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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Note that UI is efficient only if all possible equilibria achieve the highest possible output level that is technologically possible. We begin with an important benchmark:

Proposition 4 If agents are risk-neutral, the unique efficient level of UI is ze = re = 0.
Intuitively, when there is no UI, risk-neutral agents maximize the expected value of their wages, which is identical to net income. This result is a generalization of Moen (1997) and Shimer (1996) to the case in which firms choose their physical capital investments.

Proposition 5 If agents are risk-averse and ze = re = 0, output is below its maximum.
The proof of this proposition is straightforward and omitted: changes in preferences do not affect the efficient allocation, but unambiguously change the equilibrium (Proposition 2). Since ze = re = 0 maximizes output with risk-neutral agents, it yields a different, hence lower, level of output when workers are risk-averse. Proposition 5 therefore shows that labor market insurance is “inefficient”, because in the process of creating more jobs, firms reduce capital investments and labor productivity.