# EFFICIENT UNEMPLOYMENT INSURANCE: A Model of Job Search 3

Definition of Equilibrium

An allocation is a tuple {/C, W,Q,U} where /С С R+ is a set of capital investment levels,3 W : K+ =3 M+ is a set of wages offered by firms making particular capital investments, Q : K+ —>• R+ U oo is the queue length associated with each wage, and U 6 Ш+ is workers’ utility level. We also define the set of wages offered by some firm, VV = {w\w 6 W(k) for some к G /С}. Note that if w ^ W, Q(w) is not actually observed. Instead, these “off-the-equilibrium path actions” represent conjectures that help determine equilibrium behavior.

Profit Maximization ensures that given the queue length associated with each wage, firms choose wages and capital investments to maximize profits. Also, free entry drives the maximized value of profits to zero. Optimal Application ensures that workers make their application decisions to maximize utility, and imposes a form of subgame perfection: queue lengths adjust to make workers earn the maximal level of utility U* at any wage, including wages not offered along the equilibrium path.

This feature rules out situations in which firms may not deviate to a potentially profitable wage, incorrectly conjecturing that very few workers would apply. Finally, complementary slackness ensures that if a wage w does not deliver utility U* to a worker who is hired with probability 1, then no one applies for that wage, that is Q(w) — 0.

Observe that the queue length function Q* contains two other pieces of useful information: first, if w* is the unique equilibrium wage, the number of active firms is 1/Q*(w*); and second, the rate of unemployment of workers applying to a wage w* is u(w’) = 1 — fi(Q*(wt)). Clearly n(wf) is increasing in Q*(wf), since workers who apply to jobs with longer queues suffer a higher probability of unemployment.

Finally, note that at this point we do not impose a balanced budget condition. Instead, UI z and taxes т are treated as parameters. We add a balanced budget condition for the analysis of efficient and optimal UI in Sections 3 and 4.