Discussion Paper No. 246
Cyclical Growth in a Goodwin-Kalecki-Marx Model

Hiroaki SASAKI

August 2009TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND
MANAGEMENT TOHOKU UNIVERSITY

KAWAUCHI, AOBA-KU, SENDAI,

980-8576 JAPANCyclical Growth in a Goodwin-Kalecki-Marx Model
Hiroaki SASAKI∗
August 2009

Abstract
This paper presents a Kaleckian growth model that considers elements from Goodwin
and Marx. The model has a system of differential equations for the rate of utilization,
the profit share, and the rate of employment. We show that cyclical fluctuations occur
depending on the sizes of the reserve-army effect and the reserve-army-creation effect.
Moreover, we show that if the stable long-run equilibrium corresponds to the profitled growth regime, an increase in the bargaining power of workers increases the rateof unemployment; on the other hand, if the equilibrium corresponds to the wage-led
growth-regime, an increase in the bargaining power of workers decreases the rate ofunemployment.
Keywords: Goodwin-Kalecki-Marx model; cyclical growth; reserve-army-creation effects; Hopf bifurcation
JEL Classification: E12; E24; E25; E32; O41∗

Graduate School of Economics and Management, Tohoku University, 27-1 Kawauchi, Aoba-ku, Sendai
980-8576, Japan. E-mail: hsasaki@hotmail.co.jp11

Introduction

This paper presents a Kaleckian model of cyclical growth that introduces elements from
Goodwin and Marx. Using the model, we investigate how the rate of capacity utilization,
the profit share, and the rate of employment are determined, and also analyze the stability of
the long-run equilibrium.
Thus far, a number of models of cyclical growth have been developed. The point of
departure is Goodwin (1967), in which closed orbits around the equilibrium are obtainedwith regard to the rate of employment and the wage share. Goodwin’s model is considered
to capture the class-conflict over the income distribution between workers and capitalists.
The structure of Goodwin’s model is relatively simple, and as such, the model has been
modified in many different ways.1)
Shah and Desai (1981) introduce endogenous technical change into Goodwin’s model.
They assume that the growth rate of labor productivity and that of the capital-output ratio
are endogenously determined by induced technical change. Shah and Desai’s model is athree-dimensional model with regard to the rate of employment, the wage share, and thecapital-output ratio. Unlike the original Goodwin model, the equilibrium is locally stable
and cyclical fluctuations are not obtained. van der Ploeg (1987) modifies Shah and Desai’s(1981) model to allow for different savings rates between wages and profits, and the effects
of productivity growth on the wage-bargaining equation. van der Ploeg’s model is a threedimensional model with regard to the rate of employment, the wage share, and the capitaloutput ratio. Using the Hopf bifurcation theorem, he shows that there exists a stable limit
cycle.
The abovementioned models, like Goodwin’s model, are classical in the sense that savings determine investment, and accordingly, do not consider the problem of effective demand. In addition, the determination of income distribution is open to question. In these
1) Modifications of Goodwin’s (1967) model are as follows. Pohjola (1981) presents a discrete-time version of Goodwin’s model, in which a non-linearfirst-order difference equation of the rate of employment isderived,and shows that chaotic behavior occurs. Wolfstetter (1982) presentsa three-dimensional model withrespect to the rate of employment,the wage share, and the rate of capacity utilization. In Wolfstetter’smodel,
the equilibrium is either stable or unstable, and cyclicalfluctuations are not obtained. Sato (1985) develops
a two-sector Goodwin’s model with a consumption goods producing sector and a capital goods producing
sector. Sportelli (1995) presents a modified Goodwin model that generates a stable limit cycle. Choi (1995)
introduces the efficiency wage hypothesis into Goodwin’s model, and shows that if the effort level depends
positively on the real wage rate, the equilibrium is stable, whereas if the effort level depends negatively on thereal wage rate, cyclical fluctuations occur. Foley’s (2003) model can be said to be a modified Goodwin model,
in which endogenous technical change and factor substitution are considered. Ryzhenkov (2009) constructs athree-dimensional Goodwin model to analyze the Italian economy.

2

models, the equilibrium wage share is determined by only the shape of the innovation possibility frontier—only technological factors determine income distribution. Hence, otherfactors such as the bargaining between workers and capitalists never affect income distribution.
In contrast to these models, there are several models of cyclical fluctuations that consider effective demand, especially investment demand. Yoshida (1999) presents a modified
Harrodian model to analyze the dynamics of the expected real wage rate, the efficient factor intensity, and the rate of capital accumulation, and using the Hopf bifurcation theorem,
shows that there exists a limit cycle. The model of Sportelli (2000) is also a modified Harrodian model, which considers the dynamics of the natural rate of growth, the warranted
rate of growth, and the savings rate. Using a Šil’nikov scenario, he shows that the model
generates a chaotic motion. Skott (1989) combines a Kaldorian business cycle model with a
Goodwin model and using the Poincare-Bendixon theorem, shows that cyclical fluctuations
occur.
As stated earlier, we extend the Kaleckian growth model.2) In the Kaleckian framework,
several models consider cyclical fluctuations. Lima (2004) develops a Kaleckian growthmodel in which the rate of utilization is adjusted in the short run, while the wage shareand the capital-effective labor ratio are adjusted in the long run. In Lima’s model, the rateof labor-saving technological change depends non-linearly on the wage share, which cangenerate the limit cycle. Raghavendra (2006) builds a Kaleckian model that considers the
dynamics of the rate of capacity utilization and the profit share, and shows that a stable limit
cycle occurs. However, this model cannot determine the rate of employment. Moreover,the model treats the level of output, and not the growth of output. Therefore, the dynamics
obtained are not indicative of cyclical growth. Flaschel, Franke, and Semmler (2008) consider the three-dimensional dynamics of the output-capital ratio, the real wage rate, and theemployment rate.
The basic framework of our model is based on a Kaleckian model with the theory of
conflicting-claims inflation.3) In the usual Kaleckian model, the rate of capacity utilization and the rate of capital accumulation are determined with the income distribution given
exogenously. In the Kaleckian model with the theory of conflicting-claims inflation, in
contrast, the income distribution is also determined endogenously (Cassetti, 2003, 2006).
However, even such extended models do not consider the labor market satisfactorily. Existing Kaleckian growth models assume that the labor supply is unlimited and that firms
employ as many workers as they desire at the given wages. If, however, the labor supply
2) See Kalecki (1971) for his economic theory. For the basic framework of the Kaleckian growth model, see
Rowthorn (1981), Lavoie (1992), Foley and Michl (1999, ch. 10), Blecker (2002), and Taylor (2004, ch. 5).
3) The theory of conflicting-claims inflation is developed by Rowthorn (1977).

3

grows at an exogenously given rate, there is no guarantee that the endogenously determined
employment growth is equal to the exogenous labor supply growth. If labor supply grows
faster than employment, then the rate of unemployment continues to rise; this is unrealistic.
Hence, a model such as this cannot investigate long-run unemployment. Kaleckian models
with the theory of conflicting-claims inflation consider the effect of a class conflict between
workers and capitalists on income distribution, but do not consider its effect on employment.
It is interesting to investigate how the changes in the bargaining power of both classes affect
employment.
Therefore, we present a Kaleckian model in which the rate of employment is endogenously determined. For this purpose, we endogenize the growth rate of labor productivity,
which is zero or given exogenously in the usual Kaleckian model. In particular, we assume
that the growth rate of labor productivity depends positively on the rate of employment.
This assumption is based on Bhaduri (2006) and Dutt (2006).4) Given the level of output, an
increase in labor productivity lowers employment and accordingly, creates the reserve-armyof labor. Because, under our formulation, the growth rate of labor productivity increases
with the rate of employment, there exists a counterbalancing effect that lowers the increased
rate of employment. We call this effect the “reserve-army creation effect.” Marx emphasizes
the role of labor-saving technological progress in creating the reserve-army of labor (Marx,
1976, ch. 10, 13, 23). We use this term to distinguish it from the “reserve-army effect.” As
will be explained later, the reserve-army effect relates to the rate of employment and wage
rate. We show that the interaction between the reserve-army effect and the reserve-armycreation effect produces cyclical fluctuations of the endogenous variables.
As state above, our model considers the elements from Goodwin (the dynamics of therate of employment and income distribution), Kalecki (an investment function independentof savings, and mark-up pricing in oligopolistic goods markets), and Marx (the reserve-army
effect and the reserve-army-creation effect). For this reason, we call our model a GoodwinKalecki-Marx model.
The remainder of the paper is organized as follows. Section 2 presents the basic framework of the model and derives the fundamental equations for the analysis. Section 3 analyzes
the existence and the local stability of the long-run equilibrium, and analytically shows thata limit cycle can occur. Section 4 shows the occurrence of cyclical fluctuations by using numerical simulations. Section 5 conducts a comparative statics analysis. Section 6 concludes4) Based on the idea of Marx, Bhaduri (2006) states that this formulation captures the view that technologicalchange is driven by inter-class conflict over income distribution between workers and capitalists. Dutt (2006)
says that as the labor market tightens and labor shortage becomes clearer, the bargaining power of workers increases, which exerts an upward pressure on wages, leading capitalists to adopt labor-saving technicalchanges.

4the paper.

2

Basic framework of the model

Consider an economy with workers and capitalists. Suppose that workers spend all theirwages and capitalists save a constant fraction s of their profits. Let r and K be the rate ofprofit and capital stock, respectively. Then, the real savings are given by S = srK, andaccordingly, the ratio of real savings to capital stock, g s = S /K, yields
g s = sr.

(1)

We ignore capital depreciation for simplicity.
Suppose that the firms operate with the following fixed coefficients production function:
Y = min{aL, (u/k)K},(2)

where Y denotes real output; L, employment; and a = Y/L, the level of labor productivity.5)
The rate of capacity utilization is defined as u = Y/Y ∗ , where Y ∗ denotes the potential
output. The coefficient k = K/Y ∗ denotes the ratio of capital stock to potential output, whichis assumed to be constant. This assumption implies that both K and Y ∗ grow at the same rate.
Moreover, when the rate of capacity utilization is constant, the growth rates of capital stock
and actual output are the same. Accordingly, the actual output and potential output grow at
the same rate in the long-run equilibrium. To simplify the analysis, we assume k = 1 in what
follows. From this, we have r = mu, where m denotes the profit share.
Following the argument of Marglin and Bhaduri (1990), we specify the ratio of real
investment (I) to capital stock (K), gd = I/K, as follows:
gd = ψmβ uγ ,

ψ > 0, β ∈ (0, 1), γ ∈ (0, 1),(3)where ψ denotes a constant; β, the elasticity of the investment rate with respect to the profit
share; and γ, the elasticity of the investment rate with respect to the rate of capacity utilization. Equation (3) implies that the desired investment rate of firms is increasing in boththe profit share and the rate of capacity utilization. In conventional Kaleckian models, theinvestment function is assumed to depend positively on both the rate of profit and the rateof capacity utilization. Marglin and Bhaduri (1990), in contrast, argue that the profit share5) Given the fixed coefficients production function, a cost-minimizing firm operates at a point on isoquantcurves such that aL = (u/k)K, which yields a = Y/L.

5

and not the rate of profit should be a variable in the investment function.6) Our specificationof the investment function is based on Blecker (2002). Because the investment function is
not linear but Cobb-Douglas, as will be shown later, different regimes can be produced bychanging the sizes of β and γ. When β < γ, the economy is in the wage-led growth regime,
whereas when β > γ, it is in the profit-led growth regime.
An equation of motion for the rate of capacity utilization is given byu̇ = α(gd − g s ),α > 0,(4)

where α denotes the speed of adjustment of the goods market. Equation (4) shows that
excess demand leads to a rise in the rate of capacity utilization, while excess supply leads toa decline in the rate of capacity utilization.
From the definition of profit share, we have m = 1 − (wL/pY), from which we obtain thefollowing relation:7)ṗ ẇ ȧṁ
= − + ,
1−m p w a(5)

where w denotes the money wage and p, the price. To know the dynamics of m, we have to
specify the dynamics of p, w, and a.
We specify the dynamics of the money wage and price by using the theory of conflictingclaims inflation. First, suppose that the growth rate of the money wage that workers manage
to negotiate depends on the discrepancy between their target profit share and the actual
profit share. Second, suppose that the firms set their price to close the gap between theirtarget profit share and the actual profit share. From these considerations, the dynamics ofthe money wage and price can be described, respectively, as follows:ẇ= θw (m − mw ),wṗ= θ f (m f − m),pθw > 0, mw ∈ (0, 1),(6)

θ f > 0, m f ∈ (0, 1),(7)6) The reason for this is as follows. The rate of profit is equal to the product of the profit share and the rateof capacity utilization, that is, r = mu. Thus, it is plausible that a combination of high capacity utilization anda low profit share and a combination of low capacity utilization and a high profit share will produce different
levels of investment even when the rate of profit is held constant at a given level.
7) Cassetti (2002, 2003, 2006) derives an equation of motion for the profit share by specifying a pricesetting equation of firms and differentiating it with respect to time. However, this procedure is unnecessary forderiving the dynamics of the profit share, and our procedure is easier than his. Under the conflicting-claims
inflation theory, the price-setting equation in Cassetti’s model plays the role of determining the mark-up rate,
and not the price level.

6where θw and θ f denote the speed of adjustment; mw , the target profit share set by workers;
and m f , the target profit share set by firms.
We can interpret θw and θ f as the bargaining powers of the workers and firms, respectively.8) We assume θ f + θw = 1 and define θ f ≡ θ because bargaining power is a relative
concept. Then, we have θw = 1 − θ, where 0 < θ < 1.9) For example, we consider anincrease in the unionization rate as a factor in increasing the bargaining power of workers(i.e., a decrease in θ), and an increase in the market power of oligopolistic firms as a factor
in increasing the bargaining power of firms (i.e., an increase in θ).
We assume that the workers’ target mw depends negatively on the rate of employment, e.
mw = mw (e),m0w < 0,(8)

where e = L/N and N denotes the exogenous labor supply. As the rate of employment
increases, workers’ demands in the bargaining are likely to increase, which leads workers to
set a higher target wage share, and accordingly, set a lower target profit share. We consider
equation (8) as expressing the “reserve-army effect.”10) On the other hand, for simplicity,
we consider the firms’ target profit share m f as exogenously given.11) Notice the difference
between θ and equation (8). The parameter θ represents the relative bargaining power offirms (workers) and reflects the power to realize their demands. In contrast, equation (8)reflects their demands in the bargaining. To what extent their demands can be realized
depends on θ.
From equation (2), the rate of employment is given by e = uK/(aN), and hence, the rateof change in the rate of employment yieldsė u̇= + gd − ga − n,
e u

(9)

where n is the growth rate of N and given exogenously, and ga = ȧ/a.
As stated above, we assume that the growth rate of labor productivity depends positively
on the rate of employment.12)
ga = ga (e),g0a > 0, ga > 0.

(10)8) This interpretation is also adopted in Lavoie (1992, p. 393), Cassetti (2002, p. 192), and Cassetti (2003, p.
453).
9) The constraints 0 < θw , θ f < 1 are also adopted by Dutt and Amadeo (1993), who, however, do not assume
θ f + θw = 1. Even if we impose only 0 < θw , θ f < 1 and not θ f + θw = 1, we can obtain similar results.
10) Cassetti (2002, 2003) also considers such a reserve-army effect in the Kaleckian framework.
11) We can endogenize the target profit share of firms. Cassetti (2002) and Lima (2004) assume that m f is anincreasing function of u.
12) Flaschel and Skott (2006, p. 328) also use a similar specification.

7

This equation includes the reserve-army-creation effect. As the labor market tightens andlabor shortage becomes clearer, the bargaining power of workers increases, which exerts an
upward pressure on wages, leading the firms to adopt labor-saving technical changes.
In general, the natural rate of growth is defined as a sum of the growth rates of laborproductivity and labor supply. Although the growth rate of labor supply in our model isexogenously given, the growth rate of labor productivity is endogenously determined. Under
our specification, therefore, the natural rate of growth increases when business is good (i.e.,
when the employment rate is high) and it decreases when business is bad (i.e., when theemployment rate is low). The assumption that the natural rate of growth is endogenously
determined is consistent with the empirical studies of León-Ledesma and Thirlwall (2002)
and Libânio (2009).
Using equation (10), we can show that in the long-run equilibrium, the growth rate ofoutput per capita (Y/N) coincides with that of labor productivity (Y/L). In this respect,
Sedgley and Elmslie (2004) empirically show the cointegration between the log of output
per capita and the log of labor productivity. This evidence suggests that these two variables
move together in the long run.
We now focus on the derivation of the system of differential equations. First, substituting
equations (1) and (3) in equation (4), we obtain the dynamics of u. Second, substituting
equations (6) and (7) in equation (5), and substituting equations (8) and (10) in the resulting
expression, we obtain the dynamics of m. Finally, substituting the dynamics of u, equation(3), and equation (10) in equation (9), we obtain the dynamics of e.
u̇ = α(ψmβ uγ − smu),(11)ṁ = −(1 − m)[m − θm f − (1 − θ)mw (e) − ga (e)],(12)

ė = e[α(ψmβ uγ−1 − sm) + ψmβ uγ − ga (e) − n].

(13)

We now provide an explanation with regard to the structure of our model. If we introduce
mw = mw (e) with ga as exogenously given, we find that the rate of employment is endogenously determined whereas the natural rate of growth is exogenous. If we, on the other hand,
introduce ga = ga (e) with mw as exogenously given, we find that both the rate of employment and the natural rate of growth are endogenously determined. Hence, to endogenize
both the rate of employment and the natural rate of growth, we do not need to specify mw
as a function of e. Nevertheless, we use both ga (e) and mw (e). This is because we intend to
capture the interaction between the reserve-army effect and the reserve-army-creation effect.
Simply put, the reserve-army effect acts to destabilize the equilibrium, whereas the reservearmy-creation effect acts to stabilize the equilibrium. Therefore, this interaction plays animportant role in the stability analysis of the equilibrium.
8

3

Existence and stability of the long-run equilibrium

3.1 Existence of the long-run equilibrium
The long-run equilibrium is a situation where u̇ = ṁ = ė = 0. Here, we have the following
three equations:ψmβ uγ − smu = 0,(14)m − θm f − (1 − θ)mw (e) − ga (e) = 0,(15)

ψmβ uγ − ga (e) − n = 0.

(16)

These equations show that the equilibrium values do not depend on the speed of adjustment(α). From equation (14), we obtainu=1 ψ  1−γs

β−1m 1−γ .

(17)

Substituting equation (17) in equation (16), we find that the resulting expression will be an
equation of m and e. Combining this equation with equation (15), we can obtain the equilibrium values of m and e, which are substituted in equation (17) to find the equilibrium valueof u. In the following analysis, we assume that there uniquely exist long-run equilibrium
values (u∗ , m∗ , e∗ ) that satisfy equations (14), (15), and (16) simultaneously. In addition, we
assume u∗ , m∗ , e∗ ∈ (0, 1). Hereafter, the long-run equilibrium values are denoted with “∗.”
Here, we show that in the long-run equilibrium, the capital-labor ratio continues to increase. Let κ be the capital-labor ratio. Then, we have κ = K/L. The rate of change in κ isgiven by the difference between the rate of change in the capital stock and the rate of change
in the employment. Using employment, we get L = uK/a, and consequently obtainu̇κ̇= ga (e) − .
κ
u

(18)

Hence, the rate of change in the capital-labor ratio is given by the difference between thegrowth rate of labor productivity and the rate of change in the capacity utilization. In thelong-run equilibrium, we have u̇ = 0, and consequently, we obtain κ̇/κ = ga (e). Because, inthe long-run equilibrium, e is constant and hence, ga (e) is constant, the rate of change in thecapital-labor ratio is also constant and equal to the growth rate of labor productivity. This
property is similar to that of the neoclassical growth model with labor-augmenting technicalprogress. Note, however, that in the neoclassical growth model, full employment is assumed,
while in our model, full employment is not assumed.

9

3.2 Local stability of the long-run equilibrium
To investigate the local stability of the long-run equilibrium, we linearize the system ofdifferential equations (11), (12), and (13) around the equilibrium.


  
 u̇   J11 J12 0   u − u∗   ṁ =  0 J22 J23  m − m∗  ,(19)   
J31 J32 J33 e − e∗ėwhere the elements of the Jacobian matrix J are given by∂u̇= −αs(1 − γ)m < 0,
(20)
∂u
∂u̇
J12 ≡
= −αs(1 − β)u < 0,
(21)
∂m
∂ṁ
J22 ≡
= −(1 − m) < 0,
(22)
∂m
∂ṁ
= (1 − m) [(1 − θ)m0w (e) + g0a (e)] = (1 − m)Γ(e, θ) ≷ 0,
(23)
J23 ≡
|
{z
}
∂e
≡Γ(e,θ)"#∂ėα(γ − 1) + γu
J31 ≡
= sme
≷ 0,
(24)
∂u
u
∂ė
= se[α(β − 1) + βu] ≷ 0,
(25)
J32 ≡
∂m
∂ė
J33 ≡
= −eg0a (e) < 0.
(26)∂e
All elements are evaluated at the long-run equilibrium; we omit “∗” to avoid troublesome
notations.
The term Γ(e, θ) ≡ (1 − θ)m0w (e) + g0a (e) in equation (23) consists of the following threeelements: the relative bargaining power of firms θ, the extent of the reserve-army effect m0w ,and the extent of the reserve-army-creation effect g0a . Because m0w < 0 and g0a > 0, Γ can bepositive or negative. When the reserve-army effect is strong (i.e., the absolute value of m0w is
large), and the bargaining power of firms and the reserve-army-creation effect are weak (i.e.,θ and g0a , respectively, are small), we have Γ < 0. On the other hand, when the reserve-army
effect is weak, and the bargaining power of firms and the reserve-army-creation effect arestrong, we have Γ > 0. The sign of Γ plays an important role in both the stability of theequilibrium and the comparative statics analysis below.
The signs of equations (24) and (25) are indeterminate. When α is sufficiently large,
these signs are likely to be negative.
The characteristic equation of the Jacobian matrix (19) is given byJ11 ≡λ3 + a1 λ2 + a2 λ + a3 = 0,
10

(27)

where λ denotes a characteristic root. Each coefficient of equation (27) is given by
a1 = −tr J = −(J11 + J22 + J33 ),
a2 =

(28)

J22 J23
J11 0
J11 J12++= J22 J33 − J23 J32 + J11 J33 + J11 J22 ,J32 J33
J31 J33
0 J22

a3 = − det J = −J11 J22 J33 + J11 J23 J32 − J31 J12 J23 ,

(29)
(30)

where −a1 = tr J denotes the trace of J; a2 , the sum of the principal minors’ determinants;and −a3 = det J, the determinant of J.
The necessary and sufficient condition for the local stability is that all characteristicroots of the Jacobian matrix have negative real parts, which, from Routh-Hurwitz condition,is equivalent to13)
a1 > 0, a2 > 0, a3 > 0, a1 a2 − a3 > 0.

(31)

Let us examine whether these inequalities hold. We arrange the coefficients with respect toα.
First, a1 is a linear function of α.
a1 ≡ a1 (α) = s(1 − γ)m α + (1 − m) + eg0a (e) = Aα + B.
| {z }
|
{z
} +
+
≡A>0

(32)

≡B>0

Therefore, we can confirm that a1 > 0. This implies that tr J < 0, which is a necessary
condition for the local stability of the equilibrium.
Second, a2 is a linear function of α.
a2 ≡ a2 (α) = {s(1 − γ)m[1 − m + eg0a (e)] + s(1 − β)e(1 − m)Γ(e, θ)} α
|
{z
}
≡C≷0

+ e(1 − m)[(1 − βsu)g0a (e) − sβ(1 − θ)um0w (e)] = C α + D.
|
{z
} +/−
+

(33)

≡D>0

If Γ > 0, we always have C > 0. Hence, we obtain a2 > 0. If, however, Γ < 0, we do notalways have C > 0. From this, we obtain the following proposition:
Proposition 1. Suppose that C < 0. If the speed of adjustment of the goods market α issufficiently large, then the long-run equilibrium is locally unstable.
Proof. The constant term of a2 is always positive, while the coefficient of a2 can be either
positive or negative. If C is positive, we have a2 > 0. This is a necessary condition forthe local stability of the equilibrium. However, if C is negative, we have a2 < 0 when α issufficiently large. In this case, a2 > 0 is not satisfied, and hence, the long-run equilibrium
becomes locally unstable.
13) See Gandolfo (1996) for details.

11Third, a3 is a linear function of α.

"a3 ≡ a3 (α) = s(1 − γ)em(1 − m)#s(β − γ)g0a (e) −uΓ(e, θ)

|

1−γ
{z

α

}

≡Θ

= s(1 − γ)em(1 − m)Θ α = Eα.
|
{z
}

(34)

≡E

Here, we introduce the following assumption:
Assumption 1. Θ > 0.
The sign of a3 depends on the sign of Θ. If Θ > 0, we have E > 0, and consequently,
a3 > 0. This implies that det J < 0, which is a necessary condition for the local stabilityof the equilibrium. When β > γ, we always have Θ > 0 irrespective of the sign of Γ.14)
However, when β < γ, we do not always have Θ > 0. When β < γ, we always have Θ > 0
if Γ > 0. Even if Γ < 0, we can have Θ > 0. Numerical simulations, which are introduced
later, will show that there exist closed orbits around the equilibrium when both Γ < 0 andΘ > 0.
Finally, a1 a2 − a3 is a quadratic function of α.
a1 a2 − a3 ≡ φ(α) = (AC)α2 + (AD + BC − E)α + |{z}
BD .

(35)

+

At this stage, we cannot confirm whether this parabola is convex upward or convex downward. However, when α = 0, we have φ(0) = BD > 0, which shows that there exists an αsuch that a1 a2 − a3 > 0 for α > 0. From this, we obtain the following proposition:
Proposition 2. Suppose that the speed of adjustment of the goods market α is sufficiently
close to zero. Then, the long-run equilibrium is locally stable.
Proof. From the above discussion, we have a1 > 0 and a3 > 0. When α = 0, we haveφ(0) = BD > 0, that is, a1 a2 − a3 > 0. If a1 > 0, a2 > 0, and a1 a2 − a3 > 0, then a2 > 0 isnecessarily satisfied. Hence, if α = 0, the necessary and sufficient conditions given by (31)are all satisfied. Because φ(α) is a continuous function of α, even if α > 0, a1 a2 − a3 > 0 is
satisfied when α is sufficiently close to zero. Therefore, when α is sufficiently close to zero,
the necessary and sufficient conditions given by (31) are all satisfied.
14) Expanding Θ, we obtain
!
β−γ 0
β−γ
Θ = 1 − sug (e) − su
(1 − θ)m0w (e).
1−γ a
1−γ
When β > γ, (β − γ)/(1 − γ) is larger than zero and smaller than unity. From this, the first term of the right-handside is positive. The second term of the right-hand side is positive because m0w < 0. Therefore, when β > γ,
we have Θ > 0 irrespective of the sign of Γ.

12

Proposition 2 is obtained regardless of whether C > 0 or C < 0. That is, if the speedof adjustment of the goods market is very slow, the long-run equilibrium is locally stableirrespective of the size of the relative bargaining power of firms, the reserve-army effect,
and the reserve-army-creation effect.

3.3 Existence of closed orbits
As explained above, Γ > 0 implies that C > 0, which in turn implies that a2 > 0. Because
we know that a1 > 0 and a3 > 0, the long-run equilibrium is stable if a1 a2 − a3 > 0.
Proposition 3. Suppose that Γ > 0. Suppose also that the equilibrium profit share m∗ is lessthan or equal to 1/2. Then, the long-run equilibrium is locally stable irrespective of the sizeof α.
Proof. If Γ > 0, and consequently C > 0, φ(α) becomes a parabola that is convex downward.
Here, we focus on the coefficient of α in φ(α), that is, AD + BC − E. Expanding thiscoefficient, we haveAD + BC − E = s(1 − γ)m[1 − m + eg0a (e)]2 + s(1 − β)e(1 − m)Γ[eg0a (e) + (1 − m − sγum)].
{z} |{z} |
{z
}
|
{z
} |
+

+

+

≡Λ

(36)
When s, γ, and u are close to zero, Λ ≡ 1 − m − sγum will be positive because sγum will
be sufficiently small. On the other hand, when s, γ, and u are close to unity, Λ approaches
Λ = 1−2m. From this, if m∗ ≤ 1/2, we have Λ ≥ 0. When Λ ≥ 0, we have AD+ BC − E > 0,and consequently, we obtain φ(α) > 0 for α > 0. Therefore, if Γ > 0 and if m∗ ≤ 1/2, then
the necessary and sufficient conditions for local stability, that is, a1 > 0, a2 > 0, a3 > 0, anda1 a2 − a3 > 0, are all satisfied.
Proposition 3 is obtained when the reserve-army effect is weak, and the relative bargaining power of firms and the reserve-army-creation effect are strong. In general, the profit
share in the real world is considered to be less than 1/2, and hence, condition m∗ ≤ 1/2 isplausible. Note, however, that m∗ ≤ 1/2 is a sufficient and not a necessary condition fora1 a2 − a3 > 0. Moreover, m∗ depends on the parameters of the model.
Proposition 4. Suppose that C < 0. Then, cyclical fluctuations occur when the speed ofadjustment of the goods market lies within some range.
Proof. If C < 0, a2 (α) becomes a straight line whose slope is negative and intercept is
positive. This implies that there exists α1 > 0 such that a2 (α1 ) = 0. Moreover, if C < 0, φ(α)
13becomes a parabola that is convex upward. Then, the quadratic equation φ(α) = 0 has one
negative real root and one positive real root. Hence, there exists α2 > 0 such that φ(α2 ) = 0.
Let us investigate which is larger, α1 or α2 . From a2 (α1 ) = 0, we obtain α1 = −D/C > 0.
Substituting α1 in φ(α), we obtain
φ(α1 ) =

DE
<0
C

(37)

because C < 0. This implies that α2 < α1 . From this, we get that a1 > 0, a2 > 0, a3 > 0, anda1 a2 − a3 > 0 within the range α ∈ (0, α2 ), while a1 > 0, a2 > 0, a3 > 0, and a1 a2 − a3 < 0within the range α ∈ (α2 , α1 ). Consequently, a Hopf bifurcation occurs at α2 . Indeed, at
α = α2 , we obtain
a1 > 0, a2 > 0, a3 > 0, a1 a2 − a3 = 0,∂(a1 a2 − a3 )
, 0.
∂α
α=α2

(38)

That is, all conditions for the occurrence of the Hopf bifurcation are satisfied.15) Therefore,
when C < 0, there exists a continuous family of non-constant, periodic solutions of thesystem around α = α2 .
We obtain C < 0 when g0a and θ are small, and the absolute value of m0w is large. These
conditions are similar to the conditions for Γ < 0. Indeed, Γ < 0 is a necessary condition for
C < 0. From this, we can obtain Proposition 4 when the reserve-army effect is strong, andthe relative bargaining power of firms and the reserve-army-creation effect are weak.
As explained above, the long-run equilibrium can be both stable or unstable. Let us
briefly explain this mechanism. Here, we focus on the rate of capacity utilization. Supposethat the rate of capacity utilization exceeds its equilibrium value for some reason. Then, as
long as the speed of adjustment of the goods market is not extremely large, the increase in
the rate of capacity utilization induces the rate of employment to increase through equation(24). This increase in the rate of employment changes the profit share through equation (23).
The direction of the change in the profit share depends on the sign of Γ.
If Γ > 0, that is, the power of firms is relatively strong, then the profit share increases.
This increase in the profit share has two opposing effects. (1) The increase in the profit
share stimulates the investment of firms, which increases the output. (2) The increase in
the profit share increases the savings of capitalists, which decreases the output. Because theadjustment process of the goods market is stable, the latter negative effect on the output is15) For the Hopf bifurcation theorem, see Gandolfo (1996). The last condition in equation (38), that is,∂(a1 a2 − a3 )/∂α|α=α2 , 0, is equivalent to the condition that the derivatives of the real parts of the characteristicroots with respect to α are not zero when evaluated at α = α2 . For details, see Asada and Semmler (1995, p.
634–635).

14

stronger than the former positive effect, and as a result, the output and the rate of capacity
utilization decrease (see equation (21)). Therefore, here, a negative feedback effect acts on
the rate of capacity utilization, and accordingly, the long-run equilibrium will be stable.
On the other hand, if Γ < 0, that is, the power of workers is relatively strong, thenthe profit share declines, which increases the rate of capacity utilization through equation(21). Therefore, here, a positive feedback effect acts on the rate of capacity utilization, andaccordingly, the long-run equilibrium will be unstable.
m ↑ (Γ > 0) =⇒ u ↓ (stabilizing).
u ↑=⇒ e ↑ (when α is not so large) =⇒ m ↓ (Γ < 0) =⇒ u ↑ (destabilizing)
Finally, we refer to the roles of the reserve-army and reserve-army-creation effects.
When only the reserve-army-creation effect exists, that is, m0w = 0, we always have Γ =g0a > 0. In this case, from Proposition 3, the long-run equilibrium is locally stable given thatm∗ ≤ 1/2, and accordingly, the Hopf bifurcation never occurs. Therefore, the reserve-armycreation effect has a stabilizing effect.
In contrast, when only the reserve-army effect exists, that is, g0a = 0, we always have
Γ = (1 − θ)m0w < 0. In this case,Θ=−s(β − γ)u(1 − θ)m0w .
1−γ

(39)

If β > γ, then Θ > 0 necessarily holds. However, if β < γ, then Θ < 0 necessarily
holds, which contradicts Assumption 1, and consequently, the long-run equilibrium becomes
unstable. This implies that the long-run equilibrium in the wage-led growth regime is always
unstable, and also that the Hopf bifurcation never occurs. Therefore, the reserve-army effecthas a destabilizing effect.
From the above reasoning, the interaction between the reserve-army and reserve-armycreation effects plays an important role in stabilizing both the wage-led and profit-leg growth
regimes, and in the occurrence of the Hopf bifurcation.

4 Numerical simulations
In this section, we present numerical examples to show that, under plausible settings, an economically meaningful long-run equilibrium actually exists and cyclical fluctuations reallyoccur. Note, however, that the results of numerical simulations depend on the specificationof the functional form and the numerical values of the parameters.

15

For the numerical simulation, we have to specify the functional forms of equations (8)and (10). We specify these functions as follows:mw (e) = δ(1 − e),ga (e) = ηe,

δ > 0,

η > 0.

(40)(41)

In this case, Γ and Θ are respectively given byΓ = η − (1 − θ)δ,s(β − γ)Θ=η−[η − (1 − θ)δ]u.
1−γ

(42)
(43)

The equilibrium profit share m∗ satisfies the following equation:
1

γ

β−γ

ΓA 1−γ s− 1−γ m 1−γ − ηm + η[θm f + (1 − θ)δ] − Γn = 0.

(44)

From this, we obtain m∗ , which we substitute in equation (17) to determine u∗ . Furthermore,
we substitute m∗ in the following equation to determine e∗ :e∗ =m∗ − [θm f + (1 − θ)δ]
.
Γ

(45)

We consider four cases depending on which is larger, β or γ, and whether Γ is positiveor negative.

4.1 Case 1: β < γ, Γ < 0Case 1 corresponds to the case where the elasticity of the investment rate with respect to the
profit share is smaller than the elasticity of the investment rate with respect to the rate ofcapacity utilization, the reserve-army effect is strong, and the relative bargaining power offirms and the reserve-army-creation effect are weak. We set the parameters as follows:
β = 0.2, γ = 0.4, ψ = 0.15, s = 0.6, η = 0.1, δ = 0.5, θ = 0.25, m f = 0.3, n = 0.015.
(46)
In this case, the long-run equilibrium values, Γ and Θ, yield the following:16)u∗ = 0.74642, m∗ = 0.220134, e∗ = 0.835875, Γ = −0.275 < 0, Θ = 0.0589496 > 0.
(47)16) From these numerical settings, we obtain two values of m within the range m ∈ (0, 1): m∗1 = 0.220134 andm∗2 = 0.0516332. However, from m∗2 , we obtain u∗ = 5.16014 and e∗ = 1.44861, which are unrealistic. For
this reason, we do not adopt m∗2 .

16

These equilibrium values are economically meaningful. In addition, Θ > 0 is satisfied.
Figure 1 displays the solution path when the initial values and the speed of adjustment
are set as u(0) = 0.7, m(0) = 0.21, e(0) = 0.8, and α = 4. The figure shows a cyclicalfluctuation. In Figure 1, we draw the solution path from t = 100 to t = 200, and upon
performing the calculations further, we find that the solution path is not a perfect closed
orbit and that it converges to the long-run equilibrium with rotation. Moreover, if we set theinitial conditions further away from the long-run equilibrium, we find that the solution path
diverges from the equilibrium. From these observations, we confirm that in this numericalexample, the subcritical Hopf bifurcation occurs and the periodic solution is unstable.
Figure 2 projects the three-dimensional dynamics on the (u, e)-plane. The solution pathstarting from point P converges to the long-run equilibrium with rotation. In contrast, thesolution path starting from point Q diverges from the long-run equilibrium with rotation.
These phenomena correspond to the “corridor stability” of Leijonhufvud (1973).
Figure 3 shows the graphs of a2 (α) and φ(α). We find that α1 = 4.61097 and α2 =4.02288. In Figure 1, we use α = 4, from which we have 4 < α2 . Therefore, the subcritical
Hopf bifurcation certainly occurs in this case.17) Moreover, if we choose α larger than α2 ,we find that the solution path diverges irrespective of the initial conditions. This confirmsthat the subcritical Hopf bifurcation occurs at α2 .
Note, however, that the Hopf bifurcation that occurs in case 1 is not always the subcritical Hopf bifurcation. We can find a numerical example wherein the supercritical Hopf
bifurcation occurs in case 1.18)[Figures 1, 2, and 3 around here]

4.2 Case 2: β < γ, Γ > 0
Case 2 differs from case 1 in that Γ > 0, which implies that the reserve-army effect is weak
and the relative bargaining power of firms and the reserve-army-creation effect are strong.
We set the parameters as follows:
β = 0.2, γ = 0.4, ψ = 0.3, s = 0.7, η = 0.3, δ = 0.4, θ = 0.3, m f = 0.3, n = 0.015. (48)In this case, the long-run equilibrium values, Γ and Θ, yield the following:u∗ = 0.871, m∗ = 0.385, e∗ = 0.732, Γ = 0.02 > 0, Θ = 0.304 > 0.

(49)17) In this numerical example, the range of α is such that both a2 (α) > 0 and φ(α) < 0 are narrow. However,
we can widen the range by choosing different parameter settings.
18) Under the parameters β = 0.4, γ = 0.41, ψ = 0.2, s = 0.6, η = 0.1, δ = 1, and θ = 0.25, the Hopf
bifurcation point leads to α2 = 2.56626, and there exists a stable limit cycle at α > α2 .

17These equilibrium values are economically meaningful. In addition, Θ > 0 is satisfied.
Figure 4 displays the solution path when u(0) = m(0) = e(0) = 0.5 and α = 1. The figure
shows that the solution path converges stably to the long-run equilibrium. Figure 5 shows
the graphs of a2 (α) and φ(α). For α > 0, both graphs are always positive, which shows thatthe long-run equilibrium is locally stable irrespective of the size of α.
[Figures 4 and 5 around here]

4.3 Case 3: β > γ, Γ < 0Case 3 corresponds to the case where the elasticity of the investment rate with respect tothe profit share is larger than the elasticity of the investment rate with respect to the rate ofcapacity utilization, the reserve-army effect is strong, and the relative bargaining power offirms and the reserve-army-creation effect are weak. We set the parameters as follows:
β = 0.4, γ = 0.2, ψ = 0.2, s = 0.6, η = 0.1, δ = 1, θ = 0.25, m f = 0.3, n = 0.016. (50)Here, the long-run equilibrium values, Γ and Θ, yield the following:u∗ = 0.741857, m∗ = 0.238619, e∗ = 0.902125, Γ = −0.65 < 0, Θ = 0.172331 > 0. (51)These equilibrium values are economically meaningful. In addition, Θ > 0 is satisfied.
Figure 6 displays the solution path when u(0) = 0.7, m(0) = 0.2, e(0) = 0.9, and
α = 2. The figure shows a cyclical fluctuation. Using various initial values, we find thatin this case, a stable limit cycle emerges: any initial point converges to the limit cycle withrotation. Therefore, the supercritical Hopf bifurcation occurs in case 3.
Figure 7 shows the graphs of a2 (α) and φ(α). We find that α1 = 2.34506 and α2 =1.95504. In Figure 6, we use α = 2, from which we have α2 < 2. Therefore, the supercritical
Hopf bifurcation certainly occurs in this case.19)[Figures 6 and 7 around here]
Note that we were unable to find a numerical example that produced subcritical Hopf
bifurcation.

19) In this case, as in case 1, the range of α is such that both a2 (α) > 0 and φ(α) < 0 are narrow. However, we
can widen the range by choosing different parameter settings.

18

4.4 Case 4: β > γ, Γ > 0
Case 4 differs from case 3 in that Γ > 0, which implies that the reserve-army effect is weak
and the relative bargaining power of firms and the reserve-army-creation effect are strong.
We set the parameters as follows:
β = 0.4, γ = 0.2, ψ = 0.33, s = 0.6, η = 0.3, δ = 0.4, θ = 0.3, m f = 0.3, n = 0.015.
(52)
In this case, the long-run equilibrium values, Γ and Θ, yield the following:u∗ = 0.971, m∗ = 0.384, e∗ = 0.696, Γ = 0.02 > 0, Θ = 0.297 > 0.

(53)

These equilibrium values are economically meaningful. In addition, Θ > 0 is satisfied.
Figure 8 displays the solution path when u(0) = m(0) = e(0) = 0.5 and α = 1. The figure
shows that the solution path converges stably to the long-run equilibrium. Figure 9 shows
the graphs of a2 (α) and φ(α). For α > 0, both graphs are always positive, which shows thatthe long-run equilibrium is locally stable irrespective of the size of α.
[Figures 8 and 9 around here]

5

Comparative statics analysis

This section investigates the effects of the shifts in the parameters on the long-run equilibrium. To conduct a comparative statics analysis, we need the stability of the equilibrium.
For this reason, we assume Θ > 0 in the following analysis. As has been discussed above,
cases 2 and 4 are always stable but cases 1 and 3 can be unstable. Therefore, we confine
ourselves to the case where the long-run equilibrium is stable.
Table 1 summarizes the results of comparative statics in the four cases.20) In Table 1, the“+” sign indicates that the corresponding variable increases with the parameter, while the“−” sign indicates that the corresponding variable decreases with the parameter. The signs“+/−” and “−/+” indicate that an increase in the parameter either increases or decreases thecorresponding variable. The mark “†” shows that the left-hand sign applies when Γ < 0,while the right-hand sign applies when Γ > 0.
Moreover, for the effect of θ, we assume that m f − mw (e) > 0. Firms attempt to set theirtarget profit share as high as possible, whereas workers attempt to set their target profit share
as low as possible. Therefore, this assumption is reasonable.
20) See Appendix for details.

19[Table 1 around here]
Let us explain the results in Table 1. Because of the limitations of space, we focus
especially on the rate of employment.
 Savings rate
An increase in the savings rate of capitalists decreases the rate of capacity utilizationand the rate of capital accumulation. This negative effect on the growth rate is known as
the “paradox of thrift.” A rise in the savings rate decreases the rate of employment. Stockhammer (2004) also investigates the rate of employment in a Kaleckian framework. In
Stockhammer’s model, the long-run equilibrium value of the rate of employment consistsof the exogenous natural rate of growth and the parameters of the investment function andthe income distribution function, and does not depend on the savings rate. Hence, a change
in the savings rate never affects the rate of employment. In our model, on the other hand,the natural rate of growth is endogenously determined, and accordingly, the change in thesavings rate affects the rate of employment.
 Labor supply growthPrevious Kaleckian models cannot investigate the effect of supply side factors on equilibrium values. In contrast, our model can investigate these. In either case, an increase in nlowers the rate of employment. Because the relation g∗a = sm∗ u∗ − n holds in the long-run
equilibrium, an increase in n has three different effects on g∗a and consequently, on e∗ : itdirectly decreases g∗a with the coefficient of n being −1; it indirectly affects g∗a through m∗ ,which is positive when Γ < 0 and negative when Γ > 0; and it indirectly affects g∗a throughu∗ , which is negative when Γ < 0 and positive when Γ > 0. In total, the two negative effects outweigh the one positive effect, which leads to a decline in g∗a and e∗ . Stockhammer(2004) also concludes that an increase in labor supply growth leads to a decrease in the rateof employment in the profit-led growth regime (β > γ in our model), in which the long-run
equilibrium is stable. Yet, the wage-led growth regime in Stockhammer’s model (β < γin our model) is unstable, and thus, one cannot conduct a comparative statics analysis. In
contrast, even the wage-led growth regime can be stable in our model.
 Relative bargaining power
An increase in the relative bargaining power of firms either increases or decreases therate of employment depending on the size of the two elasticities of the investment function.
The rate of employment decreases when β < γ, whereas it increases when β > γ. An
increase in θ has two different effects on ga and e: it indirectly increases ga and e throughits positive effect on m and it indirectly decreases ga and e through its negative effect on u.
Whether or not the increase in θ leads to an increase in ga and e depends on which effectdominates, which in turn depends on the size of the elasticities of the investment function.
20

When β < γ, the negative effect of the capacity utilization dominates the positive effect ofthe profit share, thereby leading to a decrease in the growth rate of labor productivity and theemployment rate. When β > γ, in contrast, the positive effect of the profit share dominates
the negative effect of the capacity utilization, thereby leading to an increase in the growthrate of labor productivity and the employment rate.
Stockhammer (2004) also investigates the relationship between bargaining power andunemployment. He concludes that in the profit-led growth regime, a decrease in the bargaining power of workers leads to higher employment and lower unemployment. This result
is consistent with our results. However, in the wage-led growth regime of Stockhammer’s
model, the long-run equilibrium is unstable, and consequently, one cannot investigate therelationship between bargaining power and unemployment. In our model, in contrast, thelong run equilibrium of the wage-led growth regime can be stable. In this case, we reachthe opposite conclusion that an increase in the bargaining power of workers leads to higher
employment and lower unemployment. This result is consistent with the empirical result of
Storm and Naastepad (2007). Using data for 20 OECD countries during 1984–1997, theyshow that an increase in the bargaining power of firms because of labor market deregulation
increases the unemployment rate; this is in contrast to the view of the mainstream theory.
 Autonomous investment
We can regard the parameter ψ of the investment function as expressing a demand policy.
Setterfield (2009), for instance, relates a constant term of the investment function to a fiscal
policy and discusses the effectiveness of output targeting and inflation targeting. An increasein ψ in our model increases all equilibrium values: stimulating effective demand lowers theunemployment rate even in the long run. This implication makes a marked contrast to theimplication of the mainstream theory.

6

Concluding remarks

In this paper, we have developed a demand-led growth model that considers elements from
Goodwin, Kalecki, and Marx. In the model, we have considered the two opposing effects
caused by an increase in the rate of employment. One is the reserve-army effect: as thelabor market tightens and labor shortage becomes clearer, the bargaining power of workers
increases, which exerts an upward pressure on wages. The other is the reserve-army-creation
effect: such an upward pressure on wages leads firms to adopt labor-saving technical changes
to intentionally create the reserve-army of labor.
We have presented the four cases according to the size of parameters of the investment
function, the relative bargaining power of firms, the reserve-army effect, and the reserve21

army-creation effect. We obtain cyclical fluctuations in two of the four cases. These twocases correspond to the case where the relative bargaining power of firms is weak, thereserve-army effect is strong, and the reserve-army-creation effect is weak.
Using the model, we have analyzed how the relative bargaining power of workers andfirms affects the long-run equilibrium rate of employment. The relationship between thebargaining power and the employment rate differs depending on the regime of the long-run
equilibrium. If the long-run equilibrium is characterized as the wage-led growth regime,a rise in the relative bargaining power of workers increases the employment rate. If, on
the other hand, the long-run equilibrium is characterized as the profit-led growth regime, arise in the relative bargaining power of workers decreases the employment rate. The latter
result is also obtained in the mainstream theory, but the former result is never obtained in
the mainstream theory.
Note, however, that in the wage-led growth regime, an increase in the workers’ bargaining power leads to higher employment, but it simultaneously leads to lower profit share: theworkers’ interests interfere with firms’ interests. For this reason, it may be difficult to implement an economic policy intended to adjust the bargaining power of both classes. Even inthis case, nonetheless, the demand stimulation policy is effective. As discussed in the text,
the stimulation of effective demand brings about higher employment and accordingly, lower
unemployment. This policy implication is never obtained in the mainstream theory.

22

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25

m 0.225
0.215

0.72
0.74

u
0.76
0.78

0.86

0.84

e
0.82

Figure 1: Solution path in case 1 (α = 4)

e
e=1

0.9

Q
0.8

P

0.7

u
0.6

0.7

0.8

0.9

Figure 2: Solution paths starting from different initial values in case 1 (α = 4)26

φ(α), a2 (α)
0.1

a2
φ

0.025

α1

α

α2

2

6

-0.025-0.075Figure 3: Graphs of a2 (α) and φ(α) in case 1
0.5

m 0.45

0.6

u

0.4

0.7
0.8

0.7

e
0.6

0.5

Figure 4: Solution path in case 2 (α = 1)φ(α), a2 (α)
φ

0.8

0.6

a2
0.4

0.2

α
-1

1

2

3

4

Figure 5: Graphs of a2 (α) and φ(α) in case 2

27

0.3
0.7

m 0.25

u
0.8

0.2

1

0.95

0.9

e
0.85
0.8

Figure 6: Solution path in case 3 (α = 2)φ(α), a2 (α)
0.2

a2
0.1

φ
α1
-1

1

α2

3

4

α

-0.1

-0.2

Figure 7: Graphs of a2 (α) and φ(α) in case 3

28

0.5
0.6
0.7

u
0.8
0.9

0.7

e
0.6

0.5
0.45
0.4

m

Figure 8: Solution path in case 4 (α = 1)φ(α), a2 (α)
1

φ

0.8
0.6

a2
0.4
0.2

α
-1

1

2

3

4

Figure 9: Graphs of a2 (α) and φ(α) in case 429

Table 1: Results of comparative statics analysis
β<γ
u∗
m∗
e∗ , g∗a
g∗

s
n
θ
†
−
−/+ −
†
+/− +/−† +
−
−
−
†
−
−/+ −

mf
−
+
−
−

ψ
+‡
−/+†
+
+

β>γ
u∗
m∗
e∗ , g∗a
g∗

s
n
θ
†
−
−/+ −
+/−† +/−† +
−
−
+
†
−
+/− +

mf
−
+
+
+

ψ
+‡
−/+†
+
+

†

When Γ < 0, the left-hand sign applies,
and when Γ > 0, the right-hand sign applies.
‡
These signs are obtained by numericalcalculations.

30

A Appendix
The effects of a rise in parameters on the long-run equilibrium values are as follows.
 The rate of capacity utilizationdu∗ u∗ [sβu∗ (1 − θ)m0w (e∗ ) − (1 − sβu∗ )g0a (e∗ )]
=
< 0,
ds
s(1 − γ)Θdu∗(1 − β)u∗ Γ
=
≷ 0,
dn
(1 − γ)m∗ Θ(1 − β)u∗ g0a (e∗ )[m f − mw (e∗ )]
du∗
=−
< 0,
dθ
(1 − γ)m∗ Θ
θ(1 − β)u∗ g0a (e∗ )
du∗
=−
< 0,
dm f
(1 − γ)m∗ Θ
du∗ u∗ [(1 − γ)Θ − s(1 − β)m∗ Γ]=.
dψψ(1 − γ)2 Θ

(A-1)
(A-2)
(A-3)
(A-4)
(A-5)

 The profit shareγu∗ m∗ Γ
dm∗
=−
≷ 0,
ds
(1 − γ)Θ
dm∗Γ
= − ≷ 0,
dn
Θ
0 ∗
∗
∗
g
dm
a (e )[m f − mw (e )]
=
> 0,
dθ
Θ
dm∗ θg0a (e∗ )
=
> 0,
dm f
Θ
sm∗ u∗ Γ
dm∗
=
≷ 0.
dψ
ψ(1 − γ)Θ

(A-6)
(A-7)
(A-8)
(A-9)
(A-10)

 The rate of employmentde∗γu∗ m∗=−< 0,ds(1 − γ)Θ
1de∗= − < 0,dnΘde∗ s(β − γ)u∗ [m f − mw (e∗ )]
=
≷ 0,
dθ
(1 − γ)Θ
de∗
θs(β − γ)u∗
=
≷ 0,
dm f
(1 − γ)Θ
sm∗ u∗de∗=> 0.
dψ ψ(1 − γ)Θ
31(A-11)(A-12)(A-13)(A-14)(A-15)

 The rate of capital accumulationγm∗ u∗ g0a (e∗ )
dg∗
=−
< 0,
ds
(1 − γ)Θ
dg∗s(β − γ)u∗ Γ
=−
≷ 0,
dn
(1 − γ)Θ
dg∗ s(β − γ)u∗ g0a (e∗ )[m f − mw (e∗ )]
=
≷ 0,
dθ
(1 − γ)Θ
θs(β − γ)u∗ g0a (e∗ )
dg∗
=
≷ 0,
dm f
(1 − γ)Θ
dg∗ sm∗ u∗ g0a (e∗ )
=
> 0.
dψ
ψ(1 − γ)Θ32

(A-16)(A-17)(A-18)(A-19)(A-20)