Discussion Paper No.300
Existence and uniqueness of growth cycles
in post Keynesian systems

Hiroki Murakami
Faculty of Economics, Chuo University

February 2018

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Existence and uniqueness of growth cycles in post Keynesian systems
Hiroki Murakami∗
Faculty of Economics, Chuo University†
February 26, 2018

Abstract
In this paper, we present a post Keynesian system in which investment is determined by the expected rateof profit and explore the existence and uniqueness of a periodic orbit (or a growth cycle) in it. We find that aperiodic orbit appears if the marginal effect of the expected rate of profit on investment is strong enough andthat it is unique if the speed of revision of expectation is fast enough.

Keywords: Keynesian economics; Expected rate of profit; Growth cycle; UniquenessJEL classification: E12; E32

1

Introduction

It is variations in investment that are viewed as the primary source of business cycles in the Keynesian theory.
After the publication of Keynes’ General Theory, the foundations of the Keynesian theory of business cycles were
established by several economists such as Kalecki (1935, 1937), Harrod (1936), Samuelson (1939), Kaldor (1940),
Metzler (1941), Hicks (1950) and Goodwin (1951).1 Theories and models proposed by them have a common
characteristic in that they stand upon the Keynesian principle of effective demand (including income or quantity
adjustment induced by it) and that they emphasize variations in investment as a main factor for business cycles, butthey differ in what determines investment expenditure. Indeed, Kalecki (1935, 1937) and Kaldor (1940) postulated
that investment demand is determined by the rate of profit, which is related positively to aggregate income andnegatively to capital stock (and negatively to the rate of interest), while Harrod (1936), Samuelson (1939), Metlzer(1941), Hicks (1950) and Goodwin (1951) that investment (including inventory investment) is related positively anddirectly to a change in aggregate income. Both of these postulates can be justified from theoretical or empirical
viewpoints, but they lack some important aspect in Keynes’ (1936, chap. 11) original theory of investment. In the∗ 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) General Theory, but their theories can be classified

as “Keynesian” because they are based upon the Keynesian principle of effective demand.

1

above typical Keynesian theories and models of business cycles, the role of expectation is overlooked or slighted by
employing the current level of income or the current rate of profit as a proxy of the expected level of demand (orincome) or of the expected rate of profit, respectively.2 According to Keynes’ General Theory, however, investment
is affected positively and strongly by what he called the marginal efficiency of capital, which has the same meaning
as the expected prospected rate of profit on capital (and negatively by the rate of interest). Also, he maintainedthat business cycles are induced mainly by violent changes in the marginal efficiency of capital due to frequent
revisions of the long-term expectation. For these reasons, it may be worthwhile to take explicit account of the roleof expectation in the theory of business cycles.
The purpose of the present paper is to analyze the effect of expectation, represented by the expected rate of profit,on the occurrence (existence) and uniqueness of business cycles. In Section 2, we shall present a post Keynesiansystem of growth cycles which can describe not only business cycles but also economic growth. In particular, we shallformalize the investment (or capital formation) function by relating the rate of capital formation to the expected
rate of profit. In Section 3, we shall examine the properties of our post Keynesian system and discuss the existenceof a periodic orbit, which can be interpreted as a growth cycle, and the uniqueness of it. We shall clarify that aperiodic orbit does appear if the (marginal) effect of the expected rate of profit on capital formation is sufficiently
strong and that this periodic orbit is unique if the speed of revision of expectation is fast enough. In Section 4, weshall summarize our analysis and conclude this paper. In Appendix, we shall provide some mathematical theorem
utilized in our analysis.

2

The post Keynesian system

We shall formalize a post Keynesian system of growth cycles, which is composed of differential equations of the
expected rate of profit and of per-capita capital stock. To construct this system, we shall first give the consumptionsaving and investment functions.

2.1

Consumption and savings

We shall, following the Keynesian tradition, represent aggregate consumption by the following functional form:

C = c0 N + cY,(1)

where c and c0 are positive constants with c < 1. In (1), Y, C and N stand for aggregate income (or aggregate
output), aggregate consumption and the size of population, respectively; c is the marginal propensity to consume
and c0 represents the base or fundamental level of individual consumption. The consumption function given by (1)2 Exceptionally, Benassy (1984) put forward a Keynesian (or non-Walrasian) model of business cycles by placing much stress upon

the role of expected demand on investment.

2

can deal with the context of economic growth because aggregate base or fundamental consumption is proportionate
to population.
We can thus define the saving function in the following form:

S = Y − C = sY − c0 N,(2)where s = 1 − c ∈ (0, 1). Needless to say, s is the marginal propensity to save.3

2.2

Investment

We shall formulate the investment function reflecting Keynes’ (1936, chap. 11) theory of investment. Specifically,
the investment function is presented in the following form:

I = f (re )K.

(3)

In (3), I, K, re stand for aggregate gross investment, aggregate capital stock and the expected rate of profit,
respectively; f is the gross capital formation function. Equation (3) means that the rate of gross capital formation
I/K is affected by the expected rate of profit re (and by the rate of interest, which is assumed to be constant in our
analysis).4 Since Keynes (1936, chap. 11) defined the marginal efficiency of capital as the expected rate of profit
on capital, the investment function (3) can be viewed as consistent with his theory of investment.5
Concerning the capital formation function f, we shall make the following reasonable assumption.
Assumption 1. The real-valued function f : R+ → R is continuously differentiable for every re ∈ R+ , and thefollowing condition is satisfied for every re ∈ R+ :f 0 (re ) > 0.

(4)

Condition (4) simply means that the capital formation function f is strictly increasing in the expected rate ofprofit re . In this respect, Assumption 1 is a natural one.
3 Following Serrano (1995), Allain (2015) and Lavoie (2016) proposed the saving function in the following form:

S = sp πY − Z,
where sp and π stand for the rate of savings of capitalists and the capital share (note that workers are assumed to spend their wholeincome and to save nothing); Z represents the autonomous component of aggregate demand expenditure of capitalists, which is assumed
to change at a constant rate. In Allain (2015), Z was interpreted as government expenditure, while in Lavoie (2016), as capitalists’autonomous consumption. This saving function is identical with ours (2) if s = sp π and c0 N = Z. Note that it is not explicitly assumed
in our analysis that there are classes with different saving behavior. For implications of autonomous expenditure in Kaleckian models,see also Skott (2016).
4 It is possible to provide microeconomic foundation for the capital formation function given in (3). For details, see Murakami (2016).
5 Our investment (or capital formation) function has a lot in common with Benassy’s (1984) one because he related the level ofinvestment to the expected level of (aggregate) demand. However, his analysis of business cycles is different from ours in that he focusedon short term economic fluctuations, in which the level of capital stock remains the same over time, while we allow for changes in thelevel of capital stock (and the size of population).

3

2.3

Equilibrium in the good-service market

We shall derive the output-capital ratio, which can also be regarded as the rate of utilization, from the Keynesianprinciple of effective demand. The good-service market equilibrium condition, or the IS balance condition, implies
that aggregate income per unit of capital stock is determined by the following:

S = sY − c0 N = f (re )K = I,

or

u=


1  c0+ f (re ) .
s k

(5)

In (5), u and k stand for the output-capital ratio (or the rate of utilization) Y /K and per capita capital stock K/N,
respectively.
On the basis of the output-capital ratio u, defined in (5), we shall derive the (actual) rate of profit (on capital).
For this purpose, we shall assume that the capital share or the profit share in aggregate income, denoted byπ ∈ (0, 1), is constant.6 Since aggregate profit is equal to aggregate income multiplied by the capital share, the rateof profit can be derived as follows:

r=


πY
π  c0= πu =+ f (re ) .
K
s k

(6)

In (6), r stands for the rate of profit. Note that the actual rate of profit r is determined by the expected rate ofprofit re through investment.

2.4

Capital formation

We shall describe the process of capital formation (or of capital accumulation) through investment. Since capitalstock K is varied by investment (net of capital depreciation), this process can be represented by the following
equation: dK 
= I − δK = [f (re ) − δ]K,
K̇ =
dt

(7)

where δ is a positive constant. In (7), δ stands for the constant rate of capital depreciation.
6 According to Kalecki (1939), the capital share π is determined by the degree of monopoly. In this respect, it suffices for ourassumption to suppose that the degree of monopoly is constant.

4

2.5

Revision of expectation

We shall formalize the revision process of the expected rate of profit. The expected rate of profit re is held by firms
and not necessarily equal to the actual one r, and the former is (continually) revised on the basis of the latter.
To describe this revision process, we shall assume that re is changed in response to the difference from r in thefollowing way:
 dre 
hπc
i

0
ṙe =
= α(r − re ) = α+ f (re ) − re ,
dt
s k

(8)

where α is a positive constant. Equation (8) means that the expected rate of profit re is adaptively revised, and α
can be interpreted as the speed of revision of expectation.

2.6

Population change

To deal with economic growth, we shall take account of changes in population. To this end, we shall suppose thatthe size of population is varied at a constant rate as follows: dN 
Ṅ =
= nN,
dt

(9)

where n is a real constant. In (9), n is noting but the rate of population change, which can be zero or negative.

2.7

Full system: System (PK)We can now summarize our equations (7), (8) and (9):K̇ = [f (re ) − δ]K,

i
hπc
0
ṙe = α+ f (re ) − re ,
s k

(7)

Ṅ = nN,(9)(8)

To discuss the possibility of growth cycles, we shall consider the dynamics (differential equation) of per capita
capital stock k. The differential equation of k can easily be derived from (7) and (9) as follows: dk  K̇
K Ṅ
k̇ =
=
−
= [f (re ) − (δ + n)]k.
dt
N
NN

5

(10)

Therefore, we can complete our post Keynesian system in the following form:ṙe = α

hπc

0

s

k

i

+ f (re ) − re ,k̇ = [f (re ) − (δ + n)]k.

(8)
(10)

In what follows, we shall denote the system of equations (8) and (10) by “System (PK)” (to signify “Post Keynesian”). Note that per capita income is determined along every solution path of System (PK) through (5) (andY /N = u/k).

3

Analysis

We shall proceed to analyze our System (PK) and examine the possibility of occurrence of a limit cycle, which canbe interpreted as a persistent growth cycle.

3.1

Existence and uniqueness of equilibrium

We shall first define an equilibrium point (or a steady state) of System (PK). In our analysis, an equilibrium pointof System (PK) is defined as a point (re , k) ∈ R2++ at which we have ṙe = k̇ = 0.7 Then, an equilibrium point of
System (PK), (r∗ , k ∗ ), is derived as a solution of the following simultaneous equations.

0=π  c0+ f (re ) − re ,s k

0 = f (re ) − (δ + n).

Thus, the unique equilibrium point of System (PK), (r∗ , k ∗ ), is, provided that it exists, given by the following:(r∗ , k ∗ ) = f −1 (δ + n),πc0.
sf −1 (δ + n) − π(δ + n)

(11)

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

f (0) < δ + n < elim f (re ),r →∞ π(δ + n) f< δ + n.
s
7 Note that k = 0 is ruled out as an equilibrium value of k by this definition.

6

(12)
(13)

Under Assumption 1, condition (12) implies the existence and uniqueness of f −1 (δ + n) > 0 (by the implicit
function theorem). Also, we can see from (13) that, under Assumption 1, we haveπ(δ + n)< f −1 (δ + n),sorsf −1 (δ + n) − π(δ + n) > 0.

(14)

Then, the equilibrium point of System (PK), defined in (11), can be shown to exist on R2++ .

3.2

The positivity constraint

Before exploring the existence and uniqueness of a limit cycle, we shall make sure that every solution path of System(PK) with an initial condition (re (0), k(0)) ∈ R2++ at t = 0 stays on R2++ for all t ≥ 0. By so doing, we can ensurethat a periodic orbit, if it exists, lies entirely on R2++ , which is an economically meaningful domain. Since it is seenthat k(t) > 0 for all t ≥ 0 along each solution path with (re (0), k(0)) ∈ R2++ , it suffices for our purpose to ensurethat re (t) > 0 for all t ≥ 0 along each of such solution paths.
It follows from the continuity of solution paths of System (PK) (by Assumption 1) that if ṙe > 0 at re = 0 for
every k > 0, every solution path cannot leave the domain of R2++ , provided that it starts on (or enters) this domain.
Then, it is sufficient to assure that for every k > 0ṙe |re =0 = α

hπc

0

s

k

i
+ f (0) > 0.

(15)

For this condition to hold, we shall introduce the following assumption.
Assumption 3. The following condition is satisfied:

f (0) ≥ 0.

(16)

Combined with Assumption 1, condition (16) implies that the (gross) capital formation function takes on anonnegative value. Assumption 3 can thus be viewed as reasonable in that the rate of gross capital formation isnonnegative by definition. Note that condition (15) is fulfilled for every k > 0 under this assumption.

7

3.3

System (PK) reformulated

We shall reformulate System (PK) as a generalized Liénard system for the sake of our analysis.8 By so doing, we
can easily discuss the existence and uniqueness of a limit cycle in System (PK).
For reformulation, we shall introduce the following new variables:x = re − r∗ ,(17)

y = ln k ∗ − ln k,(18)where (r∗ , k ∗ ) is defined by (11) and ln represents the natural logarithm. Since we have shown that k ∗ is positive under Assumptions 1 and 2 and that k is always positive along every solution path of System (PK) with(re (0), k(0)) ∈ R2++ , we can take the natural logarithms of k ∗ and of k.
We shall substitute the variables defined in (17) and (18) in System (PK) to obtain the following system:ẋ = φ(y) − F (x),(19)

ẏ = −g(x),

(20)

where
g(x) = f (r∗ + x) − (δ + n)
(21)
= f (f −1 (δ + n) + x) − (δ + n),hiπ
F (x) = α x − g(x) ,ssf −1 (δ + n) − π(δ + n)φ(y) = α[exp(y) − 1].
s

(22)
(23)

In what follows, we shall denote the system of equations (19) and (20) with (21)-(23) by “System (PK*).” It is easy
to see from Zeng et al. (1994) or Xiao and Zhang (2003) that System (PK*) can be viewed as a generalized Liénard
system (under some assumptions). We shall look into System (PK*) to examine the characteristics of System (PK).

3.4

Existence of a periodic orbit

We shall now explore the existence of a periodic orbit, which can be taken as a persistent growth cycle, in System(PK*). For this purpose, we shall first confirm the local asymptotic total instability of the unique equilibrium,9
second find a (nonempty) positively invariant compact subset of R2 (with respect to System (PK*))10 and finally
8 For details on generalized Liénard systems, see, for example, Levinson and Smith (1942), Zeng et al.

(1994) or Xiao and Zhang(2003).
9 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.
10 A closed (usually compact) region D is 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 D will remain in8

apply the Poincaré-Bendixson theorem to System (PK*).11
To begin, we shall examine the local asymptotic stability of the unique equilibrium point of System (PK*),(x∗ y ∗ ) = (0, 0). One can easily derive the Jacobian matrix of System (PK*) evaluated at this equilibrium, denoted
by J ∗ , as follows:


0

0





0

−1

 −F (0) φ (0)   α[πf (f (δ + n)) − s]/sJ∗ = 
=
−f 0 (f −1 (δ + n))−g 0 (0)
0

α[sf

−1


(δ + n) − π(δ + n)]/s .
0

The trace and determinant of J ∗ are given by
πf 0 (f −1 (δ + n)) − s,s
sf −1 (δ + n) − π(δ + n)
det J ∗ = αf 0 (f −1 (δ + n))> 0,s
tr J ∗ = α

where the inequality follows from (14) (derived from Assumption 2). Then, the unique equilibrium point of System(PK*) is not a saddle point. Also, it is locally asymptotically totally unstable if the trace of J ∗ is positive.
For the local asymptotic total instability of the unique equilibrium, we shall make the following assumption.
Assumption 4. The following condition is satisfied:

f 0 (f −1 (δ + n)) >

s
.
π

(24)

Condition (24) holds if the rate of capital formation f is sufficiently elastic to a change in the expected rate ofprofit re at the unique equilibrium. In this respect, Assumption 4 may be said to reflect the argument by Keynes(1936, chap. 22) that investment is subjected to violent fluctuations due to changes in the marginal efficiency of
capital, which is represented as the expected rate of profit re , even in short run. Moreover, we can make some
remark on this assumption in terms of the so-called Keynesian stability condition (cf. Marglin and Bhaduri 1990).12
In our System (PK) or (PK*), the Keynesian stability condition is always satisfied because the marginal propensity
to invest with respect to the current income is zero. As we have observed, however, this does not imply the local
asymptotic stability of the unique equilibrium. This consequence means that if investment is influenced not by
the current rate of profit or of utilization but by the expected one, which is formed on the basis of the past andcurrent ones, the Keynesian stability condition is not directly related to the stability of equilibrium.13 Note that
Assumption 4 implies that F 0 (0) < 0.
D for ever after (i.e., for all t ≥ 0).
11 For the Poincaré-Bendixson theorem, see, for example, Coddington and Levinson (1955, chap. 16).
12 The Keynesian stability condition requires that the marginal propensity to invest be less than that to save.
13 This consequence may be consistent with Skott’s (2012) argument because he pointed out from Harrod’s (1939) viewpoint that, if
the rate of capital formation is formalized as a function not only of the current rate of utilization but also of the past ones (as well as
the past rates of capital formation), the Keynesian stability condition does not have implications for the long run stability. For recent
debates on the Keynesian stability condition, see, for example, Hein et al. (2011) or Franke (2017).

9

Next, we shall look for a (nonempty) positively invariant compact subset of R2 . For this purpose, we shall draw
the phase diagram of System (PK*). It follows from (19) and (20) that the loci of ẋ = 0 and of ẏ = 0 are givenby φ(y) = F (x) and by g(x) = 0, which can be reduced to x = 0, respectively. To illustrate the phase diagram, weshall introduce the following assumption about f.
Assumption 5. The following condition is satisfied:lim [sre − πf (re )] > sf −1 (δ + n) − π(δ + n).

r e →∞(25)

Furthermore, there exist exactly two real constants re and re with re < re such that the following conditions are
satisfied:

f 0 (re ) = f 0 (re ) =f (re ) <

s
,
π

s e
r .
π

(26)
(27)

Mathematically, condition (25) is fulfilled if the capital formation function f is bounded, and condition (26) holds
if f is a logistic (sigmoid) function (under Assumption 4).14 In this sense, these conditions are not so restrictive.
Moreover, as we shall see below, condition (27) ensures that the locus of ẋ = 0 is well-defined.
We shall examine characteristics of the locus of ẋ = 0 by making use of Assumption 5 (as well as Assumptions
1-4). To this end, we shall first take a look at the shape of the graph of F. Noting (11), (17) and (22), we can obtainthe following:

lim F (x) > 0,

x→∞

(28)

Also, it follows from Assumptions 3 and 4 thatiα h −1
sf (δ + n) − (δ + n) + sf (0) < 0,
s
i
πhs
0
− f 0 (f −1 (δ + n)) < 0.
F (0) = αs π

F (−r∗ ) = −

(29)
(30)

Since we know F (0) = 0 (from (22)), we can find from (30) that, for some sufficiently small positive ε, we have
F (−x) > 0 and F (x) < 0 for x ∈ (0, ε). Taking (28) and (29) into consideration, we can see from the continuity ofF 0 (x) (ensured by Assumption 1) that F 0 (x) = 0 has at least two roots x0 ∈ (−r∗ , 0) and x0 > 0. As for F 0 (x), on
the other hand, condition (26) in Assumption 5 is equivalent to meaning that F 0 (x) = 0 has exactly two real roots.
Hence, x0 and x0 are all the real roots of F 0 (x) = 0 and given by x0 = re − r∗ ∈ (−r∗ , 0) and x0 = re − r∗ > 0. It
then follows from the continuity of F 0 (x) and (30) that F 0 (x) < 0 for x ∈ (x0 , x0 ) and F 0 (x) > 0 for x ∈ (−r∗ , x0 )14 For a related model using a logistic capital formation function, see Murakami (2018).

10

or x > x0 and that F (x0 ) > 0 and F (x0 ) < 0. Furthermore, it is obvious from the continuity of F (x) that F (x) = 0
has two roots x0 and x0 besides x = 0 with x0 ∈ (−r∗ , x0 ) and x0 > x0 . Thus, we can draw the graph of F (x) as infigure 1.
y

x'

− r*

x0

O

x'

x0

x

Figure 1: Graph of F (x)We shall now have a closer look at the locus of ẋ = 0. It is easy to see from (22) and (23) that this locus is given
by
α

h
i
sf −1 (δ + n) − π(δ + n)π[exp(y) − 1] = α x − g(x) ,ssor

exp(y) = 1 +sx − πg(x)sf −1 (δ + n) − π(δ + n)

.

(31)

Then, for the locus of 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 x ≥ x0 .15 Since we know from figure 1 that the minimum of F (x) for x ≥ x0 is given by
F (x0 ), it suffices for the locus of ẋ = 0 being well defined that the following condition is fulfilled:1+

sx0 − πg(x0 )sf −1 (δ + n) − π(δ + n)

> 0,which can, by (21), be reduced tosre − πf (re )sf −1 (δ + n) − π(δ + n)

> 0.

This condition is satisfied due to (14) (by Assumption 1) and (27). Thus, the locus of ẋ = 0 is well defined at least
15 For our analysis, it is sufficient that the locus of ẋ = 0 is well defined for x ≥ x . This requirement is vital for verifying the existence0

and uniqueness of a limit cycle in System (PK*).

11

for x ≥ x0 .
Therefore, the phase diagram of System (PK*) can be drawn as in figure 2.

y

=0

=0

′
− *

′

O

x

Figure 2: Phase diagram of System (PK*)

Now we are in a position to detect a (nonempty) positively invariant compact subset of R2 . To begin, we shallconsider the solution path of System (PK*) with (x(0), y(0)) = (−r∗ , 0), denoted by (SP1).16 We shall below provethat (SP1) reaches the y-axis on the positive part at some (finite) nonnegative t (for the first time). To this end,
we shall first confirm that ẋ > 0 along (SP1) before it intersects with the y-axis on the positive part for the firsttime.17 It is seen from (19) and (20) that for x < 0, along the locus of ẋ = 0,dẋ|ẋ=0 = [φ0 (y)ẏ − F 0 (x)ẋ]|ẋ=0 = −g(x)φ0 (y) > 0,dt
where the inequality follows from the fact that g(x) < 0 for x < 0. It then follows from the continuity of solution
paths that solution paths of System (PK*) cannot cross the locus of ẋ = 0, if they stay on the domain of x < 0 andstart on the domain of ẋ > 0. Thus, we can confirm the desirable fact because (x(0), y(0)) = (−r∗ , 0) lies on thedomain of ẋ > 0 (cf. Assumption 3).
We shall next verify that (SP1) crosses the line of x = x0 from left to right at some nonnegative t. For thispurpose, we shall suppose, for the sake of contradiction, that (SP1) stays on the domain of x ∈ [−r∗ , x0 ] for allt ≥ 0.18 Then, we have g(x) ≤ g(x0 ) < 0 and F (x) ≤ F (x0 ) for x ∈ [−r∗ , x0 ] because g 0 (x) > 0 and F 0 (x) > 0
for x ∈ [−r∗ , x0 ]. Hence, we can obtain the following inequalities for all t ≥ 0 along (SP1) under our hypothetical
16 Note that the existence or uniqueness of (SP1) is not guaranteed at this stage. In what follows, we shall suppose that (SP1) exists
until it reaches the y-axis on the positive part for the first time, discuss characteristics of an arbitrary (SP1) and then establish theexistence and uniqueness of it.
17 Note that, at this stage, we do not rule out the possibility that (SP1) never reaches the y-axis for t ≥ 0.
18 Note that, by Assumption 3, solution paths of System (PK) cannot leave the domain of x ≥ −r ∗ once they enter this domain.

12

assumption:ẋ ≥ φ(y) − F (x0 ),(32)ẏ ≥ −g(x0 ) > 0.

(33)

Also, because of φ(0) = 0, φ0 (y) > 0 and φ(y) → ∞ as y → ∞ due to (23) and of F (x0 ) > 0, there exists a uniquepositive y1 such that φ(y1 ) = F (x0 ) + 1. It follows from (33) that along (SP1)y(t) ≥ y(0) − g(x0 )t = −g(x0 )t.

Hence, we have y(t) ≥ y1 for t ≥ t1 ≡ −y1 /g(x0 ) > 0 along (SP1). Then, we can find from (32) that for t ≥ t1
ẋ ≥ φ(y) − F (x0 ) ≥ φ(y1 ) − F (x0 ) = 1.
Since (SP1) stays on the domain of x ∈ [−r∗ , x0 ] for all t ≥ 0 (by assumption), we have x(t1 ) ≥ −r∗ andẋ(t) ≥ x(t1 ) + (t − t1 ) ≥ −r∗ + (t − t1 ).
Then, we have x(t) > x0 for t > t2 ≡ t1 + (x0 + r∗ ). But this contradicts our hypothesis. Thus, we can prove that
(SP1) must cross the line of x = x0 from left to right by the time of t = t2 .
We shall then show that (SP1) reaches the y-axis on the positive part after it crosses the line of x = x0 at t = t3
(where t3 ≤ t2 ). Since ẋ > 0 along (SP1) before it reaches the y-axis on the positive part, we have x(t) ≥ x0 fort ≥ t3 before it reaches the y-axis. Thus, we shall assume, for the sake of contradiction, that (SP1) stays on thedomain of x ∈ [x0 , 0) for all t ≥ t3 . Then, (SP1) remains on the domain of x ∈ [x0 , 0] (but never reaches the y-axis)
for t ≥ t3 . Since ẋ > 0 at t = t3 along (SP1), we have ẋ(t3 ) ≡ ∆ > 0. Because F 0 (x) ≤ 0 and ẏ = −g(x) > 0 forx ∈ [x0 , 0] (cf. figure 1), we can see that along (SP1), for t ≥ t3dẋ = φ0 (y)ẏ − F 0 (x)ẋ ≥ 0.
dt
Then, we can see that, along (SP1), for t ≥ t3

ẋ(t) ≥ ẋ(t3 ) = ∆,orx(t) ≥ x(t3 ) + ∆(t − t3 ) = x0 + ∆(t − t3 ).

13

Therefore, (SP1) must reach the y-axis (on the positive part) by the time of t = t4 ≡ t3 − x0 /∆. This is, of course,
a contradiction. Hence, (SP1) can be shown to reach the y-axis on the positive part by the time of t = t4 after itcrosses the line of x = x0 .
We shall now make sure that (SP1) uniquely exists at least until it reaches the y-axis for the first time. Becauseof 0 ≤ ẏ ≤ −g(r∗ )(> 0) along (SP1) for t ∈ [0, t4 ], we have 0 ≤ y(t) ≤ y# = −g(r∗ )t4 for t ∈ [0, t4 ], along (SP1).
Then, every (SP1), if it exists, stays on the following (nonempty) compact rectangular domain D1 for t ∈ [0, t4 ]:D1 ≡ {(x, y) ∈ R2 : x ∈ [−r∗ , 0], y ∈ [0, y# ]}.

Since g, F and φ are all continuously differentiable on D1 (by Assumption 16) and D1 is a compact convex
subset of R2 , it follows from the mean value theorem that there exists a positive constant M such that for every(x0 , y 0 ), (x00 , y 00 ) ∈ D1 19|[φ(y 00 ) − F (x00 )] − [φ(y 0 ) − F (x0 )]| + |g(x00 ) − g(x0 )| ≤ M (|x00 − x0 | + |y 00 − y 0 |).

Hence, System (PK*) satisfies the Lipschitz condition (cf. Coddington and Levinson 1955, chap. 1) on D1 (for allt ≥ 0). It then follows from the argument on continuation of solution paths (cf. Coddington and Levinson 1955, p.
15, Theorem 4.1) that the existence and uniqueness of a solution path of System (PK*) can be obtained as long as
it stays on D1 . Therefore, (SP1) exists and is unique until it reaches the y-axis for the first time. Let Q(0, yq ) be
the (first) intersection point of the unique (SP1) and the y-axis on the positive part, where 0 < yq ≤ y# . Also, letR(xr , yq ) be the intersection point of the locus of ẋ = 0 and the line through Q parallel with the x-axis.20
We shall next consider the solution path of System (PK*) with (x(0), y(0)) = (xr , 0), denoted by (SP2). By amethod similar to the above, we can verify that (SP2) exists and is unique (at least) until it reaches the y-axis on
the negative part for the first time and that it must reach the y-axis on the negative part at some finite nonnegative
time t. Let T(0, yt ) be the (first) intersection point of (SP2) and the y-axis on the negative part, where yt is anegative constant.
Now we can proceed to detect a (nonempty) positively invariant compact subset of R2 . Define the pointsP(−r∗ , 0), S(xr , 0) and U(−r∗ , yt ) and let the arcs of PQ and ST represent (SP1) and (SP2), discussed above,
respectively. Then, we can construct the (nonempty) compact subset of R2 enclosed by the arcs of PQ and ST andthe line segments of QR, RS, TU and UP, as in figure 3. We shall denote this subset by D. We can see that D ispositively invariant as follows. By the uniqueness of (SP1) and (SP2), no solution paths of System (PK*) whichstart from inside of D at t = 0 can cross the arc of PQ or of ST. Also, we have ẏ ≤ 0 along QR, ẋ ≤ 0 along
RS, ẏ ≥ 0 along TU and ẋ > 0 along UP. Then, no solution paths of System (PK*) can leave D if they enter this
19 Murakami (2014) proved in details that the Lipschitz condition is satisfied on a compact rectangular subset of Rn if all the functions
are continuously differentiable on this subset.
20 Since the locus of ẋ = 0 is given by (31), we can find from (28) (and figure 1) that such a positive x exists and is unique.
r14domain.

y
R

Q

=0

=0

D

P

− *

S

O

x

T

U

Figure 3: Positively invariant subset D

Thus, we are ready to establish the existence of a periodic orbit in System (PK) or (PK*) by way of the
Poincaré-Bendixson theorem.
Proposition 1. Let Assumptions 1-5 hold. Then, System (PK) has at least one periodic orbit on R2++ .
Proof. To prove this proposition, it suffices to verify the existence of a periodic orbit which lies entirely on thedomain of x > −r∗ in System (PK*).
To apply the Poincaré-Bendixson theorem (cf. Coddington and Levinson 1955, chap.16), we shall first confirm
the uniqueness of a solution path of System (PK*) on D.21 For this purpose, we shall construct the (nonempty)
compact rectangular domain D2 as follows:
D2 ≡ {(x, y) ∈ R2 : x ∈ [−r∗ , xr ], y ∈ [yt , yq ]},where Q(0, yq ), R(xr , yq ) are T(0, yt ) are defined above.22 Since D2 is a a compact rectangular domain, we canverify in the same way as above that System (PK*) satisfies the Lipschitz condition on D2 . Then, every solution
path of System (PK*) is unique (with respect to an initial condition) provided that it stays on D2 . Hence, theuniqueness of a solution path follows on D.
Since the unique equilibrium point of System (PK*) is locally asymptotically totally unstable (by Assumption4), we can enclose the equilibrium point by a sufficiently small rectangle such that no solution paths of System(PK*) can enter the interior of this rectangle once they leave it. We can also construct a (nonempty) positively
21 Since g, F and φ are all continuously differentiable and D is a positively invariant compact subset of R2 , we can prove that if(x(0), y(0)) ∈ D, the solution path of (PK*) exists on D for all t ≥ 0. (cf. Coddington and Levinson 1955, chap. 1).
22 As we have seen, these points are uniquely determined.

15

invariant compact domain D∗ by eliminating the interior of the small rectangle from D. Because the positively
invariant domain D∗ contains no equilibrium point, we can apply the Poincaré-Bendixson theorem (cf. Coddington
and Levinson 1955, chap.16) to System (PK*) on D∗ to draw the conclusion that a periodic orbit of System (PK*)
exists on D∗ . Due to ẋ > 0 along the line of x = −r∗ (by Assumption 3), such a periodic orbit cannot intersectwith this line and lies entirely on the domain of x > −r∗ .
Proposition 1 guarantees the existence of a periodic orbit in our post Keynesian System (PK). This implies
that a persistent and periodic “growth cycle” can be observed in System (PK). Along such a growth cycle, theexpected rate of profit re and per capita capital stock k are subjected to persistent cyclical fluctuations, and so is
the output-capital ratio u due to (5). Since the size of population N changes at a constant rate, aggregate capitalstock K = kN fluctuates around its trend value k ∗ N (t), and so aggregate income Y = ukN (t) also fluctuatesaround its trend value u∗ k ∗ N (t), where u∗ is obtained by putting re = r∗ and k = k ∗ in (5). In this respect, ourgrowth cycle may be said to describe economies growing (or shrinking) over persistent cyclical fluctuations.
We shall add some remarks on growth cycles in our System (PK) from a Keynesian viewpoint. As is wellknown, Keynes (1936, chap. 22) attributed the main cause of business cycles to violent fluctuations of the marginal
efficiency of capital, which can be identified with the expected rate of profit re . Our conclusion on the presence of
persistent growth cycles is consistent with his argument because changes in re do give rise to persistent cyclicaleconomic fluctuations. In this sense, it may be maintained that our post Keynesian system (PK) can properly
describe the phenomenon of business cycles based upon the Keynesian theory.23

3.5

Uniqueness of a limit cycle

Now that the existence of a periodic orbit is ensured in System (PK), we shall examine the “uniqueness” of it.
If this uniqueness obtains, the unique periodic orbit does represent the only (feasible) growth cycle in our system(and mathematically, it is a limit cycle).24 To derive a sufficient condition for the uniqueness, we shall continue
to investigate properties of System (PK*) instead of those of System (PK). Specifically, we shall make use of thetheorem established by Xiao and Zhang (2003), which is reproduced as Theorem 1 in Appendix.25
We can easily find that Assumptions 6 and 7, required in Theorem 1, are fulfilled in System (PK*), by setting
x = −r∗ = −f −1 (δ + n), x = ∞, y = −∞ and y = ∞ (if x0 and x0 are defined as in the previous section). Thus, to
23 Our consequence is consistent with Benassy’s (1984) one in that a persistent business cycle (or a periodic orbit) can be generated
if the effect of changes in the expected level of aggregate demand is strong enough. Note that his analysis was a short-term one, whileours is a long-term one.
24 In economics, a limit cycle is often confused with a periodic orbit. Mathematically, however, not every periodic orbit is a limit
cycle. Indeed, a limit cycle is a periodic orbit such that some other solution path has it as an α or ω limit set, i.e., that the former
converges to the latter as t → ∞ or as t → −∞. (cf. Coddington and Levinson 1955, pp. 391-392; Hirsch and Smale 1974, p. 250). For
example, all solution paths, except for those whose initial condition is an equilibrium point, of Lotka-Volterra systems, which Goodwin(1967) employed in presenting a model of growth cycle, are periodic orbits but none of them is a limit cycle (cf. Hirsch and Smale 1974,
p. 252, Theorem 3). If the number of periodic orbits is finite, on the other hand, all of them are necessarily (either stable or unstable)
limit cycles (on two-dimensional differential equations).
25 Precisely speaking, Theorem 1 is a slightly modified version of Xiao and Zhang’s (2003) theorem, which was presented by Murakami(2018).

16

guarantee the uniqueness of a limit cycle, we only have to examine under what condition the remaining hypothesis
for Theorem 1, or Assumption 8, is satisfied.
Now we shall see if Assumption 8 is fulfilled in System (PK*). As we have confirmed in the previous section, thelocus of φ(y) = F (x), given by (31), is well defined at least on [x0 , x0 ] (by Assumption 5). Then, in what follows,
we shall detect a sufficient condition for condition (44) to hold. Since we have x0 ∈ [x0 , 0] and x0 ∈ [0, x0 ], condition(44) holds if the following condition is fulfilled:26 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

(34)


 Φ(φ−1 (F (x0 ))) ≥ G(x0 ) − G(x0 ), if G(x0 ) > G(x0 ).
One can easily see that the inverse of φ is given by (23) as
φ−1 (y) = ln 1 +sysf −1 (δ + n) − π(δ + n).

It then follows from (22) that
φ−1 (F (x)) = ln 1 +sx − πg(x).
sf −1 (δ + n) − π(δ + n)

(35)

On the other hand, we can easily calculate the following:Z yΦ(y) =

φ(s)ds = α0

sf −1 (δ + n) − π(δ + n)[exp(y) − y − 1].
s(36)

By substituting (35) in (36), we obtainisf −1 (δ + n) − π(δ + n) hsx − πg(x)sx − πg(x)
−
ln
1
+
s
sf −1 (δ + n) − π(δ + n)
sf −1 (δ + n) − π(δ + n)
−1
sf (δ + n) − π(δ + n)=α[z − ln(1 + z)],s

Φ(φ−1 (F (x))) = α(37)where z = [sx − πg(x)]/[sf −1 (δ + n) − π(δ + n)]. Since sx − πg(x) = sF (x)/α is not equal to 0 for x = x0 or forx = x0 (cf. figure 1) and we have z > ln(1 + z) for z 6= 0 (provided that z > −1), we can immediately find from (37)that Φ(φ−1 (F (x0 ))) and Φ(φ−1 (F (x0 ))) are both positive. Moreover, x0 and x0 are both determined independently
from the value of α because they are roots of F 0 (x) = 0 or g 0 (x) = s/π. Hence, the left hand side of (34) is positive
26 Because of φ0 (y) > 0 (cf. (23)), the function φ−1 (F (x)) can be defined at least for x ∈ [x , x ] (by Assumption 5).
0017and proportionate to α in both cases. On the other hand, the right hand side of (34) is independent from α because
x0 and x0 are both determined, independently from α, by F (x) = 0 or sx = πg(x). Therefore, one can, by taking
sufficiently large α, make the left hand side of (34) as large as one likes while fixing the value of the right hand side.
Thus, we can argue that condition (34) is satisfied for α sufficiently large.
Now we are ready to present the following pleasant conclusion on the uniqueness of a limit cycle.
Proposition 2. Let Assumptions 1-5 hold. If α is sufficiently large, System (PK) has a unique limit cycle on R2++ ,and it is (periodically) stable.
Proof. Since we have already shown in Proposition 1 the existence of a periodic orbit on R2++ in System (PK), it
suffices to prove the uniqueness and stability of a limit cycle (or of a periodic orbit) in System (PK*). As we haveobserved above, Assumptions 6-8 are fulfilled in System (PK*) for α sufficiently large. Then, we can obtain the
conclusion of this proposition.
According to Proposition 2, the uniqueness (and stability) of a limit cycle obtains in System (PK) if α is large
enough. Put in a different way, the state of the economy under consideration, represented by (re , k), converges tothe unique growth cycle with the passage of time (provided that the initial condition of (re , k) is not (r∗ , k ∗ ) andthat variations in these variables are bounded over time).27 This is a strong conclusion in that if the revision speedof expectations α is quick enough, the asymptotic state of the (Keynesian) macroeconomic system is the uniquelydetermined growth cycle (unless the initial condition is not the equilibrium state) and so it is predictable.
We shall add some comments on the conclusion of Proposition 2. The uniqueness of a limit cycle is itself a“unique” consequence in that it has rarely been verified in economics.28 Furthermore, this consequence is significant
from a Keynesian perspective. As we have confirmed, Keynes (1936) himself emphasized sharp variations in themarginal efficiency of capital, which is conceptually identical with the expected rate of profit re , as the primaryfactor for occurrence of business cycles. In our analysis, on the other hand, if the speed of revision of expectation α
is large, which means that re is exposed to violent fluctuations (cf. (8)), the uniqueness of a growth cycle obtains.
Therefore, we may state from our present analysis that frequent and violent variations in the marginal efficiency of
capital compose a sufficient condition not only for the existence of a business cycle, as Keynes (1936) insisted, butalso for the uniqueness of it.

4

Conclusion

We shall now summarize our analysis and conclude the present paper.
27 Proposition 2 only ensures the existence and uniqueness of a limit cycle and not the convergence of every solution path to theunique limit cycle. However, it is easy to obtain this convergence by making use of the generalized Poincaré-Bendixson theorem (cf.
Coddington and Levinson 1955, pp. 394-395, Theorem 3.1) unless the solution path is unbounded.
28 There have been several exceptions such as Ichimura (1955), Lorenz (1988, 1993), Gori and Galeotti (1989), Sasakura (1996) and
Murakami (2018).

18

We have formalized a post Keynesian system by introducing the investment function as related directly tothe expected rate of profit and examined the existence and uniqueness of a periodic orbit (or of a limit cycle)
in this system. Compared with the other related Keynesian systems, our post Keynesian system may be said toconform more to Keynes’ idea that expectation, especially the long-term expectation, has a decisive influence on thedetermination of investment and aggregate income. Importantly, we have demonstrated that a periodic orbit does
exist if a change in the expected rate of profit has a strong influence on investment and that it is unique if revisions of
the expected rate of profit are frequent enough (more specifically if the speed of revision of expectation is sufficiently
fast). By so doing, we have been able to confirm that variations in expectation do give rise to a persistent businesscycle and contribute to the uniqueness of it. Our conclusion may be said to support and strengthen Keynes’ thoughton business cycles.
Throughout this paper, we have attempted to shed a new light on the role of expectation in business cycles.
Although the importance of expectation has been recognized, it has not properly been examined in theory of businesscycles. We hope sincerely that our present analysis is helpful for understanding the macroeconomic system.

Acknowledgment
This work was supported by Chuo University personal research grant.

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),(38)

ẏ = −g(x).

(39)

In what follows, the system of equations (38) and (39) is denoted by “System (L).”
Following Xiao and Zhang (2003), we shall impose the following assumptions concerning System (L).
Assumption 6. 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 ≤ ∞19and −∞ ≤ y < 0 < y ≤ ∞. Furthermore, the following conditions are satisfied:

xg(x) > 0 for x 6= 0,(40)

φ(0) = 0, φ0 (y) > 0 for y ∈ (y, y).

(41)

Assumption 7. 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

(42)
(43)


 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 8. The curve of φ(y) = F (x) is well defined for x ∈ [x0 , x0 ].29 Furthermore, the following conditionis satisfied:


 sup

x∈[0,x0 ] (G(x) + Φ(φ−1(F (x)))) ≥ G(x0 ), if G(x0 ) ≥ G(x0 ),(44) 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(y) =Ry
0φ(s)ds.

As regards the uniqueness of a limit cycle in System (L), the following theorem was verified by Xiao and Zhang(2003).
Theorem 1. Let Assumptions 6-8 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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