Quantitative Problems

1.
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Suppose it is discovered that 80% of unemployed workers experience an unemployment spell that lasts no longer than 3 weeks, but that the average duration of unemployment is 4 months. How is this possible?
2.
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Let L = E + U, and δ, μ, and a have the meanings given in the text.

(a) Determine the change in the level of unemployment, ΔU, between any two periods.

(b) What is meant by steady-state unemployment?

(c) Use your answer in part (a) to derive the steady-state unemployment rate . Depict the relationship between and μ. Explain what happens if the job-destruction rate, δ, increases.

(d) Evaluate the validity of the claim that a high rate of job destruction, δ, necessarily implies a high rate of unemployment.
3.
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Suppose that the observed steady-state level of unemployment is = 5%; 2% of jobs are destroyed every month (δ = 2%), and that 50% of unemployed workers receive job offers each month (μ = 50%). What is the probability, a, that the average worker accepts a given job offer?
4.
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In practice, new workers constantly enter the labor market and current workers exit from it. Suppose that each period a constant b new workers join the labor force (entering the unemployment pool) and that a fraction 0 ≤ ρ ≤ 1 of workers exit it. Determine the steady-state size of the labor force and the steady-state unemployment rate.
5.
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Let μ denote the probability an unemployed worker finds a job, β denote the probability an open vacancy finds a worker, and assume a = 1 (i.e., workers accept any job offer). Suppose that the matching technology takes the form .

(a) Show that the matching technology exhibits constant returns to scale (which, to recap, means that scaling U and V by the factor k > 0 results in scaling .

(b) Derive and illustrate the SS locus. What happens if m0 increases from m0 = 2 to m0 = 3?
6.
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In the text, it was assumed that wages are exogenously given, and the net value of opening a new vacancy is . Suppose, instead, that the wage depends positively on μ according to w = w(μ). Is this formulation plausible? What happens to the EE locus in this case? Use your answer to explore the effects of a sudden improvement in the matching technology on the equilibrium wage and the unemployment rate.
7.
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Despite the fact that recessions are almost always associated with an increase in the employment rate, do they potentially confer any economic benefits?

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