Key Findings:

1. Unemployment benefits increase unemployment rate.
2. Wage adjust slowly in the face of aggregate shocks.
3. Market equilibrium not always Pareto optimal, and therefore government intervention is sometimes desirable.
4. Wage distributions for identical workers can persist in equilibrium.

Basic Model

For worker:

$U=w-e$, or $\bar w$ when unemployed. Probability $b$ to be unemployed and $q$ if caught shirking.

Def: $V_E^S$, lifetime utility of shirker, and $V_E^N$ of a non-shirker. $V_u$ of lifetime unemployed individual. $r$ is discount rate.

Asset eq:

$$ rV_E^S=w+(b+q)(V_u-V_E^S)\\ rV_E^N=w-e+b(V_u-V_E^N) $$

Solving the equations, we have:

$$ V_E^S=\frac{w+(b+q)V_u}{r+b+q}\\V_E^N=\frac{(w-e)+bV_u}{r+b} $$

No-shirking condition(NSC):

$$ w\geq rV_u+(r+b+q)e/q $$

Implication: unless penalty related to being unemployed, all will shirk.

For Employers:

M identical firms, production function $Q_i=f(L_i)$, $L_i$ being effective labor force. Firm’s wage package: $w$ wage, and unemployment benefits $\bar w$.

$V_u$ is key determinant of firm behavior.

eq for $V_u$: