Discussion Paper No. 237
Endogenous Technological Change andDistribution with Inter-Class Conflict:
A Kaleckian Model of Growth
Hiroaki Sasaki
August 2008

TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND

MANAGEMENT TOHOKU UNIVERSITY

KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANEndogenous Technological Change and Distribution
with Inter-Class Conflict: A Kaleckian Model of
GrowthHiroaki SASAKI∗
August 2008

Abstract
This paper presents a Kaleckian model of growth in which both technological change
and income distribution are endogenously determined by inter-class conflict between
capitalists and workers. Considering the adjustment speed of variables, we investigate
the medium-run equilibrium and the long-run equilibrium. In the medium run, the rateof capital accumulation and the profit share are adjusted to the medium-run equilibrium.
In the long run, the normal planned rate of capacity utilization and the growth rateof labor productivity are adjusted to the long-run equilibrium. In the analysis, twoalternative investment functions are used: a Kalecki type investment function and a
Marglin and Bhaduri (1990) type investment function. We show that different results
are obtained both in the medium run and in the long run depending on which investment
function is used.
Keywords: Kaleckian model; endogenous technological change; long-run equilibrium;
wage-led; profit-led
JEL Classification: E12; E22; E25; O30; 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

The purpose of this paper is to present a Kaleckian growth model that incorporates endogenous technological change and to investigate its properties.1) Although a large numberof attempts to endogenize technical progress have been made in mainstream growth theory, relatively less attention has been paid in the post-Keynesian tradition. In mainstream
growth models, much emphasis is placed on technical progress as an engine of growth because supply-side factors determine economic growth. In contrast, since demand-side factors decide economic growth in post-Keynesian growth models, supply-side factors havenot 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 the capital-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) is incorporated into a Kaleckian
growth model. Stockhammer and Onaran (2004) also use the Kaldor-Verdoorn law to builda model based on Marglin and Bhaduri’s (1990) work, and they empirically test the modelfor 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 technological innovation depends non-linearly on thewage share, which can generate limit cycles as to the wage share and the capital-effective
labor ratio.2)
In order to endogenize technological change, this paper adopts a technique such that achange in the growth rate of labor productivity depends positively on the difference between
the growth rates of employment and labor supply. This formulation is proposed by Dutt(2006) and Bhaduri (2006).3) According to Dutt (2006), this view of technological change
differs from the mainstream endogenous growth theory in two respects. First, it draws attention on the demand side of the economy: technological change occurs in response to labor
shortage caused by the growth of employment rather than supply side which focuses on the
research and development process. Second, it stresses the process of diffusion of technological change among firms which are driven to adopt the technology by labor shortages,
rather than the process of invention. To the extent that demand-side factors and diffusion are
1) See Kalecki (1971) for his economic theory. For the fundamental Kaleckian model, see Rowthorn (1981).
Lavoie (1992, ch. 6) explains various Kaleckian models.
2) Raghavendra (2006) also obtains limit cycles in a Kaleckian model, but the method is different from thatin Lima (2004) mentioned above. In Raghavendra’s (2006) model, stagnationism and exhilarationism can be
repeated alternately, and this result greatly depends on the non-linearity of the investment function.
3) Indeed, Bhaduri (2006) proposes two specifications. One is what we employ in this paper, and the otheris that the growth rate of labor productivity is adjusted through the gap between the growth rates of real wage
and labor productivity.

2

important in the process of technological change, this approach can be argued to be a plausible one. Bhaduri (2006) states that this captures a view that 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 markup 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 income 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 labor supply growth. Thus, if the former
exceeds the latter in the steady state, then the rate of unemployment will keep on rising,but this is not realistic.4) In contrast, the steady state unemployment rate in our model remains constant because the two growth rates coincide in the long run. Stockhammer (2004)
proposes and investigates a post-Keynesian growth model in which income distribution andthe rate of unemployment are endogenously determined in the long run. In this respect, themodel is similar to our model. In Stockhammer (2004), however, technological change is
not considered, and consequently the long-run rate of capital accumulation is equal to thenatural rate of growth (i.e., the growth rate of labor supply) that is given exogenously. In
contrast, the log-run rate of capital accumulation in our model is endogenously determined.
The basic framework of our model is based on a series of Mario Cassetti’s studies (Cassetti, 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) Let us explain the theory according to Dutt (1987), Lavoie (1992,
ch. 7), and Cassetti (2002, 2003, 2006). Workers are assumed to attempt to change income
distribution in their favor by negotiating for higher money wages, while capitalists are assumed to secure their profits by setting higher prices. At a steady state, wage and profit
shares will be constant, and the growth rates of prices and money wages are equalized if the4) Cassetti (2002) also sees it as a problem that the long-run rate of employment in the conventional Kaleckian model is not constant.
5) Lima (2004), mentioned above, is also an attempt to integrate the theory of conflicting-claims inflation
with a Kaleckian growth model.

3

labor productivity is constant. The resultant inflation rate depends on the bargaining powerof workers and that of capitalists, and on the gap between the target profit share desired byworkers and the target profit share desired by capitalists. This gap is called the “aspiration
gap.”
In what follows, we analyze the medium-run equilibrium and the log-run equilibrium of
the model to consider the adjustment speed of endogenous variables. In the medium run, therate of capital accumulation and the profit share are simultaneously adjusted to the mediumrun equilibrium while, in the long run, both the normal planned rate of capacity utilizationand the growth rate of labor productivity are adjusted to the long-run equilibrium.6)
In addition to the comparison between the medium run and the long run, we use twoalternative investment functions to compare results obtained from differences between theinvestment functions. In the conventional Kaleckian model, investment is assumed to depend positively on the profit rate and the rate of capacity utilization (hereafter the Kalecki
type investment function). In contrast, Marglin and Bhaduri (1990) argue that investment
should be an increasing function of the profit share and the rate capacity utilization (hereafter the MB type investment function).7) In this regard, Cassetti (2006) conducts an analysis
that treats the medium-run equilibrium and the long-run equilibrium using the Kalecki type
investment function, but does not use the MB type investment function.
The remainder of the paper is organized as follows. Section 2 presents the basic framework of our model. Section 3 analyzes a case where capacity is fully utilized. Section 4 investigates the medium-run equilibrium and the long-run equilibrium using the Kalecki type
investment function when there is excess capacity. Section 5 conducts the same analysis as
in Section 4 using the MB type investment function. Section 6 offers results of comparative
static analysis both in the medium-run equilibrium and in the long-run equilibrium. Section
7 summarizes the paper and provides additional comments on future directions.

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 wage income and capitalists save a constant fraction s of6) For the concept of the long-run equilibrium in the Kaleckian model, see Lavoie (1995, 2003). Cassetti(2006) conducts an analysis that treats the medium-run and long-run separately as in the present paper. The
concept of the medium-run in Cassetti (2006) is equal to that in the present paper. In the long-run of Cassetti(2006) four key variables—normal capacity, the normal rate of profit, autonomous investment, and the drop-out
ratio of capital equipment—are simultaneously adjusted.
7) For this issue, see also Blecker (2002).

4their profits. Since savings S t = srt Kt equal investments It in equilibrium, the rate of capital
accumulation gt leads to
gt = srt ,0 < s ≤ 1,

(1)

where Kt is the capital stock and rt is the rate of profit.
Let us specify two alternative investment functions. In the Kalecki type investment function, the rate of capital accumulation desired by firms is assumed to depend on the rate ofprofit and the discrepancy between the actual rate of capacity utilization ut and the normalplanned rate of capacity utilization un .
Kalecki type : gdt = γ + δrt + ε(ut − un ),γ > 0, δ > 0, ε > 0, γ − εun > 0,(2)where γ is the growth rate of autonomous investment, δ is the sensitivity of investment to therate of profit, and ε is the sensitivity of investment to the rate of capacity utilization. In themedium run, un is considered to be constant. The constraint γ − εun > 0 means that gdt > 0
when rt = ut = 0.
In contrast, the MB type investment function can be described as follows:
MB type : gdt = γ + δmt + ε(ut − un ),γ > 0, δ > 0, ε > 0, γ − εun > 0,(3)where mt is the profit share.8) The constraint γ−εun > 0 means that gdt > 0 when mt = ut = 0.
Suppose that firms operate with the following fixed coefficient production function:
Yt = min{at Et , (ut /k)Kt },(4)

where Yt is real output, Et employment, and at = Yt /Et the level of labor productivity. The
rate of capacity utilization is defined as ut = Yt /Yt∗ , where Yt∗ is the potential output. Define
the ratio of the capital stock to the potential output as k = Kt /Yt∗ and suppose that k is
constant, that is, both Kt and Yt∗ grow at the same rate.9) To simplify the analysis, let k = 1.
Then, we obtain ut = Yt /Kt . Note that the relationship among the profit rate, the profit share,8) Note that the same parameters are used both in the Kalecki type and in the MB type investment functions.
To be exact, we have to use gdt = γ0 + δ0 rt + ε0 (ut − un ) for the Kalecki type investment function and gdt =γ1 + δ1 mt + ε1 (ut − un ) for the MB type investment function. In the following analysis, however, we do not
distinguish the parameters to avoid a troublesome notation.
9) Foley and Michl (1999) emphasize that, in capitalist economies, a combination of growing labor productivity and declining capital productivity (a rise in k in our model) is the typical pattern of technological change.
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, sucha biased technical change is not taken into consideration.

5and the rate of capacity utilization is given by rt = mt ut and the relationship among therate of capacity utilization, the rate of capital accumulation, and the profit share is given by
ut = gt /(smt ).
An equation of motion for the rate of capital accumulation can be formulated byġt = α(gdt − gt ),α > 0,(5)

where α is the speed of adjustment. When the profit share is constant, there is a one-to-one
relationship between the rate of capital accumulation and the rate of capacity utilization.
Thus, equation (5) implies quantity adjustment by the rate of capacity utilization in the
goods market: excess demand (gdt > gt ) leads to a rise in the rate of capacity utilization
whereas excess supply (gdt < gt ) leads to a decline in the rate of capacity utilization.

2.2 Adjustment in the profit share
In the Kaleckian tradition, firms operate with excess capacity in oligopolistic goods markets
and set their price pt with a mark-up µt on unit labor costs.
pt = (1 + µt )

wt
,
at

(6)

where wt is the money wage rate. It should be noted that mt = µt /(1 + µt ), that is, there exists
a one-to-one relationship between the mark-up and the profit share, so that mt and µt change
in the same direction. Therefore, we can rewrite equation (6) as follows:
!
1
wt
pt =
.
1 − mt at(7)

Differentiating both sides of equation (7) with respect to time yields:ṗt ẇt
ṁt
ȧt
=
+
− .
pt wt 1 − mt at(8)

Let us specify the dynamics of the money wage and price 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 the actual real wage rate.
Let ωt be the actual real wage rate. Then, the wage share is given by 1 − mt = ωt /at . Thismeans that, given the labor productivity, to determine the real wage means to determine thewage share. Therefore, setting the target real wage rate is equivalent to setting the target

6

profit share. From this the dynamics of the money wage can be described as:ẇt= θw (mt − mw ),wt

θw > 0, 0 < mw < 1,(9)where mw is the target profit share set by workers, which is assumed to be given exogenously,and θw is the speed of adjustment constant, which reflects the bargaining power of workers.
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 profit share because
the mark-up bears a one-to-one relation to the profit share.
ṗt= θ f (m f − mt ),pt

θ f > 0, 0 < m f < 1,(10)where m f is the target profit share set by firms and θ f denotes the speed of adjustment, which
can be interpreted as the bargaining power of firms.
For ease of tractability, we impose the following constraints on θ f and θw .10)
θ f + θw = 1,

0 < θ f , θw < 1.

(11)

Let us denote the bargaining power of firms by θ f ≡ θ. Then, the bargaining power ofworkers is given by θw ≡ 1 − θ. The bargaining power of firms gets stronger as θ approaches
unity while that of workers gets stronger as θ approaches zero.
Substituting equations (9) and (10) into equation (8), we obtain an equation of motion
for the profit share.
ṁt = (1 − mt )[θ(m f − mt ) − (1 − θ)(mt − mw ) + gat ],(12)

where gat is the growth rate of labor productivity. Let us now turn to the aspiration gap
mentioned above. For ease of explanation, suppose that the growth rate of labor productivity
is given exogenously as gat = ḡa . The profit share will be constant when ṁt = 0, and theequilibrium profit share m∗ is given bym∗ = A + ḡa ,where A ≡ θm f + (1 − θ)mw , 0 < A < 1.

(13)

The parameter A denotes the weighted average of the target profit shares set by firms andworkers. Substituting equation (13) into equations (9) and (10) yields the rates of change in10) The constraint 0 < θ f , θw < 1 is also adopted by Dutt and Amadeo (1993), who, however, do not assume
θ f + θw = 1.

7

the money wage and price, respectively.
ẇ= θ(1 − θ)(m f − mw ) + (1 − θ)ḡa ,wṗ= θ(1 − θ)(m f − mw ) − θḡa .
p

(14)
(15)

If ḡa = 0, then ṗ/p = ẇ/w and inflation results from the aspiration gap m f − mw .11) Whenm f > mw , the inflation rate is positive. Note that the growth rate of the real wage rateω̇/ω = ẇ/w − ṗ/p is given by ḡa . In what follows, ga will be determined endogenously inthe long-run equilibrium, and thus changes in parameters that affect ga influence the inflationrate.

2.3 Adjustment in the growth rate of labor productivity
Let us specify endogenous technological change. Following Bhaduri (2006), we describe an
adjustment in the growth rate of labor productivity as follows:ġat = β(gEt − n),β > 0,(16)

where β is the speed of adjustment, gEt ≡ Ėt /Et is the rate of change in employment, andn is the exogenously given growth rate of labor supply. Since Et = Yt /at , we have gEt =(u̇t /ut ) + gt − gat . That is, the rate of change in employment is decomposed into the rate ofchange in capacity utilization, the rate of capital accumulation, and the growth rate of laborproductivity. Note that in the following analysis ga is assumed to be constant in the medium
run and equation (16) is introduced in the long run. As shall be explained later, the rate ofcapacity utilization in the long run is constant, so that u̇t /ut = 0. Therefore, in the long run,equation (16) can be rewritten asġat = β(gt − gat − n).

3

(17)

Full-capacity equilibrium

This section analyzes a case where capacity is fully utilized, that is, ut = 1 in order tocompare the full-capacity case with demand-constrained cases introduced in Sections 4 and11) In a series of studies by M. Cassetti, m f and mw are determined endogenously. Cassetti (2002) assumes
that m f depends positively on the rate of capacity utilization while mw depends negatively on the employment
rate. Cassetti (2003) gives m f exogenously and defines mw as a decreasing function of the growth rate of employment. In Cassetti (2006), m f is given exogenously and mw is decreasing in the rate of capital accumulation.

8

5. We assume that the full-capacity equilibrium is stable and derive stability conditions.
These conditions are also imposed on the demand-constrained cases.

3.1 Medium-run analysis
When capacity is fully utilized, capital accumulation is constrained by saving, so that an
autonomous investment function does not exist. In this case, the rate of profit is given by
rt = mt , from which we have gt = smt with equation (1). Thus, an equation of motion for therate of capital accumulation leads to ġt = sṁt . The dynamics of the profit share mt is given
by equation (12).
The medium-run equilibrium is characterized by a situation in which ṁt = ġt = 0. The
equilibrium values are as follows:m∗ = A + ga ,(18)

g∗ = s(A + ga ),(19)

where g∗ is the rate of capital accumulation in the medium-run equilibrium. From equation(12), we obtain dṁt /dmt < 0. Therefore, the medium-run equilibrium is stable.

3.2 Long-run analysis
The long-run dynamics of the full capacity case is governed only by ġa = β(g∗ − ga − n) fromequation (17), and the long-run equilibrium values are given bysA − n,1−ss(A − n)
g∗∗ =
,
1−s
g∗a =

(20)
(21)

where g∗a and g∗∗ are the long-run growth rate of labor productivity and the long-run rate ofcapital accumulation, respectively. For g∗a to be positive and stable, we require sA − n > 0.
We also need the condition A − n > 0 for g∗∗ to be positive. In the following analysis, we
assume that these two conditions are met.

9

4

Analysis with the Kalecki type investment function

4.1 Medium-run analysis
The medium-run equilibrium is defined as a situation in which both the profit share and therate of capital accumulation stay constant. Substituting equations (1) and (2) into equation(5), we obtain an equation of motion for the rate of capital accumulation as follows:"ġt = α!
#εδ
+ − 1 gt + (γ − εun ) .
smt s

(22)

The dynamics of the profit share is given by equation (12).
From ṁt = 0, the profit share in the medium-run equilibrium is given by m∗ = A + ga .
The rates of capital accumulation and capacity utilization in the medium-run equilibrium aregiven byg∗ =(γ − εun )sm∗,(s − δ)m∗ − εu∗ =γ − εun.
(s − δ)m∗ − ε(23)For the medium-run equilibrium to be positive, the following condition is needed.
(s − δ)m∗ − ε > 0.

(24)This condition has the following meaning. Using r = mu, we obtain gd = δ(mu) + ε(u − un )from the investment function and g = smu from the saving function. The partial derivativeof gd with respect to u is given by ∂gd /∂u = δm + ε and that of g with respect to u is given by∂g/∂u = sm. Accordingly, equation (24) states that ∂g/∂u > ∂gd /∂u, that is, saving reacts
to the rate of capacity utilization more strongly than investment. Rearranging equation (24)gives (s − δ)m∗ > ε, which implies s > δ because ε > 0. The condition s > δ meansthat saving responds to the rate profit more strongly than investment. In addition to these
constraints, γ − εun < (s − δ)m∗ − ε is needed for ut < 1.
J12
Let us examine the stability of the medium-run equilibrium. Let J = JJ11
be the21 J2210

Jacobian matrix for this dynamical system. Then, we have∂ṁt
= −(1 − m∗ ) < 0,
∂mt
∂ṁt
J12 =
= 0,
∂gt
∂ġt
αε g∗
J21 =
=−
< 0,
∂mt
s (m∗ )2∂ġtα(γ − εun )
< 0.
J22 =
=−
∂gt
g∗
J11 =

(25)
(26)
(27)
(28)

All the elements are evaluated at m∗ and g∗ .
To know the stability, we have to know the sign of the trace of J (tr J) and that of thedeterminant of J (det J), which are respectively given byα(γ − εun )< 0,g∗
γ − εundet J = J11 J22 − |{z}J12 J21 = α(1 − m∗ )> 0.
g∗
tr J = J11 + J22 = −(1 − m∗ ) −

(29)
(30)

=0

We have both tr J < 0 and det J > 0, which imply the local stability of the medium-run equilibrium. Moreover, the discriminant of the characteristic equation for the Jacobian matrix is
given by D = (J11 − J22 )2 > 0, which shows that two roots are real and distinct. Therefore,
the medium-run equilibrium leads to a stable node.
ṁ = 0gġ = 0g = sm

g∗FC
g∗
m∗
O
1 m
Figure 1: Convergence to the medium-run equilibrium—the Kalecki caseLet us draw the medium-run phase diagram on a (m, g)-plane (Figure 1). The ṁ = 0
locus is a straight line and vertical to the horizontal line because it does not depend on g.
11

The ġ = 0 locus is a downward-sloping curve and convex to the origin becausedgsε(γ − εun )
=−
< 0,
dm
[(s − δ)m − ε]2d2 g2sε(s − δ)(γ − εun )
=
> 0.
2
dm
[(s − δ)m − ε]3

(31)
(32)

The intersection of the two lines is the medium-run equilibrium, which is stable from theanalysis above. The straight line from the origin g = sm represents a combination of mand g when u = 1, that is, capacity is fully utilized. The intersection of this straight line
and ṁ = 0 denotes the full-capacity equilibrium in the medium run and g∗FC stands for therate of capital accumulation in the full-capacity equilibrium. Excess capacity, that is, u < 1corresponds to the region below g = sm. Note that equations of motion for mt and gt differ
between the case where u = 1 and the case where u < 1. The full capacity equilibrium mustlie on g = sm and is not allowed to deviate from this line.

4.2 Long-run analysis
In the long run, the normal planned rate of capacity utilization and the growth rate of laborproductivity will be adjusted.12) Let us specify this adjustment mechanism as follows:u̇nt = φ(u∗ − unt ),φ > 0.

(33)

where φ is the speed of adjustment. From equation (17) the dynamics of the growth rate oflabor productivity follows ġat = β(gEt − n) = β(g∗ − gat − n). It should be noted that from theresult of the medium-run equilibrium we haveu∗ =γ − εunt,(s − δ)(A + gat ) − εg∗ =s(γ − εunt )(A + gat ),(s − δ)(A + gat ) − ε(34)

12) Lavoie (1996), Dutt (1997), and Cassetti (2006) consider adjustment in the growth rate of autonomous
investment (γ in our model) besides adjustment in the normal planned rate of capacity utilization in the longrun. In this case, a dynamical system composed of two differential equations for unt and γt will be a “zeroroot system,” which implies path dependency: the economy has a continuum of equilibria instead of a uniqueequilibrium in the long run and initial conditions matter.

12

which are inserted into u̇nt and ġat . The long-run equilibrium is a situation where u̇nt = ġat =0, and thus the equilibrium values u∗n and g∗a fulfill the following system of equations.
γ − εu∗n− u∗n = 0∗(s − δ)(A + ga ) − εs(γ − εu∗n )(A + g∗a )− g∗a − n = 0.
∗(s − δ)(A + ga ) − ε

(35)
(36)

From equations (35) and (36), the following solutions are obtained:13)sγ − n(s − δ)
,
s−δ
γ
u∗n =
,
sγ + (s − δ)(A − n)
sγ
g∗∗ =
.
s−δ
g∗a =

(37)
(38)
(39)

We need sγ − n(s − δ) > 0 for g∗a to be positive. From the analysis above, s > δ and A − n > 0,so that g∗∗ and u∗n are positive. For u∗n to be smaller than unity, the following condition isrequired:(s − δ) (A − n) −γ (1 − s) > 0,
| {z } | {z }| {z }
+

+

(40)

+

which is assumed to be satisfied in the following analysis.
The local stability of the long-run equilibrium is analyzed by the Jacobian matrix of thetwo-variable system:
!
"
#
∂u̇nt
∂u∗
(s − δ)m∗∗J11 ==φ− 1 = −φ
< 0,
∂unt
∂un
(s − δ)m∗∗ − ε"#(s − δ)(γ − εu∗n )
∂u̇nt
∂u∗
J12 =
=φ
= −φ
< 0,
∂gat
∂ga
[(s − δ)m∗∗ − ε]2
"
#
∂g∗
εsm∗∗
∂ġat
=β
= −β
< 0,
J21 =
∂unt
∂un
(s − δ)m∗∗ − ε!
"#sε(γ − εu∗n )
∂ġat
∂g∗
J22 =
=β
− 1 = −β+ 1 < 0,∂gat
∂ga[(s − δ)m∗∗ − ε]2

(41)
(42)
(43)
(44)

13) In fact, we can easily obtain the rate of capital accumulation in the long-run equilibrium g∗∗ without
solving equations (35) and (36) directly. Since u = un in the long-run equilibrium, we have g∗∗ = γ + δr∗∗ fromthe investment and saving functions, where r∗∗ denotes the long-run rate of profit. Substituting r∗∗ = g∗∗ /s into
this equation, we obtain g∗∗ . In addition, because g∗a = g∗∗ − n from g∗a = g∗∗ − n, we obtain g∗a .

13

where m∗∗ = A + g∗a is the profit share in the long-run equilibrium. Necessary and sufficientconditions for local stability are given by both tr J < 0 and det J > 0. It is obvious thattr J = J11 + J22 < 0. det J is given bydet J = J11 J22 − J12 J21 =φβ(s − δ)m∗∗> 0,(s − δ)m∗∗ − ε(45)which is clearly positive. Therefore, the long-run equilibrium is locally stable, and eithera stable node or a stable focus will be obtained. However, analysis with a phase diagram
introduced below will show that the long-run equilibrium is stable node.
Let us draw on a (un , ga )-plane two curves that correspond to u̇nt = 0 and ġat = 0,
respectively. The following analysis will be conducted by a linearized system around thelong-run equilibrium. The slopes of these curves are respectively given by
∂u∗

dga
∂g
= − ∂u∗ a < 0,u̇nt = 0 =⇒dun u̇n =0
−1
∂u

(46)

n

∂g∗

dga
∂u
ġat = 0 =⇒= − ∂g∗ n < 0.
dun ġa =0
−1

(47)

∂ga

Thus, both the curves are downward-sloping. Since the long-run equilibrium is stable, as
has been shown above, the slope of u̇n = 0 has to be steeper than that of ġa = 0 (see Figure2), otherwise the long-run equilibrium would be a saddle-point.
gag∗a,FCu̇n = 0ġa = 0

g∗a

u∗n
O
1 un
Figure 2: Convergence to the long-run equilibrium—the Kalecki case
In Figure 2, the horizontal line at g∗a,FC is the locus of ġa = 0 when capacity is fully
utilized. The intersection of this horizontal line and the line un = 1 corresponds to the fullcapacity equilibrium in the long run. Note that as is the case with the medium-run analysis,
14equations of motion for unt and gat differ between the case where un = 1 and the case where
un < 1. The full-capacity equilibrium must lie on the line un = 1. When un < 1, the long-run
equilibrium is determined by the intersection of u̇n = 0 and ġa = 0 as shown in Figure 2.

5

Analysis with the MB type investment function

5.1 Medium-run analysis
Substituting equations (1) and (3) into equation (5), we obtain an equation of motion for therate of capital accumulation as follows:"ġt = α!
#ε− 1 gt + δmt + (γ − εun ) .
smt

(48)

The dynamics of the profit share is given by equation (12) as before.
When ṁt = ġt = 0, equilibrium values in the medium run are obtained as follows:m∗ = A + ga ,sm∗ [δm∗ + (γ − εun )],g∗ =sm∗ − εδm∗ + (γ − εun )u∗ =.
sm∗ − ε

(49)
(50)
(51)

For g∗ to be positive, we requiresm∗ − ε > 0.

(52)

The partial derivative of gd with respect to u is given by ∂gd /∂u = ε and that of g with respect
to u is given by ∂g/∂u = sm. From this, equation (52) means that ∂g/∂u > ∂gd /∂u > 0, thatis, saving reacts to the rate of capacity utilization more strongly than investment. Moreover,for u∗ < 1 we need γ − εun < (s − δ)m∗ − ε, which is the same condition as in the analysis
with Kalecki type investment function. This condition implies s > δ because the left-handside of that condition is positive, that is, γ − εun > 0.
Let us examine the stability of the medium-run equilibrium. All elements of the Jacobian15

matrix J =

J11 J12J21 J22are as follows:∂ṁt= −(1 − m∗ ) < 0,
∂mt
∂ṁt
J12 =
= 0,
∂gt
"
#
∂ġt
sδ(m∗ )2 − εg∗
J21 =
=α
,
∂mt
s(m∗ )2!
sm∗ − ε
∂ġt
< 0.
= −α
J22 =
∂gt
sm∗
J11 =

(53)
(54)
(55)
(56)

From these equations tr J and det J are respectively given by!
sm∗ − εtr J = −(1 − m ) − α
< 0,
sm∗
!
sm∗ − ε∗det J = α(1 − m )
> 0,
sm∗
∗

(57)
(58)

which satisfy necessary and sufficient conditions for local stability. The discriminant of thecharacteristic equation for the Jacobian matrix is given by D = (J11 − J22 )2 > 0, whichmeans that two roots are real and distinct. Therefore, the medium-run equilibrium is a stablenode.
ṁ = 0gġ = 0g = smg∗a,FCg∗m∗O
1 m
Figure 3: Convergence to the medium-run equilibrium—the MB case
Figure 3 shows the medium-run phase diagram in the MB case. A major difference

16

between the MB case and the Kalecki case is the shape of ġ = 0 curve.
++z }| { z}|{dgs[δm (sm − ε) −ε (δm + γ − εun )]
=
Q 0,
dm
(sm − ε)2d2 g2sε[δε + s(γ − εun )]
=
> 0.
2
dm
(sm − ε)3

(59)
(60)

The right-hand side of equation (59) becomes positive or negative depending on the size ofm, so that dg/dm also becomes positive or negative. Figure 3 is drawn on the assumptionthat dg/dm < 0 when m is small while dg/dm > 0 when m is large. As will be shown
later, the downward-sloping part of ġ = 0 curve corresponds to wage-led growth and the
upward-sloping part of ġ = 0 curve corresponds to profit-led growth.

5.2 Long-run analysis
Long-run dynamics is described by equations (17) and (33). Substituting equations (49),(50), and (51) into equations (17) and (33) and letting u̇n,t = ġa,t = 0, we haveδ(A + g∗a ) + (γ − εu∗n )− u∗n = 0,∗s(A + ga ) − εs(A + g∗a )[δ(A + g∗a ) + (γ − εu∗n )]− g∗a − n = 0.
∗s(A + ga ) − ε

(61)
(62)

From these equations we obtain the following solutions:14)
γ − n + δA,
1−δ
γ + δ(A − n)u∗n =,s[γ + (A − n)]γ + δ(A − n)
g∗∗ =
.
1−δ
g∗a =

(63)
(64)
(65)

Because A − n > 0 from the stability condition of full-capacity case, we need 1 − δ > 0
for g∗∗ and u∗n to be positive. The condition 1 − δ > 0 states that the sensitivity of investment
to the profit share is smaller than unity. For g∗a to be positive, we require γ − n + δA > 0.

14) As in the Kalecki case, g∗a is easily obtained as follows. Since u = un in the long-run equilibrium, we haveg∗∗ = γ + δ(A + g∗a ) from the investment function. In addition, we have g∗a = g∗∗ − n from ġa,t = 0. Combining
these two equations, we get g∗a + n = γ + δ(A + g∗a ), from which g∗a is obtained.

17Moreover, For u∗n to be smaller than unity, the following condition is needed.
(s − δ) (A − n) −γ (1 − s) > 0,
| {z } | {z }| {z }
+

+

(66)

+

which is the same condition as in the Kalecki case.
Here, we examine the stability of the long-run equilibrium. We have
!
sm∗∗
∂u̇nt
= −φ
< 0,
J11 =
∂unt
sm∗∗ − ε#"δε + s(γ − εu∗n )
∂u̇nt
J12 =
< 0,
= −φ
∂gat
(sm∗∗ − ε)2
!
∂ġat
sεm∗∗
J21 =
= −β
< 0,
∂unt
sm∗∗ − ε+ or −
z }| {
 s(1 − δ)m∗∗ (sm∗∗ − 2ε) +sε(γ − εu∗ ) + ε2 
∂ġat
n
 .
= −β 
J22 =

∂gat
(sm∗∗ − ε)2



(67)
(68)
(69)

(70)

If sm∗∗ −2ε > 0 in J22 , then J22 < 0. Since g∗a is independent of ε and thus m∗∗ is independentof ε, it is possible that sm∗∗ − 2ε > 0 when ε is not large. This sufficient condition means thatinvestment does not so strongly respond to the rate of capacity utilization. In the following
analysis we assume it. Then, we have tr J < 0. The determinant of J is given bydet J =

φβs(1 − δ)m∗∗> 0.
sm∗∗ − ε(71)

Therefore, tr J < 0 and det J > 0, so that the long-run equilibrium is locally stable. The
long-run phase diagram of the MB case is similar to that of the Kalecki case, and thus it isomitted (see Figure 3).

6Effects of shifts in parameters

This section investigates the effects of shifts in parameters on the medium-run equilibrium
and the long-run equilibrium. Results are summarized in Tables 1 and 2. The +/− cases aretreated in Appendix.

18

6.1 Comparative statics in the medium-run equilibrium
Table 1 shows the results of comparative statics in the medium-run equilibrium. In theanalysis we assume m f > mw , which is relevant to the effects of θ. It is plausible to supposethat firms attempt to set m f as high as possible whereas workers attempt to set mw as low as
possible. Therefore, the assumption m f > mw is reasonable.
Table 1: Medium-run comparative statics

Full capacity
Kalecki caseMB case

∗

m
g∗
m∗
g∗
u∗
m∗
g∗
u∗

s
0
+
0
−
−
0
−
−

γ
0
0
0
+
+
0
+
+

δ
0
0
0
+
+
0
+
+

ε
mf
0
+
0
+
0
+
+/−
−
+/−
−
0
+
+/− +/−
+/−
−

mw
+
+
+
−
−
+
+/−
−

θ
+
+
+
−
−
+
+/−
−

A rise in the saving rate increases the rate of capital accumulation when capacity is fully
utilized whereas decreases when there is excess capacity. The latter effect is known as “theparadox of thrift.”
An increase in θ, that is, the bargaining power of firms leads to a rise in the profit sharein every case. However, its effects on the rate of capital accumulation are different in thethree cases: in the full-capacity case, g∗ rises; in the Kalecki case, g∗ falls; and in the MB
case, g∗ rises or falls depending on conditions. An increase in the bargaining power of firms
leads to a decline in the rate of capacity utilization both in the Kalecki case and in the MBcase.

6.2 Comparative statics in the long-run equilibrium
Table 2 shows the results of comparative statics in the long-run equilibrium. In the long run,all signs are definitely determined.
The effects of a rise in the saving rate on the rate of capital accumulation are different in
the three cases. It has a positive effect on g∗∗ when capacity is fully utilized. When there isexcess capacity, its effects are different according to the investment functions. In the Kalecki
case, a rise in the saving rate has a negative effect on g∗∗ , which suggests the paradox of thrift
as in the medium-run equilibrium. In the MB case, a rise in the saving rate has no effect on
19

Table 2: Long-run comparative statics

Full capacity

Kalecki caseMB case

∗∗

m
g∗a
g∗∗
m∗∗
g∗a
g∗∗
u∗n
m∗∗
g∗a
g∗∗
u∗n

s
+
+
+
−
−
−
−
0
0
0
−

γ
0
0
0
+
+
+
+
+
+
+
+

δ
0
0
0
+
+
+
+
+
+
+
+

ε
0
0
0
0
0
0
0
0
0
0
0

n mf
− +
− +
− +
− +
− 0
0 0
+ −
− +
− +
− +
+ +

mw
+
+
+
+
0
0
−
+
+
+
+

θ
+
+
+
+
0
0
−
+
+
+
−

g∗∗ , which is different from the MB case in the medium run.
The effect of a rise in the bargaining power of firms on the rate of capital accumulation
is interesting. In the full-capacity case, it increases g∗∗ as in the medium run. In the Kalecki
case, it has no effect on g∗∗ , which is different from the Kalecki case in the medium run. In
the MB case, it has a positive effect on g∗∗ .
Here, let us mention the differences between our model and the mainstream growthmodels. In endogenous growth models that possess “the scale effect,” an increase in thesaving rate leads to a rise in the growth rate of output on the balanced growth path.15) In
our model, on the other hand, an increase in the saving rate increases (the full-capacitycase), decreases (the Kalecki case), or does not change (the MB case) the rate of capital
accumulation in the long-run equilibrium. In semi-endogenous (non-scale) growth models,an increase in the growth rate of population leads to a rise in the growth rate of income percapita on the balanced growth path. In our model, on the other hand, an increase in thegrowth rate of labor supply leads to a decline in the growth rate of labor productivity in thelong-run equilibrium.

6.3 Discussions
Here we discuss the effects of m f and mw , which are not mentioned in the preceding subsections. The profit share in our model is an endogenous variable and not an exogenous
variable, and consequently we cannot treat the profit share as a parameter. For this reason,
15) See Jones (1999) for the scale effect, endogenous growth, and semi-endogenous growth.

20

we instead pay attention to the parameters m f and mw . Because a rise in m f (mw ) increasesm∗ and m∗∗ , that is, m f (mw ) and the profit share move in the same direction, our analysis isreasonable. Table 3 summarizes the results concerning m f and mw from Tables 1 and 2. Let
us explain each item.
In the wage-led growth regime, a rise in the profit share declines the rate of capital
accumulation.16) In the profit-led growth regime, on the other hand, a rise in the profit share
increases the rate of capital accumulation.
In the stagnationist regime (the wage-led aggregate demand regime), a rise in the profit
share leads to a decline in the rate of capacity utilization. In the exhilarationist regime (theprofit-led aggregate demand regime), a rise in the profit share increases the rate of capacity
utilization.
Table 3: Classification of regime
Kalecki caseMB case(Medium-run) (Medium-run)Wage-led growth◦◦Profit-led growth◦
Stagnationism◦◦
Exhilarationism

Kalecki case MB case(Long-run) (Long-run)◦◦◦

The medium-run equilibrium in the Kalecki case corresponds to the wage-led growth
regime and the stagnationist regime. This result is common when the Kalecki investment
function is used. If the MB type investment function is used, either the wage-led growth
regime or the profit-led growth regime is obtained. For the rate of capacity utilization, thestagnationist regime is obtained, which is the same as in the Kalecki case.
In the long-run equilibrium of the Kalecki case, neither the wage-led growth regime nor
the profit-led growth regime is obtained. For the rate of capacity utilization, we have thestagnationist regime. The long-run equilibrium in the MB case corresponds to the profit-led
growth regime and the exhilarationist regime.
In this way, different investment functions yield different regimes both in the medium
run and in the long run.
Moreover, we emphasize that the medium-run equilibrium and the long-run equilibrium
show different regimes both in the Kalecki case and in the MB case. In the Kalecki case,the medium-run equilibrium is the wage-led growth regime while the long-run rate of capital accumulation does not respond to the profit share. In the MB case, the medium-run16) Our classification of regimes is based on Blecker’s (2002) classification.

21

equilibrium is characterized either as wage-led growth or as profit-led growth whereas thelong-run equilibrium is only profit-led growth. For the rate of capacity utilization, we havestagnationist in the medium run whereas exhilarationist in the long run: a switch of regime
occurs.

7

Concluding remarks

This paper has introduced endogenous technological change into a Kaleckian growth model.
For the endogenization of technological change we assume that a change in the growth rateof labor productivity depends on the difference between the growth rates of employment
and labor supply. This adjustment process makes the rate of employment stay constant in
the long run. Using the model, we have analyzed the stability of the equilibrium and comparative statics. In addition, following the argument of Marglin and Bhaduri (1990), we
use two alternative investment functions. If we use a Kalecki investment function, then we
obtain stagnationism and wage-led growth in the medium run while we also obtain stagnationism but neither wage-led growth nor profit-led growth in the long run. If, instead, we
use a MB type investment function, then we have stagnationism and either wage-led growthor profit-led growth in the medium run while we have exhilarationism and profit-led growthin the long run.
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) theseeffects are taken into account, while in the present paper these issues are not dealt with forthe 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, mw and m f , arenot endogenized. It is evident that technological change influences the target rates if theseare endogenized. Taking these into account will be future research.

22

A Appendix
In the medium-run equilibrium of the Kalecki case, the following derivatives are worth
mentioning.
dg∗ sm∗ [γ − un m∗ (s − δ)]=,dε[(s − δ)m∗ − ε]2du∗ γ − un m∗ (s − δ).
=dε[(s − δ)m∗ − ε]2

(A-1)
(A-2)

For dg∗ /dε, dg∗ /dε > 0 if γ − un m∗ (s − δ) > 0 while dg∗ /dε < 0 if γ − un m∗ (s − δ) < 0. The
condition γ − un m∗ (s − δ) > 0 can be rewritten asm∗ <γ≡ m0 .
(s − δ)un(A-3)

In the meantime, the rate of capital utilization has to be smaller than unity, from which we
obtainm∗ >γ + ε(1 − un )≡ m1 .
s−δ(A-4)

Comparing the size of m0 with that of m1 , we havem0 − m1 =(1 − un )(γ − εun )> 0,(s − δ)un(A-5)

which shows m0 > m1 . Therefore, it is possible that dg∗ /dε > 0 or dg∗ /dε < 0 in themedium-run equilibrium. For du∗ /dε, the same reasoning holds.
In the medium-run equilibrium of the MB case, the following derivatives are worth mentioning.
dg∗ sm∗ [(δm∗ + γ − εun ) − un (sm∗ − ε)]=,dε(sm∗ − ε)2du∗ (δm∗ + γ − εun ) − un (sm∗ − ε)=,dε(sm∗ − ε)2sθ[δm∗ (sm∗ − ε) − ε(δm∗ + γ − εun )]
dg∗
=
,
dm f
(sm∗ − ε)2s(1 − θ)[δm∗ (sm∗ − ε) − ε(δm∗ + γ − εun )]
dg∗
=
,
dmw
(sm∗ − ε)2dg∗ s(m f − mw )[δm∗ (sm∗ − ε) − ε(δm∗ + γ − εun )]=.
dθ(sm∗ − ε)2

(A-6)
(A-7)
(A-8)
(A-9)
(A-10)First, we turn to dg∗ /dε and du∗ /dε. The signs of these derivatives are determined by the23sign of (δm∗ + γ − εun ) − un (sm∗ − ε). From the assumptions, we have δm∗ + γ − εun > 0,sm∗ − ε > 0, and δm∗ + γ − εun < sm∗ − ε. For this reason, these derivatives can be positiveor negative according to the size of un .
Second, we turn to dg∗ /dm f , dg∗ /dmw , and dg∗ /dθ. The signs of these derivatives are
determined by the sign of δm∗ (sm∗ − ε) − ε(δm∗ + γ − εun ). Therefore, dg∗ /dm f , dg∗ /dmw ,and dg∗ /dθ can be positive or negative.

References
Bhaduri, A. (2006) Endogenous economic growth: a new approach, Cambridge Journal of
Economics, 30, pp. 69–83.
Blecker, R. A. (2002) Distribution, demand and growth in neo-Kaleckian macro-models, in:
M. Setterfield (Ed.) The Economics of Demand-led Growth, Challenging the Supplyside Vision of the Long Run (Cheltenham: Edward Elgar).
Cassetti, M. (2002) Conflict, inflation, distribution and terms of trade in the Kaleckian
model, in: M. Setterfield (Ed.) The Economics of Demand-led Growth, Challenging
the Supply-side Vision of the Long Run (Cheltenham: Edward Elgar).
Cassetti, M. (2003) Bargaining power, effective demand and technical progress: a Kaleckian
model of growth, Cambridge Journal of Economics, 27, pp. 449–464.
Cassetti, M. (2006) A note on the long-run behaviour of Kaleckian models, Review of Political Economy, 18, pp. 497–508.
Dutt, A. K. (1987) Alternative closures again: a comment on ‘Growth, distribution andinflation’, Cambridge Journal of Economics, 11, pp. 75–82.
Dutt, A. K., Amadeo, E. J. (1993) A post-Keynesian theory of growth, interest and money,
in: M. Baranzini and G. C. Harcourt (Eds) The Dynamics of the Wealth of Nations:
Growth, Distribution and Structural Change (New York: St.Martin’s Press).
Dutt, A. K. (1997) Equilibrium, path dependence and hysteresis in Post-Keynesian models,in P. Arestis, G. Palma and M. Sawyer (Eds) Markets, Unemployment and Economic
Policy: Essays in Honour of Geoff Harcourt, Vol. 2 (London: Routledge).
Dutt, A. K. (2006) Aggregate demand aggregate supply and economic growth, International
Review of Applied Economics, 20, pp. 319–336.
Foley, D. K., Michl, T. R. (1999) Growth and Distribution (Cambridge, MA: Harvard University Press).

24

Jones, C. I. (1999) Growth: with or without scale effects? American Economic Review, 89,
pp. 139–144.
Kaldor, N. (1966) Causes of the slow rate of economic growth in the United Kingdom,
reprinted in: N. Kaldor (1978) Further Essays on Economic Theory (London: Duckworth).
Kalecki, M. (1971) Selected Essays on the Dynamics of the Capitalist Economy (Cambridge,
UK: Cambridge University Press).
Lavoie, M. (1992) Foundations of Post-Keynesian Economic Analysis (Cheltenham: Edward
Elgar).
Lavoie, M. (1995) The Kaleckian model of growth and distribution and its neo-Ricardianand neo-Marxian critiques, Cambridge Journal of Economics, 19, pp. 789–818.
Lavoie, H. (1996) Traverse, hysteresis and normal rates of capacity utilization in Kaleckian
models of growth and distribution, Review of Radical and Political Economics, 28,
pp. 113–147.
Lavoie, M. (2003) Kaleckian effective demand and Sraffian normal prices: towards a reconciliation, Review of Political Economy, 15, pp. 53–74.
Lima, G. T. (2004) Endogenous technological innovation, capital accumulation and distributional dynamics, Metroeconomica, 55, pp. 386–408.
Marglin, S., Bhaduri, A. (1990) Profit squeeze and Keynesian theory, in: S. Marglin and J.
Schor (Eds) The Golden Age of Capitalism: Reinterpreting the Postwar Experience(Oxford: Clarendon Press).
Raghavendra, S. (2006) Limits to investment exhilarationism, Journal of Economics, 87, pp.
257–280.
Rowthorn, R. E. (1977) Conflict, inflation and money, Cambridge Journal of Economics, 1,
pp. 215–239.
Rowthorn, R. E. (1981) Demand, real wages and economic growth, Thames Papers in Political Economy, Autumn, pp. 1–39.
Stockhammer, E. (2004) Is there an equilibrium rate of unemployment in the long run?
Review of Political Economy, 16, pp. 59–77.
Stockhammer, E., Onaran, Ö. (2004) Accumulation, distribution and employment: a structural VAR approach to a Kaleckian macro model, Structural Change and Economic
Dynamics, 15, pp. 421–447.

25

Verdoorn, P. J. (1949) Fattori che regolano lo sviluppo della produttività del lavoro, L’Industria,
1, pp. 14–28 (Factors that determine the growth of labour productivity, translated by
A. P. Thirlwall in: J. McCombie, M. Pugno and B. Soro (Eds) Productivity Growthand Economic Performance: Essays on Verdoorn’s Law (London: Macmillan, 2002).
You, J-I. (1994) Macroeconomic structure, endogenous technical change and growth, Cambridge Journal of Economics, 18, pp. 213–233.

26