Because the firms owners can diversify away ftrm-specific risk by holding a broad portfolio, the assumption that the ftrm is risk-neutral is reasonable for firm-speclftc shocks. For aggregate shocks, however, the assumption that the firm is less risk-averse than the workers is harder to justify. Since the main goal of the theory is to explain the effects of aggregate shocks, this is a weak point of the model. One possibility is that the owners are wealthier than the workers and that risk aversion is declining in wealth.

"If there are I workers, the representative workers hours and consumption are in fact I/I and wL/I, and so utility takes the form t/(C/I) - V{L/L). To eliminate I, define U(C) = 0(C IL) and ViL) = V{L/L).

The Model

Consider a firm dealing with a group of workers. The firms profits are

, 77 = AF{L)-wL, fC)>0, f"W<0, (10.41)

where I is the quantity of labor the firm employs and w is the real wage. A is a factor that shifts the profit function. It could reflect technology (so that a higher value means that the firm can produce more output from a given amount of labor), or economy-wide output (so that a higher value means that the firm can obtain a higher relative price for a given amount of output).

Instead of considering multiple periods, it is easier to consider a single period and assume that A is random. Thus, for example, when workers decide whether to work for the firm, they consider the expected utility they obtain in the single period given the randomness in A, rather than the average utility they obtain over many periods as their income and hours vary in response to fluctuations in A.

The distribution of A is discrete. There are possible values of A, indexed by i; p, denotes the probabihty that = ,. Thus the firms expected profits are

= X Pi [ - FiLi) - Wi Li ]. (10.42)

where L; and w, denote the quantity of labor and the real wage if the realization of A is Ai. The firm maximizes its expected profits; thus it is risk-neutral.

Each worker is assumed to work the same amount. The representative workers utility is

= U(C) - V(L), [/(•) > 0, [/"(.) < 0, VC) > 0, V"C) > 0,

(10.43)

where [!(•) gives the utility from consumption and V(*) the disutility from working. Since U"(*) is negative, workers are risk-averse.

Workers consumption, C, is assumed to equal their labor income, wL. That is, workers cannot purchase insurance against employment and wage

Wage Contracts

One simple type of contract just specifies a wage and then lets the firm choose employment once A is determined; many actual contracts at least appear to take this form. Under such a contract, unemployment and real wage rigidity arise immediately. A faU in labor demand, for example, causes the firm to reduce employment at the fixed real wage while labor supply does not shift, and thus creates unemployment (or, if all workers work the same amount, underemployment). And the cost of labor does not respond because, by assumption, the real wage is fixed.

But this is not a satisfactory explanation of unemployment and real wage rigidity. The difficulty is that this type of a contract is inefficient (Leontief, 1946; Barro, 1977b; Hall, 1980). Since the wage is fixed and the firm chooses employment taking the wage as given, the marginal product of labor is independent of A. But since employment varies with A, the marginal disutility of working depends on A. Thus the marginal product of labor is generally not equal to the marginal disutility of work, and so it is possible to make both parties to the contract better off. And if labor supply is not very elastic, the inefficiency is large. When labor demand is low, for example, the marginal disutihty of work is low, and so the firm and the workers could both be made better off if the workers worked slightly more.

Thus we can appeal to fixed-wage contracts with employment determined at the firms discretion as a potential explanation of unemployment and real wage rigidity only if we can explain why a firm and its workers would agree to such an arrangement. The remainder of this section shows, however, that our assumptions imply that they will in fact agree to a very different contract. Section 10.6 then suggests a variation on our model that could give rise to something much closer to this type of a contract.

fluctuations. In a more fully developed model, this might arise because workers are heterogeneous and have private information about their labor-market prospects. In the present model, however, the absence of outside insurance is simply assumed.

Equation (10.43) imphes that the representative workers expected utility is

E[u] = X Pi[U(Q) - Vih)]. (10.44)

There is some reservation level of expected utility, uo, that workers must attain to be willing to work for the firm. There is no labor mobility once workers agree to a contract; thus the only constraint on the contract involves the average level of utility it offers, not the level in any individual state.

1=1 \ 1=1

The hrst-order condition for , is

-p, +Ap,t/(C,) = 0,

- u()

. (10.45)

(10.46)

f/(C,)= --.

(10.47)

Equation (10.47) implies that the margmal utility of consumption is constant across states, and thus that consumption is constant across states. Thus the risk-neutral firm fully insures the risk-averse workers. The first-order condition for L, is

p,A,f(L,) = Ap,\/(I,).

(10.48)

Equation (10.47) implies A = l/U(C), where is the constant level of consumption. Substituting this fact into (10.48) and dividing both sides by p, yields

A,F{U) =

f/(C)-

(10.49)

Implications

Under efficient contracts, workers real incomes are constant, thus the model appears to imply strong real wage rigidity; in fact, because L is higher when A is higher, the model implies that the wage per hour is coun-

"•The theory of implicit contracts is due to Azariadis (1975); Baily (1974); and Gordon (1974).

Efficient Contracts

To see how it is possible to improve on wage contract, suppose that the hrm oilers the workers a contract specifying the wage and hours for each possible reahzation of . Since actual contracts do not explicitly specify employment and the wage as functions oi the state, such contracts are known as implicit contracts?

Recall that the hrm must offer the workers at least some minimum level of expected utility, uq, but is otherwise unconstrained. In addition, since I, and w, determine C,, we can think of the hrms choice variables as I and in each state rather than as I and w. The Lagrangian for the hrms problem is therefore