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B) Second hypothesis
3. The long term adjustment
5.8 Implicit contracts
Advanced Macroeconomics
176
Modern Macroeconomics
5.7.3 Criticism
This model does not explain the critical changes that impact a cyclical shift in labor demand. They are divided between employment and wages. According to Walras, the labor market adjusts immediately to a change in A. But with frictions, both permanent and temporary changes trigger complicated adjustment processes for vacancies, unemployment and wages.
Advanced Macroeconomics
177
Modern Macroeconomics We consider A = Ai and the firm maximizes profit and risk is natural. Each worker is assumed to work the same amount. The representative worker’s utility is:
U = U(c.)-V(L) (5.49)
where U = given utility from consumption V = disutility from working
Utility is negative and workers are risk averse. The workers’ consumption is assumed to equal their labor income, wL. Workers do not have insurance against unemployment and wage fluctuations. Some workers have information on the wages paid at rival firms.
A worker’s expected utility from their current job is:
1
( ) k ( ( ) ( ))
i
E u Pi U Ci V Li
=
=
∑
− (5.50)There is some expected utility and workers must be willing to work for the firm. There is no mobility if there is a contract between workers and the firm. At the time of contract, only the average level of utility is offered. This is not what individuals expect.
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Advanced Macroeconomics
178
Modern Macroeconomics
5.8.2 Wage contracts
A firm decides the contract with its workers. While under contract, the real wage and rigidity rise immediately. The fall in demand for labor causes firms to reduce employment at the fixed real wage while labor supply does not shift and there is unemployment. The cost of labor does not change because by assumption, the real wage is fixed. But it is not practical. Wages are fixed and firms choose employment..
The marginal productivity of labor is fixed. The rate of employment varies with A and the marginal disutility of workers depends on A. The marginal product of labor is not equal to the marginal disutility of work. In contrast, both parties should be better off. If the disutility of work is less, workers can work more and both workers and firms may prosper.
5.8.3 Efficient contracts
The implicit contract means the actual contract does not explicitly specify employment and the wage is determined by the state. The firms and workers draw up a contract d specifying the wage and hours for each possible realization of A.
The assumptions are that firms offer a worker at least the minimum level of expected utility u0, but are otherwise constrained. Li and Wi determine Ci. We think of the firm’s choices as variable as L and C in each state rather than as L and W. The Lagrangian of the firm’s problem is therefore:
1 1 0
( ( ) ) ({ [ ( ) ( )]} )
k k
i i i
i i
Pi AiF Li Ci
λ
P U C V L u= =
=
∑
− +∑
− −
The first order condition for Ci is
− + Pi λ PiU C '( ) 0
i=
orU C '(
i)1
= λ
(5.51)The above equation implies that the marginal utility of consumption is constant across states and thus that consumption is constant across states. Thus, the risk-neutral firm fully insures the risk adverse workers.
The first order condition for Li is P AiF Li' '( )=
λ
PiV Li'( ) (5.52) The above equation implies that1 ( ) ' C
λ = U
where C is the constant level of consumption.Substituting this into the above equation and dividing both sides by Pi yields '( ) '( )
'( ) AiF Li V Li
=U C (5.53)
Advanced Macroeconomics
179
Modern Macroeconomics