Discussion Paper No. 245
Endogenous Technological Change,
Income Distribution,
andUnemployment with Inter-Class Conflict
Hiroaki SASAKITOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND
MANAGEMENT TOHOKU UNIVERSITY

KAWAUCHI, AOBA-KU, SENDAI,

980-8576 JAPAN

Discussion Paper No. 245

Endogenous Technological Change, Income Distribution,and Unemployment with Inter-Class Conflict
Hiroaki SASAKI
June 2009

Endogenous Technological Change,
Income Distribution, and Unemployment
with Inter-Class Conflict
Hiroaki SASAKI∗
June 2009

Abstract
This paper presents a Kaleckian model of growth that incorporates endogenous technological change. The model endogenously determines the rate of capacity utilization,
the rate of economic growth, income distribution, and the employment rate besides
technological change. The paper shows that whether or not an increase in the relative bargaining power of workers raises the long-run equilibrium unemployment ratedepends on which regime is realized in the long-run equilibrium. If, for example, thelong-run equilibrium corresponds to the wage-led growth regime, a rise in the relative
bargaining power of workers leads to a decline in the unemployment rate. This result isnever obtained from the mainstream NAIRU model.
Keywords: Kaleckian model; endogenous technological change; unemployment; interclass conflict
JEL Classification: E12; E24; E25∗

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 growth that incorporates endogenous technological change and investigates the rate of economic growth, income distribution, and the employment rate.1 Although a large number of attempts to endogenize technical progress havebeen made in mainstream growth theory, relatively less attention has been paid in the postKeynesian tradition. In mainstream growth models, much emphasis is placed on technicalprogress as an engine of growth because supply-side factors determine economic growth.
In contrast, because demand-side factors decide economic growth in post-Keynesian growthmodels, supply-side factors have not been considered so much. This is not to say that there
have been no attempts to endogenize technical progress in the Kaleckian model. You (1994)
introduces into a Kaleckian model a technical progress such that the growth rate of thecapital-labor ratio depends on the rate of capital accumulation. In Cassetti (2003), induced
technical progress known as the Kaldor-Verdoorn law (Verdoorn, 1949; Kaldor, 1966) isincorporated into a Kaleckian growth model. Stockhammer and Onaran (2004) also use the
Kaldor-Verdoorn law to build a model based on Marglin and Bhaduri’s (1990) work, andthey empirically test the model for the US, UK, and France by means of a structural VAR
analysis. Lima (2004) develops a Kaleckian model in which endogenous technological innovation plays a significant role. In Lima’s model, the rate of labor-saving technologicalinnovation depends non-linearly on the wage share, which can generate limit cycles as to
the wage share and the capital-effective labor ratio.
To endogenize technological change, this paper adopts a technique such that the growthrate of labor productivity depends positively on the rate of employment. This formulation is
proposed by Dutt (2006) and Bhaduri (2006).2 According to Dutt (2006), this view of technological change differs from the mainstream endogenous growth theory in that it draws
attention on the demand side of the economy: technological change occurs in response tolabor shortage caused by the growth of employment rather than supply side which focuseson the research and development process. Bhaduri (2006) states that this captures a viewthat technological change is driven by inter-class conflict over income distribution between
workers and capitalists. Bhaduri’s (2006) model is not a Kaleckian one because income distribution is not determined by mark-up pricing. However, it bears similarity to the Kaleckian
model in that effective demand plays a crucial role in determining output. In contrast, Dutt’s(2006) model can be said to be Kaleckian, but it does not deal with such issues as income1

See Kalecki (1954, 1971) for his economic theory. For the fundamental Kaleckian model, see Rowthorn(1981), Lavoie (1992), Taylor (2004), and Lavoie (2006).
2
Bhaduri (2006) proposes two specifications. One is what we employ in this paper, and the other is that thegrowth rate of labor productivity is adjusted through the gap between the growth rates of real wage and laborproductivity.

2

distribution and inflation because its purpose is to present a simple growth model that integrates the roles of aggregate demand and aggregate supply.
Our specification of endogenous technological change has the following theoretical implication. Conventional Kaleckian growth models assume that labor supply is unlimited andthat firms employ as many workers as they desire at given wages. If, however, the labor
supply grows at an exogenously given rate, there is no guarantee that the endogenously determined growth rate of employment is equal to the growth rate of labor supply. Thus, if
the growth of labor supply exceeds that of labor demand in the steady state, then the rateof unemployment will keep on rising, but this is unrealistic.3 In contrast, the steady state
unemployment rate in our model remains constant because the two growth rates coincide in
the long run. Therefore, our model overcomes the weakness of existing Kaleckian models.
It is true that our paper is not an initial attempt to consider the determination of theemployment rate explicitly in the Kaleckian model. Stockhammer (2004) presents an augmented Kaleckian model that incorporates equations which determine employment and income distribution, and investigates the NAIRU (non-accelerating inflation rate of unemployment).4 However, our model differs considerably from Stockhammer’s model in the determination of employment and income distribution. Stockhammer (2004) uses an employment
determination equation such that a change in the unemployment rate is given by the difference between the growth rate of exogenous labor supply and the rate of capital accumulation,
and an income distribution determination equation such that the profit share depends on theunemployment rate. On the other hand, we use an employment determination equation suchthat the growth rate of labor productivity depends positively on the employment rate, andan income distribution equation which results from the theory of conflicting-claims inflation. Furthermore, our model is different from Stockhammer’s model in that what variables
are used in the investment function and whether technological progress is exogenous or endogenous. With these differences we obtain results different from Stockhammer’s results.
In Stockhammer’s model, the rate of capital accumulation (and accordingly the rate of capacity utilization) and the profit share are adjusted in the short run while the unemployment
rate is adjusted in the long run. However, employment (and accordingly unemployment)
necessarily changes with changes in the rate of capacity utilization. Hence, it is reasonable
to assume that these three variables—the rate of capacity utilization, the profit share, and theemployment rate—are adjusted at the same time. Therefore, we simultaneously analyze theadjustment process of these three variables.
The basic framework of our model is based on a series of Mario Cassetti’s studies (Cas3

Cassetti (2002) also sees it as a problem that the long-run rate of employment in the conventional Kaleckian model is not constant.
4
For the NAIRU, see also Stockhammer (2008).

3

setti, 2002; 2003; 2006). In standard Kaleckian models, the level of money wage and markup are fixed and given exogenously, so that the price level is constant. Cassetti (2002, 2003,
2006) combine a Kaleckian growth model and the theory of conflicting-claims inflation, in
which the rate of inflation is determined by negotiations between workers and capitalists(Rowthorn, 1977).5 Kaleckian models with the theory of conflicting-claims inflation consider the effect of class conflict between workers and capitalist on income distribution, but
do not consider its effect on employment. It is interesting to investigate how changes in thebargaining power of both classes affect employment.
The remainder of the paper is organized as follows. Section 2 presents the basic framework of our model. Section 3 analyzes the existence and the stability of the long-run equilibrium. Section 4 presents numerical examples to show that the long-run equilibrium actuallyexists under plausible parameter settings and that each variable in the model converges to
its long-run equilibrium value from an arbitrary initial value. Section 5 offers results ofcomparative statics analysis in the long-run equilibrium. Section 6 concludes the paper.

2

Basic framework of the model

2.1 Adjustment in the rate of capital accumulation
Consider an economy in which there are two social classes, workers and capitalists. Supposethat workers consume all their wages and capitalists save a constant fraction s of their profits.
Let r and K be the rate of profit and the capital stock, respectively. Then, the real saving isgiven by S = srK, so that the ratio of the real saving to the capital stock, g s = S /K, leads to
g s = sr,

0 < s ≤ 1.

(1)

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

where Y is real output, E employment, and a = Y/E the level of labor productivity.6 The
rate of capacity utilization is defined as u = Y/Y ∗ , where Y ∗ is the potential output. Let us
define the ratio of the capital stock to the potential output as k = K/Y ∗ and suppose that k is
5Lima (2004), mentioned above, is also an attempt to integrate the theory of conflicting-claims inflation
with a Kaleckian growth model.
6
Given the Leontief production function, a profit maximizing firm will choose employment and capacity
utilization in such a way that aE = (u/k)K, from which it follows that a = Y/E.

4

constant, that is, both K and Y ∗ grow at the same rate.7 To simplify the analysis we assume
k = 1. Then, we obtain u = Y/K. Note that the relationship among the profit rate, the profit
share (m), and the rate of capacity utilization is given by r = mu and the relationship among
the rate of capacity utilization, the rate of capital accumulation, and the profit share is given
by u = g/(sm).
Following the argument of Marglin and Bhaduri (1990), we specify the ratio of the real
investment to the capital stock, gd = I/K, as follows:
gd = Amφ uγ ,A > 0, 0 < φ < 1, 0 < γ < 1,(3)

where A denotes a constant, φ the elasticity of the investment rate with respect to the profit
share, γ the elasticity of the investment rate with respect to the rate of capacity utilization.
Equation (3) means that the desired investment rate of firms is an increasing function bothin the profit share and in the rate of capacity utilization. In conventional Kaleckian models,the investment function is assumed to depend positively on the rate of profit and the rateof capacity utilization. Marglin and Bhaduri (1990), in contrast, argue that not the rate ofprofit but the profit share should be a variable in the investment function. The reason forthis is as follows. The rate of profit is equal to the product of the profit share and the rateof capacity utilization divided by the capital output ratio at full capacity, that is, r = mu/k.
Thus, it is plausible that a combination of high capacity utilization and a low profit share anda 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.8 Our specificationof the investment function is based on Blecker (2002).9 Since the investment function is nota linear but a Cobb-Douglas form, as will be shown later, different regimes can be produced
according to the sizes of φ and γ.
An equation of motion for the rate of capacity utilization can be formulated byu̇ = α(gd − g s ),α > 0,(4)

where α is the speed of adjustment. Equation (4) represents quantity adjustment by the rate
7

Foley and Michl (1999) emphasize that, in capitalist economies, a combination of growing labor productivity and declining capital productivity (i.e., a rise in k in our model) is the typical pattern of technologicalchange. They accordingly term the pattern Marx-biased technical change. When an economy experiences
Marx-biased technical change, the rate of profit will decline as long as the profit share remains constant. In
our model, such a biased technical change is not taken into consideration.
8
For the details of the argument, see also Bhaduri and Marglin (1990).
9
If φ > 1 in this specification, we have the exhilarationist regime in which an increase in the profit shareleads to a rise in the rate of capacity utilization. However, φ > 1, as Blecker (2002) points out, is an extreme
case. Hence, we assume 0 < φ < 1 in the following analysis.

5

of capacity utilization in the goods market: excess demand (gd > g s ) leads to a rise in therate of capacity utilization whereas excess supply (gd < g s ) leads to a decline in the rate ofcapacity utilization.
Substituting equations (1) and (3) into equation (4), we finally obtain the dynamics of u
as follows:u̇ = α(Amφ uγ − smu).

(5)

2.2 Adjustment in the profit share
In the Kaleckian tradition, firms operate with excess capacity in oligopolistic goods markets
and set their prices p with a mark-up µ on unit labor costs:w
p = (1 + µ) ,a(6)where w is the money wage rate. It should be noted that m = µ/(1 + µ), that is, there exists a
one-to-one relationship between the mark-up and the profit share,10 so that m and µ change
in the same direction. Therefore, we can rewrite equation (6) as follows:
!
1
w
p=
.
1−m a

(7)

Differentiating both sides of equation (7) with respect to time yields:ṗ ẇ ȧṁ
= − + .
1−m p w a(8)

To know the dynamics of m we have to specify the rates of change 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 which workers
manage to negotiate depends on the discrepancy between their target real wage rate and theactual real wage rate. Given the labor productivity, to determine the real wage means todetermine the wage share. Therefore, setting the target real wage rate is equivalent to setting
the target profit share. Second, suppose that firms set their price to close the gap between
their target mark-up and the actual mark-up. The target mark-up corresponds to the target
10

The relationship between m and µ is given as follows:
m=1−

wE
w1
1
µ
=1−
=1−
=
.
pY
pa
1+µ 1+µ6profit share because, as stated above, the mark-up bears a one-to-one relation to the profit
share. From these the dynamics of the money wage and price can be described, respectively,
as follows:ẇ= θw (m − mw ),wṗ= θ f (m f − m),pθw > 0, 0 < mw < 1,

(9)

θ f > 0, 0 < m f < 1,(10)where θw and θ f are the speed of adjustment, mw is the target profit share set by workers, andm f is the profit share set by firms.
We can interpret θw and θ f as the bargaining power of workers and that of firms, respectively.11 We suppose θ f + θw = 1 and define θ f ≡ θ because bargaining power is a relative
concept. Then, we obtain θw = 1 − θ, where 0 < θ < 1.12 We can take an increase in theunionization rate as a factor for raising the bargaining power of workers (i.e., a decreasein θ), and an increase in the market power of oligopolistic firms as a factor for raising thebargaining power of firms (i.e., an increase in θ).
Notice the difference between θ, and m f and mw . The parameter θ represents the relative
bargaining power of firms (workers) and reflects the power to realize their demands. In
contrast, m f and mw reflect their demands in the bargaining. To what extent their demands
can be realized depends on θ.
Substituting equations (9) and (10) into equation (8), we obtain an equation of motion
for the profit share:ṁ = −(1 − m)[m − θm f − (1 − θ)mw − ga ],(11)

where ga = ȧ/a is the growth rate of labor productivity.

2.3 Adjustment in the growth rate of labor productivity
We now turn to the specification of endogenous technological change. Following Bhaduri(2006) and Dutt (2006), we describe the growth rate of labor productivity as follows:
ga = λeβ ,λ > 0, β > 0,

11

(12)

This interpretation is also adopted in Lavoie (1992, p. 393), Cassetti (2002, p. 192), and Cassetti (2003, p.
453).
12
The constraint 0 < θw , θ f < 1 is also adopted by Dutt and Amadeo (1993), who, however, do not assume
θ f + θw = 1. Even if we only impose 0 < θw , θ f < 1 and do not impose θ f + θw = 1, we obtain similar results.

7

where e is the employment rate, which is defined as e = E/N with N being the exogenous
labor supply, λ is a constant, and β is the elasticity of the growth rate of labor productivity
with respect to the employment rate.
Equation (12) shows that the growth rate of labor productivity is an increasing function of the employment rate. In general, the natural rate of growth is defined as a sum of
the growth rate of labor productivity and that of labor supply. Although the growth rateof labor supply in our model is exogenously given, the growth rate of labor productivity is
endogenously determined. Under our specification, therefore, the natural rate of growth increases while business is good (i.e., a high employment rate) and it decreases while businessis bad (i.e., a low employment rate). The assumption that the natural rate of growth is endogenously determined is consistent with empirical studies of León-Ledesma and Thirlwall(2002), Libânio (2009), and Vogel (2009).
Using equation (12), we can show that in the long-run equilibrium the growth rate ofoutput per capita (Y/N) coincides with that of labor productivity (Y/E). In this respect
Sedgley and Elmslie (2004) empirically show the cointegration between the log of output
per capita and the log of labor productivity.13 This evidence suggests that these two variables
move together in the long run.
From equation (12), the rate of change of ga is given by
ġa
ė
=β .
ga
e

(13)

From equation (2) the employment is E = uK/a, and the employment rate is e = uK/(aN),so that the rate of change of the employment rate leads toė u̇= + g − ga − n,
e u

(14)

where n is the growth rate of N and given exogenously. We should pay attention to the rateof capital accumulation g in equation (14). In our model there are two kinds of rates ofcapital accumulation, that is, gd and g s . We have g = gd = g s in equilibrium. However, wehave gd , g s out of equilibrium. Therefore, we face the problem of which rates of capital
accumulation we should adopt in equation (14). Because investment determines saving in

13

Their original purpose is to empirically test which model is more valid, the classical conventional wage
share model proposed by Foley and Michl (1999) or the classical full employment model proposed by Pasinetti(1974). The growth rate of output per capita and that of labor productivity do not coincide in the classicalconventional wage share model, whereas the two growth rates coincide in the classical full employment model.
For this reason we can know which model is more valid by examining the long-run relationship between thetwo variables.

8the model, we should adopt gd as the actual rate of capital accumulation.14
Substituting equations (3) and (5) into equation (14), and the resultant expression into
equation (13), we obtain the dynamics of ga :15ġa = βga [α(Amφ uγ−1 − sm) + Amφ uγ − ga − n].

3

(15)

Existence and stability of the long-run equilibrium

From equations (5), (11), and (15), our model consists of the following system of differential
equations:u̇ = α(Amφ uγ − smu),(16)ṁ = −(1 − m)[m − θm f − (1 − θ)mw − ga ],(17)

ġa = βga [α(Amφ uγ−1 − sm) + Amφ uγ − ga − n].

(18)

3.1 Existence of the long-run equilibrium
Long-run equilibrium is a situation in which u̇ = ṁ = ġa = 0, from which we have thefollowing three equations in order:
Amφ uγ = smu,(19)m = θm f + (1 − θ)mw + ga ,(20)

Amφ uγ = ga + n.

(21)

From equation (19) we have
1

1

φ−1

u = A 1−γ s− 1−γ m 1−γ .

(22)Substituting ga = smu − n from equations (19) and (21) into equation (20), and substituting
equation (22) into the resultant expression, we obtain
1

γ

φ−γ

A 1−γ s− 1−γ m 1−γ − m + θm f + (1 − θ)mw − n = 0.

(23)This equation determines m∗ (hereafter, equilibrium values are denoted with “∗”). Using m∗ ,we can find u∗ and g∗a .
14

See also Marglin (1984, pp. 85–88) in this respect.
We can use an equation of motion for e instead of the one for ga because e is related to ga through equation(12). Naturally, we have the same results.
15

9

Let us explain the determination of m∗ . We rewrite equation (23) as follows:
γ

1

φ−γ

A 1−γ s− 1−γ m 1−γ + B = m + n,(24)

where B ≡ θm f + (1 − θ)mw . Each side of equation (24) is a function of m, so that we canwrite each side as follows:
1

γ

φ−γ

LHS ≡ Fm (m) = A 1−γ s− 1−γ m 1−γ + B,

(25)

RHS ≡ Gm (m) = m + n.

(26)

The intersection of these curves determines m∗ . Fm (m) can be upward sloping or downward
sloping depending on the sizes of φ and γ. Gm (m) is a straight line such that the slope andthe intercept are unity and n, respectively. From this we have Figures 1 and 2. Both figures
are drawn with the condition that 0 < m∗ < 1.
[Figures 1 and 2 to be inserted here]
To obtain these figures we need the constraints B > n and Gm (1) > Fm (1). Similarly,
we can find the constraints under which economically meaningful values for u∗ and g∗a are
obtained.16 These constrains are summarized as follows:1γm∗ ∈ (0, 1) =⇒ C1 : B − n > 0 and C2 : 1 + n − B − A 1−γ s− 1−γ > 0,u∗ ∈ (0, 1) =⇒ C3 : s(1 − s)φ−1 − A(B − n)φ−1 > 0,
1

γ

φ−γ

g∗a > 0 =⇒ C4 : A 1−γ s− 1−γ B 1−γ − n > 0.
The intersection of the conditions C1–C4 is the necessary and sufficient condition for the
long-run equilibrium values to be economically meaningful. Although it is difficult to obtain
the intersection, numerical examples of parameters used in simulation below satisfy these
conditions.

3.2 Local stability of the long-run equilibrium
To analyze the local stability of the long-run equilibrium we linearize the system of differential equations (16), (17), and (18) around the equilibrium.

16

See Appendix 1 for u∗ and g∗a .

10   u̇   J11 J12 0   u − u∗   ∗ ṁ  =  0 J22 J23  m − m  ,ġa
J31 J32 J33 ga − g∗a(27)where the elements of the Jacobian matrix J are given by∂u̇= −αsm(1 − γ) < 0,∂u∂u̇
= −αsu(1 − φ) < 0,
J12 ≡
∂m
∂ṁ
J22 ≡
= −(1 − m) < 0,∂m∂ṁ
= 1 − m > 0,
J23 ≡
∂ga
∂ġa βsmga
J31 ≡=[α(γ − 1) + γu] ≷ 0,
∂u
u
∂ġa
= βsga [α(φ − 1) + φu] ≷ 0,
J32 ≡
∂m
∂ġa
J33 ≡
= −βga < 0.
∂ga

J11 ≡

(28)
(29)
(30)
(31)
(32)
(33)
(34)

All elements are evaluated at the long-run equilibrium, though we omit “∗” to avoid troublesome notations. J31 and J32 can be negative when the speed of adjustment α is large.
Let us explain the Jacobian matrix. The diagonal elements J11 , J22 , and J33 represent theown effects of endogenous variables on themselves. They are negative, which implies thatwhen a variable deviates from its long-run equilibrium value, it returns to the equilibrium
value. This negative feedback has a stabilizing effect. However, a change in one variableaffects other variables, whose effects in turn rebound on the variable in question. We takethe rate of utilization for example. J31 shows that a rise in the rate of utilization causes anincrease or a decrease in the growth rate of labor productivity. If ga increases, then from
J23 it induces a rise in the profit share. From J12 this rise in the profit share lowers the rateof capacity utilization. To sum up, this route has negative feedback, leading to stability.
In contrast, if ga decreases in J31 , then from J23 it induces a fall in the profit share, whichraises the rate of capacity utilization from J12 . This route has positive feedback, leading toinstability.
To examine the local stability of the long-run equilibrium, we have to know the properties of the characteristic roots of the Jacobian matrix. The characteristic equation of the11Jacobian matrix is given by
q3 + b1 q2 + b2 q + b3 = 0,(35)

where q denotes a characteristic root. Each coefficient of equation (35) is expressed by
b1 = −tr J = αsm(1 − γ) + (1 − m) + βga > 0,(36)

b2 = J22 J33 − J23 J32 + J11 (J22 + J33 )= βga {(1 − m)(1 − sφu) + αs[m(1 − γ) + (1 − m)(1 − φ)]}+ αsm(1 − γ)(1 − m) > 0,(37)

b3 = − det J = −J11 (J22 J33 − J23 J32 ) − J31 J12 J23= αβsm(1 − m)ga [(1 − γ)(1 − sφu) + sγu(1 − φ)] > 0,(38)

where −b1 = tr J is the trace of J, b2 is the sum of the principal minor determinants, and−b3 = det J is the determinant of J. As shown above, all signs are positive. The necessary
and sufficient condition for stability is that all characteristic roots of the Jacobian matrix
have negative real parts, which, from Routh-Hurwitz conditions, is equivalent to b1 > 0,b2 > 0, b3 > 0, and b1 b2 − b3 > 0.17 We compute b1 b2 − b3 as follows:
b1 b2 − b3 = α2 s2 m(1 − γ){βga [m(1 − γ) + (1 − m)(1 − φ)] + m(1 − γ)(1 − m)}+ β2 g2a {(1 − m)(1 − sφu) + αs[m(1 − γ) + (1 − m)(1 − φ)]}+ (1 − m)2 [βga (1 − sφu) + αsm(1 − γ)]+ αβs(1 − m)ga [2m(1 − γ) + (1 − φ) (1 − m − msuγ)].
|
{z
}

(39)

≡Θ

The sign of b1 b2 − b3 is indeterminate. Note, however, that in equation (39) all parts except
the part defined as Θ is positive. Let us turn to Θ.
Θ ≡ 1 − m − msuγ.

(40)First, when s, u, and γ are close to zero, Θ will be positive because msuγ in Θ will be
sufficiently small. Second, when s, u, and γ are close to unity, Θ approaches
Θ = 1 − m − m = 1 − 2m.

(41)

From this if m∗ ≤ 1/2, that is, the equilibrium profit share is smaller than or equal to 1/2,
17

See Gandolfo (1996) for details.

12then Θ > 0, so that b1 b2 − b3 > 0. Therefore, if m∗ ≤ 1/2, the necessary and sufficientcondition for the local stability of the long-run equilibrium is satisfied. In general the profit
share in the real world is considered to be smaller than 1/2, and thus the condition m∗ ≤ 1/2
is plausible.
Note, however, that m∗ ≤ 1/2 is a sufficient and not a necessary condition for b1 b2 − b3 >0. Moreover, note that m∗ depends on the parameters of the model.18

4

Numerical examples

This section presents numerical examples to show that the economically meaningful longrun equilibrium actually exists under plausible parameter settings and that each variable
converges to its long-run equilibrium value from an arbitrary initial value.19 Considering theshape of the investment function, we investigate two cases in what follows: φ < γ (Case 1)
and φ > γ (Case 2).
To begin with, we set parameters and initial conditions common to Cases 1 and 2 asfollows:
α = 1, β = 0.5, A = 0.3, s = 0.8, θ = 0.4, m f = 0.3, mw = 0.2, n = 0.04,
u(0) = 0.6, m(0) = 0.4, ga (0) = 0.15.
Next, we set the parameters of the investment function as follows:
Case 1 : φ = 0.2, γ = 0.3,
Case 2 : φ = 0.3, γ = 0.2.
The numerical examples above satisfy the conditions C1–C4: in Case 1, C1 : 0.20 > 0,
C2 : 0.60 > 0, C3 : 1.81 > 0, C4 : 0.20 > 0; and in Case 2, C1 : 0.20 > 0, C2 : 0.57 > 0,
C3 : 1.54 > 0, C4 : 0.16 > 0.
Table 1 shows the equilibrium values for Cases 1 and 2. From this we see that u∗ , m∗ ,and g∗a all take reasonable values.
18

The condition for m∗ ≤ 1/2 is given byGm (1/2) ≥ Fm (1/2) =⇒

! φ−γ
γ
1
1
1 1−γ
+ n ≥ A 1−γ s− 1−γ
+ B.
2
2

19

For numerical calculations, we use Mathematica 4 of Wolfram Research Inc. A Mathematica notebook
file containing the code is available from the author upon request.

13

[Table 1 to be inserted here]
Figures 3–8 show the time path (up to t = 100) of each variable. Every variable converges to its equilibrium value. In these numerical examples Cases 1 and 2 show similar
dynamics.
In the example we set the speed of adjustment of the goods market, that is, α to 1. If
we use α which is larger or smaller than 1, the speed of convergence gets faster or slower.
Nonetheless, each variable converges to its long-run equilibrium value.
We set the elasticity of the growth rate of labor productivity with respect to the employment rate, that is, β to 0.5. As can be seen from equation (18), β also represents the speed ofadjustment of ga . As is the case with α, even if we use β which is larger or smaller than 0.5,
each variable converges to its equilibrium value.
[Figures 3, 4, 5, 6, 7, and 8 to be inserted here]

5

Comparative statics analysis

This section investigates the effects of shifts in parameters on the long-run equilibrium.
Table 2 summarizes the results of comparative statics analysis.20 These results are obtained
under the assumption that the long-run equilibrium is stable. In addition, m f − mw > 0
is assumed in the analysis. It is reasonable to assume that firms attempt to set m f as highas possible whereas workers attempt to set mw as low as possible. Hence, the assumptionm f − mw > 0 can be justified. Note, finally, that g∗a and e∗ are dealt with in the same row
because these two variables move in the same direction (see equation (12)).
[Table 2 to be inserted here] Saving rate
An increase in the saving rate decreases the rate of capacity utilization, the profit share,
and the rate of capital accumulation. This negative effect on the growth rate is known as “theparadox of thrift.” An increase in the saving rate decreases the employment rate, thereby
increasing the unemployment rate. In Stockhammer (2004) the long-run equilibrium rateof unemployment consists of the exogenous natural rate of growth and parameters of theinvestment and income distribution functions, and does not depend on the saving rate. Consequently, a change in the saving rate never affects the unemployment rate. In our model, on
the other hand, g∗a and accordingly the natural rate of growth are endogenously determined,
so that a change in the saving rate affects the unemployment rate.
20

See Appendix 2 for details.

14 Labor supply growth
A rise in the growth rate of labor supply increases the rate of capacity utilization anddecreases the profit share. This mechanism is as follows. A rise in n creates excess supply
in the labor market, thereby depressing the employment rate. Because the growth rate oflabor productivity depends positively on the employment rate, the depressing effect on theemployment rate also decreases the labor productivity, which also decreases the profit sharefrom equation (20). This decrease in the profit share leads to an increase in the rate of capacity utilization through equation (22). The conventional Kaleckian model cannot investigate
the effect of supply side factors on equilibrium values. In contrast, our model can investigateit.
Let us turn to the effect on the employment rate. A rise in the growth rate of labor supply
decreases the equilibrium employment and thus increases the equilibrium unemployment
rate. Because the relation g∗a = sm∗ u∗ − n holds in the long-run equilibrium, a rise in n
has three different effects on g∗a : it directly decreases g∗a with the coefficient of n being −1;
it indirectly decreases g∗a through m∗ with m∗ decreasing in n; and it indirectly increasesg∗a through u∗ with u∗ increasing in n. In total the two negative effects outweigh the one
positive effect, which leads to a decrease in g∗a and e∗ with an increase in n. Stockhammer(2004) also concludes that an increase in the growth rate of labor supply leads to a rise in theunemployment in the profit-led growth regime where the long-run equilibrium is stable.21
Finally, a rise in the growth rate of labor supply either increases or decreases the rate ofcapital accumulation depending on the sizes of the two elasticities of the investment function. To understand it we need to remind ourselves that from equation (3), g∗ = A(m∗ )φ (u∗ )γholds in the long-run equilibrium. As stated above, a rise in n decreases m∗ and increasesu∗ . When φ < γ, the positive effect of u∗ exceeds the negative effect of m∗ , so that in total g∗increases with a rise in n. When φ > γ, the converse holds.
 Bargaining power
An increase in θ, which corresponds to an increase in the relative bargaining power offirms, brings about a decrease in the rate of capacity utilization and an increase in the profit
share. The effect on the employment rate is worth considering. A rise in θ either increases ordecreases the employment according to the sizes of φ and γ. The employment rate decreases
when φ < γ, while it increases when φ > γ. A rise in θ has two different effects on ga and e:
it indirectly increases ga and e through its positive effect on m; and it indirectly decreases ga
and e through its negative effect on u. Whether or not the rise in θ leads to an increase in ga
and e depends on which effect dominates, which in turn depends on the sizes the elasticities
21

Rowthorn (1999) reaches a similar conclusion by using different model, in which trade unions and firms
are engaged in Nash bargaining for wages.

15

of the investment function. When φ < γ, the negative effect of the capacity utilization
dominates the positive effect of the profit share, thereby leading to a decrease in the growthrate of labor productivity and the employment rate. When φ > γ, in contrast, the positiveeffect of the profit share dominates the negative effect of the capacity utilization, thereby
leading to an increase in the growth rate 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 ours. However, in the wage-led growth regime of Stockhammer’s model,
the long-run equilibrium is unstable, so that we cannot investigate the relationship between
bargaining power and unemployment. In our model, in contrast, the-long run equilibriumof the wage-led growth regime can be stable. In this case we reach the opposite conclusionthat an increase in the bargaining power of firms 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, they show that an increasein the bargaining power of firms due to labor market deregulation raises the unemployment
rate in contrast to the view of the mainstream NAIRU model.22
 Target profit share
The effect of an increase in the target profit share mw or mw is similar to that of anincrease in θ discussed above because of the structure of the model. An important issue inthe Kaleckian model is that different regimes arise due to the specification of the investment
function. According to Blecker’s (2002) classification, the stagnationist (exhilarationist)
regime is a situation where an increase in the profit share decreases (increases) the rate ofcapacity utilization. The wage-led (profit-led) growth regime is a situation where an increasein the profit share decreases (increases) the rate of capital accumulation. The profit sharein our model is not an exogenous but an endogenous variable, and thus we cannot apply
Blecker’s classification to the model as it is. However, because the actual profit share m isclosely related to the target profit shares m f and mw , these variables can be a proxy variable
for the actual profit share. An increase in m f and mw leads to a decrease in the rate ofcapacity utilization, which corresponds to the stagnationist regime. In addition, an increasein m f and mw raises the rate of capital accumulation when φ > γ while it lowers the growthrate when φ < γ: the former case corresponds to the profit-led growth regime while the lattercase corresponds to the wage-led growth regime.
 Autonomous investment

22

In Storm and Naastepad (2008) similar empirical results are obtained.

16

We can regard the parameter A 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 increase
in A in our model raises 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 NAIRU theory.

6

Concluding remarks

This paper has developed a Kaleckian model in which the rate of technological change andthe employment rate are endogenously determined. The model is based on the Kaleckian
model with the theory of conflicting-claims inflation, and is extended to incorporate endogenous technological change. Our model responds to the criticism that in the usual Kaleckian
model technological change is not considered and the long-run employment rate is not constant.
Using the model, we have analyzed how the relative bargaining power of workers andfirms affects the long-run equilibrium rate of unemployment. The relationship between thebargaining power and the unemployment rate differs depending on the regime in which
the long-run equilibrium lies. If the long-run equilibrium is characterized as the wage-led
growth regime, a rise in the relative bargaining power of firms increases the unemployment
rate. If, on the other hand, the long-run equilibrium is characterized as the profit-led growth
regime, a rise in the relative bargaining power of workers increases the unemployment rate.
The latter result is also obtained in the mainstream NAIRU model, but the former result isnever obtained in the mainstream model. Note, however, that a fall in the firms’ bargaining power, that is, a rise in the workers’ bargaining power leads to higher employment, but
it simultaneously leads to lower profit share: workers’ 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 in this case, nonetheless, demand stimulation
policy is effective. As discussed in the text, stimulation of effective demand brings about
higher employment and accordingly lower unemployment. This policy implication is neverobtainable from the mainstream NAIRU theory.
Our way of introduction of technological change is very simple. Rowthorn (1981) statesthat technical progress influences an economy in two ways. First, technical progress makes
existing equipment obsolete, and thus it will affect the rate of depreciation. Second, technicalprogress stimulates firms that undertake innovations to invest more by bringing extra profits
to them, and thus the form of investment function will be modified. In Cassetti (2003)17

these effects are taken into account, while in the present paper these issues are not dealt
with for the purpose of emphasizing the role of endogenous technological change in the
Kaleckian model of growth. For the same purpose target rates of workers and firms are notendogenized. It is evident that technological change influences the target rates if these areendogenized. Taking these into account will be future research.

A Appendix 1
A.1 Determination of the equilibrium rate of capacity utilization
Eliminating m and ga from equations (19), (20), and (21), we obtain the following equation
for u:
A(B − n)φ−1 uγ−1 = s(1 − su)φ−1 .

(A-1)

Both sides are functions of u. Thus, we can rewrite them as follows:
LHS ≡ Fu (u) = A(B − n)φ−1 uγ−1 ,(A-2)

RHS ≡ Gu (u) = s(1 − su)φ−1 .

(A-3)

The intersection of these curves determines u∗ .
[Figure 9 to be inserted here]

A.2

Determination of the equilibrium growth rate of labor productivity

Eliminating u and m from equations (19), (20), and (21), we obtain the following equation
for ga :1φ−γ

γ

A 1−γ s− 1−γ (ga + B) 1−γ = ga + n.

(A-4)

Both hands are functions of ga . Thus, we can rewrite them as follows:
1

γ

φ−γ

LHS ≡ Fga (ga ) = A 1−γ s− 1−γ (ga + B) 1−γ ,(A-5)

RHS ≡ Gga (ga ) = ga + n.

(A-6)

The intersection of these curves determines g∗a .
18

[Figures 10 and 11 to be inserted here]

B

Appendix 2

We investigate the effects of changes in parameters on the long-run equilibrium. A parameter, Ω, which will appear below, is defined asΩ≡1−(φ − γ)su
.
1−γ

(B-7)

When φ < γ, we have Ω > 0. When φ > γ, 0 < (φ − γ)/(1 − γ) < 1 because 0 < φ < 1.
Since 0 < s < 1 and 0 < u < 1, we have Ω > 0 when φ > γ. Therefore, we always haveΩ > 0 irrespective of φ < γ or φ > γ.
Moreover, B is defined as B ≡ θm f + (1 − θ)mw as in the text. A rise in θ, m f , and mw
leads to a rise in B given m f > mw .
Finally, all endogenous variables are evaluated at the long-run equilibrium.
 The rate of capacity utilization
γ−2
φ−1 1 − φsu
1
1
du∗
=−
A 1−γ s 1−γ m 1−γ
< 0,
ds
1−γ
Ω
φ+γ−2 1
1
1
du∗ 1 − φ 1−γ=
A s− 1−γ m 1−γ
> 0,
dn
1−γ
Ω
φ+γ−2 11
1
du∗
1 − φ 1−γ=−
A s− 1−γ m 1−γ
< 0,
dB
1−γ
Ω
γ
φ−1 1 − su
1
du∗
1
=
s 1−γ A 1−γ m 1−γ
> 0.
dA
1−γ
Ω

(B-8)
(B-9)
(B-10)(B-11)

 The profit share
dm∗
γ
1
=−
mu < 0,
ds
1−γ
Ω
∗
dm
1
= − < 0,dnΩ1dm∗
= > 0,
dB
Ω
dm∗
1
1
=
A−1 smu > 0.
dA
1−γΩ
 The growth rate of labor productivity
19(B-12)(B-13)(B-14)(B-15)

In the long-run equilibrium we have g∗a = m∗ − B. Hence, results for the parameters
except B are the same as in the case of m∗ . As for B, we havedg∗a φ − γ
1
=
su .
dB
1−γ
Ω

(B-16)

This derivative will be negative when φ < γ while positive when φ > γ.
 The rate of capital accumulation
In the long-run equilibrium we have g∗ = g∗a +n. Hence, results for the parameters except
n are the same as in the case of g∗a . The relationship between g∗ and m∗ is given by
γ

1

φ−γ

g∗ = A 1−γ s− 1−γ (m∗ ) 1−γ .

(B-17)As shown above, an increase in n leads to a decrease in m∗ . Therefore, we have dg∗ /dn > 0
when φ < γ while dg∗ /dn < 0 when φ > γ.

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22

Figures and Tables

RHS
Gm (1)

Gm (1)RHS

F m (1)LHS

LHS

F m (1)

B
n

n
O

m∗

m

1

O

Figure 1: Determination of m∗ (φ < γ)
u

m∗

1

m

Figure 2: Determination of m∗ (φ > γ)
ga

m

0.68
0.18

0.42

0.175

0.66
0.415

0.17
0.64

0.41

0.165
0.16

0.405

0.62

t
20

t
20

40

60

80

40

60

80

0.155

100

t

100

20

40

60

80

100

Figure 3: Time path of u Figure 4: Time path of m Figure 5: Time path of ga(φ < γ)(φ < γ)(φ < γ)
u

ga

m

0.65

0.17

0.41
0.408

0.64

0.165

0.406
0.63

0.404
0.16
0.402

0.62

t
20

0.61

40

60

80

100

0.155

0.398

t
20

40

60

80

0.396

t
20

100

40

60

80

100

Figure 6: Time path of u Figure 7: Time path of m Figure 8: Time path of ga(φ > γ)(φ > γ)(φ > γ)23

Table 1: Equilibrium values for Cases 1 and 2
Case 1Case 2

u∗
0.66
0.64

m∗
g∗a
0.42 0.18
0.41 0.17

Table 2: Results for comparative statics
analysisu∗m∗∗ga , e∗
g∗
†
‡

s
−
−
−
−

n
θ † m f , mw † A
+
−
−
+
−
+
+
+
−
−/+‡ −/+‡ ++/−‡ −/+‡ −/+‡ +We assume m f > mw .
When φ < γ, the left-hand sign applies,
and when φ > γ, the right-hand sign applies.

Gu (1)RHS

F u (1)
s
O

LHS
u∗

1

Figure 9: Determination of u∗

24

u

RHS
F ga (0)
LHS

RHS

n

O

g∗a

LHS

F ga (0)ngaOFigure 10: Determination of g∗a (φ < γ)

g∗a

ga

Figure 11: Determination of g∗a (φ > γ)25