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Heterogeneous labor and implicit contracts
Economics Letters 6 (1980) I85- 190 North-Holland Publishing Company
HETEROGENEOUS
185
LABOR AND IMPLICIT CONTRACTS
Jeff FRANK * Tulane University, University Received
New Orleans, LA 70118, USA
of Essex,Cotchester CO4 3SQ, UK 6 November
1980
The paper examines the role of layoffs under implicit contracts, when labor is heterogeneous. Wage-adjustment strategies are preferable in that employment is allocated more efficiently. The impossibility of effectively committing to employment levels may reverse the ordering.
1. Prior to the development of the implicit contracts literature, one might have argued that firms would never utilize layoffs as a policy since they could instead lower the wage, achieving the same desired workforce at lower cost. In an implicit contracts world as in Baily (1977), the argument is invalid insofar as prospective workers incorporate the wagereduction possibility into their attachment decision- the firm pays more the first period for the possibility of lowering wages the second. Baily shows, under homogeneous labor, that a firm is indifferent between wage-adjustment and layoff policies. The current note examines whether this result holds under heterogeneous labor, where workers differ in per period costs to working at a given firm. A layoff policy is generally inferior (more costly) to a wage-adjustment policy, since the latter allocates available employment more efficiently. The layoff policy, however, does allow for a commitment to future employment levels that can lead to greater efficiency, and may make that policy the more profitable one. Providing a rationale for quantity-rationing of employment behavior is important; besides implicit contracts, other models seeking to do that
* I appreciate
helpful
discussions
with Dagobert
Brito and Ron Batchelder.
J. Frank / Heterogeneow labor and implicit contracts
186
are the turnover model by Salop (1979) and the adverse of Weiss (1980).
selection
model
2. We adopt a two-period model similar to Baily (1977). The firm hires N, workers the first period and retains a state-of-nature dependent number Nz( i) in the second period, where N, L N,(i); the firm then shuts down. All workers have the same unchanging alternative to working at the firm, search with expected value V, but differ in their per period costs sustained in working at the firm. ’ Ordering these, we write the function 6’ > 0.
S(N),
There is, in addition, a fixed specific capital cost in the amount C incurred by each worker. Workers and the firm discount at the interest rate r, and are risk-neutral. For a first-period worker with per period cost value 8, to desire to remain for the second period, the wage must exceed
(1) for a worker to accept a job in the first period, that
the wages must be such
(4 where nji is the probability that workerj is employed in state-of-nature i, and where (r/( 1 + r)) V is the implicit rental on delaying search, and the bracketed term is any surplus achieved in period 2 by the worker. Proposition 1. For any employment strategy N,, N2( i), N2( i) G N,, N2 strictly less for some i, the wage-adjustment strategy:
(i)
w2(i)=
&v+ S(Nj))
’ These costs might be pecuniary (e.g., commuting from different pecuniary returns from working at the firm (e.g., job satisfaction).
distances)
or non-
187
J. Frank / Heterogeneous labor and implicil contracts
entails a lower expected
discounted labor cost than the layoff strategy
w*(i) >&V+“(N,).
(ii)
In the wage-adjustment voluntary
quits achieve
strategy, the desired
the second-period employment
worker voluntarily quits since the second-period tion level for all workers. 2 Proof
calculate
the expected
discounted
wage is set so that
level.
Under
wage exceeds
wage bill
layoffs,
no
the reserva-
W under
the two
strategies: Wage-adjustment.
a surplus w,
Since the marginal in period 2
first-period
worker
never receives
=&V+S(N,)+C.
(3)
and Ei[N2(i)]
w=Iv
l+r
Layoffs.
1
[
+r
The marginal
I
+w +[Vd+cl4
period
+
wqN2(i)lN2(i)l l+r
1 worker expects a second-period
.
surplus:
(5)
E,
so
w=rv
l+r
JqN,wl 1 +r +N,
1
G(Nl)Ei[N2(i)]
+[~(N,)+C]N,
+
Since QN,) > 6[N,(i)], with strict inequality (some i), the adjustment strategy is less costly. The result arises as it is more efficient to allocate employment ’ Condition
(ii) may be met with an equality;
in that case, the expression
(q
’
l+r
wagein the
(5) below is zero.
188
J. Frank / Heterogeneous labor and implicit contracts
second period to those workers with lower 8 values. Nor would riskaversion particularly affect this since (under wage-adjustment) the marginal first period worker is indifferent to whether he is retained since w,
=++“(N,)+c
so that his ‘investment’
costs C are covered in the first period.
3. The applicability of the above proposition is that any layoff policy with an associated employment strategy N,,&(i) is dominated by a wage-adjustment policy. This requires, however, that a binding commitment can be made by the firm to adopt the same employment strategy N1,N2(i) under the wage-adjustment policy. A difficulty arises if the state of nature is not observable by workers, so that they can not tell if the firm is meeting its commitment. In this section, we will suppose that this is the case, and since non-compliance can not be determined, workers will assume that the firm behaves in the second period in the profitmaximizing manner. This will entail an inefficient level of employment in
NI Fig. 1
J. Frank / Heterogeneous labor and implicit contracts
189
the second period (the monopsonistic solution), and can lead to a reversal in the ordering of the two policies. For the inefficient level of second-period employment to affect first period wage costs, the formulation used in section 2 must be modified. In that formulation (under wage-adjustment), the marginal worker never expected a second-period surplus, and therefore was indifferent to second-period employment. We modify that by supposing that a prospective worker does not know his period cost, 6, value until after he has invested the cost C, and that S is drawn from a probability density r(S). Under this formulation, the firm faces a horizontal supply curve in the first period, but an upward-sloping one in the second after workers learn their 6 values. 3 Under these assumptions, we have the wage bills in the two cases: Wage-adjustment
([W2W) -s,6(N2(i))(
6[ N,( i)] - S,)n( 8,) ds,] N,(i))
+E,
[
N(i)&V].
N,+E1’N2(i)1 l+r
(7)
j(f++j+CN,,
where 8 is the expected value of 6. Under this formulation, as in the formulation of section 2, for a given N,, N2(i), the wage adjustment policy is less costly. The wage bills differ only in that the layoff strategy entails paying the average period cost s for second-period workers, while the wage-adjustment entails the mean of the truncated distribution, and is less. In a more complete context, where
3 An alternative formulation that would allow for this dependence of W, upon N2( i) would be to suppose that workers engage in on-the-job search in the first period and receive different wage offers for second-period jobs.
190
J. Frank / Heterogeneous labor and implicit contracts
we define the state-of-nature by its associated revenue function, Ri, maximization with the wage function (7) leads to a higher expected value than maximization with the wage function (8). The difficulty is that the firm will not be able to maximize with the wage bill function (7) if it cannot make a firm commitment to the optimal employment function N,*(i). In (7), workers accept a lower first-period wage in anticipation of an expected surplus the second period; the firm has every reason to renege upon the optimal employment function and adopt the monopsonistic employment levels. The wage bill will be (7) with the monopsonistic employment levels N2(i) substituted. The expected value under this strategy can then fall below that of the layoff strategy since, while wage-adjustment continues to ration employment to the appropriate workers, an inefficient level of employment is chosen. Fig. 1 shows the profitability (in a particular state-of-nature) under the two schemes; the preferable scheme is wageadjustment (layoffs) as area b (a) is the greater. Hz(i) is optimal employment under the layoff strategy.
References Baily, M., 1977, On the theory of layoffs and unemployment, Econometrica. Salop, ‘5, 1979, A model of the natural rate of unemployment, American Economic Review. Weiss, A., 1980, Job queues and layoffs in labor markets with flexible wages, Journal of Political Economy.
HETEROGENEOUS
185
LABOR AND IMPLICIT CONTRACTS
Jeff FRANK * Tulane University, University Received
New Orleans, LA 70118, USA
of Essex,Cotchester CO4 3SQ, UK 6 November
1980
The paper examines the role of layoffs under implicit contracts, when labor is heterogeneous. Wage-adjustment strategies are preferable in that employment is allocated more efficiently. The impossibility of effectively committing to employment levels may reverse the ordering.
1. Prior to the development of the implicit contracts literature, one might have argued that firms would never utilize layoffs as a policy since they could instead lower the wage, achieving the same desired workforce at lower cost. In an implicit contracts world as in Baily (1977), the argument is invalid insofar as prospective workers incorporate the wagereduction possibility into their attachment decision- the firm pays more the first period for the possibility of lowering wages the second. Baily shows, under homogeneous labor, that a firm is indifferent between wage-adjustment and layoff policies. The current note examines whether this result holds under heterogeneous labor, where workers differ in per period costs to working at a given firm. A layoff policy is generally inferior (more costly) to a wage-adjustment policy, since the latter allocates available employment more efficiently. The layoff policy, however, does allow for a commitment to future employment levels that can lead to greater efficiency, and may make that policy the more profitable one. Providing a rationale for quantity-rationing of employment behavior is important; besides implicit contracts, other models seeking to do that
* I appreciate
helpful
discussions
with Dagobert
Brito and Ron Batchelder.
J. Frank / Heterogeneow labor and implicit contracts
186
are the turnover model by Salop (1979) and the adverse of Weiss (1980).
selection
model
2. We adopt a two-period model similar to Baily (1977). The firm hires N, workers the first period and retains a state-of-nature dependent number Nz( i) in the second period, where N, L N,(i); the firm then shuts down. All workers have the same unchanging alternative to working at the firm, search with expected value V, but differ in their per period costs sustained in working at the firm. ’ Ordering these, we write the function 6’ > 0.
S(N),
There is, in addition, a fixed specific capital cost in the amount C incurred by each worker. Workers and the firm discount at the interest rate r, and are risk-neutral. For a first-period worker with per period cost value 8, to desire to remain for the second period, the wage must exceed
(1) for a worker to accept a job in the first period, that
the wages must be such
(4 where nji is the probability that workerj is employed in state-of-nature i, and where (r/( 1 + r)) V is the implicit rental on delaying search, and the bracketed term is any surplus achieved in period 2 by the worker. Proposition 1. For any employment strategy N,, N2( i), N2( i) G N,, N2 strictly less for some i, the wage-adjustment strategy:
(i)
w2(i)=
&v+ S(Nj))
’ These costs might be pecuniary (e.g., commuting from different pecuniary returns from working at the firm (e.g., job satisfaction).
distances)
or non-
187
J. Frank / Heterogeneous labor and implicil contracts
entails a lower expected
discounted labor cost than the layoff strategy
w*(i) >&V+“(N,).
(ii)
In the wage-adjustment voluntary
quits achieve
strategy, the desired
the second-period employment
worker voluntarily quits since the second-period tion level for all workers. 2 Proof
calculate
the expected
discounted
wage is set so that
level.
Under
wage exceeds
wage bill
layoffs,
no
the reserva-
W under
the two
strategies: Wage-adjustment.
a surplus w,
Since the marginal in period 2
first-period
worker
never receives
=&V+S(N,)+C.
(3)
and Ei[N2(i)]
w=Iv
l+r
Layoffs.
1
[
+r
The marginal
I
+w +[Vd+cl4
period
+
wqN2(i)lN2(i)l l+r
1 worker expects a second-period
.
surplus:
(5)
E,
so
w=rv
l+r
JqN,wl 1 +r +N,
1
G(Nl)Ei[N2(i)]
+[~(N,)+C]N,
+
Since QN,) > 6[N,(i)], with strict inequality (some i), the adjustment strategy is less costly. The result arises as it is more efficient to allocate employment ’ Condition
(ii) may be met with an equality;
in that case, the expression
(q
’
l+r
wagein the
(5) below is zero.
188
J. Frank / Heterogeneous labor and implicit contracts
second period to those workers with lower 8 values. Nor would riskaversion particularly affect this since (under wage-adjustment) the marginal first period worker is indifferent to whether he is retained since w,
=++“(N,)+c
so that his ‘investment’
costs C are covered in the first period.
3. The applicability of the above proposition is that any layoff policy with an associated employment strategy N,,&(i) is dominated by a wage-adjustment policy. This requires, however, that a binding commitment can be made by the firm to adopt the same employment strategy N1,N2(i) under the wage-adjustment policy. A difficulty arises if the state of nature is not observable by workers, so that they can not tell if the firm is meeting its commitment. In this section, we will suppose that this is the case, and since non-compliance can not be determined, workers will assume that the firm behaves in the second period in the profitmaximizing manner. This will entail an inefficient level of employment in
NI Fig. 1
J. Frank / Heterogeneous labor and implicit contracts
189
the second period (the monopsonistic solution), and can lead to a reversal in the ordering of the two policies. For the inefficient level of second-period employment to affect first period wage costs, the formulation used in section 2 must be modified. In that formulation (under wage-adjustment), the marginal worker never expected a second-period surplus, and therefore was indifferent to second-period employment. We modify that by supposing that a prospective worker does not know his period cost, 6, value until after he has invested the cost C, and that S is drawn from a probability density r(S). Under this formulation, the firm faces a horizontal supply curve in the first period, but an upward-sloping one in the second after workers learn their 6 values. 3 Under these assumptions, we have the wage bills in the two cases: Wage-adjustment
([W2W) -s,6(N2(i))(
6[ N,( i)] - S,)n( 8,) ds,] N,(i))
+E,
[
N(i)&V].
N,+E1’N2(i)1 l+r
(7)
j(f++j+CN,,
where 8 is the expected value of 6. Under this formulation, as in the formulation of section 2, for a given N,, N2(i), the wage adjustment policy is less costly. The wage bills differ only in that the layoff strategy entails paying the average period cost s for second-period workers, while the wage-adjustment entails the mean of the truncated distribution, and is less. In a more complete context, where
3 An alternative formulation that would allow for this dependence of W, upon N2( i) would be to suppose that workers engage in on-the-job search in the first period and receive different wage offers for second-period jobs.
190
J. Frank / Heterogeneous labor and implicit contracts
we define the state-of-nature by its associated revenue function, Ri, maximization with the wage function (7) leads to a higher expected value than maximization with the wage function (8). The difficulty is that the firm will not be able to maximize with the wage bill function (7) if it cannot make a firm commitment to the optimal employment function N,*(i). In (7), workers accept a lower first-period wage in anticipation of an expected surplus the second period; the firm has every reason to renege upon the optimal employment function and adopt the monopsonistic employment levels. The wage bill will be (7) with the monopsonistic employment levels N2(i) substituted. The expected value under this strategy can then fall below that of the layoff strategy since, while wage-adjustment continues to ration employment to the appropriate workers, an inefficient level of employment is chosen. Fig. 1 shows the profitability (in a particular state-of-nature) under the two schemes; the preferable scheme is wageadjustment (layoffs) as area b (a) is the greater. Hz(i) is optimal employment under the layoff strategy.
References Baily, M., 1977, On the theory of layoffs and unemployment, Econometrica. Salop, ‘5, 1979, A model of the natural rate of unemployment, American Economic Review. Weiss, A., 1980, Job queues and layoffs in labor markets with flexible wages, Journal of Political Economy.