Case of small entry and exit probabilities: An example
This example arises in connection with the well-known search model of Diamond (1982), discussed in Chapter 9. This model uses an infinite number of agents with fractions of employed as deterministic state variables.
Aoki and Shirai (2000) reworked the same model with a finite number N of agents. In the model, the reservation cost for undertaking production opportunities, which arrive at a rate a to a pool of unemployed, is determined by value max-
where the subscripts e and u mean employed and unemployed, respectively Therefore, there are about 2N value functions to determine. (There are also some boundary-condition equations.) To simplify, we look for the average relation between the values of being employed and unemployed, using the change of variables
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where φ is the average fraction of employed, and ξ is a random variable to account for random fluctuation about the mean value of the fraction. We refer the reader to the cited paper for detail, and give here just a brief exposition to convey the essential points.
The approximate value functions become 
and
where r is the interest rate, and where the reservation cost is given by
Here
> is the cumulative cost up to c', and aG* is the transition rate
from the pool of unemployed to that of employed, when the fraction of employed is φ. We have ignored the effects of externalities as being small. They vanish exactly at equilibrium points φ = φe of the aggregate dynamics where 
that is, φe is a zero point of the dynamics Φ.
Solving this set of approximate value-function equations on the assumption that aG* and bG* are smaller than r, we arrive at familiar-looking expressions 
and
The reservation cost is approximately given by
These expressions give the value functions as discounted present values of the benefit and average cost streams, adjusted for the quantities a and b, which are related to the transition rates.
6.6