The Society for Economic Studies
The University of Kitakyushu
Working Paper Series No.2014-8(accepted in March 30, 2015)

Do firms consider the health status of workers in maximizing
profits? Evidence from manufacturing industry

Kazuyuki Inagaki
Graduate School of Economics
Nagoya City University
1, Yamanohata, Mizuho-cho, Mizuho-ku, Nagoya, 467-8501, Japan

Kenji Azetsu
Faculty of Economics and Business Administration
The University of Kitakyushu
4-2-1, Kitagata, Kokuraminami-ku, Kitakyushu, 802-8577, Japan

Abstract
Firms may incur non-wage costs when workers work long hours, since increased
fatigue of workers leads to additional costs such as accidents at work and a loss of
output. This paper calls these costs “the costs of fatigue” and applies this idea to ameasure for the importance of the health of workers for firms. We develop the dynamic
labor demand model in which the costs of fatigue are parameterized. This model holds
in the Japanese manufacturing industry. We find that the costs of fatigue significantly
affect the decision making of firms, and firms prefer that workers work approximately
40 hours per week, which are the same as the standard working hours in Japan. Thesefindings suggest that profit-maximizing firms have an incentive to keep the demand forhours at a level that creates healthy work environments.

Keywords: fatigue of workers; labor demand; Euler equation

11. Introduction
Long working hours generally increase the fatigue of workers and, consequently,
lead to a deterioration in their health (Ruhm, 2000). This is an important issue for firms
as well as workers, since factors such as illness and accidents at work impose additionalcosts on firms (Pencavel, 2014; Ricci et al., 2007; Stewart et al., 2003). For example,
workers with fatigue have difficulty concentrating and are more likely to make mistakes
at work. 1 In this case, firms suffer losses such as an implicit loss of output and apayment of damages. Accidents at work are likely to induce a loss of or damage togoods and services produced by oneself or others. Similarly, a loss of output occurs forthe period during which firms and/or workers deal with accidents, since they cannot
devote sufficient time and effort to production. If accidents are caused by overwork,firms may be required to pay damages. Furthermore, firms have to pay higher healthcare costs to maintain the health of workers. For convenience, this paper calls thesecosts “the costs of fatigue”, which are incurred by firms and are caused by the lack of
the careful management of health at work.
As Pencavel (2014) points out, the prevalence of long working hours leads to not
only a reduction in labor productivity, but also an increase in errors, accidents, andillness at work that impose costs on firms. Although several previous studies haveexamined non-wage costs of the number of workers such as hiring costs (e.g., Oi, 1962),
non-wage costs of hours worked discussed above seem to be not fully examined1

For example, Rogers et al. (2004) indicated that, among hospital staff nurses, the riskof making an error significantly increased when their work shifts were longer than 12hours.
2

empirically. In this respect, Pencavel (2014) suggests that the importance of the healthof workers for firms is not a new argument, whereas it seems to have been neglected incontemporary economic models. 2
The profit-maximization behavior of firms depends on the cost structure. If thecosts of fatigue are high, firms do not prefer the prevalence of long working hours.
Firms consider the health status of workers in terms of profit-maximization, and may
have an incentive to keep the demand for hours at a level that creates healthy work
environments. 3 The adequacy of this hypothesis can be examined by checking theimportance of the health of workers for firms, which is measured by the costs of fatigue.
The purpose of this paper is to examine the costs of fatigue empirically. Ourinvestigation can provide useful insights into the subjects such as the health of workers,
the labor demand, and the creation of better work environments. For example, thisinvestigation is useful in assessing the effects of labor market policies that aim to reduceoverwork and lead to better work-life balance of workers, since such policies are
particularly effective in the case where the costs of fatigue are high.
An important issue is how to analyze the costs of fatigue. For example, the costs
include implicit factors such as accidents at work and the loss of output, but theirstatistics are usually not available (Hamermesh and Pfann, 1996). To cope with this2

The relationship between population health and economic growth has been examined

empirically (e.g., Bhargava et al., 2001). On the other hand, our argument is based on
the theory of the labor market.
3

Recent studies suggest that firms have a significant influence on the determination ofhours worked. For more details, see Section 2 of this paper.
3

problem, we use the Euler equation approach (Hansen, 2002), and the costs areparameterized in our model. Specifically, we construct a dynamic labor demand modelin which firms incur non-wage costs when workers work long hours, and the adequacyof this assumption is examined by the estimation of the Euler equation. Since our model
is derived as the solution to the profit-maximization problem, its estimation results canprovide information on the costs of fatigue on the basis of the profit-maximization
behavior of firms. Therefore, the adequacy of our hypothesis that profit-maximizing
firms pay careful attention to the health of workers in determining hours worked can be
examined within the framework of our model.
This paper is organized as follows: Section 2 briefly reviews several previous
studies. In Section 3, we develop our model and explain its estimation procedure.
Section 4 details the data. Section 5 presents the empirical results. Section 6 concludes
the paper.

2. Brief review of previous studies
The empirical analysis of this paper is related to some literature, including thehealth of workers, the demand for hours worked, and labor efficiency and productivity. 4
In this section, we briefly review several previous studies.
Many studies demonstrated that long working hours, increased fatigue, and the4

Although this paper gives weight to the labor demand, many studies have also been

conducted on the relationship between the labor supply and the health of workers. For
details, see Currie and Madrian (1999).
4

resulting deterioration in the health of workers impaired the performance of workers.
For example, Dawson and Reid (1997) compared the impact of sleep deprivation on
cognitive psychomotor performance with the impact of alcohol intoxication. Using thelaboratory data, they found that the performance after 17 hours of sustained wakefulness
decreased to the level equivalent to the performance with a blood alcohol concentrationof 0.05%, the proscribed level of alcohol intoxication in many countries.
Stewart et al. (2003) examined the impact of the depression of workers on thelabor costs for firms (absenteeism and poor performance of workers). They used thesurvey data to measure the impact. The costs of lost productive time (LPT) fromdepression was defined as a difference in LPT between workers with depression andthose without. They showed that the total annual costs of workers with depression were
US$31 billion higher than those without depression.
Similarly, Ricci et al. (2007) focused on the impact of the fatigue of workers on
the costs for firms. The costs of LPT due to fatigue and the other health-related issues
were measured by the national cross-sectional telephone survey of U.S. workers. They
found that the total annual costs of workers with fatigue were US$101.0 billion higher
than those without fatigue. It was also found that the prevalence of fatigue among
workers increased the other health-related costs.
In the literature on the demand for hours worked, several empirical studies
pointed out that firms had a significant influence on the determination of hours worked.
For example, Stewart and Swaffield (1997) stated that workers could not choose theirhours worked freely since job offers typically involved hours constraint. Theydeveloped the econometric model to examine the gap between actual and desired hours
for workers. Using the British Household Panel Survey, they showed that actual hours
5

worked per week for male manual workers were 4.3 longer than their desired hours.
Hamermesh and Trejo (2000) examined the impact of overtime penalty on thelabor demand. They applied the difference-in-difference estimator to their empiricalanalysis. Using the data from the Current Population Survey, they found that theextension of the daily overtime penalty to male workers in California substantiallydecreased overtime hours and the incidence of overtime workdays for those workers.
Kuroda and Yamamoto (2013) used the friction model developed by Stewart and
Swaffield (1997) and found that hours worked in Japan were partially determined bydemand-side factors. Furthermore, they pointed out that the demand for hours worked
was an increasing function of the user costs of labor such as hiring and training costs,
and that the prevalence of long working hours was a rational strategy for Japanese firms
to protect highly skilled workers from dismissal.
The characteristics of labor efficiency have not been fully examined empirically.
The following are a few notable examples. Lee (2003) estimated the efficient working
hour function in Korea. The function was specified as a polynomial of hours worked.
Using the semiparametric vector error correction approach, he showed that the mostefficient working hours per month ranged from 180 to 190. It was also found that a
reduction in hours worked did not create new employment.
Pencavel (2014) examined the relationship between output and hours worked formunition workers. He found a non-linear relationship. Output was proportional to hours
worked for the regime in which weekly hours worked were less than 49. However, forthe regime in which weekly hours worked exceeded 49, output increased at a decreasingrate as hours worked increased.

6

3. Model and Estimation Procedure
In this section, we develop our model based on the theory of the dynamic labordemand. This model is used to examine the importance of the health of workers forfirms, which is measured by the costs of fatigue. In the literature on the dynamic labordemand, it is common to include adjustment costs of labor inputs as key variables in themodel. In this paper, this basic specification is adjusted by including the costs of fatigue.

3.1. Costs of fatigue
An increase in hours worked leads to a decrease in leisure and sleeping time forworkers, and makes it difficult for them to pursue health-promoting activities such as
exercise (Ruhm, 2000). It is widely accepted that long working hours increase thefatigue of workers and consequently have a negative impact on their health. As
mentioned in Section 2, several previous studies indicate that the costs arising frompoor health of workers amounted to hundreds of millions of dollars. Along these lines,
we assume that the costs of fatigue depend on hours worked.
As Dawson and Reid (1997), Lee (2003), and Pencavel (2014) suggest, a serious
impact of the fatigue of workers appears when hours worked exceed a certain level.
Therefore, in this paper, the costs of fatigue per worker are specified as

G( H t ) =cL(H t − H * )2 ,2

(1)

where H t denotes hours worked per worker; H * , optimal working hours per worker;
7and cL , a positive parameter. This specification assumes that the costs of fatigue
increase at an increasing rate as hours worked exceed H * , and this assumption seems
to be reasonable in describing the impact of fatigue. Let N t denotes the number ofworkers. Since N workers are employed and they work H hours, G ( H t ) N t can be
regarded as the costs at the industry or firm level (Cahuc and Zylberberg, 2004). 5This specification is in line with previous studies on the labor demand.
Specifically, G ( H t ) N t can be rewritten as

G ( H t ) N t = aN t − (bH t − cH t2 ) N t ,(2)where a = cL ( H * ) 2 2 , b = cL H * , and c = cL 2 . The first term of the right-hand side ofequation (2) represents fixed non-wage costs such as social security (Ehrenberg andSmith, 2003). The second term represents labor efficiency (Cahuc and Zylberberg,
2004), and cH t2 can be regarded as a gross impact of fatigue per worker. For H t < H * ,fixed non-wage costs are offset by the improvement of labor efficiency. However, for

H t > H * , the fatigue of workers becomes apparent and labor efficiency decreases.
Therefore, the costs of fatigue increase at an increasing rate as hours worked exceed
H* .

Our main focus is on cL and H * since the costs of fatigue are characterized bythese structural parameters. In particular, cL determines the size of the costs of fatigue.
5

Since G ( H t ) increases consistently as hours worked exceed H * , overtime

premium seems to be reflected in this term to some extent.
8Therefore, the significance of cL suggests that the health status of workers, which ismeasured by the costs of fatigue, significantly affects the decision making of firms. In
other words, our hypothesis that profit-maximizing firms pay careful attention to thehealth of workers can be examined by the estimation of cL . 6

3.2. Euler equation
Our model allows for the existence of labor adjustment costs, since it is widely
accepted that employers must incur costs when adjusting labor inputs (e.g., Hamermesh
and Pfann, 1996). Examples of labor adjustment costs include search costs (costs ofadvertising, screening, and processing new employees), severance pay, negotiation costs,
and the time taken to readjust the schedule and pattern of production. 7 Let CH (∆H t )and C N (∆N t ) be the costs of adjusting hours worked and the number of workers,
respectively, where ∆H t = H t − H t −1 and ∆N t = N t − N t −1 . In this paper, we assume
that CH (∆H t ) and C N (∆N t ) have the following quadratic forms:

6

In this paper, we adjust the basic model of the dynamic labor demand by including

equation (1). If cL is zero, our modified model is reduced to the former one. Therefore,
the adequacy of our model can be examined by checking the significance of cL .
7

Negotiation costs may arise when hours worked are adjusted, since decreased hours

lead to a reduction in the salaries of workers and increased hours lead to a reduction in
the leisure time of workers.
9C H (∆H t ) =cH(∆H t ) 22

(3)

C N (∆N t ) =cN(∆N t ) 2 ,2(4)

where cH and cN are the positive parameters that determine the size of the laboradjustment costs. 8
Let Lt = H t N t be the effective labor force. The aggregate production function isusually given by F ( Lt , ε t ) , where ∂F ∂Lt > 0 and ∂ 2 F ∂L2t < 0 , and ε t denotes a
productivity shock observed at the beginning of period t .
A risk-neutral decision maker maximizes the present discounted value of profits( V ). The optimization problem is defined as

V ( H t −1 , N t −1 ) = max F ( Lt , ε t ) − wt Lt − G ( H t ) N t− C H (∆H t ) − C N (∆N t ) + β Ε t [V ( H t , N t )] ,(5)

where Ε t denotes the expectations at the end of period t ; β , the discount factor; andwt , the real wage per hour. The first-order conditions with respect to H t and N t
8This paper assumes symmetric labor adjustment costs since this assumption enables

us to more easily estimate equation (1), which is our main focus. Although our model
can be extended to the case where labor adjustment costs are asymmetric, we leave thisissue for future research. For example, see Azetsu and Fukushige (2009) and Inagaki(2012).
10

yield the following Euler equations:

M t N t − cL ( H t − H * ) N t − cH ∆H t + β Ε t [cH ∆H t +1 ] = 0(6)cL( H t − H * ) 2 − cN ∆N t + β Ε t [cN ∆N t +1 ] = 0 ,2(7)

M t Ht +where M t = F ′( Lt , ε t ) − wt . To estimate this model, the actual values of ∆H t +1 and∆N t +1 are substituted into equations (6) and (7). Then, these equations are rewritten as

M t N t − cL ( H t − H * ) N t − cH (∆H t − β ∆H t +1 ) = u1t(8)cL( H t − H * ) 2 − cN (∆N t − β ∆N t +1 ) = u2 t ,2(9)

M t Ht −where u1t and u2t are the error terms. Combining equations (8) and (9) yields thefollowing equation:∆H t − β ∆H t +1
∆N t − β ∆N t +1
1
= λ +α+ σ 2 + ut ,2
H t NtHt
Ht(10)

where λ = − cL 2cH , α = cN cH , σ = cL ( H * ) 2 2cH , and ut = u2 t cH H t2 − u1t cH H t N t .
To estimate our model, we rewrite equation (10) as

ht = λ + α nt + σ xt + ut ,

(11)

11

where ht = (∆H t − β ∆H t +1 ) H t N t , nt = (∆N t − β ∆N t +1 ) H t2 , and xt = 1 H t2 .

3.3. Estimation Procedure
Equation (11) is used to examine the costs of fatigue. When a specific value issubstituted into the discount factor β , equation (11) can be regarded as a linear
regression model that includes three variables, ht , nt , and xt . Therefore, using timeseries data on these variables, we can estimate the structural parameters λ , σ , and α .
This estimation procedure is relatively easy to implement since it is based on a linear
regression model and requires only two types of data, H t and N t , which are used to
calculate the data for ht , nt , and xt .
The estimation procedure of our model involves four steps. The first step involves
the estimation of equation (11). Our model holds if the following conditions are
satisfied: λ < 0 , α > 0 , and σ > 0 . To correct for endogeneity and serial correlationin equation (11), we use the generalized method of moments (GMM) developed byHansen (1982).
The second step involves the test for the significance of the structural parameters

λ , σ , and α . An important point to note is that the significance of λ and σ inequation (11) is directly related to the significance of cL in equation (1). Therefore, thesignificance of the costs of fatigue described by equation (1) can be checked by theestimation of λ and σ .
The third step involves the calculation of optimal working hours per worker ( H * ).
In the framework of our model, this parameter can be calculated as follows:

12

σ
= H*.
−λ

(12)

As mentioned in Section 3.1, the main focus of this paper is on cL and H * since thecosts of fatigue are characterized by these structural parameters. Therefore, the first
three steps of this estimation procedure are particularly important for our purpose.
The fourth step involves the investigation of the dynamic adjustment process of
the labor demand in response to shocks. This information is reflected in α , which is theratio of the costs of adjusting the number of workers ( cN ) to the costs of adjusting hours
worked ( cH ), given that ∆N t = ∆H t = 1 . In our model, the long-run equilibrium of hours
worked is given by H * . However, hours worked deviate from this equilibrium in the
short-run when shocks occur, since the existence of the adjustment costs prevents
instantaneous adjustment of hours worked in response to shocks. Therefore, thecondition that α < 1 implies that the adjustment of hours worked is slower than that of
the number of workers, but the condition that α > 1 implies that the adjustment ofhours worked is quicker than that of the number of workers.

4. Data
4.1. Use of industry-level data for Japan
Given that long working hours are prevalent in a particular industry, the use ofindustry-level data seems to be helpful in examining the costs of fatigue. In the case ofJapan, the Ministry of Health, Labour and Welfare (MHLW) reported that 72.4% of the13

workers in the manufacturing industry were tired from usual work. 9 Therefore, it islikely that the impact of fatigue can be well described by using the data for the Japanesemanufacturing industry.
Figure 1 around here
Figure 1 indicates the relationship between fatigue and hours worked in Japan. It
is clear that the impact of fatigue becomes more apparent as hours worked increase.
This characteristic is consistent with the assumption of our model and is in line withprevious studies on fatigue (e.g., Dawson and Reid, 1997). Therefore, Japanese data
seems to provide useful information on the costs of fatigue.

4.2. Data explanation
We use seasonally adjusted monthly data for the Japanese manufacturing industry(Major Group 9 of the Japan Standard Industrial Classification). In this paper, we focus
on Japan mainly because hours worked in Japan tend to be longer than those in othermajor countries (Kuroda and Yamamoto 2013). Therefore, the costs of fatigue are likely
to be more significant in Japan.
The data comprise of the regular employment index (2010=100) and the hours
worked index (2010=100). We use index data because data units are different. Both data
are constructed by the MHLW. The regular employment index is constructed from thedata on the number of regular employees. The hours worked index is constructed fromthe data on total hours worked (total of scheduled and non-scheduled hours worked) per

9

The data source is the Survey on State of Employees’ Health 1997. This survey does

not report this industry-level data on fatigue after 1997.
14

regular employee.
The sample period is from January 2002 to December 2013. This paper uses thedata from 2002 since the MHLW established new recognition criteria for brain andcardiac diseases that stem from long working hours in December 2001. The data sourceis the Monthly Labor Survey, and the time series data are downloadable from thewebsite of the Japan Institute for Labor Policy and Training (JILPT).

5. Empirical Results
Before estimating equation (11), it is necessary to examine the unit root propertiesof the variables included in equation (11) since we use time series data. In this paper, we
use the unit root test developed by Phillips and Perron (1988). To check the robustnessof our results, the discount factor β is set to 0.999, 0.950, and 0.900.
Table I around here
The results are reported in Table I. The null hypothesis of a unit root is rejected
for all the variables. Therefore, we can use the GMM technique, which corrects for bothendogeneity and serial correlation in equation (11).
Table II around here
Table II reports the estimation results of equation (11). The set of instruments
includes nt and xt with 1 and 2 lags and a constant term. Similar to the above, thediscount factor β is set to 0.999, 0.950, and 0.900 to check the robustness of ourresults. The main findings are summarized as follows.
First, the conditions that λ < 0 , α > 0 , and σ > 0 are satisfied, and all the15parameters are significant at the 5 percent level. These results are consistent with thetheoretical model presented in Section 3 and are not affected by the assumption of the
discount factor β . Furthermore, the validity of overidentifying restrictions is not
rejected, and the null hypothesis of weak instruments is rejected. These diagnostic tests
suggest that equation (11) is successfully estimated. 10
Second, we find that λ and σ are significant. As mentioned in Section 3, thesignificance of these parameters directly relates to the significance of cL in equation(1), which determines the size of the costs of fatigue. Therefore, the estimation results
suggest that the costs of fatigue significantly affect the decision making of firms. In
other words, profit-maximizing firms prefer that workers work approximately H *hours.
Third, we find that H * is approximately 100. As mentioned in Section 4, ourempirical analysis uses the hours worked index (2010=100), which is constructed fromthe data on total hours worked. The annual average of total hours worked per month in2010 is 161.5 for the manufacturing industry. 11 Given that one month includes four
work weeks, our results suggest that H * is approximately 40 per week. It isnoteworthy that this value is the same as the standard working hours in Japan. Since the10

We also examine a possible impact of the U.S. financial crisis in 2008 and the Great

East Japan Earthquake in 2011. Using the test developed by Hall and Sen (1999), we
find that the structural stability of our model is not significantly affected by these events.
The results are not reported in this paper but are available from the authors uponrequest.
11

The data source is the Monthly Labor Survey.
16

standard working hours are legally established so as to maintain the health of workers,
we may reasonably state that there is evidence in favor of the hypothesis thatprofit-maximizing firms pay careful attention to the health of workers.
Finally, we find that the costs of adjusting hours worked are lower than the costsof adjusting the number of workers. Since α is significantly larger than 1, our results
indicate that cH < cN . 12 This suggests that when firms decide to change labor inputs in
response to shocks, they prefer to adjust hours worked rather than the number ofworkers since the former adjustment is less expensive than the latter. Therefore, in the
short-run, actual hours worked quickly deviate from the long-run equilibrium H * . For
example, firms are likely to reduce hours worked remarkably in response to large
negative shocks. In this respect, our finding is consistent with the conclusion of Kuroda
and Yamamoto (2013), who pointed out that in Japan, a reduction in hours worked wasa useful buffer in protecting workers from dismissal.

6. Conclusion
This paper applies the idea of the costs of fatigue to a measure for the importanceof the health of workers for firms. We develop the dynamic labor demand model inwhich non-wage costs increase as hours worked exceed a certain level. Since our model
is derived as the solution to the profit-maximization problem, its estimation results canprovide information on the costs of fatigue on the basis of the profit-maximization

12

For example, in the case where the discount factor is set to 0.999, the t-value is 2.6

and the associated p-value is less than 0.05.
17behavior of firms. Our main findings are summarized as follows: (i) our model is
empirically supported; (ii) the costs of fatigue significantly affect the decision makingof firms; and (iii) firms prefer that workers work approximately 40 hours per week,
which are the same as the standard working hours in Japan.
These findings suggest that firms have an incentive to keep the demand for hours
at a level that creates healthy work environments. Therefore, this paper concludes thatfirms pay careful attention to the health status of workers in maximizing profits, at least
during the sample period 2002–2013. As mentioned in Section 4, the MHLW
established new recognition criteria for brain and cardiac diseases that stem from long
working hours in December 2001. Therefore, firms may have been required tosuccessfully manage physical and mental health issues at the workplace during thesample period.
Finally, we discuss some implications of our findings. We find that the health ofworkers is an important factor for profit-maximizing firms, and the optimal working
hours are the same as the standard working hours. Therefore, at least in the Japanesemanufacturing industry, long working hours are inefficient in the long-run from theviewpoint of the labor demand. However, hours worked tend to be longer in Japan thanin other major countries. A possible explanation for this fact is that the short-run
adjustment of labor inputs is achieved by making changes to hours worked. This isfound from our estimation results of the labor adjustment costs. Given that the costs ofadjusting the number of workers such as hiring costs are much higher than the costs offatigue, firms allow that long working hours are prevalent in the short-run. In thisrespect, our findings are consistent with the situation in Japan. Therefore, although thehealth of workers is a significant factor for profit-maximizing firms, they attempt to18

adjust hours worked in response to shocks even though the health of workers
deteriorates in the short-run.

Acknowledgments
We would like to thank Akira Yakita, Hideki Toya, Eiji Okano, Kenji Kondoh,
Akiyoshi Furukawa, Junya Masuda, Kunihiro Hanabusa, and Taiyo Yoshimi, andseminar participants at Nagoya City University and Chukyo University for their usefulcomments. We are solely responsible for any errors.

19

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Table I
Unit root test

xt
ht

nt

–3.192*β = 0.999

β = 0.950

β = 0.900

––

––

––

–88.155**–85.467**–82.426**–32.227**–28.864**–23.546**

The unit root test includes a constant term. The spectral estimation method used is the
Bartlett kernel, and the bandwidth parameter is selected by the Newey and West (1994)
procedure. ** and * indicate significance at the 1% and 5% levels, respectively.

23

Table II
Estimation results of equation (11)β = 0.999

β = 0.950

β = 0.900

λ–0.329*[0.162]–0.355*[0.171]–0.375*[0.173]α2.828**[0.711]2.501**[0.702]2.360**[0.713]σ

3.326*[1.670]3.605*[1.758]3.813*[1.776]

H*

100.546100.776

100.8392.352(0.308)2.528(0.283)2.710(0.258)24.40223.69922.432(A) Parameter estimates(B) Diagnostic tests
J-statistic

Cragg-Donald F-statistic

The estimate of λ and its standard error is multiplied by 1,000 for convenience.
Numbers within square brackets are heteroskedasticity and autocorrelation consistent(HAC) standard errors. The spectral estimation method used is the pre-whitened Bartlett
kernel with a vector autoregression (VAR) model. The bandwidth parameter is selectedby the Newey and West (1994) procedure. ** and * indicate significance at the 1% and
5% levels, respectively.
The J-statistic is the test for the validity of overidentifying restrictions, and numbers
within parentheses are p-values. The Cragg-Donald F-statistic is the test for the null
hypothesis of weak instruments (Cragg and Donald, 1993), and the critical values aretabulated in Stock and Yogo (2002).

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Figure 1
Relationship between fatigue and hours worked in Japan
84

80

76

72

68

64

60
>6

6-7

8-9

7-8

9-10

10<

The horizontal axis shows hours worked per day, and the vertical axis shows thepercentage of the workers who are tired from work. The data source is the Survey on
State of Employees’ Health 1997.

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