# The Worker’s Problem

A worker can be employed full-time, contributing one unit of time to labor, or part-time at 1/2 unit of labor. If unemployed, the worker receives utility b > 0, representing leisure or home production. Similarly, a part-time employed worker receives utility Ếb, where I G (0,1) is the fraction of leisure or home production gained in part-time. The parameter I reflects the fact that part-time work frees up only a fraction of time for leisure, as well as the possibility that some portion of b could come from transfers or unemployment insurance benefits that would not be available to a part-time worker.

Let U (zt) be the utility value of an unemployed worker in history zt, and W(C (t)) be the present expected value of employment in contract C (t) to a worker before production occurs. Because utility is linear, the worker treats the probability of part-time work as a lottery. The employed worker’s value function satisfies the recursive equation:

An unemployed worker’s search problem involves maximizing the expected utility gain of employment, taking the matching probability and value of each contract into consideration.

Potential contracts are observed, and parameterized by the tuple (m, C(t)). Knowing that a contract yields a probability of matching for the worker of m/A(m), the worker’s search value is:

which reflects the expected probability of matching with a firm offering contract C(t) multiplied by the expected gain in the worker’s value function derived from being employed with that contract during the next period. The Bellman equation for the unemployed worker then satisfies

The unemployed worker receives constant flow utility b from leisure or unemployment benefits, and the option value of searching, p(zt). Since workers can direct their search to different contracts, their flow value from unemployment must be equal in any market that attracts workers. This implies that p(zt) is common to workers in any submarket; hence, p(zt) determines the contract value the firm must post to fill a vacancy with a positive probability m.