Home work -revised version-
Consider the following model of an one-shot DMP model. There are measure L workers, and measure N firms. While N is parameter, N is determined by the free-entry condition. The matching function is
m(L, vN ) ,
where v is the number of vacancies of each firms, and vN is then the total number of vacancies. The job-finding rate and job-filled rate can be then obtained as
m(L, vN )
L = p(θ),
and
m(L, vN )
vN = q(θ), where
θ= vN L . Utility function of a worker is
u= w if th worker is employed,
and
u= b if the worker is unemployed,
while the profit function of a firm is
π= F (l) − wl − cv − I
where I is the entry costs, c is the search costs, and l is the number of employed workers, which is determined by
l= q(θ)v,
where v is the number vacancy, and q(θ) is job-filled probability. Other notations are same as in the lecture slide. The wage is determined by the wage share rule as follows
w − b = β ∂F (l)
∂l − w + w − b
,
or equivalently,
∂F(l)
∂l − w = (1 − β)
∂F (l)
∂l − w + w − b
.
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Problem 1
Characterizing the equilibrium wage, market tightness, and the number of firms.
(Hint) The number of vacancies of each firm is determined by the following optimization problem;
maxv π, subject to l = q(θ)v,
and the number of firms is determined by the free-entry condition
π= 0.
Problem 2
Comparing the market equilibrium and the socially efficient economy. (Hint) The social planner problem is defined as
maxv,N F
m(L, vn) N
N+ (L − m(L, vn)) b − cvN − IN.
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