B.1 Shapiro-Stiglitz model with unemployment benefits

Consider an economy with a representative firm that faces perfect competition in the

goods market and sets the level of employment L, and the real wage w, to maximize

real profits

II₁ = (erLr) - wrL₁ 0 < a < 1

where e, is the employee's effort level.

The representative worker maximizes the expected value of lifetime utility

1

U = Σ; Vt

(1+p)

t=0

where p is the intertemporal discount rate (p>0) and instantaneous utility is given by

Vt = we if employed

= {"

b

if unemployed

where b is the unemployment benefit (b>0). The employee's effort level equals

{ ² >

Jē> 0 if not shirking

if shirking

0

et =

Assume that in each period, s is the probability that an employee loses the job for

exogenous reasons, q is the probability that an employee is fired at the end of the period

due to shirking, and f = f (u) is the probability that an unemployed worker finds a job,

where u is the unemployment rate, which satisfies u = s/ (s+ f) in equilibrium.

(a) Write down the expected value of lifetime utility for the worker if (i) employed and

not shirking, (ii) employed but shirking, and (iii) unemployed. Explain briefly.

(b) Derive the no-shirking condition. Provide an intuitive explanation for the relation-

ship it yields between the unemployment rate u and the real wage w.

(c) Show graphically and explain intuitively the effect on the equilibrium real wage w

and employment Lif

i. the government reduces unemployment benefits.

ii. workers become more impatient.

iii. immigration leads to an increase in the labor force.

Explain how the unemployment rate is affected in each case.