# Suppose workers can be either employed or unemployed. The job separation

rate is constant at s = 0.02 per month. Each month, a number m of unemployed workers find new jobs:

m = f u

where f is the job finding rate for unemployed workers and u is the number of unemployed workers. Equivalently, m vacant job positions are filled:

m = q v

where q is the job filling rate for firms and v is the number of vacancies posted by firms. To determine job finding rate f and job filling rate q

, we assume the number of matches is given by the following Cobb-Douglas matching function:

m = F ( u , v ) = u v

Question parts (type the answers in the box provided using the labels (a), (b), (c), (d).

(a) (2 marks) Provide formulas for the job finding rate f and job filling rate q in terms of the labor market tightness θ = v / u. Is the job finding rate increasing or decreasing in labor market tightness? Explain.

(b) (3 marks) Suppose the value to a firm of a filled job is J = 3 and the cost of posting a vacancy is c = 6. Calculate the equilibrium labor market tightness θ ∗. Using this value for θ ∗, solve for the steady-state job finding rate and steady-state unemployment rate.

(c) (2 marks) Suppose the economy goes into a recession and the value of a filled job falls to J = 2. Calculate the new equilibrium labor market tightness θ ∗. Using this value for θ ∗, solve for the steady-state job finding rate and steady-state unemployment rate. Provide intuition for your answers.

(d) (3 marks) Returning to the case of J = 3, suppose the matching function now is instead given by m = F ( u , v ) = 0.5 u v. Is the labor market more or less efficient with this matching function? Calculate the new equilibrium values of labor market tightness and unemployment with this matching function. How do these values compare to your answers from part (c)? Explain.

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