Assignment:
Problem 1: Alternative Matching Function
Consider the Mortensen-Pissarides model of unemployment where the aggregate matching function is:
H(U, V ) = A¯ (UV /U + V) with A¯ > 0;
where U denotes the number of unemployed workers and V denotes the number of vacant jobs.
(i) Does this matching function exhibit constant returns to scale? (Give a proof).
(ii) Derive the expressions for the job Önding probability (f) and the vacancy filling probability (q) as a function of market tightness, θ = V/U.
(iii) Give the expression for the equilibrium unemployment rate, u, as a function of market tightness and the separation rate, s. Give a graphical representation of the Beveridge curve.
(iv) Write down the vacancy supply condition where y denotes labor productivity, w the real wage, k the cost of opening a vacancy. Find the closed form solution for market tightness. Represent the vacancy supply condition graphically.
(v) Suppose that the new Congress votes policy measures to make it less costly to open jobs. Assume w = w¯ and use your nswers to the questions above to explain how such measures would a§ect the unemployment rate and market tightness.
Problem 2: The IS model
Consider the following IS model:
Y = C + I + G
C/Y = a¯c b¯c (R - r¯)
I/Y = a¯i b¯i (R - r¯)
G/Y = a¯g
The notations are the same as in your textbook. The only novelty is the second term on the right side of (3¯) where b¯c > 0. It says that private consumption decreases with the real interest rate, R. We interpret r as a long-run real interest rate. (Also, relative to the lecture, the marginal propensity to consume out of transitory income is x¯ = 0.)
(i) Derive the equilibrium relationship between short-run output, Y~ = (Y - Y¯)/ Y¯, and the real interest rate, R. Represent this relationship graphically.
(ii) Suppose the policymaker raises the interest rate R. What are the effects on consumption, investment, and output? Explain.