Discussion Paper No.271
The Keynesian Unemployment and Capital Shortage:
A Simple Dynamic Approach with Trade Unions
Kazuhiko Nakahira
Faculty of Economics, Meikai University
Masahiro Yabuta
Faculty of Economics, Chuo University

December 2016

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

The Keynesian Unemployment and Capital Shortage:
A Simple Dynamic Approach with Trade Unions
Kazuhiko Nakahira
Associate Professor, Faculty of Economics, Meikai University
E-mail: nakahira@meikai.ac.jp
Masahiro Yabuta
Professor, Faculty of Economics, Chuo University
E-mail: yabuta@tamacc.chuo-u.ac.jp
ABSTRACTThe purpose of this paper is to provide a dynamic analysis by utilizing the bargaining model
which incorporates the relationship between the behavior of trade union and the macroeconomic
performance in Japan. Our discussion focuses on the influence of trade union on wage andemployment through the re-examination of its behavior, although the wage and employment
effects of trade union were slighted during the stagnation of Japan’s economy as the studies ofTodate (2009) and Booth (2014). From the other aspect, the framework of today’s
macroeconomic policy by the Japanese government includes the expected roles of the prices andwages. In this sense, the effect of trade union in determining wages is worth investigating.
Based on these aspects, this paper investigates the effects of trade union on macroeconomy andthe change in the role of trade union in Japan by the theoretical and the empirical analyses.
The dynamic system of the bargaining process over wage and employment that this paper
uses is the application of the trade union model. It gives a specific definition of theunemployment. Concretely, the trade union model splits unemployment into two categories: (1)unemployment due to capital shortage, and (2) unemployment due to demand shortage.
Moreover, it analyses how the employment-wage (or the capacity utilization-wage) relation in
the short-run arises as a consequence of the wage bargaining. By introducing a particular type ofinvestment function into the dynamic model, the stability conditions are investigated and wehave the following conclusions: (1) the stability of the economy depends on the bargaining
power of the trade union, (2) the balanced growth path can occur where the union’s monopoly
power is sufficiently weak even if we have the Harrod-type investment function that is usuallyregarded as the major factor of leading the instability of the economy. In addition, we explore
the relationship between the bargaining structure in the corporatist model and the stability
condition, and then we find that a trade union’s behavior can help the capitalist economy toremain taking a balanced growth path. Furthermore, the different types of unemployment
including (a) unemployment due to capital shortage, and (b) the one due to demand shortage,
are considered.
As for the analysis concerning the impact of the wage bargaining structure on the stability ofthe economy, there are many attempts to provide empirical researches on how the wage
bargaining structure affects the macroeconomic performance. This paper conducted theestimations as the empirical researches by utilizing the “right to manage” model. Ourestimations of wage equation for manufacturing industries by OLS and GMM showed that the“right to manage” model has a certain interpretability of trade union’s behaviour in recent Japan.
However, we could not find the clue to the structural change of wage determination structure
and the strong bargaining process.
Key words: bargaining model, trade union, wage determination, GMM
JEL Classification Code: E24, J51

11. Introduction
In Japan, the impacts of wages and prices on the macroeconomy, as well as theirrelationship have been in the spotlight again recently. It is expected that inflation expectations,mainly due to monetary policy, will increase wages and aggregate demand, leading to theeconomic expansion. Under the policy framework called Abenomics, price determination
mechanism as well as the wage determination process have become an important issue.
This paper investigates the following three points: the first is to examine the wage
determination mechanism. From 1970-1990, including the stagnation period after the bubble
economy in 1986-1991, there was a growing awareness of the essential importance of the tradeunion behavior for determining wages and the macroeconomic performance of the economy.
Brunello and Wadhwani (1989) and Inoue et al. (2006) analyzed the wage equation based on thebargaining behavior between trade unions and firms. During the lost 20 years in Japan after the
bubble economy, the influence of trade unions on wage determination was believed to havebecome smaller, but then, it is beginning to be reviewed in the power of the labor unions (Noda(2004), and Boeri and Ours (2013)). In this context, this paper focuses on the trade union and its
bargaining process to determine wages.
The second is to examine the dynamic process of the bargaining process and stability of theeconomy. Bargaining theories have specified the negotiating process within the static models
rather than the dynamic framework (see, for example, McDonald and Solow (1981) and Oswald(1985). However it must be inappropriate to analyze the nature of unemployment in such a static
framework. As Klundert and Schaik (1990) has pointed out, the demand deficiency as well as
capital accumulation have an effect on the actual unemployment. It will be misleading for thegovernment to try to increase the effective demand in the case where there is no deflation gap,
and it also seems to be misleading for the government to use a demand management policy for
reducing the rate of unemployment in the case where there is a capital shortage. Thus, it shouldbe the second aim of our study to clarify the category of the unemployment. In this paper,
following and developing Klundert and Schaik (1990), the unemployment is broken down into
two parts; unemployment due to demand shortage and unemployment due to capital shortage,
where the features of the Keynesian type model with market restriction can be applied.
The third aim of the paper is to provide a dynamic analysis consistent with bargaining
theories. After examining the workings of the dynamic model, we reach the major point that thestability condition is related to the union’s bargaining behavior including the bargaining power.
Moreover, it wil1 also be proved that whether the instability principle in the Harrodian sense
can be ensured or not depends on how the trade unions behave in the wage bargaining process.
In this context, a wage equation with trade union behavior will be estimated in order to confirm
whether the bargaing power has changed or not.
The paper proceeds as follows. In section 2 we present a basic framework for the firm's
actual labor demand function and give a rigorous definition of the unemployment. Section 3
provides the “monopoly union” model which gives the wage-unemployment outcomes with a
specific union's monopoly power. In section 4, we analyse the features of the simple dynamic
model which incorporates the investment function. In section 5, we intruduce the “right tomanage” model which is applicable to the empirical study. In order to exmine the trade union
behavior empirically, the estimatation of the wage equation is conducted in section 6.
Concluding remarks are in section 7.
2. The Framework
Suppose that the production function by which national output (Y) is reated to notional labordemand(𝐿) and cpaital (𝐾) is given by a Cobb-Douglas form:(1) 𝑌 = 𝐴𝐿𝛼 𝐾𝐿1−𝛼 , 0 < 𝛼 < 1, 𝐴 > 0,2

where A represents overall TFP (total factor productivity). In the short-run, both K and A mustbe fixed at given levels. The firm’s notional real profit(π) in terms of capital stock is given by(2) 𝜋 = 𝐴𝑙 𝛼 − 𝑅𝑙 − 𝑟,𝐿where 𝑙 = 𝐾 . 𝑅 is the real wage rate and 𝑟 is the real cost of capital. Hence, the profit
maximizing the level of notional labor-capital raition (l) can be written as
𝐿(3) 𝑙 = 𝐾,𝛼where 𝑧 = 1−𝛼.
A representative firm will try to employ workers following equation (3) unless there is supply
restriction in the labor market or demand restriction in the market.
As far as demand is concerned, we adopt a simple formula:(4) 𝑋 = 𝑅𝑁 + 𝐶 + 𝐼 + 𝐺,
where X, I and G refer to total expenditure, investment and government expenditure respectively.
Assuming workers who receive wages do not save at all, total consumption equals consumption
by workers (𝑅×𝑁) and consumption by others (𝐶). According to the demand restriction given
by equation (4), the realized, not notional, output will be given by(5) 𝑋 = 𝐴𝑁 𝛼 𝐾 1−𝛼 ,where N is actual employment by firms.
The rate of capacity utilization (δ) can be defined by𝑋(6) 𝛿 ≡ 𝑌 ,the ratio of an actual putput restrcted by effective demand (X) to notional output (Y) that firms
warrant at a normal level of capacity utilixzation. From equations (1), (5) and (6) as well, we
obtain
1(7) 𝑛 = 𝛿 𝛼 𝑙,𝑁where n ( ≡ ) denotes the labor-capital ratio. As for the consumption 𝐶 , a part of total
𝐾
consumption, asseme that it is positively related to the degree of capacity utilization. Then alinear relationship between C and δ can be given as(8) 𝐶 = 𝑐𝛿𝐾, 𝑐 > 0 const.
Substitution of equations (3), (7) and (8) into equation (4) yields
1(9) 𝑥 = 𝑧𝛿 𝛼 + 𝑐𝛿 + 𝑔 + 𝑓
𝑋𝐼𝐺
where 𝑥 = 𝐾 , 𝑔 = 𝐾 , 𝑓 = 𝐾 . From equations (3), (5) and (7), taking account of the definitionof x, we obtain another formula for x:𝑧 𝛼(10) 𝑥 = 𝐴 (𝑅) 𝛿.
Hence, we use (9) and (10) to derive𝑧 𝛼1(11) 𝐴 ( ) 𝛿 − 𝑧𝛿 𝛼 − 𝑐𝛿 = 𝑔 + 𝑓,𝑅
and solving (11) for use below, we express (11) in the implicit function:(12) Ф(𝛿, 𝑅; 𝑟, 𝛼, 𝐴, 𝑐, 𝑔, 𝑓) = 0.
𝑑𝑅
𝑌
Differentiating (11) with respect to R to verify the sign of 𝑑𝛿 , we use y ≡ 𝐾 to find𝑑𝑅(13) 𝑑𝛿 =

𝑦{1−

1−𝛼
𝑐
− 𝛿 𝛼 }𝑅
𝑦

𝑧𝛿

.

𝑐
𝐶
It should be noted that 𝑦 is equal to 𝑋 , the ratio of non-worker`s consumption to actual output

and thus it is small but positive. For a given set of {𝑟, 𝑔, 𝑓}, the slope of (12) in the 𝑅 − 𝛿 space𝑐𝛼will be simply positive in the case where 𝛿 < (1 − 𝑦)1−𝛼 is assured and vice versa. Theoefore,
the alternative combinations of variables in equation (12) which will produce equilibrium in the
product market can be drawn in Figure 1 as the hump-shaped curve. It may be notable to find3

Figure 1 (𝛼 = 0.5)𝑐that when 𝛿 =1, the slope of (12) becomes − 𝑛 ， per capita non-workers’ consumption.
Moreover, it should be also noted that a decrease in g will induce a decrease in real wages andshift this curve downwards, as is shown in Figure 11.
As we have already shown, firms will estimate the level of notional labor demand so as tomake capital operate at the normal rate. However, they are bound to employ even less numberof workers than the notional level if the degree of capacity utilization is less than unity, the casewhere there is the positive output gap. Therefore, the planned employment can be realized if andonly if there is no gap between notional and actual output. It may be natural to define theemployment gap, that is unemployment, by the following simple formula:𝑀−𝑁(14) 𝑢 ≡ 𝑀 ,where M represents the labor supply in the society and u denotes the rate of unemployment. The
percentage growth rate of labor supply is assumed to be ν > 0 which is given exogenously. A
careful consideration would make it clear that if the rate of capacity utilization is less than unity
so that the notional demand for labor is greater than the actual employment, then the rate ofunemployment must be broken down into two parts, 𝑢1 and 𝑢2 , defined by
𝑀−𝐿
𝐿−𝑁(15) 𝑢 ≡ 𝑢1 + 𝑢2 ≡+, for 𝑀 ≥ 𝐿 ≥ 𝑁.
𝑀

𝑀

Using (15), the percentage contribution of 𝑢1 to u(𝑢1% ) as well as that of 𝑢2 to u(𝑢2% ) aredefined by
1𝐿𝐿(16) 𝑢1% =100× (1 − 𝑀)⁄(1 − 𝛿 𝛼 𝑀),
and
1

1

𝐿

𝐿

(17) 𝑢1% = 100× {(1 − 𝛿 𝛼 ) 𝑀} /(1 − 𝛿 𝛼 𝑀),
respectively. Let give a numerical example. Assume L/M, and 𝛿 and 𝛼 to be 0.95, 0.95 and 0.7
respectively. In this case 𝑢1 =5% and 𝑢2 =6.7% and we find that 𝑢1% becomes 27.4%. This meansthat almost three fourth of the total unemployed are attributable to a deficiency of the effective
demand.
𝑢1 can be referred as the rate of unemployment due to capital shortage because even if firms
utilize their capital stock at the normal rate, there still exists a number of unemployed. However,there also exist some unemployed caused by a lack of effective demand. In this case fromequation (7), 𝛿 <1 implies L>N and, 𝑢2 >0. Hence, 𝑢2 should be referred as the rate ofunemployment due to demand shortage or the rate of ‘keynesian unemployment’. Using
equation (7), the rate of overall unemployment can be rewritten as
1

1

𝐿

𝐾

𝑧

(18) u = (1 − 𝛿 𝛼 ) 𝑀 = 1 − 𝛿 𝛼 (𝑀)(𝑅)1

As for the locus where
that

𝑑𝑅
𝑑𝛿

𝑑𝑅
𝑑𝛿

𝑐

1−𝛼

is assured for equation (12), that is the locus of 1 − − 𝛿 𝛼 = 0, it is clear𝑦< 0 and that 𝛿 → 1 implies 𝑅 → 0.

4

In the balanced growth path, the rate of capacity utilization is equal to unity and capitalstock increases at the same rate as labor supply so that K/M will be kept constant. Therefore, theemployment rate L/M remains constant unless both z and R alter and then the rate ofunemployment would be fixed at the following level:𝐿(19) 𝑢∗ = 1 − (𝑀).
Because a steady state requires a constant employment rate which is very likely to be lessthan unity, the steady state rate of unemployment, 𝑢∗ , may be positive. Steady state also implies
L=N so that this unemployment is justified to be caused by capital shortage. Therefore, we willfind that the steady state rate of unemployment cannot be reduced by any demand-side
procedures such as the demand management policy.
We will reach a complete system made up three indigenous variables, δ, R and g after
analyzing how R as well as g are determined. These are subjects in the following sections.
3. The Monopoly Union Model
Micro-foundations related to the trade union behavior have played an important role toexplain the wage-employment settlement between trade unions and firms. Many papers tackled
this bargaining problems have categorized two dominant bargaining models: the “monopoly
union” model and the efficient bargaining model (see, for example, MacDonald and Solow(1981), Oswald (1985), Calmfors (1985) and Booth (1995)). In the sequel of this paper we will
use the “monopoly union” model in which the trade union can choose the wage by bargaining
with the firm whereas the level of employment is determined unilaterally by the firm.
Suppose that the trade union has the welfare function (W) concerning its members’ expectedutility. Using (14), we obtain the welfare function of the form
𝑁𝑁𝑁(20) 𝑊 = 𝑅 + (1 − ) 𝑅 = (𝑅 − 𝑅) + 𝑅
𝑀𝑀𝑀= (1 − 𝑢)(𝑅 − 𝑅) + 𝑅,
where R is the alternative wages (or the level of unemployment benefit) out of the union sector.
It is assumed that the union wages must be greater than the alternative wages, that is R>R, so as
to appreciate the union`s effect on the positive union-nonunion wages differential (see chapter 6
of Booth 1995, for a survey of empirical research). Introducing the union markup (m＞0) as theratio of union wages to non-union wages ,we obtain(21) 𝑅 = (1 + 𝑚)𝑅.
By substituting (18) into (20), we obtain
1𝐾𝑧(22) 𝑊 ≡ 𝑊(𝛿, 𝑅) = 𝛿 𝛼 (𝑀) (𝑅) (𝑅 − 𝑅) + 𝑅,
where K/M is kept constant in the short-run. Therefore, simple algebra of (22) shows that themarginal rate of substitution (𝑀𝑅𝑆𝑅𝛿 ) between real wages and the rate of capacity utilization is
given by the equation𝑑𝑅(23) 𝑀𝑅𝑆𝑅𝛿 ≡ 𝑑𝛿 = −(𝑅−𝑅)𝑅𝛼𝛿𝑅< 0.

As for the 𝑀𝑅𝑆𝑅𝛿 , it is clear from (23) that the absolute value of 𝑀𝑅𝑆𝑅𝛿 tends to be zero as R
decreases towards R and that it also decreases as 𝛿 increases.
Suppose that the trade union maximizes W given by (20) with subject to (11) for a given set{𝑔, 𝑓, 𝐴, 𝐾, 𝑀, 𝛼, 𝑧, 𝑅}. As far as a short-run equilibrium is concerned, the first-order condition
for the trade union’s desired wages can be written as
(24)

𝑦(1 −

1−𝛼
𝑐
− 𝛿 𝛼 )𝑅𝑦𝑧𝛿=−(𝑅−𝑅)𝑅𝛼𝛿𝑅.

(24) implies that the left- hand side which represents the slope of the labor demand schedule
given by (13) must be equal to 𝑀𝑅𝑆𝑅𝛿 , the right-hand side. Therefore, as for the point for theunion to set, as shown by A in Figure 2, the sign of left-hand side of (24) should be negative so

5Figure 2 (𝛼 = 0.5)𝑐𝑦

as to have 𝛿>(1 − )𝛼(1−𝛼) , meaning that the slope of the labor demand curve is negative2. It isconvenient to rewrite (24) by the implicit function of the form:(25) Ψ(𝛿, 𝑅; 𝑟, 𝛼, 𝐴, 𝑐, 𝑅) = 0
Equations (12) and (25) form a complete framework as for the short-run equilibrium of thesystem. Let Ω be the set which consists of these equilibrium points (𝛿, R) satisfying both (12)and (25) for various values of the rate of capital accumulation. This is also depicted in Figure 2.
We call the locus of such points the Ω-contract curve. A Ω-contract curve is positively sloped
for R ≥ R and intersects the vertical line, 𝛿=1 at 𝑅 ∗ = (1 + 𝑚∗ ) R where 𝑚∗ =𝑐𝛼 ⁄𝑧. We cansolve (11) explicitly for the equilibrium rate of capital accumulation, g*𝑧 𝛼(26) 𝑔∗ = 𝐴 (𝑅∗ ) − 𝑧 − 𝑐 − 𝑓,at the point (1, 𝑅 ∗ )∈ Ω. Therefore, g*is identical with “the warranted rate of growth” in the
Harrodian sense because this assures the normal utilization of capital (see, for example Harrod1973). Notice that there is no Keynesian unemployment on the warranted growth path and only
unemployment due to capital shortage matters. In this case, as observed before, the steady state
rate of unemployment, 𝑢∗ given in (19), is very likely to be positive but variable in the long-run.
This is because the employment rate can change if the rate of capital accumulation, 𝑔∗ is notequal to the rate of labor supply growth, v, which is referred as natural rate of growth in the
Harrodian sense. It is clear that if 𝑔∗ <v, for example, then the employment rate will fall and 𝑢∗
will rise. Policies should be directed towards raising the level of 𝑔∗ so as to avoid an increaseof 𝑢∗ .This may be achieved by some supply-side procedures, such as a decrease in R and a fallin c.
Differentiation of (12) and (25) leads in the conventional way to0𝛿Φ Φ𝑅 𝑑𝛿1(27) [ 𝛿] [ ] = [ ] 𝑑𝑔 + [𝛼𝑅 ] 𝑑𝑐 + [−Ψ ] 𝑑𝑅,Ψ𝛿 Ψ𝑅 𝑑𝑅
0
𝑅
where
𝑐

1−𝛼

Φ𝛿 = 𝑦 (1 − 𝑦 − 𝛿 𝛼 ) < 0,𝑦𝛼𝛿Φ𝑅 = − 𝑅 < 0,
2 Using the bordered Hessian matrix (𝐻 ) for his constrained optimization problem, the second-order condition can begiven by0 Φ𝛿 Φ 𝑅
det 𝐻 = |Φ𝛿 L𝛿𝛿 L𝛿𝑅 | > 0,
Φ𝑅 𝐿𝑅𝛿 𝐿𝑅𝑅
where 𝐿 = 𝑊(𝛿, 𝑅) − λΦ(𝛿, 𝑅), and λ is the Lagrangian multiplier. It can be easily shown that this condition is11−𝛼satisfied by some additional restrictions related to a small 𝛼 (𝛼 > ) and a small 𝑚 (𝑚 < 𝛿 𝛼 ).
3

6

1−2𝛼

Ψ𝛿 = −𝑦 {(1 − 𝛼)𝛿 𝛼 } 𝑅 < 01−𝛼−𝑦(1 − 𝛿 𝛼 −Ψ𝑅 =𝑅)𝛼 2 𝑅
𝛼𝑅

𝑅

and
𝑐

>0

1−𝛼

Ψ𝑅 = 𝑦 (1 − 𝑦 − 𝛿 𝛼 ) 𝛼 − 𝑧 < 0.
Subscripts denote partial differentiation. Moreover, using (26), the response of δ or R to changes
in the model parameters can be found by𝑑𝛿Ψ= 𝐽𝑅 < 0,
𝑑𝑔
(28)

−αRΦ
𝑑𝛿
= 𝐽 𝑅 < 0,
𝑑𝑐
Ψ𝑅 Φ𝑅
𝑑𝛿
= 𝐽 < 0,
𝑑𝑅

and
(29)

𝑑𝑅
Ψ
= − 𝐽𝛿 < 0,𝑑𝑔𝑑𝑅𝛼𝑅Φ𝛿 −δΨ𝛿 >
=
<0,
𝑑𝑐
𝐽
−Φ𝛿 Ψ𝑅
𝑑𝑅
𝑑𝑅

=

𝐽

> 0,

where 𝐽 ≡ (Φδ ΨR − Ψδ ΦR ) < 0.
Among the various results emerging from the comparative static, it may be inconsistent withintuitive investigations to say that an increase in g lowers both the rate of capacity utilizationand the union`s real wage rate. An increase in g itself seems to increase the effective demand so
as to raise the rate of capacity utilization. However, this is not the case. A rise in g leads to a fallin R as well as a rise in y, the firm’s production capabilities, and this inevitably causes an excess
supply for goods. Hence, the rate of capacity utilization must fall so as to keep the balance in themarket. Moreover, it is clear from (28) and (29) that an increase in alternative wages raises theunion’s wage rate but lowers the rate of capacity utilization.
However, it is also notable that the effect of a change in g has on the rate of capacity
utilization, in particular, will depend on just what type of welfare function the trade union has.
To prove this, we shall investigate the simplest example. Assume a different type of union’s
welfare function from (20), which is simply given by(30) 𝑊𝑠 = 𝑅 −R
This type of welfare function implies that the trade unions only take the union-nonunion wages
differential into consideration and that they have no need of taking care of their jobs3. From (30),
it is clear that 𝑀𝑅𝑆𝑅𝛿 (≡ 𝑑𝑅⁄𝑑𝛿 ) becomes zero. The best wage for the union to set isdetermined in the way by the tangency of an indifference curve of Ws, with the labour demand
curve as depicted in Figure 2. Let Ω𝑠 be the set of the equilibrium points in this case to comparewith Ω. The Ω𝑠 -contract curve is negatively sloped for R>R, as shown in Figure 2. Following
the same procedures as before to verify the effect of changes in g on R and δ, we obtain𝑑𝑅𝑑𝛿< 0, 𝑑𝑔 > 0, for 𝑅 > 𝑅.
𝑑𝑔
It should be noted here that if the union’s welfare is given by (30) instead of (20), then the effectof a rise in g on the rate of capacity utilization becomes opposite to that given in (29). The
implication of the difference in the sign of 𝑑𝛿 ⁄𝑑𝑔 for the dynamic property of the system in thelong-run will be discussed in the next section.

3

In the seniority model, it is proved that there is a seniority rule among union members by which theirjobs are ensured at least in the short-run and the union is bound to bargain over the wages. Therefore,
this case may be classified as a trade union model in line with such a seniority rule (see Oswald 1984).

7

4. A Dynamic System
In the case where the output gap arises either from the excess demand for goods or from thedeficiency of effective demand and, hence, 𝛿<> 1, the gap will closed over time by two operating
mechanisms: the price mechanism by which prices change with the degree of capacity
utilization and the investment mechanism by which the stock of capital also changes with thedegree of capacity utilization. We assume that prices are fixed and that there is no Pigou effect,with the objectives of clarifying the trade union’s behavior represented by a welfare function
defined in real terms. Assuming a price rigidity along traditional lines will make the analysis
concerning the effect of investment on the growth path of the economy be more prominent.
As for the investment mechanism, we assume that the rate of capital accumulation changes
as a function of the rate of capacity utilization:(31) 𝑔̇ = 𝛽(𝛿 − 1), 𝛽 > 0,
where dot (∙) denotes the change in variables related over time. It is notable that even if demand
shortage causes unemployment in the society, there is no incentive for firms to dissolve such an
unemployment problem due to demand shortage. Equation (31) implies that if firms are boundto employ more (less) workers than those who they plan to employ in order to attain the
optimality, that is 𝛿 > 1 (𝛿 < 1 ), then firms are sure to enhance (decrease) their production
capacity to avoid capital shortage (superflux)and it is referred as the Harrod type of investment
function (see Harrod (1973)). It is easy to show that the system for {𝛿, 𝑅, 𝑔} completed by (12),(25) and (31), can be summed up by the following reduced equation;(32) 𝑔̇ = 𝛽(𝛿(𝑔) − 1), 𝛿′ < 0
Therefore, as far as the Ω-contract curve is concerned, the stability of the system can be ensured
without any additive conditions 𝑑𝑔̇ ⁄𝑑𝑔 = 𝛽𝛿′ < 0 . As we have already mentioned in theformer section, this system is related to the short-run stability because the equilibrium rate ofcapital accumulation must warrant the normal rate of capacity utilization so that the actual
growth path becomes identical with the warranted growth path whereas it does not mean that theeconomy actually grows at the natural rate, that is the growth rate of labor supply.
It should be noted that the system developed here offers a remarkable contrast to the
Instability Principle that the Harrodian models have proved. The stability results of the system,
therefore, may be attributable not only to the type of investment function but also to the shapeof the union’s welfare function.
Moreover, it is also notable that if there is a competitive labor market for the non-union
sector and the non-union wages can be affected by the demand-supply conditions of the labormarket, then the long-run stability, concerning the relationship between the warranted growth
path and the natural growth path, can be also ensured. To account for this aspects, we shallfeature the following adjustment equation:(33) 𝑅̇ = 𝛾(𝑔 − 𝑣), 𝛾 > 0
It is acceptable that if 𝑔 > 𝑣, for example, then the employment rate increases so that u* in (19)will decrease on the warranted growth path. Hence, the lowering unemployment rate will lead toa rise on the non-union wages. Elimination of (12) and (25) and substitution into (31) and (33)give a reduced form of the dynamic system in the rate of capital accumulation and the nonunion wages:
𝑔̇ = 𝛽(𝛿(g, R) − 1), 𝛿𝑔 < 0, 𝛿𝑅 < 0,(34)
𝑅̇ = 𝛾(𝑔 − 𝜈).
As can be easily checked, the dynamic system presented by (34) is stable because as for theJacobi matrix of the system (JΩ) evaluated at the steady state, we obtain trace of 𝐽𝛺 = 𝛽𝛿𝑔 < 0.
Therefore, it follows that the dynamic path in the Ω-contract curve will possibly converge indue course of the time to the steady state where the actual growth path is identical both with thewarranted growth path and with the natural growth path. Hence, the long-run steady state can be8Figure 3 (𝛼 = 0.5)𝑐𝛼attained when 𝑔∗ = 𝑣, 𝛿 ∗ = 1, 𝑅 ∗ = (1 + 𝑧 ) 𝑅 ∗ are assured and It has no unemployment due
to the demand shortage.
What we have just mentioned the stability of the system is obviously conditional on theunion’s bargaining behavior, that is presented as the MRSRS evaluated at the contract curve. As
for the Ωs-contract curve, for example, using (12), (13) and (31), the short-run dynamic systemin the Harrodian sense is given by the following equations:
𝑅 = 𝑅(𝛿), 𝑅 ′ < 0,(35) 𝜑(𝛿, 𝑅; 𝑟, 𝛼, 𝐴, 𝑐, 𝑔, 𝑓) = 0, 𝜑𝛿 < 0, 𝜑𝑅 < 0, 𝜑𝑔 = −1 < 0,𝑔̇ = 𝛽(𝛿 − 1),
where the first equation of (35) is derived from taking the right-hand side of (13) to be zero.
(35) can be easily summarized by(36) 𝑔̇ = 𝛽(𝛿(𝑔) − 1), 𝛿 ′ > 0
to compare with (32). Therefore, the short-run instability (not stability), that is the Instability
Principle, can be ensured in the case of Ωs-contract curve.
As sketched out above it is clear whether the system becomes stable or not depends on the
shape of the contract curve given by (24). As Johnson and Layard (1986) have proved, themonopoly power of the trade union can be defined by the elasticity of demand for labor withrespect to the union wages and its reciprocal is measured by the left-hand side of (24). The
monopoly power is also proved to be positively related to the union markup (see, for example,
Yabuta (1993)). Moreover, it is clear from (24) that on the contract curve the slope of the labordemand schedule is equal to MRSRδ. Therefore, it follows that the smaller MRSRδ is, the higher
the union markup and the greater the union’s bargaining power. As we have already shown, theΩs-contract curve is the case where the trade unions have the smallest MRSRδ, that is zero.
Hence, if the bargaining power of the monopoly union is strong enough, meaning that the union
markup is sufficiently large, then the system with the Harrod type of investment function may
be unstable. On the contrary, the stability of the economy can be ensured in the case where themonopoly power of the trade union is weak enough so that the contract curve can intersect theline 𝛿=1 at the point E in Figure 3. Moreover, if there is a mechanism by which the non-union
wages continuously are adjusted to clear the labor market, then the balanced growth path will beattained in due course of time. It is notable that the stability condition depends on the union's
bargaining power.
It is clear that the trade union behavior seems to be of central importance because it
determines the system’s stability. Therefore, some issues may still remain concerning how thetrade union’s behavior as well as its bargaining structure is fixed in the economy; why theydiffer from nation to nation and how they affect the dynamic growth path. A possible way totackle these issues seems to be the theories of Corporatism, which have argued the institutional

9structure of wage bargaining over the 1980s (see, for example, Calmfors and Driffill (1988),
Moene et al. (1993) or Booth (1995) for a survey). One of the major results of the corporatist
model is that the confederation of trade unions knows the trade-off between the higher wages
and the lower employment so that it tends to constrain its wage demand with the intention oftaking into account the impact of higher union wages on the overall unemployment. Taking ourterms used in this paper, this means that the centralized wage setting leads to the lower union
markup or the higher MRSRδ4. Therefore, it is possible to conclude that the trade unions behavior
with the centralized wage bargaining might have helped the capitalist economy to keep thebalanced growth path stable.
5. The Model for Our Empirical Analysis
In the previous sections, the dynamic analysis with the “monopoly union” model which
incorporates the relationship between the behavior of trade union and the macroeconomic
performance is examined. However, it is hard to conduct empirical analysis for Japan byutilizing the “monopoly union” model because of the difficulty in obtaining the data set. Instead,the “right to manage” model, which also includes the framework of bargaining structure, can beapplied for our empirical estimation. It was developed mainly by Nickell (1982) and Nickell andAndrews (1983) in an early stage. In this section, the basic structure of the “right to manage”
model is introduced based on Yabuta (1989).
The “right to manage” model assumes that employment is determined unilaterally by thefirm but the wage level is decided by bargaining process between the trade union and the firm.
The firm is assumed to choose an optimal employment level so as to maximize its profits once
wages are determined through negotiation between the unions and the firm. The utility function(U) of the union isw(37) U = U(Pe , L, h),where w is the hourly wage rate, Pe is the price of consumption goods presumed by workers, L
is the level of employment, H is the number of working hours in a set if period. On the other
hand, utility function (V) of the representative firm is given byw
P(38) V = f(L, h, X, δK) − ( ) hL − ( x ) X − rK,PP
where f is a production function, X is resources, K is the capital stock and it is in operation at
δ% of its capacity as well as labor-related inputs (L and h), Px is the price of raw materials, r is
the rental cost of capital. If we represent the outcome of the bargaining process by anasymmetric Nash bargaining solution, this model is expressed as:(39) max. λlogU + (1 − λ)logVw
subject to VL = fL − ( P ) h = 0,
where λ (1 ≥ λ > 0) represents the bargaining power to the union. By some calculations, wedwknow that dδ becomes positive if the union’s preference takes priority over the wages rather
than employment with a given level of λ, and vice versa.
The solution of the model can be written as(40) w = w(p, h, r, δ, X, Px , K),+0−? − − −where the sign pattern of (40) is realized when the economy is in the optimal condition. The w
also depends on various factors such as the bargaining power to the trade union although theydw
do not appear explicitly in this equation. However, as mentioned above, the sign of dδ would
play the key role in our analysis.
4

Because the level of wage markup is low in the system of centralized wage setting in Nordic countries,
the economy will become more stable.

106. Empirical Results
This section is for our empirical study of the wage equation derived through “right tomanage” model explained in section 5 for manufacturing industries in Japan. Our analysis aims
not only to assess the statistical properties of the model but also to investigate the stability of thewage determination process in Japan. If there were a structural change in Japanese trade unions’behavior, we would be able to find the instability through the econometric test for structural
break.
The sample period of estimation is 2008:M2 (February) − 2016:M3 (March). Our dataset isconstructed by the following variables.5
w: Real Wage Index (total cash earnings, for Manufacturing, for establishments with 30
employees or more, 2010 average = 100)
P: Corporate Goods Price Index (major group, manufacturing industry products, CY2010
average = 100)
H: Hours Worked Indices (total hours worked, manufacturing, 2010 average = 100)
R: Average Contracted Interest Rates on Loans and Discounts (outstanding loans and bills
discounted, long-term loans, domestically licensed banks) This is the proxy variable forrental cost of capital.
D: Indices of Operating Ratio (indices of industrial production, original indices by industry,
manufacturing, 2010 = 100)
X: Index of Producer's Shipments (indices of industrial production, original Indices byindustry, manufacturing, 2010 = 100) This is the proxy variable for index of raw
materials consumption.
Px: Corporate Goods Price Index for Raw Materials (CY2010 average=100)
K: Gross Capital Stock of Private Enterprises (preliminary quarterly estimates, released atAugust 15, 93SNA, construction in progress, tangible fixed assets, all industries)
All items except “K” are monthly. The quarterly data on “K” is converted into monthly.6 The
rate of change from the previous month of real wage index, which is the dependent variable ofour estimation, is based on “w”.
The specification for our estimation is constructed as:log 𝑊 = α0 + α1 logP + α2 logH + α3 logR + α4 logD + α5 logX + α6 logPx + α7 logK,where W is the wage inflation. As we have studied in the previous section, they theoretically
imply at the micro-level that an increase in input costs, such as resource price or cost of capital,
should bring the decrease in wage rate since it reduces the profit, and an increase in output priceshould cause wages to increase because of union’s preference for raising real wages.
The result of the estimation of wage equation for manufacturing industries by OLS
methodology is indicated in Table 1. The standard errors and covariance are computed based on
the Newey-West Heteroscedasticity and Autocorrelation Consistent (HAC) weighting matrix.
The coefficients on hours worked index (for total hours worked), index of operating ratio(indices of industrial production), index of corporate goods price index for raw materials, andgross capital stock of private enterprises are significantly estimated at 1% level. The coefficienton the average contracted interest rates on loans and discounts as the proxy variables for therental cost of capital is barely significant at 10%. With these estimated coefficients, one
5

The data on “Real Wage Index,” and “Hours Worked Indices” can be retrieved from the Website of the
Ministry of Health, Labour, and Welfare. The “Indices of Operating Ratio,” and “Index of Producer's
Shipments” can be obtained from the website of the Ministry of Economy, Trade and Industry. The“Corporate Goods Price Index,” “Corporate Goods Price Index for Raw Materials,” and “AverageContracted Interest Rates on Loans and Discounts” are available at the website of the Bank of Japan.
The data on “Gross Capital Stock of Private Enterprises” can be obtained from the Economic and Social
Research Institute, Cabinet Office’s website.
6
The quarterly data on “K” is converted into monthly by simply allocating same figures on the threemonths in the corresponding quarter.

11

Table 1: OLS Estimation
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
269.8705
172.6532
1.563079
0.1215log P
2.485260
1.843363
1.3482210.1810log h6.038523
0.676269
8.9291720.0000log r-2.505047
1.434348-1.746471 0.0841
log δ-3.322653
1.039340-3.196887 0.0019
log X
2.594196
0.917030
2.828912
0.0058log Px-0.034892
0.176233-0.197989 0.8435
log K-14.53909
8.393850-1.732112 0.0867
Adjusted R-squared 0.423510
F-statistic
11.17993
S.E. of regression
0.306571
Durbin-Watson stat. 2.172364
Notes: Dependent Variable: Change from the previous month of real wage
index. Sample (adjusted): 2008M02 - 2016M03. Included observations = 98
(after adjustments). Standard errors and covariance computed using HAC
weighting matrix (Bartlett kernel, Newey-West fixed bandwidth = 4).
Convergence achieved after 32 weight iterations.

Table 2: Quandt-Andrews Unknown Breakpoint Test
Statistic
Value
Prob.
Maximum LR F-statistic(2011M05)
1.075253
0.9767
Notes: Null Hypothesis: No breakpoints within 15% trimmed data.
Equation Sample: 2008M02 2016M03. Test Sample: 2009M05
2015M01. Number of breaks compared: 69. Probabilities calculatedusing Hansen's (1997) method.

important topic that we have to consider through the examination of the estimated coefficient on
the index of operating ratio is the bargaining position of the Japanese trade unions. The sign ofthe coefficient is apparently negative, and it might be interpreted that the trade unions haveweak bargaining positions or preferences for employment rather than wages. The other topic
that we should pay attention is
the structural change of wage determination structure. In order to examine this problem, threekinds of test are implemented.
Table 2 displays the result of the Quandt-Andrews Unknown Breakpoint Test. The test
statistic denies the possibility of the structural break. On the other hand, Figure 4 and 5 describe
the results of CUSUM test and CUSUM of Squares Test, respectively. They also cannot find theclue to the structural change. Further, as to serial correlation problem, the Durbin-Watson test
statistic indicated in Table 1 falls in the ambiguous region of the Durbin-Watson table of criticalvalues. Thus, we cannot draw a conclusion with respect to serial correlation. In this respect, it isinstructive for us to try another way of estimation.
As the next step, we examine the result of estimation by utilizing GMM (Generalized
Method of Moments) by considering the correlation (between the explanatory variables and theerror term) and the endogeneity of the variables. The GMM estimation is implemented with the
Newey-West Heteroscedasticity and Autocorrelation Consistent (HAC) weighting matrix todeal with the possibility of serial correlation. (Details of this topic with respect to our estimation
are described in the note under the Table 3.) Since we should take a critical stance toward
estimation by GMM, the Hansen’s test for over-identification, the C-test for instrumental
variable’s orthogonality, endogeneity test for variables, weak identification and bias, and thetest utilizing Cragg-Donald statistic and Stock-Yogo critical values are implemented. Our GMM
estimation is conducted with the assumption that the rental cost of capital and the gross fixed
capital stock are endogenous.
Table 3 reports the result of GMM estimation. Considering the diagnostic test, the null
hypotheses of over-identification for GMM estimation cannot be rejected by the Hansen’s test,
supporting the validity of the moment conditions, as shown by J-statistic and p-value in the12Figure 4: CUSUM Test

Figure 5: CUSUM of Squares Test

table. With regard to the estimated parameters, all coefficients except the one on corporate
goods price index for raw materials are significant at 1% level. In particular, the coefficient on
the operating ratio is significant with negative sign. It implies the trade unions in Japan haveweak bargaining positions or preferences for employment rather than wages.
Next, we inspect our estimation specification. Table 4 displays the results of theorthogonality C-tests for each instrumental variable (except constant term). Namely, the test
detects whether each instrumental variable satisfies the orthogonality condition. The
instrumental variables used in our GMM estimation are the constant term and the variables
indicated in the column of test instrument in Table 4. The results suggest that the null
hypotheses of respective instrumental variable’s orthogonality to the error term cannot berejected at the conventional level for all cases. Moreover, the result of the endogeneity test forvariables - rental cost of capital and gross fixed capital stock - is shown in Table 5. The null
hypothesis of exogeneity cannot be rejected at the conventional level by the test statistic.
Further, to investigate the weak identification problem pointed out by some studies
including Mavroeidis (2004), we use the Cragg and Donald (1993) statistic and Stock and Yogo
(2005) critical values 7 . In Table 5, the Cragg-Donald F-statistic is larger than Stock-Yogo
critical values for both relative bias and size. It means the null hypothesis of weak identificationis rejected. Therefore, the test result implies our instrumental variables are not weak.
On the whole, our estimations of wage equation for manufacturing industries by OLS and
GMM methods show that the “right to manage” model has a certain interpretability of the
behaviour of trade unions in recent Japan. However, we cannot find the clue to the structuralchange of wage determination structure and the strong bargaining process.

7

See Cragg and Donald (1993), Stock, Wright, and Yogo (2002), and Stock and Yogo (2002) in details.

13

Variable
C
log P
log h
log r
log δ
log X
log Pxlog K
S.E. of regression
Instrument rank

Table 3: GMM Estimation
Coefficient
Std. Error
t-Statistic
683.6721
225.91953.026176
9.279352
2.813324
3.298359
7.930461
0.838313
9.460025-6.236308
1.868021-3.338456-9.118244
1.760089-5.180558
6.126551
1.327947
4.613551
0.239692
0.243115
0.985919-35.69489
11.10495-3.214321
0.409368
J-statistic13
Prob(J-statistic)

Prob.
0.00320.00140.00000.00120.00000.00000.32680.00186.643991
0.248492

Notes: Dependent Variable: Change from the previous month of real wage index. Sample(adjusted): 2008M02 - 2016M03. Included observations = 98 (after adjustments).
Estimation weighting matrix: HAC (Bartlett kernel, Newey-West fixed bandwidth = 4).
Standard errors and covariance computed using HAC weighting matrix (Bartlett kernel,
Newey-West fixed bandwidth = 4). Convergence achieved after 39 weight iterations.
Instrument specification: Constant, log P, log P(-1), log h, log h(-1), log r(-1), log δ, log δ(1), log X, log X(-1), log Px, log Px(-1), log K(-1).

Table 4: Orthogonality C-Test for Instrumental Variables
Difference in J-stats
Restricted Unrestricted
Test instruments
J-statistic
J-statistic
Value
d.f.
p-valuelog P
0.357786
10.54976.643991
6.286205log P(-1)
0.328929
10.56636.643991
6.315062
log h
4.793504
10.02866.643991
1.850488
log h(-1)
2.26981710.13196.643991
4.374174
log r(-1)
0.3281011
0.56686.643991
6.315891
log δ
2.019456
10.14646.643991
4.534535
log δ(-1)
0.314783
10.57486.643991
6.329209
log X
2.18941010.13906.643991
4.454581
log X(-1)
0.615661
1
0.43276.643991
6.028331
log Px10.410505
10.52176.643991
6.233486
log Px(-1)
0.342635
10.55836.643991
6.301356
log K(-1)
0.222161
1
0.64746.643991
6.421830
Table 5: Endogeneity Test for Variables
Difference in J-stats
Test instruments
Value
d.f.
p-valuelog r, log k
3.81829710.1482
Table 6: Weak Instrument Diagnostics
Cragg-Donald F-stat5%
Stock-Yogo critical values
10%(relative bias)
20%
30%
10%
Stock-Yogo critical values
15%(size)
20%
25%

Restricted Unrestricted
J-statistic
J-statistic
7.103874
3.285577
77.39638
16.88
9.92
6.16
4.76
23.72
13.34
9.77
7.91

7. Concluding Remarks
This paper tried to explore the dynamic system with bargaining process over wages between
the monopoly union and the firm. It also provided a rigorous definition of the unemployment
broken down into two categories: unemployment due to capital shortage and unemployment due
to demand shortage. Moreover, we considered how the employment-wage (or the capacity
utilization-wage) outcomes in the short-run arise as a consequence of the wage bargaining. By

14

introducing a specific type of investment function into the dynamic model, the stability
conditions were investigated and we reached the following conclusions: the stability of theeconomy depends on the bargaining power of the monopoly union; and the balanced growth
path can occur where the union’s monopoly power is sufficiently weak even if the investment
function is of the Harrod-Okishio type which is usually regarded as the major factor that causes
the instability of the economy. Furthermore, we also investigated the relationship between thebargaining structure in the corporatist model and the stability condition, and then we reached aconclusion that a trade union's behavior can help the capitalist economy to remain on thebalanced growth path.
As for the analysis concerning the impact of the wage bargaining structure on the stability ofthe economy, there are many attempts to provide empirical researches on how the wage
bargaining structure affects the macroeconomic performance. This paper conducted theestimations as the empirical researches by utilizing the “right to manage” model. Ourestimations of wage equation for manufacturing industries in Japan by OLS and GMM methods
showed that the “right to manage” model has a certain interpretability of trade union’s
behaviour in recent Japan. However, we could not find the clue to the structural change of wage
determination structure and the strong bargaining process.
Acknowledgements
The authors would like to thank Professor Yasuhide TANAKA (Okayama Shoka University)
for useful comments on the topic treated in this paper at the presentation in the 15th
International Conference of the Japan Economic Policy Association held on October 29, 2016.
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