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TRAINING, JOB SECURITY AND INCENTIVE WAGES
MARGARITA KATSIMI
CESIFO WORKING PAPER NO. 955
CATEGORY 4: LABOUR MARKETS
MAY 2003
An electronic version of the paper may be downloaded
• from the SSRN website: www.SSRN.com
• from the CESifo website: www.CESifo.de
CESifo Working Paper No. 955
TRAINING, JOB SECURITY AND INCENTIVE WAGES
Abstract
This paper considers the optimal level of firm-specific training by taking into account the
positive effect of training on the expected duration of workers’ current employment. In the
framework of an efficiency wage model, a short expected job tenure represents a disamenity
that reduces the penalty from shirking. As this disamenity increases, workers have an
incentive to continue providing a positive level of effort only if they are compensated by a
higher wage. We endogenize the employment separation rate by introducing firm-specific
training. Firm-specific training creates a rent that is lost if the worker is separated from the
firm. As a result, the firm will be more reluctant to fire its trained workforce in a recession.
This implies that firm-specific training can decrease current wages as it implies a credible
commitment to lower future labour turnover.
JEL Code: J41, J33, J24.
Keywords: efficiency wages, firm-specific training.
Margarita Katsimi
Dept of Int.and European Economic Studies
Athens University of Economics and Business
76, Patision Avenue
10434 Athens
Greece
Mkatsimi@aueb.gr
I am indebted to Apostolis Philippopoulos for many helpful discussions. I would like to thank
the participants of the CESifo workshop on Employment and Social Protection, June 2001,
and in particular my discussant Josef Falkiner, for valuable suggestions for improvement. I
am also grateful to Gylfi Zoega and Alison Booth for valuable comments and suggestions on
an earlier version of this paper. Finally, I would like thank two anonymous referees for their
insightful comments.
1 Introduction
As early developments of the human capital theory suggest, worker’s uncertainty about future
employment has a negative impact on investment in …rm-speci…c training. According to Becker
(1962, 1964) and Oi (1962), workers will be reluctant to bear the cost of this investment if there is
a positive probability of being dismissed and not bene…t from investment returns. In Booth and
Chatterji (1989) workers will be induced to undertake speci…c training in sectors where there is a
positive probability of being redundant if a redundancy payment is part of the …rm’s contract. This
paper argues that the relationship between …rm-speci…c training and the level of job security is twofold:
1 Firstly, investment in …rm-speci…c training creates a surplus in employment relationships.
The …rm can enjoy part or all of this rent as long as the worker remains employed. This implies
that in a recession the optimal policy of the …rm will be to retain the employment relationship
even if the wage paid to the worker exceeds his marginal product in an alternative job. As a
result, investment in …rm-speci…c training becomes a credible commitment to lower future labour
turnover. Secondly, a credible promise for high future employment will decrease the probability
of being dismissed in a downturn and, in the framework of an e¢ciency wage model, it will lower
wages. The ability of …rm-speci…c training to provide this commitment mechanism will have a
positive e¤ect on the level of …rm-speci…c training. Speci…cally, the …rm will invest in …rm speci…c
training even if marginal cost exceeds expected marginal return since training ‘buys’ the …rm
a commitment technology. This implies that the marginal value of training exceeds expected
marginal return.
The idea that job security will have a negative impact on e¢ciency wages has been analyzed
by Katsimi (1995), Saint-Paul (1996) and Fella (2000).2 All authors base their arguments on
di¤erent versions of the Shapiro and Stiglitz (1984) model which shows that in an environment
1 In this pap er job security is de…ned as a negative function of the probability of being …red in a recession.
2 Empirical evidence suggesting a wage premium based on the job’s unemployment risk has been found by
Abowd and Ashenfelter (1981), Adams (1985) and Li (1986). Research on this topic dates back to the pioneering
work of Hall (1970, 1972).
1
where employers are unable to monitor workers’ on-the-job e¤ort costlessly, unemployment can
induce workers to provide a positive level of e¤ort. In this framework, a short expected job tenure
represents a disamenity that reduces the penalty for shirking. As this disamenity increases, workers
will have an incentive to continue providing a positive level of e¤ort only if they are compensated
by an increase in the wage rate. The optimal incentive wage will increase with the risk of being
…red in a recession. In other words, the cost of labour decreases when the …rm is expected to
retain its workforce in a recession. This, however, requires either that the …rm can commit to a
low …ring policy or that workers expect that low …ring is the optimal future employment policy of
the …rm. In Saint-Paul (1996), …rms’ commitment to a future level of employment is represented
by the fact that employment is set one period in advance. In Katsimi (1995) and Fella (2000), the
imposition of redundancy payments by the government leads to a higher level of expected future
employment in a recession.
This paper considers the case where the above commitment mechanism is endogenous. In the
absence of an imposed turnover cost such as a redundancy payment, a pro…t-maximizing …rm
may have an incentive to voluntarily increase the cost of dismissing the existing workforce in a
recession. In that context, the level of …rm-speci…c training can be viewed as a ‘commitment’
to higher future employment in a recession. It is assumed that speci…c human capital is a team
investment concerning all workforce and involves costs and returns only for the …rm.3 Forward
looking workers perceive that a …rm who has undertaken investment in …rm-speci…c training will
…re less in a recession. Since workers are uniformly trained at an optimal level, …ring workers
in a recession implies the loss of some of the return to training. If a recession occurs in the
period when the investment return is realized, a reduction in employment implies the loss of the
return from the human capital acquired by the dismissed workers. As a result, a positive level of
training in the previous period will increase optimal employment in a recession. A higher level of
expected employment in a downturn of the economy will decrease the probability of being …red
3 For simpli…cation we neglect the uncertainty stemming from quitting.
2
in a recession for each worker. In other words, …rms’ decision to undertake …rm speci…c training
improves the ‘security’ of workers’ current employment and increases the penalty of shirking with
a negative impact on incentive wages. This wage-reducing e¤ect increases the value of human
capital investment leading to a higher level of …rm-speci…c training. Ceteris paribus the level of
investment is higher the higher is the elasticity of the …ring probability with respect to the level
of training.
The rest of the paper is organized as follows: Section 2 presents the basic arguments of the
paper in the framework a two-period model. In section 2.1, wage and employment determination
is discussed. Firm’s decision to invest in …rm-speci…c human capital is described in section 1.3.
Section 3 extends the basic model to an in…nite horizon framework. The last section concludes.
2 A two-period model of employment and training
2.1 Wage and employment determination
We …rst present a two-period model that builds on Shapiro and Stiglitz (1984) shirking model by
endogenizing workers’ horizon on their job and allowing for investment in …rm-speci…c training. We
will …rst consider wage and employment determination in the absence of training. The economy
consists of a continuum of …rms indexed by j, where j 2 [0; 1]. Each …rm produces output by the
following production function in each period:
Yj;t = egi
j;tNa
j;t (1)
where Y is the level of output, N denotes the level of employment, e is the level of e¤ort
produced by each worker and and the subscript t denotes the time period, t = 1; 2. Output
depends on a productivity shock that can be either high, gH, or low, gL; gH > gL. We will
assume that g takes the high value in the …rst period. In the second period, g will take the low
value with probability h whereas it will take the high value with probability 1 ¡h:4
4 This assumption is consistent with the two-state Markov process assumed for the in…nite horizon case later
on.
3
Firms set wages unilaterally in order to maximize pro…ts. There is asymmetric information
between employers and employees about the on-the-job e¤ort of the latter: the monitoring technology
is imperfect. Workers are risk neutral and their utility depends on the amount of goods
they can consume with their wage, w, and on their job e¤ort, e. The instantaneous utility, u; of
the representative employed worker in each period is given by:
ut = wt ¡ e (2)
In the Shapiro and Stiglitz (1984) model, wages are paid before e¤ort is observed and the
penalty for shirking is the loss of the worker’s job. However, in the framework of a …nite horizon
model this would imply no punishment for shirking in the last period. In the presence of a last
period it is clear that …rms should hold some of the workers pay for the end of the period in order
to prevent workers from shirking. Thus, although in the in…nite horizon version presented in a
later section we will retain the timing of the Shapiro and Stiglitz (1984) model, in this section
we will modify the timing along the lines of the simple life cycle incentive model presented by
Carmichael (1989) in order to ensure enforceability. Speci…cally, we will assume that workers’ pay
consists of two parts: a legal minimum wage, wm, that is paid to all workers independently from
their level of e¤ort and a bonus payment, Bt, that is paid after e¤ort has been monitored only to
non-shirkers.5
wt = wm + Bt (3)
5 Note that the presence of a legal minimum wage is crucial. In the absence of a legal minimum wage the cost
minimizing wage paid by the …rm will be the same in each p eriod, workers enjoy no rents and the threat of being
…red is ine¤ective [Carmichael (1989)].
4
Table 1 summarizes the timing of events.
Table 1
Period 1
* the prod. shock g1 is realized, g1 = gH
* …rms set wages, w1 and employment, N1
* workers choose e¤ort, e and production takes place
* workers who chose e = 0 get …red and receive wm with prob. p
* workers who chose e > 0, receive wm and a bonus B1
Period 2
* the prod. shock g2 is realized
* …rms set wages, w1and employment, N1 :
-workers can lose their job with prob. hq
-unemployed can get a job with prob. ’
* workers choose e¤ort, e and production takes place
* workers who chose e = 0 receive wm with prob. p
* workers who chose e > 0 receive wm and a bonus B2
At the beginning of the …rst period, …rms set employment and announce a bonus payment by
taking into account the optimal behaviour of workers in the next stage. Then, each worker decides
either to shirk and provide minimal e¤ort (e = 0) or not to shirk in which case he provides some
…xed positive level (e = ¹e). If a worker chooses to shirk, there is a probability p that his e¤ort
will be monitored in which case will not receive the bonus payment and he will be dismissed.
In the second period, …rms observe the productivity shock, g2; and reset wages and the bonus
payment while adjusting their workforce so that the marginal productivity condition for pro…t
maximization is satis…ed. Again, there is a probability p that the e¤ort level of each worker will
5
be monitored, in which case if e = 0 the worker will only receive the minimum wage.
All employed workers may lose their job in the second period if the …rm switches to a low state
with probability q. Thus, uncertainty in the …rst period results from the unknown value of the
productivity shock in the next period. We denote by q the probability that a worker will lose his
job in a low state in period 2:
q =
NH
1 ¡ NL
2
NH
1
(4)
Clearly, the lower the expected level of employment in the low state, NL
2 , the higher the
probability that a worker will be …red.6
Both workers and employers are forward-looking: workers decide on their level of e¤ort by
taking into account …rms’ current and future optimal behaviour and …rms decide on the level of
training, wages and employment by taking into account workers’ current and future incentives.
Thus, we solve our model by backward induction.
2.1.1 The last period
Firms set the value of the bonus payment so as to minimize labour cost subject to being able to
prevent workers from shirking. Workers will provide the amount of e¤ort that maximizes their
utility. After the realization of the productivity shock in period 2, there is no uncertainty for
workers who do not shirk. The value of not shirking in the last period is given by equation (2)
when e = ¹e.
V NS
2 = wm +B2 ¡ ¹e (5)
Since shirking involves the risk of being caught and receiving only the minimum wage with
probability p; the value of shirking in the second period is given by
6 We assume that concerns about their reputation prevents …rms from falsely claiming that cheating has occured
in order to avoid the bonus payment. Although we do not model this incentives, young workers may provide e¤ort
at a higher wage if they suspect that they may be trated badly in the future.
6
EV S
2 = wm + (1¡ p)B2 (6)
Each worker will choose a positive level of e¤ort as long as the value of not shirking exceeds
the value of shirking:
V N
2 ¸ EV S
2 (7)
Solving expression (7) as an equality gives the lower level of real wage at which workers will not
have an incentive to shirk, wE
2 :
wE
2 = wm +
¹e
p
(8)
Each pro…t maximizing …rm will increase its workforce until the real wage equals the marginal
product of labour :
wE
2 = a¹egi2
Na¡1
2 (9)
Solving the pro…t maximization condition for N, gives the equilibrium level of employment:
N2 = (
pwm + ¹e
pgi
2a
) 1
a¡1 (10)
2.1.2 The …rst period
At the beginning of the …rst period, each worker selects an e¤ort level in order to maximize his
expected utility in the two periods by taking into account optimal policies in the next period. A
worker who decides not to shirk in the …rst period will always face a probability of being …red in
the next period in the presence of a negative productivity shock. Assuming a zero discount rate,
the value of not shirking is given by
EV N
1 = wm +B1 ¡ ¹e+ (1 ¡ hq)max[V NS
2 ;V S
2 ] +hqU2 (11)
7
where U2 is the value of being unemployed in period 2, h is the probability of a recession, and
q is the probability of being …red in a recession de…ned by equation (4).
At the beginning of the second period, workers may …nd themselves out of job either due to
shirking in the …rst period or due to the …rm’s transition to a low state. The value from shirking
in the …rst period can be written as:
EV S
1 = (1 ¡ p)fwm + B1 +(1 ¡hq)max[V NS
2 ;V S
2 ] + hqU2g + p(wm + U2) (12)
The terms in the squiggled brackets on RHS of equation (12) states the expected utility of a
shirker who does not get monitored in the …rst period and the remaining terms give his expected
utility when he gets caught and …red in period 1.
Assuming no unemployment bene…t, and denoting with ’ the probability that a worker …nds
a job in the second period, the value of being unemployed is given by:
U2 = ’maxfEV S
2 ;V NS
2 g (13)
By using equations (11) and (12), we derive that the worker will provide a positive level of
e¤ort in the …rst period as long as
B1 ¸
¹e
p ¡ (1 ¡hq)(1 ¡’)max[UNS
2 ; US
2 ] (14)
Substituting (14) into (13) gives the lower wage at which workers will provide a positive level
of e¤ort:
wE1
= (wm +
¹e
p
)(’+ hq(1 ¡ ’)) (15)
The wage that satis…es the NSC will depend positively on the probability of exiting unemployment
since a high ’ decreases the penalty from shirking. A better monitoring technology
8
re‡ected in a high p will increase the unemployment probability for each worker who decides to
shirk, leading to a lower expected utility of shirking. Similarly, a decrease in the disutility of
e¤ort, ¹e, will imply a lower opportunity cost of not shirking, which will lower w1. Finally, the
e¢ciency wage will depend positively on the probability of being …red for cyclical reasons. As the
probability of being …red in a low state increases, expected job tenure falls and so does workers’
expected utility from being employed. Workers view a long job tenure as a form of compensation.
Thus, if for any reason expected job tenure decreases, wages should increase for workers’ expected
utility to remain constant and vice versa. In the context of this model, a long job tenure implies
a low value of q and h. One can see from equation (4) that the probability of being …red in a
recession, q, depends negatively on the expected level of employment in the low state, NL
2 . In
Figure 1, the non-shirking condition is satis…ed at a wage rate equal to w¤. As expected job
tenure increases, the expected utility from not shirking will now exceed the expected utility from
shirking. This implies that the NSC is now satis…ed for a lower level of wage. The NSC shifts to
the left and the level of the e¢ciency wage decreases to w0 .
Employment in the …rst period can be derived by the marginal productivity condition:
wE
1 = ¹eagH1
Na¡1
1
2.2 Firm-speci…c training
We will now consider the …rms’ decision to invest in speci…c training. We assume that in the
…rst period the …rm decides to invest on b units of …rm-speci…c training for each worker. Firms
bear the total cost of training , c(b)N1: We assume that the workers’ e¤ort required by training
is perfectly observable so that shirking during training is not an option for workers in the …rst
period.7 Training is characterized by diseconomies of scale which implies that c(b) is a twice
7 Therefore, even if the cost of training represents an extra payment to workers it will not a¤ect the incentive
wage since all workers will receive this payment in the …rst period. Alternatively one can assume that training
requires e¤ort that is not perfectly observable. In that case, the cost would take the form of higher wages in the
…rst p eriod due to a higher e. However, this assumption would complicate the algebra without altering any of the
results.
9
w’ w* w’’ w
0
EVs-EVns
Figure 1: The non-shirking condition
di¤erentiable, strictly convex function. Returns to training are realized in the second period. We
assume that these returns by the …rm are not threatened by the possibility of a premature quit. b
units of training in the …rst period will increase productivity by r(b) in the next period. r(b) is a
twice di¤erentiable, strictly concave function that can represent an index of worker quality.8 This
implies that uncertainty over investment returns stems only by the uncertainty of the economic
environment.9 It is assumed that the expected return is a linear function of employment so that
output in period 2 is de…ned as:
Y2 = gi(Ni2
)a + r(b)minfN1;N2g (16)
We assume that training improves only …rm-speci…c skills so that in period 2 the productivity
8 According to Hart and Moutos (1995), this index may represent the speed at which the product is produced or
the number of saleble units achieved at a given rate of production so that more investment leads to more saleable
units.
9 Hashimoto (1979, 1981) among others investigates the imp ortance of uncertainty over investment returns to
speci…c training when there is uncertainty over r(b).
10
in …rm k of a worker who worked in …rm k in the previous period exceeds the productivity in …rm
k of a worker employed in …rm l in the previous period by r(b).The introduction of training makes
the pro…t maximization decision of the …rms dependent across the two periods. Expected pro…ts
in the two periods are given by
E(¦) = ¦1 + E(¦2)
= Y1 ¡ wE1
N1 ¡c(b)N1 + E[Y2 ¡ wE2
N2] (17)
In period 2, there is no uncertainty and employment in the low state must satisfy the following
FOC:
gLaNa¡1
2 +r(b) = w2 (18)
Solving equation (18) for N2 gives the equilibrium level of employment in the low state in the
second period.
NL
2 = (
pwm + ¹e ¡ r(b)p
pga
) 1
a¡1 (19)
From equation (19), it is easy to see that employment in a recession will depend positively
on the level of b so that the introduction of training will lead to higher employment in period 2,
#NL
2
#b > 0. Firms will be more reluctant to …re workers after a negative productivity shock, since
this is associated with losing part of the rent created by training. Note that in our analysis we
assumed that monitoring takes place in per capita terms. If one were to assume monitoring per
units of output, then one cannot exclude the possibility that the probability of being monitored
in equation (20) would be a negative function of r(b). However, the positive e¤ect of training on
NL
2 would still carry through if one assumes economies of scale in monitoring technology.10 One
10 For example, a camera can be used for monitoring the production of more than one unit of output. A proper
assessment of the e¤ect of productivity on the monitoring probability would, however, require endogenizing p: One
11
can see from equation (20) that an increase in NL
2 will have a negative impact on the e¢ciency
wage in the …rst period.
In period 1, …rms choose the level of employment and training that maximizes ex-ante pro…ts,
given the level of the real wage that satis…es the NSC. The …rst order condition with respect to
employment from equation (17) is:
gaNa¡1
1 +(1 ¡h)r(b) = w1 +
#w1
#N1
N1 + c(b) (20)
Solving for employment in the …rst period from equation (20), gives:
N1 = (
(pwm + ¹e)[’ +h(1 ¡ ’)] + p[c(b) ¡ (1 ¡ h)r(b)]
pagH
)
1
a¡1 (21)
The …rm’s choice variable for investment in speci…c training is b. After some rearrangements,
the …rst order condition with respect to b can be written as:
F = h
#NL
2
#b
[r(b) + (wm +
¹e
p
)(1 ¡’)]
¡ fc(b)bN1 ¡ r(b)b[hNL
2 + (1 ¡ h)N1]g = 0 (22)
The …rst part of equation (22) re‡ects the decrease in labour cost due to the reduction of
the dismissal probability for cyclical reasons and the remaining part represents the di¤erence
between the marginal cost of training and the expected marginal return of investment in the second
period. Solving equations (21) and (22) for N1 and b, gives the optimal values of employment and
investment in …rm-speci…c training.
can assume that p = f(m) where m denotes the level of monitoring and f (m)0 > 0: Let us introduce a monitoring
cost that is linear to the level of monitoring cm and solve for the optimal level of m. If the …rm monitors the
level of output, it would maximize the following pro…t function with respect to N2 and m in the second period:
[g(Ni2
)a + r(b)N2](1 ¡ cm) ¡ w(m)N2, where #w
#m < 0:Taking the two FOCs and combining them gives us the
following condition for optimal m: cgNa¡1
2 [r(b);m] + cr(b) = ¡#w
#m . After checking the e¤ect of r(b) on the
optimal level of m by taking the total di¤erential we can conclude that its sign and magnitude is ambiguous.
12
Proposition 1:
Assuming that b0 is the level of training that equalizes marginal cost with expected marginal
return to training, …rms will choose a higher level of …rm speci…c training, b¤ > b0; due to the
positive impact of b on workers’ expectations for future employment.
Proof: Assume that b0 is the level of training which equalizes marginal cost with expected
marginal return to investment. If one substitutes b0 in equation (22), the second term of the
equation will equal zero. Given that the remaining part of equation (22) is positive for positive
values of b, b = b0 does not satisfy equation (22). F = 0 will be satis…ed for a higher level of b if
Fb < 0. The derivative of equation (22) with respect to b is negative since E(¦)bb < 0.11
Finally, we want to compare the level of b derived by equations (21) and (22) with the optimal
level of training in the absence of uncertainty, h = 0.
Proposition 2:
Assume that ¹b is the level of training that equalizes marginal cost with marginal return to
training if the probability of shifting to a low state is zero. The optimal level of investment in
…rm-speci…c training , b¤, will exceed this level, b¤ >¹b, if the elasticity of the …ring probability with
respect to the level of training is signi…cantly high.
Proof:
If h = 0, then equation (22) becomes ¹ F = r(b)b ¡ c(b)b = 0. Assume that ¹b satis…es ¹ F = 0.
By substituting ¹b in equation (21) and then in (22), we obtain:
F(¹b) = w1b ¡hr(¹b)b[N1(¹b ) ¡NL
2 (¹b)] +hr(¹b)
#NL
2
#b
(23)
Given that Fb < 0, b¤ >¹bif F(¹b) > 0.
11 The stability condition E(¦)N1N1 < 0 and E(¦)N1N1E(¦)bb¡E(¦)N1bE(¦)bN1 > 0 is satis…ed for reasonable
parameter values.
13
2.3 Firm-speci…c training as a commitment technology
Several shirking models including the original Shapiro and Stiglitz (1984)model show the negative
e¤ect of labour turnover on incentive wages. However, the usual assumption in the literature is
that this turnover is exogenous to the …rm’s decision. Endogenizing the expected duration of
employment is a natural step since …rms can a¤ect labour turnover in various ways. One obvious
way is …rm-speci…c training. As we showed above, training increases the probability of remaining
employed in a low state, thereby reducing incentive wages. This mechanism has a positive impact
on the optimal level of training. Furthermore, it is important to emphasize the credibility of
training as a mechanism for reducing labour turnover. It is clear that if workers were not forward
looking at the beginning of period 1, the …rm would have an incentive to guarantee employment
continuation in the low state (q = 0). This would imply lower wages from equation (15). However,
such a promise would be time-inconsistent since the …rm would have an incentive to deviate from
that promise in the next period and set employment according to equation (10). Forward looking
workers will choose their level of e¤ort by taking into account the …rm’s optimal policy in the
next period. By its decision to invest in b units of …rm-speci…c training, the …rm commits to the
employment level de…ned by equation (19) in the low state. This commitment is credible since the
…rm bears the cost of training in the …rst period. Alternative ways of reducing labour turnover
such as the imposition of …ring costs or hiring costs are less credible since they impose a cost that
the …rm will bear in a future period. This implies that after workers make their e¤ort choice,
the …rm will always have an incentive to try to reduce this cost. For example, since e¤ort is a
nonveri…able variable, there is always the possibility that the …rm will falsely claim that workers
cheated in order to avoid …ring costs.
3 An in…nite horizon model of employment and training
Next we wish to extend our model to an in…nite horizon framework in order to see whether our
main results depend crucially on the two-period framework of the previous analysis. The only
14
di¤erence with the timing assumptions of the previous section is that wages are now paid at the
beginning of each period after the realization of the productivity shock. This implies that the new
setup is consistent with the e¢ciency wage model of Shapiro and Stiglitz.
3.1 Workers’ behaviour
We assume that workers behave as in the shirking model of Saint-Paul (1996)12 . Fella (2000)
makes similar assumptions in a continuous time model. Each worker will provide a positive level
of e¤ort in period t only if the expected present discounted income of being employed in period
t+1, EV , exceeds the one of being unemployed in period t+1, EU, by a markup that equals the
current level of e¤ort divided by the monitoring probability p:
EVt+1 = EUt+1 +
¹e
p
(24)
If workers are risk neutral with a discount factor ±; the present discounted value of being employed
in the two states can be written as:
V H
t = wH
t + ±[(1 ¡h)EV H
t+1 +h
NL
t+1
NH
t
EV L
t+1 + h(1 ¡
NL
t+1
NH
t
)EUt+1)] (25)
and
V L
t = wLt
+ ±[(1 ¡h)EV L
t+1 +hEV H
t+1] (26)
Equations (25) and (26) say that the value of being employed in the current state equals the
current wage plus the expected present discounted value of the next period. When the …rm is
in the high state, the expected present discounted value will depend on the probability that the
worker will retain his position if the …rmshifts to the low state, NL
t+1
NH
t
. This probability is equivalent
to probability q de…ned by equation (4). A worker who is employed in the low state will remain
employed in the next period. Inserting (24) into (25) and (26) and after some manipulations we
can solve for e¢ciency wages in the two states:
12 We borrow the basic setup regarding workers’ behaviour from chapter 7 of Saint-Paul (1996).
15
wH
t = Ut ¡ ±EUt+1 +
¹e
p
[1 ¡ ± +±h(1 ¡
NL
t+1
NH
t
)] (27)
wLt
= Ut ¡ ±EUt+1 +
¹e
p
(1 ¡ ±) (28)
Assuming zero unemployment bene…ts, the value of being unemployed can be written recursively
as:
Ut = ±[(1 ¡’)EUt+1 + ’(EUt+1 +
¹e
p
)] (29)
where ’ is the exit rate from unemployment. Firms take the exit rate from unemployment
as given since, under our assumptions about g, in the steady state the exit rate is a function of
average employment in each …rm.13
Inserting (29) in (27) and (28) we get:
wHt
= [1 ¡ ±(1 ¡ ’) + ±h(1 ¡
NL
t+1
NH
t
)]
¹e
p
(30)
wL
t = [1 ¡ ±(1 ¡ ’)]
¹e
p
(31)
As expected, wages in both states will increase in the exit rate of the unemployment and in
the required level of e¤ort. A higher monitoring probability will induce workers not to shirk at a
lower wage. Future employment in the low state will have a negative e¤ect on e¢ciency wages as
in the two-period model of the previous section.
3.2 Employment and training
Let us re-write the in…nite horizon equivalent of the production function, equation (1), assuming
non-linear returns to training:
Yj;t = gi
j;tb¯
j;t¡1(Ni
j;t)1¡¯ (32)
13 This argument is demonstrated in Fella (2000) under the same assumptions about the stochastic environment.
16
where i takes a high or a low value. Equation (32) implies that output increases with training of
the previous period, bt¡1.14 Output depends on a productivity shock, g, that follows a two-state
Markov process with symmetric transition probabilities:
P(gj;t+1 = gH=gj;t = gL) = P(gj;t+1 = gL=gj;t = gH) = h
and
P(gj;t+1 = gH=gj;t = gH) = P(gj;t+1 = gL=gj;t = gL) = 1 ¡ h
For simpli…cation we assume that …rms never train their workforce in the low state, so that bt = 0
in the low state. After the realization of the current period state, …rms set the level of employment
and wages and the level of …rm speci…c training in the high state. In the low state, …rms set only
the level of employment and wages. The expected discounted value of the …rm’s pro…t at the
beginning of period t in the high and low state are given respectively by:
V H(bt¡1; gt) = maxfY (gH
t ; bt¡1;NH
t ) ¡ wH
t (NH
t ;NL
t+1)NH
t ¡ c(bt)NH
t +
±[(1 ¡h)V H(bt; gH t+1) +hV L(bt; gL t+1)]g
(33)
and
V L(bt¡1; gt) = maxfY (gLt
; bt¡1;NL
t ) ¡ wLt
NL
t +
±[(1 ¡ h)V L(0; gL
t+1) +hV H(0; gH
t+1)]g (34)
From equation (30), we can see that wages in the high state will depend on the probability
of being …red if there is a switch to the low state. Therefore, future employment in the low state
will a¤ect current wages. In line with Obstfeld (1991), we de…ne employment in a recession as a
14 For simpli…cation, we will assume that the training cost does not include payment to workers.
17
function of the state variables NL
t = f(bt¡1; gt) which implies that future employment in the low
state in (33) can be de…ned as a function of current training15 :
NL
t+1 = f(bt; gt+1) (35)
The …rst-order conditions for pro…t maximization in the high state can be written as:
(1 ¡ ¯)gHb¯
t¡1(NH
t )¡¯ = [1 ¡ ±(1 ¡ ’) + ±h]
¹e
p
+ c(bt) (36)
c(bt)bNH
t = ±[(1 ¡h)
#V H(bt; gH t+1)
#bt
+h
#V L(bt; gL t+1)
#bt
]
¡
#wHt
#NL
t+1
#NL
t+1
#bt
NH
t (37)
#V H(bt¡1; gHt
)
#bt¡1
= ¯gHb¯¡1
t¡1 (NH
t )1¡¯ (38)
Similarly, the …rst-order conditions in the low state are:
(1 ¡¯)gLb¯
t¡1(NL
t )¡¯ = [1 ¡ ±(1 ¡ ’)]
¹e
p
(39)
#V L(bt¡1; gL
t )
#bt¡1
= ¯gLb¯¡1
t¡1 (NL
t )1¡¯ (40)
By using (38) and (40) and the envelope theorem, we can rewrite equation (37) as:
±[(1 ¡ h)¯gHb¯¡1
t (NH
t+1)1¡¯ +h¯gLb¯¡1
t (NL
t+1)1¡¯] +
¹e
p
h
#NL
t+1
#bt
= c(bt)bNH
t (41)
Employment in the two states is derived by solving (36) and (39) for NH
t and NL
t :
15 This assumption is consistent with subgame perfection.
18
NH
t = [
(1 ¡ ¯)gH
[1 ¡±(1 ¡’) + ±h] ¹e
p +c(bt)
]
1
¯ bt¡1 (42)
NL
t = [
(1 ¡ ¯)gL
[1 ¡ ±(1 ¡ ’)] ¹e
p
]
1
¯ bt¡1 (43)
Equation (43) suggests that employment in the low state increases with training in the previous
period and allows us to get a functional form for future employment in the low state de…ned by
equation (35).
Proposition 3: The marginal value of training is higher than its expected marginal return.
Proof: The marginal value of training is given by the LHS of equation (41). Our proposition
holds since as one can see from equations (35) and (43) #NL
t+1
#bt
> 0:
Proposition 3 implies that an additional unit of training decreases the compensation workers
require in order to provide a positive level of e¤ort by improving their job security.
Finally, we substitute (42) and (43) in (41) which gives us the FOC for the optimal level of
training in the current period:
c(bt)bX
1
¯
1 = [¯±(1 ¡ h)gHX
1¡¯
¯
2 + ¯±hgLX
¯¡1
¯
3 +
h¹eX
1
¯
3
p
]b¡1
t¡1 (44)
where X1(bt) = (1¡¯)gH
[1¡±(1¡’)+±h] ¹e
p+c(bt ) > 0, X2(bt+1) = (1¡¯)gH
[1¡±(1¡’)+±h] ¹e
p+c(bt+1) > 0 and X3 =
(1¡¯)gL
[1¡±(1¡’)] ¹e
p
> 0:
Equation (44) is a second order non-linear di¤erence equation. Since one can get solutions for
the key economic variables from equations (42), (43) and (44), we do not specify the properties
of the value function in the Bellman equations.16 In order to check for stability, we linearize
(44) by taking a Taylor expansion around the steady state level of b:17 Next, we de…ne the
16 Sp eci…cally, one should substitute the solution for the control variables back into the Bellman equations and
specify the properties of the assumed value function so as to equate the RHS and the LHS.
17 We assume that c(b) = b° , where ° > 1:The derived steady state level of training b0 is b0 = [ ¯±(1¡±)¹e
p[°(1¡¯)¡¯±] ]
1
° :
19
characteristic equation and solve for the two roots. The expression for the two roots appears to
be very complicated and hence we can check for stability only by numerical simulation. We can
conclude that for a wide range of reasonable parameter values we get one stable and one unstable
root. Given that we have a backward looking solution, the condition for saddle path stability is
satis…ed for these values.18
4 Conclusions
The model presented above examines the decision to invest in …rm-speci…c training in a two-state
incentive wage model. It investigates a neglected positive aspect of job security provisions: the
wage reduction associated with the lower unemployment risk. Several shirking models, including
the original Shapiro and Stiglitz (1984) model, show the negative e¤ect of the job separation rate
on incentive wages. However, the usual assumption in the literature is that this rate is exogenous
to the …rm’s decision. However, …rms may a¤ect labour turnover in various ways. We showed
that …rms can improve workers’ expected job tenure by investing in …rm speci…c training since
…ring a trained worker implies the loss of some rent. Forward looking workers expect that the
probability of being …red in a recession will be lower the higher is the level of …rm-speci…c training
undertaken by the …rm. This implies that training is a credible commitment to lower …ring in a
low state. In an e¢ciency wage framework, a longer expected job tenure implies a higher penalty
for shirking. As a result, …rms’ investment in …rm-speci…c training will have a negative e¤ect
on the e¢ciency wage. The ability of …rm-speci…c training to provide a credible commitment to
a lower job separation rate thereby reducing wage cost, will increase human capital investment
above the level that equates marginal cost to expected marginal return.
18 Results available uppon request.
20
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21
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