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

The Macroeconomic Effects of the Wage Gap between
Regular and Non-Regular Employment and Minimum Wages

Hiroaki Sasaki, Jun Matsuyama, and Kazumitsu Sako

Discussion Paper No. E-12-003

Research Project Center
Graduate School of Economics
Kyoto University
Yoshida-Hommachi, Sakyo-ku
Kyoto City, 606-8501, JapanMay 2012

The Macroeconomic Eﬀects of the Wage Gap
between Regular and Non-Regular Employment andMinimum Wages∗
Hiroaki Sasaki†

Jun Matsuyama‡

Kazumitsu Sako§

Abstract
We analyze the eﬀect of the wage gap between regular and non-regular employment
on a macroeconomy by using a Keynesian dynamic model. If the steady state equilibrium exhibits the stagnationist regime, the size of the reserve army eﬀect aﬀects thestability of the equilibrium. On the other hand, if the steady state equilibrium exhibits
the exhilarationist regime, an increase in the wage gap destabilizes the equilibrium. The
introduction of the minimum wage is desirable in that it mitigates fluctuations of business cycles. However, the introduction of an inappropriate minimum wage policy leads
to the real wage and employment rate that are lower than the steady state values.
Keywords: wage gap; regular and non-regular employment; minimum wage; demandled growth model
JEL Classification: E12; E24; E25; E32; J31; J83∗

We would like to thank Peter Skott, Naoki Yoshihara, Takashi Ohno, Shinya Fujita, and seminar participants at Kyoto University, Nihon University, and Hitotsubashi University for their useful comments andsuggestions.
†
Corresponding author. Graduate School of Economics, Kyoto University‡
Institute of Economic Research, Hitotsubashi University, JSPS Research Fellowships for Young Scientists(PD)§
Department of Economics, North Asia University

11 Introduction
Quite a long time has passed since the debate on the so-called regular and non-regular employment has begun in Japan. It has become an increasingly urgent problem.1 We can find,as reasons for the aggravation of this situation, not only the increase of non-regular employment due to a prolonged economic depression but also the prevalence of a discriminative
treatment based on the employment pattern, that is, regular employment and non-regular
employment.2 The progress of such an undesirable situation leads us to pay attention to thequestions: how the expansion of the wage gap between regular and non-regular employees has an influence on a macroeconomy; and what a consequence of a greater increase in
non-regular employees is. In order to investigate the questions, we build a model in which
regular and non-regular workers coexist. Further, we consider the minimum wage policy as
one of the policies taken by a government to alleviate the situation. We analyze how the
introduction of the minimum wage policy aﬀects the economy.
Flaschel and Greiner (2009) have justified a minimum wage policy based on humanrights.3 Flaschel and Greiner’s model is an application of Goodwin’s (1967) growth cycle
model, which assumes the two-class economy that consists of working and capitalist classes.
They show the existence of an endogenous and perpetual business cycle with respect to theemployment rate and wage share. They also show that the introduction of the minimum
wage within certain bounds below the steady state wage share can increase the stability of theeconomy. In their model, the minimum wage means the minimum wage share exogenously
determined by the government because labor productivity is constant. Based on Flascheland Greiner (2009), a line of research has been developed. Flaschel and Greiner (2011)
and Flaschel et al. (2011) introduce into the Goodwin model heterogeneous labor, that is,skilled and unskilled workers. The labor market consists of the first and second markets: theformer is the market only for the skilled workers and the latter is one for both the unskilledworkers and the skilled workers who do not find a job in the first market. It is assumed thatthe skilled workers do not lose their job because they can always find a job in the second
labor market even if they do not find a job in the first one. Thus, the excess supply of laborin the first labor market means that some of the unskilled workers lose their jobs with the1

For instance, according to the labor force survey February 2012 by the Ministry of Internal Aﬀairs
and Communications (http://www.stat.go.jp/data/roudou/longtime/03roudou.htm), the ratio of non-regular employee to the whole employee excluding executive of company or corporation of 2011 October-December average is 35.7 %, the highest record ever registered. Especially, that of the younger generation from 15 to 34 is
33.1%.
2
Japan does not ratify the International Labor Organization (ILO) convention 111, which prescribes fora prohibition of discriminating in respect of employment and occupation, in particular, a discrimination ofregular and non-regular employee.
3
As a theoretical contribution to researches on human rights, see, for example, Sen (2004).

2

assumption that labor supply is abundant. In addition, it is also assumed that only the skilledworkers in the first labor market are involved in participating in the wage bargain. Thismeans that wages of the workers in the second labor market depend on the bargain by theskilled workers. Thus, the skilled workers have a large influence on the unskilled workers
in the second labor market with respect to their wage determination and employment. Then,they show, like Flaschel and Greiner (2009), that endogenous business cycles can occur,and that the introduction of the minimum wage in the second labor market can alleviate
fluctuations of the economy.
The model developed in the present paper can be seen as a line of the above contributions.
We consider a two-class economy, which is the same as Flaschel and Greiner (2009). In
addition, two kinds of heterogeneous labor, that is, regular and non-regular employments
are introduced within the working class. Thus, our model is the two-class economy thatconsists of three units, which is similar to Flaschel and Greiner (2011). We analyze theeﬀect of the wage gap between regular and non-regular employments on the stability of thesteady state equilibrium by using a Kaleckian model. We also analyze how business cycles
are aﬀected by the introduction of the minimum wage.
The so-called Kaleckian model is a kind of Keynesian dynamic models based on the principle of the eﬀective demand, whose basic framework was developed by Rowthorn (1981).
In his model, two kinds of labor, that is, direct labor and indirect labor, which are emphasized by Kalecki himself, are considered. Direct labor refers to variable labor in accordancewith a change in output, while indirect labor refers to fixed labor regardless of the changeof output. In this sense, we can interpret direct (indirect) labor as regular (non-regular) one.
Although Rowthorn (1981) emphasized two kinds of labor, surprisingly, almost all Kaleckian models do not consider it.4 In particular, only a few contributions analyze the eﬀect of
direct and indirect labor on the stability of the steady state equilibrium. Raghavendra (2006),
one of the exceptional cases, extends Rowthorn’s model and presents a model in which an
endogenous business cycle with respect to the two variables (the capacity utilization rate
and profit share) occurs. To be concrete, although the income distribution (the wage andprofit shares) is exogenously given in Rowthorn’s model, it is endogenously determined in
Raghavendra’s one. In place of the investment function in Rowthorn’s model, which is anincreasing function of the capacity utilization and the profit rate, he introduces an investment function presented by Marglin and Bhaduri (1990), which is an increasing function of
the capacity utilization and the profit share. Moreover, he adopts a non-linear investment
function.
4

For the literature considering the two kinds of labor in a Kaleckian model, see Lavoie (1992). Lavoie(2009) is a recent contribution to the field.

3

In Raghavendra’s model, a limit cycle occurs under certain conditions due to the interaction between the nonlinearity of the investment function and increasing returns to scalecaused by the existence of fixed labor. That is to say, the occurrence of an endogenous
and perpetual business cycle is proved by Poincaré-Bendixson theorem. This suggests analternation of the exhilarationist and stagnationist regimes, where the exhilarationist (stagnationist) regime means that an increase in the profit share increases (decreases) the capacity
utilization rate.5 In this paper, fixed and variable labor in Raghavendra model are interpreted as regular and non-regular employments, respectively. Furthermore, we present theextended model in some respects, and analyze the stability of the steady state equilibrium
and the eﬀect of an introduction of minimum wages on a business cycle.
There are two diﬀerences between our model and Raghavendra one. First, an assumption
concerning wage diﬀers. Whereas Raghavendra (2006) assumes an equal pay between fixedlabor and variable one, we assume that regular workers earn a higher wage than non-regular
workers. Second, a formulation regarding dynamics of income distribution diﬀers. In his
model, the dynamics of income distribution (the profit share) are derived from the dynamic
markup pricing rule of firms and the wage curve.6 In contrast, we adopt the conflictingclaims theory of inflation in order to obtain the dynamics of income distribution.7 In particular, we introduce the reserve army eﬀect that the target profit share of labor unions is adecreasing function of the capacity utilization rate.
We analyze the dynamics of the capacity utilization rate and profit share by using theextended model. In the case where the steady state equilibrium is unstable in the exhilarationist regime, under certain conditions, an endogenous and perpetual business cycle occurs
by the Hopf bifurcation theorem. Generally, the longer a downturn of a business cycle is, theworse a non-regular worker becomes. One of the policies that a government should take toalleviate such an undesirable situation is a minimum wage policy, which puts lower bounds
to the wage share, that is, upper bounds to the profit share.
Let us explain the minimum wage policy. Here, the minimum wage is introduced by setting the upper bound of the profit share. Such formalization can be regarded as the minimum
wage in our model because the determination of the maximum profit share is equivalent tothat of the minimum wage of the non-regular employment. To be concrete, the reason is asfollows. First, by definition, the maximum profit share is equal to the minimum wage share.
Second, given the labor productivity, the determination of the minimum wage share is equiv5

See Blecker (2002) for further discussion concerning various regimes in the Kaleckian model.
The wage curve in Raghavendra (2006) means that a level of wage is an increasing function of the capacity
utilization rate.
7
Flaschel and Greiner (2009) present an extended version of the Goodwin model, where the dynamics ofincome distribution are given by the way that the dynamic markup pricing rule and a rate of change in wage
depends on both capacity utilization and inflation rate.
64

alent to that of the minimum average wage of the whole economy. Third, the average wageof the whole economy is proportional to the wage of the non-regular employment.8 Thus,
from the three facts, the equivalence between the determinations of the maximum profit
share and the minimum wage of the non-regular employment holds.
In our analysis, we focus on the case where the steady state equilibrium is locally unstable. The government does not accurately know the equilibrium profit share. Thus, theminimum wage determined by the government does not necessarily coincide with it. In thiscase, the following results are obtained by our analysis.
Under the exhilarationist regime, setting a higher profit share, that is, a lower wage one
relative to the equilibrium, diminishes the amplitude of the business cycle. In contrast, setting a lower profit share, that is, a higher wage one relative to the equilibrium, the economy
converges to a point that is diﬀerent from the equilibrium. In this case, we are faced with the
problem with respect to the comparison between the steady state without minimum wage
and the equilibrium after introducing it. We can compare the two equilibria by means of an
iso-real wage curve. Each value of the two equilibria depends on the shape of the curve.
Under the stagnationist regime, for any minimum wage relative to the steady state profit
share, the economy converges to the steady state equilibrium (as long as the minimum andmaximum wages are simultaneously introduced). In this case, in accordance with a minimum wage being put above, below, or equivalent to the steady state profit share, the economy
converges to the corresponding value. In comparing among the three equilibria, contrary to
the above case, for any shape the iso-real wage curve has, we can completely order the threeequilibria.
The rest of the paper is organized as follows. Section 2 provides our model. Section
3 analyzes the properties of the steady state equilibrium and its stability. Section 4, given
tha the steady state equilibrium is locally unstable, analyzes how the introduction of theminimum wage policy aﬀects the economy. Section 5 concludes.

2 The model
We consider an economy in which there exist two heterogeneous workers (regular and nonregular employments) and capitalists. Workers’ income consists of wages only, and they
spend all their income for consumption. On the other hand, capitalists obtain their income
from profits only, which is saved at a constant rate s. Then, the saving function is given by
g s = sr,
80 < s < 1,

See equations (7) and (8) below.

5(1)where g s = S /K is the ratio of real savings S to the capital stock K, and r is the profit rate.
Following Marglin and Bhaduri (1990), we assume that the investment function of firms
is an increasing function of the capacity utilization rate u and the profit share m.9
gd = gd (u, m),

gdu > 0, gdm > 0,(2)where gd = I/K denotes the ratio of the real investment I to the capital stock, gdu denotes
the partial derivative of the investment function with respect to the capacity utilization, andgdm denotes the partial derivative of the investment function with respect to the profit share.
For simplicity, we do not consider capital depreciation.
Let us assume that the ratio of the potential output Y F to the capital stock is technically
fixed, that is, always constant. Then, the capacity utilization rate can be represented asu = Y/K, where Y denotes the actual output. Note that the relation r = mu holds among
the profit rate, the profit share, and the capacity utilization rate. In the following analysis,
without loss of generality, we assume that the ratio of the potential output to the capital stock
is equal to unity.
We assume that the regular employment Lr is related to the potential output while thenon-regular employment Lnr is related to the actual output.
Lr = αY F ,Lnr = βY,

α > 0,
β > 0,(3)(4)

where α and β are positive constants. The ratio of the regular employment to the non-regular
employment leads to Lr /Lnr = α/(βu). This means that an increase in the capacity utilization
rate leads to a decrease in this ratio. In other words, relatively many non-regular workers
are employed with a rise in the capacity utilization rate. Once the capacity utilization rateis determined at the steady state equilibrium, the ratio of the regular employment to the
non-regular employment is also determined.
We assume that a quantity adjustment that the capacity utilization rate increases (decreases) in accordance with an excess demand (supply) in the goods market.
u̇ = ϕ(gd − g s ),ϕ > 0,(5)

where the parameter ϕ denotes the speed of adjustment of the goods market.
We define the level of the average labor productivity of the economy as a = Y/L, where L
9In the typical Kaleckian model, the investment function is formalized as an increasing function of the
profit rate and the capacity utilization rate. On the contrary, Marglin and Bhaduri (1990) assert that one of thevariables of the investment function should be the profit share rather than the profit rate.

6

denotes the aggregate employment, that is, L = Lr + Lnr . From this, we obtain the following:a=Y
αY F + βY=u.
α + βu(6)

This implies that the average labor productivity is an increasing function of the capacity
utilization rate, that is, increasing returns to scale prevail. Since u is constant at the steadystate equilibrium, the corresponding average labor productivity is also constant. Thus, there
is no perpetual technical progress in our model.
The nominal wage of the regular employment wr is supposed to be higher than that of
the non-regular employment wnr at a certain rate γ.10
wr = γwnr ,γ > 1.

(7)

From these, the average wage of the economy leads to[]γα + βu
Lnr
α
βu
Lr
wr +
wnr =
wr +
wnr =
wr .
w=
L
L
α + βuα + βuγ(α + βu)(8)

The average wage is given by the weighted average between the wages of regular and nonregular employments. Each weight corresponds to the corresponding employment share.
Whereas the weight of the regular employment is a decreasing function of the capacity
utilization rate, the weight of the non-regular employment is an increasing function of thecapacity utilization. The component in the square bracket is a decreasing function of thecapacity utilization rate.
Next, we formalize the equation of the price of goods p and the equation of the wageof the regular employment by using the theory of conflicting-claims inflation.11 First, firms
set their price so as to narrow the gap between the target profit share of firms m f and theactual profit share, and accordingly, the price changes. Second, labor unions negotiate so
as to narrow the gap between the target profit share of labor unions mw and the actual profit
share, and accordingly, the nominal wage of the regular employment changes. The two
assumptions can be written as follows:ṗ= θ(m f − m), 0 < θ < 1, 0 < m f < 1,pẇr
= (1 − θ)(m − mw ), 0 < mw < 1,
wr
10

(9)
(10)

A similar formalization is adopted by Lavoie (2009).
The theory of conflicting-claims inflation is presented by Rowthorn (1977). For previous studies on
Kaleckian models with this theory, see, for example, Dutt (1987) and Cassetti (2003).
11

7

where θ is a positive parameter. We interpret θ and 1 − θ as the bargaining power of firms
and that of labor unions, respectively. Further, by taking the reserve army eﬀect into consideration, we assume that mw is a decreasing function of the capacity utilization rate.
mw = mw (u),m′w < 0.

(11)

Since the profit share is given by m = 1 − (wL/pY) by definition, taking the derivative ofm with respect to time, we obtainṁṗ ẇ ȧ= − + .
1−m p w a

(12)

From equations (8) and (10), the rate of change of the average wage in the whole economy
is given byẇ
(γ − 1)αβ=−· u̇ + (1 − θ)[m − mw (u)].
w(γα + βu)(α + βu)(13)

From equation (6), the rate of change of the labor productivity is given by
α
ȧ
=
· u̇.
a (α + βu)u(14)

By substituting equations (1) and (2) in equation (5), and also equations (9), (12), and(13) in (11), we can obtain the following dynamic equations with respect to the capacity
utilization rate and the profit share.
u̇ = ϕ[gd (u, m) − smu],

ϕ > 0,(15)ṁ = −(1 − m)[m − θm f − (1 − θ)mw (u) − f (u)u̇],f (u) =αγ
, f ′ (u) < 0.
(γα + βu)u(16)

Firms set m f as large as possible. Conversely, labor unions set mw as small as possible.
Hence, we can assume that m f > mw (u).

8

3 Characteristics of the steady state equilibrium and its
stability
3.1 Characteristics of the steady state equilibrium
The steady state equilibrium is an equilibrium such that u̇ = ṁ = 0. From this, we have thesimultaneous equation with respect to u∗ and m∗ .
gd (u∗ , m∗ ) = sm∗ u∗ ,(17)m∗ = θm f + (1 − θ)mw (u∗ ).

(18)

In the following analysis, we assume that there exists a unique pair of u∗ ∈ (0, 1) and m∗ ∈(0, 1) that simultaneously satisfies equations (17) and (18).
The capacity utilization rate and the profit share at the steady state equilibrium dependon the bargaining power, the target profit share of firms, and the target profit share of labor
unions. However, the steady state equilibrium does not depend on the four parameters, ϕ, γ,α and β. This property will be used in the analysis of the next section.

3.2 The stability of the steady state equilibrium
In order to investigate the stability of the steady state equilibrium, we analyze the Jacobianmatrix of the system of the diﬀerential equations. Let the Jacobian matrix be J. The matrix
J is a 2 × 2 matrix, and its components are given as follows:∂u̇= ϕ(g∗du − sm∗ ),
∂u
∂u̇
J12 =
= ϕ(g∗dm − su∗ ),
∂m
∂ṁ
J21 =
= (1 − m∗ )[(1 − θ)m′w (u∗ ) + f (u∗ )J11 ],∂u∂ṁ= −(1 − m∗ )[1 − f (u∗ )J12 ].
J22 =
∂m
J11 =

(19)
(20)
(21)
(22)

Each component of J is evaluated at the steady state equilibrium.
Let us assume the following condition:
Assumption 1. sm∗ > g∗du .
This means that the response of savings to the capacity utilization rate is larger than thatof investments. This assumption makes the quantity adjustment of the goods market stable.
9Assumption 1 is sometimes called the Keynesian stability condition (Marglin and Bhaduri,1990), which is often imposed in Kaleckian models. With Assumption 1, we can obtain
J11 < 0.
Let us classify the regime according to the eﬀect of an increase in the profit share on thecapacity utilization.
Definition 1. If the relation g∗dm < su∗ holds, the steady state equilibrium is called thestagnationist regime. If, on the other hand, the relation g∗dm > su∗ holds, the steady state
equilibrium is called the exhilarationist regime.12
If the response of investment to the profit share is less than that of saving, then the steadystate equilibrium exhibits the stagnationist regime. On the other hand, if the response of theinvestment to the profit share is more than that of saving, then the steady state equilibrium
exhibits the exhilarationist regime. Depending on which regime is realized in the steadystate equilibrium, the stagnationist regime or the exhilarationist regime, we have J12 < 0 or
J12 > 0.
The steady state equilibrium is locally stable if and only if the determinant of the Jacobian matrix J is positive and the trace is negative. Let us confirm this fact in our model.
First, calculating the determinant, we havedet J = −(1 − m∗ )[J11 + (1 − θ)m′w (u∗ )J12 ].

(23)

When the steady state equilibrium exhibits the exhilarationist regime, that is, J12 > 0, det J >0 always holds. On the contrary, when the steady state equilibrium exhibits the stagnationist
regime, that is, J12 < 0, det J < 0 holds if the absolute value of m′w (u) is suﬃciently large.
Further, when det J < 0, the discriminant of the characteristic equation is always positive.
Hence, we obtain the two distinct real roots.
Second, calculating the trace, we havetr J = J11 − (1 − m∗ ) + (1 − m∗ ) f (u∗ )J12 .

(24)

By assumption, J11 < 0. However, since J12 can be positive or negative, tr J is not always
negative. When the steady state equilibrium exhibits the stagnationist regime, that is, J12 <0, tr J < 0 always holds. On the contrary, when the steady state equilibrium exhibits theexhilarationist regime, that is, J12 > 0, tr J > 0 can hold.
12

The exhilarationist regime is generally defined as a regime such that an increase in the exogenously given
profit share leads to a rise in the capacity utilization rate at the steady state equilibrium. However, we cannot
apply this definition to our model because the profit share at the steady state equilibrium is an endogenous
variable. Here, we follow the definition of Raghavendra (2006).

10

From the above analysis, we obtain the following propositions.
Proposition 1. Suppose that the steady state equilibrium exhibits the stagnationist regime.
If the reserve army eﬀect is small, then the steady state equilibrium is locally stable. On theother hand, if the reserve army eﬀect is large, then the steady state equilibrium is locallyunstable.
Let us explain Proposition 1 intuitively. This proposition means that, in the stagnationist
regime, the higher reserve army eﬀect has unstable eﬀects on the economy. Consider thecase where the capacity utilization rate deviates from the steady state capacity utilization
rate due to the occurrence of an exogenous shock. Suppose that the capacity utilization rateis less than the steady state equilibrium, for instance. In this case, there exist two opposing eﬀects. First, the capacity utilization rate increases due to J11 < 0, which is called the
direct stable eﬀect. Second, the fall in the capacity utilization rate leads to an increase in
the profit share due to J21 < 0. As a consequence, due to J12 < 0, the capacity utilization
rate decreases, which is called the indirect unstable eﬀect. Thus, in the stagnationist regime,
whereas the direct eﬀect is stable, the indirect eﬀect is unstable. If the reserve army mechanism eﬀect is large, the profit share fairly decreases due to a strong eﬀect of J12 < 0. As aresult, the capacity utilization rate largely increases via an eﬀect of J12 . That is, the indirect
unstable eﬀect works intensely. In the unstable case, the steady state equilibrium is a saddle
point.
Proposition 2. Suppose that the steady state equilibrium exhibits the exhilarationist regime.
Then, the steady state equilibrium can be locally unstable.
This proposition means that, in the exhilarationist regime, the higher response of thecapacity utilization rate to a change in the profit share can destabilize the economy. First, inthe exhilarationist regime, if the capacity utilization rate is less than the steady state value,then it increases due to J11 < 0. This is the direct eﬀect. Second, if the capacity utilization
rate is less than the steady state value, then the profit share increases J21 < 0, which increases
the capacity utilization rate due to J12 > 0. This is the indirect eﬀect. In the exhilarationist
regime, both the direct and indirect eﬀects seem stable. However, if the indirect eﬀect is toolarge, that is, if the response of the capacity utilization rate to the profit share is suﬃciently
large, the capacity utilization rate fairly increases due to the strong eﬀect of J12 > 0. In thiscase, the capacity utilization rate increases too much and deviates from the steady state, andconsequently, the steady state equilibrium will be unstable.
Now, we show that endogenous and perpetual business cycles occur. By rearranging tr J,11we obtain:
tr J = ϕA − (1 − m∗ ),A ≡ (g∗du − sm∗ ) + (1 − m∗ ) f (u∗ )(g∗dm − su∗ ).

(25)

The term A does not depend on the parameter ϕ that represents the speed of adjustment of
the goods market. Both the capacity utilization rate and the profit share at the steady state
equilibrium do not also depend on ϕ. Thus, we can choose ϕ as a bifurcation parameter.
Proposition 3. Suppose that the steady state exhibits the exhilarationist regime and that theterm A is positive. Then, a limit cycle occurs when the speed of adjustment of the goods
market lies within some range.
Proof. Put ϕ0 = (1 − m∗ )/A > 0. Taking a positive value ϕ arbitrarily, we have tr J = 0 for
ϕ = ϕ0 , tr J < 0 for ϕ < ϕ0 , and tr J > 0 for ϕ > ϕ0 . Thus, ϕ = ϕ0 is a Hopf bifurcation point.
That is, there exists a continuous family of non-stationary, periodic solutions of the system
around ϕ = ϕ0 .
This proposition means that, as the speed of adjustment of the goods market becomes
faster, there appears a point at which the stable steady state equilibrium switches to the unstable one. As Raghavendra (2006) has shown, we also have a similar result: with regard
to the phase such that both the profit share and the capacity utilization rate simultaneously
increase (decrease), and the phase such that one increases while the other decreases, the former phase alternates with the latter. This means an apparent alternation of the exhilarationist
and stagnationist regimes.
Next, we consider the parameter γ that represents the wage gap between regular andnon-regular employments. Note that γ does not aﬀect the equilibrium. Therefore, γ appears
only in the part f (u∗ ) of tr J. Furthermore, we find that∂ f (u; γ)
αβ
=
> 0.
∂γ
u(γα + βu)2(26)This means that, in the case where the steady state equilibrium exhibits the exhilarationist
regime, a rise in γ amplifies the instability of the steady state equilibrium.
Proposition 4. Suppose that the steady state equilibrium exhibits the exhilarationist regime.
Then, an expansion of the gap between regular and non-regular employments makes thesteady state equilibrium unstable.
Furthermore, we obtain the following proposition.

12

Proposition 5. Suppose that the steady state equilibrium is unstable in the exhilarationist
regime. Then, a limit cycle occurs within a certain range of the wage gap between regularand non-regular employments.
Proof. Putγ0 =β(u∗ )2 [(1 − m∗ ) − J11 ]·> 0.
α {[J11 − (1 − m∗ )]u∗ + (1 − m∗ )J12 }Taking a positive value γ arbitrarily, we have tr J = 0 for γ = γ0 , tr J < 0 for γ < γ0 ,and tr J > 0 for γ > γ0 . Thus, γ = γ0 is a Hopf bifurcation point. That is, there exists a
continuous family of non-stationary, periodic solutions of the system around γ = γ0 .
As the wage gap expands, there appears a point at which the stable steady state equilibrium switches to the unstable one. Thus, an increase in the wage gap makes the economy
more unstable. In order to stabilize the economy, it is desirable to shrink the wage gap
between regular and non-regular employments.
Note that the steady state capacity utilization rate does not depend on parameters α and
β. Given the steady state capacity utilization rate, we consider the eﬀect of a rise in β/α on
the stability of the steady state equilibrium. Then, a rise in β/α means that relatively many
non-regular employees are employed compared to regular employees. The ratio β/α appears
only in the part f (u∗ ) of tr J. From equation (16), we find that∂ f (u; β/α)
u2
=−
< 0.
∂(β/α)u[γ + (β/α)u]2(27)This means that, in the case where the the steady state equilibrium exhibits the exhilarationist
regime, the higher the ratio β/α is, the smaller the tr J is. From this, we obtain the following
proposition.
Proposition 6. Suppose that the steady state equilibrium exhibits the exhilarationist regime.
Then, a rise in relatively many non-regular employees compared with regular employees
makes the steady state equilibrium stable.
This proposition shows a counter-intuitive result that an increase in relatively many nonregular employees will have an eﬀect of making the economy unstable. Thus, if the government tries to decrease relatively many non-regular employees, the economy becomes
unstable. However, we cannot conclude that a government should take an employment policy to increase non-regular employment in order to stabilize the economy. Rather, frompropositions 4 and 6, we emphasize a policy mix between wage and employment policies;
13whereas the economy becomes unstable due to a decrease in non-regular employees, a government makes an eﬀort to stabilize the economy by conducting a policy to shrink the wage
gap between regular and non-regular employments.

4 Introduction of minimum wages
As one of the policies taken by the government, we consider a minimum wage policy. The
minimum wage is introduced by setting the upper bound of the profit share. We consider
such formalization to be justifiable because the determination of the maximum profit shareis equivalent to that of the minimum wage of the non-regular employment. The equivalence is obtained from the following three facts. First, by definition, the introduction of
the maximum profit share (mmax ) is equal to that of the minimum wage share (1 − mmax ).
Second, given the labor productivity (a = Y/L), the determination of the minimum wage
share is equivalent to that of the minimum average wage of the whole economy, that is,
1 − mmax = w/(pa). Third, by substituting equation (7) in (8), the average wage of the whole
)
(
wnr ,
economy is proportional to the wage of the non-regular employment, that is, w = γα+βu
α+βu
where (·) > 1 for all u ∈ (0, 1). Note that the minimum average wage of the economy wmincorresponding to mmax lies between wnr and wr from equation (8) and the third fact, that is,
wr > w > wmin > wnr .13
Here, in the same way as Flaschel and Greiner (2009, 2011) and Flaschel et al. (2011),
we analyze how the introduction of the minimum wage aﬀects the dynamics of the profit
share and the capacity utilization rate.
In this section, in order to facilitate our analysis, numerical simulations will be used. For
this purpose, we need to specify the investment function and the target profit share of labor
unions. Here, we assume the following functional forms:
gd = ψuδ mε ,mw = σ0 − σ1 u,

0 < δ < 1, ε > 0,0 < σ0 < 1, σ1 > 0.

(28)
(29)

If Okun’s law, that is, a positive correlation between the employment rate and the capacity utilization rate, holds, we can regard the capacity utilization rate as a proxy variable ofthe employment rate. Such a method is adopted by Tavani et al. (2011).14
13

To be concrete, the value of the bracket in equation (8) is smaller than unity for anymeans that the( u. This)
relation w ≤ wr holds. Equation (8) holds in the case where u = 0. Again, the value γα+βuofthethird fact is
α+βu
greater than unity for any u, which means that the relation w > wnr holds.
14
For Kaleckian models that consider the endogenous determination of the employment rate, see Sasaki(2010, 2011, 2012a, 2012b)

14

4.1 The case of the exhilarationist regime
In this subsection, we assume that the steady state equilibrium exhibits the exhilarationist
regime and that a limit cycle occurs.
We set the parameters as follows:ψ = 0.8, δ = 0.2, ε = 2, ϕ = 8, s = 0.6, θ = 0.4,m f = 0.7, σ0 = 0.2, σ1 = 0, α = 1, β = 1.2, γ = 2, mmax = 0.6.
Calculating the steady state with the above parameters, we obtain u∗ = 0.46 and m∗ = 0.4.
Given the arbitrary initial value, a limit cycle occurs (see Figure 1). The limit cycle
is stable. That is, if the initial value is put far away from the steady state, it converges tothe limit cycle. Even if the initial value is put at the neighborhood of the steady state, it
converges to the limit cycle.




























Figure 1: The occurrence of the limit cycle

Here, we introduce mmax . Depending on how the government sets mmax , we obtain different results.
First, we consider the case where exogenously given mmax is more than the maximum
value (i.e., the top) of the limit cycle without introducing minimum wages (for instance,
mmax = 0.8). From the phase diagram, we can see that the initial value in this case converges
15

to the limit cycle faster than the initial value in the case without mmax .
Second, we consider the case where mmax is set between the maximum value of the limit
cycle and the steady state equilibrium. Then, from the analysis of the phase diagram, wefind that the size of the limit cycle becomes smaller. This means that the size of fluctuations
decreases. In other words, by introducing the minimum wage, the size of business cycles
can be reduced. Figure 2 shows the limit cycles in the case where mmax = 0.6. Figure 3
shows the limit cycles of these two cases before and after introducing the minimum wage.
Clearly, we can see that the size of business cycles diminishes.











 





 











Figure 2: Introduction of the minimum wage

Third, we consider that mmax is exactly the same as the steady state profit share. Then,from the phase diagram, the economy converges to the steady state almost without fluctuations. That is, if the government knows the steady state equilibrium in advance, by putting
the level of the minimum wage at the same level of the steady state, it can lead the economy
to converge to the steady state.
Fourth, we consider the case where mmax is set less than the steady state equilibrium.
If the government does not actually know the equilibrium, it is significant to consider sucha case. Then, from the phase diagram, the economy converges to a point on line u̇ = 0
almost without fluctuations. That is, although fluctuations can be diminished, both the profit
share and the capacity utilization rate obtained in the long run become smaller than those
16










 













Figure 3: Comparisons of the two cycles

obtained before the introduction of the minimum wage policy. Figure 4 shows the casewhere mmax = 0.35.
Let us compare the real wage of point E and that of point P. The real wage of regular
employment and that of non-regular employment are given by using the capacity utilization
rate and the profit share as follows:15
wrγu=(1 − m),p
γα + βuwnr
u=(1 − m).
p
γα + βu(30)(31)

From this, the real wage is an increasing function of the capacity utilization rate, and adecreasing function of the profit share. By rearranging the formulae, we obtain the iso-realwage curves in the (u, m)-space.
)
1 wr ( αγ
+β ,
m=1−
γ p u

(32)

nr /p)Taking the derivatives of equations (30) and (31) with respect to γ, we obtain ∂(w∂γr /p) > 0 and ∂(w∂γ< 0,respectively. This means that an expansion of wage gap between regular and non-regular employments leads
to an increase in real wage of regular employees and a decrease in real wage of non-regular employees.

15

17






  





 


 

  












Figure 4: Introduction of the minimum wage that is larger than the equilibrium

m=1−)
wnr ( αγ
+β .
p u

(33)

Each curve becomes a rectangular hyperbola. In the first quadrant of the (u, m)-space, thelower right-hand side these curves are located in, the higher real wages are associated with.
Which point E or P gives a higher real wage depends on two things: the shape of the
iso-real wage curve and relative positions between points E and P. Iso-real wage curves andpoints E and P are shown in Figure 5. The real wage associated with the dotted iso-realwage curve is higher than that associated with the solid one. In the left figure, point E gives
a higher real wage compared to point P. The employment rate (capacity utilization rate) in E
is also high compared to its counterpart because u in E is higher than in P. On the contrary,in the right figure, a higher real wage is given in point P compared to point E. However, theemployment rate in P is low compared to that in E because u in P is smaller than u in E.

4.2 The case of the stagnationist regime
We consider the case where the steady state equilibrium exhibits the stagnationist regime.
First, we consider the case where the reserve army eﬀect is small and consequently, thesteady state equilibrium is stable. Figure 6 shows the phase diagram in the stagnationist
regime case.
18

















Figure 5: Iso-real wage curves and the two equilibria 
  


Figure 6: Phase diagram in the case where the stable steady state equilibrium exhibits thestagnationist regime

19

Next, we consider the case where the reserve army eﬀect is large and consequently, thesteady state equilibrium is unstable. Steady state E is the saddle point in Figure 7. Thus,
if the economy starts from the initial value above the stable arm, for instance, point A0 , itconverges to a point where the capacity utilization rate is zero and the profit share is unity.
If the economy starts from the initial value below the stable arm, for instance, point B0 , itconverges to a point where the capacity utilization rate is unity and the profit share is zero.
Here, we introduce mmax and mmin . If the government does not actually know the steadystate equilibrium, the minimum wage given by the government does not necessarily coincidewith the steady state profit share. The interval [mmin , mmax ] represents the feasible combinations of the profit share determined by the government as the minimum wage policy. Then,as shown in Figure 7, starting from point A0 , the economy converges to point A, while starting from point B0 , it converges to point B. If we compare point A with point B, point B issuperior to point A because the real wage associated with B is higher than that associatedwith A. Further, the employment rate in B is also high compared to that in A because u in B
is higher than u in A.

  

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


Figure 7: Phase diagram in the case where the unstable steady state equilibrium exhibits thestagnationist regime

5 Concluding remarks
In this paper, we have developed a Kaleckian model in which two types of labor (regularand non-regular employments) are incorporated. We have analyzed how the expansion of
the wage gap between regular and non-regular employments aﬀects the economy.
20

First, if the steady state equilibrium exhibits the stagnationist regime, an increase in thewage gap does not aﬀect the stability of equilibrium. In this case, the size of the reserve
army eﬀect aﬀects the stability of the equilibrium. If the reserve army eﬀect is strong, thesteady state equilibrium will be unstable. However, even if the steady state equilibriumis unstable, the introduction of the minimum and/or maximum wage prevents the capacity
utilization and profit share from diverging. In this case, it is possible that depending on
conditions, we obtain the real wage and the employment rate that are higher than the steadystate equilibrium values.
Second, if the steady state equilibrium exhibits the exhilarationist regime, an increase in
the wage gap destabilizes the equilibrium. It is possible that depending on conditions, anincrease in the wage gap produces endogenous and perpetual business cycles. In addition,
an increase in the non-regular employment relative to the regular employment stabilizes thesteady state equilibrium. However, we must note that the government should not increase
the non-regular employment literally. On the contrary, the government should adopt a policy
mix that decreases both the non-regular employment and the wage gap. The introduction of
the minimum wage is desirable in that it mitigates fluctuations of business cycles. However,
the introduction of an inappropriate minimum wage policy that sets a minimum wage higher
than the steady state equilibrium value consequently leads to the real wage and employment
rate lower than the steady state values.
Finally, we must note that our model is a short-run one, and not a long-run one. As
stated in the text, if we assume Okun’s law, the capacity utilization rate has a one-to-one
relationship with the employment rate. However, these two variables are strictly diﬀerent.
In this paper, for convenience, we identify the capacity utilization rate with the employment
rate, but we cannot in a strict sense. To build a long-run model that distinguishes between thecapacity utilization rate and the employment rate and investigate the employment in detailwill be left for future research.

References
Blecker, R. A. (2002) “Distribution, Demand and Growth in Neo-Kaleckian MacroModels,” in: M. Setterfield (ed.), The Economics of Demand-led Growth, Challenging
the Supply-side Vision of the Long Run, Cheltenham: Edward Elgar, pp. 129–152.
Cassetti, M. (2003) “Bargaining Power, Eﬀective Demand and Technical Progress: A
Kaleckian Model of Growth,” Cambridge Journal of Economics, 27 (3), pp. 449–464.

21

Dutt, A. K. (1987) “Alternative Closures Again: A Comment on Growth, Distribution and
Inflation,” Cambridge Journal of Economics, 11 (1), pp. 75–82.
Flaschel, P. and Greiner, A. (2009) “Employment Cycles and Minimum Wages: A MacroView,” Structural Change and Economic Dynamics, 20 (4), pp. 279–287.
Flaschel, P., and Greiner, A. (2011) “Dual Labor Markets and the Impact of Minimum
Wages on a Typical Employment,” Metroeconomica 62 (3), pp. 512–531.
Flaschel, P., Greiner, A., Logeay, C., and Proano, C. (2011) “Employment Cycles, Low Income Work and the Dynamic Impact of Wage Regulations. A Macro Perspective,” Journalof Evolutionary Economics, doi: 10.1007/s00191-011-0236-2.
Goodwin, R. (1967) “A Growth Cycle,” in C. H. Feinstein (ed.) Socialism, Capitalism andEconomic Growth, Cambridge: Cambridge University Press.
Lavoie, M. (1992) Foundations of Post-Keynesian Economic Analysis, Cheltenham: Edward Elgar.
Lavoie, M. (2009) “Cadrisme within a Post-Keynesian Model of Growth and Distribution,”
Review of Political Economy, 21 (3), pp. 369–391.
Marglin, S. and Bhaduri, A. (1990) “Profit Squeeze and Keynesian Theory,” in: S. Marglin
 Golden Age of Capitalism: Reinterpreting the Postwar Experience,
and J. Schor (eds.), The
Oxford: Clarendon Press, pp. 153–186.
Raghavendra, S. (2006) “Limits to Investment Exhilarationism,” Journal of Economics, 87(3), pp. 257–280.
Rowthorn, R. E. (1977) “Conflict, Inflation and Money,” Cambridge Journal of Economics,
1 (3), pp. 215–239.
Rowthorn, R. E. (1981) “Demand, Real Wages and Economic Growth,” Thames Papers in
Political Economy, Autumn, pp. 1–39.
Sasaki, H. (2010) “Endogenous Technical Change, Income Distribution, and Unemployment with Inter-class Conflict,” Structural Change and Economic Dynamics, 21 (2), pp.
123–134.
Sasaki, H. (2011) “Conflict, Growth, Distribution, and Employment: A Long-run Kaleckian Model,” International Review of Applied Economics, 25 (5), pp. 539–557.

22

Sasaki, H. (2012a) “Cyclical Growth in a Goodwin-Kalecki-Marx Model,” Journal of Economics, forthcoming.
Sasaki, H. (2012b) “Is the Long-run Equilibrium Wage-led or Profit-led? A Kaleckian
Approach,” Structural Change and Economic Dynamics, forthcoming.
Sen, A. (2004) “Elements of a Theory of Human Rights,” Philosophy and Public Aﬀairs,
32 (4), pp. 315–356.
Tavani, D., Flaschel, P., and Taylor, L. (2011) “Estimated Non-Linearities and Multiple
Equilibria in a Model of Distributive-demand Cycles,” International Review of Applied
Economics 25 (5), pp. 519–538.

23