introduce simultaneously a bargaining power for workers and an EW component. Ž
. Lindbeck and Snower 1991 study the interactions between the EW and insider–
outsider theories but do not formalize the recruiting costs as the result of matching Ž
. frictions. Manning and Thomas 1997 build a partial equilibrium model based on
job search theory in the presence of incentives problems. This paper is organized as follows. Section 2 presents the model and its main
assumptions. The steady-state equilibrium and the two regimes are studied in the third section. The model is extended in Section 4 to incorporate an endogenous
monitoring technology. Section 5 introduces heterogeneity among workers, and Section 6 concludes.
2. The model
Ž .
The basic framework is similar to that of Pissarides 2000, Chapter 1 except that unobservable shirking is allowed.
2.1. Main assumptions Consider an economy composed of a continuum of infinitely lived homoge-
neous workers of measure equal to one, and of a continuum of identical firms holding at most one vacancy. Time, denoted by t, is continuous.
Workers and firms are risk neutral. The utility function of an individual whose Ž .
q
4 Ž .
q
4 consumption and effort trajectories are c t , t g
R and e t , t g
R is:
exp yrt c t y e t
d t ,
Ž .
Ž . Ž .
H
q
R
where r g R
q
is the rate of time preference. Ž
. Ž
Following Shapiro and Stiglitz 1984 , there are two levels of work intensity e .
or 0 and workers’ effort is imperfectly observed by employers. The productivity per unit of time of a job-worker match is y if the work intensity is e, and 0
otherwise. Inspections obey a Poisson process with arrival rate l g R
q
and the punishment for shirking is to be fired.
The labour market has matching frictions. The flow of job creations per unit of Ž
. time is m u, Õ where u is the number of unemployed, Õ the number of vacancies
Ž . and m .,. exhibits constant returns to scale. Let u be defined as Õru, which is
commonly referred to as labour market tightness. Further, if q denotes the rate at which a vacancy fills, it is given by
m u,Õ 1
Ž .
q s s x
,1 ,
ž
Õ u
Ž . Ž .
Ž and hence q s q u with q’ . - 0. As the exit rate out of unemployment is m u,
. Ž .
Õ ru, note that it equals u q u .
Each filled job is destroyed according to a Poisson process with arrival rate
q
Ž .
Ž . s g
R . As the flow out of unemployment is m u, Õ suu q u , while the flow in Ž
. Ž
. is s 1 y u , the steady state which requires that these two flows are equal is
characterized by an unemployment rate equal to: s
u s .
1
Ž .
s q u q u
Ž .
2.2. The Õalue functions Ž
Since there is zero productivity while shirking, an active equilibrium with .
positive production is such that the incentive-compatibility constraint is always fulfilled.
A non-shirker is a worker who chooses not to shirk in all periods while attached to the current job. The lifetime expected utility of a non-shirker who earns wage w
Ž . is denoted by W w and obeys the following asset pricing equation:
E
rW w s w y e q s W y W
w ,
2
Ž . Ž .
Ž .
Ž .
E U
E
where W is the lifetime expected utility of an unemployed worker and w the real
U
Ž . wage. If W
represents the AassetB value of employment, Eq. 2 simply states
E
that the opportunity cost of holding it, rW , is equal to the current income flow w
E
minus the disutility of effort e plus the expected capital loss flow, the third term of the RHS.
The lifetime expected utility of a currently employed worker who chooses to
S
Ž . Ž
. shirk during a spell of length d t, denoted by W
w; d t S for shirking , satisfies:
E S
W w ;d t s wd t q exp yrd t
P min t ,t Fdt W
Ž .
Ž .
Ž .
E s
l U
q 1 y P min t ,t Fdt W w ,
3
Ž .
Ž . Ž .
4
Ž .
s l
E
Ž where t is the length of time until the next inspection an exponential distribution
l q
. Ž
of parameter l g R
and t the random duration of a job an exponential
s q
.
3
Ž . distribution of parameter s g
R . According to Eq. 3 , during the time interval of length d t, the worker receives the real wage wd t and suffers no disutility; he
loses his job if he is caught shirking or if his job is destroyed by an idiosyncratic shock. If neither of these two events occurs during the time interval of length d t,
the employed worker stops shirking in all the subsequent periods: his lifetime
3
Ž .
The random variable min t , t is characterized by an exponential distribution of parameter
s l
sq l.
Ž . Ž .
expected discounted utility is equal to W w . After some manipulation, Eq. 3
E
yields: W
S
w ;d t s wd t q 1 y rd t s q l d tW
Ž .
Ž . Ž
.
E U
q 1 y s q l d t W w
q o d t ,
4
Ž .
Ž . Ž .
E
s W w q ed t y 1 y rd t ld t W
w y W q o d t ,
4
Ž . Ž
. Ž .
Ž .
E E
U
Ž . with lim
o d t rd t s 0. A worker who draws a benefit from cheating his
d t ™ 0
employer over a period of time of length d t, chooses to shirk all the time. When d t approaches zero, the worker’s optimal strategy is not to shirk if and only if
S
Ž .
Ž . w
Ž Ž .
.x W
w; d t y W w , e y l W w y W d t F 0, that is
E E
E U
e W
w y W G .
4
Ž . Ž .
E U
l To prevent the employed worker from shirking, he must get a rent at least equal to
erl. Indeed, a shirker saves the disutility of effort e but bears a capital loss Ž .
Ž W w y W
if he is dismissed an event which occurs with an instantaneous
E U
. probability l .
Ž .
As in Pissarides 2000 , recruiting is costly. The instantaneous advertising costs Ž
. are g . Thus, the expected profits of a vacancy W
satisfy the following Bellman
V
equation in continuous time: rW s yg q q u
W y W ,
5
Ž . Ž .
Ž .
V J
V
where W is the value of a filled job.
J
Ž .
Finally, the value functions of an unemployed worker W and of a filled job
U
Ž .
W obey the following asset pricing equations:
J
rW s b q u q u W y W
, 6
Ž . Ž .
Ž .
U E
U
rW w s y y w q s W y W w ,
7
Ž . Ž .
Ž .
Ž .
J V
J
Ž . where b represents unemployment benefits and u q u
the exit rate out of Ž .
unemployment. Note that, according to Eq. 6 , an unemployed worker who finds a job becomes a non-shirker.
Under a free-entry condition, the expected profits of a vacant job are zero Ž
. Ž .
Ž . W s 0 . Thus, from Eq. 5 , W s grq u . By substituting this expression into
V J
Ž . Eq. 7 , we get the Õacancy supply condition which gives a decreasing relation
Ž .
between labour market tightness u s Õru and the real wage: g
y y w s r q s .
8
Ž .
Ž .
q u
Ž .
2.3. The bargaining process The wage flow w that will be paid to the worker during the lifetime of the job
is determined through bilateral bargaining.
4
The outcome of the bargaining
4
We rule out upward-sloping age–earnings profile. A review of other devices to motivate Ž
. employees is offered by Ritter and Taylor 1997 .
process corresponds to the asymmetric Nash solution with threat points equal to the employer’s and worker’s respective values of continued search. The worker’s
Ž .
bargaining power is denoted by b g 0, 1 . Furthermore, the NSC gives the minimum wage that must be paid by an employer to prevent his worker from
shirking. Indeed, an employer will never agree to pay a wage which does not satisfy the incentive constraint. The Nash program can be written as follows:
b 1yb
w s arg max W
w y W W w y W
,
Ž . Ž .
E U
J V
Ž .
wg 0 , y
Ž . subjected to Eq. 4 .
Ž . Ž .
The outcome of the bargaining is represented in Fig. 1. From Eqs. 2 and 7 , the Pareto frontier of the bargaining set is given by:
y y e y rW
U
W w y W q W s
.
Ž .
E U
J
r q s The Nash solution is efficient; it is the highest level of the asymmetric Nash
product situated on the Pareto frontier. Two cases can be distinguished: a binding and a non-binding NSC. In Fig. 1a, the NSC is not binding, and the asymmetric
Nash solution is given by the tangency point between the Pareto frontier of the
Ž bargaining set and the highest Nash product curve
the locus of the pairs Ž
. .
W y W , W associated with the same Nash product . In Fig. 1b, the NSC is
E U
J
binding, and the bargaining outcome is the point of intersection of the NSC and the Pareto frontier of the bargaining set.
2.3.1. The NSC is not binding The threat to shirk is not credible and the equilibrium wage, denoted by w, is
˜
the wage that would be obtained in the absence of unobservable shirking, that is: 1 y b
W w y W
s b W w y W ,
9
Ž .
Ž . Ž .
Ž . ˜
˜
E U
J V
Ž . Ž .
From Eqs. 2 and 7 , the wage expression is given by: w s 1 y b
rW q e q b y. 10
Ž .
Ž .
Ž .
˜
U
The FNW is a weighted mean of the workers’ productivity and the workers’ reservation wage.
2.3.2. The NSC is binding The FNW violates the incentive constraint: the threat to shirk is credible.
Employers must use the wage as a motivation device. The EW, denoted by w,
ˆ
Ž . Ž .
satisfies Eq. 4 at equality. Using Eq. 2 , we find: e
w s r q s q e q rW .
11
Ž .
Ž . ˆ
U
l
Fig. 1. The bargaining solution.
Ž .
The EW has two components: the reservation wage e q rW
and a linear
U
function of effort. It depends on workers’ productivity only through rW .
U
Ž .
Finally, note that the employees’ rent W y W vanishes if the inspection rate
E U
is infinite and if workers have no bargaining power. The wage is then equal to the reservation wage.
lim w s lim w s e q rW .
ˆ ˜
U
l ™q`
b ™0
3. Steady-state equilibrium
