GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.129

Unemployment and Workplace Safety in a Searchand Matching Model

Masaru Sasaki

April 2010

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Unemployment and Workplace Safety in a Searchand Matching Model
Masaru Sasakiy
April 2, 2010

Abstract
Are recessions really good for workplace safety? This paper develops a modelwith search to consider the determinants of workplace safety and then investigates
the relationship between unemployment and the incidence of work-related injury.
There is a view following Arai and Thoursie (2005), Ruhm (2000) and Boone andvan Ours (2006) that the rate of work-related injury is procyclical. However, datafrom several countries do not necessarily support this view. This paper considers analternative approach to support the countercyclical variation in the rate of workrelated injury in which the …rm bargains about the optimal input for workplacesafety.
JLE Classi…cations: J64, J65, J81
Keywords: Job Search, Workplace Safety, Work-Related Injury andIllness, Unemployment

The author is grateful to the participants in Search Conference at Osaka University, FESAM2008
and the 2009 EALE Annual Meeting. I also thank seminar participants in the Kansai Labor Seminar,
Nihon University , GRIPS, and Tohoku University. Part of this research is …nancially supported by theJapan Society for the Promotion of Science, No. 20530196. All errors are mine.
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

Are recessions really good for workplace safety? This paper develops a model with search
to consider the determinants of workplace safety and to explore the relationship between
unemployment and the incidence of work-related injury. Assuming that the probabilityof a worker being injured in a workplace depends negatively on the amount of input forworkplace safety purchased by …rms, there is a trade-o¤ for …rms between the cost of the
input and the risk of losing a worker.
There is a view that the rate of workplace injury is procyclical, thereby indicatingthat workplaces are safer during recessions. For instance, Arai and Thoursie (2005)
argue that during an economic boom when labor demand exceeds labor supply, …rms
hired even inexperienced workers. These were more likely to be injured in the workplace,and therefore both employment and the ‡ow of employed workers absent because of workrelated injuries were higher during economic upturns. Put di¤erently, employment andwork-related injuries are lower during recessions. Therefore, the unemployment rate andthe ‡ow rate of absent employed workers are negatively correlated after controlling forthe labor force.
Further, Ruhm (2000) shows that the mortality rate was also procyclical, indicatingthat people were healthier during recessions. He argued that workers’health was sappedby the deterioration of working conditions, increased workload and work-related stresscaused by longer working hours during a short-lasting economic boom. Boone and vanOurs (2006) also suggest that the rate of work-related injury was procyclical, but provided
an alternative explanation to Ruhm (2000) as follows. Consider the situation where theextent of injury incurred by a worker is asymmetric; that is, an employer cannot observe
the worker’s injury. The worker is then less likely to report his or her accident andattempt to keep working during a recession when unemployment is high if he or she
believes that workers who report accidents and take sick leave are more likely to be …redby the employer. Hence, even though working conditions remain unchanged irrespectiveof the business cycle, the rate of work-related injury is lower during recessions.
However, there is an opposing view that the rate of work-related injury is not nec2

essarily procyclical. For example, Ussif (2004) undertook an international comparative
study using time-series data between 1970 and 1999 from several countries and found the
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 after controlling for the labor force. Additionally,
Ussif (2004) found that these rates move in the same direction after controlling for any
time trend. On this basis, Ussif (2004) concludes that the number of work-related injuries
declined because of technical advancement in workplace devices and the environment as
captured by the time trend.
To the best of my knowledge, we have not so far considered other elements determining the number of injuries in the workplace, even though it is obviously of great
importance from the policymakers’point of view. Many other factors a¤ect the rate ofwork-related injury, including employer practices at the workplace, employee training,
the role of unions, and the provision of safety mandates. This paper focuses attention on
the determinants of the level of workplace safety and develops a model that endogenizes
the probability of a worker being subject to a work-related injury. In this model, the …rm
bargains on how much input to purchase for workplace safety with its worker. We thenexplore the optimal safety level that determines the number of injuries at the workplacein response to exogenous shocks. Our contribution is to provide an alternative approach
to explain the relationship between the unemployment rate and the incidence of workrelated injury by incorporating the determinants of the input for workplace safety into
a search and matching model. However, this paper does not discuss the role of mandates in keeping workplaces safe by, for example, the Occupational Safety and HealthAdministration (OSHA) in the US and its e¤ect on labor market conditions.1
There are already a few theoretical studies in this …eld. For example, Engström andHolmlund (2007) construct a general equilibrium model with search by incorporating
absenteeism from work as an additional state.2 They derived the optimal compensation
1

Jolls (2008) surveys both theoretical and empirical studies on the e¤ects of OSHA and the compensation programs for work-related injuries.
2
Barmby et al. (1994) presents a model in which the wage is endogenously determined within ane¢ ciency wage setting. They show that the wage is adjusted to a¤ect the decision on absence from3

package to maximize the expected pro…t a¤ected by the number of job applications andthe determinants of absenteeism of sick workers from work under the condition that accidents randomly took place. They also provide a welfare analysis and compared alternativesocial insurance policies. They focus on the individual worker’s decision on labor supply
and sickness absenteeism. Instead, our paper focuses on the determinants of the amountof input purchased by …rms for workplace safety to reduce the risk of losing employedworkers from work-related injuries.
Alternatively, using time series data from 16 OECD countries, Boone and van Ours
(2006) present empirical evidence that the procyclical variations in workplace accidents
are more attributable to a reluctance to report accidents during recessions than to changes
in working conditions and the composition of experienced and inexperienced workers.
Other empirical studies in this area have thus far explored the e¤ect of OSHA on workrelated injuries using state-, industry-, and plant-level data from the US.3 Overall, thee¤ect of OSHA enforcement on the rate of work-related injury was found to be modest(Viscusi 1979 1986, Bartel and Thomas 1985). In contrast, Scholz and Gray (1990) founda signi…cant relationship between OSHA enforcement and the rate of work-related injury
using plant-level data of …rms that were frequently inspected. According to a recent study
by Mendelo¤ (2005), this signi…cant relationship was still observed in the early 1990s but
disappeared thereafter.
Our …ndings are summarized as follows. To start with, productivity improvement
encourages …rms to enter the labor market, which makes it more competitive for …rms
to hire workers. An increase in competitiveness among …rms then lowers the marginal
gain of operating an actively occupied position and its marginal cost with respect to thelevel of safety, and the latter e¤ect overwhelms the former in an environment where both
the wage and the amount of input for workplace safety are bargained over. Therefore,…rms have an incentive to improve safety conditions in the workplace. In addition, theproductivity improvement leads to an increase in pro…t, which implies an increase insickness. Garibaldi and Wasmer (2005) also construct a multistage model where the wage is endogenously
determined.
3
Smith (1992) surveyed these empirical studies in the early 1990s.

4the opportunity cost that …rms would have incurred if their workers had been injured.
Firms are then induced to purchase more input for workplace safety to reduce the riskof work-related injury. Overall, the productivity improvement unambiguously raises theamount of input used for workplace safety. The risk of work-related injury that depends
negatively on the input for workplace safety is then countercyclical.
Considering the impact on the unemployment rate, the productivity improvement
encourages …rm entry, which increases the tightness of the labor market and thereby
lowers the rate of unemployment. An increase in the amount of input for workplacesafety then increases active employment but reduces absenteeism. The ‡ow rate to theunemployment pool is then larger from the active employment pool but smaller from theabsenteeism pool. We …nd that the latter e¤ect exceeds the former, and therefore an
increase in the input for workplace safety lowers the unemployment rate. The overalle¤ect of the productivity improvement is then negative on the unemployment rate, andthis is consistent with the implications found in the extant literature. Our model of thedeterminants of workplace safety then shows that both the risk of work-related injury andthe unemployment rate are countercyclical, and this contrasts with the …ndings in Araiand Thoursie (2005), Ruhm (2000), and Boone and van Ours (2006), but is consistent
with Ussif (2004). We also considers the model in which the cyclical shock arrives at theeconomy according to the Poisson process and obtain the same implication derived fromthe steady state equilibrium model.
We then extend the model to analyze the e¤ects of several policy parameters (namely,
sickness and unemployment bene…ts). Among these, the e¤ect of unemployment bene…ts
is particularly noteworthy. To the best of our knowledge, unemployment bene…ts andworkplace safety have thus far been discussed separately, and we show that a strong policy
linkage exists between them. Namely, an increase in unemployment bene…ts discourages…rms from entering the labor market, which makes it less competitive for …rms to hire
workers. An increase in the unemployment bene…t then raises wages and lowers pro…ts,
implying a decrease in the opportunity cost …rms would have incurred if their workers
had been injured. These …rms are then less concerned about reducing the risk of work-5related injury. Put di¤erently, an increase in unemployment bene…ts raises the valueof unemployment, which makes the state of unemployment more favorable for workers.
To attract unemployed workers, …rms raise the expected value of employment for an
active worker and for an absent worker by reducing the risk of losing the wage because ofwork-related injury. Firms are therefore encouraged to increase the input for workplacesafety. The overall impact of unemployment bene…ts is then unambiguous on the riskof work-related injury. However, if the …nal e¤ect is dominant, we can conclude thatunemployment bene…ts are a valid tool to reduce the risk of injury in the workplace.
The organization of the rest of this paper is as follows. Section 2 illustrates the empirical data. Section 3 presents a matching model with an endogenous input determinant forworkplace safety. Section 4 provides comparative statics exercises and discusses the determinants of the relationship between unemployment and the incidence of work-related
injury through a productivity improvement. The e¤ects of sickness and unemployment
bene…ts are analyzed in Section 5. We discuss several miscellaneous issues in Section 6,
and the …nal section provides some concluding remarks.

2

Graphical Inspection

In this section, we show the empirical relationships existing between the rate of unemployment and the rate of work-related injury. For visual inspection, we compare the relative
variations in the unemployment rate and the rate of nonfatal injury. Figure 1 displays
the annual growth rates of these variables using time-series data for selected countries
in Europe, North America, and Asia. The data are from LABORSTA, the International
Labour O¢ ce (ILO) database on labor statistics. Based on this data, it appears thata negative relationship between the unemployment rate and the rate of nonfatal injury
exists in Poland, Romania, Spain, Sweden, Canada, and the US. These data then support
the view of Arai and Thoursie (2005), Ruhm (2000), and Boone and van Ours (2006)
that while the unemployment rate is countercyclical, the injury rate is procyclical because
inexperienced workers (who are more likely to be hired during an economic boom) are6more likely to be injured in the workplace. However, we cannot discern a clear negative
relationship in the other sample countries and, in fact, there is a positive relationship during some subperiods. For example, in Italy both the unemployment rate and the nonfatal
injury rate decline in the 2000s, while in the UK both have moved in the same direction
since the late 1990s. In Japan, both the unemployment rate and the nonfatal injury rate
have also moved in the same direction during the period of the bubble economy (the late1980s and the early 1990s) and again after 2005.
This evidence helps illustrate that there are many other mechanisms or factors that
determine the relationship between the unemployment rate and the injury rate. We
choose to focus attention on the labor demand side through the …rm’s optimal choice of
input for workplace safety to prevent accidents from occurring.

3

The Model

3.1

Steady States

We consider a continuous-time model with matching in which there are a continuum ofrisk neutral workers and a continuum of risk neutral …rms. The measure of workers isnormalized to one. Workers are in…nitely lived and homogeneous with respect to theirpreferences for work. At any point in time, a worker is either employed or nonemployed.
In turn, there are two states of employment: an active working state and an absent statebecause of work-related injury. Nonemployment meanwhile consists of an unemployment
state in which workers actively search for a job and a nonlabor participation state in which
injured workers recuperate at home and thus cannot engage in job search. The injured
workers in the absent state can automatically return to the same workplace once they
are well, while those in the state of nonlabor force participation enter the unemployment
pool and begin looking for a job after making a recovery.
An employed worker is injured in the workplace and is absent from work at a Poisson
rate (k), where k represents the safety and health input, with its price normalized toone, purchased by a …rm to improve workplace safety conditions. To improve working
7

conditions, …rms usually incur legal welfare expenses to comply with labor standards andnonlegal welfare expenses for housing, food, and work uniforms along with the costs ofsickness and injury. Work-related injuries are de…ned here as an immediate health hazard
incurred during work that forces workers to be absent from work for treatment. They
potentially include lower back pain, cuts, bruises, broken bones, falls, being struck by
objects, mental illness, and so on, but not long-term latent health hazards from work
such as pneumoconiosis. We assume that (k) is characterized by(0) =1 and limk!1 (k) =0( ) < 0;00( ) > 0;0. As a …rm buys input to improve workplace safety

conditions, the likelihood that an employed worker is injured is reduced at a decreasingrate. The absent employed worker heals and immediately returns at the same workplaceat an exogenous Poisson rate .4 We thus assume that k a¤ects the accident rate in theworkplace but not the extent or duration of injury.
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 unemployed
and v is a 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 encountersan unemployed worker is computed by m(u; v)=v = m(u=v; 1)

q( ) wherev=u isthe tightness of the labor market, while the rate at which an unemployed worker matcheswith a job vacancy is represented by q( ): Note that q( ) is decreasing in ; that is,
q 0 ( ) < 0.
A job is separated at an exogenous Poisson rate , regardless of whether an employedworker actively works or is absent from work. While the active worker then becomes
unemployed and begins to search for a job, the absent worker loses his or her employment
status and is removed from the labor force because of the treatment of injury. We assume
that in a similar manner as the absent employed worker, nonlabor participants get wellagain at the exogenous Poisson rate :4

In fact, the rate of return to work is not exogenous as it largely depends on the amount of compensation as well as the extent of the injury or illness. Unfortunately, it is di¢ cult to observe whether absent
workers heal from their injuries or get well from sickness. Meyer, Viscusi, and Durbin (1995) undertook anatural experiment and found that an increase in the compensation received by absent employed workers
extended the duration of absence. Similar results are obtained in Ehrenberg (1988) and Krueger (1990).

8

Both workers and …rms discount the future at the common rate r. Before various
value functions are developed, the timing of decisions needs to be de…ned. We assume
that a …rm recruits a worker and then simultaneously bargains over both the wage and theamount of input for workplace safety k with the worker. In this setup, the wage and theamount of input are not the predetermined variables, being allowed to instantaneously
change in response to exogenous shifts. Di¤erent timings of the decisions and di¤erent
ways of decision making on the wage and the input for workplace safety are possible,
which we discuss later.
We begin with the value for an employed worker of engaging actively in work as
follows:rW = w + (k)(Wa

W ) + (U

W ):

(1)

The instantaneous utility is linear with earnings. The second term on the right-hand sideof eq.(1) represents the expected capital loss incurred by being injured, and the third
term indicates the expected capital loss of being unemployed. We assume that a newly
injured worker does not choose to quit a job to be out of labor force participation; that is,Wa

N; the condition to meet hereinafter described. In a similar manner, the value foran employed worker of being absent from work because of work-related injury is de…nedby:rWa = (WWa ) + (N

Wa ):

(2)

We assume in the benchmark case that if absent from work because of injury in theworkplace, the worker is not recompensed at all for his or her earnings w. Note thatthe disutility incurred by the absent employed worker is ruled out in this model without
loss of generality. The …rst term on the right-hand side shows the expected capital gainof making a recovery, and the second is the expected capital loss of losing the status of
employment at , where N represents the value of being out of the labor force becauseof the treatment of injury.

9The value of unemployment is as usual given by:

rU = q( )(W

(3)

U ):

At any point in time, an unemployed worker who is assumed to receive no instantaneous
utility meets a …rm with a vacant job at the transition rate q( ). The value of being outof the labor force is:rN = (UA nonlabor force participant heals at rate

(4)

N ):

and becomes unemployed, and is thenlooking for a job.
The di¤erence between eqs.(1) and (2) is given by:

W

Wa =

w + (U N ):r + + + (k)This shows that active employment is more favorable for the worker than absenteeism
from work as a result of work-related injury because the absent worker is not compensatedand is more likely to be out of the labor force.
Substituting this above equation and eqs.(4) into eq.(1) yields the worker’s surplus:

W
We assume W

U=

r+ +
w
(r + )(r + + + (k))

r+

+ (k)
rU :
r+

(5)

U and therefore obtain:

w

r+

+ (k)
rU:
r+

The wage has to equal or exceed the reservation wage (rU ) plus the expected injury
risk premium ( r+(k) rU ). In a similar manner, we obtain:Wa

N=(r + )(r ++ + (k))

w

r+

+ (k)
rU :
r+

The optimal choice for a newly injured worker is to stay employed but not quit the10current job, again if the wage is large enough to satisfy w

r+ + (k)rU:r+

Next, we discuss the value functions for a …rm. We consider a …rm to be a collectionof individual jobs. At any point in time, jobs are either occupied, un…lled, or inactive
because employed workers are absent as a result of work-related injuries. We assume
that …rms operate under a constant returns-to-scale production technology with respect
to the labor input. This assumption ensures that jobs in the …rm are independent of one
another.
The value of a job being actively occupied is:rJ = pw

k + (k)(JaJ) + (VJ):(6)

A matched pair produces p instantaneously. The second term on the right-hand sideof eq.(6) represents the expected capital loss of a job being inactive because a worker isabsent from work owing to a work-related injury. The third term indicates the expected
capital loss of a job being separated.
Similarly, the value of an occupied job being inactive because of work-related injury
is:rJa = (JThe job turns out to be active at rateJa ) + (V

Ja ):

(7)

and separated at rate . We assume for sim-

plicity that there are no disability insurance programs. However, in reality many …rms
join federal or state disability insurance programs with compulsory payroll deductions.
If their own employees are then injured in the workplace, they are compensated throughtheir program. Because the disability insurance program is mainly …nanced by …rms, wesuggest that …rms indirectly bear the burden of compensation. Further, because premiums depend positively on their safety record, these …rms have an incentive to improve
workplace conditions, which is di¤erent from our motivation in the sense that …rms areencouraged to improve workplace conditions to reduce the likelihood that workplace accidents cause a fall behind in production. Furthermore, many …rms have their own absence11

leave programs with payment. According to a survey conducted by the Japanese Ministryof Health, Labour and Welfare (MHLW) in January 2008, 58.6% of the surveyed …rms
have their own absence leave programs and 41.1% keep paying an average 85.8–93.6% ofsalaries to absent employees. For the surveyed …rms with more than 1,000 employees,
85.3% have their own absence leave programs and 56.8% pay 88.5–91.8% of their salaries
to absent workers.5
The value of a vacancy is given as usual by:rV =+ q( )(JA vacancy is incurred the instantaneous cost

(8)

V ):

and …lled at the transition rate q( ).

The free-entry condition ensures V = 0 in equilibrium.
Eqs.(6) and (7) give the following equations:J=(r + + )(p w k)(r + )(r + + + (k))

0;

Ja =

(p w k)(r + )(r + + + (k))

0:

(9)

and

Ja

V = 0 if the instantaneous pro…t equals or exceeds zero, so the inactive …rm

does not choose to …re the absent worker.
A …rm and a worker consummate a match if and only if the joint surplus gained
through this match is nonnegative, and then the wage and the amount of k are simultaneously chosen to maximize the weighted product of the net return from the job match
according to the Nash bargaining rule. Assuming that the worker’s share of the surplus
is de…ned by

2 [0; 1], the conditions to determine w and k are given by:(1)(WU ) = J;

(10)

5
See Rodo Sinbun (Labour Newspaper) No. 2688 (July 14, 2008) in Japanese published by Rodo
Sinbunsha.

12

and

(1

)(W

U)

@J
@k

=

J

@(W U )
@k

(11)

:

Substituting eqs.(5) and (9) into eq.(10) gives:

w = (pk) + (1

)

r+

+ (k)
rU:
r+

The wage is determined by the weighted-average of the instantaneous net production
and the reservation wage rU , after accounting for the risk of injury. From eqs.(3), (8),and (10), the reservation wage rU can be expressed as rU = 1. Substituting this

into the above wage equation yields:

w=

p

k+

r+

+ (k)
r+

(12)

:

The wage depends negatively on k. An increase in k lowers the risk of injury in theworkplace, which means that it is more favorable for the worker to be employed thanunemployed. This reduces the worker’s bargaining position, as directly resulting in alower wage. In addition, an increase in k lowers expected pro…t, leading to a further
decrease in the wage. Recall that w

r+ + (k)rU to ensure Wr+

U and Wa

N . Wethus have:(1)(r + )(p k):(r + + (k))

The term on the right-hand side either increases or decreases with k. If labor market
tightness is lower than the right-hand side term, conditions such as W

U and WaN

are satis…ed.
The problem regarding the Nash bargaining determination over the input for workplace safety is solved in a similar manner. Using eqs.(5), (9) and (12), eq.(11) can be
rewritten as:

13

0

r+

(k)
(1
+ + (k)

)(p

r+

k)

+ (k)
r+

=1+

1+

0

(k)
r+

:
(13)

The term on the left-hand side represents the marginal gain of operating an actively
occupied job with respect to the input for workplace safety while the term on the righthand side is its marginal cost. The parentheses in the marginal gain term represent the
instantaneous pro…t (pwk);which is assumed to be zero or positive for all k. Usingeq.(12), the condition pw

k

0 is rewritten as:(1)(r + )(p k):(r + + (k))

This condition coincides with that needed to ensure W

U and Wa

N: The left-

hand side of eq.(13) is positive, as must be the right-hand side. The marginal cost consistsof the direct cost of k with its price normalized to one plus the marginal wage cut withrespect to k; implying @w=@k(< 0): The marginal gain is decreasing with k while themarginal cost is increasing in k. The second-order condition ensures that the optimal
amount of input maximizes the weighted product of the net return from the job match.
Eq.(13) is depicted as an upward sloping curve in kspace. Here, a higher

lowersthe marginal gain, shifting the curve to the left and therefore lowering k. Meanwhile, ahighershifts the marginal cost curve to the right, thereby increasing k. We recognizethat the latter e¤ect dominates the former because both w and k are simultaneously
bargained. k is then determined through the Nash bargaining rule, taking into accountthe e¤ect of k on w.
Substituting eq.(9) into eq.(8) yields the free-entry condition:q( )(r + + )(1(r + )(r + + + (k)))(p

Eq.(14) is represented as a horizontal line in k

k)

r+

+ (k)
r+

space; that is,

= :

(14)

is constant irrespec-tive of k because the partial di¤erential of the left-hand side with respect to k turns out to14

be zero according to eq.(13). An increase in k lowers (k) and thereby the likelihood thata worker is injured in the workplace, which induces …rms to enter the market. However,an increase in k simultaneously raises the costs of operation, which eventually negates
the e¤ect on entry at the optimal level. Both eqs.(13) and (14) are shown in Figure 2.
We next illustrate the steady-state conditions needed to derive the unemployment
rate and the fraction of absent employed workers. Let u be the fraction of unemployedworkers, and a and n denote the fraction of employed workers who are absent from workand the fraction of nonlabor force participants who cannot search for a job because of
the treatment of injury, respectively. The remainder is then considered active employed.
The steady-state conditions require:q( )u =

(1

n =

a;

and ( + )a =uan) + n;

(k)(1

u

a

n):

Then, we obtain:

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

) (k)
) (k)
) (k)

(15)

;
;
:

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

e

1

u

a

n=

( + )( + q( ))( + )( + (k))( + q( )) + (1As one would expect, greater labor market tightness

15

) (k)

:

(16)

lowers the fraction of unem-

ployed workers u but raises both a and n because the measure of employed e is larger
in size. An increase in the input for workplace safety k lowers both a and n because
workplace safety conditions are improved. An increase in k raises u through the channelof an increase in e, but on the other hand, lowers u through the channel of a decrease inn. Overall, the latter e¤ect dominates the former: that is, an increase in k lowers u.
The unemployment rate is then computed by:

u
e

u
1

n

=

(1
) (k)
( + + (k))( + q( )) + (1

) (k)

:

Similar to u, the unemployment rate depends negatively on k and :Eqs.(13), (14) and (15) provide a complete description of the equilibrium used tosolve for the vector (k; ; u; a; n).

For convenience, these equilibrium conditions are

summarized below.
(i) Condition to determine k (rewritten version of eq.(13))
0

(k)
r+

= (1

)[ 0 (k)(pk) + (r ++ + (k))](ii) Free-entry condition (eq.(14))
q( )(r + + )(1(r + )(r + + + (k))

)(p

k)

r+

+ (k)
r+

=

(iii) Steady-state conditions (eq.(15))

(1
) (k)
( + )( + (k))( + q( )) + (1(k)( + q( ))a =( + )( + (k))( + q( )) + (1(k)( + q( ))and n =( + )( + (k))( + q( )) + (1u =Eq.(13) is upward sloping while eq.(14) is horizontal in k

) (k)
) (k)
) (k)
space. These two

equations ensure the existence of a unique equilibrium for k and , and then it is possible16

to …nd the optimal value set (u; a; n) from eq.(15). We investigate the characterizationsof the equilibrium by examining the comparative statics in the following section.
Our focus is on the relationship between the unemployment rate and the incidence of
work-related injury via exogenous parameter changes. The main purpose is to illustrate
the changes in workplace safety and labor market conditions in response to a productivity
improvement. We here focus on the steady-state equilibrium, not the equilibrium where
the aggregate productivity shock is anticipated (Mortensen and Pissarides 1994), whichis considered latter. Using (i) …rst-order conditions and (ii) free-entry conditions, thecomparative statics system is described. The appendix provides analytical detail.
Proposition 1 The comparative statics analysis provides the following characterizations:(1) an increase in productivity raises both the amount of input for workplace safety andlabor market tightness:dkd> 0; and> 0;dpdp(2) an increase in productivity lowers the unemployment rate (e
u):
de
u
=
dp

@e
u
dk
@e
u
d
+
< 0;
@k
dp
@
dp
| {z }| {z } | {z } | {z }++and (3) an increase in productivity raises the employment rate (e) but exerts an ambiguous e¤ect on the fractions of absent workers (a) and nonlabor force participants (n):

da
=
dp

@a
dk
@a
d
+
7 0;
@k
dp
@
dp
| {z }| {z } | {z } | {z }
+

dn
=
dp

+

@n
dk
@n
d
+
0 7 0;@kdp@
dp| {z }| {z } | {z } | {z }
+

de
=
dp

+

+

+

@e
dk
@e
d
+
> 0:
@k
dp
@
dp
| {z }| {z } | {z } | {z }
+

+

+

+

As depicted in Figure 3, an increase in productivity p shifts eq. (13) to the right and17eq. (14) upward, thus leading to unambiguous increases in k and . More …rms enter
the labor market and create vacancies because the productivity improvement leads to anincrease in pro…t, thus resulting in an increase in . As shown in eq. (13), an increase inlowers the marginal gain in operating an actively occupied job and the marginal costwith respect to k, the latter e¤ect overwhelms the former, and therefore …rms have an
incentive to increase the amount of input for workplace safety. Additionally, an increasein productivity p directly raises the marginal gain, implying an increase in the pro…t that…rms would have earned without worker absenteeism. An increase in the opportunity
cost incurred by forcing the worker to be absent induces the …rms to increase the amountof input for workplace safety k to prevent employed workers from being injured in theworkplace, leading to a lower rate of injury (k). When combined with the above e¤ects,
the productivity improvement unambiguously exerts a positive impact on the input ofworkplace safety k.
We recognize that an increase in productivity lowers the unemployment rate throughthe channels of the increased amount of input for workplace safety k and greater tightness
in the labor market . These results imply that both the unemployment rate and the rateof injury in the workplace are countercyclical. This view di¤ers from Arai and Thoursie(2005), Ruhm (2000), and Boone and van Ours (2006), but supports Ussif (2004), whoshows that despite a steady increase in the number of employed workers, the numberof work-related injuries declined from 1970 to 1999 using time-series data from selected
countries. That is, he implied a positive relationship between the unemployment rate andthe rate of injury in the workplace. Our …ndings are then at least partially consistent
with data from some countries, including Italy, the UK, and Japan, as displayed in Figure
1.
Productivity improvement provides ambiguous e¤ects on the fractions of absent workers (a) and nonlabor force participants (n). An increase in k reduces the ‡ow of absent
workers from e to a, but on the other hand, greater labor market tightness increases
employment e, which thereby leads to an increasing ‡ow of absent workers from e to a,even though the rate of injury (k) remains …xed. The same intuition can be applied in18the case of n.
The following exercise explores the e¤ect of the healing rateon labor market con-ditions and helps illuminate di¤erences in the rate of injury by occupational type.
Proposition 2 The comparative statics analysis provides the following characterizations:(1) The higher rate of healing

increases labor market tightness but has an ambiguouse¤ect on the amount of input for workplace safety; that is:dkd
7 0; and> 0;dd(2) an increase in the healing rate either raises or lowers the unemployment rate (e
u):
de
u
=
d

@e
u
dk
@e
u
d
+
7 0:
@k
d
@
d
| {z }| {z } | {z } | {z }++=As the healing rate is higher, that is, workers are not seriously injured and can return
to work sooner, more …rms enter the market and create vacancies because the loss thatthe …rm would incur by their own workers’ absenteeism is relatively small. Therefore,eq.(14) shifts upwards. This implies that jobs are more likely to be created in sectors
in which workers’injuries are generally not severe, such as the retail sales and services
sectors. In contrast, …rms are discouraged from creating jobs in sectors where the extent
of work-related injury is usually severe, such as construction, transportation, and mining.
The higher healing rate has an ambiguous e¤ect on the amount of input for workplacesafety k. Because eq.(14) is shifted up, the higher by …rm entry induces …rms to purchasemore input for workplace safety k according to eq.(13). In addition, the higher healing
rate shifts eq.(13) to the left, leading directly to a decrease in k. If injured workers
can return to work sooner, work-related injury is a trivial issue; …rms do not have an
incentive to buy input k to prevent accidents from occurring in the workplace. On theother hand, the higher healing rate raises the values of employment, W (w) and Wa (w).
This means that the employment state is more attractive for the unemployed than thestate of unemployment, thus lowering the reservation wage and thereby the wage. This19

increases the marginal gain and shifts the curve to the right, and …rms are therefore
encouraged to purchase input for workplace safety k. As shown in Figure 4, we conclude
that the amount of input for workplace safety and the healing rate are not monotonicallycorrelated, so …rms do not necessarily practice to keep workplaces safe by purchasing k
in sectors with lower healing rates

3.2

where the extent of work-related injury is severe.

Cyclical Changes

The previous section has shown the e¤ects of productivity on the labor market at thesteady states, and therefore it did not allow for the cyclical implications of productivity
change. To capture the characterizations of cyclical productivity shocks, the model isextended to the case in which productivity shock arrives according to the Poisson process
in this section. Productivity takes either a high value ph during an economic boom ora low value pl during a recession at a Poisson rate . Incorporating this Poisson rate
into the model allows the economy to switch back and forth between economic boom
and recession probabilistically. We then compare the equilibrium when the economy isin a recession and when the economy is in an economic boom and explore the cyclicalpatterns of the rate of injury in the workplace and unemployment rate. Furthermore, we
focus attention on the di¤erences in terms of the relationship between the rate of injury
in the workplace and the unemployment rate between the steady state equilibrium andthe equilibrium of the model into which the cyclical productivity shock is incorporated.
The values for a worker in each state are rewritten as:

rWi = wi + (ki )(Wai

rWai = (WiWi ) + (UiWai ) + (NirNi = (UiWai ) + (WajNi ) + (Nj20

Wi ) + (Wj

Ni );

Wi );

Wai );

(17)

(18)

(19)

and

rUi =

i q( i )(WiUi ) where i; j = h; l, i 6= j:Ui ) + (Uj(20)

The last terms on the right-hand side of the above equations represent the expected
capital change of the value because of the arrival of the productivity shock.
In a similar manner, the values for a job in each state are given by:

rJi = piwi

ki + (ki )(Jai

rJai = (JiJai )Jai + (JajJi )Ji + (Jj

(21)

Ji );

and

Jai ) where i; j = h; l, i 6= j:(22)

Because the free-entry condition ensures the value of a vacancy to be zero in equilibrium, we obtain:q( i )(23)= Ji :

Similarly to Section 3.1, a …rm and a worker match if and only if the joint surplus
gained through this match is nonnegative, and then the wage and k are simultaneouslychosen to maximize the joint surplus according to the Nash bargaining rule. The condition
to determine the wage is given by:

(1

)(Wi

Ui ) = Ji where i = h; l:(24)

For simpli…cation of the computation, we assume that there exist only two amountsof the input for workplace safety, k and k (k >k), and a …rm and a worker choose eitherof them to maximize the joint surplus of job match according to the following rule.

ki =k if (Wi (k)Ui ) Ji (k)1

k otherwise, where i = h; l:21(Wi (k)Ui ) Ji (k)1

;

(25)

There are four possible patterns of the input for workplace safety: (kh ; kl ) = (k; k), (k;k), (k; k) and (k; k). We choose one of these patterns that satis…es with the equilibrium
conditions, eq.(17) - eq.(25).
Because there are too many equilibrium conditions to solve for the variables analytically, we have to rely on the numerical analysis to compare the equilibrium variables
between a recession and an economic boom. We employ two steps to solve for the model.
The …rst step is to solve for 16 variables {Wh , Wah , Nh , Uh , Jh , Jah , wh ,Ul , Jl , Jal , wl ,h ,Wl , Wal , Nl ,l } from eq.(17) - eq.(24) for each pattern of the input for the workplacesafety. The second step is to calculate the joint surplus in each economic environment
for each pattern of the workplace safety, (Sh (k), Sl (k)), (Sh (k), Sl (k)), (Sh (k), Sl (k)),
and (Sh (k), Sl (k)), where Si (k) represents the joint surplus, i = h; l: The choice of the
input for the workplace safety when the economy is in a recession or an economic boom
is determined from this 2

2 one-stage simultaneous game.

We assume that k = 0:2 and k = 0:1, and furthermore, the rate of injury is de…nedby (k) = 0:001=k. Various parameter values are presented in Table 1. Table 2 displays
numerical results. The …rst row considers the case where k = 0:2 is chosen during an
economic boom while k = 0:1 is chosen during a recession. The second row shows thecase where k = 0:2 is chosen, irrespective of the business cycle, followed by k = 0:1
irrespective of the business cycle in the third row and k = 0:1 during an economic boom
and k = 0:2 during a recession in the …nal row. We now choose one of the four patterns.
According to the 2 2 one-stage simultaneous game, when the economy is in an economic
boom (p = ph ), k = 0:2 is the dominant strategy. It implies that a matching pair of a…rm and a worker choose k = 0:2 during the economic boom, regardless of which one ischosen, k = 0:1 or k = 0:2 when the economy becomes downturn. In contrast, whenthe economy is in a recession (p = pl ), k = 0:1 is the dominant strategy. k = 0:1
is chosen during the recession, irrespective of which one is chosen when the economy
becomes upturn. Therefore, the …rst row is a solution; that is, k = 0:2 is chosen during
an economic boom while k = 0:1 is chosen during a recession.
Looking at the …rst row of Table 1, as the economy takes a cyclical upturn from pl

22

to ph , unemployment rate decreases from 0.0878 to 0.0234, and the amount of the input
for workplace safety increases, thereby leading to lower rate of injury from 0.01 to 0.005.
This result implies that not only the unemployment is countercyclical, but that the rateof injury is countercyclical; that is, we obtain the same implication as the steady state
equilibrium obtained in Section 3.1 of a positive relationship between the unemployment
rate and the rate of injury in the workplace.

3.3

Discussion

We have so far shown a positive relationship between the unemployment rate and therate of injury in the workplace, incorporating the determinants of the input for workplacesafety into an otherwise standard matching model. This mechanism may help explain
why both the unemployment rate and the injury rate move countercyclically, as observedin the time-series data for some countries in Figure 1. However, as pointed out by Araiand Thoursie (2005), Ruhm (2000), and Boone and van Ours (2006), it is recognizedthat while the unemployment rate is countercyclical, the injury rate in the workplace isprocyclical in some other countries according to Figure 1. How do we revise this model
to capture the procyclical movement of the injury rate in the workplace?
Arai and Thoursie (2005) argue that …rms hired even inexperienced workers who weremore likely to be injured in the workplace during an economic boom, thus leading to theprocyclical movement of the injury rate. One extension is to incorporate heterogeneous
workers into the model. For example, there are two types of workers: veterans andbeginners. We assume that veterans are less likely than beginners to be injured in theworkplace. Firms initially attempt to match with veteran workers because the expected
costs of loss incurred by workplace injury are low. As overall productivity rises during
an economic boom, more …rms enter the market, and then …rms start to hire beginners
as well as veterans because it is more di¢ cult to meet and match with workers. The
increase in employed beginners then drives up the injury rate in the workplace, in whichcase the injury rate is procyclical.
An alternative extension is much easier; that is, the injury rate is revised to the23increase in productivity such that (p; k) and @ =@p > 0: The intuition to support thisrevision is as follows. During an economic boom, workers spend long hours at work
under the condition of high labor adjustment costs, which increases the risk of attention
problems in the workplace for workers and thereby the likelihood of injury. An increasein p induces …rms to increase k and then lower the injury rate as shown in Section 3.1,
but additionally, an increase in p directly raises the injury rate in the workplace. If thelatter e¤ect dominates the former, the injury rate becomes procyclical, and otherwise
countercyclical.

4

Miscellaneous Issues

4.1

Unemployment and Sickness Bene…ts

This section returns to the steady states model introduced in Section 3.1 and incorporates
sickness and unemployment bene…ts into the model to present their policy e¤ects. These
impacts then help illuminate the …rms’ incentives for the determinants of job creationand the amount of input for workplace safety in response to the policy changes. Here theemphasis lies on the impact of unemployment and sickness bene…ts on workplace safety. It
appears that the linkage between these bene…ts and workplace safety has hitherto received
little attention. The comparative statics exercise contributes to a better understandingof this linkage.
An employed worker who is actively engaged in work instantaneously earns w as
before, but an employed worker who is absent from work because of work-related injury
is recompensed for his or her loss through sickness bene…ts b. Additionally, an unemployedworker now receives unemployment bene…ts z. A nonlabor force participant who cannot
look for a job because of the treatment of a work-related injury also receives the sameamount of sickness bene…ts b. The value functions for a worker are modi…ed by:

rW = w + (k)(Wa24

W ) + (UW );rWa = b + (WWa ) + (UrU = z + q( )(W

Wa );

U );

and

rN = b + (U

(26)

N ):

Because workers are risk neutral, sickness and unemployment bene…ts are merely
considered subsidies. The value functions for a …rm remain the same: eqs.(6), (7), and(8). We assume for simplicity that …rms do not take out disability insurance; that is,…rms do not pay the insurance premium and therefore do not receive disability insurance
bene…ts faced by absent workers who are injured in the workplace.
Using these value functions and the free-entry condition, the wage is solved according
to the Nash bargaining rule:

w=

p

k+

r+

+ (k)
r+

+ (1

)

r+

+ (k)(z
r+

b)

:

(27)

Note that this wage equation is reduced to eq.(12) in the case of b = z = 0. It
is appropriate to assume that sickness bene…ts are more generous than unemployment
bene…ts; that is, z < b, in the case where an increase in the input for workplace safety kexerts an unambiguous impact on the wage. An increase in k lowers (p k), leading to thelower wage. In addition, an increase in k lowers the likelihood that an employee is injuredin the workplace and raises the expected value of being employed relative to the value ofbeing unemployed. The state of employment is more attractive, thus resulting in decreases
in the reservation wage and thereby the wage. Alternatively, if b > z, an increase in klowers the value of employment because employees are less likely to be injured and lose
the chance of receiving a generous b. This makes the state of employment less attractive
for workers, which raises the reservation wage and thereby the wage. The wage depends

25negatively on sickness bene…ts b. As an absent worker is recompensed more generously,
the employment state is more favorable than the unemployment state, thus leading to
the lower reservation wage and thereby the wage. In contrast, the wage increases with
unemployment bene…ts z, which can be explained in the opposite way.
Similarly to Section 3.1, the nature of the equilibrium is characterized by (i) the
condition to determine k through the Nash bargaining rule, (ii) the free-entry condition,and (iii) the steady-state conditions. The condition to determine k and the free-entry
condition are then given by:

0

(k)
r+

= (1

)

0

(k)(p

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

0

(k)

z

(r +
r+

+ )b

;
(28)

and

q( )(r + + )(r + )(r + + + (k))

(1

) p

r+

k

+ (k)(z
r+

b)

r+

+ (k)
r+

= :
(29)

Eq.(28) is either upward or downward sloping in kspace, depending on the ini-tial values of the parameters. To satisfy the second-order condition that ensures the
maximization of the weighted product of the net return from the job match, eq.(28) isrepresented as an upward sloping curve in kspace. Meanwhile, eq.(29) is depicted asa horizontal line, similarly to eq.(14). The comparative statics show the characterizationsof the equilibrium in response to the changes in the policy parameters.
We next explore the e¤ects of the policy parameters (sickness and unemployment
bene…ts) on the input for workplace safety and labor market tightness. According to thecomparative statics analysis, we obtain the following results:
Proposition 3 The comparative statics analysis provides the following characterizations:(a) sickness bene…t:dkd
7 0 and
> 0;
db
db
26

and (b) unemployment bene…t:ddk? 0 and< 0:dz
dz
The appendix provides further analytical detail.
(a) Sickness Bene…t
Figure 5 illustrates the case in which eq.(28) is upward sloping as before. An increasein sickness bene…ts b shifts eq.(29) up and eq.(28) to the left, thus leading to a decreaseinbut an ambiguous change in k. An increase in sickness bene…ts b encourages …rms

to enter the market because the wage is lower according to eq.(27), thereby leading to ahigher .
An increase in labor market tightness changes the …rm’s behavior toward the determinant of the input for workplace safety. There are two e¤ects on k to be considered. First,
according to eq.(28), an increase inlowers the marginal gain of operating an actively

occupied job and the marginal cost with respect to k, the latter e¤ect overwhelms theformer, and therefore …rms have an incentive to increase the amount of input for workplace safety. The second e¤ect is described as follows. As b increases more, the di¤erence
between the value for an active employed W and the value for an absent employed Wa
is smaller, which implies that workers faced with a larger b are more likely to accept therisk of injury. Therefore, …rms are discouraged from buying k to keep their workplaces
safe.
The overall e¤ect of sickness bene…ts on workplace safety is then ambiguous. However,
it is possible that sickness bene…ts exert a positive e¤ect on the amount of workplace safety
purchased by …rms. One possible policy implication from this exercise is to increase
sickness bene…ts, thereby inducing …rms to pay more attention to workplace safety.
(b) Unemployment Bene…t
The linkage between unemployment bene…ts and workplace safety is explored here.
The intuitive explanations are completely opposite to those for the e¤ect of sickness bene…ts discussed above. As unemployment bene…ts z increase, eq.(29) is shifted downward27

while eq.(28) is shifted to the right. This comparative statics exercise shows a decrease in
but an ambiguous change in k. As in standard matching models, an increase in unemployment bene…ts z lowers labor market tightness . The intuition behind this result isthat an increase in z raises the reservation wage of workers and thereby the wage, which
encourages …rms to exit.
There are two di¤erent e¤ects on k. First of all, according to eq.(28), a decrease inraises the marginal gain of operating an actively occupied job and the marginal costwith respect to k. The latter e¤ect dominates the former, so …rms are encouraged todecrease the amount of the input for workplace safety. Secondly, an increase in z raises
the value of unemployment, making the state of unemployment more favorable. To raise
the expected value of employment for both an active worker and an absent worker, …rms
are induced to increase the amount of input for workplace safety to reduce the likelihoodthat the active employed is injured in the workplace and loses the wage.
The overall impact of unemployment bene…ts on the input for workplace safety isunambiguous. The graphical exercises are illustrated in Figure 6. This is with theemphasis on the novel result that unemployment bene…ts can have a positive, althoughindirect, e¤ect on the input for workplace safety, implying a decrease in the risk of being
injured in the workplace. Unemployment bene…ts may then encourage …rms to improve
working conditions.

4.2

Social E¢ ciency

This section shows the socially optimal solution and compares this with the solution of
the decentralized economy presented above. The social planner optimally chooses labormarket tightness

and the amount of input for workplace safety k to maximize thediscounted present value of net output. The social planner cannot a¤ect the matchingprocess and therefore shares the same matching constraints with workers and …rms. The
social planner’s problem is illustrated below:

28

max

Z 1

[(1

u

a

n)(p

k)

]e rt dt;u0

subject to the ‡ow conditions (15). In Appendix, we solve for the social planner’s
problem using the dynamic programming method. The socially optimal values of k andare solved by the following two equations:0(k)(r + )=1+ ( )
q( )(r + + )0(r + ) +
r+

(k)

;

and

q( )(r + + )(1(r + )(r + + + (k))
where ( )( ))(pk) + ( )

r+

+ (k)
r+

= ;

q 0 ( ) =q( ). The …rst equation represents the condition to determine

k while the second shows the free-entry condition. These two equations correspond toeqs.(13) and (14) in the decentralized economy. Using eq.(14), eq.(13) can be rewrittenas:0(k)(r + )=1+
q( )(r + + )(r + ) +
r+

0

(k)

;

For early reference, eq.(14) is:q( )(r + + )(1(r + )(r + + + (k))

)(p

k) +

r+

+ (k)
r+

= :

Comparing the socially optimal conditions with the conditions for the decentralized
economy, we obtain the familiar result that the social optimal allocation forand kcan be achieved in the decentralized economy if the Hosios condition is satis…ed; thatq 0 ( ) =q( )

is,

=

dk=d

7 0 and d =d( ): According to the comparative statics analysis, we …nd< 0 (shown in the Appendix and Figure 7). An increase in

reduces the share of the surplus obtained by …rms, thus discouraging them from entering
the market. Therefore,

is lower. Because eq.(13) is represented as an upward sloping29

curve in kspace, a lower

leads to a decrease in k. Meanwhile, an increase inlowersnot only the marginal gain of operating an actively occupied job but also the marginal
cost with respect to k, and the e¤ect on the marginal cost is dominant because both kand w are simultaneously bargained. This implies that …rms are induced to purchasemore of the input for workplace safety k, shifting eq.(13) to the right. The overall e¤ect
on k is thus ambiguous.
Suppose that the socially optimal solutions are represented by bk and b while thesolutions derived in the decentralized economy are represented by k andwe …nd that …rms are underentered (b >. If> (b);); but it is not clear whether the amount of

workplace safety is over- or underpurchased in the decentralized economy (bk 7 k ):

4.3

Timing and Decision Methods

This section discusses the di¤erent timings and decision methods other than that thus
far presented. Two alternatives are to be considered. The …rst alternative is that a …rm
recruits a worker and chooses the amount of input for workplace safety, and then bargains
the wage with the worker. The second is that …rms simultaneously bargain about thewage and choose the amount of input for workplace safety after the worker is hired.
In the …rst case, k is a predetermined variable but w is not. The wage is continuously
and instantaneously bargained in response to any change of k. The wage is thus considereda priori function of k, w(k). We solve backwards for this problem. A …rm bargains thewage w(k) with an employee, and then chooses the optimal value of k to maximize thevalue of the job being actively occupied J, evaluated at w(k): It is well-recognized inthis case that we obtain the same result as that derived in Section 3, in which a …rm
simultaneously bargains w and k with an employee after recruitment.
The second case is di¤erent from the …rst in terms of the timing of the choice ofk. Because a …rm simultaneously bargains the wage and chooses the amount of input
for workplace safety, the wage is no longer a priori function of k. We …nd that the …rmchooses the lower amount of input for workplace safety in this timing of events than in theprevious two frameworks. The …rm decides how much to purchase k, facing a marginal30

cost of one, that is, a normalized price of k. On the other hand, in the previous twocases, …rms have a marginal cost with a normalized price plus w0 (k)(< 0). In this case,…rms simultaneously bargain w and choose k, not taking into account the marginal e¤ectof w(k): The analytical details are presented in the Appendix.

5

Concluding Remarks

This paper allows for decisions on the amount of input for workplace safety, and the tradeo¤ between its cost and the risk of employed workers being absent from work because ofwork-related injuries.
By incorporating decisions on the amount of input for workplace safety in a search
and matching model, we investigated the relationship between unemployment and theincidence of work-related injury. Productivity improvement encourages …rm entry andraises labor market tightness. Greater competitiveness in …nding an unemployed worker
induces …rms to increase the amount of input for workplace safety. Additionally, theproductivity improvement raises pro…ts, so the …rms are induced to increase the input
for workplace safety to prevent the potential loss of pro…t. Overall, the e¤ect of the
productivity improvement is positive on the amount of input for workplace safety and, inother words, negative on the risk of work-related injury. As for the impact on the unemployment rate, …rm entry increases tightness in the labor market and thereby lowers theunemployment rate. The lower risk of injury in the workplace increase active employment but reduces absenteeism. The ‡ow rate to unemployment is thus larger from active
employment but smaller from absenteeism. The comparative statics analysis documentsthat the latter e¤ect dominates the former, and therefore that an increase in the input
for workplace safety lowers the unemployment rate. The overall e¤ect of the productivity
improvement is then negative on the unemployment rate through the input for workplacesafety and labor market tightness.
This exercise shows that both the risk of injury and the unemployment rate arecountercyclical. This implication may not be supported according to some data …ndings.
31

However, we clearly observe this relationship from other data sources. For example, therisk of injury along with the unemployment rate is countercyclical during at least somerecent subperiods in Italy, the UK and Japan.

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[16] Ussif, A-A. (2004) “An International Analysis of Workplace Injuries,”Monthly Labor
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[17] Viscusi, W. K. (1979) “The Impact of Workplace Safety Regulation,” Bell Journalof Economics, 10(1): pp. 117–40.
[18] Viscusi, W. K. (1986) “The Impact of Workplace Safety Regulation,” 1973–1983
RAND Journal of Economics, 17(4): pp. 567–80.

AppendixComparative Statics E¤ect (Section 3.1)
In this appendix, we use comparative statics analysis to explore the e¤ects of productivity p on the input for workplace safety k and labor market tightness , using eq.(13)33

and eq.(14). The comparative statics system is given by:
2

32

3

2

3

2

3

6 dp 7
7
A
A
dk
B
B
B
11
12
11
12
13
6
76
7 6
76
7
4
54
5=4
56
6 d 7;
5
A21 A22d
B21 B22 B23 4dwhere:

00

A11 =

(k)

r+

+ (1

)(p

0

(k)
r+

A12 =

k) > 0;< 0;

A21 = 0 from the …rst-order condition (eq.(13)),A22 =q 0 ( )(p w k)r + + + (k)

r+ +
r+

B11 =

B12 =

B13 =

0

q( )(r + + (k))< 0;(r + )(r + + + (k))) 0 (k) > 0;

(1

0

(k)
(r + )2(1) < 0;(k)r++ [ 0 (k)(pB21 =q( )(r + + )(1)< 0;(r + )(r + + + (k))k) + (r +

34

+ +

0

(k)] > 0;

B22 =

q( ) (k)(p w k)(r + )(r + + + (k))2(r + + )q( )(r + )(r + + + (k))(k)(r + )2< 0;

andq( )(r + + )(p(r + )(r + + + (k))
hip k + r+ r++ (k):B23 =where w =The Jacobian determinant is rA

k) +

r+

+ (k)
r+

> 0;

A12 A21 < 0. Then we …nd:

A11 A22ddk> 0; and
> 0;
dp
dp

dk
d
? 0; and
> 0;
da
da
and
dk
d
? 0; and< 0:ddComparative Statics E¤ect (Section 4.1)
The condition to determine k (eq. (28)) and the free-entry condition (eq. (29)) are
rewritten here:

0

(k)
r+

= (1

)

q( )(r + + )(r + )(r + + + (k))

0

(k)(p

k) + (r +

(1

) p

k

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

0

(k)

b)

z

(r +
r+

r+

+ )b

+ (k)
r+

;

= :

The comparative statics analysis is provided below to investigate the e¤ects of the35policy parameters:
2

32

2

3

32

3

6 C11 C12 7 6 dk 7 6 D11 D12 7 6 db 7
4
54
5=4
54
5;
C21 C22d
D21 D22dz

where:

C11 =

00

(k)
r+

) 00 (k) (p

+ (1

k) +

0

(k)
r+

C12 =

z

(r +
r+

+ )b

7 0;

< 0;

C21 = 0 from the …rst-order condition (eq.(28)),

C22 =

r+ +
q 0 ( )(p(r + )(r + + + (k))

D11 =

(1

D12 =

D21 =

w

) 0 (k)(r +r+(1q( )(r + + (k))
r+

k)

+ )

< 0;

< 0;

) 0 (k)> 0;
r+(1)q( )(r +(r + )(r + )(r ++ ) (k)< 0;+ + (k))

and

D22 =

(1

)q( )(r + + )(r + + (k))> 0:(r + )(r + )(r + + + (k))

If C11 > 0; the second-order condition is satis…ed such that the optimal amount of
input maximizes the weighted product of the net return from the job match.

36

Social E¢ ciency (Section 4.2)
We attempt to solve for the social optimal allocation of (k; ) using the dynamic
programming method. The following value V (u; a; n) is maximized with respect to k and:rV (u; a; n) = maxf(1

u

a

n)(p

k)

+V2 [ (k)(1ua( + )a] + V3 [ an)u+ V1 [ (1uan) + nq( )u]n]g;

where:

V1 =

@V
@V
@V
; V2 =; V3 =:@u@a@n

We surmise that the value takes a linear form in u; a; and n and verify that ourestimate is correct. Assuming V = P0 + P1 u + P2 a + P3 n, where Pi s are parameters, it isthen recognized that V1 = P1 ; V2 = P2 and V3 = P3 : The …rst-order conditions are given:=( )] where ( ) =P1 q( )[1q 0 ( ) =q( );P2 0 (k) = 1:

(30)

(31)

The envelope conditions are obtained by:

(p

k) +

+ P1 (r + + q( )) + P2 (k) = 0;(pk) + P1 + P2 (r +

(p

k)

P1 (

(32)

P3 = 0;(33)) + P2 (k) + P3 (r + ) = 0:(34)

+ + (k))Eqs. (32) - (34) solve for:37h+ ) (p(r + )(r +
P1 =

P2 =

(r + )(r + + q( ))(r +k) + r+ r++ (k)+ + (k)) +i(k) q( );[(r + )(r + ) + q( )(r + + )](p k) + (r + ):(r + )(r + + q( ))(r + + + (k)) + (k) q( )Substituting P1 into eq. (30) yields:

q( )(r + + )(1(r + )(r + + + (k))( ))(pk) + ( )

r+

+ (k)
r+

= :

(35)

Recall the free-entry condition in the decentralized economy (eq. (14)):q( )(r + + )(1(r + )(r + + + (k))
If the Hosios condition (

=

)(p

k) +

r+

+ (k)
r+

= :

( )) is satis…ed, the entry level coincides with thesocially optimal level.
In a similar manner, substituting P2 into eq. (31) gives:
0(k)f[(r + )(r + ) + q( )(r + + )](p k) + (r + )(r + )(r + + q( ))(r + + + (k)) + (k) q( )

g

= 1:

Using eq. (35), we obtain:0(k)(r + )=1+ ( )
q( )(r + + )0(r + ) +
r+

(k)

:

(36)

Correspondingly, eq. (13) is rewritten using eq. (14):0(k)(r + )=1+
q( )(r + + )(r + ) +
r+

0

(k)

:

If the Hosios condition is satis…ed, the amount of input for workplace safety coincides
with the socially optimal amount.

Timing of Decisions and Methods (Section 4.3)
38

(a) A …rm recruits a worker, chooses the amount of input for workplace safety k andthen bargains the wage with the worker.
A …rm chooses the optimal value of k to maximize the value of a job being actively
occupied J (eq. (6)):

max rJ = pw(k)k + (k)(JaJ) + (VJ);

where:

w(k) =

p

k+

r+

+ (k)
r+

:

This wage takes the same form as that derived in Section 3 (eq. (12)). Because k is apredetermined variable, but w is not, the wage is bargained instantaneously in responseto any change of k. Therefore, the wage is a priori function of k, w(k). The …rm choosesk, taking into account that the wage varies with k. The …rst-order condition is given by:0(k)(JaJ) = 1 + w0 (k):Using eq. (9) and (12), we obtain:

0

r+

(k)
(1
+ + (k)

)(p

k)

r+

+ (k)
r+

=1+

1+

0

(k)
r+

;

which coincides with eq. (13), the condition to determine k through the Nash bargaining solution.
(b) A …rm recruits a worker, and then simultaneously chooses the amount of input
for workplace safety k and bargains the wage with the worker.
Because the choice of k and the bargaining over w are made simultaneously, a …rm
chooses k, taking w as given. The …rm’s objective function is:

max rJ = pw

k + (k)(Ja39J) + (VJ):

The …rst-order condition is:0(k)(JaJ) = 1:Substituting eq. (9) and (12) rewrites the …rst-order condition as:
0

r+

(k)
(1
+ + (k)

)(p

k)

r+

+ (k)
r+

= 1:

Comparing with eq. (13), the marginal cost with respect to k is overstated by0

1 + r+(k)(= w0 (k)): This implies that the …rm purchases the lower amountof input for workplace safety in this framework.

40

Table 1: Paramtersparameter

values



2



1
0.5
0.001



0.02
0.8
0.01
0.005



0.2



0.1
()

0.005

()

0.01

Table2: Numerical results









()



0.2 0.005 1.6858 0.0234 1.771 1.039



0.1 0.01



0.2 0.005 1.7012 0.0233 1.771 1.043



0.2 0.005 0.7949 0.0654 0.795 0.356



0.1 0.01

1.5789 0.0365 1.869 1.005



0.1 0.01

0.2771 0.0827 0.891 0.421



0.1 0.01

1.5925 0.0363 1.869 1.010



0.2 0.005 0.2345 0.0604 0.794 0.387

0.2435 0.0878 0.893 0.395

Note that we de…ne () = 0001 in this numerical exercise.

41

Figure 1: Injury Rate and Unemployment
PolandRomania

0.4

0.35
0.3

0.3

0.25
0.2

0.2

0.15
0.1

0.1
0.05

0
1994

1996

1998

2000

2002

2004

0

2006

1994
-0.05

-0.1

1996

1998

2000

2002

2004

2006

-0.1
-0.2

-0.15
-0.2

-0.3

unemployment rate

unemployment rate

injury rate

injury rate

Sweden

Spain

0.25

0.3

0.2

0.2

0.15
0.1

0.1

0.05

0

0

1991

1993

1995

1997

1999

2001

2003

2005

2007

1995
-0.05

1997

1999

2001

2003

2005

-0.1
-0.1
-0.15

-0.2

-0.2

-0.3

-0.25

unemployment rate

injury rate

unemployment rate

injury rate

Germany

Finland

0.2

0.1
0.15

0.05
0.1

0
1993

0.05

1995

1997

1999

2001

2003

2005

2007
0
1995

-0.05

1996

1997

1998

1999

2000

2001

2002

2003

-0.05

-0.1
-0.1

-0.15

-0.15

unemployment rate

injury rate

unemployment rate

42

injury rate

2004

2005

2006

Hungary

Greece

0.15

0.25
0.2

0.1
0.15
0.05
0.1
0
1994

0.05
1995

1996

1997

1998

1999

2000

2001

2002

2003

2004
0

-0.05

1994

1996

1998

2000

2002

2004

2006

-0.05
-0.1
-0.1
-0.15

-0.15

unemployment rate

injury rate

unemployment rateItaly

injury rate

United Kingdom

0.15

0.2

0.15

0.1

0.1

0.05
0.05

0
1993

1995

1997

1999

2001

2003

2005

2007

0
1991

-0.05

1993

1995

1997

1999

2001

2003

2005

-0.05

-0.1

-0.1

-0.15

-0.15

unemployment rate

injury rate

unemployment rate

injury rate

CanadaMexico

0.3

0.3

0.25

0.2

0.2
0.1

0.15
0
1995

0.1
1997

1999

2001

2003

2005

2007

0.05

-0.1

0
-0.2

1987
-0.05

-0.3

1989

1991

1993

1995

1997

1999

2001

2003

-0.1
-0.15

-0.4

unemployment rate

unemployment rate

injury rate

43

injury rate

2005

2007

JapanUnited States

0.3

0.25
0.2

0.2

0.15
0.1

0.1

0.05
0
1976

1981

1986

1991

1996

2001

0
-0.051979

2006

-0.1

1989

1999

-0.1
-0.2

-0.15

-0.3

unemployment rate

unemployment rate

injury rate

44

Injury rate

Figure 2: EquilibriumEq.(13)

Eq.(14)k
Figure 3: An Increase in Productivity

Eq.(13)

Eq.(14)k45

Figure 4: An Increase in Healing RateEq.(13)



Eq.(14)k

Figure 5: An Increase in Sickness BenefitEq.(28)Eq.(29)k46

Figure 6: An Increase in Unemployment BenefitEq.(28)Eq.(29)k

Figure 7: An Increase in 

Eq.(13)



Eq.(14)k47