# EFFICIENT UNEMPLOYMENT INSURANCE: A Model of Job Search 6

When UI increases, workers wish to apply to higher wages which axe associated with higher unemployment risk. Firms once more cater to these preferences, and wages, unemployment and capital intensity increase. Underlying this labor market adjustment is market generated moral hazard (distinct from conventional moral hazard discussed in Section 5). As is common, the insurer would like to prevent workers from taking on more risk. The inability to control agents’ actions directly is the essence of the moral hazard problem (e.g. Holmstrom, 1982).

In our model, the insurer would like the worker not to apply to higher wages after receiving UI. But, since we assume that the worker’s application decision is private information, insurance cannot be conditioned on it. We call this market generated moral hazard, because the response of the labor market is crucial: if firms did not change their wage offers, workers would be unable to apply to higher wages itat on.

Figure 3: This depicts the equilibrium under the frictionless matching technology rf ,/iF. It is unaffected by unemployment benefits and preferences.
A corollary of Proposition 2 is that the equilibrium is generically unique. For example, holding all the other parameters fixed, the equilibrium is unique for almost every level of UI. To see why, consider the correspondence K(z), defined as the set of all к that solve the constrained optimization problem in Proposition 1. The third part of Proposition 2 implies К is strictly monotonic, and so there are at most a countable number of 2 for which К is not a singleton. At all other z, the equilibrium is unique. A similar argument implies generic uniqueness in an appropriate space of utility functions.

We conclude by observing that with the frictionless matching technology described above, Proposition 2 does not hold. If rjF(q) = min(l,g) and fiF(q) = min(l/g, 1), the unique equilibrium is {k, w, q} = {к, z, 1} for any degree of risk aversion and any UI z < z. Workers gain nothing by applying for a job with queue length less than 1, and firms gain nothing by offering a wage that yields a queue length greater than 1.

Therefore, indifference curves and the zero profit condition are kinked at q — 1, ensuring that this is the point of “tangency” (see Figure 3). Then q = rjF(q) = 1 implies f(k) = 1, and hence к = k. Finally, (4) yields w = f(k) — к = z. Nevertheless, with any continuously differentiable approximation to rf and jiF, the comparative statics results in Proposition 2 obtain. The fact that our results limit to the competitive equilibrium and fail to hold at this limit point (with r)F and fj,F) is reassuring: frictions, as well as incomplete insurance, are crucial for our conclusions.