Discussion Paper No.369
On growth cycles in a stochastic post-Keynesian model
Hiroki Murakami
Faculty of Economics, Chuo University
May 2022

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, JapanOn growth cycles in a stochastic post-Keynesian model
Hiroki Murakami*
Faculty of Economics, Chuo University

Abstract
This paper evaluates the impact of stochastic disturbances on a deterministic limit cycle in a stochastic
post-Keynesian model. It presents an approximation formula for solution paths near the limit cycle and derives
an approximated stationary distribution of the limit cycle.

Keywords: Limit cycles; Keynesian theory; Stationary distribution; Stochastic differential equations
JEL classification: C62; E12; E32

1

Introduction

It is well known that there are two approaches in the theory of business cycles: the “endogenous” and “exogenous”
approaches.1 The endogenous approach attributes the mechanism of business cycles to internal (endogenous) factors
in economic systems (e.g., characteristics of consumption or investment behavior). In this approach, the phenomenaof business cycles are usually explained by the existence of periodic motions in nonlinear economic models.2 The
exogenous approach, on the other hand, takes business cycles as phenomena induced by external (exogenous) factors
outside economic systems (e.g., shocks in demand or supply). As such, this approach focuses on the propagation
mechanism of random shocks in economic systems.3 Although both of the approaches provide insightful expositions
for business cycles, there have not been many attempts to integrate them until recently.
In recent years, several theoretical works have studied the response of deterministic business cycles to random
shocks. Bashkirtseva et al. [6], Bashkirtseva et al. [7] and Bashkirtseva et al. [4] examined stochastic versions of
Kaldor’s [17] model;4 Li et al. [24], Li et al. [22], Li et al. [23], Nguyen Huu and Costa-Lima [35], Lin et al. [25] andJungeilges and Ryazanova [15, 16] explored stochastic versions of Goodwin’s [9, 10] models. In these studies, theinfluence of stochastic disturbances is mostly evaluated by the properties of approximated stationary distributions.
* E-mail: hmura@tamacc.chuo-u.ac.jp
 742-1 Higashi-Nakano, Hachioji, Tokyo 192-0393, Japan.
1 For details on them, see also Murakami and Zimka [33].
2 Since the classical contributions of Kaldor [17], Hicks [13] and Goodwin [9], nonlinearity has been considered the most essential
factor for verifying the existence of periodic motions in economic dynamics.
3 Slutzky [38] is one of the classic works in this approach.
4 Kosobud and O’Neill [21] and Grasman and Wentzel [11] are classical works studying the implications of stochastic disturbances in
Kaldor’s [17] model.

1The stochastic sensitivity function approach of Bashkirtseva and Ryashko [5] is, among others, a popular method
for this purpose. It is a useful and convenient way to assess stochastic implications because it can eschew complexity
associated with the Fokker-Planck equations making appropriate approximations for them.5 Without relying on the
Fokker-Planck equations, however, it is possible to evaluate the effect of stochastic disturbances. Indeed, making
proper approximations for stochastic models themselves, we can gain much more information on the characteristics
of stochastic models.
The purpose of this paper is to explore the influence of random shocks on deterministic business cycles in astochastic version of Murakami’s [29, 32] post-Keynesian model of growth cycles. For this purpose, we employ
Bonnin’s [8] method to evaluate the response of the unique limit cycle to stochastic disturbances. By so doing, we
aim to pave a road to the synthesis of the endogenous and exogenous approaches in the theory of business cycles.
This paper is organized as follows. Sect. 2 sets up a stochastic post-Keynesian model based on Murakami[29, 32]. The model consists of stochastic differential equations. Sect. 3 analyzes our post-Keynesian model. Basedon Murakami [32], we first confirm that our model, without stochastic disturbances, possesses a unique limit cycle.
Following Bonnin [8], we then present an explicit approximation formula for solution paths near the unique limit
cycle and derive an approximated stationary distribution of the limit cycle in our model with small stochastic
disturbances. The (approximation) formula for solution paths is a distinguished feature of our analysis becausesuch a formula has not been derived in the related literature. Sect. 4 performs numerical simulations. Sect. 5
concludes this paper.

2

The modelThis section presents a stochastic post-Keynesian model. The model is based on Murakami [29, 32], but it consistsof stochastic differential equations.

2.1

Aggregate saving

Following the post-Keynesian theory of income distribution (cf. Kaldor [18]; Pasinetti [36]), we postulate that
capitalists have a higher propensity to save than workers. Specifically, aggregate saving is assumed to be written
as follows:

S = sc Π + sw (Y − Π),(1)where sc and sw are, respectively, a positive constant and a nonnegative constant with sw ≤ sc < 1. In (1), Y, S
and Π stand for aggregate income, aggregate saving and aggregate profits (capitalists’ income), respectively; sc and
5 For the Fokker-Planck equations in stochastic differential equations, see Arnold [3, sect. 9.4] for instance.

2

sw represent the (average) propensities to save of capitalists and of workers, respectively.6
Reflecting Kaldor’s [19] stylized fact (cf. Jones [14]), aggregate share of capital (or the ratio of aggregate profits
to aggregate income) Π/Y is assumed to be constant as follows:Π = πY,(2)

where π is a positive constant less than unity which represents aggregate share of capital.
It follows from (1) and (2) that aggregate saving can be written as follow:

S = sΠ,(3)

where s is a positive constant defined as follows:

s=

2.2

1−π
sc π + sw (1 − π)
= sc + sw.
ππ

Aggregate investment

Based on Keynes’ [20, chap. 11] theory of investment, we postulate that the rate of gross capital formation (or theratio of investment to capital stock) is positively influenced by the expected rate of profit on capital, which can be
identified with the marginal efficiency of capital, in the following fashion: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.7
The following is a reasonable assumption on the capital formation function. (cf. Murakami [29, 32]).
Assumption 1. The nonnegative-valued function f : R+ → R+ is twice continuously differentiable withf 0 (re ) > 0

for every re ∈ R+ .8
6 It was empirically confirmed by Naastepad and Storm [34] that the (average) propensity to save of capitalists was higher than thatof workers in eight developed countries in the period of 1960-2000.
7 For this type of capital formation (or investment) function, it is possible to provide a microeconomic foundation from a viewpointof dynamic optimization (cf. Murakami [28]).
8 The twice continuous differentiability of f is assumed for the application of Ito’s formula to our model, but only the continuous
differentiability of f suffices for the proof of Theorem 1 (cf. Murakami [32]).

3

2.3

The rate of profit

According to the Keynesian principle of effective demand, aggregate saving is adjusted to the sum of aggregate
investment and aggregate autonomous demand (or the exogenous component of effective demand) (cf. Hicks [13];Serrano [37]; Allain [1]) in the following way:9

S = I + A.

(5)

In (5), A stands for aggregate autonomous demand, which includes government expenditure and net exports.
It follows from (3) and (4) that the rate of profit can be given as follows:r=1[f (re ) + a].
s(6)

In (6), r and a stand for the rate of profit (or the ratio of profits to capital stock) Π/K and the demand-capitalratio (or the ratio of autonomous demand to capital stock) A/K, respectively.

2.4

Revisions of the expected rate of profit

We postulate that the expected rate of profit is adaptively revised in response to the (realized) rate of profit but that
this revision is subjected to stochastic disturbances. Specifically, the revision process is assumed to be described bydre = α(r − re )dt + σe dze .

(7)

n1h
o
1i
f (re ) +− re dt + σe dze ,
s
k

(8)

Due to (6), this reduces todre = αwhere α and σe are a positive constant and a nonnegative constant, respectively; dze is a Wiener process (or aBrownian motion) such that, for t1 > t0 ,E[ze (t1 ) − ze (t0 )] = 0,
E[(ze (t1 ) − ze (t0 ))2 ] = t1 − t0 .

Equation (8) is a stochastic differential equation in Ito’s sense.10
In (8), α represents the (mean) speed of revisions of the expected rate of profit, while σe measures the size
9 Equation (5) is equivalent to

Y = C + I + A,
where C stands for aggregate consumption.
10 For stochastic differential equations, see Arnold [3] for example.

4

of volatility in the expectations formation. In the right-hand side, the first term expresses the usual (deterministic) adaptive expectations formation, while the second term describes stochastic disturbances in the expectations
formation.
One reasonable interpretation can be given to (8) as follows.11 Suppose that there exist N firms in the economy
and that they are identical, and denote each of them by i, i = 1, ..., N. Firm i is assumed to revise its expected
rate of profit rie during the short time interval (t, t + h) in the following way:rie (t + h) − rie (t) = α[r(t) − rie (t)]h + σe e(t) + νηi (t),(9)

where ν is a nonnegative constant; e(t) and ηi (t), i = 1, ..., N, are random variables such that for every t and every
i and j with i 6= jE[e(t)] = E[ηi (t)] = E[e(t)ηi (t)] = E[ηi (t)ηj (t)] = 0,E[e2 (t)] = E[ηi2 (t)] = h.

Firm i’s adaptive expectations formation is expressed by the first term of the right-hand side of (9), while the second
and third terms describe the macro (economy-wide) shock occurring to firm i and the shock idiosyncratic to firm i,
respectively. The expected rate of profit can be defined as follows:
N

re =

1 X e
r .
N i=1 iThen, the conditional mean and variance of the change in the expected rate of profit at t can be given by

Et [re (t + h) − re (t)] = α[r(t) − re (t)]h,
ν2 Vart [re (t + h) − re (t)] = σe2 +h.
NEquation (8) is thus obtained as the limit of (9) as N → ∞ and h → 0.

2.5

Aggregate capital formation

Aggregate capital formation is assumed to be carried out through aggregate investment without stochastic disturbances in the following fashion:d(ln K) =IK− δ dt,11 Similar interpretations can be found in Merton [26] and Yoshikawa [39].

5which, due to (4), reduces tod(ln K) = [f (re ) − δ]dt,(10)

where δ is a positive constant which represents the constant rate of capital depreciation.12

2.6

The rate of change in autonomous demand

We postulate that the natural logarithm of autonomous demand changes in the following fashion:d(ln A) = µdt + σd dzd ,(11)

where µ and σd are a real constant and a nonnegative constant, respectively; dzd is a Wiener process such that, for
t1 > t0 ,E[zd (t1 ) − zd (t0 )] = 0,
E[(zd (t1 ) − zd (t0 ))2 ] = t1 − t0 .

Equation (11) is also a stochastic differential equation in Ito’s sense.
In (11), µ represents the (mean) rate of change in autonomous demand, while σd measures the size of “(aggregate)
demand shock.”13 The first and second terms of the right-hand side characterize the deterministic (steady) andstochastic (random) natures of the rate of change in autonomous demand.
Due to (10), we can obtain the following stochastic differential equation for the demand-capital ratio:d(ln a) = [δ + µ − f (re )]dt + σd dzd .

(12)12 Equation (10) is equivalent to the following ordinary differential equation:K̇= f (re ) − δ.
K
We may introduce stochastic disturbances (due to unpredicted capital depreciation etc.) in capital formation in the following fashion:d(ln K) = [f (re ) − δ]dt + σs dzs ,where σs is a nonnegative constant and dzs is a standard Wiener process. In this case, equation (12) becomesd(ln a) = [δ + µ − f (re )]dt + σd dzd − σs dzs .
The qualitative nature of our analysis will, however, be unaltered even in this case.
13 If σ = 0, equation (11) reduces to
d
Ȧ
= µ.
A

6

2.7

Full model: Model (K)The post-Keynesian model to be analyzed can be summarized as follows:

dre = αn1so[f (re ) + a] − re dt + σe dze ,d(ln a) = [δ + µ − f (re )]dt + σd dzd .

(8)
(12)

In what follows, this model is denoted by “Model (K)” (to symbolize “Keynesian”).14

3

Analysis

This section analyzes Model (K) to derive an approximation formula for solution paths near the unique limit cycle
observed in the deterministic case with σe = σd = 0 and an approximated stationary distribution of the limit cycle.
By so doing, we evaluate the impact of random shock on deterministic business cycles.

3.1

Existence and uniqueness of equilibrium

We define (deterministic) equilibrium of Model (K) as (re , a) = (r∗ , a∗ ) ∈ R2++ that satisfies ṙe = ȧ = 0. It can also
be defined as a solution of the following simultaneous equations:

0 = f (re ) + a − sre ,0 = δ + µ − f (re ).

Then, the unique equilibrium of Model (K), if it exists, can be given as follows:(r∗ , a∗ ) = (f −1 (δ + µ), sf −1 (δ + µ) − (δ + µ)).

(13)This unique equilibrium can be shown to exist by the following assumption together with Assumption 1 (cf. Murakami [29, 32]).
Assumption 2. The following conditions are satisfied:

f (0) < δ + µ < elim f (re ),r →∞δ + µf< δ + µ.
s
14 Model (K) can be regarded as a stochastic version of Murakami’s [29, 32].

7

3.2

Reformulation of Model (K)For our analysis, we reformulate Model (K) introducing the following variables:x = re − r∗ ,(14)

y = ln a − ln a∗ .

(15)dx = [φ(y) − F (x)]dt + σe dze ,(16)

dy = −g(x)dt + σd dzd ,(17)g(x) = f (f −1 (δ + µ) + x) − (δ + µ),ih1
F (x) = α x − g(x) ,
s
h
δ
+ µi
φ(y) = α f −1 (δ + µ) −[exp(y) − 1].
s(18)Model (K) can then be transformed as follows:

where

(19)
(20)

In what follows, Model (K) is redefined as the system of equations (16) and (17) with (18)-(20), equivalent to theoriginal Model (K).15
It is easily seen from (14) and (15) that (the redefined) Model (K) possesses unique (deterministic) equilibrium
at the origin (0, 0).

3.3

Existence, uniqueness and stability of a limit cycle

To establish the existence, uniqueness and stability of a limit cycle in the deterministic case (σe = σd = 0) of Model(K), we make the following assumptions (cf. Murakami [32]).
Assumption 3. The following condition is satisfied:

f 0 (f −1 (δ + µ)) > s.

Assumption 4. The following condition is satisfied:lim [sre − f (re )] = ∞.

r e →∞

15 The redefined Model (K) can be taken as a stochastic version of a generalized Liénard system. For applications of (deterministic)
generalized Liénard systems to economics, see Murakami [29, 30, 31, 32].

8

The following equation (with respect to re ) possesses exactly two nonnegative roots re and re with re < re :f 0 (re ) = s.

Furthermore, the following condition is satisfied:f (re ) < sre .

Assumption 3 says that the unique equilibrium is locally asymptotically totally unstable in the deterministic
case.16 In this sense, this assumption is concerned with a local nature (at the equilibrium value r∗ ) of the capitalformation function f . Assumption 4 is, on the other hand, related to the global nature of this function. Indeed,
the function f can be illustrated as in figure 1 under Assumptions 1-4. Thus, Model (K), with σe = σd = 0, hasquite a lot in common with Kaldor [17] (cf. Murakami [27]) in that the investment (or capital formation) function
has a sigmoid shape.

f

s
O

𝑟

r*

𝑟

𝑟

Figure 1: f (re )We can also explore the implications of Assumptions 3 and 4 in (the reformulated) Model (K) (cf. Murakami[32]). Assumption 3 is equivalent to
F 0 (0) < 0.
Assumption 4, together with Assumption 3, implies thatlim F (x) = ∞,x→∞

16 In this paper, the unique equilibrium is said to be “locally asymptotically totally unstable” if it is an unstable node or an unstablefocus.

9and that F 0 (x) = 0 has exactly two real roots x = re − r∗ < 0 and x = re − r∗ > 0 with

F (x) < 0.

The following theorem ensures the existence, uniqueness and stability of a limit cycle in the deterministic case.
Theorem 1. Let Assumptions 1-4 hold. Assume that α is sufficiently large and that σe = σd = 0. Then, for every
initial condition (x(0), y(0)) ∈ D \ (0, 0), there exists a unique solution path of Model (K), (x(t), y(t)), for all t ≥ 0,
and it converges to a unique and stable limit cycle on D as t → ∞, where17

D = {(x, y) ∈ R2 : x > −r∗ }.

Proof. See Murakami [32].
The unique and stable limit cycle established in Theorem 1 can be taken as the unique growth cycle in theeconomy without stochastic disturbances. Thus, this theorem implies that the economy, in the absence of random
shocks, almost surely undergoes persistent cyclical fluctuations along the convergence to the unique growth cycle.
This consequence may be interpreted to mean that persistent business cycles are inevitable in capitalist economies,
characterized by frequent revisions of prospected profits, and taken as a strong theoretical support for Keynes’ [20]view on business cycles (cf. Murakami [32]).

3.4

Small stochastic disturbances

To examine in detail the influence of stochastic disturbances on the unique limit cycle, we consider the case in which
the sizes of them, σe and σd , are small enough. Specifically, we assume that they can be written as follows:

σe = σ̃e ,σd = σ̃d ,where σ̃e and σ̃d are positive constants;  is a small positive constant. In this case, Model (K) reduces to thefollowing:dx = [φ(y) − F (x)]dt + σ̃e dze ,(21)

dy = −g(x)dt + σ̃d dzd .

(22)

In what follows, the system of equations (21) and (22) with (18)-(20) is denoted by Model (K*).
17 It is seen from (14) and (15) that (x, y) ∈ D is equivalent to (r e , a) ∈ R2 .
++

10

3.5

Behavior near the unique limit cycle

We introduce local coordinates in a neighborhood of the unique limit cycle for Model (K*) to explore the behaviorof solution paths in the neighborhood.18 To this end, denote the unique limit cycle by Γ and let (x∗ (t), y ∗ (t)) and
T > 0 be the solution path and (smallest) period of Γ (as in Proof of Theorem 1), respectively. We make thefollowing transformation in a neighborhood of Γ:x = x∗ (θ) + g(x∗ (θ))R,(23)

y = y ∗ (θ) + [φ(y ∗ (θ)) − F (x∗ (θ))]R.

(24)

Noting that (x − x∗ , y − y ∗ ) is orthogonal to the vector field along Γ or (φ(y ∗ ) − F (x∗ ), −g(x∗ )), it can be seenthat θ is the phase (time) of the point on Γ nearest to (x, y) and that the absolute value of R corresponds to the“distance” between (x, y) and Γ.19 Thus, every point (x, y) near the limit cycle Γ can be represented by the “phase”
θ and “amplitude” R.
It is seen from Bonnin [8, pp. 638-639, Theorem 1] that the following stochastic differential equations of θ andR can be derived from (16), (17), (23) and (24):dθ = [1 + aθ (θ, R) + 2 ãθ (θ, R)]dt + [σ̃e b̃e (θ, R)dze + σ̃d b̃d (θ, R)dzd ],(25)

dR = [L(θ)R + aR (θ, R) + 2 ãR (θ, R)]dt + [σ̃e be (θ, R)dze + σ̃d bd (θ, R)dzd ],(26)

where L, aθ , aR , ãθ and ãR are continuously differentiable and be , bd , b̃e and b̃d are twice continuously differentiable(due to Assumption 1) withdln([φ(y ∗ (θ)) − F (x∗ (θ))]2 + g 2 (x∗ (θ))).
dθ∂aRaθ (θ, 0) = aR (θ, 0) =(θ, 0) = 0,∂Rg(x∗ (θ)),
be (θ, 0) =∗[φ(y (θ)) − F (x∗ (θ))]2 + g 2 (x∗ (θ))φ(y ∗ (θ)) − F (x∗ (θ))
bd (θ, 0) =.
[φ(y ∗ (θ)) − F (x∗ (θ))]2 + g 2 (x∗ (θ))L(θ) = −F 0 (x∗ (θ)) −

(27)
(28)
(29)
(30)

The behavior of solution paths of Model (K*) near the unique limit cycle Γ can be described by (25) and (26).
18 In Andronov et al. [2, chap. V, sect. 7] and Hale [12, chap. VI], similar methods can be found for analyzing the behavior of solution
paths near periodic orbits in ordinary differential equations. Our method, following Bonnin [8], may be taken as a stochastic differential
equation version of them.
19 Note that the actual distance between (x, y) and Γ, denoted by ρ, is given by

|R|
ρ= p
.
φ(y ∗ (θ)) − F (x∗ (θ))]2 + g 2 (x∗ (θ))

11

3.6

Approximation of solution paths

We consider the deviation of R around the unique limit cycle Γ. Since R = 0 on Γ due to (23) and (24), we may
write

R = 0 + R̃ = R̃.

(31)

Then, equations (25) and (26) can be reduced to the following stochastic differential equations of θ and R̃:dθ = [1 + aθ (θ, R̃) + 2 ãθ (θ, R̃)]dt + [σ̃e b̃e (θ, R̃)dze + σ̃d b̃d (θ, R̃)dzd ],ih1dR̃ = L(θ)R̃ + aR (θ, R̃) + ãR (θ, R̃) dt + σ̃e be (θ, R̃)dze + σ̃d bd (θ, R̃)dzd .
It is seen from (28) that letting  → 0, they can be approximated as follows:20

dθ = dt,(32)

dR̃ = L(θ)R̃dt + σ̃e be (θ, 0)dze + σ̃d bd (θ, 0)dzd .

(33)

Solving (32) and (33), we can establish the following fact on solution paths near the unique limit cycle.
Theorem 2. Let Assumptions 1-4 hold. Assume that α is sufficiently large and that  is sufficiently small. Assumingthat θ(0) = 0 (without loss of generality), solution paths of Model (K*) in a neighborhood of the unique limit cycle
can be approximated as follows:

θ(t) = t,Z t
R(t) = R(0)P (t) exp(λt) + σ̃e

P (t)exp(λ(t − s))be (t, 0)dze + σ̃d
P
(s)
0

(34)
Z t

P (t)
exp(λ(t − s))bd (t, 0)dzd ,
P
(s)
0

(35)

where21
1
λ=−
T

Z T

F 0 (x∗ (θ))dθ,

(36)

0

Z t

P (t) = exp[L(θ) − λ]dθ0Z t
t Z T[φ(y ∗ (0)) − F (x∗ (0))]2 + g 2 (x∗ (0))
0 ∗
0 ∗
=
exp
F
(x
(θ))dθ
−F(x(θ))dθ
.
[φ(y ∗ (t)) − F (x∗ (t))]2 + g 2 (x∗ (t))
T 0
020 Note, in particular, that due to (28),

i
1
1h
∂aR
1
aR (θ, R̃) =aR (θ, 0) +(θ, 0) + o() = o() → 0,∂Ras  → 0.
21 It is seen from (27), (34), (36) and (37) that P (t) > 0 and P (t + T ) = P (t) for all t ≥ 0.

12

(37)

Proof. The solution of (32) with θ(0) = 0 is obviously (34). After substitution of (34), we can solve (33) to obtain
Z t
R̃(t) = R̃(0)P (t) exp(λt) + σ̃eP (t)exp(λ(t − s))be (θ(t), 0)dze + σ̃d
P
(s)
0

Z t

P (t)
exp(λ(t − s))bd (θ(t), 0)dzd .
P(s)0This implies (35) due to (31).
Theorem 2 presents an explicit (approximation) formula for solution paths near the unique limit cycle. Sincesuch a formula has not been derived in the related literature, this theorem may be taken as one of the notable
contributions of our study.22

3.7

Stationary distribution for the unique limit cycle

We derive the (approximated) stationary distribution of solution paths starting on the unique limit cycle. It follows
from (32) and (33) thatdE[R̃] = L(θ)E[R̃]dθ,

(38)

Z θ

E[R̃] = R̃(0) exp
L(θ0 )dθ0 = R̃(0)P (θ) exp(λθ).

(39)

With θ(0) = 0, this be solved as follows:0This is the (approximated) mean of R̃.
It is seen from (32), (33) and (38) thatd(R̃ − E[R̃]) = L(θ)(R̃ − E[R̃])dθ + σ̃e be (θ, 0)dze + σ̃d bd (θ, 0)dzd .

It follows from Ito’s formula thatd(R̃ − E[R̃])2 = {2L(θ)(R̃ − E[R̃])2 + B(θ)}dθ + σ̃e be (θ, 0)dze + σ̃d bd (θ, 0)dzd ,which implies thatdVar[R̃] = {2L(θ)Var[R̃] + B(θ)}dθ,(40)

22 In Bashkirtseva et al. [6], Bashkirtseva et al. [7], Bashkirtseva et al. [4] and Jungeilges and Ryazanova [15, 16], the stochastic
sensitivity approach of Bashkirtseva and Ryashko [5] is employed to discuss the (approximated) stationary distribution of deterministic
stable limit cycles (for the stationary distribution in Model (K*), see Theorem 3), but explicit (approximated) solutions are not derived.

13whereB(θ) = σ̃e2 b2e (θ, 0) + σ̃d2 b2d (θ, 0) =σ̃e2 g 2 (x∗ (θ)) + σ̃d2 [φ(y ∗ (θ)) − F (x∗ (θ))]2.
{[φ(y ∗ (θ)) − F (x∗ (θ))]2 + g 2 (x∗ (θ))}2(41)

Thus, we havesZ θσ(θ) ≡ Var[R̃] = Z θB(θ0 ) exp 2
L(θ00 )dθ00 dθ0 =
θ0

0

s

Z θ
0

B(θ0 )Z θ

P 2 (θ)exp(2λ(θ − θ00 ))dθ00 dθ0 .
2 00
θ 0 P (θ )(42)This is the (approximated) variance of R̃.
The following theorem characterizes the (approximated) stationary probability distribution of solution paths ofModel (K*) starting on the unique limit cycle Γ.
Theorem 3. Under the hypotheses of Theorem 2, the stationary probability density function for solution paths ofModel (K*) with the initial conditions on the unique limit cycle (i.e., with (x(0), y(0)) ∈ Γ), denoted by p∗ (θ, R),
can be approximated as follows:p∗ (θ, R) = √


1
R2 
exp − 2 2.
2 σ (θ)
2πσ(θ)(43)

Proof. It can be found from (39)-(42) that if (x(0), y(0)) ∈ Γ, i.e., if R(0) = 0 or R̃(0) = 0, the stationary distributionof (θ, R̃) can be characterized by the following probability density function:23p(θ, R̃) = √


R̃2 
1
exp − 2,2σ (θ)
2πσ(θ)

because E[R̃] = 0. Then, the probability density function (43) follows from (31).
Theorem 3 provides the distribution of the unique growth cycle (limit cycle) when the economy is exposedto small random shocks (stochastic disturbances). In this sense, the stationary probability density function p∗
evaluates the impact of external shocks and the robustness of (deterministic) business cycles.24
23 This function can be obtained as a stationary solution of the Fokker-Planck equation for (32) and (33):

∂p
∂p
∂
1 ∂2
(θ, R̃, t) = − (θ, R̃, t) −[L(θ)p(θ, R̃, t)] +[B(θ)p(θ, R̃, t)].
∂t∂θ
2 ∂ R̃2∂ R̃
For details, see Bonnin [8].
24 In the stochastic sensitivity function approach of Bashkirtseva and Ryashko [5], the stationary probability density function of stable
limit cycles, corresponding to our p∗ (θ, R), is discussed under the restriction of p∗ (θ + T, R) = p∗ (θ, R) because the variance Var[R̃(θ)],
as a solution of (40), is evaluated under the constraint of Var[R̃(T )] = Var[R̃(0)]. The difference between their approach and ours is due
to whether the phases θ and θ + T are identified or not; in our Theorems 2 and 3, they are distinguished.

14

4

Numerical analysis

This section performs numerical simulations to confirm the analytical results in Sect. 3. To this end, we specify
the parameter values and the capital formation function based on the empirical data in Japan.25

4.1

Setup

Based on the average value of the ratio of aggregate saving S to aggregate profits Π, we set the propensity to saves to26s = 0.7.

(44)

To formulate the capital formation function f consistent with Assumptions 1-4, we assume that it is written in
the following logistic form:27
I
βl + βu exp(β0 + βre )= f (re ) =,K
1 + exp(β0 + βr re )where β0 and βr are, respectively, a real constant and a positive constant; βl and βu are a real constant and anonnegative constant with βl < βu . Note that the upper and lower limits of f are βu and βl , respectively. It canthus be seen that
 I/K − β lln= β0 + βr r e .
βu − I/K
For estimation, we may write (I/K) − β tl= β0 + βr rte .
lnβu − (I/K)t(45)

In (45) and what follows, t represents the year of the data.
Since the expected rate of profit re is unobservable, we conduct estimations for (45) using the rate of profitr = Π/K, which is observable. For this purpose, we write a discrete version of (7), without stocahastic disturbances,in the following form:eerte − rt−1
= α(rt−1 − rt−1),
25 For details on the data, see Appendix.
26 It is seen from (3) that s can be obtained from S and Π alone. The empirical average of S/Π is 0.718.
27 For logistic capital formation functions, it is possible to provide a theoretical foundation from a viewpoint of econophysics (cf.

Murakami [30, 31]).

15

It then follows from (45) thatln

 (I/K) − β tlβu − (I/K)t (I/K)
t−1 − βl− (1 − α) ln= α(β0 + βr rt−1 ),βu − (I/K)t−1(46)

which implies that
I
K

=
t

βl [βu − (I/K)t−1 ]1−α + βu exp(α(β0 + βr rt−1 ))[(I/K)t−1 − βl ]1−α.
[βu − (I/K)t−1 ]1−α + exp(α(β0 + βr rt−1 ))[(I/K)t−1 − βl ]1−α(47)

We estimate the parameters in (47) by way of the method of nonlinear least squares.28 The estimates are given
as follows:29(α̂, β̂0 , β̂r , β̂l , β̂u ) = (0.59683, −2.03917, 4.90721, −0.08947, 0.61235).
Then, we set the parameters as follows:(α, β0 , βr , βl , βu ) = (0.6, −2.0, 5.0, −0.08, 0.6).

(48)

The capital formation function can thus be formalized as follows:f (re ) =−0.08 + 0.6 exp(−2.0 + 5.0re ).
1 + exp(−2.0 + 5.0re )(49)

Based on the average of the ratio of aggregate depreciation of capital to aggregate stock of capital,30 we set therate of capital depreciation δ to

δ = 0.1.

(50)

Finally, we set the mean rate of change in autonomous demand µ to

µ = 0.03.

(51)

It can be confirmed that Assumptions 1-4 are all satisfied with (44) and (49)-(51).
In what follows, Model (K*) with (44) and (48)-(51) is denoted by “Model (NK*).”
28 To determine the parameter values, it is possible to estimate (46) by the method of maximum likelihood. But the computationalload is much higher in this method.
29 For the nonlinear least squares estimation, we set the initial values to(α, β0 , βr , βl , βu ) = (0.6, −2.8, 8.2, 0, 0.4).
30 The empirical average of this ratio is 0.127.

16

4.2

Simulations

We first confirm that Model (NK*) possesses a (deterministic) unique limit cycle if  = 0. The red and blue curves
in figure 2 illustrate the solution paths of Model (NK*) with  = 0, respectively, for (x(0), y(0)) = (0.1, 0) and for(x(0), y(0)) = (−0.2, 0). It can be seen that there exists a unique limit cycle in Model (NK*) with  = 0.

y
1

-0.2

0.1

-0.1

0.2

x

-1

-2

-3

-4
Figure 2: Model (NK*) with  = 0

We then examine Model (NK*) in the case of  > 0. In figure 3, the blue curve describes a sample solution
path of Model (NK*) with σ̃e = σ̃d = 0.01 for (x(0), y(0)) = (0, −1.863703), which lies on the unique limit cycle
illustrated by the dashed black curve. In figures 4 and 5, respectively, the blue curves describe the time series of x
and y along the same sample solution path, and the area enclosed by the two red curves represent the approximated
95% confidence intervals of x and y calculated by (14), (15) and (43) in Theorem 3, while the dashed black curves
correspond to the unique limit cycle paths. It is seen that the prediction based on our analysis shows a reliable
performance.

17

y

0.5

-0.2

0.1

-0.1

0.2

x

-0.5
-1.0
-1.5
-2.0

Figure 3: Model (NK*) with σ̃e = σ̃d = 0.01

x
0.6
0.4
0.2
10

20

30

40

50

-0.2
-0.4

Figure 4: Sample path and 95% confidence interval of x

18

t

y
0.5
10

20

30

40

50

t

-0.5
-1.0
-1.5
-2.0
-2.5
Figure 5: Sample path and 95% confidence interval of y

5Conclusion

This paper has evaluated the impact of stochastic disturbances on deterministic limit cycles in a stochastic postKeynesian model of growth cycles. We have employed a new method for the evaluation, different from those adopted
in the related literature, and presented an explicit approximation formula for the stochastic model. By so doing, wehave attempted to integrate the endogenous and exogenous approaches in the theory of business cycles. We hopethat our analysis is helpful for better understanding of business cycles.

Acknowledgments
This work was supported by Chuo University Personal Research Grant and JSPS KAKENHI (Grant Numbers
18K12748 and 21K13262).

Appendix: Data
For the estimation of the propensity to save s, we use the data taken from Annual Report on National Accounts,
Cabinet Office, Japan, for the period of FY1980-FY2020. Aggregate gross saving S and aggregate gross profits Π
are defined as follows:
S = “Saving, gross”,
Π = “Operating surplus and mixed income” + “Consumption of fixed capital”.
For the estimation of the capital formation function f, we use the data taken from Financial Statements Statics
of Corporations by Industry, Ministry of Finance, Japan, for the period of FY1962-FY2020. Aggregate stock of19

capital K, aggregate gross investment I and aggregate profits Π are defined as follows:
K = the average of “Other tangible fixed assets” + “Construction in progress” at the beginning and end of theperiod,I = the change in “Other tangible fixed assets” + “Construction in progress” during the period + “Depreciation
expenses”,
Π = “Operating profits” + “Depreciation expenses”.

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