Value functions
Let We(k, t) be the present discounted value of the lifetime utility of an employed person in state k at time t. Similarly let Wu (k, t) be that of an unemployed person when the state is k.
Because k is a random variable, we take the expectation of these random value functions later, after we derive the stationary distribution of k. We drop t from the argument of the value functions, because dynamic programming involves an infinite horizon and the problem is time-homogeneous.Denote the discount rate by r. Value functions depend on the fraction k/n rather than on k directly. For shorter notation, however, we denote them by We (k) and Wu (k) for the employed and unemployed when the number of the employed is k. For an employed agent, we obtain the relation for the value functions as 
for k between 3 and n — 1, and for an umemployed agent5
for k = 2, 3,..., n — 1. There are boundary relations, but we do not use them, so they are not mentioned here. See Aoki and Shirai (2000) for their expressions.
We next take the expected values of these value functions with respect to the stationary distributions of k.
Because the set of value functions is huge, we solve for them approximately, by first obtaining the expressions for the expected values of the value functions and then the expression for the reservation costs as functions of φ up to terms of O(1/n).
9.5.1 Expected-value functions
Define
and
The expressions in the square brackets of the value function relations in the previous section become
and
respectively Noting that Eξ = 0 and
the expected-value function
becomes, after dropping terms of order 1 /n or smaller, 
See Aoki and Shirai (2000) for the details of the calculations.
These two equations correspond with (4a) and (4b) in Diamond and Fudenberg (1989).
Making use of the fact that Φ(φe) = 0, where φe denote locally stable equilibrium points, these two equations yield equilibrium value functions
where r = r∕a, and
analogy with the case of an infinite number of agents holds.
We can actually see that this choice of c' is optimal by differentiating the expected-value functions with respect to c*, noting that b(φ) is exogenously specified and its derivative with respect to c* is zero. Solving for the derivatives of the expected-value functions with respect to c*, we see that they are both zero. This is the first-order condition for optimality. The second-order condition may be shown to hold by taking derivatives once more.
9.6