Productivity Growth and the female labor supply in Japan
Yoko Furukawa*
Tomohiko Inui∗
Abstract:
In the 1990s, Japanese economy has experienced a surge in the unemployment rate,
and this is due partly to the recent change in the structure of the female labor supply.
In the past recessions, female workers tended to be discouraged from participating in
the labor market and this made the Japanese unemployment rate substantially lower
than that of the other industrialized countries. In the current prevailing economic
stagnation, female workers tended to stay in the labor market, and this caused the
economy to experience a higher natural unemployment rate. We develop a one-sector
model with two factors of production, and the female worker chooses either to
participate the labor market or to supply housework services. Our model shows that
the change in the degree of the elasticity of consumption will contribute to the increase
in the female labor participation ratio under the economy’s stagnation, and suggests
that the role of housework services in each household has changed in recent years.
JEL classification: E24, J21, O41, O53.
Key words: female labor participation, productivity, added worker effect, discouraged
worker effect.
Yoko Furukawa: The official name is Yoko Tsuda.
Graduate school of Economics, University of Tokyo.
7-3-1 Hongo, Bunkyo, Tokyo, Japan. 113-0033.
E-mail: ee27017@mail.ecc.u-tokyo.ac.jp
∗
Tomohiko Inui
College of Economics, Nihon University
1-3-2 Misakicyou Ciyoda-ku, Tokyo, Japan, 101-8380
E-mail:inui@eco.nihon-u.ac.jp
*
1
Ⅰ.
Introduction
In the 1990s, the Japanese economy has experienced the prolonged economic
stagnation and the highest level of the unemployment rate. Though it is natural that
the unemployment rate rises in the economic stagnation, the unemployment rate in
Japan stayed at a lower level in the past recessions, compared to other industrialized
countries. This is partly due to the structure of the female labor supply. In Japan, the
number of the discouraged workers was large, and discouraged workers mostly
consisted of females 1 . The female workers had a tendency to be discouraged from
participating in the labor market in the past recession and this made the Japanese
unemployment rate substantially lower than that of the other industrialized countries.
However, in the current prevailing economic stagnation, the female workers tend to
stay in the labor market and the unemployment rate begin rising rapidly. One question
arises now. In the past recession, female workers tend to leave the labor market and
engage in housework. Why does the net discouragement effect become smaller in the
recent recession? In other words, what causes the added workers effect larger that the
discouraged workers effect in the recent stagnation than the cases in the previous
recessions in Japan?
The controversy against the added and discouraged workers effects was shown in
Cain (1966). He provided the empirical analysis of the net effect of unemployment by
using data from each census year for 1940, 1950, and 1960 for the U.S. His empirical
test suggested that the discouraged workers effect exceeded the added workers effect in
the United States. A similar result appeared in Higuchi (2001), that the discouraged
workers effect was larger than the added workers effect in the past recessions in Japan.
However, Higuchi also pointed that recently the female labor participation rate does
not fall compared to the past recession although the labor demand declined in the
recent recession. We will provide why the there is a net effect of unemployment
changes from past recessions and the recessions in 1990s.
Zabalza (1983) estimated the female labor participation in Britain by using the CES
utility function. He showed the household decision of the female labor supply by
1
See OECD (1993).
2
considering the choice between housework and labor market participation. Then he
provided a function of the female labor participation rate that allows for easier
empirical analysis.
However, most of the researches about the female labor supply examined only the
supply side of female labor. It means that the linkage between the business cycle and
female labor participation has not been examined theoretically. In this paper, we try to
analyze the relationship between the productivity of the production sector and the
household decision of the female labor supply theoretically and empirically. We will
show that the decision of the female labor supply determines the role of housework in
each household, and suggests that the change of the role of housework may be one of
the reasons that the discouraged worker effect is smaller in the recent recession.
We had to develop a model that shows the relationship between female labor
participation and the growth of the economy. In this paper, we develop a one-sector
model with two factors of production, male-female labor. The framework heavily
depends on the analysis provided in Wong (1995). It shows how the growth of various
factors contributes the economic environment in a small and open economy by using a
two-factor, two-sector model.
The explanation of the Japanese economy’s recession in terms of productivity was
formalized by Hayashi and Prescott (2000) with one-sector aggregate model. They
argued that slackening productivity growth is the cause of the Japanese economic
recession. Thus, we consider the sluggish productivity growth to be the cause of the
stagnation of the economy, and examine the relationship between female labor
participation and the growth of the Japanese economy.
Section 2 provides the basic model and its implications. The concluding remarks and
suggestions of further research are given in Section 3.
Ⅱ. The Model
The economy is endowed with fixed amounts of two types of homogeneous labors,
labor 1 and labor 2. Denote the endowments of labor 1 and 2 by L1 and L2 respectively.
Labor 1 is thought to be a highly skilled worker, and labor 2 is thought to be a
low-skilled worker, mostly consisting of female workers. All highly skilled workers are
3
assumed to participate in the labor market, and low-skilled workers choose labor
market participation or housework.
The aggregate production function is given as;
Q = AF ( L1 , L2 ) ,
(1)
where Q is aggregate output, and A is the productivity which represent levels of
technology, knowledge capital, and etc. Productivity is treated as an exogenous
parameter. The production function is increasing, concave, linearly homogeneous, and
differentiable in inputs up to the necessary order.
It is assumed that markets are perfectly competitive, and prices are perfectly flexible.
Let me denote the wage rate of labor i by wi, and the equilibrium conditions for the two
types of labor become as follows.
wi = A
∂F
,
∂Li
(i=1, 2)
(2)
Under perfect competition, firms producing positive outputs earn zero economic profit,
which means,
Q = w1 L1 + w2 L2 ,
(3)
Let yˆ ≡ dy y represent the rate of growth of any variable y. Differentiate equation (1),
(2), and (3), and rearrange terms to obtain
Qˆ = Aˆ + θLˆ1 + (1 − θ ) Lˆ 2 ,
0 = − Aˆ + θwˆ 1 + (1 − θ ) wˆ 2 ,
(4)
(5)
where θi is labor i’s share in the production technology, that means
assumption of a zero-economic profit condition requires
θ i ≡ wi Li Q . The
θ 1 + θ 2 = 1 . Define s as the
elasticity of technical substitution of the aggregate production function 2 , then the
changes of each wage rate can be written as follow;
wˆ 1 = Aˆ + (1 − θ )( Lˆ1 − Lˆ 2 ) s ,
wˆ = Aˆ − θ ( Lˆ − Lˆ ) s .
2
1
2
(6)
(7)
Now we turn to the household problem. There is a continuum of households of total
measure one. Each household has L1 , amount of labor 1, and L2 , amount of labor 2.
The household derives utility from family income (or goods consumption) and
2The
elasticity of technical substitution is negative in a usual production function.
4
housework. Define I as the family income and h as hours of housework by labor 2. Each
household preference is given by
 ε h
U i = I i +  i
α

1
 ε
 ,

α>0, ε<1,
(8)
where α and ε are parameters. The parameter α expresses the weight on
housework relative to income. The budget constraint of each household can be written
as follows.
I i ≤ w1 L1 + w2 ( L2 − hi ) .
(9)
To simplify the problem, saving will not be considered in this economy. Labor 1 workers
are considered to be firmly attached to the labor force over time, although labor 2
workers, who are low-skilled and mostly consist of females, are considered to allocate
their time to work or to housework. Then, maximizing (8) subject to (9), we have
1
hi
= αw2 ε −1 .
Ii
(10)
The behavior of each labor 2 is characterized in terms of a critical housework-income
ratio. An individual who is labor 2 will go to work if the housework-income ratio is
smaller than L2 w1 L1 . Let us introduce a stochastic variation of the model in the
parameterα. Since α must be a positive parameter, we define
α = exp( X ) ,
(11)
where X is a random variable, which is assumed to be normally distributed with zero
mean and variance
σ 2 , X ~ N (0, σ 2 ) . Now, α can also be thought as the parameter
which expresses the income or wage distribution. It is natural to consider the income or
wage distribution to analyze the decision of household labor supply, because usually,
the wives’ decision to be housewives or to work may depend on the levels of household
income.
An individual who is labor 2 will not participate in the labor market if her
housework-income ratio, assuming that the wage rates are given for each household, is
bigger than L2 w1 L1 . Therefore, denoting P(NP) as the non-participation rate of labor
2, we have
P ( NP ) = prob[(hi I i ) ≥ (L2 w1 L1 )].
(12)
5
Substituting (10) and (11) into (12), the non-participation rate can be written as
P ( NP ) = 1 − F ( X NP ) ,
(13)
where F(·) is the standardized cumulative normal distribution function, and XNP is
defined as
X NP =

1
 1 
ln L2 + 
 ln w2 − ln w1 − ln L1  .

σ
1− ε 

(14)
The change of the non-participation rate can be written as follows:
dP( NP ) = f ( X NP )

1  1 
 wˆ 2 + wˆ 1  ,
− 
σ  1− ε 

where f ( X ) = dF ( X ) dX . From the assumption of ε<1, it is easy to see that the
non-participation rate is increasing in w1 and decreasing in w2. Now we would like to
investigate the relationship between the swings in the business cycle and the supply of
labor 2 by combining the production sector and the household sector.
To examine the relationship between the growth of the productivity in the economy
and the participation rate of labor 2, we have to calculate the aggregate labor supply.
The aggregate supply of labor 1 is L1 because all the individuals who are labor 1 are
thought to spend all of their time working to earn income. Thus, Lˆ1 = 0 in this model.
The aggregate amount of supply of labor 2 should be equal to the demand in the
production sector, and it is given as follows,
L2 = ∫
X NP
−∞
F ( X )dX .
(15)3
Then, using Leibniz’ rule, the increase of the aggregate supply of labor 2 is,
dL2 = F ( X NP )dX NP ,
and, defining
λ ≡ L2 L2 , the change rate can be written as,
ˆ


1
(1 − P( NP) ) 1  w2 − wˆ 1  .
Lˆ 2 =
λL2

σ  1 − ε
(16)
By using (6), (7), and (16), we have
3
To be exact, the aggregate supply of labor 2 can be written as follows,
L2 = ∫ hi f ( X (hi ) )dhi = [(L2 − hi )F ( X (hi ) )]0 2 − ∫ F ( X (hi ) )dhi = ∫
L2
L
0
L2
0
6
X NP
−∞
F ( X )dX .
ηLˆ 2 =
where
ε
1− ε
Aˆ ,
η ≡σ −
(17)
1
(1 − P( NP) )1 − ε (1 − θ ) and η > 0 .4
s (1 − ε )
λL2
Also, the relationship between the non-participation rate of labor 2 and the growth of
productivity can be solved by using (16) and (17), that is,


L2 λLˆ 2
dP( NP ) = f ( X NP )−

 η (1 − P( NP ) ) 
=−
 f ( X NP )λL2 
Aˆ 
.
1 − ε η (1 − P( NP ) ) 
ε
(18)
What is interesting is that the relationship between the participation of labor 2 and
the growth of the productivity is determined by the parameter ε, which indicates the
elasticity of substitution between goods consumption and housework, that is, 1/(1-ε).
In the case where 0<ε<1, the goods consumption and housework are substitutes, and
the positive growth rate of the productivity in the economy leads to a increase in the
labor 2 involving in the labor market. If ε is negative, and the goods consumption and
housework are less substitutes for the household, the growth of productivity in the
economy leads to an increase in the number of “housewives”. In other words, if ε is
positive, the discouraged workers effect exceeds the added workers effect, and if ε<0,
the added workers effect is larger than the discouraged workers effect.
The implication behind this fact is as follows. In this model, all the income is
consumed for goods consumption. Thus, the consumption-housework ratio allows us to
examine the labor resource allocation in the household over the business cycle. If ε is
positive and the goods consumption and housework are substitutes for the household,
the consumption-housework ratio will decline when the wage level falls compared to
the price level when the economy is in the recession. As the result, the participation
rate of labor 2 decreases, and the labor force moves toward the household sector to
equalize the marginal utilities. Thus, the discouraged workers effect appears. However,
if ε is negative, to sustain the income level is important when the economy’ is in the
4
Remember s<0, 0<θ<1, and ε<1.
7
recession, then low-skilled workers tend to participate in the labor market rather than
conducting housework. Therefore, the consumption-housework ratio increases and the
added workers effect appears in this case.
It is notable that the relationship between the low skilled or female labor supply and
the business cycle is determined by the elasticity of goods consumption and housework.
In this case, the elasticity can be considered to be the role of housework in the
household. To describe it simply, the household can derive utility from either goods
consumption or housework services if 0< ε <1. However, if ε <0, both goods
consumption and housework service are necessary to derive utility. Therefore, the role
of housework becomes important in examine the issue of the female labor supply.
The discouraged workers effect lowers unemployment rate in the recession. In
contrast, the unemployment rate will be higher when there is more added workers
effect because the labor supply expands although the labor demand may shrink
because of the stagnation in the economy. The model in this paper provides that the
elasticity of substitution between housework and goods consumption determines the
net effect of the female labor participation. This suggests that changes in the role of
housework services in each household also bring about a change in the net effect of the
female labor supply. Then, the tendency of the unemployment is affected by the role of
housework services.
Also, notice that the parameter σ affects the participation rate of labor 2. The
participation rate will change more when σ is small and less if σ is large. This
parameter represents the degree of the income or wage distribution in this model. For
instance, when ε is positive, the discouraged worker effect becomes large in a
recession if the income difference is slight (σ is small), and in contrast to that, the
discouraged workers effect becomes small if the income difference is large.
This is because the number of so called the middle class households decreases when
the income variance becomes large, and the middle class households are the most
strongly affected by the change of the income. The choice of the middle class household
is the threshold that determines the participation rate of labor 2.
Ⅲ.
Concluding remarks
8
This paper has provided a theoretical analysis on the female labor supply and the
growth of productivity in a closed economy. The role of housework services is the key to
determine the female labor participation in the swings of the economy.
The discouraged workers effect is often observed in the Japanese economy and it leads
to a low unemployment rate during the past recessions. Recently in Japan, however,
the unemployment rate has risen gradually because of Japanese economy has
stagnated in the 1990s. This is partly because the labor participation rate, especially
the female labor participation rate, did not decline in the 1990s, that means, the
discouraged workers effect becomes smaller, different from the previous recessions.
According to the model provided in this paper, it might be possible to state that the role
of housework services becomes less substitutable for goods consumption. Although no
empirical analysis is provided, the theoretical framework provided in this paper will
help us to understand the female labor supply and the recent high unemployment rate
in Japan.
In addition, this paper shows that the income distribution affects the degree of the net
effect of the female labor supply. The net effect, either added workers effect or
discouraged workers effect becomes insignificant when the income variance is small.
There is an observation that income difference becomes large in recent Japan. It also
might be the reason why the female workers tend to stay in the labor market in
contrast to the tendency in the past recessions.
In the next paper, we will empirically examine high unemployment rate, the female
labor supply, and the change of the role of housework services in Japan. Further
analysis of these issues will be shown together in the next paper.
References
G. G. Cain (1966) Married Women in the Labor Force: An Economic Analysis, The
University of Chicago Press.
F. Hayashi and E. Prescott (2002) The 1990s in Japan: A lost decade, Review of Economic
Dynamics, 5
Y.
Higuchi
(2001)
Economics
of
Employment
Nihonkeizaishinbunsha Tokyo. (in Japanese)
9
and
Unemployment,
K. Matsuyama (1992) Agricultural Productivity, Comparative Advantage, and
Economic Growth, Journal of Economic Theory, 58.
OECD. 1993. Employment Outlook, Paris.
R. H. Topel. (1997) Factor Proportions and Relative Wages: The Supply-Side
Determination of Wage Inequality, Journal of Economic Perspectives, vol. 11, no.
2.
K. Wong (1995) International Trade in Goods and Factor Mobility, MIT Press.
A. Zabalza (1983) The CES Utility Function, Non-Linear Budget Constraints and
Labor Supply: Result on Female Participation and Hours, The Economic
Journal, 93.
10