Kyoto University,
Graduate School of Economics
Research Project Center Discussion Paper Series

Is the Long-run Equilibrium Wage-led or Profit-led?
A Kaleckian Approach

Hiroaki Sasaki

Discussion Paper No. E-11-002

Research Project Center
Graduate School of Economics
Kyoto University
Yoshida-Hommachi, Sakyo-ku
Kyoto City, 606-8501, Japan

April 2011

Is the Long-run Equilibrium Wage-led or Profit-led?
A Kaleckian Approach
Hiroaki SASAKI
Graduate School of Economics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto
606-8501, Japan; e-mail: sasaki@econ.kyoto-u.ac.jp

Abstract
This paper presents a Kaleckian growth model in which (i) the rate of capacity utilization, the profit share, and the rate of employment are adjusted in the medium run,and (ii) the normal rate of capacity utilization and the expected rate of growth are adjusted in the long run. Both the Kalecki type and the Marglin-Bhaduri type investment
functions are introduced. Using the model, we examine which regime is obtained in thelong-run equilibrium, the wage-led regime or the profit-led regime.
Keywords: Kaleckian model; long-run equilibrium; wage-led; profit-led
JEL Classification: E12; E24; O41

1 Introduction
How do changes in wages aﬀect the performance of economy? On the one hand, wages arecosts: an increase in wages lowers the profit of the firms, and thus, decreases the investmentof firms, which consequently, has a negative eﬀect on production and employment. On theother hand, wages are income: an increase in wages stimulates consumption demand, andthus, induces the investment of firms, which consequently, has a positive eﬀect on productionand employment. The eﬀect of an increase in wages on the performance of the economy
diﬀers depending on which eﬀect dominates.
In post-Keynesian models, especially in Kaleckian models, these diﬀerences are explained by the diﬀerences in regimes.1 Here, the regime is a characteristic of the equilibrium. In particular, the regime is defined by the relationship between the profit share (i.e.,
the ratio of profit income to nominal national income) and the rate of capacity utilization
and by the relationship between the profit share and the rate of capital accumulation. If an
increase in the profit share decreases (increases) the rate of capacity utilization and the rateof capital accumulation, then we call it the wage-led (profit-led) regime. In the wage-led
regime, an increase in the wage share has a positive eﬀect on production and employment,
while in the profit-led regime, it has a negative eﬀect on production and employment.
Many early studies investigate which regime is obtained under what circumstances.
Which economic policy is eﬀective depends on a kind of regime. Therefore, regime analysis
is important.
Almost all existing studies consider the regime in the short-run or the medium-run equilibrium. Here, the short run or the medium run relates to the period in which the variablesof the model such as the rate of capacity utilization, the rate of accumulation, and the profit
share are adjusted. For this reason, the objective of these existing studies concerns the effectiveness of an economic policy in the short run or the medium run.
However, the long-run regime has been seldom discussed. Accordingly, the existing
theory cannot address the eﬀectiveness of an economic policy in the long run. To consider along-run situation, we need a long-run theory. Here, we define the long run as the period inwhich the normal rate of capacity utilization and the expected rate of growth are adjusted.
These two variables are treated as constants in the short run and the medium run. The normal
rate of capacity utilization is the rate of capacity utilization that firms determine throughconvention, production experience, and strategic considerations. In the short run and the
medium run of the Kaleckian model, the actual rate of capacity utilization diverges from thenormal rate of capacity utilization even in the equilibrium. However, in the long run, such1

For the framework of the Kaleckian model, see Rowthorn (1981) and Lavoie (1992). For various regimes
in the Kaleckian model, see Marglin and Bhaduri (1990), Bhaduri and Marglin (1990), and Blecker (2002).

1

divergence will be adjusted. In addition, it is reasonable to consider that the expectation isadjusted.
Some early studies investigate such a long-run adjustment in the Kaleckian model. Thesestudies sought to challenge the criticism of Marxists and Sraﬃans. These studies claim that
the Kaleckian model is a short-run/medium-run model and not a long-run model (Auerbachand Skott, 1988; Duménil and Lévy, 1999; Park, 1997). Critics assert that in the long run,the divergence between the actual rate of capacity utilization and the normal rate of capacity
utilization should vanish: all variables are fully adjusted; consequently, we have the returnof classical economics.
In contrast to this criticism, Lavoie (1995, 2003), Dutt (1997), and Cassetti (2006) introduce the long-run adjustment process of the normal rate of capacity utilization in the
Kaleckian model. In such extended models, Kaleckian results are obtained even in the longrun. For example, “the paradox of thrift” and “the paradox of cost” hold even in the longrun. Here, the paradox of thrift means that an increase in the capitalists’ savings rate lowers
the rate of capital accumulation. The paradox of cost means that an increase in the real wage
leads to an increase in the realized rate of profit.
However, even such extended models do not satisfactorily examine which regime holds
in the long run.
It is well known that in the short-run and the medium-run analyses, the use of Marglin
and Bhaduri’s (1990) investment function (MB type investment function) produces various
regimes depending on the conditions. In the MB type investment function, the profit shareand the rate of capacity utilization determine investment. We apply this MB type investment
function to the long-run analysis, and investigate which regime is obtained in the long-run
equilibrium. For comparison, we also use the conventional Kalecki type investment function, in which the rate of profit and the rate of capacity utilization determine investment. It is
known that in the short-run and the medium-run equilibria with the Kalecki type investment
function, both the stagnationist regime and the wage-led growth regime hold. However, thelong-run equilibrium with the Kalecki type investment function, too, has not been discussed
in detail.2 To sum up, this paper contributes to the long-run analysis.
Another contribution of this paper concerns the analysis of the rate of employment. The
existing Kaleckian models do not satisfactorily investigate the rate of employment. Thesemodels consider that labor supply is unlimited and that the firms employ as many workers
as they desire at the given wages. However, it is reasonable to think that labor supply isconstrained in the long run. In this case, the equalization of labor demand growth andlabor supply growth is an accidental occurrence. Thus, if the labor supply growth steadily
2

Cassetti (2006) uses the Kalecki type investment function in the long-run analysis.

2exceeds the labor demand growth, then the rate of employment will be zero; however, thisis unrealistic. Therefore, to determine the long-run rate of employment, we need a model inwhich the economically meaningful employment rate is endogenously determined.
Some papers present Kaleckian models in which the rate of employment is endogenously
determined (Dutt, 1992; Lima, 2004; Stockhammer, 2004; Sasaki, 2010). However, fromour viewpoint, these models are only concerned with the medium-run analysis, that is, they
do not consider the long-run analysis. In our model, the rate of employment is endogenously
determined in both the medium run and the long run. From this, we can see the factor thataﬀects the rate of employment. For this purpose, we introduce into the Kaleckian model, a
specification relating the rate of employment with the growth rate of labor productivity, as
in Bhaduri (2006), Dutt (2006), and Flaschel and Skott (2006).
The remainder of the paper is organized as follows. Section 2 conducts a mediumrun analysis. In the medium run, the rate of capacity utilization, the profit share, and theemployment rate are adjusted and endogenously determined. Section 3 conducts a longrun analysis. In the long-run, the normal rate of capacity utilization and the expected rateof growth are adjusted and endogenously determined. Moreover, this section investigates
which regime is obtained in the long-run equilibrium. Section 4 concludes the paper.

2 Medium-run equilibrium
Consider an economy with workers and capitalists. Suppose that the workers consume all
their wages and the capitalists save a fraction s of their profit: only the capitalists save. Then,the ratio of real savings S to the capital stock K, that is, g s = S /K leads to
g s = sr,

0 < s ≤ 1,

(1)

where r denotes the rate of profit.
Suppose that the firms operate with the following fixed coeﬃcient production function:
Y = min{aL, (u/k)K},(2)

where Y denotes the real output; L, employment; and a = Y/L, the level of labor productivity.
The rate of capacity utilization u is defined as u = Y/Y ∗ , where Y ∗ denotes the potential
output. The coeﬃcient k = K/Y ∗ denotes the ratio of the capital stock to the potential
output, which is assumed to be constant. This assumption implies that both K and Y ∗ growat the same rate. Moreover, when the rate of capacity utilization is constant, the growth rateof capital stock and that of the actual output will be the same. Accordingly, the actual output
and the potential output grow at the same rate in the equilibrium where the rate of capacity3

utilization is constant. To simplify the analysis, in what follows, we assume that k = 1.
From this, we have r = mu, where m denotes the profit share.
Let us specify the firms’ planned investment. As stated above, we introduce two kindsof investment function.
For the Kalecki type investment function, we specify the ratio of the real investment I to
the capital stock, gd = I/K as follows:
Kalecki type : gd = γ + δr + ε(u − un ),γ ∈ (0, 1), δ ∈ (0, 1), ε ∈ (0, 1), γ > εun ,(3)where γ denotes a constant term capturing the expected rate of growth, and un , the normal
rate of capacity utilization. Note that the investment function is increasing in both the rateof profit and the gap between the actual rate of capacity utilization and the normal rateof capacity utilization. The specification that u − un is an explanatory variable is adopted
by Amadeo (1986) and Lavoie (2006). If the actual rate of capacity utilization is equal
to the normal rate of capacity utilization (u = un ), the firms expand plants at the rate of
gd = γ + δr. If, however, the actual rate of capacity utilization is less than the normal rateof capacity utilization (u < un ), the firms consider themselves as facing excess capacity, andthey decrease the rate of capital accumulation to a level lower than in the case of u = un . If,
on the other hand, the actual rate of capacity utilization exceeds the normal rate of capacity
utilization, the firms increase the rate of capital accumulation to a level higher than in thecase of u = un . Finally, the constraint γ > εun means that gd > 0 even if both u and r are
zero.
For the MB type investment function, we specify gd as follows:
MB type : gd = γ + δm + ε(u − un ),γ ∈ (0, 1), δ ∈ (0, 1), ε ∈ (0, 1), γ > εun .

(4)Marglin and Bhaduri (1990) state that it is not the rate of profit but the profit share thatshould be an explanatory variable of the investment function. In this case, it is well known
that we obtain diﬀerent regimes in the equilibrium depending on which eﬀect dominates,
the eﬀect of the profit share or the eﬀect of the rate of capacity utilization.
Now that we present the basic framework of the model, we turn to the derivation of the
dynamical equations for the medium-run analysis.
To begin with, we specify the adjustment of the rate of capacity utilization. In the goods
market, excess demand leads to a rise in the rate of capacity utilization while excess supply
leads to a decline in the rate of capacity utilization.
u̇ = α(gd − g s ),α > 0,

where α denotes the speed of adjustment of the goods market.
4(5)

From the definition of the profit share, we have m = 1 − (wL/pY), where w denotes themoney wage, and p, the price of goods. Diﬀerentiating the profit share with respect to time,
we obtain the following relationshipṁṗ ẇ ȧ= − + .
1−m p w a

(6)

To know the dynamics of the profit share, we must specify the dynamics of p, w, and a ofequation (6).
We specify the dynamics of the money wage and price by using the theory of conflictingclaims inflation.3 First, suppose that the growth rate of the money wage that the workers
manage to negotiate depends on the discrepancy between their target profit share and theactual profit share. Second, suppose that the firms set their price to close the gap between
their target profit share and the actual profit share. From these considerations, the dynamics
of the money wage and price can be described, respectively, as follows:ẇ= (1 − θ)(m − mw ), mw ∈ (0, 1),wṗ= θ(m f − m), θ ∈ (0, 1), m f ∈ (0, 1),
p

(7)
(8)

where θ is a positive parameter, which captures the bargaining power of the firms, and 1 − θis the bargaining power of the workers. The parameter mw denotes the target profit share set
by the workers, and m f , the target profit share set by the firms.4 In the following analysis,
we assume that m f > mw . The firms attempt to set their targets as high as possible whereas
the workers attempt to set their targets as low as possible. Therefore, the assumption thatm f > mw is reasonable.
We now turn to the specification of the endogenous technological change. As statedabove, we assume that the growth rate of labor productivity ga depends positively on therate of employment e.
ga = λeβ ,λ > 0, β > 0,

3

(9)

The theory of conflicting-claims inflation was originally developed by Rowthorn (1977). For the Kaleckian models with conflicting-claims inflation, see also Dutt (1987), Lavoie (1992), and Cassetti (2002, 2003,
2006).
4
In our model, mw and m f are given exogenously. Naturally, we can endogenize them. For example, it
is possible that mw is a decreasing function of ga whereas m f is an increasing function of ga . The former
specification captures the eﬀect wherein the labor unions attempt to reflect the fruit of an increase in laborproductivity in higher wages, whereas the latter specification captures the eﬀect wherein the firms attempt toreflect the fruit of an increase in labor productivity in higher profit. As will be shown by equation (9), ga is
an increasing function of e, and hence, mw is a decreasing function of e while m f is an increasing functionof e. With these modifications, the positive eﬀect of m′f (e) > 0 has an eﬀect of stabilizing the medium-run
equilibrium while the negative eﬀect of m′w (e) < 0 has a destabilizing eﬀect on the medium-run equilibrium.

5where ga = ȧ/a. The rate of employment is defined as e = L/N, where N denotes theexogenous labor supply. The coeﬃcient λ denotes a positive constant, and β, the elasticityof the growth rate of labor productivity with respect to the rate of employment. A rise in theemployment rate has an upward pressure on wages. The firms facing this upward pressure
tend to adopt labor-saving techniques.5 We use the above specification to explicitly solve forthe rate of employment. Note, however, that our results do not depend on this specification
as long as ga is an increasing function of e.6
Let us derive an equation of motion for the rate of employment. From equation (2), therate of employment is given by e = uK/(aN), from which the rate of change in e leads toė u̇= + gd − ga − n,e uwhere n denotes the growth rate of N and is given exogenously.

(10)

2.1 Medium-run with Kalecki type investment function
In conventional Kaleckian models, the variables u and m are adjusted in the short run or
medium run. In our medium-run model, in contrast, the three variables u, m, and e areadjusted. Substituting equations (1), (3), (7), (8), and (9) in equations (5), (6), and (10), we
obtain the following dynamical system:u̇ = α[γ + δmu + ε(u − un ) − smu],(11)ṁ = −(1 − m)[m − Γ − ga (e)], where Γ ≡ θm f + (1 − θ)mw ,{}α[γ + δmu + ε(u − un ) − smu]ė = e+ γ + δmu + ε(u − un ) − ga (e) − n .
u

(12)
(13)

Here, we assume that Γ > n. This assumption means that the weighted average of thetwo groups’ target profit shares is larger than the growth rate of exogenous labor supply.
Given that the size of n is about 10 percent at most, this assumption is plausible. Because
Γ = θm f + (1 − θ)mw , an increase in m f or mw corresponds to an increase in Γ. Moreover,because m f > mw , an increase in θ corresponds to an increase in Γ. Furthermore, we assume
that s > δ. This means that the capitalists’ savings rate is larger than the sensitivity ofinvestment with respect to the rate of profit, which is called the Keynesian stability condition.
The medium-run equilibrium is a situation wherein u̇ = ṁ = ė = 0; using this, we obtainthe following quadratic equation for the profit share:(s − δ)m2 − Θm + ε(Γ − n) = 0,

where Θ ≡ s(γ − εun ) + (s − δ)(Γ − n) + ε > 0.

5

(14)

For a Kaleckian model that considers such an eﬀect, see also Sasaki (2010).
Indeed, the stabilization analysis and the comparative static analysis that will be shown later are conductedwith the general form of ga = ga (e).
66From equation (14), we obtain two real and distinct roots. However, the larger root corresponds to the medium-run equilibrium value.7√
Θ2 − 4ε(s − δ)(Γ − n)
Θ
+
m∗ =
.
(15)
2(s − δ)Hereafter, the medium-run equilibrium values are denoted with “∗.” Using equation (15),
we obtain the rates of capacity utilization and employment in the medium-run equilibrium:γ − εun,(s − δ)m∗ − ε)1( ∗m −Γ β
∗
e =
.
λ

u∗ =

(16)
(17)

For the rate of capacity utilization to be positive, we need (s − δ)m∗ − ε > 0, which can be
rewritten as m∗ > ε/(s − δ). In addition, we assume that m∗ , u∗ , and e∗ are larger than zeroand less than unity.
To analyze the local stability of the medium-run equilibrium, we linearize equations (11),(12), and (13) around the equilibrium, and investigate the corresponding Jacobian matrix.
From this procedure, we obtain the following proposition:
Proposition 1. Suppose that the coeﬃcients of the investment function with respect to therate of profit and the rate of capacity utilization are not extremely large. Then, the mediumrun equilibrium is locally stable.


Proof. See Appendix C.

In the following analysis, we assume that the coeﬃcients of the investment function are
not extremely large.

2.2 Medium-run analysis with MB type investment function
Following the same procedure as in the case with the Kalecki type investment function, we
obtain the following dynamical system:7u̇ = α[γ + δm + ε(u − un ) − smu],(18)ṁ = −(1 − m)[m − Γ − ga (e)], where Γ ≡ θm f + (1 − θ)mw ,{}α[γ + δm + ε(u − un ) − smu]ė = e+ γ + δm + ε(u − un ) − ga (e) − n .
u(19)

See Appendix A.

7

(20)

The medium-run equilibrium is a situation wherein u̇ = ṁ = ė = 0; using this, we obtainthe following quadratic equation for the profit share:s(1 − δ)m2 − Θm + ε(Γ − n) = 0,

where Θ ≡ s(γ − εun ) + s(Γ − n) + ε > 0.

(21)

From equation (21), we obtain two real and distinct roots. However, the larger root corresponds to the medium-run equilibrium value.8√Θ+Θ2 − 4sε(1 − δ)(Γ − n)
m∗ =
.
(22)
2s(1 − δ)Using equation (22), we obtain the rates of capacity utilization and employment in themedium-run equilibrium:δm∗ + (γ − εun ),sm∗ − ε( ∗)1m −Γ β
∗
e =
.
λ

u∗ =

(23)
(24)

For the rate of capacity utilization to be positive, we need sm∗ −ε > 0, which can be rewritten
as m∗ > ε/s. In addition, we assume that m∗ , u∗ , and e∗ are larger than zero and less thanunity.
By examining the system of diﬀerential equations (18), (19), and (20), we obtain thefollowing proposition with regard to the stability of the medium-run equilibrium.
Proposition 2. Suppose that the coeﬃcient of the investment function with respect to the rateof capacity utilization is not extremely large. Then, the medium-run equilibrium is locallystable.


Proof. See Appendix D.

In the following analysis, we assume that the coeﬃcient of the investment function is not
extremely large.

2.3 Medium-run comparative statics analysis
2.3.1 Kalecki type investment function
Table 1 summarizes the results of the comparative statics analysis in the medium-run equilibrium with the Kalecki investment function.9 These results are the same as those obtained
8
9

See Appendix B.
We omit the partial derivatives because of spatial limitations.

8

from the conventional Kaleckian model except for the results of the rate of employment thatis not analyzed in the conventional models. For example, a rise in the capitalists’ savings
rate lowers the rate of capital accumulation, which is also known as the paradox of thrift. In
our model, the profit share is an endogenous variable, and thus, we cannot directly relate arise in m to the changes in u and g. Accordingly, we relate a rise in m f (mw ) to the changes
in u and g. A rise in m f (mw ) increases the profit share and lowers the rate of utilization andthe rate of capital accumulation. In this sense, the medium-run equilibrium corresponds toboth the stagnationist regime and the wage-led growth regime.
Table 1: Medium-run comparative statics analysis—Kalecki type investment function

u∗
m∗
e∗
g∗

s n m f , mw , θ
− +
−
− −
+
− −
−
− +
−

We now look at the eﬀect of a rise in θ on the rate of employment. In the Kalecki type
investment function, an increase in the bargaining power of the firms decreases the rate of
employment in the medium-run equilibrium. In other words, an increase in the bargaining
power of the workers decreases the rate of unemployment. This result contrasts sharply with
the neoclassical assertion that lowering the bargaining power of the labor union decreases
the rate of unemployment.
2.3.2 MB type investment function
Table 2 summarizes the results of the comparative statics analysis in the medium-run equilibrium with the MB type investment function.10 The columns that contain two opposing
signs indicate that two results are obtained depending on the conditions.
Using the MB type investment function, we obtain results that are diﬀerent from theresults in the Kalecki type investment function.
As in the case with the Kalecki type investment function, we define the regime using m f
and mw . First, a rise in m f (mw ) decreases the rate of capacity utilization, which means thatthe medium-run equilibrium is the stagnationist regime.11 Second, a rise in m f (mw ) either
10

We omit the partial derivatives because of spatial limitations.
When using a linear MB type investment function like in the present paper, we cannot obtain the exhilarationist regime. For details, see Blecker (2002).
11

9

Table 2: Medium-run comparative statics analysis—MB type investment function

∗

u
m∗
e∗
g∗

s
−
−
−
−

n
m f , mw , θ
+
−
−
+
−
−/+
+/−
−/+

decreases (the wage-led growth regime) or increases (the profit-led growth regime) the rateof capital accumulation.
When the medium-run equilibrium is the profit-led growth regime, a rise in the bargaining power of the firms increases the rate of employment. In contrast, when the medium-run
equilibrium is the wage-led growth, a rise in the bargaining power of the firms decreases therate of employment. Therefore, the relationship between the bargaining power and the rateof employment diﬀers between the two regimes.12

3 Long-run equilibrium
In the long run, the normal rate of capacity utilization and the expected rate of growth areadjusted. We assume that in the long run, the medium-run equilibrium is always attained:
the medium-run equilibrium values u∗ , m∗ , and e∗ correspond to the actual values in thelong run. Following Dutt (1997), Lavoie et al. (2004), and Cassetti (2006), we describe thelong-run dynamics.13
First, we assume that the normal rate of capacity utilization is adjusted according to thegap between the actual rate of capacity utilization and the normal rate of capacity utiliza-

12

Similar results are also obtained in Sasaki (2010).
In the long run as considered in Cassetti (2006), in addition to the normal rate of capacity utilizationand the expected rate of growth, the normal rate of profit and the drop-out ratio of capital equipment are also
adjusted: the four variables are adjusted in the long run. In the present paper, we consider only the adjustmentsof the normal rate of capacity utilization and the expected rate of growth because we think that these twovariables are essential in the long run.
13

10

tion.14u̇n = ϕ(u∗ − un ),ϕ > 0,(25)

where ϕ denotes the speed of adjustment. Note that we have u∗ = u∗ (un , γ) from the mediumrun analysis.
Second, we assume that the expected rate of growth is adjusted according to the gap
between the actual growth rate and the expected rate of growth:γ̇ = η(g∗ − g∗d ),η > 0,(26)

where η denotes the speed of adjustment. Note that when u = un , we have g∗d = γ + δr∗ forthe Kalecki type investment function and g∗d = γ + δm∗ for the MB type investment function.
Using this, for both the Kalecki type and the MB type investment functions, equation (26)can be rewritten as follows:γ̇ = ηε(u∗ − un ).

(27)

3.1 Kalecki type investment function
The system of diﬀerential equations composed of equations (25) and (27) takes a special
form called a zero-root system.15 In the zero-root system, one eigenvalue of the Jacobianmatrix is zero.16
The long-run equilibrium is a situation wherein u̇n = γ̇ = 0. From equations (25) and(27), we obtain u∗ (un , γ) = un , which yields the following relationship between un and γ:17γ=(s − δ)(Γ − n)un.
1 − sun(28)

14

The empirical validity of the simultaneous adjustments of the normal rate of capacity utilization and the
expected rate of growth is supported by Lavoie et al. (2004). In contrast, Skott (2010) criticizes Lavoie et al.
(2004): their estimated empirical hysteresis equation bears no relation to the theoretical model. For debateswith regard to the long-run adjustment of the normal rate of capacity utilization, see also Hein, Lavoie, andvan Treeck (2010a, b) and Skott and Zipperer (2010).
15
For solutions to the zero-root system, see Giavazzi and Wyplosz (1985).
16
The other eigenvalue is the trace of the Jacobian matrix.
17
Using u∗ (un , γ) = un and (16), we obtainγ − εun= un .
(s − δ)m∗ (un , γ) − ε
Because m∗∗ = [sγ + (s − δ)(Γ − n)]/(s − δ) in the long-run equilibrium, by replacing m∗ (un , γ) with m∗∗ =[sγ + (s − δ)(Γ − n)]/(s − δ) and rearranging the resultant expression, we obtain (28).

11

On the un -γ plane, this is an upward-sloping curve through the origin with un = 1/s being
an asymptote. The points on this curve correspond to the long-run equilibria. However, not
all the points on this curve are long-run equilibria and some constraints are imposed.18
From the two adjustment equations (25) and (27), we obtain the relationship γ̇ = (ηε/ϕ)u̇n ,from which we have the constant ratio γ = (ηε/ϕ)un along the transitional process. Hence,with initial conditions γ(0) and un (0), the transitional process is given by the following
upward-sloping straight line:γ(t) =ηεηε
un (t) + γ(0) −un (0).
ϕϕ(29)

The intersection of equations (28) and (29) yields the long-run equilibrium. It follows
from this that the long-run equilibrium depends on the initial conditions γ(0) and un (0):diﬀerent initial conditions produce diﬀerent long-run equilibria. Therefore, in this sense,
the long-run equilibrium shows path-dependency.
Figure 1 shows the long-run phase diagram. First, we consider an economy that starts
from point A. In this case, the long-run equilibrium is given by point E1 . Next, we consider
an economy that starts from point B. In this case, the long-run equilibrium is given by pointE2 




      

!




Figure 1: Long-run phase diagram

Strictly speaking, we need additional conditions: u∗ , m∗ , and e∗ should be more than zero and less thanunity in the medium run, un should be more than zero and less than unity in the long run, etc. In the following
analysis, we omit those conditions for the sake of simplicity.
18

12

When γ∗∗ is determined, the other long-run equilibrium values are determined as follows:
sγ∗∗
,
s−δ
sγ∗∗ − n(s − δ)
g∗∗
=
,
a
s−δ
sγ∗∗ + (s − δ)(Γ − n)
m∗∗ =
,
s−δ
γ∗∗
u∗∗ = ∗∗,
sγ + (s − δ)(Γ − n)[ ∗∗]1sγ − n(s − δ) β∗∗e =.
λ(s − δ)
g∗∗ =

(30)
(31)
(32)
(33)
(34)

The long-run equilibrium values are denoted with “∗∗.” Note that in addition to un (0) andγ(0), γ∗∗ depends on the initial conditions.
The long-run equilibrium is not always stable. If the coeﬃcient ηε/ϕ, the slope of thepath, is extremely large, then the economy cannot reach the long-run equilibrium. In addition, even if the size of ηε/ϕ is modest, it is possible that the economy does not reachthe long-run equilibrium depending on the initial condition. Let us return to figure 1. The
solution path denoted by the broken line starting from point A crosses the locus u̇n = γ̇ = 0
at a point where un exceeds unity, and consequently, the path is divergent. Moreover, thesolution path denoted by the broken line starting from point B does not cross the locusu̇n = γ̇ = 0. Therefore, both the size of adjustment and the initial condition are crucial forstable convergence to the long-run equilibrium.
We conduct a comparative statics analysis in the long-run equilibrium by assuming thatthe equilibrium is stable. Here, we investigate un and γ. First, when s or Γ increases, thelocus u̇n = γ̇ = 0 rotates counterclockwise around the origin. Because the solution path isunaﬀected by the change in s or Γ, the intersection of the solution path and the locus move∗∗toward the lower left. Therefore, both u∗∗n and γ decrease. Second, when n increases, thelocus rotates clockwise. Because the solution path is unaﬀected by the change in n, both u∗∗nand γ∗∗ increase. These results are summarized in table 3.
Table 3: Long-run comparative statics analysis—Kalecki type investment function

∗∗

γ
u∗∗

s
−
−

n m f , mw , θ+−+−

We investigate the regime obtained in the long-run equilibrium. As in the medium-run
13equilibrium, we define the regime using m f and mw . Depending on whether or not the
expected growth rate γ is adjusted, we obtain diﬀerent results.19
Diﬀerentiating u∗∗ and g∗∗ with respect to Γ, we obtain∗∗dγdu∗∗ (s − δ)(Γ − n) dΓ − γ(s − δ)
=
< 0,
dΓ
[sγ + (s − δ)(Γ − n)]2dg∗∗
s dγ∗∗=≤ 0.
dΓ
s − δ dΓ

(35)
(36)

First, when the expected rate of growth is not adjusted, we have dγ∗∗ /dΓ = 0, andconsequently, a rise in m f decreases the rate of capacity utilization but does not aﬀect the rateof capital accumulation. Therefore, the long-run equilibrium is the stagnationist regime andnot the wage-led growth regime or the profit-led growth regime. These results are explained
as follows. When γ is not adjusted, we can regard γ as an exogenous variable. g∗∗ depends on
γ but γ does not depend on m f , which implies that g∗∗ does not depend on m f . Therefore, thelong-run equilibrium is neither the wage-led growth regime nor the profit-led growth regime.
On the other hand, u∗∗ depends on m f through Γ. Therefore, even if γ is an exogenous
variable, a rise in m f decreases u∗∗ .
Second, when the expected rate of growth is adjusted, we have dγ∗∗ /dΓ < 0, and consequently, a rise in m f decreases both the rate of capacity utilization and the rate of capital
accumulation in the long-run equilibrium. Therefore, the long-run equilibrium is both thestagnationist regime and the wage-led growth regime. In other words, if the expected rate of
growth is adjusted, even in the long run, we obtain properties that are typical of the Kaleckian model. The reasoning is explained as follows. Both g∗∗ and u∗∗ depend on γ∗∗ . When the
expected rate of growth is adjusted, γ becomes an endogenous variable, and hence, γ∗∗ isdecreasing in m f . Therefore, a rise in m f leads to a decrease in γ∗∗ , which leads to a decreasein both u∗∗ and g∗∗ .
We now focus on the eﬀect of a rise in the capitalists’ savings rate on the rate of capacity
utilization and the rate of capital accumulation in the long-run equilibrium. Diﬀerentiatingu∗∗ and g∗∗ with respect to s, we obtain the following equations:∗∗dγdu∗∗ (s − δ)(Γ − n) ds − γ[γ + (Γ − n)]
=
< 0,
ds
[sγ + (s − δ)(Γ − n)]2dg∗∗
s dγ∗∗
δγ∗∗
=
−
< 0.
ds
s − δ ds(s − δ)2

(37)
(38)

When γ is fixed and only un is adjusted in our model, the resultant long-run equilibrium is always stable.
On the other hand, when un is fixed and only γ is adjusted, the resultant long-run equilibrium is necessarily
unstable. Therefore, we do not examine this latter case.
1914While u∗∗ and g∗∗ depend directly on s, these values depend indirectly on s through γ∗∗ .
From this, we can see that both u∗∗ and g∗∗ are aﬀected by a change in the savings rateirrespective of whether the expected rate of growth is adjusted or not. From equations (37)and (38), we find that a rise in s decreases both u∗∗ and g∗∗ . That is, a rise in the capitalists’savings rate lowers both the rate of capacity utilization and the rate of capital accumulationin the long-run equilibrium.20 Therefore, even in the long run, the paradox of thrift stillholds, which is a characteristic of the Kaleckian growth model.
In turn, we investigate the long-run relationship between the rate of employment and the∗∗bargaining power. Since, in the long-run equilibrium, the relation g∗∗
a = g − n holds andga is increasing in e, the eﬀect of Γ (i.e., the eﬀect of θ) on e∗∗ is the same as the eﬀect of
Γ on g∗∗ . First, when the expected rate of growth is not adjusted, the bargaining power ofthe firms is unrelated to the rate of employment because θ does not appear in equation (34).
Second, when the expected rate of growth is adjusted, a rise in the bargaining power of thefirms lowers the rate of employment in the long-run equilibrium, which is the same result asthat in the medium-run equilibrium.

3.2 MB type investment function
In the long-run equilibrium, both un and γ satisfy the following relationship:21γ=(Γ − n)(sun − δ).
1 − sun(39)

As in the Kalecki type investment function, this is an upward-sloping curve.
The long-run solution path satisfies the following relationship:
γ(t) =ηεηε
un (t) + γ(0) −un (0).
ϕϕ(40)This is identical to the solution path in the Kalecki type investment function. The intersectionof (39) and (40) determines the long-run equilibrium.

20

We must pay attention to the case where the expected rate of growth is not adjusted. Note that in the present
paper, both the rate of profit and the rate of capacity utilization are explanatory variables of the investment
function. In contrast, if the investment function includes only the rate of capacity utilization as the explanatory
variable, that is, if δ = 0, we have g∗∗ = γ∗∗ . If the expected rate of growth is not adjusted, that is, if γ is fixed,
then g∗∗ is independent of s, and consequently, the paradox of thrift is not obtained. Therefore, the results inthe present paper depend on the specification of the investment function.
21
Equation (39) can be obtained by the same procedure as that explained in footnote 17.

15

When γ∗∗ is determined, the other long-run equilibrium values are determined as follows:γ∗∗ + δ(Γ − n),1−δγ∗∗ − n + δΓ
g∗∗
=
,
a
1−δ
γ∗∗ + Γ − n
m∗∗ =
,
1−δ
γ∗∗ + δ(Γ − n)∗∗u =,s[γ∗∗ + (Γ − n)][ ∗∗]1
γ − n + δΓ β∗∗e =.
λ(1 − δ)
g∗∗ =

(41)
(42)
(43)
(44)
(45)

The long-run phase diagram is almost the same as that of the Kalecki type investment
function, and thus, we omit it. Therefore, the long-run equilibrium is stable in some cases
and unstable in others. Table 4 shows the results of the long-run comparative statics analysis
given the stability of the long-run equilibrium. The reasons why we obtain such results are
the same as those in the Kalecki type investment function case. A rise in s or Γ rotates thelocus of u̇n = γ̇ = 0 counterclockwise, and a rise in n rotates the locus clockwise. In either
case, the solution path does not shift. Therefore, we obtain the results shown in table 4.
Table 4: Long-run comparative statics analysis—MB type investment function

∗∗

γ
u∗∗

s
−
−

n m f , mw , θ+−+−

Let us investigate the long-run equilibrium regime. Diﬀerentiating u∗∗ and g∗∗ with respect to Γ, we obtain∗∗dγdu∗∗ s(1 − δ)(Γ − n) dΓ − sγ(1 − δ)
=
< 0,
dΓ
[sγ + s(Γ − n)]2
( ∗∗
)
1
dγ
dg∗∗
=
+δ .
dΓ
1 − δ dΓ

(46)
(47)

First, we consider the case where the expected rate of growth is not adjusted. Since γ
is fixed and constant in this case, we obtain dγ∗∗ /dΓ = 0. From this, we get that a rise inm f decreases u∗∗ but increases g∗∗ . Therefore, the long-run equilibrium is the stagnationist
regime and the profit-led growth regime. Recall that in the medium-run equilibrium of the
MB type investment function, we obtain either the wage-led growth regime or the profit-led
16

growth regime depending on the conditions. In contrast, we obtain only profit-led growth in
the long-run equilibrium when the expected rate of growth is not adjusted.
Next, we consider the case where the expected rate of growth is adjusted. In this case, we
obtain dγ∗∗ /dΓ < 0. However, the sign of equation (47) is undetermined: the sign depends
on the size of δ of the investment function (4). If δ is smaller than the absolute value ofdγ∗∗ /dΓ, the long-run equilibrium is the wage-led growth regime. On the other hand, if δ
is larger than the absolute value of dγ∗∗ /dΓ, then the long-run equilibrium is the profit-led
growth regime. Note that when δ is small (large), the medium-run equilibrium tends tobe the wage-led (profit-led) growth regime. This fact suggests that when the medium-run
equilibrium is the wage-led (profit-led) growth regime, the long-run equilibrium is also thewage-led (profit-led) growth regime.22
We now focus on the eﬀect of a rise in the capitalists’ savings rate on the rate of capacity
utilization and the rate of capital accumulation in the long-run equilibrium. Diﬀerentiatingu∗∗ and g∗∗ with respect to s, we obtain the following equations:∗∗dγdu∗∗ s(1 − δ)(Γ − n) ds − [γ + δ(Γ − n)][γ + (Γ − n)]
=
< 0,
ds
[sγ + s(Γ − n)]2dg∗∗
1 dγ∗∗=≤ 0.
ds
1 − δ ds

(48)
(49)

First, when γ is not adjusted, a rise in the savings rate lowers the rate of capacity utilization. However, the rate of capital accumulation in the long-run equilibrium is independentof s when dγ∗∗ /ds = 0, and consequently, the paradox of thrift does not hold. Second, when
γ is adjusted and accordingly, dγ∗∗ /ds < 0, a rise in the savings rate increases both therate of capacity utilization and the rate of capital accumulation. Therefore, the paradox ofthrift holds. In the Kalecki type investment function case, we obtain the paradox of thrift
irrespective of whether or not the expected rate of growth is adjusted. However, in the MB
type investment function case, we obtain the paradox of thrift only when the expected rateof growth is adjusted.
We consider the long-run relationship between the rate of employment and the bargaining power. From the structure of the model, the eﬀect of a rise in m f on the rate of employment is identical to the eﬀect of a rise in θ on the rate of employment. As a result, when
the long-run equilibrium is the wage-led (profit-led) growth regime, a rise in the bargaining
power of the firms decreases (increases) the rate of employment. This result is the same asthat obtained in the medium-run equilibrium.

22

We can show that such a relationship actually holds using numerical examples. For this, see Appendix E.

17

3.3 Summary
Finally, we summarize the results obtained in the medium-run and long-run equilibria.
Table 5: Classification of the regimes in the medium-run and long-run equilibria
Kalecki(medium-run)MB(medium-run)

◦
◦
×

◦
◦
◦

Stagnationist
Wage-led growthProfit-led growth

Kalecki(long-run)
γ: fixed; adjusted
◦, ◦
×, ◦
×, ×

MB
(long-run)
γ: fixed; adjusted
◦, ◦
×, ◦
◦, ◦

In table 5, the symbol “◦” indicates that the corresponding regime is obtained whereas
the symbol “×” indicates that the corresponding regime is not obtained. For example, in themedium-run equilibrium with the Kalecki type investment function, both the stagnationist
and the wage-led growth regimes are obtained whereas the profit-led growth regime is not
obtained. In the columns headed with “long-run,” the left cell corresponds to the case where
the expected rate of growth is fixed while the right cell corresponds to the case where theexpected rate of growth is adjusted. For example, in the long-run equilibrium with the MB
type investment function, the wage-led growth regime is not obtained when γ is fixed, while
the wage-led growth regime is obtained when γ is adjusted.

4 Conclusions
This paper presents a long-run Kaleckian model, and investigates which regime is obtainedin the long-run equilibrium. For this purpose, we use the Kalecki type investment function
and the MB type investment function. Moreover, we discuss the equilibrium rate of employment that is not considered in detail in the existing Kaleckian models. The results are
summarized as follows.
When the expected rate of growth is not adjusted, the long-run equilibrium with the
Kalecki type investment function is neither the wage-led growth regime nor the profit-led
growth regime, while the long-run equilibrium with the MB investment function is the profitled growth regime.
When the expected rate of growth is adjusted, the long-run equilibrium with the Kalecki
type investment function is the wage-led growth regime; on the other hand, the long-run
equilibrium with the MB type investment function is the wage-led growth regime if themedium-run equilibrium is the wage-led growth regime, but is the profit-led growth regime18

if the medium-run equilibrium is the profit-led growth regime. Therefore, we find that theimportant characteristics of the Kaleckian model are also observed in the long run.
The relationship between the rate of employment and the bargaining power of the firms
is as follows. In the medium-run equilibrium, a rise in the bargaining power of the firms
decreases the rate of employment if the equilibrium is the wage-led growth regime, but
increases the rate of employment if the equilibrium is the profit-led growth regime. In thelong-run equilibrium with the Kalecki type investment function, the rate of employment isfree from the bargaining power of the firms if the expected rate of growth is not adjusted.
From these observations, we find that the adjusting of the expected rate of growth isdecisive in determining whether or not the characteristics of the medium-run equilibrium
are reflected in the characteristics of the long-run equilibrium. Naturally, the adjustmentof the normal rate of capacity utilization also aﬀects the results. Therefore, in the future,
we intend to focus on the empirical tests for the validity of these long-run adjustments. In
addition, since the type of the investment function aﬀects the results, the empirical tests forthe investment function will also be looked into.

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21

Appendix
A Medium-run equilibrium profit share: Kalecki type investment function
Let m1 and m2 be the roots of the quadratic equation (14). Calculating the discriminant of
the equation, we find it to be positive. Hence, m1 and m2 are the two real distinct roots. Let
m1 < m2 . We have
Θ
> 0,
s−δ
ε(Γ − n)m1 m2 => 0.
s−δm1 + m2 =

(A-1)
(A-2)

That is, both m1 and m2 are positive. The condition for the medium-run equilibrium valueof the rate of capacity utilization to be positive is given by (s − δ)m − ε > 0. From this, we
obtain m > ε/(s − δ), which is a constraint on m∗ . Let f (m) be the left-hand side of equation(14). Substituting m = ε/(s − δ) in f (m), we obtainf (ε/(s − δ)) = −εs(γ − εun )
< 0.
s−δ

(A-3)

This means that m1 is smaller than ε/(s − δ) and that m2 is larger than ε/(s − δ). Therefore,
m2 is the medium-run equilibrium value.

B Medium-run equilibrium profit share: MB type investment function
Let m1 and m2 be the roots of the quadratic equation (21). Calculating the discriminant of
the equation, we find it to be positive. Hence, m1 and m2 are the two real distinct roots. Let
m1 < m2 . We haveΘ> 0,s(1 − δ)ε(Γ − n)m1 m2 => 0.
s(1 − δ)m1 + m2 =

(B-4)
(B-5)

That is, both m1 and m2 are positive. The condition for the medium-run equilibrium valueof the rate of capacity utilization to be positive is given by sm − ε > 0. From this, we
obtain m > ε/s, which is a constraint on m∗ . Let f (m) be the left-hand side of equation (21).
22Substituting m = ε/s in f (m), we obtainε2− (γ − εun ) − (1 − ε)(Γ − n) < 0.
(B-6)
sThis means that m1 is smaller than ε/s and that m2 is larger than ε/s. Therefore, m2 is themedium-run equilibrium value.
f (ε/s) = −C Local stability of the medium-run equilibrium: Kalecki
type investment function
To investigate the local stability of the medium-run equilibrium, we linearize the system ofdiﬀerential equations (11), (12), and (13) around the equilibrium.
  


 u̇   J11 J12 0   u − u∗   
 

(C-7)
ṁ =  0 J22 J23  m − m∗  .
  
 


ė
J31 J32 J33 e − e∗
The elements of the Jacobian matrix J are given by∂u̇= −α[(s − δ)m − ε] < 0,
(C-8)
J11 ≡
∂u
∂u̇
J12 ≡
= −α(s − δ)u < 0,
(C-9)
∂m
∂ṁ
J22 ≡
= −(1 − m) < 0,
(C-10)∂m∂ṁ
J23 ≡= (1 − m)g′a (e) > 0,
(C-11)∂e∂ė
αe[(s − δ)m − ε]J31 ≡=−+ e(δm + ε),(C-12)
∂u
u
∂ė
J32 ≡
= −e[α(s − δ) − δu],(C-13)∂m∂ė
J33 ≡= −eg′a (e) < 0.
(C-14)∂e
All elements are evaluated at the equilibrium value; we omit “∗” to avoid troublesome notations.
The characteristic equation of J is given by
q3 + a1 q2 + a2 q + a3 = 0,

(C-15)

where q denotes a characteristic root. Each coeﬃcient of equation (C-15) is given by
a1 = −tr J = −(J11 + J22 + J33 ),a2 =(C-16)J22 J23
J11 0
J11 J12++= J22 J33 − J23 J32 + J11 (J33 + J22 ),J32 J33
J31 J33
0 J22
23(C-17)

a3 = − det J = −J11 (J22 J33 − J23 J32 ) − J31 J12 J23 ,(C-18)

where −a1 = tr J denotes the trace of J; a2 , the sum of the principal minors’ determinants;and −a3 = det J, the determinant of J.
The necessary and suﬃcient conditions for the local stability are a1 > 0, a2 > 0, a3 > 0,
and a1 a2 − a3 > 0. Accordingly, we need to examine these inequalities.
a1 = [(s − δ)m − ε] α + (1 − m) + eg′a (e) = Aα + B > 0,
|
{z
}
|
{z
} +
+
≡A>0

(C-19)≡B>0

a2 = {[(s − δ)m − ε][(1 − m) + eg′a (e)] + (s − δ)(1 − m)eg′a (e)} α + (1 − δu)(1 − m)eg′a (e)
|
{z
}
|
{z
}
≡C>0

≡D>0

= C α + D > 0,+

(C-20)+

a3 = (1 − m)eg′a (e){[(s − δ)m − ε] + εsu} α = E α > 0,
|
{z
}
+

(C-21)

a1 a2 − a3 = |{z}AC α + (AD + BC − E)α + |{z}BD .

(C-22)

≡E>0

2

+

+

Here, we haveAD + BC − E = [(s − δ)m − ε][(1 − m) + eg′a (e)]2
|
{z
}
+

+ (s − δ)(1 − m)eg′a (e)[eg′a (e) + 1
δum − εu
| − m −{z
}].
|
{z
}

(C-23)≡Λ+Λ will be positive if δ and ε are not extremely large. When Λ > 0, we obtain AD+BC−E > 0,
which leads to a1 a2 − a3 > 0. Therefore, if Λ > 0, the necessary and suﬃcient conditions forthe local stability are satisfied. Note that Λ > 0 is a suﬃcient condition for a1 a2 − a3 > 0.

D Local stability of the medium-run equilibrium: MB type
investment function
The elements of the Jacobian matrix J that consists of equations (18), (19), and (20) are
given by∂u̇
= −α(sm − ε) < 0,
∂u
∂u̇
J12 ≡
= −α(su − δ),∂m∂ṁ
= −(1 − m) < 0,
J22 ≡
∂m

J11 ≡

24

(D-24)(D-25)(D-26)

∂ṁ= (1 − m)g′a (e) > 0,∂e∂ėαe(sm − ε)
J31 ≡
=−
+ εe,
∂u
u
∂ė
e[α(su − δ) − δu]
J32 ≡
=−
,
∂m
u
∂ė
J33 ≡
= −eg′a (e) < 0.
∂e
J23 ≡

(D-27)(D-28)(D-29)(D-30)All elements are evaluated at the equilibrium value.
The coeﬃcients of the characteristic equation are given by
a1 = (sm − ε) α + (1 − m) + eg′a (e) = Aα + B > 0,
(D-31)| {z }
|
{z
} +
+
≡A>0
≡B>0
{
}
(1 − m)eg′a (e)(su − δ)′a2 =+ (sm − ε)[(1 − m) + ega (e)] α + (1 − δ)(1 − m)eg′a (e)
|
{z
}
u
|
{z
}
≡D>0
≡C>0

= C α + D > 0,

(D-32)++

a3 = (1 − m)eg′a (e)[(1 − δ)(sm − ε) + ε(su − δ)] α = E α > 0,
|
{z
}
+

(D-33)

a1 a2 − a3 = |{z}AC α + (AD + BC − E)α + |{z}BD .

(D-34)

≡E>0

2

+

+

Here, we haveAD + BC − E = (sm − ε)[(1 − m) + eg′a (e)]2
|
{z
}
+


≡Λ

z }| {  eg′ (e) + 1 − m − εu  .
+ (1 − m)eg′a (e)(su − δ)  a

|
{z
} 
u


+

(D-35)

Λ is positive if ε is not extremely large. When Λ > 0, we obtain AD + BC − E > 0, whichleads to a1 a2 − a3 > 0. Therefore, if Λ > 0, the necessary and suﬃcient conditions for the
local stability are satisfied. Note that Λ > 0 is a suﬃcient condition for a1 a2 − a3 > 0.

25

E Numerical examples in the long run: MB type investment function
Table 6 shows the parameters used for the numerical example. The initial values are set toun (0) = 0.5 and γ(0) = 0.15 (common to all cases).23 In these examples, only the sizes of
the coeﬃcients ε and δ are diﬀerent. The coeﬃcient ε is larger (smaller) than the coeﬃcientδ in the wage-led (profit-led) growth regime example.
Table 6: Parameters used in the numerical examples
θ m f mwsnεδλMB wage-led growth 0.3 0.3 0.2 0.75 0.05 0.08 0.05 0.15
MB profit-led growth 0.3 0.3 0.2 0.75 0.05 0.04 0.09 0.15

ψ
1
1

α
1
1

ϕ η
1 1
1 1

Table 7 shows the results when the medium-run equilibrium is the wage-led growth
regime. We increase the target profit share of the firms m f from 0.3 to 0.31. From this,we have that a rise in m f lowers both the rate of capacity utilization and the rate of capital
accumulation in the long-run equilibrium. In other words, in this numerical example, thelong-run equilibrium is the stagnationist regime and the wage-led growth regime. Therefore,
the long-run equilibrium regime is the same as the medium-run equilibrium regime.
Table 7: Example where the long-run equilibrium is the wage-led growth regime (themedium-run equilibrium is the wage-led growth regime)
u
g
m
e
un
γ
ga
Benchmark0.670 0.182 0.362 0.878 0.670 0.164 0.132
m f : 0.3 → 0.31 0.664 0.181 0.364 0.875 0.664 0.163 0.131
Table 8 shows the results when the medium-run equilibrium is the profit-led growth
regime. From this, we have that a rise in m f decreases the rate of capacity utilization andthe rate of capital accumulation in the long-run equilibrium. In other words, in this numerical example, the long-run equilibrium is both the stagnationist regime and the profit-led
growth regime. Therefore, the long-run equilibrium regime is the same as the medium-run
equilibrium regime.

23

Since the long-run equilibrium has a path-dependent property, diﬀerent initial conditions produce diﬀerent
long-run equilibrium values.

26

Table 8: Example where the long-run equilibrium is the profit-led growth regime (themedium-run equilibrium is the profit-led growth regime)
u
g
m
e
un
γ
ga
Benchmark0.6861 0.1908 0.3708 0.9388 0.6861 0.1574 0.1408
m f : 0.3 → 0.31 0.6807 0.1909 0.3739 0.9392 0.6807 0.1572 0.1409

27