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Job security in an efficiency wage model
WILLIAM SJOSTFtOM University
College,
Cork Ireland
Job Security in an Efficiency Wage Model* In a Shapiro-Stiglitz efficiency wage model, restrictions to fire employees caught shirking are shown to reduce voluntary unemployment.
on the ability employment
of employers and raise in-
1. Introduction In efficiency wage models, employers use wage premiums to lower monitoring costs. Shirking is prevented by the threat of being fired and therefore losing the wage premium. For a recent survey of these models, see Weiss (1990). In this paper, I examine how employment and unemployment change when job security rules lessen the probability that an employer may fire an employee caught shirking. Such rules have proliferated, for example, in Europe (Nickel1 1979). The model is an atemporal version of Shapiro and Stiglitz (1984) in which risk neutral employees either shirk or do not shirk. I do not ask whether efficiency wages are optimal. Instead, I explore the implications of employers paying an efficiency wage in a Shapiro and Stiglitz model. This means that the employer sets a wage sufficiently high that, given the probability of being caught shirking and successfully fired, no employee will shirk. It should be pointed out that a model in which effort is a continuous variable might produce different comparative statics.
2. The Model The employer pays a wage, W, which by assumption will be an efficiency wage. The employee gets that wage unless the employer catches him shirking and successfully fires him, in which case he does not get paid. The value of on-the-job leisure time to be had by shirking is L. L is not necessarily equal to, and is probably less than, the value of leisure time if the employee were not in the *I am indebted to the particularly careful work of two anonymous workshop participants at Northern Illinois University, and to Ardeshir simple proof that the objective function has a global maximum.
Journal of Macroeconomics, Winter 1993, Vol. Copyright 0 1993 by Louisiana State University 0164-0704/93/$1.50
15, No. Press
1, pp.
183-187
referees, to Dalal for a
183
William
Sjostrom
labor force. The probability that an employee will be caught shirking and be successfully fired is p = p(R, M). M is the level of monitoring chosen by the employer. The price of a unit of monitoring is implicitly set at one. R is a measure of the difficulty of firing an employee caught shirking. It reflects rules about job security, such as due cause firing, and is therefore a parameter for the employer. Because the model is atemporal, it does not reflect rules related to job seniority. The probability, p, is increasing in M and decreasing in R. So, if the employee shirks, he gets L if he is caught, and W + L otherwise. The employee’s expected gain from shirking is (W + L)[ 1 - p(R, M)] + p(R, M)L = W[l
- p(R, M)] + L .
(1)
His expected gain if he does not shirk is W. The employer sets a wage high enough so that W exceeds Equation (I), which implies that the lowest wage that prevents employees from shirking is W = L/p(R,
M) .
(2)
The wage set by the employer depends on the level of monitoring expenses he chooses. The employer maximizes his profits by selecting the number of homogeneous employees (N) and the level of monitoring per employee (M). Th e assumption of homogeneous employees is important because, along with the assumption that effort is a zero-one choice, it implies that employers cannot substitute higher quality employees for a higher quantity. Maxn=g(N)-N.
[L/p(R,M)]-N-M,
(3)
N,M
where g(N) is the production function, and the price of output is normalized to equal one. Diminishing returns to labor and to monitoring imply that g”(N) < 0 and PMM(R, M) < 0 (that is, both functions are concave). The profit function can be rewritten as
Max 7~= g(N)- N’M$ W/p@%WI f N
M}) ,
(4)
which makes it clearer that the firm has a two-step selection process. First, it selects M to minimize the cost it incurs per employee, and then it chooses N to maximize profits given the cost per employee. It is straightforward to show that the function has a unique global maximum. The concavity of p(R, M) in M implies that the 184
Job Security
in an Efficiency
Wage Model
cost per worker, L/p + M, is convex in M, implying function (3) can be rewritten as ye
that the profit
WI + WI .
7~ = g(N) + N-{tLIp(R
(5)
Because g(N) is concave and the negative of L/p + M is concave, it follows that the profit function is concave. Because effort is discontinuous in this model, and the wage is assumed to be an efficiency wage so that workers do not shirk, only the quantity of labor (N) enters the production function g(N). If effort were a continuous variable, the level of effort would also enter the production function, This would mean that the firm could substitute between effort and quantity, a substitution that is not possible in this model. The first-order conditions are adaN
= g’(N) - [L/p@,
anr/aM = (NL/p2).
M)] - M = 0 ;
(ap/aM)
- iv
= zv{(L/~~)~ (ap/aM) The second-order
conditions
a27r/alv2 = f(N); a2T/aM2 = (m/p2) The second-order dM2 are negative, (W
3. Implications
(64
(W
- i} = 0
for a maximum
are
(a2m/aivaM) = (L/p2). (ap/aM) {a2p/aM2 - (2/p) . (ap/aM)2}
- i ; .
(7)
conditions are satisfied because a2n/aiv2 and a2-rr/ and a2n/aNaM is zero by the first-order condition
and Conclusions
The effect of a change in R on the two choice variables is aN/aR
= [(-L/p2).
ap/aR]
aM/aR
= -[(NL/p2)
* (azp/aRaM)
- (ap/aR) . (ap/aM)]
+ a2p/alv2 ;
(84
- (2NL/p3)
+ a%/aM2
.
(W 185
William
Sjostrom
The sign of (Sa) is negative, implying that employment falls when job security rules are stricter. This is because N is the only argument in the production function, so anything that increases costs will lower the demand for labor. The sign of (Bb) is ambiguous, meaning that the level of monitoring can rise or fall. Stricter job security rules have a clear effect on demand, but their effect on supply is less clear. The model assumes that all employees receive the same value of leisure from shirking, but nothing in the model requires that their reservation wages for entering the labor force be identical. If their reservation wages are not identical, then the supply curve of labor will be upward sloping. The effect of a change in R on the wage received by employees is
aw/aR = (--L/p2) [(ap/aR) + (ap/aM) . (aM/aR)] .
(9)
Because the sign of aM/aR is ambiguous, (9) cannot be signed. Although (Sa) implies that employment has fallen, we cannot conclude that there is increased unemployment. Equations (Bb) and (9) can be signed by imposing a simple restriction on the form of the probability function p(R,M). Let p(R,M) = f(R) * h(M), with f’ < 0 and h’ > 0. h(M) is the probability an employee will be caught shirking, which depends on the level of that an employee may be monitoring, and f(R) is the probability fired if he is caught shirking, which depends on the rules about job security. The concavity of p(R,M) in M implies that h” < 0. Then (Bb) reduces to aM/aR
= [(df/dR)
* (dh/dM)
- NL/p’]
+ a2n/aM2 ,
00)
which is negative, implying that (9) is positive; that is, the efficiency wage rises with R. When R increases, employees earn a higher wage, increasing the quantity of labor supplied, and employment falls off. There is an increase in unemployment. This efficiency wage model generates two straightforward implications. When rules change so that it becomes more difficult to fire an employee for shirking, employment falls and involuntary unemployment rises. Received: June 1991 Final oersion: March
186
1992
Job Security in an Efficiency
Wage Model
References Stephen. “Unemployment
Nickell,
Carnegie-Rochester
Conference
and the Structure
of Labor Costs.”
Series on Public Policy 11 (1979):
187-222. Shapiro, Carl, and Joseph Stiglitz. “Equilibrium Unemployment as a Worker Discipline Device. ” American Economic Review 74 (June 1984): 433-44. Weiss, Andrew. Efficiency Wages: Models of Unemployment, Layoffs, and Wage Dispersion. Princeton: Princeton University Press, 1999.
Appendix List of Variables W = L = p = R = M = N =
efficiency wage. value of on-the-job leisure from shirking. probability o f ca t ch ing and firing a shirking employee. parameter measuring difficulty of firing shirking employee. monitoring effort. employment level.
187
College,
Cork Ireland
Job Security in an Efficiency Wage Model* In a Shapiro-Stiglitz efficiency wage model, restrictions to fire employees caught shirking are shown to reduce voluntary unemployment.
on the ability employment
of employers and raise in-
1. Introduction In efficiency wage models, employers use wage premiums to lower monitoring costs. Shirking is prevented by the threat of being fired and therefore losing the wage premium. For a recent survey of these models, see Weiss (1990). In this paper, I examine how employment and unemployment change when job security rules lessen the probability that an employer may fire an employee caught shirking. Such rules have proliferated, for example, in Europe (Nickel1 1979). The model is an atemporal version of Shapiro and Stiglitz (1984) in which risk neutral employees either shirk or do not shirk. I do not ask whether efficiency wages are optimal. Instead, I explore the implications of employers paying an efficiency wage in a Shapiro and Stiglitz model. This means that the employer sets a wage sufficiently high that, given the probability of being caught shirking and successfully fired, no employee will shirk. It should be pointed out that a model in which effort is a continuous variable might produce different comparative statics.
2. The Model The employer pays a wage, W, which by assumption will be an efficiency wage. The employee gets that wage unless the employer catches him shirking and successfully fires him, in which case he does not get paid. The value of on-the-job leisure time to be had by shirking is L. L is not necessarily equal to, and is probably less than, the value of leisure time if the employee were not in the *I am indebted to the particularly careful work of two anonymous workshop participants at Northern Illinois University, and to Ardeshir simple proof that the objective function has a global maximum.
Journal of Macroeconomics, Winter 1993, Vol. Copyright 0 1993 by Louisiana State University 0164-0704/93/$1.50
15, No. Press
1, pp.
183-187
referees, to Dalal for a
183
William
Sjostrom
labor force. The probability that an employee will be caught shirking and be successfully fired is p = p(R, M). M is the level of monitoring chosen by the employer. The price of a unit of monitoring is implicitly set at one. R is a measure of the difficulty of firing an employee caught shirking. It reflects rules about job security, such as due cause firing, and is therefore a parameter for the employer. Because the model is atemporal, it does not reflect rules related to job seniority. The probability, p, is increasing in M and decreasing in R. So, if the employee shirks, he gets L if he is caught, and W + L otherwise. The employee’s expected gain from shirking is (W + L)[ 1 - p(R, M)] + p(R, M)L = W[l
- p(R, M)] + L .
(1)
His expected gain if he does not shirk is W. The employer sets a wage high enough so that W exceeds Equation (I), which implies that the lowest wage that prevents employees from shirking is W = L/p(R,
M) .
(2)
The wage set by the employer depends on the level of monitoring expenses he chooses. The employer maximizes his profits by selecting the number of homogeneous employees (N) and the level of monitoring per employee (M). Th e assumption of homogeneous employees is important because, along with the assumption that effort is a zero-one choice, it implies that employers cannot substitute higher quality employees for a higher quantity. Maxn=g(N)-N.
[L/p(R,M)]-N-M,
(3)
N,M
where g(N) is the production function, and the price of output is normalized to equal one. Diminishing returns to labor and to monitoring imply that g”(N) < 0 and PMM(R, M) < 0 (that is, both functions are concave). The profit function can be rewritten as
Max 7~= g(N)- N’M$ W/p@%WI f N
M}) ,
(4)
which makes it clearer that the firm has a two-step selection process. First, it selects M to minimize the cost it incurs per employee, and then it chooses N to maximize profits given the cost per employee. It is straightforward to show that the function has a unique global maximum. The concavity of p(R, M) in M implies that the 184
Job Security
in an Efficiency
Wage Model
cost per worker, L/p + M, is convex in M, implying function (3) can be rewritten as ye
that the profit
WI + WI .
7~ = g(N) + N-{tLIp(R
(5)
Because g(N) is concave and the negative of L/p + M is concave, it follows that the profit function is concave. Because effort is discontinuous in this model, and the wage is assumed to be an efficiency wage so that workers do not shirk, only the quantity of labor (N) enters the production function g(N). If effort were a continuous variable, the level of effort would also enter the production function, This would mean that the firm could substitute between effort and quantity, a substitution that is not possible in this model. The first-order conditions are adaN
= g’(N) - [L/p@,
anr/aM = (NL/p2).
M)] - M = 0 ;
(ap/aM)
- iv
= zv{(L/~~)~ (ap/aM) The second-order
conditions
a27r/alv2 = f(N); a2T/aM2 = (m/p2) The second-order dM2 are negative, (W
3. Implications
(64
(W
- i} = 0
for a maximum
are
(a2m/aivaM) = (L/p2). (ap/aM) {a2p/aM2 - (2/p) . (ap/aM)2}
- i ; .
(7)
conditions are satisfied because a2n/aiv2 and a2-rr/ and a2n/aNaM is zero by the first-order condition
and Conclusions
The effect of a change in R on the two choice variables is aN/aR
= [(-L/p2).
ap/aR]
aM/aR
= -[(NL/p2)
* (azp/aRaM)
- (ap/aR) . (ap/aM)]
+ a2p/alv2 ;
(84
- (2NL/p3)
+ a%/aM2
.
(W 185
William
Sjostrom
The sign of (Sa) is negative, implying that employment falls when job security rules are stricter. This is because N is the only argument in the production function, so anything that increases costs will lower the demand for labor. The sign of (Bb) is ambiguous, meaning that the level of monitoring can rise or fall. Stricter job security rules have a clear effect on demand, but their effect on supply is less clear. The model assumes that all employees receive the same value of leisure from shirking, but nothing in the model requires that their reservation wages for entering the labor force be identical. If their reservation wages are not identical, then the supply curve of labor will be upward sloping. The effect of a change in R on the wage received by employees is
aw/aR = (--L/p2) [(ap/aR) + (ap/aM) . (aM/aR)] .
(9)
Because the sign of aM/aR is ambiguous, (9) cannot be signed. Although (Sa) implies that employment has fallen, we cannot conclude that there is increased unemployment. Equations (Bb) and (9) can be signed by imposing a simple restriction on the form of the probability function p(R,M). Let p(R,M) = f(R) * h(M), with f’ < 0 and h’ > 0. h(M) is the probability an employee will be caught shirking, which depends on the level of that an employee may be monitoring, and f(R) is the probability fired if he is caught shirking, which depends on the rules about job security. The concavity of p(R,M) in M implies that h” < 0. Then (Bb) reduces to aM/aR
= [(df/dR)
* (dh/dM)
- NL/p’]
+ a2n/aM2 ,
00)
which is negative, implying that (9) is positive; that is, the efficiency wage rises with R. When R increases, employees earn a higher wage, increasing the quantity of labor supplied, and employment falls off. There is an increase in unemployment. This efficiency wage model generates two straightforward implications. When rules change so that it becomes more difficult to fire an employee for shirking, employment falls and involuntary unemployment rises. Received: June 1991 Final oersion: March
186
1992
Job Security in an Efficiency
Wage Model
References Stephen. “Unemployment
Nickell,
Carnegie-Rochester
Conference
and the Structure
of Labor Costs.”
Series on Public Policy 11 (1979):
187-222. Shapiro, Carl, and Joseph Stiglitz. “Equilibrium Unemployment as a Worker Discipline Device. ” American Economic Review 74 (June 1984): 433-44. Weiss, Andrew. Efficiency Wages: Models of Unemployment, Layoffs, and Wage Dispersion. Princeton: Princeton University Press, 1999.
Appendix List of Variables W = L = p = R = M = N =
efficiency wage. value of on-the-job leisure from shirking. probability o f ca t ch ing and firing a shirking employee. parameter measuring difficulty of firing shirking employee. monitoring effort. employment level.
187