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

International Competition and Distributive Class
Conflict in an Open Economy Kaleckian Model
Hiroaki Sasaki, Ryunosuke Sonoda, and Shinya Fujita

Discussion Paper No. E-12-005

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

July 2012International Competition and Distributive Class Conflict in an Open
Economy Kaleckian Model
Hiroaki Sasaki∗
Graduate School of Economics, Kyoto University

Ryunosuke Sonoda
Graduate School of Economics, Kyoto University

Shinya Fujita
Graduate School of Economics, Nagoya University
Abstract
This paper constructs an open economy Kaleckian model in which the international
competition affects the bargaining process between firms and workers, and investigates theeffect of such bargaining on macroeconomy. If the real exchange rate has little impact on the
trade balance, the economy is stable, whereas if it has a larger impact on the trade balance, theeconomy is unstable. Moreover, we show that the effect of a change in the bargaining poweron aggregate demand depends not only on the demand regimes but also on which agent bears
the burden arising from the international price competition.
Keywords: Kaleckian model, Open economy, International price competition, Wagebargaining
JEL Classification: E12; F43; J50∗

Corresponding author. Graduate School of Economics, Kyoto University. Yoshida-Honmachi, Sakyo-ku,
Kyoto 606-8501, JAPAN. E-mail: sasaki@econ.kyoto-u.ac.jp. Phone: +81-(0)75-753-3446

1

1. Introduction
Thus far, a number of Kaleckian models have been developed, and the relation between
income distribution and aggregate demand has been investigated.1 From those models, wederived a familiar result that there are two types of demand regime, stagnationist (wage-led)and exhilarationist (profit-led), according to the parameter constellation of investment andsaving functions.2 The stagnationist regime indicates an economy in which a rise in the profit
share (a fall in the wage share) decreases aggregate demand and the exhilarationist regime
indicates an economy in which a rise in the profit share increases aggregate demand.
Most of the Kaleckian models that investigate the relation between income distribution andaggregate demand assume a closed economy and abstract from international trade. The real
economy is nevertheless an open economy, and accordingly there are complications inapplying analytical results based on the assumption of a closed economy to the real world. It isnecessary to construct an open economy model and derive effective implications for the real
economy.
Needless to say, some of the Kaleckian models take into account an open economy case.3 A
pioneering work that introduces international price competition into the Kaleckian model is
Blecker (1989). By adding net export into demand components, he shows that an increase in
the wage share can lower international price competitiveness and has a negative impact on
investment, which implies that the stagnationist regime is hard to obtain. Another novel ideaof Blecker (1989) is the assumption that firms restrain their prices in terms of international
price competition, whereas normal Kaleckian models assume that firms determine their priceas they like. This modification implies that international competition strictly affects income
distribution in the domestic country.
Blecker (1989) nevertheless assumes that the measure of international price competition(i.e., the ratio of domestic unit labor cost to import prices) is exogenously given and hence
1

For the so-called Kaleckian model, see Rowthorn (1981) and Lavoie (1992). As for the empirical studies
on the relation between income distribution and aggregate demand based on the Kaleckian framework, see
Stockhammer and Onaran (2004), Barbosa-Filho and Taylor (2006), Naastepad and Storm (2007), Heinand Vogel (2008), Stockhammer, Onaran, and Ederer (2009), and Stockhammer, Hein, and Grafl (2011).
2
For a theoretical explanation of the demand regime, see Blecker (2002).
3
Cordero (2002) constructs an open economy Kaleckian model that integrates the theory of
conflicting-claims inflation. This model assumes, however, that price is determined in the international
goods market and formalizes only workers’ bargaining process. La Marca (2010), using an open economy
version of the stock-flow consistent model, considers the dynamics of the rate of capacity utilization, profit
share, and trade balance. Von Arnim (2011) constructs an open economy Kaleckian model and considers theeffect of the wage policy on growth and distribution by using Monte Carlo simulation.

2

does not explicitly consider the process in which international competition influences income
distribution. We therefore need to endogenize income distribution to improve the open
economy model.
A representative way to endogenize the process of income distribution in a Kaleckian model
is to use the theory of conflicting-claims inflation developed by Rowthorn (1977).4 This
theory assumes that both firms and workers have their own target value of each distributive
share and then negotiate over the price and the nominal wage in response to the gap between
the actual share and their own target.
The Cassetti (2002) model is one that takes into account the open economy case using thetheory of conflicting-claims inflation. He considers the situation in which inflation ofdomestic products due to class conflict changes the real exchange rate, and this change affects
the growth rate of the economy through exports and imports. His model is important in that it
simultaneously introduces international price competition and the determination of income
distribution.
Cassetti’s (2002) model nevertheless leaves room for further investigation. When extendinga Kaleckian model with conflict-inflation to an open economy model, he uses equations that
determine the wage and price dynamics in closed economy without any modifications. In
other words, he assumes that in the face of international price competition in the international
goods market, firms and workers never consider the competition, which is unrealistic. To
investigate income distribution under an open economy, we must consider the effect ofinternational price competition on the conflict between firms and workers.
A study that considers this aspect is Missaglia (2007). He assumes that the price equation offirms depends on the real exchange rate; when the terms of trade deteriorate, firms restrain
their prices. Missaglia’s (2007) approach is still unsatisfactory because he does not consider
the effect of international price competition on the wage bargaining of workers. The sameproblem holds for Blecker (1998), which is an extension of Blecker (1989).
With the ongoing globalization of the real world, workers cannot demand a wage increase
without considering international competition as well as firms. Increasing wages recklessly inopen economy causes a decrease in the price competitiveness of the domestic industry and a
decline in market share on the international market, which in turn causes a fall in domestic
4

For closed economy Kaleckian models with the theory of conflicting-claims inflation, see Cassetti (2003)
and Dutt (1987).

3

labor demand and leads to the loss of workers. In addition, if international capital flow isallowed, firms facing losses due to international price competition transfer their production
base to foreign countries to seek cheaper labor. Accordingly, the pressure on domestic workers
will intensify. It is therefore reasonable to suppose that workers who engage in wage
bargaining pay attention to the relative price of domestic and foreign products.5
When both firms and workers consider international price competition, the effect of pricepressure on income distribution has two meanings. Whether the wage share or the profit share
increases when international competition intensifies depends on how much firms and workers
share the burden arising from price restraint.
The Blecker (2011) model accordingly assumes that international price competition affects
both firms and workers. In this model, the target profit share of firms depends on the realexchange rate, and moreover, the rate of change in the nominal wage that is determined by
wage bargaining depends on the real exchange rate. In his approach, however, the target profit
share of workers is assumed to be constant. It is because the price of imports is indexed to thenominal wage and not because the international price competition affects the target of workers
that changes in the real exchange rate affect the nominal wage. As stated above, however, it ispossible that under severe international competition, labor unions revise the target downward.
It is therefore necessary to build a conflict model that considers this possibility.
In addition, the Blecker (2011) model separates the determination of output (capacity
utilization), the determination of income distribution and the real exchange rate are separated.
In other words, the goods market has nothing to do with the profit share and the real exchange
rate that are determined in the labor market; there is a feedback from the labor market to thegoods market, but not from the goods market to the labor market.
In the present paper, we therefore present a Kaleckian model in which international pricecompetition affects both firms’ decision and workers’ decisions. In addition, there is bilateral
feedback from the labor market to the goods market and from the goods market to the labormarket. Using this model, we investigate the stability of the steady state equilibrium and theeffect of international price competition on the equilibrium values. From our analysis, we
obtain the following new results.
5

For a model that assumes the effect of international price competition on the wage bargaining of workers,
see Blecker (1996). He also introduces a real world example that supports this assumption: the labor unionof Xerox Corporation accepts a sharp wage cut to prevent workers from moving abroad in the face ofinternational price competition.

4

First, with regard to the stability of the equilibrium, as long as the effect of the realexchange rate on trade balance is sufficiently small, the equilibrium is likely to be stableirrespective of which regime is realized in the equilibrium, the stagnationist regime or theexhilarationist regime. By contrast, if the real exchange rate effect is large, then theequilibrium is likely to be unstable, and depending on conditions, cyclical fluctuations canoccur.
Second, in regard to the comparative statics analysis, if we focus on the bargaining powers
of firms and workers, we obtain the following results. Even if the domestic economy ischaracterized as an exhilarationist regime, unlike in a closed economy, a rise in the bargaining
power of firms can depress domestic business. Moreover, even if the domestic economy ischaracterized as a stagnationist regime, unlike in a closed economy, a rise in the bargaining
power of workers can depress domestic business.
The remainder of the paper is organized as follows. Section 2 presents our model. Section 3
investigates the dynamics of the model. Section 4 conducts comparative statics analysis.
Section 5 is the conclusion.
2. ModelConsider a small open economy in which workers and capitalists coexist. Workers consume all
their wages and capitalists save a fraction s of their profits. Workers and capitalists consume
both domestic and foreign goods. The goods market is imperfectly competitive, and hence,firms set prices according to a mark-up pricing rule. Moreover, firms have an investment
function independent of savings. Investment is conducted by only domestic goods.
2.1 Dynamics of the capacity utilization
Suppose that firms operate with a fixed coefficient production function. The ratio of thepotential output Y F to the capital stock K is assumed to be constant. We can then write therate of capacity utilization as u = Y / K , where Y denotes the actual output.6 From this, wehave r = πu , where r and π denote the profit rate and the profit share, respectively.
Following Marglin and Bhaduri (1990), we assume that the firms’ investment function isincreasing in both the rate of capacity utilization and profit share:
6F

The rate of capacity utilization is defined as Y / Y . Since we have Y / Y

can use Y / K as the rate of capacity utilization as long as K / Y

5

F

F

= (Y / K ) /( K / Y F ) , weis technologically fixed and constant.

g d = g d (u , π ), g du > 0, g dπ > 0 ,(1)where g du denotes the partial derivative with respect to the rate of capacity utilization andg dπ the partial derivative with respect to the profit share.
Let us specify consumption demand. Total domestic consumption demand consists ofdemand for domestic goods and foreign goods. First, we assume that total nominal
consumption of domestic goods is a constant fraction α of total nominal consumption
expenditure.
p (C wD + C cD ) = α [ wE + (1 − s )rpK ] ,(2)

where p denotes the price of domestic goods, C wD workers’ real consumption of domestic
goods, C cD capitalists’ real consumption of domestic goods, w the nominal wage, and E
employment. Next, we assume that total nominal consumption of foreign goods (i.e., total
nominal imports) is a constant fraction 1 − α of total nominal consumption expenditure.
ep ∗ (C wM + C cM ) = (1 − α )[ wE + (1 − s )rpK ] ,(3)

where e denotes the nominal exchange rate in terms of home currency, p ∗ the price of
foreign goods that is exogenously given, C wM workers’ real consumption of foreign goods,
and C cM capitalists’ real consumption of foreign goods.
Following Cassetti (2002), we assume that the expenditure coefficient for domestic goods isa function of the real exchange rate ε = ep ∗ / p .

α = α (ε ), 0 < α (⋅) < 1,(4)

where p ∗ denotes the price of foreign goods. How the expenditure coefficient changes when
the real exchange rate changes depends on the elasticity of substitution ρ between domestic
goods and foreign goods. If ρ is less than unity, then α is a decreasing function of ε . On
the other hand, if ρ is more than unity, then α is an increasing function of ε .
We specify demand for exports. Nominal exports equal real exports multiplied by the priceof domestic goods. We assume that real exports are increasing in both the real exchange rate
and foreign real incomes Y ∗ .
p ⋅ EX = p ⋅ EX (ε , Y ∗ ) = p ⋅ ex(ε ) ⋅ Y ∗ , ex ′ > 0
For simplicity, we assume that real exports are linear in foreign real incomes.
The goods market clearing condition leads to
6(5)

pY = pC + pI + pEX − ep ∗ M= ( pC D + ep ∗ C M ) + pI + pEX − ep ∗ M ,(6)= pC D + pI + pEX

where C denotes total consumption made up of the consumption of domestic goods C D
and that of foreign goods C M , and M denotes imports. Dividing both sides of equation (6)by pK and substituting equations (2), (3), (4), and (5) in the resultant expression, we obtainY∗.
g d = [1 − α (ε )]u + α (ε ) suπ − ex(ε )K(7)

If we denote the right-hand side of equation (7) as g , the excess demand of the goods market
is given by g d − g . Here, for simplicity, we assume that Y ∗ / K is constant and unity.7
We assume that in the goods market, quantity adjustment prevails.
u = φ ( g d − g ), φ > 0 ,(8)where φ denotes the speed of adjustment of the goods market. Equation (8) shows that
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. Substituting equation (1) and the right-hand side ofequation (7) in equation (8), we obtain the dynamics of the rate of capacity utilization.
u = φ {g d (u , π ) − [1 − α (ε )]u − α (ε ) suπ + ex(ε )}.

(9)

2.2 Dynamics of the profit share
Differentiating the definition of the profit share π = 1 − [ w /( pa )] with respect to time, we
obtainπ
1−π= pˆ − wˆ + aˆ ,(10)where xˆ = x / x denotes the rate of change in a variable x and a the level of laborproductivity. In this subsection, we specify each term in the right-hand side of equation (10).
We specify changes in the domestic price and the nominal wage by using Rowthorn’s(1977) conflicting-claims theory of inflation. First, suppose that firms set their price to close
the gap between their target profit share π f and the actual profit share. Second, suppose thatthe growth rate of the nominal wage that workers manage to negotiate depends on the gap
7

We assume that the economy is in the short run or medium run wherein capital stock remains constant.
Hence, this assumption is reasonable.

7between their target profit share π w and the actual profit share.
pˆ = θ f (π f − π ), π f = π f (ε ), π ′f > 0, θ f > 0, 0 < π f < 1 ,(11)wˆ = θ w (π − π w ), π w = π w (ε , u ), π wε < 0, π wu < 0, θ w > 0, 0 < π w < 1 .

(12)Here, we assume that π f > π w . Firms attempt to set their targets as high as possible whereas
workers attempt to set their targets as low as possible. This assumption is therefore reasonable.
We can interpret θ f and θ w as the bargaining power of firms and that of workers,
respectively. We assume that θ f + θ w = 1 and define θ f = θ ( 0 < θ < 1 ), because bargaining
power is a relative concept. We can consider an increase in the unionization rate as a factor for
raising the bargaining power of workers (i.e., a decrease in θ ), and an increase in the market
power of oligopolistic firms as a factor for raising the bargaining power of firms (i.e., anincrease in θ ).
In our model, the two target profit shares are determined endogenously.
First, we assume that the target profit share of firms is an increasing function of the realexchange rate. This means that domestic firms set their price by considering international
price competition with foreign firms. When the price competitiveness of domestic firms
lowers, domestic firms cut their target profit share and hence their price to defend their market
share in the international goods market.
Second, we assume that the target profit share of workers is decreasing in both the realexchange rate and the rate of capacity utilization. A decrease in the price of the domestic
goods has a negative effect on employment and workers thus set their target profit shareconsidering the price decrease. When the real exchange rate decreases and pricecompetitiveness worsens, workers therefore compromise to lower the target profit share. Let
us identify an increase in the rate of capacity utilization with an increase in the rate of
employment. When the rate of employment increases, workers’ attitude in bargaining becomesstrong, leading them to seek a higher target wage share, that is, a lower target profit share.
This is known as the “reserve army effect.”
The growth rate of labor productivity is determined endogenously. Here, we assume thatlabor productivity growth is an increasing function of the rate of capacity utilization.
aˆ = g a (u ), g a′ > 0(13)This specification is similar to the “reserve-army creation effect” described in Sasaki (2011,8

2012), where the growth rate of labor productivity is an increasing function of the rate of
employment. If the rate of employment is positively related with the rate of capacity
utilization, that is, if Okun’s law holds, we can use capacity utilization in place of theemployment rate (Tavani, Flaschel, and Taylor, 2011). As the rate of employment (capacity
utilization) increases and the labor market tightens, the bargaining power of workers increases,
which exerts an upward pressure on wages, leading capitalists to adopt labor-saving technicalchanges. In other words, capitalists intentionally create unemployment (Bhaduri, 2006; Dutt,
2006; Flaschel and Skott, 2006; Sasaki, 2010, 2011).
Substituting equations (11), (12), and (13) in equation (10), we obtain the dynamics of the
profit share.

π1−π= θ [π f (ε ) − π ] − (1 − θ )[π − π w (ε , u )] + g a (u ) .

(14)

2.3 Dynamics of the real exchange rate
We specify the dynamics of the real exchange rate. The rate of change in the real exchange is
given asεˆ = eˆ + pˆ ∗ − pˆ .

(15)

Following Blecker and Seguino (2002) and Blecker (2011), we introduce a crawling peg
system in regard to the nominal exchange rate:eˆ = λ (ε − ε ), ε > 0, λ > 0 ,(16)

where λ denotes the speed of adjustment. The currency authority has a target level of thereal exchange rate ε and adjusts the nominal exchange rate according to the gap between thetarget and the actual levels.
Substituting equations (11) and (16) in equation (15), we obtain the following equation of
the dynamics of the real exchange rate:εˆ = λ (ε − ε ) + pˆ ∗ − θ [π f (ε ) − π ] .

(17)

3. Dynamics of the model
From the above analysis, the dynamics of the rate of capacity utilization, profit share, and realexchange rate are given as9u = φ {g d (u , π ) − [1 − α (ε )]u − α (ε ) suπ + ex(ε )},(18)π = (1 − π ){θ [π f (ε ) − π ] − (1 − θ )[π − π w (ε , u )] + g a (u )},(19)ε = ε {λ (ε − ε ) + pˆ ∗ − θ [π f (ε ) − π ]}.

(20)

The steady state is a situation where u = π = ε = 0 . We let the steady state values bedenoted as u ∗ , π ∗ , and ε ∗ . In the following analysis, we assume that there exist steady statevalues such that 0 < u ∗ < 1 , 0 < π ∗ < 1 , and ε ∗ > 0 . As the numerical simulations inAppendix B show, such a steady state actually exists though this result depends on functional
specifications.
The elements of the Jacobian matrix J that corresponds to the system of the differential
equations are given by as follows:∂u= φ{g du − [1 − α (ε )] − α (ε ) sπ } ,
∂u

(21)

∂u
= φ[ g dπ − α (ε ) su ] ,
∂π

(22)

∂u
= φ{α ′(ε )u (1 − sπ ) + ex ′(ε )} ,
∂ε

(23)

∂π
= (1 − π )[(1 − θ )π wu + g a′ (u )] = (1 − π )Ω, where Ω ≡ (1 − θ )π wu + g a′ (u ) ,
∂u

(24)

∂π
= −(1 − π ) < 0 ,
∂π

(25)

∂π
= (1 − π )[θπ fε + (1 − θ )π wε ] ,
∂ε

(26)

∂ε
= 0,
∂u

(27)

∂ε
= θε > 0 ,
∂π

(28)

∂ε
= −ε (λ + θπ fε ) < 0 .
∂ε

(29)

J 11 ≡J 12 ≡J 13 ≡J 21 ≡J 22 ≡J 23 ≡J 31 ≡J 32 ≡J 33 ≡All the elements are evaluated at the steady state values. In what follows, we explain elements
whose signs are ambiguous.
To conduct the analysis further, we introduce the following assumption.
Assumption 1. The condition g du − [1 − α (ε )] − α (ε ) sπ < 0 holds.
10This condition is an open economy version of the Keynesian stability condition in which we
assume that the quantity adjustment in the goods market is stable. We then have J 11 < 0 .
Next, we introduce the following definition.
Definition

1.

We

defineg dπ − α (ε ) su < 0

as

the

stagnationist

regimeandg dπ − α (ε ) su > 0 as the exhilarationist regime.
The element J 12 shows the effect of an increase in the profit share on the rate of capacity
utilization. If the sign is negative, the economy is in the stagnationist regime, and if the sign ispositive, the economy is in the exhilarationist regime.
The element J 13 shows the effect of an increase in the real exchange rate on the tradebalance (normalized by capital stock) TB = ex(ε ) − [1 − α (ε )](1 − sπ )u . If ρ is more thanunity, we have α ′(ε ) > 0 , which leads to J 13 > 0 . This corresponds to ∂TB / ∂ε > 0 , whichmeans that the Marshall-Lerner condition (ML condition, hereafter) is satisfied. If ρ is lessthan unity, we have α ′(ε ) < 0 , which leads to J 13 < 0 , depending on conditions. In this case,
the ML condition is not satisfied.
The element J 21 shows the effect of an increase in the rate of capacity utilization on the
profit share. If the reserve army effect exceeds the reserve army creation effect, we have
Ω < 0 , leading to J 21 < 0 . This corresponds to the case where the profit share iscounter-cyclical to the rate of capacity utilization. On the contrary, if the reserve army creation
effect exceeds the reserve army effect, we have Ω > 0 , leading to J 21 > 0 . This corresponds
to the case where the profit share is pro-cyclical to the rate of capacity utilization.
The element J 23 shows the effect of an increase in the real exchange rate on the profit
share. If firms are more responsive than workers, that is, if the absolute value of π fε isgreater than that of π wε , then J 23 > 0 . In contrast, if workers are more responsive than firms,
that is, if the absolute value of π wε is greater than that of π fε , then J 23 < 0 . If firms are
responsive, firms bear the burden arising from international price competition more thanworkers, but if workers are responsive, workers bear the burden arising from international
price competition more than firms.
The characteristic equation that corresponds to the Jacobian matrix J is given as
q 3 + a1 q 2 + a 2 q + a3 = 0 ,

11

(30)

where q denotes a characteristic root. The coefficients of equation (30) are given by
a1 = − trJ = −( J11 + J 22 + J 33 ) > 0 ,(31)

a 2 = ( J 22 J 33 − J 23 J 32 ) + ( J 11 J 33 ) + ( J 11 J 22 − J 12 J 21 ) ,(32)

a3 = − det J = − J 11 ( J 22 J 33 − J 23 J 32 ) + J 21 ( J 12 J 33 − J 13 J 32 ) ,(33)

where trJ denotes the trace of J and det J the determinant of J .
The necessary and sufficient conditions for the local stability of the steady state equilibrium
are given by a1 > 0 , a 2 > 0 , a3 > 0 , and a1 a 2 − a3 > 0 . We investigate whether or not these
conditions are satisfied. In this case, because of the property of our model, the coefficients ofthe characteristic equation are linear functions of the adjustment speed of φ and hence,
a1 a 2 − a3 is a quadratic function of φ , where8

a1 = ∆ 1φ + ∆ 2 ,(34)

a 2 = ∆ 3φ + ∆ 4 ,(35)

a3 = ∆ 5φ ,(36)

a1 a 2 − a3 = f (φ ) ≡ (∆ 1 ∆ 3 )φ 2 + (∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 )φ + ∆ 2 ∆ 4 .

(37)

In the following analysis, we explain the coefficients ∆ 1 , ∆ 2 , ∆ 3 , ∆ 4 , and ∆ 5 .
To begin with, the signs of ∆ 1 , ∆ 2 , and ∆ 4 are immediately obtained as∆ 1 = [1 − α (ε )] + α (ε ) sπ − g du > 0 ,(38)∆ 2 = (1 − π ) + ε (λ + θπ fε ) > 0 ,(39)∆ 4 = ε (1 − π )[λ + θ (1 − θ )(π fε − π wε )] > 0 .

(40)∆ 1 > 0 is obtained from Assumption 1, and ∆ 4 > 0 is obtained from both π fε > 0 andπ wε < 0 .
Next, the coefficient ∆ 3 leads to∆ 3 = {[1 − α (ε )] + α (ε ) sπ − g du }[(1 − π ) + ε (λ + θπ fε )] − (1 − π )[ g dπ − α (ε ) su ]Ω .

(41)

We use the fact that the speed of adjustment of the goods market φ affects the dynamic process of themodel but not the steady state values.

8

12

From

this,

if

both

g dπ − α (ε ) su < 0

and

Ω>0

hold,

then

∆3 > 0 ,andif

g dπ − α (ε ) su > 0 and Ω < 0 hold, then ∆ 3 > 0 . Under other combinations, however, thesign of ∆ 3 is ambiguous. We therefore assume these two combinations in the following
analysis.
Assumption 2. Both g dπ − α (ε ) su < 0 and Ω > 0 hold.
Assumption 3. Both g dπ − α (ε ) su > 0 and Ω < 0 hold.
Assumption 2 corresponds to the case where the economy is in the stagnationist regime andthe reserve army creation effect exceeds the reserve army effect. Then, we have both J 12 < 0
and J 21 > 0 . Assumption 3 corresponds to the case where the economy is in theexhilarationist regime and the reserve army effect exceeds the reserve army creation effect.
We then have J 12 > 0 and J 21 < 0 .
The coefficient ∆ 5 leads to∆ 5 = ε (1 − π ){[1 − g du − α (ε )(1 − sπ )][λ + θ (1 − θ )(π fε − π wε )]− Ω{[ g dπ − α (ε ) su ](λ + θπ fε ) + θ [α ′(ε )u (1 − sπ ) + ex ′(ε )]}}

.

(42)

The first line of the right-hand side of equation (42) is always positive. Let us focus on thesecond line.
First, when Assumption 2 holds and the ML condition is satisfied, the sign of the secondline depends on the absolute size of the following term:− θΩ[α ′(ε )u (1 − sπ ) + ex ′(ε )] .

(43)

If equation (43) is small, we have ∆ 5 > 0 , leading to a3 > 0 . This effect is small when

α ′(ε ) > 0 is small and ex ′(ε ) > 0 is small, that is, when the expenditure coefficient fordomestic goods is not so responsive to the real exchange rate and when export demand is not
so responsive to the real exchange rate. In other words, when the trade balance is not so
responsive to the real exchange rate, then ∂TB / ∂ε > 0 is small. By contrast, if Assumption 2
holds and the ML condition is not satisfied, ∆ 5 > 0 necessarily holds.
Second, when Assumption 3 holds and the ML condition is satisfied, ∆ 5 > 0 necessarily
holds. By contrast, when Assumption 3 holds and the ML condition is not satisfied, we have
∆ 5 > 0 as long as the absolute value of ∂TB / ∂ε < 0 is small, which is to say, close to zero.
Finally, we investigate the sign of ∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 . If the sign is positive, we always
13

have a1 a 2 − a3 > 0 . On the other hand, if the sign is negative and its absolute value is large,
we have a1 a 2 − a3 < 0 .

∆1∆ 4 + ∆ 2 ∆ 3 − ∆ 5= [(1 − π ) + ε (λ + θπ fε )]2 (1 − α + αsπ − g du )

.

(44)

− (1 − π )Ω{(1 − π )( g dπ − αsu ) − θε [α ′(ε )u (1 − sπ ) + ex ′(ε )]}The first line of the right-hand side of equation (44) is always positive. Let us focus on thesecond line.
When Assumption 2 holds and the ML condition is satisfied, the second line of equation(44) is always positive and hence, we have ∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 > 0 , leading to a1 a 2 − a3 > 0 .
By contrast, when Assumption 2 holds and the ML condition is not satisfied, we have
∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 > 0 as long as the absolute value of ∂TB / ∂ε < 0 is small, which is to say,close to zero.
When

Assumption

3

holds

and

the

ML conditionissatisfied,

we

have

∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 > 0 if ∂TB / ∂ε > 0 is small because the second line is positive. By
contrast,

when

Assumption

3

holds

and

the

ML

conditionisnotsatisfied,∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 > 0 necessarily holds.
From these analyses, we obtain the following proposition:
Proposition 1. Suppose that the effect of the real exchange rate on the trade balance is small,that is, the absolute value of ∂TB / ∂ε is small. Then, irrespective of whether or not the ML
condition is satisfied, the combination of either the stagnationist regime and Ω > 0 or theexhilarationist regime and Ω < 0 makes the steady state equilibrium stable.
Proof. See Appendix A.1.
As Appendix C shows, in a closed economy, the combination of the stagnationist regimeand Ω > 0 or the exhilarationist regime and Ω < 0 is a stability condition. In the open
economy, if the effect of the real exchange rate on the trade balance is small, these
combinations are also the stability conditions.
We also obtain the two further propositions:
Proposition 2. Suppose that the effect of the real exchange rate on the trade balance is large,
14

that is, the absolute value of ∂TB / ∂ε is large. Then, if the ML condition is satisfied, thecombination of the stagnationist regime and Ω > 0 makes the steady state equilibrium
unstable. On the other hand, if the ML condition is not satisfied, under the combination of thestagnationist regime and Ω > 0 , limit cycles occur when the speed of adjustment of the goods
market lies within some range.
Proof. See Appendix A.2.
Proposition 3. Suppose that the effect of the real exchange rate on the trade balance is large,
that is, the absolute value of ∂TB / ∂ε is large. Then, if the ML condition is satisfied, under
the combination of the exhilarationist regime and Ω < 0 , limit cycles occur when the speed ofadjustment of the goods market lies within some range. On the other hand, if the ML condition
is not satisfied, the combination of the exhilarationist regime and Ω < 0 makes the steadystate equilibrium unstable.
Proof. See Appendix A.3.
We now explain why there occur limit cycles when the speed of adjustment of the goods
market is medium. First, when φ is sufficiently close to zero, the rate of capacity utilization
is not adjusted and the analysis of the dynamic system therefore amounts to the analysis of thesubsystem that consists of the profit share and the real exchange rate. After some calculations,
we find that the dynamics of the subsystem are stable. Second, when φ is sufficiently large,
the rate of capacity utilization is adjusted immediately and the effect of the rate of capacity
utilization on the dynamical system therefore does not last. In this case, too, the analysis of the
dynamic system amounts to the analysis of the subsystem that consists of the profit share andthe real exchange rate. When φ takes an intermediate value, the rate of capacity utilization
changes with lags. The rate of capacity utilization therefore has a lasting effect on the
dynamical system and, accordingly, cyclical fluctuations occur.
Propositions 1, 2, and 3 are summarized in Figure 1.
[Figure 1 around here]15

4. Comparative statics analysis
This section examines the effect of changes in the saving rate of capitalists, the bargaining
power of firms, and the target value of the real exchange rate on the steady state values of therate of capital utilization, profit share, and the real exchange rate.9 Here, we only consider the
stable steady state: Assumption 1 holds, either Assumption 2 ( J 12 < 0 and J 21 > 0 ) or
Assumption 3 ( J 12 > 0 and J 21 < 0 ) holds, and the trade balance does not respond much to
the real exchange rate (i.e., the absolute value of ∂TB / ∂ε is sufficiently small). In addition,
we suppose here that the ML condition is always satisfied, that is, ∂TB / ∂ε > 0 holds.10
4.1 Saving rate of capitalists
We represent the effect of a change in the saving rate of capitalists on the rate of capacity
utilization as follows:[]

du αε (1 − π )πu λ + θ (1 − θ )(π fε − π wε )
=
< 0.
ds
det J

(45)

Note here that the stable steady state always requires det J < 0 . Equation (45) thus means thatthe paradox of thrift holds even in an open economy version of the Kaleckian model: an
increase in the saving rate of capitalists reduces the rate of capacity utilization.
The effects of a change in the saving rate of capitalists on the profit share and the realexchange rate are given bydπ − απuJ 21 J 33
.
=
det J
ds

(46)

dε απuJ 21 J 32
.
=
ds
det J

(47)

As described above, an increase in the saving rate of capitalists decreases the rate of capacity
utilization. When the reserve army creation effect exceeds the reserve army effect ( J 21 > 0 ), a
decline in the rate of capacity utilization reduces labor productivity growth and thus the profit
share ( dπ / ds < 0 ). Furthermore, a fall in the profit share decreases the real exchange rate
( dε / ds < 0 ) because a loss of profitability urges capitalists to raise the price of domestic
goods.
9

The current analysis abstracts from the effect of the price of foreign goods on the steady state values. Thisis because, for the home country, a rise in the target value of the real exchange rate is equivalent to a rise in
the price of foreign goods in the sense that they both produce the same results.
10
The results of the comparative statics analysis in the case where the ML condition does not hold is more
complicated than those in which the condition holds. We therefore do not investigate the former case.

16

On the other hand, if the reserve army effect is stronger than the reserve army creation
effect ( J 21 < 0 ), a rise in the saving rate of capitalists leads to a rise in the profit share( dπ / ds > 0 ). This is because a fall in the rate of capacity utilization decelerates nominal wage
growth. An increase in profitability, moreover, leaves room for lowering the price of domestic
goods. Consequently, an increase in the saving rate of capitalists raises the real exchange rate
( dε / ds > 0 ).
4.2 Bargaining power of firms
The next task is to investigate the effects of the bargaining power of firms on the steady statevalues. Because it is difficult to derive purely analytical results from our model, we here
confine ourselves to pointing out that the propositions obtained in a closed version of a
Kaleckian model may not be applied to an open economy case.
We represent an effect of an increase in the firms’ bargaining power on the rate of capacity
utilization as follows:
du − (π f − π w )J 13 J 32 + (π f − π w )J 12 J 33 − (π f − π )J 13 J 22 + (π f − π )J 12 J 23
.
=
dθ
det J

(48)

In a closed economy, a rise in the bargaining power of firms has a strictly positive impact on
the rate of capacity utilization under the exhilarationist regime. 11 In an open economy,
however, this may not occur.
Assume that the economy exhibits the exhilarationist regime ( J 12 > 0 ). The first and second
terms on the right-hand side of equation (48) are thus positive, whereas the third and fourthterms are negative in the case where π f − π > 0 and J 23 > 0 are satisfied.12 If the latter
effects are larger than the former, a rise in the bargaining power of firms reduces the rate ofcapacity utilization even under the exhilarationist regime. The story behind such a situation is
explained as follows. An increase in the firms’ bargaining power puts upward pressure on theprice of domestic goods under π f − π > 0 , which in turn leads to a decline in the realexchange rate. A direct effect of a decrease in the real exchange rate is that it worsens the tradebalance and the rate of capacity utilization (the third term); its indirect effect is that it reduces
11

See Appendix C.
Our model endogenizes the growth rate of labor productivity. If the reserve army creation effect issufficiently strong, labor productivity grows rapidly and the profit share accordingly reaches a high level. In
this case, π f − π < 0 is likely to be obtained in the steady state. On the contrary, if the reserve army

12effect is sufficiently large, π f − π > 0 is likely to be obtained because the reserve army effect squeezes
the profit share.

17the profit share and thus shrinks domestic demand under the exhilarationist regime if firms
bear the burden arising from international price competition more than workers (the fourthterm).
The fourth term also shows negative sign if π f − π < 0 and J 23 < 0 are satisfied. An
increase in the firms’ bargaining power lowers the price of domestic goods under π f − π < 0and raises the real exchange rate. If workers who face a rising real exchange rate demandhigher nominal wage growth, the profit share decreases and domestic demand contracts underthe exhilarationist regime. This indicates that strengthening the bargaining power of firms
under the exhilarationist regime does not necessarily raise the rate of capacity utilization.
A well-known implication of equation (48) is that the stagnationist regime may not allow
firms to weaken their bargaining power.13 The first term in the right-hand side of equation(48) implies that a decrease in the firms’ bargaining power reduces its profitability and forcesthem to push up the price of domestic goods; a rise in the price of domestic goods, in turn,decreases the rate of capacity utilization by deteriorating terms of trade. This gives reason tojustify strengthening the bargaining power of firms that confront international pricecompetition, even under the stagnationist regime.
The impact of a change in the firms’ bargaining power on the profit share is given asdπ − (π f − π w )J 11 J 33 + (π f − π )J 13 J 21 − (π f − π )J 11 J 23
.
=
dθ
det J

(49)

It is likely that an increase in the bargaining power of firms raises the profit share, but theresult is not so simple in an open economy framework. For instance, if π f − π > 0 , J 21 > 0 ,and J 23 > 0 are satisfied, both the second and third terms in the right-hand side of equation(49) show a negative sign, which implies that there exists a possibility of increasing
bargaining power of firms to reduce the profit share. Strengthening its bargaining power raises
the price of domestic goods under π f − π > 0 and decreases the real exchange rate. A decline
in the real exchange rate,, in turn, leads to decreases in the rate of capacity utilization andprofit share if the reserved army creation effect is sufficiently large (the second term). It also
causes a decrease in the profit share if firms bear the burden arising from international pricecompetition more than workers (the third term).
Furthermore, in the case of π f − π < 0 , J 21 < 0 , and J 23 < 0 , both second and third terms
in the right-hand side of equation (49) show a negative sign. Strengthening the bargaining
13

This point is stressed by Blecker (2011) and Cassetti (2012). They show that cutting down a mark-up rateof domestic goods as well as decreasing nominal wage causes higher growth under the stagnationist regime.

18

power of firms reduces the price of domestic goods under π f − π < 0 and increases the realexchange rate. A rise in the real exchange rate, in turn, leads to a rise in the rate of capacity
utilization but reduces the profit share if the reserved army effect is sufficiently large (thesecond term). It also suppresses the profit share if workers request higher nominal wage
growth (the third term).
We represent the effect of a change in the firms’ bargaining power on the real exchange rate
as follows:dε (π f − π )( J 11 J 22 − J 12 J 21 ) + (π f − π w )J 11 J 32
.
=
dθ
det J

(50)

Under either Assumption 2 or Assumption 3, the sign of the first term in the right-hand side ofequation (50) is negative in the case where π f − π > 0 holds, whereas the sign of the second
term is positive. The result is therefore ambiguous. If π f − π < 0 holds, however, the firstterm becomes positive as well as the second term so that increasing the firms’ bargaining
power has a positive impact on the real exchange rate and thus improves the terms of trade.
4.3 Target value of the real exchange rate
Governments often intend to raise the real exchange rate to improve the trade balance andoutput. To begin with, we consider the effect of an increase in the target value of the exchange
rate on its steady state value:dε − λ ( J 11 J 22 − J 12 J 21 )
=
>0.
dε
det J

(51)

Because dε / dε > 0 is obtained from J 11 J 22 − J 12 J 21 > 0 , raising the target value of the realexchange rate increases its steady state value.
Next, we investigate whether the attempt to depreciate the exchange rate (i.e., raising thetarget value of the real exchange rate) succeeds in stimulating output.
du λJ 13 J 22 − λJ 12 J 23
.
=
dε
det J

(52)

The sign of the first term in the right-hand side of equation (52) is positive, whereas the signof the second term is ambiguous. The depreciation policy therefore may not work. The story
behind the failure of that policy is explained as follows.
Assume that the economy exhibits the exhilarationist regime and workers are more
responsive to the change of the real exchange rate than firms. Under these assumptions, thesign of the second term in the right-hand side of equation (52) is negative. A rise in the real19

exchange rate caused by the depreciation policy triggers workers to demand higher nominal
wage, which causes a decline in the profit share and stagnation of domestic demand. The
depreciation policy stimulates foreign demand as long as the ML condition is met, but it
reduces domestic demand. The total effect is therefore ambiguous. To succeed in the
depreciation policy under the exhilarationist regime, it is necessary for workers to bear theburden arising from international price competition.
Next, we assume that the stagnationist regime is realized and firms are more responsive tothe change of the real exchange rate than workers. In this case, currency depreciation may notwork because the sign of second terms in the right-hand side of equation (52) is negative. A
rise in the real exchange rate increases the profit share by increasing the price of domestic
goods, which in turn causes a decline in domestic demand. Under the stagnationist regime, thenominal wage must therefore rise higher than the price of domestic goods to stimulate
aggregate demand by means of the depreciation policy.
Finally, we represent the effect of a change in the real exchange rate on the profit share as
follows:
dπ − λJ 13 J 21 + λJ 11 J 23
.
=
dε
det J

(53)

If the reserve army creation effect is stronger than the reserve army effect, the depreciation
policy raises the rate of capacity utilization by improving the trade balance, and increases the
profit share (i.e., the sign of the first term in the right-hand sided of equation (53) is positive).
On the other hand, if the reserve army effect is stronger than the reserve army creation effect,
the policy decreases the profit share (i.e., the sign of the first term is negative).
In addition, when firms are more responsive to the change of the real exchange rate thanworkers, the depreciation policy raises the price of domestic goods and increases the profit
share (i.e., the sign of the second term is positive). When workers are more responsive,
however, this policy reduces the profit share through higher nominal wage growth (i.e., thesign of the second term is positive).
4.4 Summary
Table 1 shows the results of comparative statics analysis in the case where the stagnationist
regime is realized and the reserve army creation effect is larger than the reserve army effect,
while Table 2 shows the results of the case in which the exhilarationist regime is realized and20the reserve army effect is larger than the reserve army creation effect.
Let us review the key points obtained from our analysis. First, the paradox of thrift holds
even in the open economy Kaleckian model. Second, the effect of an increase in thebargaining power of firms on the rate of capacity utilization hinges on which demand regime
is realized and which agent, firms or workers, bear the burden arising from the international
price competition. Depending on the combination of these two factors, there exist various
scenarios that might unfold under international competition. Third, to succeed in the
depreciation policy, it is necessary for the government to consider both the demand regime inthe domestic economy and how to bear the burden of international price competition.
[Tables 1 and 2 around here]
5 Conclusions
This paper has presented an open economy Kaleckian model, considering the process in whichinternational price competition affects wage bargaining between firms and workers and theeffect of such bargaining on the stability and steady state values. In particular, we intend to
make it clear that some of the propositions obtained from a closed economy Kaleckian model
do not hold in an open economy model. Our results are summarized as follows.
Stability analysis. (1) The stability conditions in the closed economy case (i.e., thecombination of either the stagnationist regime and a larger reserve army creation effect or theexhilarationist regime and a larger reserve army effect) are accepted in the open economy casewhere a change in the real exchange rate has little impact on the trade balance. (2) By contrast,
if a change in the real exchange rate has a larger impact on the trade balance, the steady state
becomes unstable. (3) Moreover, if the ML condition is satisfied under the combination of theexhilarationist regime and a larger reserve army effect, limit cycles occur as long as the speedof adjustment of the goods market lies within some range.
Comparative statics analysis. (4) The paradox of thrift is true even in the open economy
setting. (5) Strengthening the firms’ bargaining power may depress the economy that exhibits
the exhilarationist regime but may not reduce the rate of capacity utilization under thestagnationist regime. We therefore conclude that not only the demand regimes but also theagent, whether firms or workers, that bears the burden arising from the international pricecompetition determines the effect of the bargaining power on the aggregate demand. (6)21Furthermore, the success of the depreciation policy depends on how to spread the burden ofinternational price competition between firms and workers as well as demand regime.

22

Appendix A: Proofs of propositions
A.1 Proof of proposition 1
Irrespective of the sign of ∂TB / ∂ε , if the absolute value of ∂TB / ∂ε is small, we have
∆ 5 > 0 and ∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 > 0 . From this, we have a1 > 0 , a 2 > 0 , a3 > 0 , anda1 a 2 − a3 > 0 . All the necessary and sufficient conditions are therefore satisfied.
A.2 Proof of proposition 2First part: If ∂TB / ∂ε > 0 is large, we have ∆ 5 < 0 and ∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 > 0 . Then, we
have a1 > 0 , a 2 > 0 , a3 < 0 , and a1 a 2 − a3 > 0 , which means that one condition is notsatisfied. The steady state equilibrium is therefore unstable.
Second part: If the absolute value of ∂TB / ∂ε < 0 is large, we have ∆ 5 > 0 and∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 < 0 . If the absolute value of ∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 < 0 is large, it is possiblethat the sign of f (φ ) = (∆ 1 ∆ 3 )φ 2 + (∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 )φ + ∆ 2 ∆ 4 alternates. The quadratic
function f (φ ) is convex downwards and its intercept is positive. If the discriminant off (φ ) = 0 is positive, the equation f (φ ) = 0 has two positive real roots: for φ ∈ (0, φ1 ) , wehave a1 > 0 , a 2 > 0 , a3 > 0 , and a1 a 2 − a3 > 0 ; for φ ∈ (φ1 , φ 2 ) , we have a1 > 0 , a 2 > 0 ,a3 > 0 , and a1 a 2 − a3 < 0 ; and for φ > φ 2 , we have a1 > 0 , a 2 > 0 , a3 > 0 , anda1 a 2 − a3 > 0 . There thus occurs the Hopf bifurcation at φ = φ1 and φ = φ 2 . Indeed, at

φ = φ1and

φ = φ 2 , we havea1 > 0 ,a2 > 0 ,a3 > 0 ,a1 a 2 − a3 = 0 ,and∂ (a1a2 − a3 ) / ∂φ φ =φ or φ ≠ 0 , which mean that all the conditions for the Hopf bifurcation are
12

satisfied. There therefore exists a continuous family of non-constant, periodic solutions of thesystem around φ = φ1 and φ = φ 2 .
A.3 Proof of proposition 3
First part: If ∂TB / ∂ε > 0 is large, we have ∆ 4 > 0 and ∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 < 0 . When theabsolute value of ∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 < 0is large, it is possible that the sign off (φ ) = (∆ 1 ∆ 3 )φ 2 + (∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 )φ + ∆ 2 ∆ 4 alternates. The quadratic function f (φ ) isconvex downwards and its intercept is positive. If the discriminant of f (φ ) = 0 is positive,the equation f (φ ) = 0 has two positive real roots: for φ ∈ (0, φ1 ) , we have a1 > 0 , a 2 > 0 ,a3 > 0 , and a1 a 2 − a3 > 0 ; for φ ∈ (φ1 , φ 2 ) , we have a1 > 0 , a 2 > 0 , a3 > 0 , anda1 a 2 − a3 < 0 ; and for φ > φ 2 , we have a1 > 0 , a 2 > 0 , a3 > 0 , and a1 a 2 − a3 > 0 . Therethus occurs the Hopf bifurcation at φ = φ1 and φ = φ 2 . Indeed, at φ = φ1 and φ = φ 2 , we23

have a1 > 0 , a 2 > 0 , a3 > 0 , a1 a 2 − a3 = 0 , and ∂ (a1a2 − a3 ) / ∂φ φ =φ or φ ≠ 0 , which means
12that all the conditions for the Hopf bifurcation are satisfied. There therefore exists a
continuous family of non-constant, periodic solutions of the system around φ = φ1 andφ = φ2 .
Second part: If the absolute value of ∂TB / ∂ε < 0 is large, we have ∆ 5 < 0 and∆ 1 ∆ 4 + ∆ 2 ∆ 3 − ∆ 5 > 0 . We then have a1 > 0 , a 2 > 0 , a3 < 0 , and a1 a 2 − a3 > 0 , whichmeans that one condition is not satisfied. The steady state equilibrium is therefore unstable.
Appendix B: Numerical simulations
Using numerical simulations, we show that the Hopf bifurcation actually occurs. For thispurpose, we have to specify functional forms.
Investment function: g d = γ 0 u γ 1 π γ 2 , γ 0 > 0, 0 < γ 1 < 1, γ 2 > 0 .

(B1)Here, following Blecker (2002) and Sasaki (2010), we use the Cobb-Douglas investment
function. Roughly speaking, the parametric restriction 0 < γ 2 < 1 corresponds to thestagnationist regime while γ 2 > 1 corresponds to the exhilarationist regime.
Firms’ target profit share: π f = α 0 + α 1ε , 0 < α 0 < 1, α 1 > 0 ,(B2)Workers’ target profit share: π w = β 0 − β1u − β 2 ε , 0 < β 0 < 1, β1 > 0, β 2 > 0 ,(B3)Labor productivity growth: g a = ηu , η > 0 ,(B4)Expenditure coefficient: α (ε ) = B0 ε ρ −1 , B0 > 0, 0 < ρ < +∞substitution,
Export demand function: ex(ε ) = A0 ε ψ ,ρ , the elasticity of(B5)

A0 > 0, 0 < ψ < +∞export demand.

ψ , the price elasticity of(B6)

In what follows, we present a numerical example that corresponds to the case where Ω < 0 ,the economy is in the exhilarationist regime, and ∂TB / ∂ε > 0 is large. First, we set theparameters as follows:s = 0.7 , γ 0 = 0.2 , γ 1 = 0.2 , γ 2 = 1.7 , α 0 = 0.3 , α 2 = 0.01 , β 0 = 0.3 , β1 = 0.2 ,24

β 2 = 0.01 , A0 = 0.4 , ψ = 1500 , θ = 0.3 , η = 0.01 , λ = 1 , ε = 1 , p ∗ = 0.01 , B0 = 0.3 ,ρ = 150 .
In this numerical example, the open economy version of the Keynesian stability condition
holds, workers are more responsive than firms, the two endogenously determined target profit
shares are more than zero and less than unity, the inequality π f > π w holds, and theendogenously determined expenditure share is 0 < α (ε ∗ ) < 1 .
We set initial conditions to u (0) = 0.15 , m(0) = 0.25 , and ε (0) = 0.98 . As figure B1
shows, there exist two Hopf bifurcation points.

Figure B1: Existence of two Hopf bifurcation points
Using φ = 1 as the speed of adjustment of the goods market, we obtain the following
figures with regard to the time series of the endogenous variables (figures B2-B5).

Figure B2: Dynamics of the capacity utilization
25

Figure B3: Dynamics of the profit share

Figure B4: Dynamics of the real exchange rate

Figure B5: Dynamics of the trade balance
Appendix C: Closed economy model
We briefly explain the closed economy model, which is a model that removes import and26

export demand as well as the effect of the real exchange rate from the open economy model.
In this case, the dynamics of the rate of capacity utilization and the profit share are as follows:u = φ{g d (u , π ) − suπ } ,(C1)π = (1 − π ){θ (π f − π ) − (1 − θ )[π − π w (u )] + g a (u )} .

(C2)

The steady state equilibrium is given by u = π = 0 .
The elements of the Jacobian matrix are given by
J 11 = φ ( g du − sπ ) ,(C3)J 12 = φ ( g dπ − su ) ,(C4)

J 21 = (1 − π )[(1 − θ )π w′ (u ) + g a′ (u )] = (1 − π )Ω ,(C5)

J 22 = −(1 − π ) < 0 .

(C6)

All the elements are evaluated at the steady sate equilibrium values.
Similar to the open economy model, we use the following three assumptions.
Assumption 1’. g du − sπ < 0 .
Assumption 2’. g dπ − su < 0 and Ω > 0 .
Assumption 3’. g dπ − su > 0 and Ω < 0 .
From assumption 1’, we have trJ < 0 . The determinant is given bydet J = φ (1 − π )[( sπ − g du ) − ( g dπ − su )Ω] .

(C7)Under assumption 2’, we have det J > 0 . In addition, under assumption 3’, we have det J > 0 .
If assumptions 1’ and 2’ hold simultaneously or if assumptions 1’ and 3’ hold simultaneously,
we have both trJ < 0 and det J > 0 , which thus satisfies the necessary and sufficientconditions for the local stability of the equilibrium.
We investigate the effect of an increase in θ (i.e., the bargaining power of firms) on theequilibrium rate of capacity utilization. Totally differentiating the equilibrium conditions, we
obtaindu ∗ φ (1 − π )(π f − π w )( g dπ − su )
.
=
det J
dθ

(C8)

As in the open economy model, we assume that π f > π w . We have that det J > 0 , from thestability condition. Then, if g dπ − su > 0 , we have du ∗ / dθ > 0 , and if g dπ − su < 0 , we have27

du ∗ / dθ < 0 . Consequently, if the economy is in the exhilarationist regime, an increase in thebargaining power of firms increases the rate of capacity utilization, whereas if the economy isin the stagnationist regime, and an increase in the bargaining power of firms decreases the rateof capacity utilization.

28

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31

Figure and Tables

Figure 1: Diagram of stability analysis
Table 1: Results for comparative statics analysis under assumption 2

u

π

ε

s

−-

−

−

θ

+ or −+ or −+ or −ε+ or −+ or −+

Table 2: Results for comparative statics analysis under assumption 3
u

π

ε

s

−

+

+

θ

+ or −+ or −+ or −ε+ or −+ or −+

32