Unemployment and Workplace Safety in a Searchand Matching Model
Masaru Sasakiy
January 30, 2009

Abstract
This paper develops a model with search to examine the determinants of theamount of capital purchased by a …rm for workplace safety and investigates a relationship between unemployment and the incidence of work-related injury or illness.
Productivity improvement encourages …rms’ entry and therefore lowers the unemployment rate but instead raises the employment rate, including the fractionof absent workers (scale e¤ ect). On the other hand, productivity improvement
encourages …rms to buy more capital for workplace safety (capital e¤ ect). Thisbiases the distribution of employed workers toward non-injured workers, lowering
the fraction of absent workers. These two e¤ects determines the relationship between unemployment and the incidence of work-related injuries or illnesses. The
empirical exercises con…rm that the scale e¤ect had been dominant over the capital e¤ect over the sample period in the US; that is, the unemployment rate andthe ‡ow rate of work-related injured workers moved in the opposite direction, butthat a movement in the same direction is observed only from the late 1980s to themid 1990s, so called the period of economic slowdown after the burst of "Bubble
Economy" in Japan.
JLE Classi…cations: J64, J65, J81
Keywords: Job Search, Workplace Safety, Work-Related Injury andIllness, and Unemployment

I thank Jun-ichiro Ishida, Jun-nichi Goto, Fumio Ohtake, Kei Sakata, Kengo Yasui and the participants in FEMES 2008 and the seminars of the Kansai Labor Economics Workshop and Policy Modelling
Workshop (PMW) at GRIPS. All errors belong to me. This paper was substantially modi…ed the previous version entitled, "The E¤ect of Disability Insurance on Occupational Safety and Health" written
in Japanese. Part of this research is …nancially supported by the Japan Society for the Promotion of
Science, No. 18730164.
yInstitute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka, 5670047, Tel: +81-6-6879-8560, Fax: +81-6-6878-2766, E-mail: sasaki@econ.osaka-u.ac.jp.

11Introduction

This paper develops a model with search to consider …rms’determinants of the amountof capital for workplace safety and to explore a relationship between unemployment andthe incidence of work-related injury or illness. Assuming that the probability of a worker
being injured or taken ill at work sites depends negatively on the amount of capital for
workplace safety purchased by …rms, there is trade-o¤ for the …rms between the cost of
its capital and the risk of losing a worker.
According to Robinson (1988) and Poteet and Didonato (2001), the number of workrelated injuries is positively associated with employment size. They argued that during an
economic boom in which labor demand exceeds labor supply, …rms hire even inexperienced
workers who are more likely to be injured at the work sites, and therefore that both
employment size and ‡ow of absent employed workers because of work-related injuries
are large. In other word, the unemployment rate and the ‡ow rate of absent employedworkers are negatively correlated, controlling for the labor force.
However, Ussif (2004) undertook an international comparative study using time-series
data between 1970 and 1999 from several countries and found an opposite relationship;that is, as employment size increased, the number of work-related injuries decreased. In
other words, the unemployment rate and the ‡ow rate of absent employed workers are
positively correlated, controlling for the labor force. Additionally, he found movementsof these two rates in the same direction if a time trend was controlled, which turns out tobe consistent with the evidence from Robinson (1988)and Poteet and Didonato (2001).
Ussif (2004) concluded that the number of work-related injuries had declined because of
the technical advancement of workplace devices and environments captured by the timetrend.
Using time-series data from the US and Japan, the …rst graphs of Figures 1 and 2
present movements of relative deviations of the unemployment rate and the fraction ofwork-related injured workers from their corresponding trends detrended by the Hodrick
and Prescott …lter. For the US, the relative deviations moved in the opposite direction
over the sample period (1980-2001), implying a negative correlation between the unem2

ployment rate and a ‡ow rate of work-related injured workers. Although not as clearlyas the movement in the US, the same pattern can be seen in Japan except the late 1980s
and the mid 1990s, following the burst of "Bubble Economy", but during which the relative deviations moved in the same direction. It means a positive correlation between theunemployment rate and the ‡ow rate of work-related injured workers.
These results showed that during the stagnation period after the burst of the bubble
economy in Japan, the number of work-related injuries was not determined only by thee¤ect of employment size. We have not so far considered other elements determining thesize of injuries at the work site, but to do so is of importance from policy makers’pointof view as well as academic economists. There are many other factors a¤ecting the rateof work-related injury, including employer practice at the work site, employee training,
the role of unions, the technical advancement of work goods and environments as pointed
out by Ussif (2004), the provision of mandates for safety, and investment in safety. This
paper focuses attention on the last element among them, that is, investment in safetyand develops a model that endogenizes the probability for a worker being subject to awork-related injury or illness in which a …rm decides how much capital to purchase forworkplace safety. Then we explore the optimal decision of …rms that determines thenumber of injuries at the work site in response to exogenous shocks. Our contribution isto provide a new dimension to explain the relationship between the unemployment rate
and the incidence of work-related injury or illness by incorporating the determinants ofcapital for workplace safety into a search-matching model. This paper does not discuss
the role of mandates to keep workplaces safe by, for example, the Occupational Safety
and Health Administration (OSHA) in the United States and its e¤ect on labor market
conditions.1
There are a few theoretical studies in this …eld. Holmlund (2005) presented a modelwith individual search behavior and a decision on sickness absence with the framework of
the stochastic utility function of sickness and analyzed the impact of social insurance on
a worker’s labor supply decision. He focused on an individual worker’s decision on labor
1

Jolls (2008) surveyed both theoretical and empirical studies on the e¤ects of OSHA and compensation
programs of work-related injuries.

3

supply and sickness absence. Our paper instead focuses on the determinant of the amountof capital purchased by …rms for workplace safety to reduce the risk of losing employedworkers because of work-related injuries or illnesses. One of the main …ndings in Holmlund(2005) is that an increase in sickness bene…ts raises the value of participating in the labor
force, thereby resulting in an increase in employment size. Engström and Holmlund(2007) extended to a general equilibrium model with search by incorporating absencefrom work as an additional state.2 They derived the optimal compensation package tomaximize the expected pro…t a¤ected by the number of job applications and sick workers’
determinants of absence from work under the condition that accidents randomly arrived
at workers. They provided the welfare analysis and compared with alternative social
insurance policies.
Empirical studies in this area have thus far explored the e¤ect of OSHA on workrelated injuries using state-level, industry-level or plant-level data from the US.3 Overall,the e¤ect of OSHA enforcement on the rate of work-related injuries was modest in the
US (Viscusi 1979 1986, Bartel and Thomas 1985). In contrast, Scholz and Gray (1990)
found a signi…cant relationship between OSHA enforcement and the rate of work-related
injuries using plant-level data of …rms that were frequently inspected. According to therecent study by Mendelo¤ (2005), its signi…cant relationship was observed in the early
1990s but it disappeared afterward.
Our …ndings are summarized below. Productivity improvement encourages …rms toenter the labor market, which makes more competitive for …rms to hire a worker. Firms
that buy capital for workplace safety before meeting unemployed workers are then discouraged from doing so. On the other hand, productivity improvement leads to an increasein pro…t. In an environment in which …rms have to recompense for a loss from a worker’s
absence, an increase in pro…t implies an increase in the opportunity cost that …rms would
have incurred if their workers had been injured or taken ill, which induces the …rms topurchase more capital for workplace safety to reduce the risk of work-related injury or
2

Barmby et al. (1994) presented a model in which the wage is endogenously determined within ane¢ ciency wage setting. They showed that the wage is adjusted to a¤ect the decision on sickness absence.
Garibaldi and Wasmer (2001) also built a multistage model where the wage is endogenously determined.
3
Smith (1992) surveyed empirical studies in the early 1990s.

4

illness. Which e¤ect is more dominant is theoretically ambiguous. If the capital elasticity of the injury rate is su¢ ciently inelastic, the letter e¤ect is more dominant overthe former e¤ect, in case of which productivity improvement raises the amount of capitalfor workplace safety and then lowers the fraction of absent employed workers. This isreferred to as the capital e¤ect. The amount of capital for workplace safety determines
the distribution of employees between active and absent employed workers, but does nota¤ect the unemployment rate.
There is another e¤ect of productivity improvement through the labor market conditions. Productivity improvement encourages …rms’ entry as mentioned before, whichraises labor market tightness and thereby lowers the unemployment rate. The larger
employment size raises the number of workers who are absent from work because ofwork-related injuries or illnesses, holding the transition rate of being injured or taken ill…xed. This is referred to as the scale e¤ect.
Overall, productivity improvement unambiguously lowers the unemployment rate but
provides an ambiguous e¤ect on the fraction of absent employed workers to the labor
force. If the scale e¤ect is dominant over the capital e¤ect, the unemployment rate
and the fraction of absent employed workers move in opposite directions in response to
productivity improvement. However, in a case in which the capital e¤ect is dominant, thesame decreasing trends in the unemployment rate and the fraction of absent employedworkers are observed as productivity improves.
The same argument can be applied to the ‡ow of absent employed workers as well. In
a case in which productivity improvement leads to an increase in the amount of capitalfor workplace safety, the transition rate from active employment to absence is lower, thus
resulting in the lower ‡ow rate of absent employed workers, holding the employment
rate …xed (capital e¤ect). On the other hand, the larger employment rate led to by…rms’ entry raises the ‡ow rate of absence employed workers (scale e¤ect) holding thetransition rate of being injured or taken ill …xed. If the scale e¤ect is dominant over thecapital e¤ect, the unemployment rate and the ‡ow rate of absent employed workers arenegatively correlated as argued by Robinson (1988) and Poteet and Didonato (2001). In

5

the reverse case, these two rates are positively correlated, which is consistent with Ussif(2004).
The amount of capital for workplace safety purchased by …rms is a key variable that
determines the relationship between unemployment and the incidence of work-related injury or illness. The puzzling empirical results could be partially explained by endogenizing
the determinant of capital for workplace safety.
The model is extended to analyzing e¤ects of various policy parameters (replacement
rate, disability insurance payment and premium, and unemployment bene…t). Amongthem, the e¤ect of unemployment bene…ts is particularly noteworthy. To the best of
our knowledge, unemployment bene…t and workplace safety have been thus far discussed
separately. We here show the strong linkage in policy between them. An increase in theunemployment bene…t raises a worker’s reservation wage and thereby the wage because
the state of unemployment is more attractive for workers. It implies an increase in the costthat …rms would have incurred if their workers had been injured or taken ill at the worksite. Therefore, …rms are encouraged to increase capital for workplace safety to reduce
the likelihood of injury or illness at the work site. However, there is the opposite view;that is, the increasing wage lowers pro…t gained by …rms, which discourages them frompurchasing capital for workplace safety. Alternatively, a decrease in the pro…t encourages…rms to exit, and less competitiveness then provides …rms with incentives to raise capitalfor workplace safety because the probability of meeting a worker is higher. If the positivee¤ects are dominant over the negative one, we recognize the novel result that an increasein unemployment bene…t raises the amount of capital for workplace safety and thereby
lowers the likelihood that employees are injured or taken ill at the work site.
We compare the e¤ects on the ‡ow rate of employees who become absent from work
because of work-related injuries through changes in capital for workplace safety (capitale¤ect) and labor market tightness (scale e¤ect), using time-series data from the US andJapan.
The next section presents a matching model with an endogenous determinant of capital for workplace safety. Section 3 illustrates comparative statics exercises and discusses

6

a determinant of a relationship between unemployment and the incidence of work-related
injury or illness by productivity improvement. The e¤ects of various social insurance programs are analyzed in Section 4. Some evidences are presented in Section 5 to con…rm
implications obtained from the model. The …nal section provides concluding remarks.

2

The Model

We consider a continuous-time model with matching in which there are a continuumof risk-neutral workers and a continuum of risk-neutral …rms. The measure of workers
is normalized to one. Workers are in…nitely lived and homogeneous with respect topreferences to work. At any moment, a worker is either unemployed, employed or absent
from work because of work-related injury or illness. An employed worker is injured or
taken ill at the work site and absent from work at a Poisson rate (k), where k represents
capital, with its price normalized to one, purchased by a …rm to improve conditions ofworkplace safety. Work-related injuries and illnesses are de…ned here by immediate healthhazard in the course of work that forces workers to be absent from work for treatment
such as lower-back pain, cuts, bruises, broken bone, falls, struck by objects, mental illness
and so on, but not by long-term latent health hazard from work such as pneumoconiosis.
Following Acemoglu and Shimer (1999), a …rm that creates job vacancies through freeentry decides on how much capital k to buy for each job vacancy before meeting an
unemployed worker.4 We deal with a situation in which devices for workplace safety
are durable enough to exceed an employment duration of a …rst worker. An idea of ex
ante investment in capital for workplace safety is then more favorable.5 A construction
company installs handrails and safety net for falls at a construction site before hiring
workers. A manufacturing company complies an instruction manual for workplace safety
and health and purchases safety machines and devices when constructing a factory. These
items are ususally in long-lasting use. We assume that4is characterized by

0( ) < 0;Acemoglu and Shimer (1999) determined how much capital to buy for production in a search model.
In the reverse case, that is, if the devices for workplace safety are not durable enough, it is favorable
to assume that capital for workplace safety is purchased ex posit.
5

7

00

( ) > 0; (0) =1 and limk!1 (k) =0. As a …rm buys capital to improveconditions of workplace safety, the probability of an employed worker being injured or
taken ill is reduced at a decreasing rate.
There is search and matching friction. The unemployed and job vacancies are matchedrandomly according to a matching function, m(u; v) where u is the number of the unemployed and v is the measure of job vacancies across all …rms. The matching function
is assumed to exhibit constant returns to scale, implying that the rate at which a vacancy encounters an unemployed worker is computed by m(u; v)=v = m(u=v; 1)
where

q( )

v=u is labor market tightness, while the rate at which an unemployed worker

matches with a job vacancy is represented by q( ): Note that q( ) is decreasing in ;that is, q 0 ( ) < 0.
A job is destroyed at an exogenous Poisson rate , and then a worker becomes unemployed and begins to search for a job. Note here that a …rm loses even a vacancy when
the job is destroyed, which is di¤erent from the standard model with matching. Bothworkers and …rms discount the future at the common rate r.
Various value functions are developed below. We begin with the value for an employedworker of engaging actively in work as follows:rW (w) = w + (k)[Wa (w)W (w)] + [U

W (w)]:(1)

The instantaneous utility is linear with earnings. The second term on the right-handside of equation (1) represents the expected capital loss incurred by being injured or
taken ill, and the third term indicates the expected capital loss of being unemployed. Ina similar manner, the value for an employed worker of being absent from work becauseof work-related injury or illness is de…ned by:rWa (w) = w + [W (w)

Wa (w)] + [U

Wa (w)]:(2)

For convenience, a worker and …rm have a contract that if being absent from work due
to injury at the work site, the worker is compensated in full for her earning w. The absent

8employed worker heals and returns to work at an exogenous Poisson rate .6 Note that
disutility incurred by the absent employed worker is ruled out in this model without lossof generality. Equations (1) and (2) thus show that the employed worker is indi¤erent
between working and absence from work due to work-related injury or illness because
the absent worker is fully compensated. Therefore, these values can be expressed in astandard form:rW (w) = rWa (w) = w + [U

W (w)]:

A worker’s surplus is then calculated by:

W (w)

U=

w rU
:
r+

(3)

The value of unemployment is given as usual by:

rU = q( )[W (w)

(4)

U ]:

At any moment, an unemployed worker, who is assumed to receive no instantaneous
utility, meets a …rm with a vacant job at the transition rate q( ).
Next we discuss value functions for a …rm. We consider a …rm to be a collectionof individual jobs. At any time in point, jobs are either occupied, un…lled, or inactive
because employed workers are absent due to work-related injuries or illnesses. We assume
that …rms operate under constant-returns-to-scale production technology with respect tolabor input. This assumption assures that jobs are independent of one another.
The value of a job being occupied and active can be expressed as:rJ(w; k) = p

w + (k)[Ja (w; k)J(w; k)]J(w; k):(5)

A matched pair produces p instantaneously. The second term on the right-hand side of6

In fact, the rate of return to work is not exogenous. It depends largely on the amount of compensations as well as the extent of injury or illness if it is di¢ cult to observe whether or not absent workers heal
from injury or get well from sickness. Meyer, Viscusi, and Durbin (1995) undertook a natural experimentand found that an increase in compensations received by absent employed workers extends the durationof absence. Similar results are obtained in Ehrenberg (1988) and Krueger (1990).

9equation (5) represents the expected capital loss of a job being inactive because a worker
is absent from work owing to work-related injury or illness. The third term indicates the
expected capital loss of a job being destroyed.
Similarly, the value of an occupied job being inactive because of work-related injury
or illness is:

rJa (w; k) =w + [J(w; k)Ja (w; k)]Ja (w; k):(6)

We then assume that there are no disability insurance programs, and therefore that the…rm has to compensate the absent worker for her/his wage according to the contract.
It implies that renegotiation over the wage is not allowed between a worker and a …rm
after the injury or illness shock. We believe that this assumption is acceptable. In
reality, many …rms join the federal or state disability insurance programs with compulsory
payroll deductions. If own employees are injured or taken ill at the work site, they arecompensated through its program.7 Because the disability insurance program is mainly…nanced by …rms, however, it is interpreted that …rms indirectly bear the burden ofcompensation. Even though own employees become absent from work because of injury
or illness that occurs out of the work site, many …rms provide them with absence leave
with payments. According to the survey conducted by Japanese Ministry of Health,Labour and Welfare (MHLW) in January 2008, 58.6% of surveyed Japanese …rms haveown absence leave programs, and 41.1% of them keep paying average 85.8-93.6% of salaries
to absent employees. Putting down to surveyed …rms over 1,000 employees, 85.3% haveown absence leave programs, and 56.8% pays 88.5-91.8% of salaries to absent workers.8
The job turns out to be active at the rate . Equations (5) and (6) give the following
equation:J(w; k) =(r ++ )p (r + + + (k))w:(r + )(r + + + (k))7(7)

We consider these e¤ects on capital for workplace safety and labor market tightness later
See Rodo Sinbun (Labour Newspaper) No. 2688 (July 14, 2008) in Japanese published by Rodo
Sinbunsha.
810The value of a vacancy is given by:rV (k) = q( )[J(w; k)

V (k)](8)

V (k):

A vacancy is …lled at the transition rate q( ), but destroyed at the rate , similar toan occupied job.
The free entry condition ensures that the value of a vacancy equals k in equilibrium(V (k) = k). Using equation (8), the labor demand equation is thus given by:k=q( )J(w; k):r + + q( )(9)A …rm and a worker consummate a match if and only if the joint surplus gained
through this match is nonnegative, and then they share the joint surplus according to
the Nash bargaining rule. Assuming that the worker’s share of the surplus is de…ned by
2 [0; 1], we have:(1)[W (w)

U ] = [J(w; k)(10)V (k)]:

Because renegotiation over the wage is not allowed after the injury or illness shock,the wage is determined, taking account of the possibility of a worker being injured in the
future. Substituting equations (3) and (7) into equation (10) gives:

w=

r+ +
p
r + + + (k)(r + )k + (1r+ +p is the expected productivity, and (r +r+ + + (k))rU:)k is the cost of capital for work-

place safety, accounting for the future discount rate r and the rate of destroying . According to the Nash bargaining rule, the wage is determined by the weighted-averageof the expected pro…t (the …rst parentheses on the right-hand side) and the reservation
wage rU . From equations (4), (9) and (10), the reservation wage rU can be expressed asrU = 1(r + )k . Substituting this into the above wage equation yields:

11

w=

r+ +
p
r + + + (k)

(1

(11)

)(r + )k :There are three e¤ects of capital for workplace safety k on the wage. An increase ink raises the expected productivity r+ r++ ++ (k) p because of the lower likelihood that theemployed worker is injured or taken ill at the work site and absent from work.9 (r+ )k is
the total cost of capital for workplace safety per unemployed worker incurred by the …rm.
An increase in k provides the worker with extra bargaining power, directly re‡ecting thehigher wage. These two e¤ects therefore lead to an increase in the wage. On the other
hand, an increase in k raises the cost of capital (r + )k and thereby lowers the expected
pro…t, leading to a decrease in the wage. As seen in the wage equation (11), if

1;

that is, the measure of vacancies is larger than the number of unemployed workers, thepositive e¤ects are unambiguously dominant over the negative e¤ect.
Substituting this wage equation (11) into (7) gives the value for an occupied job of
being active:J(k; ) =(1)(r + + )p(1)(r + )k+:(r + )(r + + + (k))
r+

The …rst term on the right-hand side shows that an increase in capital for workplacesafety k raises the expected productivity and thereby the value for an occupied job of
being active. The second term shows that an increase in k lowers the value of an occupied
job if1 because the higher wage is burdensome for …rms. If< 1, an increase in kstrictly raises the value of an occupied job. The free entry condition can be rewritten by
substituting this into equation (9):q( )(1)(r + + )p(1)(r + )k+= k:
r + + q( ) (r + )(r + + + (k))
r+
9

(12)

Suppose that renegotiation is allowed after the injury or illness shock. The wage for an active worker
is computed by w = [p (1)(r + )k] while the wage for an absent worker because of injury orillness is w =(1)(r + )k: As we would expect, the wage is higher when the renegotiation isallowed than when it is not. The wage for an absent worker turns to be negative because …rms have torecover the cost for ex ante investment in capital for workplace safety from the absent worker. Unlessthe expected capital gain of the absent worker by recovering from injury or illness to the work site issu¢ ciently large, it is better o¤ for the absent worker to quit and become unemployed.

12

We move to the problem regarding the optimal choice of capital for workplace safety.
A …rm chooses the optimal capital for workplace safety per job vacancy to maximize theexpected value of a vacancy:

maxV (k)k =) max

k

k

q( )
J(k; )
r + + q( )k:The …rst-order condition yields:q( )r + + q( )(1)(r + + ) 0 (k)p+ (1(r + )(r + + + (k))2(13)) = 1:The left-hand side term represents the expected marginal value of an occupied job
with respect to the capital for workplace safety while the right-hand side term is its
marginal cost, that is, the price of capital normalized to one.
According to the second-order condition, the slope of the marginal value of an occupied
job described above is computed as:@ 2 V (k)
=
@k 2

If

00

q( )
@ 2 J(k; )q( )=r + + q( )@k 2
r + + q( )00(k)(r + + + (k)) 2( 0 (k))2:(r + + + (k))3(k)(r + + + (k))

(1

)(r +
r+

+ )p

2( 0 (k))2 > 0 for all k, then this is su¢ cient to show thatthe optimal capital level maximizes the value of a vacancy.10
Next, steady-state conditions are illustrated to derive the unemployment rate and therate of absent employed workers. Let u and a denote fractions of unemployed workers andemployed workers who are absent from work because of work-related injuries or illnesses,
respectively. The steady-state conditions require …rst of all that the in‡ow rate to theunemployment pool equals the out‡ow rate from it, and secondly that the in‡ow rate to
the absent pool equals the out‡ow rate from it,
10

If (k) = 1=(1 + k), for example,

00(k)(r +

+ + (k))13

2( 0 (k))2 = 2(r ++ )=(1 + k)3 > 0

q( )u = (1ua) + a = (1and ( + )a = (k)(1

u

u);

a):

Then we obtain:u =and a =;+ q( )(k)+ + (k)(14)q( )+ q( ):

The fraction of employed workers who are actively engaged in work is therefore computed by:

1

u

a=

+
+ + (k)q( )+ q( )As one would expect, the higher labor market tightness

(15)

:

lowers the unemployment

rate, u but raises the fraction of absent employed workers a because the number of
employed workers is larger. An increase in capital for workplace safety k lowers thefraction of absent employed a because workplace safety conditions are improved, but kdoes not a¤ect u. A change of k just in‡uences the distribution between active and absent
employed workers.
The nature of equilibrium is characterized by the free entry condition, the …rst-ordercondition to determine the optimal level of capital, and the steady-state conditions. Equations (12), (13) and (14) provide a complete description of equilibrium to solve for thevector (k; ; u; a). For convenience, these equilibrium conditions are summarized below.
(i) First-order condition (equation (12))
q( )r + + q( )(1)(r + + ) 0 (k)p+ (1(r + )(r + + + (k))214) = 1;

(ii) Free entry condition (equation (13))(1)(r + + )p
q( )
+ (1
r + + q( ) (r + )(r + + + (k)))k = k;and (iii) Steady-state conditions (equation (14))

u =and a =;+ q( )(k)+ + (k)q( )+ q( ):

We investigate the characterizations of the equilibrium by examining the comparative
statics in the next section.

3

The Relationship between Unemployment and theIncidence of Injury or Illness

Our concerns focus on a relationship between unemployment rate and the incidence of
work-related injury or illness via exogenous parameter changes. We …rst pick up productivity p as a key parameter. This exercise helps provide an explanation about the
puzzling results obtained from data …ndings.
Does productivity improvement encourage …rms to raise the amount of capital for
workplace safety as well as to enter the labor market? The main purpose in this subsection
is to illustrate changes in workplace safety and labor market conditions in response to
productivity improvement. Using (i) …rst-order conditions and (ii) free entry conditions,the comparative statics system is described. The appendix section shows the analyticaldetails.
Proposition 1 The comparative statics analysis provides the following characterizations:dkd
Q 0; and
> 0:
dp
dp
15

As one would expect, an increase in productivity p raises labor market tightness. More …rms enter the labor market and create vacancies because the productivity
improvement leads to an increase in pro…t, thus resulting in an increase in . If

= 1;that is, there is no bargaining power over the wage determination for …rms, …rms do nothave incentives to enter the labor market despite the fact that productivity improves,and therefore d =dp = 0.
An increase in productivity p has two e¤ects on the optimal capital for workplacesafety k. On the one hand, because more vacancies created newly due to productivity
improvement make it more di¢ cult for …rms to …nd unemployed workers, …rms that buy
capital for workplace safety k before matching are discouraged from doing so.11 On theother hand, an increase in productivity implies an increase in pro…t that …rms would haveearned without workers’absence from work. It therefore encourages …rms to increase theircapital for workplace safety k to prevent employed workers from being injured or taken ill
at the work sites. Which e¤ect is dominant is theoretically ambiguous. According to thecomparative statics analysis shown in Appendix, we …nd that the positive e¤ect on capitalis more likely to dominate as the injury rate (k) is more inelastic; that is, as an marginal
decrease in (k) is larger, …rms are encouraged to buy more capital for workplace safety k
to reduce the likelihood that accidents occur. If= 1, both opposite e¤ects are canceled

out, and therefore, productivity improvement does not a¤ect the amount of capital for
workplace safety (dk=dp = 0).
We next consider the e¤ect on the unemployment rate and the fraction of workers who
are absent from work because of work-related injuries or illnesses. Using (iii) steady-state
conditions (equation (14)), the total derivatives of these equations are taken with respect
to p:
du
=
dp
and

|

[ + q(@ q( )
@

)]2
{z

d
6 0;
dp
}| {z }

(16)

+

11

In a case that …rms buy capital for workplace safety after hiring (ex post investment in safety), thisnegative e¤ect of productivity on capital for workplace safety disappear.

16

da
=
dp

( + ) 0 (k)[ + + (k)]2
|
{z
+
|

(k)
+ + (k)q( )+ q( )dk
dp}| {z }

(17)+=

[ + q({z+)]2@ q( )@d:dp}| {z }+The term on the right-hand side in equation (16) and the second term on the righthand side in equation (17) show that higher labor market tightness caused by productivity
improvement lowers the unemployment rate u but raises the fraction of absent employedworkers a. The entry of more …rms raises the likelihood for unemployed workers ofobtaining a job and thereby the employment rate. An increase in the employment rate
results directly in an increase in the fraction of absent employed workers because of workrelated injuries or illnesses, holding the transition rate of work-related injury or illness…xed. This is referred to as the scale e¤ect.
Looking at the …rst term on the right-hand side in equation (17), in a case thatthe productivity improvement raises the amount of capital for workplace safety k; thefraction of absent employed workers is lower. This is because an increase in k reduces thelikelihood that employed workers are injured or taken ill at the work sites and absent fromwork (k). In the opposite case, productivity improvement leads to the higher fraction ofabsent employed workers through a decrease in the amount of capital for workplace safety.
This e¤ect through capital for workplace safety is referred to as the capital e¤ect. It causes
a transfer between active employed workers (1ua) and absent employed workers ain the distribution of employees, holding the transition rates between employment andunemployment …xed. As seen in equation (16), there is no capital e¤ect on u. Its e¤ect
a¤ects only the distribution of employees between active workers and absent employedworkers.
The implications are summarized below:
Proposition 2 As productivity improves,(a) (scale e¤ect) the unemployment rate is lower,
but the fractions of absent and active employed workers are higher; (b) (capital e¤ect 1)
17

if …rms are encouraged to purchase more (less) capital for workplace safety, the fractionof absent employed workers is lower (higher), but the fraction of active employed workers
is higher (lower); and (c) (capital e¤ect 2) there is no e¤ect on the unemployment rate
through a change of capital for workplace safety.
Although the scale e¤ect of productivity improvement on u is unambiguously negative,
the combined capital and scale e¤ects on a cannot be determined with certainty if dk=dp >0 from equation (17). If the capital e¤ect is dominant over the scale e¤ect, the overalle¤ect of productivity is negative on a. Therefore, u and a are positively correlated throughproductivity improvement. On the other hand, if the scale e¤ect is dominant over thecapital e¤ect, u and a are negatively correlated. What are the factors determining themagnitude relation between the capital and scale e¤ects? One of them is the elasticity of(k); that is, as (k) is inelastic with respect to k, it is more likely that the capital e¤ect
dominates the the scale e¤ect, thus resulting in the positive relationship between u anda. These implications are summarized.
Proposition 3 Suppose dk=dp > 0. As productivity improves, (a) if the capital e¤ect isdominant over the scale e¤ect, the unemployment rate, u and the fraction of workers who
are absent from work because of work-related injuries or illnesses, a move downward in thesame direction; and (b) in the reverse case, the unemployment rate, u moves downward
while the fraction of workers who are absent from work because of work-related injuries,
a moves upward.
We have so far paid attention on the stock of injured employed, but our concern shifts
to an impact of production on the ‡ow of injured employed and its relationship with
unemployment rate. It has been argued by policy makers and OSH specialists that the‡ow of employed who become injured or ill at the work sites is positively associated withthe employment size according to the scale e¤ect. This view was supported by Robinson(1988) and Poteet and Didonato (2001). In other words, it implies that the ‡ow rate of
employed workers who become injured or ill and the unemployment rate are negatively
correlated, holding the labor force …xed.
18However, Ussif (2004) showed an opposite view that despite a steady increase in thenumber of employed workers, the number of work-related injuries had declined from 1970
to 1999 using time-series data from selected countries (Canada, Finland, France, US, andSweden). That is, he implied a positive relationship between the unemployment rate andthe ‡ow rate of employed workers who become injured.
What are those arguments omitting to be unable to explain this relationship between
the unemployment rate and the ‡ow rate of absent employed found in the data? Ourmodel incorporating the capital e¤ect through a determinant of capital for workplacesafety k in the search and matching framework helps provide some implications consistent
with the data. Capital and scale e¤ects on u and the ‡ow rate of absent employed workers
(k)(1

u

) alike illustrate a di¤erent relationship between them.

According to equation (15), the ‡ow rate of absent employed workers because ofwork-related injuries or illnesses, y can be expressed as,

y = (k)(1u++ + (k)a) = (k)q( )+ q( ):

Then, the total derivative of this ‡ow equation are taken with respect to p:

dy
=
dp

0

|

(k)

+ (k)
|

+
+ + (k){z++ + (k)2q( )+ q( )dk
dp}| {z }

(18)+=

[ + q({z+)]2@ q( )@d:dp}| {z }+The …rst term on the right-hand side of equation (18) shows the capital e¤ect; that is,
in a case that …rms buy more capital for workplace safety k in response to productivity
improvement (dk=dp > 0), the probability of being injured or taken ill at the work site(k) is lower, resulting in a decrease in the ‡ow rate of absent employed workers becauseof work-related injuries or illnesses to the absent employed pool. The same argument canbe applied in the reverse case. The second term reveals the scale e¤ect; that is, as more19

…rms enter the market and create vacancies due to productivity improvement, the ‡ow
rate of absent employed workers increases because of the increasing number of employedworkers.
Con…ning our argument to dk=dp > 0, if the scale e¤ect is dominant over the capitale¤ect, the productivity improvement leads to a decrease in the unemployment rate u but
an increase in the ‡ow rate of absent employed workers (k)(1 u a): These two rates arenegatively correlated, as argued by Robinson (1988) and Poteet and Didonato (2001). If
productivity is pro-cyclical, this implication is consistent with the empirical …nding from
Arai and Skogman Thoursie (2005) showing that the extent of sickness absence is strongly
pro-cyclical. If the scale e¤ect is instead dominant, throughout a change in capital for
workplace safety k in response to a productivity shock u and (k)(1ua) alike move

downward in the same direction. This implication is consistent with the data …nding
from Ussif (2004). The sign of y depends entirely on which e¤ect is more dominant, thecapital e¤ect or the capital e¤ect. The next proposition summarizes:
Proposition 4 Suppose dk=dp > 0. As productivity improves, (a) if the capital e¤ect
is dominant over the scale e¤ect, the unemployment rate, u and the ‡ow rate of workers
who are absent from work because of work-related injuries or illnesses, (k)(1ua)

move downward in the same direction; and (b) in the reverse case, the unemployment
rate, u moves downward while the ‡ow rate of workers who are absent from work becauseof work-related injuries or illnesses, (k)(1ua) moves upward.

In Section 5, some evidences are provided.

4

Applications

This section presents the e¤ects of various policy parameters: replacement rate, disability
insurance payment, disability insurance premium, and unemployment bene…ts. These
impacts illuminate …rms’incentives for determinants of job creation and the amount of
capital for workplace safety in response to policy changes. Here is with the emphasis on
the impact of unemployment bene…ts on workplace safety. It appears that the linkage
20between unemployment bene…ts and workplace safety has not at least thus far beenargued. This comparative statics exercise contributes to illuminating its linkage.

4.1

The Extended ModelAn employed worker who is actively engaged in work instantaneously earns w as heretofore, but an employed worker who is absent from work because of work-related injury orillness is recompensed not fully but partially for her/his loss by w where

represents

the replacement rate ( 2 (0; 1)). Additionally, an unemployed worker now receives theunemployment insurance bene…ts z. The value functions for a worker are modi…ed by:rW (w) = w + (k)[Wa (w)W (w)] + [U

W (w)];rWa (w) = w + [W (w)

Wa (w)] + [U

Wa (w)];

andrU = z + q( )[W (w)

(19)

U ]:

We assume that the compensation for work-related injury or illness w exceeds theunemployment insurance bene…ts z, which is more realistic. Since workers are riskneutral, unemployment bene…ts are considered just as a subsidy. In a similar manner,the value functions for a …rm are:rJ(w; k) = p(1 + t)w + (k)[Ja (w; k)J(w; k)]J(w; k);rJa (w; k) =w+ w

Ja (w; k)]Ja (w; k);

tw + [J(w; k)

andrV (k) = q( )[J(w; k)
21

V (k)]

V (k):(20)

Firms that hire workers take out disability insurance and pay the insurance premium
tw. A …rm has to recompense an injured worker for her/his loss by part of the wage w
but receives w in disability insurance bene…ts. We assume 1 >>> t > 0; thatis, the disability insurance program does not fully cover a …rm’s loss, and the insurance
premium is lower than the insurance payment.
Using equations (19) and (20) and the free entry condition (V (k) = k), the wage issolved according to the Nash bargaining rule:

w=

[(r +

+ ]p (1[(1 + t)(r +)(r + )(r + + + (k))k] + (1)[r + + + (k)]z:+ ) + (k)(t +)] + (1)[r + + + (k)]Be aware that this wage equation is reduced to equation (11) in a case of= t = z = 0. Increases in the replacement rate

= 1 andand the disability insurance premium

rate t encourage …rms to exit the labor market, thus lowering labor demand and thereby
the wage. In contrast, an increase in the disability insurance payment rate

raises labordemand by encouraging …rms’entry, leading to an increase in the wage. For simplicity,
we consider the take-it-or-leave-it wage ( = 0) from now on12 :

w=

[r +
r+

+ + (k)]z:+ + (k)(21)

This is the same as the reservation wage. An increase in capital for workplace safety,
k lowers the wage. An increase in k lowers the likelihood that an employee is injured
or taken ill at the work site and raises the expected value of being employed relative tothe value of being unemployed. The state of employment is more attractive, and …rms
are allowed to capture the whole surplus produced by matched pairs, thus resulting indecreases in the reservation wage and thereby the wage13 .
12

The holdup problem does not arise because there is no bargaining power over the wage determinantion
for workers.
13
If = 1 as in the basic model described in Section 2, the wage is reduced to z and does not dependon the amount of capital for workplace safety, k. Because the value for a worker being actively employed
and the value of being absent from work due to work-related injury or illness are the same in this case,the likelihood of being injured or taken ill at the work site, which depends on k, does not a¤ect thereservation wage.

22

From equations (20) and (21), we obtain the value of a job being occupied and active:

J(k) =(r + + )p(r + )[r + + + (k)][(1 + t)(r + + ) + (k)(t +(r + )[r + + + (k)]

)]z

:

(22)

Similarly to Section 2, the nature of equilibrium is characterized by (i) the …rst-ordercondition (V 0 (k) = 1), (ii) the free entry condition (V (k) = k) and (iii) the steady-state
conditions:(i) the …rst-order condition
q( )J 0 (k)r + + q( )1 = 0;

q( )J(k)r + + q( )k = 0;(ii) the free entry conditionand (iii) Steady-state conditions

u =
a =

;
+ q( )(k)+ + (k)q( )+ q( ):The …rst-order and the second-order conditions are presented in the appendix section.
The steady-state conditions are the same as those derived in Section 2. The comparative
statics studies show the characterizations of the equilibrium in response to the changesof policy parameters.

4.2

Comparative Statics E¤ects

We next explore the e¤ects of policy parameters (replacement rate, disability insurance
payment and premium, and unemployment bene…ts) on capital for workplace safety and23

labor market tightness.
According to the comparative statics analysis, we obtain the following results:
Proposition 5 (a) replacement ratedkd
7 0 and7 0;dd(b) disability insurance payment ratedkd
< 0 and> 0;dd(c) disability insurance premium ratedkd
> 0 and< 0;dtdt
and (d) unemployment bene…tddk
7 0 and< 0:dz
dz
The appendix section shows analytical details.
(a) Replacement Rate
There are two e¤ects ofon k andto be considered: various direct e¤ects andindirect e¤ects via w. We begin the direct e¤ects. As the replacement rate

is higher,…rms have to bear the more burden of the recompense for losses from work-related injuries
and illnesses w, which encourages …rms to exit the labor market, lowering . Less
competitiveness induces …rms to buy more capital for workplace safety because it is more
likely for …rms to meet unemployed workers. At the same time, it implies an increase in
the cost that …rms would have incurred if accidents had occurred to employed workers
at the work sites. Firms thus have more incentives to increase the amount of k to reduce
the likelihood of being injured or taken ill at the work site.

24

The indirect e¤ects show the opposite story. An increase inlowers the wage accord-

ing to equation (21). As absent employed workers are more su¢ ciently recompensed fortheir losses, the state of employment is more attractive for workers. In a case in which
workers are allowed to have power over wage bargaining, unemployed workers then accept
the lower wage. Firms bear the less burden of the recompense for losses from work-related
injuries and illnesses w, encouraging …rms to enter and thereby raising . More severe
competitiveness discourages …rms from buying capital for workplace safety. A decrease in
the wage lowers the cost that …rms would have incurred if accidents had occurred, which
also discourages …rms from buying capital for workplace safety. In addition, an decreasein the wage raises the pro…t, and therefore …rms are induced to raise capital for workplace safety because of an increase in the marginal return to capital for workplace safety.
Putting di¤erently, an increase in the pro…t encourages …rms’entry, and an increase in
the number of rivals lowers the amount of capita for workplace safety.
(b) Disability Insurance Payment Rate
An increase in the disability insurance payment rate

reduces the burden of therecompense for losses from work-related injuries and illnesses, which therefore induces…rms to lower the amount of k and encourages …rms’entry to the market. An increase injob vacancies due to …rms’entry discourages …rms from purchasing capital for workplacesafety.
(c) Disability Insurance Premium Rate
An increase in the disability insurance premium rate, t implies an increase in the costthat …rms would have incurred if accidents had occurred, which thereby induces …rms
to increase the amount of capital for workplace safety to prevent accidents at the worksites from the value of remaining an occupied job inactive because the worker is absent.
Putting di¤erently, an increase in t lowers the pro…t, which instead induces …rms to lower
the amount of capital for workplace safety because of a decrease in its marginal returnfrom the value of remaining an occupied active. According to the comparative static
analysis shown in Appendix, the former e¤ect is dominant over the latter e¤ect.
Firms are encouraged to exit the labor market by an increase in the recompense for25

workplace injury and illness and a decrease in the pro…t. Less competitiveness allows incumbent …rms to provide more chances to meet unemployed workers. Ex ante investment
in workplace safety is less likely to be idle, therefore providing …rms with the incentive
to purchase more capital for workplace safety k.
(d) Unemployment Bene…t
As seen in standard matching models, an increase in unemployment bene…ts z lowers
labor market tightness . The intuition behind this result is that an increase in z raises
the reservation wage of workers and thereby the wage, which encourages …rms to exit.
The linkage between unemployment bene…ts and workplace safety is considered here.
The comparative statics study shows that capital for workplace safety k is positively
or negatively associated with unemployment bene…ts z. According to equation (21), an
increase in z raises the wage. Because the state of unemployment is more attractive forworkers, unemployed workers then raises their reservation wage. It implies an increasein the cost that …rms would have incurred if their employed workers had been injured
or taken ill at the work sites and been absent from work because the compensation ispositively proportional to the wage ( w). Firms have more incentives to increase capitalfor workplace safety to reduce the likelihood of being injury or taken ill at the work site.
However, an increase in the wage implies a decrease in the pro…t, lowering the expected
gain of investing in workplace safety. Therefore, it leads to a decrease in the amount of
capital for workplace safety. Putting oppositely, a decrease in the pro…t discourages …rms
from entering the labor market, and less competitiveness provides …rms with incentives
to purchase capital for workplace safety. If the positive e¤ects on k are dominant over thenegative e¤ect, the novel result is that the unemployment bene…ts have a positive e¤ect
on capital for workplace safety, implying a decrease in the likelihood of being injured
or taken ill at the work site. The unemployment bene…ts encourage …rms to improve
working conditions.
(e) Capital and Scale E¤ects
How do these policy parameters a¤ect the unemployment rate and the fraction ofworkers who are absent from work because of work-related injuries or illnesses? The total
26

derivatives of equation (14) are taken with respect to x = f ; ; t; zg:
du
=
dx

and

da
=
dx

[ + q( )]2{z|( + ) 0 (k)[ + + (k)]2
|
{z
+
|

@ q( )@q( )+ q( )(k)+ + (k)[ + q(
{z

}

)]2

d
dx

}

;

dk
dx
@ q( )@+

x = f ; ; t; zg:

}

d
dx

;

Comparative statics e¤ects are summarized in Table 1. The scale e¤ects of the disability insurance premium rate t and unemployment bene…ts z are unambiguously positiveon the unemployment rate u but negative on the fraction of absent employed workers
a. The disability insurance payment rate
respectively. The replacement rate

has the opposite scale e¤ects on u and a,

has ambiguous scale e¤ects on both u and a.

Table 1: Summary of Comparative Statics E¤ects

Capital E¤ect
Scale E¤ectaau

The capital e¤ect of(a) Replacement
Rate(b) Disability
Insurance
Payment Rate(c) Disability
Insurance
Premium Rate(d) Unemployment
Insurance

?
?
?

+
+
-

+

?
+

on the fraction of absent employed workers a is unambiguously

positive through an decrease in capital for workplace safety k. The opposite capital e¤ect
of t is then obtained on a. The capital e¤ects ofand z are ambiguous on a because

it is unclear whether …rms raise or lower the amount of capital for workplace safety inresponse to increases in these parameters.

27

5

Simple Empirical Evidence

In this section, we attempt to show that, although indirectly, implications of a relationshipbetween unemployment and the number of work-related injured workers are supported bysimple empirical studies. Because the amount of capital for workplace safety purchasedby …rms cannot be observed, there is no way but the capital e¤ect is identi…ed indirectlyfrom a comparison of the e¤ects of labor productivity on unemployment and the numberof work-related injured workers.
For visual inspections, we compare relative variations of unemployment rate, thenumber of workers who have to be absent from work for non-fatal injury to the labor force,and labor productivity per person with respect to the corresponding trends detrended by
the Hodrick and Prescott …lter method. Figures 1 and 2 display relative deviations ofthese variables from their trends, using time-series data from two countries, the US andJapan, and Table 2 shows correlation coe¢ cients of these variables over the whole sample
period and by sub-periods.
Table 2: Correlations of Unemployment Rate, Labor Productivity andFraction of Injured Workers
unemployment unemployment
injuryvs. injury
vs. productivity vs. productivity
USoverall-0.790-0.9060.800
Japan
overall-0.484-0.942-0.4801980-1988-0.820-0.981
0.812
1989-1995
0.556-0.856-0.1291996-2006-0.784-0.9230.679
Figure 1 shows graphs using the data from the US covering from 1980 to 2001. The
data of unemployment rate and the labor force are obtained as usual from the Labor Force
Survey, the US Department of Labor, and the ‡ow rate of work-related injured workers
is de…ned by the number of non-fatal reported injuries occurred in the private sector
divided by the labor force where the data of non-fatal reported injuries are obtained fromthe Labor-Related Establishment Survey, the US Department of Labor.
The …rst graph of Figure 1 displays that the unemployment rate and the ‡ow rateof work-related injured workers are negatively correlated over the sample period (-0.790
28

from Table 2). The second graph compares relative deviations of the unemployment rateand labor productivity de…ned by GDP per employed person evaluated at the 1990 US$
at PPP from KILM, ILO. As one would expect, the correlation between these variables isnegative over the sample period (-0.906 from Table 2), which strongly supports a view that
productivity improvement raises job opportunities, thereby lowering the unemployment
rate. The third graph of Figure 1 makes the same comparison of the relative deviationsof labor productivity and the ‡ow rate of work-related injured workers. The correlation
between labor productivity and the ‡ow rate of work-related injured workers is positive
over the sample period (0.800 from Table 2); that is, as productivity improved, the ‡ow
rate of work-related injured workers had risen. These results imply that the scale e¤ectof labor productivity is dominant over the capital e¤ect: as labor productivity improves,more jobs are created, and therefore employment rate rises, which leads to an increase in
the number of workers who become injured. We predict that an increase in the numberof work-related injured workers is attributable mainly to the larger employment size in
the US.
Figure 2 shows relative deviations of the variables using the Japanese data from 1980
to 2006. We observe di¤erent patterns in movements of the deviations between the period
after the burst of the bubble economy (the late 1980s and the early 1990s) and otherperiods. The data source for unemployment rate and the labor force is the Labour Force
Survey, the Japanese Ministry of Internal A¤airs and Communications (MIC). The ‡ow
rate of work-related injured workers is de…ned by the number of non-fatal injury workers
who become absent from work for four days or more, divided by the labor force. The
data of non-fatal injury cases come from the Labour Inspectorate Records, the Japanese
Ministry of Health, Labour and Welfare (MHLW).
The correlations between the unemployment rate and the ‡ow rate of work-related
injured workers except the stagnation period after the collapse of the bubble economy(from the late 1980s to the mid 1990s) are similar to that of the US; that is, these two
deviations moved in the opposite direction. On the other hand, an interesting observation is that these relative deviations moved in the same direction during the stagnation29

period after the burst of the bubble economy, implying a positive correlation between
the unemployment and the ‡ow rate of work-related injured workers. The same results
are con…rmed from Table 2; that is, the correlation coe¢ cients on 1980-1988 and 19962006 are negative (-0.820 and -0.784), but the one on the period of economic slowdown
after the collapse of the bubble economy (1989-1995) becomes positive (0.556). Labor
productivity, de…ned in a similar way with that of the US from the same data source,
and the unemployment rate move oppositely over the sample period, which is the sameas in a case of the US. The correlation between labor productivity and the ‡ow rate ofwork-related injured workers is di¤erent during the period of economic slowdown after the
burst of the bubble economy from during the other periods; that is, as labor productivity
improved, the ‡ow rate of work-related injured workers increased as well before the late1980s and after the mid 1990s, but the opposite pattern is seen following the collapse of
the bubble economy. Table 2 exhibits that the correlation coe¢ cients on 1980-1988 and1996-2006 are positive (0.812 and 0.679), but that the one is negative between 1998 and1995 (-0.129).
This implies that similar to the US, the scale e¤ect is dominant except the period of
weak economy after the burst of the bubble economy, but during which the capital e¤ect
is dominant over the scale e¤ect. We predict that increasing productivity encouraged…rms to purchase more capital for workplace safety to reduce the number of work-related
accidents following the collapse of the bubble economy. It might be interrupted that the
number of work-related injured workers would rather be determined by the amount of
capital for workplace safety than by the employment size from the late 1980s to the mid
1990s.
We estimate equations for a ‡ow rate of work-related injured workers and the unemployment rate to reinforce the results obtained from visual inspections. The purposeof this empirical works is to, although indirectly, extract the capital e¤ect in the sensethat productivity improvement encourages …rms to purchase capital for workplace safety,
leading to a decrease in the number of work-related injured workers.
Tables 3 and 4 present empirical results from the US and Japan, respectively. The

30

AR(1) method is used for any estimations because the null hypothesis of no autocorrelation is signi…cantly rejected. As the independent variables for the estimate of a ‡ow
rate of work-related injured workers, the one-period lagged unemployment rate, the oneperiod lagged productivity and a time trend term are chosen. To control simultaneous
bias, the one-period lagged variables are used here. Inclusion of the one-period lagged
unemployment rate allows us to control the scale e¤ect of productivity via the unemployment rate, which means that the coe¢ cient on the one-period lagged productivity
represents the capital e¤ect on the ‡ow rate of work-related injured workers. To explore
the scale e¤ect of productivity, the unemployment rate is estimated using the one-period
lagged productivity.
Table 3: Estimates of Growth Rates of Work-Related Injured Workers andthe Unemployment Rate (US)
(1)
(2)
AR(1)
AR(1)
ln(injury rate)ln(unemployment rate)
Coef.
Std. Err.
Coef.
Std. Err.
ln(unemployment rate(-1))
0.126(0.239)(-)
1.618(1.478)-5.449(0.703)ln(productivity(-1))-0.043(0.026)0.084(0.015)
Trend
Constant-19.283(13.960)
50.938(6.887)
Value
P-value
Value
P-Value
DW Statistic
0.976
0.579
Alt. DW Statistic (F value)
4.764(0.044)10.498(0.005)
Breusch-Godfrey (F-value)
4.818(0.043)8.017(0.012)F value250.25(0.000)232.54(0.000)
2
R
0.978
0.963
Sample2121* 1%, ** 5%, *** 10% signi…cant.
According to the …rst column of Table 3, the coe¢ cient on productivity is positiveand insigni…cant, which implies that the capital e¤ect is trivial in the US. This result
is consistent with the visual inspection from Figure 1. Looking at the second column,
productivity has a signi…cantly negative impact on the unemployment rate, which is theone we would expect from Figure 1. However, the impact of the unemployment rate on
the ‡ow rate of work-related injured workers is statistically insigni…cant, so I am afraid
to say that the capital e¤ect of productivity through a channel of the unemployment rate31

is weak on the ‡ow rate of work-related injured workers.
Table 4: Estimates of Growth Rates of Work-Related Injured Workers andthe Unemployment Rate (Japan)
(1)
(2)
AR(1)
AR(1)
ln(injury rate)ln(unemployment rate)
Coef.
Std. Err.
Coef.
Std. Err.
ln(unemployment rate(-1))-0.278(0.060)(-)
ln(productivity(-1))-1.518(0.211)-3.628(0.651)
Trend-0.007(0.006)0.103(0.016)
Constant
8.143(1.837)30.551(6.189)Value
P-value
Value
P-Value
DW Statistic
1.083
0.358
5.637
(0.027)34.239(0.000)
Alt. DW Statistic (F value)
5.502(0.029)15.829(0.001)
Breusch-Godfrey (F-value)
2244.34(0.000)157.81(0.000)F value
2
R
0.997
0.932
26
26
Sample* 1%, ** 5%, *** 10% signi…cant.
Results are di¤erent for Japan from for the US. According to the …rst column of Table
4, the coe¢ cient on productivity is signi…cantly negative; that is, productivity improvement lowers a ‡ow rate of work-related injury workers. This result is consistent with thetheoretical implication of the capital e¤ect described in the previous section. From thesecond estimation, productivity has a signi…cantly negative e¤ect on the unemployment
rate, and in addition, the impact of unemployment is signi…cantly negative on the ‡ow
rate of work-related injured workers. These results imply that the scale e¤ect of productivity through a channel of unemployment is signi…cantly positive on the ‡ow rateof work-related injured workers. Productivity improvement not only created more job
vacancies to enlarge the employment size but also induced …rms to purchase more capital for workplace safety to reduce the probability of losing a productive worker due towork-related injury. This mechanism operates both an increase and decrease in the ‡ow
rate of work-related injured workers.

32

6Concluding Remarks

This paper allows for …rms’ decisions on the amount of capital for workplace safety,
and trade-o¤ between its cost and the risk of employed workers being absent from work
because of work-related injuries or illnesses.
By incorporating …rms’ decisions on the amount of capital for workplace safety ina search and matching model, we investigated the relationship between unemployment
and the incidence of work-related injury or illness. Productivity improvement encourages…rms’entry and then raises labor market tightness. It lowers the unemployment rate but
instead raises the employment rate, including the fraction of absent employed workers.
This scale e¤ect implies a negative relationship between the unemployment rate and thefraction of absent employed workers. However, productivity improvement changes …rms’
behavior toward a determinant of the amount of capital for workplace safety. Potential …rms are induced to enter the market as productivity improves, which lowers thelikelihood that …rms meet unemployed workers. Firms are thereby discouraged frompurchasing capital for workplace safety ex ante. On the other hand, it is possible that
productivity improvement induces …rms to buy more capital for workplace safety becauseof an increase in the opportunity cost of losing workers due to work-related injuries orillnesses. This capital e¤ect determines the distribution of employed workers between active and absent employed workers but does not a¤ect the unemployment rate. The signof the relationship between the unemployment rate and the ‡ow rate of absent employedworkers depends on which is more dominant, the scale e¤ect or the capital e¤ect. The
determinant of the amount of capital for workplace safety helps provide an explanation
about the puzzling results regarding the relationship between the unemployment rate andthe number of absent employed workers who are injured or taken ill suggested by Ussif(2004).
Using the time-series data from the US and Japan, the empirical exercises showedthat the scale e¤ect had been dominant over the capital e¤ect over the sample period
in the US; that is, the correlation between unemployment rate and a ‡ow rate of workrelated injured workers is negative, but that the one turns out to be negative only from33

the late 1980s to the mid 1990s, following the collapse of "Bubble Economy" in Japan.
Additionally, to distinguish these two e¤ects, we estimated equations for the ‡ow rateof work-related injured workers and the unemployment rate. In a case of Japan, as wewould expect, productivity improvement negatively a¤ects the number of worked-related
injuries but yields a positive e¤ect on it through a channel of the unemployment rate.
We now brie‡y discuss social e¢ ciency. According to this model setting, …rms buy
capital for workplace safety before meeting workers in their search activity, and therefore
it is well-known that ex ante investment in workplace safety causes a hold-up problem asmentioned by Acemoglu and Shimer (1999). It is expected that the optimal capital level
for workplace safety in the decentralized environment is lower than the one solved by
the social planner14 . It implies that the enforcement of mandates of workplace safety by
OSHA is justi…ed to improve social welfare in an environment where a hold-up problem
arises.

References[1] Acemoglu, D. and R. Shimer (1999) “Holdups and E¢ ciency with Search Frictions,”
International Economic Review, 40(4): pp. 827-49.
[2] Arai, M. and P. Skogman Thoursie (2005) “Incentive or Selection in Cyclical Absenteeism,”Labour Economics, 12: pp. 269-80.
[3] Barmby, T., J. Sessions, and J. Treble (1994) “Absenteeism, E¢ ciency Wages andShirking,”Scandinavian Journal of Economics, 96: pp. 561-66.
[4] Bartel, A. P. and L. G. Thomas (1985) "Direct and indirect e¤ects of regulation: a
new look at OSHA7s impact," Journal of Law and Economics, 28(1): pp.1-25.
14

The hold-up problem does not arise in a case of ex post investment in workplace safety. Because
there still exists a search externality, however, the optimal amount of capital for workplace safety solvedin the model with ex post investment is not usually socially e¢ cient.

34

[5] Ehrenberg, R. G. (1988) “Workers’Compensation, Wages, and the Risk of Injury,”
in John F. Burton, Jr., ed., New perspectives in workers’compensation. Ithaca, NY:
ILPR Press: pp. 71-96.
[6] Engströn, P. and B. Holmlund (2007) "Worker Absenteeism in Search Equilibrium,"
Scandinavian Journal of Economics, 109(3): pp.439-67[7] Garibaldi, P. and E. Wasmer (2001) “Labor Market Flows and Equilibrium Search
Unemployment,”IZA Discussion Papers, No. 406.
[8] Gray, W. B. and J. M. Mendelo¤ (2005) "The Declining E¤ects of OSHA Inspections
on Manufacturing Injuries, 1979-1998," Industrial and Labor Relations Review, 58(4):pp.571-87.
[9] Holmlund, B. “Sickness Absence, Search Unemployment and Social Insurance,”
mimeo.
[10] Joll, C. (2008) "Employment Law," in K. Arrow and M. Inrilligator, ed., Handbookof Law and Economics, Vol. 2, Ch. 17: North-Holland.
[11] Krueger, A. B. (1990) “Incentive E¤ects of Workers’Compensation Insurance,”Journal of Public Economics, 41(1): pp. 33-50.
[12] Meyer, B. D., W. Kip Viscusi, and D. L. Durbin (1995) “Worker’ Compensation
and Injury Duration: Evidence from a Natural Experiment,” American Economic
Review, 85(3): pp. 322-40.
[13] Poteet, C., and T. Didonato (2001) Journal of Workers’Compensation, pp. 72-85.
[14] Robinson, J. C. (1988) "The Rising Long-Term Trend in Occupational Injury Rates,"
American Journal of Public Health, 78(3): pp.276-81.
[15] Smith, R. S. "Have OSHA and workers’compensation made the workplace safer? in
D. Lewin, O. S. Mitchell, and P. D. Sherer, ed., Research Frontiers in Industrial Relations and Human Resources: pp.557-86, Industrial Relations Research Association,
Madison.
35

[16] Scholz, J. T., and W. B. Gray (1990) "OSHA enforcement and workplace injuries:
a behavioral approach to risk assessment," Journal of Risk and Uncertainty, 3(3):pp.283-305[17] Ussif, A-A. (2004) “An International Analysis of Workplace Injuries,”Monthly Labor
Review, March: pp.41-51.
[18] Viscusi, W. K. (1979) "The impact of workplace safety regulation," The Bell Journalof Economics, 10(1): pp.117-40.
[19] Viscusi, W. K. (1986) "The impact of workplace safety regulation, 1973-1983" The
RAND Journal of Economics, 17(4): pp.567-80.

AppendicesComparative Statics E¤ect (Section 3)
In this appendix, we use comparative statics analysis to explore the e¤ects of productivity p on capital for workplace safety k and labor market tightness , using the…rst-order condition (equation (13)) and the free entry condition (equation (12)). The
comparative statics system is given by:
2

32

dk
dp

3

2

3

6 A11 A12 7 6
7 6 B1 7
4
54
5=4
5;
d
A21 A22B2
dp
where:

A11 =

(1

)(r +
r+

00

+ )p

(k)(r + + + (k)) 2( 0 (k))260(r + + + (k))3from the second-order condition,

A12 =

+

(r + )q 0 ( )< 0;[q( )]236

(1)(r + + ) 0 (k)p+ (1(r + )(r + + + (k))2

A21 =

)

r + + q( )
= 0;
q( )

from the …rst-order condition,

A22 =

k+

(r + )q 0 ( )k 6 0;[q( )]2(1)(r + + ) 0 (k)
B1 =
6 0;
(r + )(r + + + (k))2
and

B2 =

(1
)(r + + )6 0:
(r + )(r + + + (k))

The Jacobian determinant is rA

dk
=
dp

and

1
rA

+
|

A12 A21 > 0. Then we …nd:

A11 A220(k)k + r + + + (k)(r + )q 0 ( )(1)(r + + );2
(r + )(r + + + (k))
r + + + (k)[q( )]
{z
}|
{z
}
{z
}|
+

+=

d
A11 B2 A21 B1
=
> 0:
dp
rA
As the elasticity of (k) ( 0 (k)k= (k)) is much lower, it is more likely that dk=dp > 0:Comparative Statics E¤ect (Section 4)
The …rst-order condition and the free entry condition are rewritten as:

J 0 (k)r + + q( )
= 0;
q( )

37

and J(k)r + + q( )k = 0;q( )where :

J(k) =(r + + )p(r + )[r + + + (k)][(1 + t)(r + + ) + (k)(t +(r + )[r + + + (k)])]z:

Since J 0 (k) > 0, we recognize that the marginal value of vacancy is positive for anyk (V 0 (k) > 0). If00(k)(r +

2( 0 (k))2 > 0 for any k, J 00 (k) < 0, which is+ + (k))su¢ cient to show that the optimal capital level maximizes the value of a vacancy.
The comparative statics analysis is provided below to investigate the e¤ects of thepolicy parameters:
2

32

3

2

3

6 C11 C12 7 6 dk 7 6 D11 D12 D13 D14 7
4
54
5=4
5
d
D21 D22 D23 D24
C21 C22dwhere:

C11 = J 00 (k) < 0
from the second-order condition,

C12 =C21 = J 0 (k)(r + )q 0 ( )< 0;[q( )]2

r + + q( )=0
q( )from the …rst-order condition,

38

d

dt dz

;

C22 =

D11 =0(k)(r + + )z
r+

t(r +

+ + (k)) + 2 (k)((1(r + + + (k))3(k)[t(r + + ) + (k)(t(r + + + (k))2

z
r+

D21 =

(r + )q 0 ( )k
< 0;
[q( )]2

)]

7 0;

0

D12 =

(k)(r + + )z
> 0;
(r + )(r + + + (k))2

D22 =

(k)z
(r + )(r + + +
0

(k))

< 0;

D13 =

(k)(r + + )(1(r + )(r + + +D23 =(r + + + (k))z> 0;(r + )(r + + + (k))2

D14 =assuming

)t

0

)z
< 0;
(k))2

(k)(r + + )((1(r + )(r + + +

6= (1

)t
)
> 0;
2
(k))

)t;

and

D24 =
assuming(1 + t)(r + + ) + (k)(t +(r + )(r + + + (k))

6= t + :

39

)

> 0;

)

7 0;

The Jacobian determinant is rC

C11 C22

cations described in proposition 5.

40

C12 C21 > 0. We then obtain the impli-

Figure 1 US
Relative Deviation of Unemployment Rate and Injury
Rate
0.1

0.3
0.2
0.1
0
-0.11979
-0.2
-0.3

0.05
0
1984

1989

1994

1999

-0.05
-0.1
-0.15

unemployment

injury rate

The left vertical line measures unemployment rate.
The right vertical line meaures flow rate of injured workers.

Relative Deviation of Unemployment Rate andProductivity
0.3
0.2
0.1
0
-0.11979
-0.2
-0.3

1984

1989

unemployment

1994

1999

0.04
0.02
0
-0.02
-0.04
-0.06

productivity

The left vertical line measures unemployment rate.
The right vertical line meaures labor productivity..

Relative Deviation of Injury Rate and Productivity
0.1

0.04

0.05

0.02

0
-0.051979

0
1984

1989

1994

1999

-0.02

-0.1

-0.04

-0.15

-0.06
injury rate

productivity

The left vertical line measures flow rate of injured workers..
The right vertical line meaures labor productivity..

Figure 2 Japan
Relative Deviation of Unemployment Rate and Injury
Rate
0.2
0.1
0
-0.11979

1984

1989

1994

1999

2004

-0.2
-0.3
unemployment

0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06

injury rate

The left vertical line measures unemployment rate.
The right vertical line meaures flow rate of injured workers.

Relative Deviation of Unemployment Rate andProductivity
0.2

0.06

0.1

0.04

0
-0.11979

0.02
1984

1989

1994

1999

2004

0

-0.2

-0.02

-0.3

-0.04
unemployment

productivity

The left vertical line measures unemployment rate.
The right vertical line meaures labor productivity..

Relative Deviation of Injury Rate and Productivity
0.08
0.06
0.04
0.02
0
-0.021979
-0.04
-0.06

0.06
0.04
0.02
0
1984

1989

1994

1999

2004

-0.02
-0.04

injury rate

productivity

The left vertical line measures flow rate of injured workers..
The right vertical line meaures labor productivity..