EFFICIENT UNEMPLOYMENT INSURANCE: Risk-A version

This section demonstrates that if workers are risk-averse, a moderate amount of UI funded by lump-sum taxation will raise output. Since this says nothing about the riskiness of consumption, it is not a normative statement. Nevertheless, it distinguishes our approach from existing theories of UI and implicit contracts more generally.
Suppose workers are homogeneous, and all firms invest offer a common wage w, and attract an average queue length of q. Net output in such an economy is given by
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The number of jobs is equal to the measure of workers who find a job, /x(g), times the output of each filled job, f{k). From this we subtract investment expenditures, which is the measure of firms, 1 /q, times capital expenditure, k. If firms choose different queue lengths and capital, net output would be a weighted average of each firm’s expected output. We suppress this nongeneric possibility to simplify our notation Reading here.
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EFFICIENT UNEMPLOYMENT INSURANCE: A Model of Job Search 7

Worker Heterogeneity

We have simplified our analysis by assuming that all workers have the same level of assets, the same utility function, and receive the same level unemployment benefits. Our results generalize to an environment in which workers differ with respect to all of these features. In particular, suppose that there are 5 = 1,2,.. .,5 types of workers, where type s has utility function ua, after-tax asset level As, and unemployment benefit zs. Let U now be a vector in R5, and assume, for simplicity, that zs < z for all s. Then:
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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.
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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.
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