Discussion Paper No.334
The unique limit cycle in post Keynesiansystems

Hiroki Murakami
Faculty of Economics, Chuo University

September 2020

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

The unique limit cycle in post Keynesian systems
Hiroki Murakami∗
Faculty of Economics, Chuo University†

Abstract
This paper establishes not only the existence and uniqueness of a limit cycle (representing persistent growth
cycles) but also the convergence of every non-equilibrium solution path (describing the state of the economy) tothe unique limit cycle in a post Keynesian system.

Keywords: Convergence; Growth cycle; Keynesian theory; Unique limit cycle
JEL classification: C62; E12; E32; E52

1

Introduction

Cyclical economic fluctuations, referred to as business cycles or growth cycles (business cycles around the trend of
economic growth), have been one of the major phenomena which call for proper theoretical expositions. Soon after
the publication of Keynes’ General Theory, the foundations of the Keynesian theory of business cycles were laid by,for instance, Kalecki (1935, 1937), Harrod (1936), Samuelson (1939), Kaldor (1940), Metzler (1941), Hicks (1950)
and Goodwin (1951).1 In the Keynesian theory of business cycles, aggregate income is determined by aggregate
effective demand composed mainly of consumption and investment (through the principle of effective demand) andfluctuations in the former are brought about by variations in the latter, especially by those in investment (throughthe multiplier process). For this reason, the Keynesian theory of business cycles focuses mainly on the mechanismof cyclical changes in investment. Indeed, Keynes (1936, chap. 22) attributed the main cause of business cycles tofluctuations in investment on account of variations in the marginal efficiency of capital (or the expected rate of profit
on capital). Although Keynes (1936) himself emphasized the effect of expectations (or the long-term expectation) on
investment, most Keynesian models of business cycles, including those in the aforementioned literature, slight thiseffect, postulating that investment is determined by past and current levels of income, profits and other variables.2∗ E-mail: hmura@tamacc.chuo-u.ac.jp† 742-1, Higashi-Nakano, Hachioji, Tokyo 192-0393, Japan
1 Precisely speaking, Kalecki (1935) and Harrod (1936) preceded Keynes (1936), but their theories stand on Keynesian perspectives(cf. Kalecki 1937).
2 The dominant principles of investment in the Keynesian theory are the followings: the profit principle (cf. Kalecki 1935, 1937;
Kalecki 1940), the acceleration one (cf. Harrod 1936; Samuelson 1939; Metzler 1941; Hicks 1950; Goodwin 1951) and the utilization one(cf. Steindl 1952, 1979; Rowthorn 1981; Dutt 1984). Investment is determined by the current aggregate profits or rate of profit in the1

To understand the mechanism of business cycles in depth, it may be necessary to explore the role of expectationsin investment, following Keynes’ (1936) view.
Recently, Murakami (2018, 2020b) examined the effect of expectations on investment and established the existence and uniqueness of a limit cycle, which is regarded as persistent growth cycles, in a post Keynesian system.3
It was found in Murakami (2018, 2020b) that the existence and uniqueness of a limit cycle is obtained under reasonable assumptions if investment (or the rate of capital formation) is highly elastic to the expected rate of profit
and the latter is frequently revised in response to the realized rate of profit. In particular, the uniqueness (as well
as existence) of a limit cycle is the distinguished contribution of this paper because it has rarely been exploreduntil the recent studies by Murakami (2018, 2019, 2020a) due to technical difficulty.4 In Murakami (2018, 2020b),
however, the existence and uniqueness of a limit cycle was only confirmed but the convergence of solution paths,
which describe the states of the macroeconomic system, to the unique limit cycle was not discussed.
The purpose of this paper is to generalize the conclusion of Murakami’s (2018) analysis verifying that, if therevision speed of the expected rate of profit is high enough, every solution path, except for the one starting at
the long-run equilibrium, converges to the unique limit cycle, regardless its initial condition. By so doing, we shalldemonstrate the inevitability of persistent business cycles in capitalist economies. This paper is organized as follows.
In Section 2, we shall formalize a post Keynesian system. In Section 3, we shall analyze the post Keynesian system
to establish not only the existence and uniqueness of a limit cycle but also the convergence to the unique limit cycle.
In Section 4, we shall conclude this paper. In Appendix, we shall present the mathematical theorem employed in
our analysis.

2

The post Keynesian system

In this section, we shall formalize a post Keynesian system. In what follows, we shall make use of the following
notations; let Rn , Rn+ and Rn++ denote the n-dimensional Euclidean space, the subspace of Rn composed of allnonnegative vectors and the subspace of Rn composed of all strict positive vectors, respectively; let v̇ stand for thetime derivative of v, i.e., v̇ ≡ dv/dt.
first, by the current changes in aggregate income in the second and by the current rate of utilization, which is usually identified withthe ratio of aggregate income to stock of capital, in the last. Unlike Keynes’ (1936, chap. 11) theory of investment, these principles do
not put much stress on the role of expectations in decision makings on investment.
3 Benassy (1984) and Franke (2012, 2014) verified the existence of a periodic orbit in Keynesian models in which investment is affectedby expected aggregate demand or the expected rate of profit, but they failed to examine the uniqueness of it.
4 Until Murakami (2018, 2019, 2020a), the uniqueness of a limit cycle in models of business cycles was investigated by Ichimura(1955), Kosobud and O’Neill (1972), Lorenz (1986, 1993), Galeotti and Gori (1989) and Sasakura (1996) alone, as far as we know.

2

2.1

Aggregate consumption and saving

We shall assume that aggregate saving is determined in the following way:S = sc Π + sw (Y − Π) − A,(1)where sc and sw are, respectively, positive and nonnegative constants with sw ≤ sc < 1. In (1), Y, S, Π and A standfor, respectively, aggregate income, aggregate saving, aggregate profits (capitalists’ income), aggregate autonomous
consumption, which shall be referred to simply as “autonomous demand” in this paper;5 sc and sw represent themarginal propensities to save of capitalists and workers, respectively.6
We shall also postulate that aggregate share of capital (or aggregate share of profits) is constant.7 Aggregate
profits can then be written asΠ = πY,(2)

where π is a positive constant less than unity. In (2), π represents aggregate share of capital. Thus, we can express
aggregate saving in the following way:S = sΠ − A,(3)

where s is a positive constant defined by8

s=

2.2

sc π + sw (1 − π)
1−π
= sc + sw.
ππ

Aggregate investment

In the light of Keynes’ (1936, chap. 11) theory of investment, we shall postulate that the (gross) rate of capitalformation (or the ratio of investment to capital stock) is positively influenced by the expected rate of profit on
5 Our aggregate autonomous demand A can be regarded as aggregate base (or fundamental) consumption. It can also be viewed as
autonomous government expenditure or autonomous exports as in the literature on autonomous demand (cf. Trezzini 1995; Serrano
1995; Allain 2015; Lavoie 2016; Dutt 2019).
6 In this paper, we shall follow the post Keynesian (or Kaleckian) hypothesis that workers and capitalists may differ in spending
behavior (cf. Kaldor 1955-1956; Pasinetti 1962).
7 This postulate is consistent with Kaldor’s (1961) stylized fact. According to Karabarbounis and Neiman (2014) and Jones (2016),
the U.S. share of capital was almost constant (about 34.2 percent) until around 2000, while it has recently been rising (to 38.7 percent
by 2012). In this respect, the postulate does not reflect the recent trend, but it works well as a first approximation to the reality because
the recent rise in aggregate share of capital is only slight (at least in most developed countries).
8 The saving function (3) is similar to Kalecki’s (1935), but we allow for the possibility that workers also save. As we shall see, if
the marginal propensities to save are the same between capitalists and workers (or if sc = sw ), our post Keynesian system reduces to
Murakami’s (2018).

3

capital in the following way:9

I = f (re )K.

(4)

In (4), I, K and re stand for aggregate gross investment, aggregate capital stock and the expected rate of profit on
capital, respectively; f is the gross capital formation function, whose characteristics shall be discussed below. The
investment function formalized reflects Keynes’ (1936) theory of investment because it emphasizes the influence ofthe expected rate of profit (or the marginal efficiency of capital) on investment.10
Concerning the capital formation function f, we shall make the following reasonable assumption.
Assumption 1. The nonnegative-valued function f : R+ → R+ is continuously differentiable for every re ∈ R+ ,and the following condition is satisfied for every re ∈ R+ :f 0 (re ) > 0.

(5)

Assumption 1 can be seen as reasonable because condition (5) simply implies that the rate of gross capitalformation is nonnegative, which is required by its definition, and is a strictly increasing function of the expected
rate of profit re .

2.3

The rate of profit

The Keynesian principle of effective demand implies that aggregate saving is adjusted to aggregate investment in
the following way:

S = I,

which determines aggregate profits as follows:Π=1[f (re )K + A].
s

The (actual or realized) rate of profit can be derived by dividing both sides by aggregate capital stock as follows:

r=

1h
1i
Π
= f (re ) + .
K
s
k

(6)

In (6), r and k stand for the rate of profit Π/K and the ratio of aggregate stock of capital to autonomous demand
K/A, respectively. This equation implies that the actual rate of profit r is determined by the expected rate of profit
9 The marginal efficiency of capital, defined by Keynes (1936), is nothing but the expected rate of profit on capital determined by
the long-term expectation.
10 The investment function (4) is consistent with Robinson’s (1962) theory of investment.

4

re through investment.11

2.4

Revisions of the expected rate of profit

It is seen from (6) that the actual or realized rate of profit r is not necessarily equal to the expected one re . It isnatural to think that the latter is changed based on the former. To describe the revision process, we shall assume
that the expected rate of profit re is adaptively revised reflecting on the actual or realized rate of profit r in thefollowing fashion:12ṙe = α(r − re ),orṙe = αn1hsf (re ) +o1i
− re ,
k(7)

where α is a positive constant. In (7), α can be regarded as the speed or frequency of revisions of the long-term
expectation, based on which the expected rate of profit is determined.

2.5

Aggregate capital formation

Aggregate capital formation process can be described as follows:K̇ = I − δK,orK̇ = [f (re ) − δ]K,

(8)

where δ is a positive constant. In (8), δ represents the constant rate of capital depreciation.

2.6

Changes in autonomous demand

To allow for changes in autonomous demand A, we shall assume that it varies at a constant rate as follows:13

Ȧ = aA,(9)

11 The ratio of aggregate income to stock of capital Y /K is proportionate to the rate of profit Π/K due to (2).
12 For a justification of adaptive expectation formations in nonlinear systems including ours, see Murakami (2018).
13 In the literature on autonomous demand (cf. Trezzini 1995; Serrano 1995; Allain 2015; Lavoie 2016; Dutt 2019), its rate of change
is assumed to be exogenously given (this is the reason for the name of “autonomous” demand). We can also suppose that autonomouscomponent of aggregate consumption A changes at the same rate as the rate of change in population (cf. Murakami 2018).

5

where a is a real constant. As we shall see, the rate of change in autonomous demand a can be zero or negative
provided that δ + a > 0 (cf. Assumption 2). Note that with the initial value of A given, equation (9) is equivalenttoA(t) = A(0) exp(at) > 0,(10)where A(0) is a positive constant which represents the (positive) initial value of A for t = 0 and the inequality holds
for all t ≥ 0.
In the light of (8) and (9), the dynamics of the ratio of capital stock to autonomous demand k can be represented
as follows:

k̇ =

K̇
K Ȧ
−
,
A
AA

or

k̇ = [f (re ) − (δ + a)]k.

2.7

(11)

Full system: System (K)We are now ready to formalize our Keynesian system in the following way:ṙe = αn1hsf (re ) +o1i
− re ,kk̇ = [f (re ) − (δ + a)]k.

(7)
(11)

In what follows, we shall denote the system of (7) and (11) by “System (K).”14

3

Analysis

We shall now analyze our System (K) to examine the existence and uniqueness of a limit cycle, which can be viewed
as persistent growth cycles, and the convergence of every non-equilibrium solution path to the unique limit cycle.

3.1

Existence and uniqueness of equilibrium

We shall now define an equilibrium point of System (K). A point (re , k) ∈ R2++ is said to be an equilibrium pointof System (K) if ṙe = 0 and k̇ = 0 at this point.15 Then, an equilibrium point of System (K), denoted by (r∗ , k ∗ ),14 As we have observed, System (K) is a generalized version of Murakami’s (2018) post Keynesian system (System (PK)) in that themarginal propensities to save may differ between workers and capitalists.
15 By this definition, k = 0 is ruled out as an equilibrium value of k.

6can be defined as a solution of the following simultaneous equations:

0=

1i
1h e
f (r ) +− re ,sk

0 = f (re ) − (δ + a).

The unique equilibrium point of System (K), if it exists, is given as follows:(r∗ , k ∗ ) = f −1 (δ + a),1.
sf −1 (δ + a) − (δ + a)

(12)

To ensure the existence and uniqueness of an equilibrium point of System (K), we shall make the following
assumption.
Assumption 2. The following conditions are satisfied:

f (0) < δ + a < elim f (re ),r →∞δ + af< δ + a.
s

(13)
(14)

Under Assumption 1, condition (13) ensures the existence and uniqueness of f −1 (δ + a) > 0. Also, conditionsand (13) and (14) imply that0<

δ+a< f −1 (δ + a),sorsf −1 (δ + a) − (δ + a) > 0.

(15)

Thus, Assumptions 1 and 2 guarantee the existence of the unique equilibrium point of System (K), defined in (12).

3.2

System (K) reformulated

We shall reformulate System (K) as a generalized Liénard system.16 The reformulation facilitates our analysis of
System (K).
For reformulation, we shall introduce the following new variables:x = re − r∗ ,(16)

y = ln k ∗ − ln k,(17)

16 For a brief exposition of generalized Liénard systems, see Appendix. For other applications of theory of generalized Liénard systems

to economic theory, see Murakami (2018, 2019, 2020a).

7

where (r∗ , k ∗ ) is defined by (12) and “ln” represents the natural logarithm (note that k ∗ is positive under Assumption
2).
Substituting the variables defined in (16) and (17) in System (K), we can reduce System (K) to the followingone:ẋ = φ(y) − F (x),(18)

ẏ = −g(x),(19)

g(x) = f (f −1 (δ + a) + x) − (δ + a),ih1
F (x) = α x − g(x) ,shδ + ai−1
φ(y) = α f (δ + a) −[exp(y) − 1].
s

(20)

where

(21)
(22)

In what follows, we shall denote the system of equations (18) and (19) with (20)-(22) by “System (K*).” It isseen from Appendix that System (K*) can be classified as a generalized Liénard system. Note that the uniqueequilibrium of System (K*) is (x∗ , y ∗ ) = (0, 0), which corresponds to the unique one of System (K), (r∗ , k ∗ ).

3.3

Existence and uniqueness of a solution path

Before exploring the characteristics of System (K), we shall confirm that for every initial condition (re (0), k(0)) ∈R2++ , System (K) has a unique solution path (re (t), k(t)) ∈ R2++ for all t ≥ 0. For this purpose, it is necessary
and sufficient to verify that for every initial condition (x(0), y(0)) ∈ D, System (K*) has a unique solution path(x(t), y(t)) ∈ D for all t ≥ 0, where D is defined as follows:

D = {(x, y) ∈ R2 : x > −r∗ }.

(23)

We shall now make two additional assumptions for our analysis. The first assumption is given as follows.
Assumption 3. The following condition is satisfied:

f 0 (f −1 (δ + a)) > s.

(24)

Assumption 3 means that investment or capital formation is sufficiently elastic to changes in the expected rateof profit at the unique equilibrium. In this respect, it is consistent with Keynes’ (1936) view that variations in
the expected rate on profit (the marginal efficiency of capital) give rise to violent changes in investment.17 It also
17 As Murakami (2018) argued, Assumption 3 does not violate the so-called Keynesian stability condition that the marginal propensity

8

concerns the (in)stability of the unique equilibrium of System (K) or (K*). The Jacobian matrix of System (K*)
evaluated at the unique equilibrium is given as follows:0 −F (0)J∗ = 
−g 0 (0)

0





0

−1

φ (0)   α[f (f (δ + a))/s − 1] α[f=0−f 0 (f −1 (δ + a))−1

(δ + a) − (δ + a)/s] .
0The trace and determinant of J ∗ are given bytr J ∗ = α

h f 0 (f −1 (δ + a))i− 1 > 0,sh
δ + aidet J ∗ = αf 0 (f −1 (δ + a)) f −1 (δ + a) −> 0,swhere these inequalities hold under Assumptions 2 and 3. The unique equilibrium of System (K*) is then locallyasymptotically totally unstable.18
The second assumption is concerned with the form of the capital formation function f .19
Assumption 4. The following condition is satisfied:lim [sre − f (re )] = ∞.

r e →∞(25)

There exist exactly two nonnegative constants re and re with re < re such that the following condition is satisfied:

f 0 (re ) = s.

(26)

f (re ) < sre .

(27)

Furthermore, the following condition is satisfied:

As we shall see, Assumption 4 plays a pivotal role for analysis. In particular, condition (25) is vital for establishing
the existence and uniqueness of a solution path and of a limit cycle in System (K) or (K*). This assumption is,on the other hand, reasonable because it holds if f is a logistic function with empirically plausible parameters asshown in Murakami (2019, 2020a).
Under Assumptions 1-4, the capital formation function f can be drawn as in figure 1. It can be seen from thisfigure that f has a sigmoid shape as in Kaldor’s (1940) investment function.
to save is larger than that to invest (with respect to the current income) (cf. Marglin and Bhaduri 1990). This condition does notnecessarily guarantee the local asymptotic stability of the long-run equilibrium, if investment is influenced by the expected rate of profit.
For debates on the Keynesian stability condition, see, for instance, Hein et al. (2011) or Skott (2012).
18 We mean by the term “local asymptotic total instability” that the equilibrium point under consideration is either an unstable nodeor an unstable focus, i.e., that the trace and determinant of the Jacobian matrix evaluated at this equilibrium point are both positive.
19 It is conceptually identical with Murakami’s (2018, p. 298, Assumption 4), whose error was corrected by Murakami (2020b).

9

f

s
O
Figure 1: Capital formation function f

We shall take a close look at the phase diagram of System (K*) to examine the characteristics of solution pathsof System (K). Since the locus ẏ = 0 is given by x = 0 due to (19) and (20), we shall look into that of ẋ = 0. Since
the locus ẋ = 0 is given by φ(y) = F (x), we shall now have a close look at F (x). Because of (12), (16) and (21),condition (25) implies20

lim F (x) = ∞.

x→∞

(28)

Also, it follows from Assumptions 1-3 (and (15)) thath
δ + a f (0) iF (−r∗ ) = −α f −1 (δ + a) −
+
< 0,
s
s
h
f 0 (f −1 (δ + a)) i
F 0 (0) = α 1 −
< 0.
s

(29)
(30)

Due to F (0) = 0 (by (21)), it is seen from (30) that, there exists a sufficiently small positive ε such that F (−x) > 0
and F (x) < 0 for every x ∈ (0, ε). It then follows from the continuity of F 0 (x) (by Assumption 1) and from (28) and(29) that F 0 (x) = 0 has at least one root both in (−r∗ , 0) and in (0, ∞), respectively. Since F 0 (x) = 0 is equivalent
to f 0 (r∗ + x) = s for x ≥ −r∗ , it is known from Assumption 4 that F 0 (x) = 0 has exactly two roots and and thatthe smaller one x0 = re − r∗ lies in (−r∗ , 0), while the bigger one x0 = re − r∗ in (0, ∞). We can also find from (30)that F (x0 ) > 0 and F (x0 ) < 0 are a local maximum and a local minimum, respectively. Moreover, we can see fromthe continuity of F that F (x) = 0 has exactly two roots x0 and x0 , besides x = 0, in (−r∗ , ∞) with −r∗ < x0 < x0
and x0 > x0 . Thus, the graph of F (x) can be drawn as in figure 2.
20 As we shall see, condition (28) plays a vital role for our analysis.

10

F( x )

−r *

x0
−

−
x'
O

−x '

−
x0

x

Figure 2: Graph of F (x)Now we shall turn to the locus ẋ = 0 or φ(y) = F (x). It follows from (20)-(22) that the locus is given byhhδ + aif (f −1 (δ + a) + x) − (δ + a) iφ(y) = α f −1 (δ + a) −[exp(y) − 1] = α x −= F (x),ssor

exp(y) = 1 +sF (x)
α[sf −1 (δ + a) − (δ + a)]=s(r∗ + x) − f (r∗ + x).
sf −1 (δ + a) − (δ + a)

(31)

Thus, for the locus ẋ = 0 to be well-defined at least for x ≥ x0 , it is necessary and sufficient that the right-handside of (31) is positive for every x ≥ x0 .21 Since it is known from figure 2 that the minimum of F (x) for x ≥ x0 is
given at x = x0 = re − r∗ , the following condition is sufficient for the locus ẋ = 0 to be well-defined:1+

sF (x0 )α[sf −1 (δ + a) − (δ + a)]=s(re ) − f (re )sf −1 (δ + a) − (δ + a)

> 0.

This is fulfilled by condition (27) in Assumption 4. Also, it is seen from that for every real y,ẋ|x=−r∗ = [φ(y) − F (x)]|x=−f −1 (δ+a) = αnhf −1 (δ + a) −o
δ + ai1exp(y) + f (0) > 0,
s
s

(32)

where the inequality follows from f (0) ≥ 0 and (15) (due to Assumptions 1 and 2). This indicates that the line
x = −r∗ is wholly located on the region ẋ > 0 or φ(y) > F (x) (and that the locus ẋ = 0 never intersects with theline x = −r∗ ). Therefore, the phase diagram of System (K*) can be drawn as in figure 3.22
21 For our analysis, it is vital for proving the existence and uniqueness of a solution path of System (K) or (K*) that the locus ẋ = 0or φ(y) = F (x) is well-defined for x ≥ x0 (cf. Appendix).
22 Note that Assumption 4 (or condition (28)) ensures that, for every (especially, large) y > 0, there exists a unique x such that thelocus ẋ = 0 or (31) passes through (x, y); if condition (25) is dropped, there may exist a (sufficiently large) y > 0 such that no x satisfies(31) because the left-hand side of (31) tends to ∞ as y → ∞. As we shall see, this fact is vital for the existence and uniqueness of asolution path, as well as of a limit cycle, in System (K*).

11yD1 =0 D2 =0
′
− *

′

x

O

D4

D3

Figure 3: Phase diagram of System (K*)

For our analysis, we shall divide the region D, excluding the origin, into the following four parts (cf. figure 3)
to examine non-equilibrium solution paths of System (K*), where D is defined by (23):23

D1 = {(x, y) ∈ D : ẋ > 0, ẏ > 0} = {(x, y) ∈ R2 : −r∗ < x < 0, φ(y) > F (x)},(33)

D2 = {(x, y) ∈ D \ (0, 0) : ẋ ≥ 0, ẏ ≤ 0} = {(x, y) ∈ R2 \ (0, 0) : x ≥ 0, φ(y) ≥ F (x)},(34)

D3 = {(x, y) ∈ D : ẋ < 0, ẏ < 0} = {(x, y) ∈ R2 : x > 0, φ(y) < F (x)},(35)

D4 = {(x, y) ∈ D \ (0, 0) : ẋ ≤ 0, ẏ ≥ 0} = {(x, y) ∈ R2 \ (0, 0) : −r∗ < x ≤ 0, φ(y) ≤ F (x)}.

(36)

We shall now verify that, for initial condition (re (0), k(0)) ∈ R2++ , there exists a unique solution path of System(K), (re (t), k(t)) ∈ R2++ , for all t ≥ 0. Because of (16) and (17) (and of the equivalence of Systems (K) and (K*)),
it is (necessary and) sufficient for our purpose to prove the existence and uniqueness of a solution path of System(K*), (x(t), y(t)) ∈ D, for all t ≥ 0, with every initial condition (x(0), y(0)) ∈ D.
To begin, we shall claim that every solution path of System (K*) with its (arbitrary) initial condition (x(0), y(0)) ∈D remains on D all the time (as long as it exists).24 For the sake of contradiction, we shall assume that a solution
path of System (K*) with (x(0), y(0)) ∈ D eventually does not lie on D.25 By the continuity of solution pathsof System (K*) (due to Assumption 1), such a solution path must eventually reach the line x = −r∗ . But it is
known from the continuity of f and (32) that for every real y, there exists a (small) positive ε such that ẋ > 0
for every x ∈ [−r∗ , −r∗ + 2ε) and that no solution path passing through (x, y) = (−r∗ + ε, y) can cross the line
23 In this paper, we mean by a “non-equilibrium solution path” a solution path whose initial condition is not an equilibrium point.
24 At this stage, of course, we may not assume that solution paths of System (K*) exist for all t ≥ 0.
25 In this paper, we mean by “eventually” that the phenomenon under consideration occurs for a positive time t (excluding the caseof t = ∞).

12x = −r∗ + ε > −r∗ (because ẋ > 0 for x = −r∗ + ε). This contradicts our hypothesis. Our claim can thus beverified.
Next, we shall take an arbitrary point (xi , yi ) on Di , for i = 1, 2, 3, 4, defined by (33)-(36), and consider thesolution path of System (K*) with (x(0), y(0)) = (xi , yi ), denoted by SP-i. We shall first claim that SP-1 eventuallyenters D2 and that SP-3 eventually enters D4 . In what follows, we shall only prove the latter; one can prove theformer in a similar way.26 To this end, it is now confirmed that SP-3 never crosses the locus ẋ = 0 until it leaves
D3 for the first time (if it does).27 It is seen from (35) that x(t) ∈ [0, x3 ] along SP-3 due to ẋ < 0 on D3 and that
there exists an upper limit of F 0 (x(t)), denoted by µ, along SP-3 because of the continuity of F 0 (x). Letting u = −ẋand noting that φ0 (y) > 0 and ẏ < 0 on D3 , it follows from (18) that until SP-3 leaves D3 for the first time,u̇ = −dẋ = −[φ0 (y)ẏ + F 0 (x)ẋ] ≥ −µu,dtoru̇ + µu ≥ 0.

Multiplying both sides by exp(µt) > 0, we have(u̇ + µu) exp(µt) ≥ 0.

Integrating both sides by t in the interval [0, t],28 we obtainu(t) exp(µt) − u(0) ≥ 0,or−ẋ(t) = u(t) ≥ u(0) exp(−µt) = −ẋ(0) exp(−µt).
Hence, it is seen that along SP-3,ẋ(t) ≤ ẋ(0) exp(−µt) = [φ(y3 ) − F (x3 )] exp(−µt) < 0,where the inequality holds for all t due to (x(0), y(0)) = (x3 , y3 ) ∈ D3 . It is then known from the continuity ofsolution paths that SP-3 never crosses the locus ẋ = 0, if it exists and stays on D3 , and from (33) and (34) that
26 For a proof of a similar fact, see Murakami (2018).
27 We do not exclude the possibility that SP-3 never leaves D .
3
28 Note thatd(u exp(µt)) = (u̇ + µu) exp(µt).
dt13

SP-3 never enters D1 or D2 before entering D4 .
We shall then claim that SP-3 eventually enters D4 , crossing the locus ẏ = 0 or the y-axis. It is now confirmedthat if x3 ≥ x0 , SP-3 eventually crosses the line x = x0 , as long as it exists. For this purpose, it is assumed for thesake of contradiction that SP-3 remains the region x ≥ x0 , as long as it exists. We can find from (20) and (35) thatalong SP-3, as long as it exists and remains on the region x ≥ x0 in D3 , we have x(t) ∈ [x0 , x3 ] andẏ = −g(x) ≤ −g(x0 ) < 0,ory(t) ≤ y(0) − g(x0 )t = y3 − g(x0 )t.
Since there exists a real y 0 such that φ(y 0 ) < F (x0 ) (because φ−1 (F (x0 )) is well-defined by Assumption 4), it is seenthat y(t) ≤ y 0 for t ≥ t̃1 ≡ max[0, (y3 − y 0 )/g(x0 )] along SP-3 as long as it exists and remains on the region x ≥ x0 .
If x3 ≥ x0 , we can then find from (35) that for t ≥ t̃1 along SP-3, as long as it exists and remains on the regionx ≥ x0 ,ẋ = φ(y) − F (x) ≤ φ(y 0 ) − F (x0 ) < 0,orx(t) ≤ x(0) + [φ(y 0 ) − F (x0 )]t = x3 + [φ(y 0 ) − F (x0 )]t
because F (x) ≥ F (x0 ) for x ≥ x0 (cf. figure 2). It then follows that if x3 ≥ x0 , SP-3 enters the region x < x0 ,crossing the line x = x0 , by the time t = t̃2 ≡ t̃1 + (x3 − x0 )/[F (x0 ) − φ(y 0 )] (as long as it exists until this time). It
is shown below that if x3 < x0 , SP-3 eventually enters D4 , crossing the y-axis, as long as it exists. To this end, it
is assumed for the sake of contradiction that SP-3 with x3 < x0 remains on D3 , as long as it exists. We can thensee from (35) that if x3 < x0 , along SP-3 as long as it exists, 0 ≤ x(t) ≤ x3 < x0 andẋ = φ(y) − F (x) ≤ φ(y3 ) − F (x3 ) < 0,orx(t) = x3 + [φ(y3 ) − F (x3 )]t.
because F (x) ≥ F (x3 ) for 0 ≤ x ≤ x3 < x0 . Hence, we have x(t) ≤ 0 for t ≥ t̃3 = x3 /[F (x3 ) − φ(y3 )] along SP-3,

14

as long as it exists and remains on D3 , and this is a contradiction. It then follows that if x3 < x0 , SP-3 enters D4 ,crossing the y-axis by the time t = t̃3 (as long as it exists by this time). Thus, we have shown that SP-3 enters
D4 , crossing the y-axis for a positive time t = T1 ≤ t̃2 + t̃3 (for the first time), as long as it exists for t ∈ [0, T1 ]. It
remains to prove that SP-3 exists (at least) until it enters D4 for t = T1 (for the first time). It is known from (20)and (35) that for t ∈ [0, T1 ] along SP-3 (on D3 ) as long as it exists, (x(t), y(t)) ∈ D, especially, x(t) ∈ [0, x3 ] and−g(x3 ) ≤ ẏ = −g(x) ≤ 0,ory3 − g(x3 )T1 ≤ y(t) ≤ y3 .
Then, it can be verified from the argument on continuation of solution paths (cf. Coddington and Levinson 1955,
chap. 1) that SP-3 exists for t ∈ [0, T1 ]. Also, it can be shown that SP-3 uniquely exists at least for t ∈ [0, T1 ]because System (K*) satisfies the Lipschitz condition (cf. Coddington and Levinson 1955, chap. 1) on the following(nonempty) compact rectangular (convex) region D̃1 :29

D̃1 = {(x, y) ∈ D : 0 ≤ x ≤ x3 , y3 − g(x3 )T1 ≤ y ≤ y3 }.

Therefore, we can verify that SP-3 eventually enters D4 (and uniquely exists until entering D4 for the first time)and (in a similar manner) that SP-1 eventually enters D2 (and uniquely exists until entering D2 for the first time).
Note that it is possible to prove in a similar manner that for every real y, a solution path of System (K*) with
(x(0), y(0)) = (−r∗ , y) eventually enters D2 (and uniquely exists until entering D2 for the first time) because System(K*) is defined for x = −r∗ and such a solution path cannot leave the region x ≥ −r∗ due to (32).
We shall second claim that SP-2 eventually enters D3 and that SP-4 eventually enters D1 . We shall only prove
the former; one can prove the latter in a similar manner. It is now confirmed that SP-2 never crosses the locusẏ = 0 or the y-axis until it leaves D2 for the first time. It is seen from (19) that if SP-2 reaches the y-axis on D2
at (x, y) = (0, ỹ), we have ỹ > 0 (because (x, y) = (0, ỹ) is known (from figure 3) not to lie on D2 if ỹ ≤ 0) andẋ|(x,y)=(0,ỹ) = φ(ỹ) − F (0) = φ(ỹ) > 0,

where the inequality follows from (22). It follows from the continuity of solution paths that SP-2 never crosses they-axis until it leaves D2 for the first time and never enters D1 or D4 before entering D3 .
29 Since g, F and φ are all continuously differentiable on D by Assumption 1, it can be shown by the mean-value theorem that there
exists a positive M such that for every (x0 , y 0 ), (x00 , y 00 ) ∈ D̃1 ,|[φ(y 00 ) − F (x00 )] − [φ(y 0 ) − F (x0 )]| + |g(x00 ) − g(x0 )| ≤ M (|x00 − x0 | + |y 00 − y 0 |).
This implies that System (K*) satisfies the Lipschitz condition on D̃1 . See Murakami (2014), for details of the proof that the Lipschitz
condition is satisfied on a compact rectangular subset of Rn if all the functions are continuously differentiable on this set.

15

We shall proceed to prove that SP-2 eventually enters D3 , crossing the locus ẋ = 0. It is assumed for the sakeof contradiction that SP-2 remains on D2 as long as it exists. Let (x, y) = (x̃, y2 ) be the unique intersection of theline y = y2 and the locus ẋ = 0 on the region x ≥ x0 .30 If x2 = 0 (i.e., if ẏ = 0 for t = 0), we can find from (34) thatẋ > 0 for t = 0 along SP-2 (because y2 > 0) and that there exists a (small) positive t̃4 such that x(t) > 0 for t = t̃4 .
It then follows that if x2 = 0, SP-2 (immediately) enters the region x > 0 (in D2 ). If x2 > 0, we can see from (20)and (34) that along SP-2 as long as it exists, x(t) ∈ [x2 , x̃] andẏ = −g(x) ≤ −g(x2 ) < 0,ory(t) ≤ y(0) − g(x2 )t = y2 − g(x2 )t.
Hence, we have y(t) ≤ y 0 < φ−1 (F (x0 )) for t ≥ t̃5 ≡ (y2 − y 0 )/g(x2 ) > 0 along SP-2 as long as it exists and remains
on D2 , where y 0 has been defined above (and confirmed to exist). This is a contradiction because y ≥ φ−1 (F (x0 ))
if (x, y) ∈ D2 (cf. figure 3). It is then seen that if x2 > 0, SP-2 enters D3 by the time t = t̃5 (as long as it exists bythis time). Thus, we have proved that SP-2 enters D3 , crossing the locus ẋ = 0, for a positive time t = T2 ≤ t̃4 + t̃5(for the first time), as long as it exists at least for t ∈ [0, T2 ]. Since (x(t), y(t)) ∈ D (because x(t) ∈ [x2 , x̃] andy(t) ∈ [φ−1 (F (x0 )), y2 ]) for t ∈ [0, T2 ] along SP-2 as long as it exists, it can be proved from the argument on
continuation of solution paths (cf. Coddington and Levinson 1955, chap. 1) that SP-2 exists for t ∈ [0, T2 ]. Also,
it can be confirmed that SP-2 uniquely exists at least for t ∈ [0, T2 ] because System (K*) satisfies the Lipschitz
condition on the following (nonempty) compact rectangular (convex) region D̃2 , defined byD̃2 = {(x, y) ∈ D : x2 ≤ x ≤ x̃, φ−1 (F (x0 )) ≤ y ≤ y2 }.

Therefore, we can verify that SP-2 eventually enters D3 (and uniquely exists until entering D3 for the first time)and (in a similar manner) that SP-4 eventually enters D1 (and uniquely exists until entering D1 for the first time).
Now we have established the fact that all solution paths of System (K*) starting on D1 , D2 , D3 and D4 for t = 0
eventually enter D2 , D3 , D4 and D1 , respectively.
Finally, we shall make sure that for every initial condition (x(0), y(0)) ∈ D, a unique solution path of System (K*)
exists (or is defined) for all t ≥ 0.31 To this end, we shall claim that there exists a (nonempty) positively invariant
30 We can ensure by Assumption 4 that such a positive x̃ uniquely exists.
31 By a method similar to ours, Hirsch and Smale (1974, p.

220, Proposition 2) proved the existence and uniqueness of a solution
path with respect to every initial condition for Van der Pol’s system, which is a special case of generalized Liénard systems and given
as follows:ẋ = y − (x3 − x)ẏ = −x.
They also gave a proof of the convergence of non-equilibrium solution paths to the unique limit cycle in this system (cf. Hirsch and16compact region with respect to System (K*)32 and then that every solution path starting on D eventually enters
the positively invariant region. As we have confirmed, a solution path of System (K*) with (x(0), y(0)) = (−r∗ , 0)
eventually enters D2 , and it can be similarly shown to eventually enter D2 , D3 and D4 in order and then eventually
return to D1 . It is also seen that a solution path with (x(0), y(0)) = (−r∗ , 0) uniquely exists until it returns to D1
for the first time. It then follows that the solution path starting at P(−r∗ , 0), crosses the y-axis and the ẋ = 0 for
the first time at Q(0, yq ) and R(xr , yr ), respectively, and it crosses the y-axis and the ẋ = 0 for the second timeat S(0, ys ) and T(xt , yt ), respectively (cf. figure 4).33 Taking the point U(−r∗ , yt ), we have a (nonempty) compact
region enclosed by the arc PQRST and the segments TU and UP and denote it by D0 . By the uniqueness of thesolution path starting at P (until it reaches T), no solution path of System (K*) starting from the inside (or outside)of D0 can cross the arc PQRST; no solution path starting from the inside of D0 can cross the segment TU or UP
because ẏ ≥ 0 along TU and ẋ > 0 on UP. It then follows that D0 is a positively invariant region with respect toSystem (K*). It can thus be verified from the argument on continuation of solution paths that every solution pathwith (x(0), y(0)) ∈ D0 uniquely exists on D0 for all t ≥ 0.34 Since every solution path starting on D is alreadyknown to remain on D as long as it exists, it can also be proved that for every (x(0), y(0)) ∈ D0 ∩ D, a uniquesolution path exists on D0 ∩ D for all t ≥ 0. Moreover, it can be shown from the above argument that every solution
path starting on the region D not contained in D0 eventually enters the region y < yt (below the segment UT) in
D1 (because of the uniqueness of the solution path, it never meets or crosses the arc PQRST) and then D0 ∩ D
before entering D2 again (and that it uniquely exists until it enters D0 ∩ D). Therefore, we have established thefact that for every (x(0), y(0)) ∈ D, a unique solution path of System (K*) exists on D for all t ≥ 0, which implies
the following proposition.
Proposition 1. Let Assumptions 1-4 hold. Then, for every initial condition (re (0), k(0)) ∈ R2++ , there exists a
unique solution path of System (K), (re (t), k(t)) ∈ R2++ , for all t ≥ 0.
Proof. It is straightforward to draw the conclusion from the above argument (as well as from (16) and (17)).
Smale 1974, p. 218, Theorem). Their method of proof was, however, different from ours.
32 A closed (usually compact) region is R said to be positively invariant with respect to the dynamical system under consideration ifevery positive semi-trajectory (i.e., every solution path for t ≥ 0) of this system which starts at an arbitrary point in R will remain in
R for ever after (i.e., for all t ≥ 0).
33 Points Q, R, S and T are, of course, uniquely defined. Note that Assumption 4 plays a vital role for R to be properly defined.
34 The uniqueness of a solution path on D follows because it can be shown by the continuity of g, F and φ and the mean-value
0
theorem that System (K*) satisfies the Lipschitz condition on the following (nonempty) compact rectangular set, which includes D0 :D0∗ = {(x, y) ∈ R2 : −r∗ ≤ x ≤ xr , yt ≤ y ≤ yq }.

17

y
Q

=0

=0
R

D0

P
O

− *

U

x

T

S
Figure 4: Positively invariant region D0

3.4

Convergence to a periodic orbit

We shall now examine the existence of a periodic orbit, which may be regarded as persistent business cycles, inSystem (K) or (K*). For this purpose, we shall make use of the Poincaré-Bendixson theorem (cf. Coddington
and Levinson 1955, chap. 16) to ensure the existence of a periodic orbit. In so doing, we shall also confirm theconvergence of every non-equilibrium solution path to a periodic orbit.35 Since it has already been confirmed in thelast subsection that the unique equilibrium (0,0) is locally asymptotically totally unstable and that every solution
path of System (K*) starting on D eventually enters the positively invariant compact set D0 , it is straightforward
to present the following theorem.
Theorem 1. Let Assumptions 1-4 hold. Then, for every initial condition (re (0), k(0)) ∈ R2++ with (re (0), k(0)) 6=(r∗ , k ∗ ), the unique solution path of System (K), (re (t), k(t)) ∈ R2++ , either is a periodic orbit on R2++ or converges
to a periodic orbit on R2++ as t → ∞.
Proof. Because of (16) and (17), it suffices for the proof of this proposition to verify that every non-equilibrium
solution path of System (K*) starting on D either is or converges to a periodic orbit on D.
It has been confirmed in the last subsection (especially in Proposition 1) that every solution path of System (K*)
with (x(0), y(0)) ∈ D uniquely exists on D for all t ≥ 0 and eventually enters the positively invariant set compact
D0 (cf. figure 4). Also, the unique equilibrium of System (K*) has been shown to be locally asymptotically unstable(by Assumption 3). We can then enclose the unique equilibrium by a sufficiently small rectangle such that every
non-equilibrium solution path starting on its interior eventually leaves the interior and never enters the interior
35 In a related post Keynesian system, Murakami (2018, p. 301, Proposition 1; 2020b) failed to ensure the convergence to a periodic
orbit.

18

again ever after. We can construct a (nonempty) positively invariant compact set, denoted by D0∗ , eliminating theinterior of the small rectangle from D0 . It follows that every non-equilibrium solution path starting on D eventuallyenters the positively invariant compact set free of the unique equilibrium D0∗ and remains on D0∗ ever after. We canthus apply the Poincaré-Bendixson theorem (cf. Coddington and Levinson 1955, chap. 16, especially pp. 391-392,
Theorem 2.1) to conclude that every non-equilibrium solution path starting on D, uniquely defined for all t ≥ 0,
either is or converges to a periodic orbit on D0∗ as t → ∞. Also, each of such periodic orbits is entirely locatedon D0∗ ∩ D, because every solution path starting on D remains on D for all t ≥ 0. It then follows that every
non-equilibrium solution path starting on D either is or converges to a periodic orbit on D.
Theorem 1 confirms not only the existence of a periodic orbit, which may be viewed as persistent growth cycles,
but also the convergence of every non-equilibrium solution path to a periodic orbit. It is then implied from thistheorem that unless the macroeconomic system happens to be in the (long-run) equilibrium state, it is surely
subjected to persistent cyclical fluctuations along (the way of convergence to) a periodic orbit. We can also draw
from this theorem the economic implication that the macroeconomic system (with Keynesian features) necessarily(precisely speaking, almost surely) undergoes persistent business cycles around the trend path determined by therate of change in autonomous demand a. In this respect, it may be stated that business cycles or growth cycles are
an inevitable phenomenon in capitalist economies, which are characterized by investment behavior highly responsive
to prospected profits or to the long-term expectation (as expressed by Assumption 3). Note that along each of thegrowth cycles, aggregate income also undergoes cyclical fluctuations around its trend level r∗ k ∗ A(t)/π, proportional
to autonomous demand A, because it is determined by Y (t) = r(t)k(t)A(t)/π (cf. (2), (6) and (10)).

3.5

Convergence to the unique limit cycle

We shall proceed to establish not only the uniqueness of a periodic orbit (or a limit cycle in this case) and but
also the convergence of every non-equilibrium solution to the unique periodic orbit in System (K) with the revision
speed of expectations α sufficiently large.36 To this end, we shall utilize the theorem of Xiao and Zhang (2003),
which is reproduced as Theorem 3 in Appendix.
It is easily seen that Assumptions 5 and 6, required in Theorem 3, are fulfilled in System (K*), if x = −r∗ , x =∞, y = −∞ and y = ∞ (and if x0 and x0 are defined as in figure 2). Because of Theorem 1, it suffices for our
purpose to make sure that the remaining hypothesis for Theorem 3, Assumption 7, is satisfied if α is sufficiently
large.
Now we shall take a look at Assumption 7. Since we have x0 ∈ [x0 , 0] and x0 ∈ [0, x0 ] (cf. figure 2), condition
36 The argument that follows draws on Murakami (2018). Note that Murakami (2018) only proved the uniqueness of a limit cycle but
not the convergence of every non-equilibrium solution path to it.

19(49) in Assumption 7 holds if the following condition is fulfilled:37 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

(37)


 Φ(φ−1 (F (x0 ))) ≥ G(x0 ) − G(x0 ), if G(x0 ) > G(x0 ).
It is straightforward to obtainZ x

Z x
g(s)ds =G(x) =f (f −1 (δ + a) + τ )dτ − (δ + a)x,00which implies that0

G(x0 ) − G(x ) =G(x0 ) − G(x0 ) =

Z x0
x0

Z x0

f (f −1 (δ + a) + τ )dτ − (δ + a)(x0 − x0 ),(38)

f (f −1 (δ + a) + τ )dτ − (δ + a)(x0 − x0 ).

(39)

x0

It is seen from (21) and (22) that

φ−1 (F (x)) = ln

 s(r∗ + x) − f (r∗ + x) ,
sf −1 (δ + a) − (δ + a)h
δ + aiΦ(y) =
φ(τ )dτ = α f −1 (δ + a) −[exp(y) − y − 1],s0
Z yIt follows from (12) thath
 s(r∗ + x) − f (r∗ + x) iδ + a ih sx − f (r∗ + x) + δ + a−ln
Φ(φ−1 (F (x))) = α f −1 (δ + a) −s
sf −1 (δ + a) − (δ + a)sf −1 (δ + a) − (δ + a)hi
δ+a
= α f −1 (δ + a) −[z(x) − ln(1 + z(x))],s
wherez(x) =

sx − f (r∗ + x) + f (r∗ )s(r∗ + x) − f (r∗ + x)
−1=.
−1
sf (δ + a) − (δ + a)sf −1 (δ + a) − (δ + a)(40)

37 Due to φ0 (y) > 0 (by (22)), the function φ−1 (F (x)) can be defined at least for x ∈ [x , x ] because φ(y) = F (x) is well-defined for00x ∈ [x0 , x0 ] (by Assumption 4).

20

Then, we haveh
δ + ai[z(x0 ) − ln(1 + z(x0 ))] > 0,
Φ(φ−1 (F (x0 ))) = α f −1 (δ + a) −sh
δ + ai[z(x0 ) − ln(1 + z(x0 ))] > 0,
Φ(φ−1 (F (x0 ))) = α f −1 (δ + a) −
s

(41)
(42)

where the inequalities hold because it is seen from (21), (31) and (40) that z(x) 6= 0 and z(x) > −1 for x = x0
and for x = x0 (by Assumption 4). Since x0 and x0 are the roots of F (x) = 0 or sx = f (r∗ + x) and x0 and x0
are the roots of F 0 (x) = 0 or f 0 (r∗ + x) = s, all of them are determined independently from the value of α. Thus,
we can find from (38), (39), (41) and (42) that the left-hand side of (37) is positive and proportional to α while
the right-hand is fixed and independently from α and that if α is large enough, condition (37) is fulfilled (in eithercase).
Therefore, we can present the main theorem, which sharpens Theorem 1 for the case of α being large enough.38
Theorem 2. Let Assumptions 1-4 hold. Assume that α is sufficiently large. Then, for every initial condition(re (0), k(0)) ∈ R2++ with (re (0), k(0)) 6= (r∗ , k ∗ ), the unique solution path of System (K), (re (t), k(t)) ∈ R2++ , either
is the unique (and periodically stable) limit cycle on R2++ or converges to the unique limit cycle on R2++ as t → ∞.
Proof. If α is sufficiently large, the periodic orbit (or limit cycle) is confirmed by the above argument to be uniquein System (K*), and it follows from Theorem 1 that every non-equilibrium solution path of System (K) either is orconverges to the unique limit cycle, which corresponds to the one in System (K*). It is obvious from Theorem 1
that the unique limit cycle of System (K) lies entirely on R2++ .
Theorem 2 establishes not only the existence and uniqueness of a limit cycle but also the convergence of every
non-equilibrium solution path to the unique limit cycle in the case of the speed of revisions of expectations α being
large enough. The main economic implication from this theorem is that if expectations on the future rate of profit
are frequently revised (or if α is large enough), the macroeconomic system with Keynesian features asymptoticallyapproaches the unique business cycle around its trend level determined by autonomous demand, regardless its
initial condition (except for the case in which it is the unique equilibrium). We may then state that the ultimate (orfinal) state of capitalist economies, where prospected profits (or the long-term expectation) are frequently revised
in response to realized ones, is persistent cyclical fluctuations with constant period and amplitude along the uniquegrowth cycle. Also, we can confirm that Theorem 2 strengthens Keynes’ (1936) view that violent fluctuations in
the marginal efficiency of capital (or the expected rate of profit) are the essential cause of business cycles.39
38 The conclusion of Theorem 2 is stronger than Murakami’s (2018) related proposition (p. 302, Proposition 2) because the convergenceof every non-equilibrium solution path to the uniqueness is obtained in the former but not in the latter.
39 Theorem 2 sharpens Keynes’ (1936) view on business cycles more than Murakami’s (2018, p. 302, Proposition 2) related proposition
because the former guarantees the inevitability of the unique growth cycle irrespective of initial conditions.

21

4Conclusion

In this paper, we have verified that every non-equilibrium solution path converges to the unique limit cycle (if therevision speed of the expected rate of profit is high enough) in a post Keynesian system which emphasizes the role ofthe expected rate of profit. By so doing, we have demonstrated that persistent cyclical fluctuations are inevitable if
the expected rate of profit (or the marginal efficiency of capital) is frequently revised and given a strong theoreticalsupport to Keynes’ (1936) view on business cycles. We hope that our present analysis is helpful for understanding
the mechanism of business cycles.

Acknowledgment
This work was supported by Chuo University Personal Research Grant and by JSPS KAKENHI (Grant Number
18K12748).

Appendix: Generalized Liénard systems
We shall introduce the theorem by Xiao and Zhang (2003) on the uniqueness of a (stable) limit cycle in generalizedLiénard systems.
We shall consider the following generalized Liénard system:ẋ = φ(y) − F (x),(43)

ẏ = −g(x).

(44)

In what follows, the system of equations (43) and (44) is denoted by “System (L).”
Following Xiao and Zhang (2003), we shall impose the following assumptions concerning System (L).
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,(45)φ(0) = 0, φ0 (y) > 0 for y ∈ (y, y).

(46)

22

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

(47)
(48)


 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 ].40 Furthermore, the following conditionis satisfied:


 sup

x∈[0,x0 ] (G(x) + Φ(φ−1(F (x)))) ≥ G(x0 ), if G(x0 ) ≥ G(x0 ),(49) supx∈[x ,0] (G(x) + Φ(φ−1 (F (x)))) ≥ G(x0 ), if G(x0 ) > G(x0 ),
0

where
Z x
G(x) =g(τ )dτ,Z 0yΦ(y) =

φ(τ )dτ.
0

As regards the uniqueness of a limit cycle in System (L), the following theorem was established by Xiao andZhang (2003).
Theorem 3. Let Assumptions 5-7 hold. Then, System (L) 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).

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23

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