 ) + fp=
− δ.
K
1 + exp(ηu u − ηr r − η0 )(12)

16 Since firms determine their level of production on the basis of their (current) level of capacity, it is reasonable to think that therate of utilization, proportionate to the output-capital ratio, is an adjusting variable in firms’ daily production process. Indeed, theformalization in (9) has been adopted in a lot of studies of the post Keynesian or Kaleckian system (cf. Chiarella and Flaschel 2000;
Sasaki 2014; Murakami and Asada 2018).

6

2.6

Population changes

To allow for economic growth or decline due to demographic changes, I assume that the size of population N varies
at a constant rate as follows:
Ṅ
= ν,
N

(13)

where ν is a real constant. For the constant rate of population change ν, it is assumed that fp < δ + ν < fo .
By combining (12) and (13), I can obtain the following differential equation of per capita capital stock k = K/N :k̇
ϕo exp(ηu u − ηr r − η0 ) − ϕp=,k
1 + exp(ηu u − ηr r − η0 )(14)

where ϕo = fo − (δ + ν) > 0 and ϕp = δ + ν − fp > 0.

2.7

Full system: System (K)The dynamical system to be analyzed is given as follows:u̇ = αh f exp(η u − η r − η ) + f
o

u

r

0

p

− σu +aik

1 + exp(ηu u − ηr r − η0 )ϕo exp(ηu u − ηr r − η0 ) − ϕpai=α δ+ν+− σu + ,1 + exp(ηu u − ηr r − η0 )
k

(10)

h

ϕo exp(ηu u − ηr r − η0 ) − ϕpk̇=.
k
1 + exp(ηu u − ηr r − η0 )(14)

The system of (10) and (14) is denoted by “System (K)” (to signify “Keynesian”) in what follows.

3

Analysis

To figure out the effect of counter-cyclical monetary policy, I examine first the case in which the rate of interest r
is constant and then the one in which r is adjusted in response to the gap between the rate of utilization and that
corresponding to full utilization, denoted by u − uf .

7

3.1

The constant rate of interest: System (K*)

To begin, I consider the case in which the rate of interest r is set to a constant by the monetary authority. In thiscase, System (K) can be written in the following form:hϕo exp(ηu u − ηr r∗ − η0 ) − ϕpai,u̇ = α δ + ν +−σu+
1 + exp(ηu u − ηr r∗ − η0 )
k

(15)

k̇
ϕo exp(ηu u − ηr r∗ − η0 ) − ϕp=,k
1 + exp(ηu u − ηr r∗ − η0 )(16)

where r∗ is a nonnegative constant set by the monetary authority. In what follows, the system of (15) and (16), or
System (K) with r = r∗ , is denoted by “System (K*).”I define an equilibrium point of System (K*) as a point (u, k) ∈ R2++ at which u̇ = k̇ = 0. Then, an equilibrium
point of System (K*), (u∗ , k ∗ ), can be redefined as a solution of the following simultaneous equations:ϕo exp(ηu u − ηr r∗ − η0 ) − ϕpa
− σu + ,1 + exp(ηu u − ηr r∗ − η0 )k
ϕo exp(ηu u − ηr r∗ − η0 ) − ϕp.
0=
1 + exp(ηu u − ηr r∗ − η0 )0=δ+ν+The unique equilibrium point of System (K*), provided it exists, can easily be calculated as follows:(u∗ , k ∗ ) = η + η r∗ + ln ϕ − ln ϕ
0

r

p

o

ηu

,


ηu a
.
σ(η0 + ηr r∗ + ln ϕp − ln ϕo ) − ηu (δ + ν)(17)

To make sure that the unique equilibrium point (u∗ , k ∗ ) lies on the economic meaningful domain of R2++ , I impose
the following assumption.
Assumption 1. The following condition is satisfied:r∗ >i
1 h ηu (δ + ν)
+ (ln ϕo − ln ϕp ) − η0 .
ηr
σ

(18)

Condition (18) holds if the rate of interest r∗ is sufficiently large or if the parameter ηr is sufficiently large (i.e.,
if investment is elastic enough to a change in the rate of interest). Under Assumption 1, I can ensure the existence
and uniqueness of an economically meaningful equilibrium point of System (K*).
I confirm that the values of u(t) and of k(t) are positive all the time for every solution path of System (K*) with
its initial condition (u(0), k(0)) ∈ R2++ . As for k(t), it is easily known from (14) to be positive for all t ≥ 0 provided
that k(0) > 0. Then, it suffices to check that u(t) is positive all the time along every solution path of System (K*).
For this purpose, we only have to see that u̇ > 0 for u = 0 and for every k > 0:u̇|u=0 = α

h f + f exp(η r∗ + η )
o

p

r
0
1 + exp(ηr r∗ + η0 )

8

+

ai
> 0.
k

This inequality turns out to hold because 0 ≤ fp < fo . I can thus find that (u(t), k(t)) ∈ R2++ along every solution
path of System (K*) with (u(0), k(0)) ∈ R2++ .
To facilitate the analysis, I reformulate System (K*) by introducing the following new variables x and y:x = u − u∗ ,(19)

y = ln k ∗ − ln k,(20)

where (u∗ , k ∗ ) is the unique equilibrium defined by (17). Substituting (19) and (20) in System (K*), I can obtainthe following system:ẋ = φ(y) − F (x),(21)

ẏ = −g(x),

(22)

where

g(x) =ϕo ϕp[exp(ηu x) − 1],
ϕo + ϕp exp(ηu x)(23)

F (x) = α[σx − g(x)],φ(y) = α

hσ
ηu

(24)

i
(η0 + ηr r∗ + ln ϕp − ln ϕo ) − (δ + ν) [exp(y) − 1].

(25)

In what follows, the system of (21) and (22) with (23)-(25) is denoted by System (L*).17 To investigate the properties
of System (K*), I have a closer look at System (L*) instead of System (K*). Note that the unique equilibrium pointof System (L*) is given by (x∗ , y ∗ ) = (0, 0).
Now I explore the possibility of emergence of persistent growth cycles, represented by a periodic orbit, in System(K*). To begin, I turn to local asymptotic stability of the unique equilibrium in System (L*). The Jacobian matrixof System (L*) evaluated at the unique equilibrium (0, 0), denoted by J ∗ , is given by∗ α[ηu ϕo ϕp /(ϕo + ϕp ) − σ] α[σ(η0 + ηr r − ln ϕo + ln ϕp )/ηu − (δ + ν)] J∗ = .
−ηu ϕo ϕp /(ϕo + ϕp )0
The trace and determinant of J ∗ are given as follows:

tr J ∗ = α
det J ∗ = αη ϕ ϕuop

ϕo + ϕp


−σ ,

ϕo ϕp
[σ(η0 + ηr r∗ − ln ϕo + ln ϕp ) − ηu (δ + ν)] > 0,
ϕo + ϕp

17 System (L*) can be regarded as a generalized Liénard system. For generalized Liénard systems, see, for instance, Xiao and Zhang(2003).

9where the inequality follows from Assumption 1. Then, the unique equilibrium point is not a saddle point. For the
emergence of a periodic orbit, I make the following assumption.
Assumption 2. The following condition is satisfied:ηu ϕo ϕp> σ.
ϕo + ϕp(26)

The economic implication of this assumption can be expounded as follows. The left hand side of (26) is themarginal propensity to invest, while the right hand side is that to save with income tax taken into consideration. In
this sense, condition (26) requires that the marginal propensity to save be smaller than that to invest at the uniqueequilibrium, i.e., that the Keynesian stability condition (cf. Marglin and Bhaduri 1990) be violated at this point.
As Kaldor (1940) insisted, the violation of the Keynesian stability condition plays a pivotal role for the emergenceof persistent business cycles in the Keynesian system.18
Next, I detect a (nonempty) compact and positively invariant region in System (L*). To this end, I draw thephase diagram of System (L*). First, I have a look at the loci of ẋ = 0 and of ẏ = 0 of System (L*). As regards
the locus of ẏ = 0, I can see from (22) and (23) that it is given by x = 0. As for the locus of ẋ = 0, it is givenby φ(y) = F (x). To scrutinize this locus, I first consider the characteristics of F (x). I can easily calculate F 0 (x) as
follows:n
F 0 (x) = α σ − ηu ϕo ϕp (ϕo + ϕp )oexp(ηu x).
2[ϕo + ϕp exp(ηu x)](27)

I can then calculate the two roots of F 0 (x) = 0, denoted by x0 and x0 , as follows:p
1  ϕo {ηu (ϕo + ϕp ) − 2σ − ηu (ϕo + ϕp )[ηu (ϕo + ϕp ) − 4σ]} 
x =
ln
< 0,
ηu
2σϕp
p
1  ϕo {ηu (ϕo + ϕp ) − 2σ + ηu (ϕo + ϕp )[ηu (ϕo + ϕp ) − 4σ]} 
x0 =
ln
> 0,
ηu
2σϕp
0

(28)
(29)

where the inequalities follow from Assumption 2. It then follows that F 0 (x) > 0 for x < x0 or x > x0 and F 0 (x) < 0
for x ∈ (x0 , x0 ). Thus, I have F (x0 ) > F (0) = 0 > F (x0 ). Since F (±∞) = ±∞, I can find that F (x) = 0 has exactly
two roots x0 and x0 besides x = 0 with x0 < x0 < 0 < x0 < x0 . Thus, I can draw the graph of F (x) as in figure 2.
18 Recently, Skott (2012) and Murakami (2018b) reported that the marginal propensity to save is less than that to invest in reality.

10

F(x)

x'

x0
O

x'

x0

x

Figure 2: Graph of F (x)Second, I ensure that the locus of ẋ = 0 or of φ(y) = F (x) is well defined at least for x ∈ [x0 , x0 ].19 It is easy tosee from figure 2 that, for x ∈ [x0 , x0 ], the minimum of F is given by F (x0 ). According to the definition of φ givenby (25), φ is strictly increasing in y and satisfies φ(∞) = ∞. Thus, I impose the following condition for the locusof ẋ = 0 to be well defined for x ∈ [x0 , x0 ]:F (x0 ) > φ(−∞).

(30)

For this condition to hold, I make the following additional assumption.
Assumption 3. The following condition is satisfied:pηu (ϕo + ϕp )[ηu (ϕo + ϕp ) − 4σ]2σ η (ϕ + ϕ ) + pη (ϕ + ϕ )[η (ϕ + ϕ ) − 4σ]

u
o
p
u
o
p
u
o
p
− ln
−1 .
2σ

η0 >

ηu (2δ + 2ν + ϕo − ϕp ) +(31)

Condition (31) is satisfied if η0 is relatively large. Condition (30) is actually fulfilled under Assumption 3. Under
Assumptions 2 and 3 along with Assumption 1, the phase diagram can be drawn as in figure 3. Note that the locusof ẋ = 0 is wholly located on the right side of the line of x = −u∗ , which corresponds to u = 0 in System (K*),
because u̇ > 0 for every u = 0 and k > 0.
19 This is one of the requirements of Theorem 1, which is used for the uniqueness of a limit cycle in System (K*).

11

y

=0

=0

′
− *

′

O

x

Figure 3: Phase diagram of System (L*)

Now I am ready to detect a positively invariant region with respect to System (L*) by making use of the phase
diagram illustrated in figure 3. To this end, I consider the solution path of System (L*) with the initial condition(x(0), y(0)) = (−u∗ , 0). I can see that it starts at P(−u∗ , 0) for t = 0 and intersects with the y axis on the positive
part at Q(0, yq ) for the first time. I denote by R(xr , yq ) the intersection point of the locus of ẋ = 0 and the line
through Q and parallel with the x axis. I next consider the solution path of System (L*) with the initial condition(x(0), y(0)) = (xr , 0). I can find that it starts at S(xr , 0) for t = 0 and intersects with the y axis on the positive
part at T(0, yt ) for the first time. I denote by D the region enclosed by the arcs of PQ and of ST and by the line
segments of QR, of RS, of SU and of UP, where U(−u∗ , yt ) (cf. figure 4). Since the arc of PQ or of ST turns out
to be uniquely determined as a solution path of System (L*),20 no solution path can cross either of them. Then, it
is easily seen from the phase diagram drawn in figure 4 that D is a positively invariant compact region.
20 It is possible to check that the Lipschitz condition (cf. Coddington and Levinson 1955, chap. 1) holds to guarantee the uniquenessof a solution path of System (L*) with respect to an initial condition. For the Lipschitz condition in related Keynesian models, see
Murakami (2014, 2017, 2018a).

12

y
R

Q

𝑥̇ = 0𝑦̇ = 0

D

P

−𝑢*

U

S

O

x

T

Figure 4: Positively invariant region D

I am in a position to verify the existence of a periodic orbit in System (K*) or (L*) with the help of the
Poincaré-Bendixson theorem.
Proposition 1. Let Assumptions 1-3 hold. Then, System (K*) has at least one periodic orbit on R2++ .
Proof. It follows from Assumptions 1 and 2 that the unique equilibrium of System (L*), (0, 0), is locally asymptotically totally unstable. It is then possible to construct a new positively invariant compact set, denoted by D0 ,by eliminating the interior of a sufficiently small rectangle surrounding the unique equilibrium from the positively
invariant compact set D illustrated in figure 4. I can apply the Poincaré-Bendixson theorem (cf. Coddington andLevinson 1955, chap. 16) to the compact set D0 to conclude that there exists at least one periodic orbit of System(L*) on this set. Since ẋ > 0 for x = −u∗ and every real y, such a periodic orbit is wholly located on the domainof x > −u∗ . I can thus draw the conclusion of this proposition due to (19) and (20).
Proposition 1 implies that persistent growth cycles, represented by a periodic orbit, are observed in System(K*).
Now I proceed to discuss the uniqueness of a periodic orbit (or of persistent growth cycles) in System (K*).
For this purpose, I make use of the theorem established by Xiao and Zhang (2003), reproduced as Theorem 1 inAppendix.
To establish the uniqueness of a periodic orbit (or of a limit cycle), I examine whether System (L*) satisfies therequirements of Theorem 1, Assumptions 5-7 in Appendix. First, it is easy to find from (23)-(25) and figure 2 that
Assumptions 5 and 6 are fulfilled. Second, the former half of Assumption 7 has already been proved to hold by
Assumption 3. In what follows, I check if the latter half of Assumption 7 is met in System (L*).

13

Now I claim that the latter half of Assumption 7, condition (56), holds if α is sufficiently large. To see this, I begin
with confirming that condition (56) is satisfied if the following condition is fulfilled because x0 < x0 < 0 < x0 < x0 :
 G(x0 ) + Φ(φ−1 (F (x0 ))) ≥ G(x ) if G(x ) > G(x0 ),00 G(x0 ) + Φ(φ−1 (F (x0 ))) ≥ G(x0 ) if G(x0 ) > G(x0 ),or Φ(φ−1 (F (x0 ))) ≥ G(x ) − G(x0 ) if G(x ) > G(x0 ),
0

0

(32)


 Φ(φ−1 (F (x0 ))) ≥ G(x0 ) − G(x0 ) if G(x0 ) > G(x0 ).
Next, I confirm that Φ(φ−1 (F (x))) is positive and proportional to α for x ∈ (x0 , 0) or x ∈ (0, x0 ), whereRyΦ(y) = 0 φ(s)ds. It is easy to see from (25) that the inverse function of φ is given as follows:21

φ−1 (y) = ln 1 +


ηu
y
.
α[σ(η0 + ηr r∗ + ln ϕp − ln ϕo ) − ηu (δ + ν)]

Also, I haveZ yΦ(y) =

φ(s)ds = α0

σ(η0 + ηr r∗ + ln ϕp − ln ϕo ) − ηu (δ + ν)[exp(y) − y − 1].
ηuThen, I find thatσ(η0 + ηr r∗ + ln ϕp − ln ϕo ) − ηu (δ + ν)[v(x) − ln(1 + v(x))],
ηu

(33)

ηu
[σx − g(x)].
σ(η0 + ηr r∗ + ln ϕp − ln ϕo ) − ηu (δ + ν)(34)

Φ(φ−1 (F (x))) = α

wherev(x) =

Since F (x) = α[σx − g(x)] 6= 0 for x ∈ (x0 , 0) or x ∈ (0, x0 ) and v(x) > ln(1 + v(x)) for v(x) 6= 0, it is easily seenfrom (33) that Φ(φ−1 (F (x))) is positive and proportional to α for x ∈ (x0 , 0) or x ∈ (0, x0 ).
Finally, I verify that condition (32), a sufficient condition for (56), holds if α is sufficiently large. To this end,
it suffices to confirm that the left hand side of (32) is positive and proportional to α and the right hand side isindependent from α. As is easily seen from (28), (29), (33) and (34), the left hand side of (32) is positive andproportional to α in each of the cases, while the right hand side is independent from α because x0 and x0 are
independent from α. Thus, I can assert that condition (32) is fulfilled for α large enough.
With the help of Theorem 1, I can establish the uniqueness of a periodic orbit (or of a limit cycle) in System
21 It follows from Assumption 3 that φ−1 (F (x)) is well defined for x ∈ [x , x ].
0014

(K*) or (L*).
Proposition 2. Let Assumptions 1-3 hold. Then, System (K*) has a unique and (periodically) stable limit cycle
on R2++ , if α is sufficiently large.
Proof. As I have confirmed above, Assumptions 5-7 are all satisfied if α is large enough. It then follows from
Theorem 1 that System (L*) possesses a unique limit cycle and so does System (K*).
This proposition indicates the fact that there exists a “unique” growth cycle in System (K*), in which the rateof interest is kept constant by the monetary authority, if the speed of quantity (or utilization) adjustment α issufficiently large. Since it is fast quantity adjustment that is one of the distinguished features of the Keynesian
theory (cf. Leijonfuhvud 1968; Tobin 1993), the conclusion of Proposition 2 implies that a persistent and uniquegrowth cycle is a universal phenomenon in the Keynesian system (when the rate of interest is kept constant).

3.2

The monetary policy rule for the rate of interest: System (K)Now I explore the possibility that the monetary authority can mitigate persistent economic fluctuations along theunique growth cycle. For this purpose, I first define the “natural” rate of interest as one of the criteria in monetary
policy implementations and then formalize a monetary policy rule for the rate of interest.
To begin, I define the natural rate of interest rn as the rate of interest for which the equilibrium value of therate of utilization of System (K*), u∗ , is equal to the level of full utilization uf . It then follows from (17) that thenatural rate of interest rn is defined as follows:rn =1(ηu uf − η0 + ln ϕo − ln ϕp ),
ηr

(35)

where uf is a positive constant that stands for the output-capital ratio (rate of utilization) in full utilization.
Next, I turn to the monetary authority’s counter-cyclical policy rule. Specifically, I postulate that the monetary
authority adopts the following rule in setting the rate of interest r:

r = rn + β(u − uf ),(36)

where β is a positive constant. In (36), β is a policy parameter that represents the willingness of the monetary
authority to pursue the level of full utilization. Equation (36) means that the monetary authority follows a kind of
feedback policy rule similar to Taylor’s (1993) one.22 In this section, I examine the effect of changes in the monetary
policy parameter β on the stability of the unique equilibrium and on the unique growth cycle in my post Keynesiansystem.
22 Unlike in Taylor’s (1993) rule, the rate of inflation is not taken into consideration because price changes are not considered in myanalysis.

15

I can thus complete our post Keynesian system by substituting (36) in (10) and (14):hai
ϕo exp(ηu u − ηr [rn + β(u − uf )] − η0 ) − ϕp
− σu + ,u̇ = α δ + ν +1 + exp(ηu u − ηr [rn + β(u − uf )] − η0 )
k

(37)

k̇
ϕo exp(ηu u − ηr [rn + β(u − uf )] − η0 ) − ϕp=.
k
1 + exp(ηu u − ηr [rn + β(u − uf )] − η0 )(38)

In what follows, the system of equations (37) and (38) is redefined as “System (K).”An equilibrium point of System (K), (u∗ , k ∗ ), is defined as a point (u, k) ∈ R2++ at which we have u̇ = k̇ = 0 oraϕo exp(ηu u − ηr [rn + β(u − uf )] − η0 ) − ϕp
− σu + ,1 + exp(ηu u − ηr [rn + β(u − uf )] − η0 )k
ϕo exp(ηu u − ηr [rn + β(u − uf )] − η0 ) − ϕp0=.
1 + exp(ηu u − ηr [rn + β(u − uf )] − η0 )0=δ+ν+

Then, the unique equilibrium point of System (K), provided that it exists, can be given as follows:(u∗ , k ∗ ) = uf ,a.
σuf − (δ + ν)(39)

To guarantee the existence of the unique equilibrium point on R2++ , I make the following assumption.
Assumption 4. The following condition is satisfied:

uf >

δ+ν
.
σ

(40)

Condition (40) requires that the rate of full utilization uf be large enough. In this respect, Assumption 4 is nota stringent assumption.
As in the previous subsection, I re-formalize System (K) by introducing the variables x and y defined by (19)and (20), where the equilibrium values, u∗ and k ∗ , are redefined by (39). Making use of these x and y, System (K)can be transformed into the following system:ẋ = φ(y) − F (x),(41)

ẏ = −g(x),(42)

where F is defined by (24) and g and φ are redefined byg(x) =ϕo ϕp[exp((ηu − βηr )x) − 1],
ϕo + ϕp exp((ηu − βηr )x)φ(y) = α[σuf − (δ + ν)][exp(y) − 1].

16

(43)
(44)

In what follows, the system of (41) and (42) with (24), (43) and (44) is denoted by System (L). The uniqueequilibrium point of System (L) is, of course, given by (x∗ , y ∗ ) = (0, 0).
I am now in a position to examine the stability of the unique equilibrium of System (L). I can easily calculate
the Jacobian matrix of System (L), denoted by J, as follows:23



n
o
2
α[σuf − (δ + ν)] exp(y)  α [ϕo ϕp (ϕo + ϕp )(ηu − βηr ) exp((ηu − βηr )x)]/[ϕo + ϕp exp((ηu − βηr )x)] − σJ =.
−[ϕo ϕp (ϕo + ϕp )(ηu − βηr ) exp((ηu − βηr )x)]/[ϕo + ϕp exp((ηu − βηr )x)]20
The trace and determinant of J are as follows:on (η − βη )ϕ ϕ (ϕ + ϕ ) exp((η − βη )x)
u
r
o p
o
p
u
r
−
σ
,
[ϕo + ϕp exp((ηu − βηr )x]2
ϕo ϕp (ϕo + ϕp ) exp((ηu − βηr )x + y)
det J = α(ηu − βηr )[σuf − (δ + ν)].
[ϕo + ϕp exp((ηu − βηr )x]2
tr J = α

(45)
(46)

It is easily seen from (46) that if β is large enough to satisfy β > ηu /ηr , the determinant of J is negative and theunique equilibrium is a saddle point and locally asymptotically unstable. If β is close to 0, on the other hand, I canfind from (45) that the unique equilibrium is locally asymptotically totally unstable and from the conclusion of theprevious section that a periodic orbit is generated.
I explore the possibility that the monetary authority can provide the unique equilibrium of System (L) with
“global” asymptotic stability by choosing a suitable value of the policy parameter β.24 According to the wellestablished Olech theorem, the unique equilibrium of System (L) is globally asymptotically stable if the following
conditions are satisfied for every (x, y):25n (η − βη )ϕ ϕ (ϕ + ϕ ) exp((η − βη )x)
o
u
r
o p
o
p
u
r
−
σ
< 0,
[ϕo + ϕp exp((ηu − βηr )x]2
ϕo ϕp (ϕo + ϕp ) exp((ηu − βηr )x + y)
det J = α(ηu − βηr )[σuf − (δ + ν)]> 0.
[ϕo + ϕp exp((ηu − βηr )x]2
tr J = α

(47)
(48)

I can immediately know that condition (48) holds for every (x, y) if β < ηu /ηr . In what follows, I examine the casein which condition (47) is satisfied. As for this condition, it is seen that−1exp((ηu − βηr )x)
ϕo
=
2ϕ
ϕ
+
+
ϕ
exp((η
−
βη
)x)
o p
p
u
r
[ϕo + ϕp exp((ηu − βηr )x)]2
exp((ηu − βηr )x)
−11≤ 2ϕo ϕp + 2ϕo ϕp=,
4ϕo ϕp23 Note that the Jacobian matrix J is not confined to the unique equilibrium.
24 The unique equilibrium point of the dynamical system under consideration is said to be globally asymptotically stable if every

solution path with an arbitrary initial condition converges to this point as → ∞. The concept of global asymptotic stability is, of course,
stronger than that of local asymptotic stability.
25 Olech’s (1963, p. 395, Theorem 4) theorem also requires that either the product of the (1,1) and (2,2) elements or that of the (1,2)
and (2,1) elements of the Jacobian matrix J is nonzero for every (x, y), but this requirement is met in our case if the determinant is
positive.

17where the equality in the above inequality holds if x = (ln ϕo − ln ϕp )/[2(ηu − βηr )]. Then, we obtainϕo + ϕp(ηu − βηr )ϕo ϕp (ϕo + ϕp ) exp((ηu − βηr )x)≤(ηu − βηr ).
[ϕo + ϕp exp((ηu − βηr )x]2
4
I can thus find that conditions (47) and (48) are both fulfilled for every (x, y) if
1
4σ 
ηu
ηu −
<β<
.
ηr
ϕo + ϕpηr(49)

It is thus possible to draw the following conclusion on the stability of the unique equilibrium of System (K).
Proposition 3. Let Assumptions 3 and 4 hold. Then, the unique equilibrium point of System (K) is globally
asymptotically stable if β satisfies (49).
Proof. I have seen that, if condition (49) holds, the unique equilibrium of System (L) possesses global asymptotic
stability and then so does that of System (K).
Proposition 3 implies that the monetary authority has to set the monetary policy parameter β, measuring theintensiveness of its monetary policy, to a “medium” value, not too small or too large, so as to achieve the objectiveof global asymptotic stability. I can thus assert that the monetary authority has to conduct a “moderate” countercyclical policy to attain the economic stability in our post Keynesian system, i.e., that the monetary authority
should not be too passive or too active in its monetary policy implementations.

4

Conclusion

I am now in a position to summarize my analysis.
I have set up a post Keynesian system to discuss the existence and uniqueness of a growth cycle and evaluate theeffectiveness of monetary policy for macroeconomic stability. Following Murakami (2018b), I have first demonstrated
that a persistent growth cycle, represented by a limit cycle, uniquely exists in our post Keynesian system if the speedof quantity adjustment is high enough. I have also clarified that the monetary authority can mitigate (or eliminate)
cyclical fluctuations along the unique growth cycle by its “moderate,” not too active or too passive, counter-cyclicalpolicy. These consequences indicate that the (post) Keynesian system, characterized by fast quantity or utilization
adjustment, is, without policy interventions, exposed to persistent economic fluctuations (with constant amplitudes)
induced by the unique growth cycle but that this system can be stabilized (in the global sense) by the monetary
authority’s proper policy implementations. In this respect, it is possible to state that our analysis confirms thevulnerability of the laissez-faire regime and the need for policy interventions.

18

Acknowledgment
I would like to thank Yoshiyuki Arata, Marwil J. Dávila-Fernández, John Barkley Rosser Jr., Serena Sordi andAnna M. Variato (alphabetical order) for their helpful comments on an earlier version of this paper. Needless tosay, I am solely responsible for all the possible remaining errors. This work was supported by Chuo University
Personal Research Grant and by JSPS KAKENHI (Grant Number 18K12748).

Appendix
I briefly introduce the theorem established by Xiao and Zhang (2003) on the uniqueness of a (stable) limit cycle ingeneralized Liénard systems.
I consider the following generalized Liénard system:ẋ = φ(y) − F (x),(50)

ẏ = −g(x).

(51)

In what follows, the system of equations (50) and (51) is denoted by System (GL).
Following Xiao and Zhang (2003), I make the following assumptions concerning System (GL).
Assumption 5. The real valued functions g(x) and F (x) are, respectively, continuous and continuously differentiable on (x, x), and the real valued function φ(y) is continuously differentiable on (y, y) with −∞ ≤ x < 0 < x ≤ ∞and −∞ ≤ y < 0 < y ≤ ∞. Furthermore, the following conditions are satisfied:

xg(x) > 0 for x 6= 0,(52)φ(0) = 0, φ0 (y) > 0 for y ∈ (y, y).

(53)

Assumption 6. There exist x0 and x0 with x < x0 < 0 < x0 < x such that the following conditions are satisfied:

F (x0 ) = F (0) = F (x0 ) = 0,
 xF (x) ≤ 0 for x ∈ (x , x0 ),
0

(54)
(55)


 xF (x) > 0, F 0 (x) ≥ 0 for x ∈ (x, x0 ) or x ∈ (x0 , x).
Furthermore, F (x) is not identically equal to 0 for x sufficiently close to 0.
Assumption 7. The curve of φ(y) = F (x) is well defined for x ∈ [x0 , x0 ].26 Furthermore, the following condition
26 Xiao and Zhang (2003) assumed that the curve of φ(y) = F (x) is well defined for x ∈ (x, x), but my assumption suffices for the
proof of their theorem (cf. Xiao and Zhang 2003, pp. 1187-1190).

19

is satisfied:


 sup

x∈[0,x0 ] (G(x) + Φ(φ−1(F (x)))) ≥ G(x0 ) if G(x0 ) ≥ G(x0 ),(56)

 supx∈[x ,0] (G(x) + Φ(φ−1 (F (x)))) ≥ G(x0 ) if G(x0 ) > G(x0 ),0where G(x) =Rx0g(s)ds and Φ(x) =Rx0

φ(s)ds.

Assumption 5 is a typical one in the theory of generalized Liénard systems, while Assumptions 6 and 7 concern
the shape of F (x) and those of g(x) and φ(x), respectively.
Concerning the uniqueness of a periodic orbit (or of a limit cycle) in System (GL), the following theorem wasestablished by Xiao and Zhang (2003).
Theorem 1. Let Assumptions 5-7 hold. Then, System (GL) has at most one limit cycle, and it is (periodically)
stable if it exists.
Proof. See Xiao and Zhang (2003, p. 1187, Theorem 2.2).

References
Allain, O., 2015. Tacking the instability of growth: a Kaleckian-Harrodian model with an autonomous expenditurecomponent. Cambridge Journal of Economics 39 (5), 1351-1371.
Aoki, M., Yoshikawa, H., 2007. Reconstructing Macroeconomics: A Perspective from Statistical Physics andCombinatorial Stochastic Processes. Cambridge University Press, Cambridge.
Asada, T., 2014. Japanese contributions to dynamic economic theories from the 1940s to the 1970s: A historicalsurvey. Asada, T. (Ed.), The Development of Economics in Japan: From the Inter-war Period to the 2000s.
Routledge, London, pp. 61-89.
Benhabib, J., Miyao, T., 1981. Some new results on the dynamics of the generalized Tobin model. International
Economic Review 22 (3), 589-596.
Benhabib, J., Nishimura, K., 1979. The Hopf bifurcation and the existence and stability of closed orbits in multisector models of optimal economic growth. Journal of Economic Theory 21 (3), 421-444.
Chang, W. W., Smyth, D. J., 1971. The existence and persistence of cycles in a non-linear model: Kaldor’s 1940
model re-examined. Review of Economic Studies 38 (1), 37-44.
Chenery, H. B., 1952. Overcapacity and the accelerator principle. Econometrica 20 (1), 1-28.
Chiarella, C., Flaschel, P., 2000. The Dynamics of Keynesian Monetary Growth: Macro Foundations. Cambridge
University Press, Cambridge.

20

Coddington, E. A., Levinson, N., 1955. Theory of Ordinary Differential Equations. McGraw Hill, New York.
Flaschel, P., 1984. Some stability properties of Goodwin’s growth cycle a critical elaboration. Journal of Economics
44 (1), 63-69.
Franke, R., 2014. Aggregate sentiment dynamics: A canonical modelling approach and its pleasant nonlinearities.
Structural Change and Economic Dynamics 31, 64-72.
Franke, R., 2017. A simple approach to overcome the problems arising from the Keynesian stability condition.
European Journal of Economics and Economic Policies: Intervention 14 (1), 48-69.
Franke, R., Westerhoff, F., 2017. Taking stock: A rigorous modelling of animal spirits in macroeconomics. Journalof Economic Surveys 31 (5), 1152-1182.
Galeotti, M., Gori, F., 1989. Uniqueness of periodic orbits in Liénard-type business cycle models. Metroeconomica
40 (2), 135-146.
Goodwin, R. M., 1951. The non-linear accelerator and the persistence of business cycles. Econometrica 19 (1),1-17.
Goodwin, R. M., 1967. A growth cycle. In: Feinstein C. H. (ed.), Socialism, Capitalism and Economic Growth.
Cambridge University Press, Cambridge, pp. 54-58.
Harrod, R. F., 1936. The Trade Cycle: An Essay. Clarendon Press, Oxford.
Hein, E., Lavoie, M., van Treeck, T., 2011. Some instability puzzles in Kaleckian models of growth and distribution:a critical survey. Cambridge Journal of Economics 35 (3), 587-612.
Hicks, J. R., 1950. A Contribution to the Theory of the Trade Cycle. Clarendon Press, Oxford.
Ichimura, S., 1955. Toward a general nonlinear macrodynamic theory of economic fluctuations. In: Kurihara, K.
K. (Ed.), Post Keynesian Economics. New York, Wolff Book Manufacturing, pp. 192-226.
Jones, C. I., 2016. The facts of economic growth. In: Taylor, J. B., Uhlig, H., (Eds.), Handbook of Macroeconomics,
Vol. 2. North Holland, pp. 3-69.
Kaldor, N., 1940. A model of the trade cycle. Economic Journal 50 (192), 78-92.
Kaldor, N., 1951. Mr. Hicks on the trade cycle. Economic Journal 61 (244), 833-847.
Kaldor, N., 1961. Capital accumulation and economic growth. In: Lutz, F. A., Hague, D. C. (Eds.), The Theoryof Capital: Proceedings of a Conference held by the International Economic Association. Macmillan, London, pp.
177-222.

21

Kalecki, M., 1939. Essays in the Theory of Economic Fluctuations. George and Unwin, London.
Kalecki, M., 1971. Selected Essays on the Dynamics of the Capitalist Economy 1933-1970. Cambridge University
Press, Cambridge.
Keynes, J. M., 1936. The General Theory of Employment, Interest and Money. Macmillan, London.
Kosobud, R. F., O’Neill, W. D., 1972. Stochastic implications of orbital asymptotic stability of nonlinear tradecycle model. Econometrica 40 (1), 69-86.
Lavoie, M., 2016. Convergence towards the normal rate of capacity utilization in neo-Kaleckian models: the roleof non-capacity creating autonomous expenditures. Metroeconomica 67 (1), 172-201.
Leijonhufvud, A., 1968. On Keynesian Economics and the Economics of Keynes: A Study in Monetary Theory.
Oxford University Press, Oxford.
Lorenz, H. W., 1986. On the uniqueness of limit cycles in business cycle theory. Metroeconomica 38 (3), 281-293.
Lorenz, H. W., 1993. Nonlinear Dynamical Economics and Chaotic Motion, 2nd ed. Springer-Verlag, Berlin Heidelberg.
Marglin, S. A., Bhaduri, A., 1990. Profit squeeze and Keynesian theory. In: Marglin, A., Schor, J. B. (Eds.), The
Golden Age of Capitalism: Reinterpreting the Postwar Experience. Clarendon Press, Oxford, pp. 153-186.
Marsden, J. E., McCracken, M., 1976. The Hopf Bifurcation and its Application. Springer-Verlag, New York.
Morishima, M., 1959. A contribution to the nonlinear theory of the trade cycle. Journal of Economics 18 (4),165-173.
Murakami, H., 2014. Keynesian systems with rigidity and flexibility of prices and inflation-deflation expectations.
Structural Change and Economic Dynamics 30, 68-85.
Murakami, H., 2016. A non-Walrasian microeconomic foundation of the “profit principle” of investment. In: Matsumoto A., Szidarovszky, F., Asada, T. (Eds.), Essays in Economic Dynamics: Theory, Simulation Analysis, andMethodological Study. Springer, Singapore, pp. 123-141.
Murakami, H., 2017. Time elements and oscillatory fluctuations in the Keynesian macroeconomic system. Studies
in Nonlinear Dynamics and Econometrics 17 (2).
Murakami, H., 2018a. Existence and uniqueness of growth cycles in post Keynesian systems. Economic Modelling,
forthcoming.
Murakami, H., 2018b. A note on the “unique” business cycle in the Keynesian theory. Metroeconomica, forthcoming.

22

Murakami, H., Asada, T., 2018. Inflation-deflation expectations and economic stability in a Kaleckian system.
Journal of Economic Dynamics and Control 92, 183-201.
Olech, C., 1963. On the global stability of an autonomous system on the plane. In: Lasalle, J. P., Diaz, J. B.
(Eds.), Contributions to Differential Equations 1 (3), pp. 389-400.
Peixoto, M. M., 1962. Structural stability on two-dimensional manifolds. Topology 1 (2), 101-120.
Rose, H., 1967. On the non-linear theory of the employment cycle. Review of Economic Studies 34 (2), 153-173.
Rowthorn, R. E., 1981. Demand, real wages and economic growth. Thames Papers in Political Economy 22, 1-39.
Samuelson, P. A., 1939. Interactions between the multiplier analysis and the principle of acceleration. Review of
Economics and Statistics 21 (2), 75-78.
Sasaki, H., 2014. Growth, Cycles, and Distribution: A Kaleckian Approach. Kyoto University Press, Kyoto.
Sasakura, K., 1996. The business cycle model with a unique stable limit cycle. Journal of Economic Dynamics andControl 20, 1763-1773.
Skott, P., 1989. Conflict and Effective Demand in Economic Growth. Cambridge University Press, Cambridge.
Skott, P., 2012. Theoretical and empirical shortcomings of the Kaleckian investment function. Metroeconomica 63(1), 109-138.
Steindl, J., 1952. Maturity and Stagnation in American Capitalism. Basil Blackwell, Oxford.
Steindl, J., 1979. Stagnation theory and stagnation policy. Cambridge Journal of Economics 3 (1), pp. 1-14.
Stiglitz, J. E., 1967. A two-sector two class model of economic growth. Review of Economic Studies 34 (2), 227-238.
Taylor, J. B., 1993. Discretion versus policy rules in practice. Carnegie-Rochester Conference Series on Public
Policy 39, 195-214.
Tobin, J., 1993. Price flexibility and output stability: an old Keynesian view. Journal of Economic Perspectives 7(1), 45-65.
Velupillai, K. V., 1979. Some stability properties of Goodwin’s growth cycle. Journal of Economics 39 (3-4),
245-257.
Velupillai, K. V., 2008. Japanese contributions to nonlinear cycle theory in the 1950s. Japanese Economic Review
59 (1), 54-74.
Xiao, D., Zhang, Z., 2003. On the uniqueness and nonexistence of limit cycles for predator-prey systems. Nonlinearity 16 (3), 1185-1201.

23

Yasui, T., 1953. Nonlinear self-excited oscillations and business cycles. Cowels Commission Discussion Papers:
Economics 2065.
Yoshikawa, H., 2015. Stochastic macro-equilibrium: a microfoundation for the Keynesian economics. Journal of
Economic Interaction and Coordination 10 (1), 31-55.

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