Question

unemployment and the labor market — end of chapter problem
consider a special case of the model of labor - market dynamics in section 8 - 3. suppose the matching function is
new hires = fu
where f is the exogenous fraction of unemployed people who find a job in any given period, called the job finding rate.
a. in the steady state, the number of people separating from a job equals the number of separations, which is se. thus,
the condition for steady - sate unemployment is fu = se. the steady - state unemployment rate u* expressed as a function
of the job finding rate f and the job separation rate s can thus be written:
b. suppose that 1 percent of those who are employed lose their jobs each month (s = 0.01). this means that the average spell of
employment lasts \\( \frac{1}{0.01} \\), or 100 months, about 8 years. suppose further that 20 percent of those who are unemployed find a job
each month (f = 0.2), so that the average spell of unemployment lasts 5 months. calculate the steady - state rate of
unemployment u*.
u* = %

Explanation:

Step1: Express \( E \) in terms of \( U \)

In the labor - market, \( L=U + E \), so \( E = L - U \).

Step2: Substitute \( E \) into the steady - state condition

The steady - state condition is \( fU=sE \). Substitute \( E = L - U \) into it: \( fU=s(L - U) \).

Step3: Expand and solve for \( \frac{U}{L} \)

Expand \( fU=sL - sU \). Then \( fU+sU=sL \). Factor out \( U \): \( U(f + s)=sL \). So \( \frac{U}{L}=\frac{s}{s + f} \).

Step4: Calculate \( U^{*} \)

Given \( s = 0.01 \) and \( f=0.2 \), substitute into \( U^{*}=\frac{s}{s + f}\times100\% \).

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Answer:

\( 4.76\% \)