Discussion Paper No.281

A two-sector Keynesian model of business cycles

Hiroki Murakami
Faculty of Economics, Chuo University

July 2017

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, JapanA two-sector Keynesian model of business cycles
Hiroki Murakami∗
Faculty of Economics, Chuo University†
July 5, 2017

Abstract
In this paper, we examine the effect of sectoral interactions on business cycles in a simple Keynesian model.
As a first step for introducing viewpoints of multiple sectors in the context of business cycles, we consider adual economy in which there are only two kinds of goods: the consumption good and the investment good. By
examining a two-sector Keynesian model, we intend to take a look at some phenomena induced by interactions
between the consumption good sector and the investment good sector, which cannot be observed in one-sector
models. We then find that the stability of equilibrium and the possibility of emergence of a periodic orbit
depend upon whether the Keynesian stability condition holds or not and that the consumption good sector lags
behind the investment good sector along the periodic orbit (business cycles). Also, we supplement the analysis
by performing numerical simulations.

Keywords: Business cycles; Keynesian economics; Quantity adjustment; Two-sector analysis
JEL classification: E12; E32; E37

1

Introduction

It is no exaggeration to say that one of the main goals of macroeconomic studies is to explain the mechanismof business cycles. Soon after the basis of macroeconomics was established by Keynes’ General Theory, a lot oftheories of business cycles were proposed from the late 1930s to the 1950s. For example, Kalecki (1935, 1937) andKaldor (1940) put forward models of business cycles by synthesizing the Keynesian multiplier theory and the profit
principle of investment, while Harrod (1936), Samuelson (1939), Metzler (1941), Hicks (1950) and Goodwin (1951)
initiated the so-called multiplier-accelerator model of business cycles by combining the multiplier theory and theacceleration principle of investment.1 2 These classic models of business cycles can be characterized by the following∗ 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 they can be included in the classic works of theKeynesian theory of business cycles because their models were based upon the Keynesian principle of effective demand.
2 The profit principle of investment states that investment demand is determined by the level of income (or of profit) and by thevolume of capital stock, while the acceleration principle means that investment demand is determined by changes in the level of income.
These two principles are often confused with each other, but they are different.

1

“Keynesian” features: (i) quantity or income adjustment governed by the principle of effective demand prevails:3(ii) variations in investment are the main source of business cycles.4 5 In their models, the mechanism of businesscycles is explained as follows: investment is linked to aggregate income and capital stock and aggregate income andcapital stock are varied through the multiplier process and capital formation induced by investment.
It is true that a lot of models of business cycles, including the aforementioned ones, can describe some aspectsof actual business cycles, but a certain important viewpoint is missing in them: the role of sectoral interactions
in business cycles. This aspect is lacking in one-sector or one-commodity models of business cycles, but it should
not be ignored in discussing actual business cycles. It goes without saying that propagations of shocks from one
industry to another do enhance economic fluctuations in reality. Unfortunately, there have been only a few attempts
made to study the role of sectoral interactions in business cycles from theoretical points of view.6 7 To understandthe mechanism of actual business cycles in depth, it is necessary to present a theoretical foundation for multi-sector
analysis of business cycles.
The purpose of this paper is to examine the effect of sectoral interactions on business cycles in a simple Keynesianmodel. As a first step for introducing viewpoints of multiple sectors in the context of business cycles, we consider
a dual economy in which there are only two kinds of goods: the consumption good and the investment good. By
examining a two-sector Keynesian model, we intend to take a look at some phenomena induced by interactions
between the consumption good sector and the investment good sector, which cannot be observed in one-sector
models.
This paper is organized as follows. In Section 2, we set up a two-sector model by decomposing the economy
into the consumption-good sector and the investment-good sector. This two-sector model describes the dynamics
of the quantities of output and stocks of capital of the two sectors. In Section 3, we study the characteristics ofthe two-sector model. In particular, we see that the stability of equilibrium and the possibility of emergence of aperiodic orbit depend upon whether the Keynesian stability condition holds or not and that the consumption good
sector lags behind the investment good sector along the periodic orbit (business cycles). In Section 4, we perform
numerical simulations to confirm the validity of our analysis. In Section 5, we summarize our analysis and conclude
this paper.
3 We mean by the term “quantity adjustment” that firms adjust the quantity of their products so that it would be equal to thedemand for their products. This is what the principle of effective demand (Keynes, 1936, chap. 3) implies. The opposite concept is“price adjustment,” which means that firms respond to excess demand or supply by raising or lowering the price of their products.
4 Keynes (1936, chap. 22) attributed the main cause of business cycles to violent fluctuations of the marginal efficiency of investment(i.e., profitability of investment).
5 Precisely speaking, Metzler (1941) emphasized variations in inventory investment rather than those in fixed investment as a causeof economic fluctuations.
6 In the context of international trade, Lorenz (1987a, 1987b), Asada et al. (2001), Asada et al. (2003), Asada (2004) and Chiarella
and Flaschel (2000) presented multi-country open-economy Keynesian models of business cycles. However, their concern was withinteractions between countries, not with those between domestic sectors.
7 In the field of economic growth, Okishio (1967, 1968) extended the Harrod-Domar model (e.g., Harrod 1939; Domar 1946) totwo-sector ones, while Dutt (1987), Lavoie and Ramı́rez-Gastón (1997), Franke (2000) and Nishi (2014) extended the Kaleckian model(e.g., Kalecki 1971; Asimakopulos 1975; Rowthorn 1981) to two-sector ones, but they only focused on the stability or instability of
economic growth and did not examine the possibility of business cycles. In the neoclassical two-sector model of economic growth (e.g.,
Shinkai 1960; Uzawa 1961-1962, 1963), on the other hand, Inada (1963) and Stiglitz (1967) examined the possibility of emergence of
cyclical fluctuations, but their studies are based upon the full-employment assumption, and so cycles in their models cannot correspondto actual business cycles.

2

2

The model

In this section, we set up a two-sector Keynesian model of business cycles.
We suppose that there are only two kinds of goods that differ in property and name each of them the consumption
good and the investment good, respectively.8 The consumption good is supposed to be used for consumption
alone and not for production, while the investment good is assumed to be used for production alone and not for
consumption.9 We also suppose for simplicity that the prices of the consumption good and of the investment good
are fixed.10
First, we consider the demand for the consumption good. We assume that the demand for the consumption good
C is dependent upon both the income of the consumption good and that of the investment good.11 Specifically, C
is assumed to be represented by12

C = C(yc , yi ),(1)

where C is twice continuously differentiable with

0 < Cyc ≡∂C∂C< 1, 0 < Cyi ≡
< 1.
∂yc
∂yi

(2)

In (1), yc and yi stand for the income or output of the consumption good and that of the investment good,
respectively. Condition (2) means that the marginal propensity to consume with respect to each sector’s income
Cyc or Cyi lies between 0 and 1.
Second, the output (supply) of the consumption good yc is assumed to be varied in response to the existing
8 Traditionally, the consumption good and the investment good have been interpreted as the output in the agricultural sector andthat in the manufacturing sector, respectively, but in a modern way of thinking, the consumption good may be regarded as the output
in the service sector.
9 It is possible to extend our analysis to a more general situation in which the investment good is also enjoyed for consumption. For
simplicity, however, we make the above assumption about the investment good.
10 It is quantity or income adjustment governed by Keynes’ (1936, chap. 3) principle of effective demand that is the most distinguishedfeature of the Keynesian theory. This point was stressed by Leijonhufvud (1968) as follows:

In [neoclassical] general equilibrium flow models, prices are the only endogenous variables which enter as arguments
into the demand and supply functions of individual households. Tastes and initial resource endowments are parametric.
In “Keynesian” flow models the corresponding arguments are real income and the interest rate. Of these, real income is ameasure of quantity, not of prices. On a highly abstract level, the fundamental distinction between general equilibrium andKeynesian models lies in the appearance of this quantity variable in the excess demand relation to the latter. The differenceis due to the assumptions made about the adjustment behavior of the two systems. In the short run, the “[Neo]Classical”
system adjusts to changes in money expenditures by means of price-level movements; the Keynesian adjusts primarily by
way of real income movements. (p. 51)
In the Keynesian theory, quantity or income, rather than prices, plays the primary role in dynamic adjustment (see also Tobin (1993)
or Yoshikawa (1984)). To focus on the importance of quantity adjustment, price adjustment is ignored in our analysis.
11 Since the prices of the consumption good and of the investment good are assumed to be fixed, we can use the terms “income” and“output” interchangeably.
12 If consumption demand can be assumed to depends upon aggregate income, the consumption function may be expressed as
C = C(yc + yi ),with
0 < Cy ≡∂C< 1.
∂(yc + yi )Equation (1) is, of course, a more general formulation than this.

3

excess demand or supply. Specifically, yc is assumed to obey the following process:ẏc = αc [C(yc , yi ) − yc ],(3)where αc is a positive parameter which stands for the speed of adjustment. It is assumed in (3) that the adjustmentof yc is proportional to the existing excess demand or supply, represented by C(yc , yi ) − yc . Equation (3) expresses
the Keynesian quantity adjustment process in the consumption good sector.13
Third, we take a look at the demand side of the investment good. For each sector, the consumption good sector
or the investment good sector, the demand for the investment good (including the replacement demand for theinvestment good) is assumed to be dependent upon the output of the sector and upon the existing stock of theinvestment good (i.e., the existing stock of capital) of the sector. Specifically, we assume that the gross investment
functions of the consumption good sector Ic and of the investment good sector Ii are represented by

Ic = Ic (yc , kc ),(4)

Ii = Ii (yi , ki ),(5)where Ic and Ii are twice continuously differentiable with∂Ic
∂Ic
< 1, Ikc ≡
< 0,
∂yc
∂kc
∂Ii
∂Ii
< 1, Iki ≡< 0.
0 < Iyi ≡∂yi∂ki

0 < Iyc ≡(6)(7)

In (4) and (5), kc and ki denote the existing stocks of the investment good of the consumption good sector andof the investment good sector, respectively. Equations (4) and (5) with (6) and (7) are consistent with the “profit
principle” of investment.14 It is also assumed in (6) and (7) that the marginal propensity to invest with respect
to each sector’s income Iyc or Iyi lies between 0 and 1.15 In this paper, we assume for simplicity that the rate ofinterest r is fixed by the monetary authority. By so doing, we can omit the dependence of Ic or of Ii on r.16
Fourth, we turn to the output (supply) of the investment good yi . As in the consumption good sector, yi is
13 It is implicitly assumed that the existing excess demand or supply is absorbed as unintended inventory investment.
14 Strictly speaking, the profit principle argues that the more the rate of profit is, the more demand for investment is, but, under theassumption of fixed price and wage, the (gross) rate of profit for each sector can be shown to be increasing in the level of output anddecreasing in the stock of capital. The profit principle in this form was used first by Kalecki (1937) and Kaldor (1940). For discussions
on microeconomic foundation of this principle in the context of fixed investment, see Murakami (2016a), and, for that in the context of
R&D investment, see Murakami (2017b).
15 The usual profit principle does not require that the marginal propensity to invest be less than unity, but this assumption helps tosimplify our analysis, especially the stability analysis in Section 3.
16 We may include the rate of interest r in the investment functions I and I , but if r can be represented as a function of y and yccii(as the Keynesian liquidity preference theory implies) and the interest elasticity of liquidity preference is sufficiently large (as in theKeynesian liquidity trap), our argument shall not dramatically be changed.

4

assumed to be changed in response to the existing excess demand or supply in the following way:ẏi = αi [Ic (yc , kc ) + Ii (yi , ki ) − yi ],(8)where αi is a positive parameter which represents the speed of adjustment. Equation (8) is, of course, the Keynesianquantity adjustment process in the investment good sector.
Fifth, we describe the capital formation process of each sector. The demand for the investment good of eachsector Ic or Ii is assumed to be realized as the gross increment of the stock of the investment good in the sector kc
or ki , respectively, in the following way:k̇c = Ic (yc , kc ) − δkc ,(9)k̇i = Ii (yi , ki ) − δki ,(10)where δ is a positive constant which stands for the rate of capital depreciation. In (9) and (10), it is assumed that all
the existing excess demand (resp. supply) is absorbed as unintended inventory decumulation (resp. accumulation).17
Thus, our two-sector model can be completed as follows:ẏc = αc [C(yc , yi ) − yc ],(3)

ẏi = αi [Ic (yc , kc ) + Ii (yi , ki ) − yi ],(8)k̇c = Ic (yc , kc ) − δkc ,(9)k̇i = Ii (yi , ki ) − δki .

(10)

In what follows, the system of equations (3), (8), (9) and (10) is simply called “System.” The structure of System
is similar to that of Kaldor’s (1940) model,18 but the notable difference from his is that our System consists of two
sectors.
Since System deals with medium-run economic fluctuations but not with long-run economic growth, the levels
of income and capital stocks of both sectors, yc , yi , kc and ki , can be assumed to be bounded.19 For this reason,
17 More generally, the demand for the investment good of the investment good sector may not be realized as net capital accumulation
due to the existence of excess demand or supply. To describe this more general situation, it may be better to formalize the capitalformation process in the investment good sector as follows:k̇i = Ii (yi , ki ) − δki − θ[Ic (yc , kc ) + Ii (yi , ki ) − yi ],where θ ∈ [0, 1]. This kind of formalization can be found in, for instance, Stein (1969) or Fischer (1971). For the sake of simplicity inanalysis, however, equation (10) is adopted in our analysis.
18 For extensions of Kaldor’s (1940) model of business cycles (in continuous-time formalizations), see, for example, Chang and Smyth(1971), Varian (1979), Semmler (1986, 1987), Asada (1987, 1995, 2004), Lorenz (1987), Skott (1989), Chiarella and Flaschel (2000),
Asada et al. (2003), Chiarella et al. (2013) or Murakami (2014, 2015, 2016b).
19 In reality, the levels of income and capital stocks of both sectors grow with some trend due to population growth and technicalprogress, which implies that they cannot be considered to be bounded. If our variables yc , yi , kc and ki can be viewed as cyclicalcomponents (or detrended versions) of their actual time series, however, we can, without so much difficulty, extend our analysis even in
the context of long-run economic growth.

5we can restrict the domain of the variables in System as follows:D ≡ {(yc , yi , kc , ki ) ∈ R4++ : yc ∈ [y c , y c ], yi ∈ [y i , y i ], kc ∈ [k c , k c ], ki ∈ [k i , k i ]},where xj and xj are positive constants with xj < xj for x = y, k and for j = c, i. For x = y, k and for j = c, i,we may regard xj and xj as the maximum and minimum values of xj in the medium-run context. In this sense,
the compact domain D can be viewed as the set of combinations of yc , yi , kc and ki economically feasible in themedium run.20
For convenience, we summarize the assumptions made thus far as follows.
Assumption 1. The real valued functions C, Ic and Ii are twice continuously differentiable everywhere on D andthe following conditions are satisfied everywhere on D:21

3

0 < Cyc < 1, 0 < Cyi < 1,

(2)

0 < Iyc < 1, Ikc < 0,

(6)

0 < Iyi < 1, Iki < 0.

(7)

Analysis

In this section, we analyze System, which consists of (3), (8), (9) and (10). First, we verify the existence anduniqueness of equilibrium in System. Second, we examine the stability of this equilibrium. Third, we explore thepossibility of persistent business cycles being generated.

3.1

Existence and uniqueness of equilibrium

To begin, we define an equilibrium point of System. An equilibrium point of System is defined as a point at which
we have ẏc = ẏi = k̇c = k̇i = 0. We can easily see that an equilibrium point of System, (yc , yi , kc , ki ) ∈ D, is given
as a solution of the following simultaneous equations:

yc = C(yc , yi ),(11)

yi = Ic (yc , kc ) + Ii (yi , ki ),(12)δkc = Ic (yc , kc ),(13)

δki = Ii (yi , ki ).

(14)

20 For more detailed economic interpretations of a related compact domain, see Murakami (2014).
21 The assumption of twice-continuous differentiability is made for the application of the Hopf bifurcation theorem in our analysis.
For details on this theorem, see Marsden and McCracken (1976).

6

In order to establish the existence and uniqueness of an equilibrium point of System, we make the following
assumptions.
Assumption 2. The following conditions are satisfied:C(y c , y i ) − y c < 0 < C(y c , y i ) − y c ,(15)

Ic (y c , k c ) + Ii (y i , k i ) − y i < 0 < Ic (y c , k c ) + Ii (y i , k i ) − y i ,(16)

Ic (y c , k c ) − δk c < 0 < Ic (y c , k c ) − δk c ,(17)

Ii (y i , k i ) − δk i < 0 < Ii (y i , k i ) − δk i .

(18)Assumption 3. The following conditions is satisfied everywhere on D:(1 − Cyc )Ikc Iki > [(1 − Cyc )(1 − Iyi )Ikc + (1 − Cyc − Cyi Iyc )Iki ]δ − [(1 − Cyc )(1 − Iyi ) − Cyi Iyc ]δ 2 .

(19)

Regarding Assumption 2, conditions (15)-(18) may seem technical, but it is possible to provide economic interpretations for them under Assumption 1. Indeed, Assumption 1 and conditions (15)-(18) imply that for every(yc , yi , kc , ki ) ∈ D

C(y c , yi ) − y c ≤ C(y c , y i ) − y c < 0 < C(y c , y i ) − y c ≤ C(y c , yi ) − y c ,(20)

Ic (yc , kc ) + Ii (y i , ki ) − y i ≤ Ic (y c , k c ) + Ii (y i , k i ) − y i < 0 < Ic (y c , k c ) + Ii (y i , k i ) − y i ≤ Ic (yc , kc ) + Ii (y i , ki ) − y i ,(21)
Ic (yc , k c ) − δk c ≤ Ic (y c , k c ) − δk c < 0 < Ic (y c , k c ) − δk c ≤ Ic (yc , k c ) − δk c ,(22)

Ii (yi , k i ) − δk i ≤ Ii (y i , k i ) − δk i < 0 < Ii (y i , k i ) − δk i ≤ Ii (yi , k i ) − δk i .

(23)

Condition (20) or (21) means that, when income of each sector yc or yi is sufficiently large (resp. sufficiently small)or at the maximum y c or y i (resp. at the minimum y c or y i ), this sector is in excess supply (resp. in excess demand);condition (22) or (23) means that, when capital stock of each sector kc or ki is sufficiently large (resp. sufficiently
small) or at the maximum k c or k i (resp. at the minimum k c or k i ), net investment in this sector is negative (resp.
positive). Conditions (20) and (21) may be consistent with the reality in that the propensity to consume and thatto invest tend to decline as the level of income rises (cf. Keynes 1936, chap. 10), while conditions (22) and (23)can be seen as realistic because net investment demand may be considered to be positive (resp. negative) whenthe volume of capital stock is sufficiently small (resp. sufficiently large). Therefore, Assumption 2 can be justified
from economic viewpoints. Moreover, mathematically, conditions (20)-(21) imply that the compact domain D ispositively invariant with respect to System.22 This property is helpful for verifying the existence of an equilibrium
22 A (nonempty) closed domain is said to be positively invariant if every solution of the dynamical system under consideration whichstarts from this domain at t = 0 will remain in the same domain for all t ≥ 0.

7

point of System.23
As for Assumption 3, condition (19) is satisfied under Assumption 1 if the constant rate of capital depreciation
δ is sufficiently small. Since δ is annually around 0.09 in reality (in Japan), condition (19) is highly likely to hold.
In this respect, Assumption 3 is realistic.
Now we are ready to establish the existence and uniqueness of a solution path and of an equilibrium point of
System.
Proposition 1. Let Assumptions 1-3 hold.
(i) For every initial condition (yc (0), yi (0), kc (0), ki (0)) ∈ D, there exists a unique solution path of System,(yc (t), yi (t), kc (t), ki (t)) ∈ D, for all t ≥ 0.
(ii) There exists a unique equilibrium point of System, (yc∗ , yi∗ , kc∗ , ki∗ ) ∈ D◦ , where D◦ denotes the interior of D.
Proof. See Appendix A.

3.2

Stability and instability

Next, we study the stability of System. For this purpose, we have a look at the Jacobian matrix, denoted by J ∗ ,evaluated at the unique equilibrium:





∗
J =




αc (Cy∗c − 1)

αc Cy∗i00

αi Iy∗c

αi (Iy∗i − 1)

Ik∗c

Ik∗i

Iy∗c

0

Ik∗c − δ

0

0

Iy∗i

0

Ik∗i − δ






,




where ∗ signifies the value evaluated at the unique equilibrium point of System. The characteristic equation
associated with J ∗ is given by

λ4 + a1 λ3 + a2 λ2 + a3 λ + a4 = 0,(24)

23 Assumption 2 may be viewed as “boundary conditions” for existence of equilibrium in our two-sector Keynesian model. For boundary
conditions in related one-sector Keynesian models, see, for instance, Murakami (2014, 2017a).

8wherea1 = (1 − Cy∗c )αc + (1 − Iy∗i )αi + 2δ − Ik∗c − Ik∗i > 0,
a2 = [(1 − Cy∗c )(1 − Iy∗i ) − Cy∗i Iy∗c ]αc αi + (1 − Cy∗c )(2δ − Ik∗c − Ik∗i )αc

(25)

(26)

+ [(1 − Iy∗i )(2δ − Ik∗c ) − Ik∗i ]αi + (δ − Ik∗c )(δ − Ik∗i ),a3 = {2[(1 − Cy∗c )(1 − Iy∗i ) − Cy∗i Iy∗c ]δ − (1 − Cy∗c )(1 − Iy∗i )Ik∗c − (1 − Cy∗c − Cy∗i Iy∗c )Ik∗i }αc αi(27)+ (1 − Cy∗c )(δ − Ik∗c )(δ − Ik∗i )αc + [(1 − Iy∗i )δ − Ik∗c ](δ − Ik∗i )αi ,a4 = {(1 − Cy∗c )Ik∗c Ik∗i(28)

− [(1 − Cy∗c )(1 − Iy∗i )Ik∗c + (1 − Cy∗c − Cy∗i Iy∗c )Ik∗i ]δ + [(1 − Cy∗c )(1 − Iy∗i ) − Cy∗i Iy∗c ]δ 2 }αc αi > 0.
The inequalities follow from Assumptions 1 and 3.
For the unique equilibrium of System to possess the local asymptotic stability, it is necessary and sufficient that
all the roots of the characteristic equation (24) have negative real parts. According to the Routh-Hurwitz criterion,
the following set of conditions is necessary and sufficient:

a1 > 0, a3 > 0, a4 > 0, (a1 a2 − a3 )a3 > a21 a4 ,which are, under Assumption 1, equivalent to24{[(1 − Cy∗c )(1 − Iy∗i ) − Cy∗i Iy∗c ]δ − (1 − Cy∗c )(1 − Iy∗i )Ik∗c − (1 − Cy∗c − Cy∗i Iy∗c )Ik∗i }αc αi(29)+(1 − Cy∗c )(δ − Ik∗c )(δ − Ik∗i )αc + [(1 − Iy∗i )δ − Ik∗c ](δ − Ik∗i )αi > 0,f (αc , αi )(≡ (a1 a2 − a3 )a3 − a21 a4 )= [(1 − Cy∗c )(1 − Iy∗i ) − Cy∗i Iy∗c ]{[(1 − Cy∗c )(1 − Iy∗i ) − Cy∗i Iy∗c ]δ − (1 − Cy∗c )(1 − Iy∗i )Ik∗c − (1 − Cy∗c − Cy∗i Iy∗c )Ik∗i }× [(1 − Cy∗c )αc + (1 − Iy∗i )αi ]αc2 αi2+ (1 − Cy∗c )3 (δ − Ik∗c )(δ − Ik∗i )(2δ − Ik∗c − Ik∗I )αc3
+ (1 − Iy∗i )[(1 − Iy∗i )δ − Ik∗c ][(1 − Iy∗i )(2δ − Ik∗c ) − Ik∗i ](δ − Ik∗i )αi3
+ b1 αc3 αi + b2 αc αi3 + b3 αc2 αi2 + b4 αc2 αi + b5 αc αi2 + b6 αc2 + b7 αi2 + b8 αc αi+ (δ − Ik∗c )(δ − Ik∗i )2 (2δ − Ik∗c − Ik∗i ){(1 − Cy∗c )(δ − Ik∗c )αc + [(1 − Iy∗i )δ − Ik∗c ]αi } > 0.

To discuss the stability of System, we impose the following realistic assumption.
24 The exact values of b for l = 1, ..., 8 are omitted because they are irrelevant to our analysis.
l

9

(30)

Assumption 4. The following condition is satisfied:

2[(1 − Cy∗c )(1 − Iy∗i ) − Cy∗i Iy∗c ]δ > (1 − Cy∗c )(1 − Iy∗i )Ik∗c + (1 − Cy∗c − Cy∗i Iy∗c )Ik∗i .

(31)

Condition (31) is not so restrictive under Assumption 1 because the right hand side is likely to be negative due
to Cyc , Cyi , Iyc , Iyi ∈ (0, 1) and Ikc , Iki < 0 and the rate of capital depreciation δ is sufficiently small (around0.09) in reality. Therefore, Assumption 4 can safely be made from realistic viewpoints. Note that condition (29) issatisfied under Assumptions 1 and 4.
As regards the local asymptotic stability, we can state the following fact.
Proposition 2. Let Assumptions 1-4 hold.
(i) The unique equilibrium point of System is locally asymptotically stable if αc and αi are sufficiently small.
(ii) The unique equilibrium point of System is locally asymptotically stable if αc is sufficiently large and αi issufficiently small.
(iii) The unique equilibrium point of System is locally asymptotically stable if αc is sufficiently small and αi issufficiently large.
(iv) Assume that the following condition is satisfied:(1 − Cy∗c )(1 − Iy∗i ) > Cy∗i Iy∗c .

(32)Then, the unique equilibrium point of System is locally asymptotically stable if αc and αi are sufficiently large.
(v) Assume that the following condition is satisfied:(1 − Cy∗c )(1 − Iy∗i ) < Cy∗i Iy∗c .

(33)Then, the unique equilibrium point of System is locally asymptotically unstable if αc and αi are sufficiently large.
Proof. See Appendix B.
Proposition 2 says that the (local) stability of System is gained when the quantity adjustments in both sectors
are relatively slow or when the quantity adjustment in one of the sectors is comparatively fast while that in theother is comparatively slow. This proposition also states that when the quantity adjustments in both sectors are
rapid enough, the (local asymptotic) stability of System rests upon whether condition (32) holds or not and thestability is lost if the reverse condition (33) is satisfied. Since the situation in which both αc and αi are large may
be taken as the Keynesian case (cf. Leijonhufvud 1968; Tobin 1993; Yoshikawa 1984), we may state that conditions(32) and (33) are, respectively, the stability condition and the instability one in our Keynesian model.
We can find that our stability results are related to the well-established Keynesian stability condition. As is well10

known, the Keynesian stability condition, which states that the sum of the marginal propensity to consume andthat to invest is less than unity, plays a vital role in stability in one-sector Keynesian models.25 In our two-sector
model, the Keynesian stability condition (at the equilibrium) may be written for each sector as follows:Cy∗c + Iy∗c < 1,

(34)

Cy∗i + Iy∗i < 1.

(35)

It is seen that condition (32) holds under Assumption 1 and these conditions. Then, it follows from Proposition
2 (iv) that the local asymptotic stability of System obtains for αc and αi sufficiently large. In other words, the“two-sector” version of Keynesian stability condition, (34) and (35), can assure the stability in our two-sector
System if the quantity adjustment is rapid enough in each sector. On the contrary, if the reverse inequalities hold
both in (34) and in (35), condition (33) is satisfied and the instability arises for αc and αi sufficiently large in
our two-sector System (cf. Proposition 2 (v)). Recently, some economists have cast doubt on the validity of theKeynesian stability condition and maintained that the reverse condition is realistic.26 Thus, chances are not low
that, in practice, the reverse inequalities may be fulfilled both in (34) and in (35) and that condition (33) holds. In
this way, the (counterpart of) Keynesian stability condition also plays a key role in the stability in our two-sector
Keynesian model.

3.3

Existence of a periodic orbit

Now we examine the phenomenon of the unique equilibrium of System turning from stable to unstable (or fromunstable to stable), i.e., that of “stability switching” occurring to System, through changes in the parameters αc
and αi .
It follows from the above argument that the stability switching occurs only if the following condition is satisfied:27f (αc , αi ) = 0.

(36)Furthermore, according to Asada and Yoshida (2003, p. 527, Theorem 3), a non-constant periodic orbit is generated
by a Hopf bifurcation if either of the following conditions as well as (36) holds:

fαc (αc , αi ) 6= 0,(37)

fαi (αc , αi ) 6= 0.

(38)

25 The term of “Keynesian stability condition” was coined by Marglin and Bhaduri (1990). To our best knowledge, Samuelson (1941,
p. 117) was the first economist who found that this condition ensures the (local asymptotic) stability in the Keynesian system.
26 For instance, Skott (2012) judged from his empirical study that the Keynesian stability condition is far from realistic.
27 Due to (28), the characteristic equation (24) cannot possess any zero root.

11

To discuss the possibility of Hopf bifurcations, we make the following assumption.
Assumption 5. The following condition is satisfied:(1 − Cy∗c )(1 − Iy∗i ) < Cy∗i Iy∗c .

(33)

According to Proposition 2 (v), condition (33) is a sufficient condition for the instability of System for αc andαi sufficiently large. As we have stated above, the validity of (33) may be verified on empirical grounds. In Section
4 and Appendix E, we shall see that condition (33) was empirically satisfied in Japan. In this respect, Assumption
5 can be justified.
Concerning the existence of a periodic orbit, we can establish the following fact.
Proposition 3. Let Assumptions 1-5 hold.
(i) Assume that αi is fixed at some sufficiently large value. Then, there exists at least one positive value αc∗ suchthat the unique equilibrium of System is locally asymptotically stable (resp. unstable) for αc < αc∗ (resp. αc > αc∗ )and that at least one (non-constant) periodic orbit is generated by a Hopf bifurcation for αc sufficiently close to αc∗ .
(ii) Assume that αc is fixed at some sufficiently large value. Then, there exists at least one positive value αi∗ suchthat the unique equilibrium of System is locally asymptotically stable (resp. unstable) for αi < αi∗ (resp. αi > αi∗ )and that at least one (non-constant) periodic orbit is generated by a Hopf bifurcation for αi sufficiently close to αi∗ .
Proof. See Appendix C.
Proposition 3 states that there can be a periodic orbit generated by Hopf bifurcations through changes in αc orin αi . Note that it does not tell us about the stability of the periodic orbit. As is well known, there is a criterion todiscern whether a Hopf periodic orbit is stable or not, i.e., whether a Hopf bifurcation is supercritical or subcritical(cf. Marsden and McCracken 1976, sect. 4). However, it is usually hard to derive economic implications fromthis criterion because it requires third order partial derivatives of the relevant functions, which usually do not haveeconomic meanings, and so we do not analytically investigate the periodic stability.28

3.4

Characteristics of the Hopf periodic orbit

We investigate the properties of a periodic orbit generated by the aforementioned Hopf bifurcation, which are simply
called a “Hopf periodic orbit” in what follows.
To begin, we calculate the period of a Hopf periodic orbit. When a Hopf bifurcation occurs to System, condition
28 There have been some attempts made to derive the condition for stability of a Hopf periodic orbit in economics. For example, Lorenz(1993), Matsumoto (2009) and Dohtani (2010) provided periodic stability conditions in their systems of two-dimensional differential
equations.

12(36) is fulfilled and the characteristic equation (24) has a pair of purely imaginary root ±iω with
r
ω=
s

=

a3
a1

∗ )(1−I ∗ )−C ∗ I ∗ ]δ−(1−C ∗ )(1−I ∗ )I ∗ −(1−C ∗ −C ∗ I ∗ )I ∗ }αc α +(1−C ∗ )(δ−I ∗ )(δ−I ∗ )αc +[(1−I ∗ )δ−I ∗ ](δ−I ∗ )α{2[(1−Cyiiyiyc
yi yc ki
yi kc
yc
yc
yi
yi yc
kc
ki
kc
ki
c
∗ )αc +(1−I ∗ )α +2δ−I ∗ −I ∗
(1−Cy
i
yi
kc
ki
c

.

According to the Hopf bifurcation theorem (cf. Marsden and McCracken 1976, sect. 5), the period of a Hopfperiodic orbit can be approximated by
T =

2π
ω
v
u
u

(39)

(1−C ∗ )α +(1−I ∗ )α +2δ−I ∗ −I ∗

c
i
yi
yc
kc
ki
t
= 2πu
.
{2[(1−C ∗ )(1−I ∗ )−C ∗ I ∗ ]δ−(1−C ∗ )(1−I ∗ )I ∗ −(1−C ∗ −C ∗ I ∗ )I ∗ }αc αi +(1−C ∗ )(δ−I ∗ )(δ−I ∗ )αc +[(1−I ∗ )δ−I ∗ ](δ−I ∗ )αi
yc

yi

yi yc

yc

yi

kc

yc

yi yc

ki

yc

kc

ki

yi

kc

ki

Next, we examine the movements of the variables yc , yi , kc and ki along Hopf periodic orbits. A simple way
to look at what happens along Hopf periodic orbits is to draw phase diagrams. In figure 1, the phase diagramof System is illustrated on the yc -yi plane with kc = kc∗ and ki = ki∗ (it is drawn under Assumptions 1-5). By
applying the conclusion of the Poincarè-Hopf theorem (cf. Milnor 1965, p. 35), we can find that the point (yc∗ , yi∗ )is, under Assumption 5, a saddle point on the yc -yi plane with kc = kc∗ and ki = ki∗ and that the loci of ẏc = 0 andof ẏi = 0 intersect at least two points other than (yc∗ , yi∗ ) on this plane (note that, letting the other point be, say,(yc∗∗ , yi∗∗ ), the point (yc∗∗ , yi∗∗ , kc∗ , ki∗ ) is not an equilibrium point of the four-dimensional System).29 Since (yc∗ , yi∗ )is a saddle stationary point, however, we cannot directly gain from this simple method some useful information on
the movements of yc and of yi along Hopf periodic orbits.
29 For k = k ∗ and k = k ∗ , the domain D ∗ ≡ {(y , y ) ∈ R2
c
c i
i
c
++ : yc ∈ [y c , y c ], yi ∈ [y i , y i ]} is positively invariant with respect to theisubsystem of (3) and (8). By the same logic as used in the proof of Proposition 1 (cf. Appendix A), we can see from the Poincarè-Hopftheorem that the sum of the indices of all the stationary points of (3) and (8) is +1 on D∗ . Since the index of (yc∗ , yi∗ ) is −1 because it
is a saddle point on the yc -yi plane, we can conclude that at least other two stationary points exist on D∗ .

13

yi
yi

0

yi*

0

yi
O yc

yc*

yc

yc

Figure 1: Phase diagram on yc -yi plane

Another way to get information on the properties of Hopf periodic orbits is to make use of the linearized versionof the original System. The linearized version of System, which is linearized around the unique equilibrium point(yc∗ , yi∗ , kc∗ , ki∗ ), can be written in the following form:ẏc = αc [(Cy∗c − 1)(yc − yc∗ ) + Cy∗i (yi − yi∗ )],(40)

ẏi = αi [Iy∗c (yc − yc∗ ) + (Iy∗i − 1)(yi − yi∗ ) + Ik∗c (kc − kc∗ ) + Ik∗i (yi − yi∗ )],(41)k̇c = Iy∗c (yc − yc∗ ) + (Ik∗c − δ)(kc − kc∗ ),(42)k̇i = Iy∗i (yi − yi∗ ) + (Ik∗i − δ)(ki − ki∗ ).

(43)

A Hopf bifurcation is a local phenomenon in our analysis, a Hopf periodic orbit of System can be approximated by
the solution path of the system of (40)-(43) with (36). In what follows, we denote the system of equations (40)-(43)with (36) by “System (L).”14

A (purely) periodic solution of System (L) can be written in the following form:30
 2π(t − θ ) yc,
yc = yc∗ + ayc sinT 2π(t − θ ) yiyi = yi∗ + ayi sin,T 2π(t − θ ) kc
kc = kc∗ + akc sin,T 2π(t − θ ) kiki = ki∗ + aki sin
,
T

(44)
(45)
(46)
(47)

where T is the constant number defined by (39); axj is a positive constant (amplitude) and θxj is a real constantwith θxj ∈ [−T /2, T /2], for x = y, k and j = c, i, where axj and θxj are dependent upon an initial condition.
To study the movements of variables along Hopf periodic orbits, we have a closer look at the phase differences(i.e., the differences in θxj ). By looking into the phase differences, we can gain helpful information about the leadand lag between the four variables yc , yi , kc and ki along Hopf periodic orbits. If we have θi < θc , for instance, we
may say from figure 2 that yi reaches its peak earlier than yc does along the periodic orbit of System (L), i.e., thatthe movement of yi precedes that of yc (or that the latter lags behind the former). Then, we can approximately
make a judgment on the lead-and-lag relationship between variables along Hopf periodic orbits of (the original)
System.

yc - yc*
yi - yi*

O

time

θi
θc

Figure 2: Phase difference

Concerning the phase differences, we can obtain the following proposition.
30 The general solution of System (L), x , for x = y, k and j = c, i, can be written in the form ofj

xj = x∗j + axj sin 2π(t − θx ) jT+ bxj eλ1 t + cxj eλ2 t ,where λ1 and λ2 are the negative roots of (24); axj , bxj and cxj are constants that depend upon an initial condition. The pure periodic
solution can be obtained if the initial condition satisfied bxj = cxj = 0. Note however that, due to λ1 < 0 and λ2 < 0, the above generalsolution converges to a periodic orbit with the passage of time.

15

Proposition 4. Let Assumptions 1-5 hold. Let a (purely) periodic solution path of System (L) be represented by(44)-(47).
(i) Assume thatαc (1 − Cy∗c ) + Ik∗i − δ ≤ 0.

(48)

θyi < θki ≤ θyc < θkc .

(49)

αc (1 − Cy∗c ) + Ik∗i − δ > 0.

(50)

Then, the following condition holds:(ii) Assume that

Then, either of the following conditions holds:

θyi < θyc < θkc ≤ θki ,(51)

θyi < θyc < θki < θkc .

(52)

Proof. See Appendix D.
We can find from Proposition 4 that the relationships θyi < θyc , θyc < θkc and θyi < θki always hold but thatwhich of θyc and θki is greater is conditional.31 It can thus be said that the ups and downs of yi firstly occurs andthose of yc and ki follow them (those of yc are followed by those of kc .) along (purely) periodic orbits of System(L). In particular, if condition (48) holds, the ups and downs of ki precede those of yc , while otherwise (if condition(50) is fulfilled), the reverse is true. Since Hopf periodic orbits of System can be approximated by periodic orbits
of System (L), we can infer that, along Hopf periodic orbits of System, the movements of yc and ki lag behind thatof yi and which of yc and ki precedes the other depends upon which of conditions (48) and (50) holds. Moreover,the lead and lag between the four variables may imply that variations in the output of the investment good sector
play a leading role in business cycles and that the consumption good sector passively responds to changes in theinvestment good sector. These lead and lag relationships reflect not only the Keynesian multiplier process but also
the Keynesian view that what causes economic fluctuations is volatility of investment demand.
31 It is also conditional which of θ
kc and θki is larger. To avoid complexity in calculation, we do not dwell on the conditions uponwhich θkc is larger than θki and which the reverse is true.

16

4

Numerical analysis

We proceed to perform some numerical simulations to check if the conclusions drawn from our analysis are valid.
For this purpose, we need information about the consumption and investment functions.

4.1

Specification

To begin, we specify the investment functions of the consumption good and investment good sectors, Ic and Ii
based upon the empirical data in Japan. We first estimate the gross investment functions of both sectors and thenallow for depreciation of capital. To do so, we make the simplifying assumption that the shape of the gross capital
accumulation function, which relates the rate of gross capital accumulation (i.e., the ratio of gross investment tocapital stock) to the ratio of gross output to capital stock, is common for both sectors. Specifically, we assume thatthe gross capital accumulation functions are formalized as follows:
y 
Ic
c
=g
,
kc
kc


Ii
yi
=g
,
ki
ki

(53)
(54)

where g is the common gross capital accumulation function. By adopting this simplifying assumption, we give up
formalizing the investment function for each sector. The reason for choosing this approach is that it is difficult
to obtain separately information about the consumption good and investment good sectors. Indeed, to our best
knowledge, there have been only a few attempts made to divide aggregate date into these two sectors (cf. Kuga
1967; Takahashi, et al. 2012). By making use of input-output tables, Kuga (1967) and Takahashi et al. (2012)
estimated the volume of capital stock and the number of employed workers for each of the sectors, but because of thelack of reliable input-output tables, however, they were only able to obtain these data in every five years.32 Since
we have difficulty in getting input-output tables for every year and we want to estimate the investment functions
for annual investment behaviors, we make the estimation of the investment function for each sector by assumingthat both of these functions are common.
To formalize the gross capital accumulation function, we suppose that there are (only) two types of firms interms of attitudes towards investment: “optimists” and “pessimists.”33 Optimists and pessimists are assumed tocarry out the “optimistic” and “pessimistic” plans of investment, respectively, where the optimistic and pessimisticplans of investment are to spend 100g% and 100g%, respectively, of the (evaluated) value of capital stock on new(gross) investment with g < g. In other words, all firms are assumed to choose the binary options on investment
plans, one of which realizes a higher rate of capital accumulation g and the other of which a lower one g. Under
32 In Japan, in particular, input-output tables are issued in every five years. What is worse, these tables may not be comparable witheach other due to changes in details.
33 This kind of assumption was made in Lux (1995) or Brock and Hommes (1997) to describe market sentiments. Also, Franke (2012,2014) made use of Lux’s (1995) framework to model the transition dynamics of “animal spirits” (cf. Keynes 1936, chap. 12) in thecontext of business cycles. Our model is the same in spirit as Franke’s (2012, 2014), but, of course, the details are different.

17this hypothetical setting, the rates of gross capital accumulation are the highest g (resp. the lowest g), when all
firms are optimists (resp. pessimists). Our binary-choice assumption may seem artificial but it can be justified
in terms of aggregate analysis. Since the number of firms is almost infinite in reality, each firm’s behavior cannot
directly be observed. In this sense, aggregate data as macroeconomic or collective behavior should be interpreted
in terms of “distributions.”34 The binary-choice treatment is one of the simplest ways to describe a macroeconomic
phenomenon as a distribution.35 Binary-choice settings may seem to describe only a few limited cases of phenomena
due to the number of options, but this is not correct. Indeed, in our setting, if g and g are regarded as the maximum
and minimum values of rate of gross capital accumulation, every number g ∈ [g, g] can be realized as a rate of grosscapital accumulation when the share of optimists among all firms is (g − g)/(g − g). We may thus argue that theabove binary-option assumption is not inappropriate as a treatment in our analysis.
Now we proceed to give a specific shape to the gross capital accumulation function g. On the basis of the above
argument, we set g in the following form:g = pg + (1 − p)g(55)

where p stands for the probability that a firm is an optimist or the share of optimists. Since all firms are assumed tochoose the binary alternatives, optimists and pessimists, and the current economic condition, especially the current(gross) output-capital ratio y/k, can be considered to affect positively the share (probability) of optimists p, we canestablish the following relationship:
 p 
y
ln
= β0 + βu .
1−pkwhere βu is a positive constant. This is nothing but a logistic formulation of the probability p. Making use of (55),
we obtain
g − gyln= β0 + βu .
g−g
k

(56)

By setting g and g, we can estimate the parameters βu and β0 in (56). According to Appendix E, the rate of grosscapital accumulation was annually around 0.3 at most (in Japan during 1970-2014). Therefore, we may set the
maximum value of g or g as 0.3. Also, since we know from Appendix E that the estimated constant term is negativeat the 1% confidence level in naive ordinary least squares relating g to y/k, we may conjecture that the minimum
value of g or g is 0, which is required by the definition of the rate of gross capital accumulation, and so g is set as
34 Aoki and Yoshikawa (2007, chaps. 3 and 4) and Yoshikawa (2015) emphasized distributions as collective behavior of heterogeneous
agents or of the macro economy and introduced the concept of “stochastic macro-equilibrium” borrowing concepts from statisticalphysics.
35 Income distributions among workers and capitalists are one of the classic and typical examples of distributions of binary options(classes). In the two-class income distributions, the share of wage income or that of profit income determines the macroeconomic
distribution.

18

0. It thus follows from (56) that we have

ln

y
g 
= β0 + βu .
0.3 − gk

(57)

From the results of regression analysis of (57) in Appendix E, we know that the estimated values of βu and β0 areβ̂u = 9.38 and β̂0 = −5.81 and that they are significant at the 0.1% confidence level. Hence, we can set βu = 9.4
and β0 = −5.8. We can thus formalize the gross capital accumulation function g as

g=

0.3
.
1 + exp(5.8 − 9.4(y/k))Utilizing (53) and (54), we can obtain the following investment functions of both sectors:
0.3kc,
1 + exp(5.8 − 9.4(yc /kc ))
0.3ki
Ii =
,
1 + exp(5.8 − 9.4(yi /ki ))

Ic =

(58)
(59)

Note that the investment functions (58) and (59) have sigmoid shapes with respect to the level of output as in
Kaldor’s (1940) model of business cycles.
Next we specify the consumption function C on the basis of empirical data in Japan. For simplicity, we assume
that C is expressed in the following form:

C = α0 + αy Y

(60)

where Y is the total (national) income, which is equal to the sum of yc and yi . Note that equation (60) satisfies
Assumption 1 if we have αy ∈ (0, 1).
Since our analysis only deals with business cycles and not with economic growth, we should remove the trend
component from raw data of national income and consumption expenditure. The standard way to do this is tomake use of the Hodrick-Prescott filter suggested by Hodrick and Prescott (1997) or the band-pass filter proposedby Christiano and Fitzgerald (2003), which is more sophisticated than the Hodrick-Prescott one. Employing the
band-pass filter, the resulting de-trended data will dramatically be changed if a different band of frequencies ischosen. So we adopt another simpler method. In our analysis, we get rid of the trend component from data by
replacing aggregate income Y and consumption expenditure C with per-capita income and per-capita consumption
both measured in the efficiency unit, respectively. Letting N and A be the number of population and the total factor
productivity, respectively, we define per-capita income y and per-capita consumption measured in the efficiency unit
as y = Y /AN, c = C/AN. According to the results of simple ordinary least squares relating c to y given in Appendix
E, we can obtain the estimated values of the marginal propensity to consume α̂y = 0.532 and of the fundamental

19

consumption α̂0 and these values are significant at the 0.1% confidence level. Thus, we may set αy = 0.53. Also,
since the investment functions Ic and Ii , given in (53) and (54), are both homogeneous of degree one in the levelsof output and capital stock and the consumption function is given in the form of (60), the equilibrium values ofyc , yi , kc and ki are homogeneous of degree one in that of α0 . Hence, we may normalize α0 as α0 = 1. Therefore,
we may make the following formulation of C:

C = 1 + 0.53(yc + yi ).

(61)

Finally, we determine the value of the rate of capital depreciation δ. Empirically, according to Appendix E, theaverage rate of capital depreciation, denoted by δ̂, was about 0.0873 in Japan (in the period of 1970-2014). Then,
we may set δ = 0.087.
By substituting (58), (59) and (61) in System, we can obtain the following system:ẏc = αc [1 + 0.53(yc + yi ) − yc ],
h
i
0.3kc
0.3ki
ẏi = αi+
− yi ,
1 + exp(5.8 − 9.4(yc /kc )) 1 + exp(5.8 − 9.4(yi /ki ))0.3kc− 0.087kc ,k̇c =1 + exp(5.8 − 9.4(yc /kc ))0.3ki− 0.087ki .
k̇i =1 + exp(5.8 − 9.4(yi /ki ))

(62)
(63)
(64)
(65)

In what follows, the system of equations (62)-(65) is redefined as “System.”
In System, there is a unique equilibrium point (yc∗ , yi∗ , kc∗ , ki∗ ) represented as follows:(yc∗ , yi∗ , kc∗ , ki∗ ) = (2.748, 0.550, 5.266, 1.054)(66)

It is not difficult to verify that all the conditions in Assumptions 1 and 3 are satisfied at least in a neighborhoodof the unique equilibrium point given by (66) and that Assumptions 4 and 5 are fulfilled. Note that, since theexistence and uniqueness of an equilibrium point is already established numerically, Assumption 2, which is madefor the existence, is unnecessary in our numerical analysis.36

4.2

Existence of a Hopf periodic orbit

We discuss periodic orbits generated by Hopf bifurcations in System. To do so, we first explore for what valuesof αc and αi periodic orbits are generated by Hopf bifurcations. We have already known from Section 3 that
Hopf bifurcations occur to System if we have (36). Under the specifications of (62)-(66), the curve of (36) can beillustrated in the following figure. Along the curve in figure 3, System undergoes Hopf bifurcations. That is, for
36 It follows that we do not need to dwell upon the domain D of System.

20

every (αc , αi ) on the curve, a Hopf bifurcation happens to System. In what follows, the curve is denoted by the“stability switching curve.” Note that the unique equilibrium point is locally asymptotically stable (resp. unstable)
if (αc , αi ) is located on the left (resp. right) side of the stability switching curve.
10

8

unstable region
6αi4stable region

2

0
0

2

4

αc

6

8

10

Figure 3: Stability switching curve

To see if a periodic orbit is actually generated by a Hopf bifurcation, we set the values of (αc , αi ) sufficiently
near the stability curve and perform a numerical simulation in of System. For instance, (αc , αi ) = (4, 3.292) is on
the stability switching curve. Then, for our simulation, we set(αc , αi ) = (4, 3.293).

(67)

Finally, we set the initial condition for our simulation as follows:(yc (0), yi (0), kc (0), ki (0)) = 2.748, 0.6, 5.266, 1.054)(68)

We are now ready to conduct a numerical simulation of System. In figure 4 described is the solution path of
System (yc (t), yi (t), kc (t), ki (t)) with (67) and (68). We can see from figure 4 that the solution path of System isperiodic with (67). We can thus confirm the validity of Proposition 3.

21

yi

0.4

2.6

0.5

2.7

2.8

0.6

2.9

0.7

3.0

0.8

yc

9920

9940

9960

9980

10000

9900

9920

9940

time

time

kc

ki

9960

9980

10000

9960

9980

10000

5.1

1.0

5.2

1.1

5.3

1.2

5.4

5.5

1.3

9900

9900

9920

9940

9960

9980

10000

9900

9920

9940

time

time

Figure 4: Periodic orbit

In Figure 5 depicted are the time paths of yc − yc∗ , yi − yi∗ , kc − kc∗ and ki − ki∗ of System with (67) and (68).
This figure illustrates the lead and lag relationships between yc , yi , kc and ki . We can find from figure 5 that theups and downs of yi come first and then those of yc , ki and kc follow in order. This consequence is consistent with

- 0.2

- 0.1

0

0.1

0.2

0.3

Proposition 4.37

9900

9920

9940

time

9960

9980

Figure 5: Lead and lag(—-: yc − yc∗ , —-: yi − yi∗ , —-: kc − kc∗ , —-: ki − ki∗ )37 Since System with (67) satisfies (50), condition (51) holds.

22

10000

5

Concluding remarks

We are in a position to summarize our analysis.
As a first step for multi-sectoral analysis of business cycles, we have put forward a two-sector model of businesscycles by disaggregating the economy into two sectors: the consumption good sector and the investment good sector.
We have examined the stability of equilibrium and the possibility of existence of a periodic orbit in our two-sector
model. As a result, we have revealed that the counter part of the Keynesian stability condition plays a key role inthe stability in our two-sector model and that a periodic orbit may arise by way of a Hopf bifurcation if the stability
condition is not satisfied. We have also observed that the consumption good sector lags behind the investment
good sector along the periodic orbit and that interactions between these two sectors do play a significant role inbusiness cycles. Furthermore, we have numerically investigated the properties of the periodic orbit generated by
Hopf bifurcations in our two-sector model. By conducting numerical simulations, we have verified that periodic
orbits are actually generated by Hopf bifurcations in our two-sector model.
Both consumption and investment are components of aggregate expenditure. As Keynes (1936) argued, however,they differ on the demand side in that the former responds passively to aggregate income (through the “consumptionfunction”) while the latter has a decisive role in determination of aggregate income (through the “multiplier process”)
and is the main cause of economic fluctuations. As Harrod (1939) noticed, they also differ on the supply side inthat the former is not utilized as a factor of production while the latter is durable and functions as a factor of
production (the “duality of investment”). These differences cannot properly be described in one-sector models, but
we believe that our two-sector model may shed a new light on them.
Our two-sector model is only a first step for developing a more general multi-sector model of business cycles.
The dichotomy between consumption and investment in our model is artificial in that a lot of goods function both asa consumption good and as an investment good in reality. Also, our assumption of rigid prices is stringent because
prices fluctuate over business cycles in reality. The removal of these unrealistic points may be left for future study.

Acknowledgments
The author would like to thank Toichiro Asada, Shinya Fujita, Gang Gong, Miki Kobayashi, Kazuhiro Kurose,
Masao Ishikura, Akio Matsumoto, Tadasu Matsuo, Yasutaka Niisato, Kenshiro Ninomiya, Takashi Ohno, Tamotsu
Onozaki, Hiroshi Osaka, Yasuhiro Sakai, Hiroaki Sasaki, Ryunosuke Sonoda, Hiroyuki Yoshida and Naoki Yoshihara(alphabetical order) for their helpful comments on an earlier version of this paper. Needless to say, the author issolely responsible for all the possible remaining errors in this paper. This work was supported by Chuo University
Personal Research Grant and JSPS KAKENHI Grant Number 14J03350.

23

AppendicesA

Proof of Proposition 1

Proof. (i)38 Since System is (twice) continuously differentiable on D and D is a compact convex set, it follows fromthe mean value theorem that there exist a positive constant K such that for every (yc0 , yi0 , kc0 , ki0 ), (yc1 , yi1 , kc1 , ki1 ) ∈ D,
we have|αc [C(yc1 , yi1 ) − yc1 ] − αc [C(yc0 , yi0 ) − yc0 ]| + |αi [Ic (yc1 , kc1 ) + Ii (yi1 , ki1 ) − yi1 ] − αi [Ic (yc0 , kc0 ) + Ii (yi0 , ki0 ) − yi0 ]|+ |[Ic (yc1 , kc1 ) − δkc1 ] − [Ic (yc0 , kc0 ) − δkc0 ]| + |[Ii (yi1 , ki1 ) − δki1 ] − [Ii (yi0 , ki0 ) − δki0 ]|≤ K(|yc1 − yc0 | + |yi1 − yi0 | + |kc1 − kc0 | + |ki1 − ki0 |).

Hence, the Lipschitz condition (Coddington and Levinson 1955, chap. 1) is satisfied on D in System. Thus, forevery initial condition (yc (0), yi (0), kc (0), ki (0)) ∈ D, there exists a nonnegative t0 such that the solution path of
System uniquely exists for t ∈ [0, t0 ]. Such a constant t0 is called the end point below.
Since conditions (20)-(23) (deduced from Assumptions 1 and 2) imply that D is positively invariant with respect
to System, we can, by applying the argument on the continuation of solution paths of differential equations (cf.
Coddington and Levinson 1955, chap. 1), conclude that, for every initial condition (yc (0), yi (0), kc (0), ki (0)) ∈ D,
the end point t0 can be extended to any positive number.
(ii) Since D is positively invariant with respect to System (by Assumptions 1 and 2), we can apply the PoincarèHopf theorem (cf. Milnor 1965, p. 35) to find that the sum of the indices of all equilibrium points of System is +1,
which already implies the existence of an equilibrium point on D. Moreover, we know from Assumption 3 that thedeterminant of the Jacobian matrix J of System is positive everywhere on D, especially at every equilibrium point:

det J = αc αi {(1 − Cyc )Ikc Iki − [(1 − Cyc )(1 − Iyi )Ikc + (1 − Cyc − Cyi Iyc )Iki ]δ + [(1 − Cyc )(1 − Iyi ) − Cyi Iyc ]δ 2 } > 0.

By definition, we find that the index of each equilibrium point of System is +1. Hence, the number of equilibrium
points has to be one.39 Thus, the existence and uniqueness of an equilibrium point of System (on D) can beestablished.
It follows from (20)-(23) that xj or xj cannot be the equilibrium value of xj for x = y, k and j = c, i. Then, theunique equilibrium point (yc∗ , yi∗ , kc∗ , ki∗ ) must be on the interior D◦ .
38 The method of proof is the same as used in Murakami (2014, 2017a).
39 By the same method, Varian (1979) showed the existence and uniqueness of equilibrium of Kaldor’s (1940) model of business cycles.

24

B

Proof of Proposition 2

Proof. (i) If αc and αi are sufficiently small, the dominant term of f is given by40
(δ − Ik∗c )(δ − Ik∗i )2 (2δ − Ik∗c − Ik∗i ){(1 − Cy∗c )(δ − Ik∗c )αc + [(1 − Iy∗i )δ − Ik∗c ]αi } > 0.

The inequality follows from Assumption 1. Hence, condition (30) holds if αc and αi is sufficiently small.
(ii) If αc is sufficiently large and αi is sufficiently small, the dominant term of f is given by(1 − Cy∗c )3 (δ − Ik∗c )(δ − Ik∗i )(2δ − Ik∗c − Ik∗I )αc3 > 0,

where the inequality holds under Assumption 1. Then, the same logic as in (i) applies.
(iii) If αc is sufficiently small and αi is sufficiently large, the dominant term of f is
(1 − Iy∗i )[(1 − Iy∗i )δ − Ik∗c ][(1 − Iy∗i )(2δ − Ik∗c ) − Ik∗i ](δ − Ik∗i )αi3 > 0,

where the inequality holds under Assumption 1. The same logic as in (i) applies.
(iv) If αc and αi are sufficiently large, the dominant term of f is[(1 − Cy∗c )(1 − Iy∗i ) − Cy∗i Iy∗c ]{[(1 − Cy∗c )(1 − Iy∗i ) − Cy∗i Iy∗c ]δ − (1 − Cy∗c )(1 − Iy∗i )Ik∗c + (1 − Cy∗c − Cy∗i Iy∗c )Ik∗i }×[(1 − Cy∗c )αc + (1 − Iy∗i )αi ]αc2 αi2 > 0,

where the inequality holds under Assumptions 1 and 4 and condition (32). Then, the same logic as in (i) applies.
(v) If αc and αi are sufficiently large, the dominant term of f is[(1 − Cy∗c )(1 − Iy∗i ) − Cy∗i Iy∗c ]{[(1 − Cy∗c )(1 − Iy∗i ) − Cy∗i Iy∗c ]δ − (1 − Cy∗c )(1 − Iy∗i )Ik∗c + (1 − Cy∗c − Cy∗i Iy∗c )Ik∗i }×[(1 − Cy∗c )αc + (1 − Iy∗i )αi ]αc2 αi2 < 0.

The inequality holds under Assumptions 1 and 4 and condition (33). Thus, condition (30) is violated if both αcand αi are sufficiently large.

C

Proof of Proposition 3

Proof. (i) If αi is sufficiently large, the dominant term of f (0, αi ) is
(1 − Iy∗i )[(1 − Iy∗i )δ − Ik∗c ][(1 − Iy∗i )(2δ − Ik∗c ) − Ik∗i ](δ − Ik∗i )αi3 > 0,
40 We mean by a “dominant term” the one which is sufficiently large compared with the other terms in absolute values.

25while we have
lim f (αc , αi ) = −∞,αc →∞under Assumptions 1 and 5. Then, equation (36) has at least one positive root αc∗ for αi sufficiently large. Sincef (αc , αi ) turns from positive to negative at αc = αc∗ for αi sufficiently large value, we have (37) at αc = αc∗ . It
follows from Asada and Yoshida’s (2003) theorem that a periodic orbit is generated by a Hopf bifurcation.
(ii) Fix αi at a sufficiently large value. Then, the dominant term of f (αc , 0) is(1 − Cy∗c )3 (δ − Ik∗c )(δ − Ik∗i )(2δ − Ik∗c − Ik∗I )αc3 > 0,

while we have
lim f (αc , αi ) = −∞,αi →∞under Assumptions 1 and 5. Hence, the same logic as in (i) applies.

D

Proof of Proposition 4

Proof. (i) We substitute (44)-(47) in System (L) to obtain the following: 2π(t − θ ) h 2π(t − θ ) i
 2π(t − θ ) 
2πayc
yc
yi
yc
cos
= αc (Cy∗c − 1)ayc sin+ Cy∗i ayi sin
,
T
T
T
T
 2π(t − θ )  2π(t − θ ) h 2π(t − θ ) 
2πayi
yi
yi
yc
cos
= αi Iy∗c ayc sin+ (Iy∗i − 1)ayi sinTTTT 2π(t − θ ) i
 2π(t − θ ) kikc+ Ik∗i aki sin,+ Ik∗c akc sin
T
T





2π(t − θkc )2π(t − θyc )
2π(t − θkc ) 
2πakc
cos
= Iy∗c ayc sin+ (Ik∗c − δ)akc sin
,
T
T
T
T






2πaki
2π(t − θki )
2π(t − θyi )
2π(t − θki )cos= Iy∗i ayi sin+ (Ik∗i − δ)aki sin
.
T
T
T
T
Equations (69), (71) and (72) can be transformed into the following:r 2π(t − θ )  2π 2  2π[t − θ + (θ − θ )] 
yc
yc
yi
yi
αc2 (1 − Cy∗c )2 +sin
= αc Cy∗i ayi sin
,
T
T
T
r
 2π 2  2π[t − θ + (θ − θ )] 
 2π(t − θ ) 
kc
kc
yc
yc
akc (Ik∗c − δ)2 +sin= Iy∗c ayc sin
,
T
T
T
r
 2π 2  2π[t − θ + (θ − θ )] 
 2π(t − θ ) 
ki
ki
yi
yi
sin
= Iy∗i ayi sin,aki (Ik∗i − δ)2 +
T
T
T

ayc

26

(69)

(70)

(71)
(72)

where we can find that 2π 2 2π(θ − θ )  2π .r
yc
yi
=
> 0,
αc2 (1 − Cy∗c )2 +
sin
T
T
T
r
 2π(θ − θ ) .
 2π 2yc
yicos= αc (1 − Cy∗c )> 0,αc2 (1 − Cy∗c )2 +TTr
 2π(θ − θ )  2π .
 2π 2
kc
yc
=
> 0,
sin
(Ik∗c − δ)2 +TTTr
 2π(θ − θ ) .
 2π 2kc
yccos= (δ − Ik∗c )> 0,(Ik∗c − δ)2 +TTr 2π(θ − θ )  2π .
 2π 2kiyisin=(Ik∗i − δ)2 +
> 0,
T
T
T
r
 2π(θ − θ ) .
 2π 2kiyicos= (δ − Ik∗i )(Ik∗i − δ)2 +
> 0.
T
T

(73)

(74)

(75)

Since we have θx0j0 − θxj ∈ [−T, T ], for x, x0 = y, k and j, j 0 = c, i, it follows from (73)-(75) thatT,4T
0 < θkc − θyc < ,4T
0 < θki − θyi < .
4
0 < θyc − θyi <

(76)
(77)
(78)

Under Assumption 1 and (48), it is seen from (73) and (75) that
 2π(θ − θ )  2π(θ − θ ) 
ki
yi
yc
yi
sin
≤ sin
.
T
T
Taking account of (76) and (78), this leads us to the following:

0 < θki − θyi ≤ θyc − θyi <T.
4

Combined with (77), this implies (49).
(ii) By the same argument as made in (i), the following holds under (50):

0 < θyc − θyi ≤ θki − θyi <T.
4Since we have (76)-(78) even under (50), condition (51) or (52) follows.

E

Data and estimations

The data on the levels of GDP (Y ), consumption expenditure (C), investment expenditure (I), private capital stock(K) and private capital depreciation (D) are taken from Annual Report on National Accounts, issued by Cabinet

27Office, Japan. Those on the number of population (N ) and TFP (A) are taken from Population Estimates, issued
by Statistics Bureau, Ministry of Internal Affairs and Communications, Japan and from Japan Main Productivityindicators database, issued by Japan Productivity Center. Utilizing the data, we calculate the following variables:
Yt
,
At Nt
Ct
ct =
,
At Nt
Yt
,
ut =
Kt
It
gt =
Kt−1
Dt
δt =
Kt−1
yt =

The time series of these variables are presented in the following tables.

Year t
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984

yt
52447.3506
55013.68838
57051.96345
58857.36951
61051.11748
64366.22915
67253.57605
69996.92529
73190.66295
76795.21284
80267.59664
85122.8436
89969.47849
95430.56721
100619.3079
107199.8807
112666.7582
116869.9426
122953.1865
123443.5707
124399.5791
129684.7461
133693.7278
137986.3474
143074.6262
146750.7696
147392.5621
149528.3098
151563.7855
154098.0531

Table 1: Data on consumption
ctYear tyt
34078.17756
1985
153864.6269
36252.61448
1986
156125.3148
37738.24008
1987
160426.0343
38997.9724
1988
164549.1842
40095.62564
1989
168585.833
41542.98532
1990
171990.8208
42877.80707
1991
176003.1668
44211.67471
1992
177044.9549
46248.96586
1993
175630.2417
48392.68646
1994
176118.6834
50603.66094
1995
175845.3864
53539.14072
1996
182791.9129
56223.1002
1997
186026.6593
57869.02348
1998
181965.3922
60123.4434
1999
182625.5191
62343.77079
2000
182809.3908
66164.99703
2001
180842.1034
68892.10944
2002
178970.9385
72984.65676
2003
178867.2086
74273.6232
2004
179031.4318
75746.47524
2005
178190.8678
78144.77558
2006
179831.7834
80248.55244
2007
182361.6312
82740.0092
2008
178401.1968
86525.30545
2009
172986.7644
87371.41093
2010
177937.5003
86463.48082
2011
176238.1682
88612.99813
2012
173891.8425
90633.92518
2013
173969.832
90951.2639
2014
175641.2031

28

ct
89699.92055
91544.53121
93905.94445
95469.94741
97651.33145
98974.13324
100020.5226
101452.4419
101480.57
103152.4538
103596.6206
105245.92
105731.2342
105323.1219
102412.1456
101142.4088
100756.6888
100721.831
99765.28645
99019.39508
99257.60558
100116.0771
100462.7929
100210.9704
101109.3692
103898.083
105074.604
103461.576
103845.6398
104349.2264

Year t
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992

ut−1
0.920540411
0.861842909
0.816810644
0.746888181
0.656871046
0.630956113
0.645727528
0.641003805
0.668038815
0.684368258
0.637740358
0.620540451
0.630148695
0.62946249
0.638086432
0.647524516
0.642018714
0.656702825
0.638795784
0.639862382
0.616123054
0.60327599
0.593849223

Table 2: Data on investment
gt
Year tut−10.2981151
1993
0.586731409
0.240154214
1994
0.580881778
0.229252888
1995
0.582359019
0.243385516
1996
0.583256859
0.202108329
1997
0.588555194
0.163178192
1998
0.575262762
0.162766744
1999
0.570474009
0.150631948
2000
0.521879901
0.151325608
2001
0.525945192
0.161263592
2002
0.528027288
0.152370902
2003
0.528254957
0.141145521
2004
0.524990784
0.135851506
2005
0.525998606
0.128638911
2006
0.518379046
0.136205692
2007
0.509119624
0.143041077
2008
0.503120972
0.138712231
2009
0.482781297
0.147648958
2010
0.473082549
0.15672139
2011
0.490865069
0.164925227
2012
0.483356602
0.166810789
2013
0.49127967
0.159280002
2014
0.485373411
0.140180568

Table 3: Data on capital depreciation
Year
δt
Year
δt
1970 0.127447471 1993 0.085978657
1971 0.113426883 1994 0.085382263
1972 0.115418444 1995 0.08625172
1973 0.110588994 1996 0.088598348
1974 0.093032028 1997 0.088046291
1975 0.079194407 1998 0.085682152
1976 0.079162266 1999 0.096150697
1977 0.077645084 2000 0.086042349
1978 0.078456826 2001 0.085210528
1979 0.080642677 2002 0.085517862
1980 0.077040678 2003 0.084997226
1981 0.076879552 2004 0.084576848
1982 0.077099521 2005 0.084427673
1983 0.077819367 2006 0.085566701
1984 0.08060643
2007 0.085775444
1985 0.08465134
2008 0.085740372
1986 0.084312335 2009 0.082681698
1987 0.088034464 2010 0.083173753
1988 0.088379331 2011 0.082248902
1989 0.092058328 2012 0.081851671
1990 0.090016885 2013 0.083653054
1991 0.089415745 2014 0.083354347
1992 0.087763779

29

gt
0.124239961
0.117256516
0.11697812
0.125143077
0.124770106
0.107852181
0.101667537
0.095604815
0.091719391
0.086284506
0.088409658
0.089575079
0.092253872
0.094179211
0.093929706
0.089334967
0.072974815
0.07450905
0.077885821
0.080982492
0.083401591
0.084423299

Based upon the argument in Section 4, we make use of the following equations for our estimations.

lnct = α0 + αy yt + εc ,= β0 + βu ut−1 + εg ,gt
0.3 − gt

(79)
(80)

where εc and εg obey N (0, σc2 ) and N (0, σg2 ), respectively. In (79), the level of consumption is assumed to be
dependent upon that of income in the year, while in (80), the level of investment is assumed to be dependent uponthat of income in the previous year. This difference reflects the reality that the Robertsonian consumption lag (i.e.,
the time lag between earnings of income and consumption expenditure) is shorter than the investment decision lag(i.e., the time lag between earnings of income and investment expenditure).41
The results of estimations on (79) and (80) are given in the following tables.
Table 4: Estimations
(79)
(80)
yt
0.5318∗∗∗ut−1
9.3798∗∗∗(0.005691)(0.6899)
Constant
7675∗∗∗
Constant−5.8092∗∗∗(829.9)(0.4212)
Observations
60
Observations
45
Adjusted R2
0.9933Adjusted R2
0.8069Standard errors in parentheses*** p < 0.001

The average rate of capital depreciation can directly be calculated from table 3 as follows:δ̂ = 0.0873.

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