# 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

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.

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.