1
Labor Market Matching Models
•
The Hansen lotteries model is not a great framework to understand per
sistent equilibrium unemployment (the lottery is iid).
•
The seminal work to understand equilibrium unemployment is Mortensen
(1982) and Pissarides (1990). The material here is closer to Cole and
Rogerson (1999) and Ljunqvist and Sargent (Ch. 26.3).
•
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duction of a matching technology that takes job seekers (those who are
unemployed) and vacancies as inputs and the output is a match. The typ
ical technology assumed is CRS, so matches are increasing in job seekers
and vacancies.
1.1
MortensenPissarides
1.1.1
Simplest Environment
•
Pop: A unit measure of workers and a continuum of managers.
•
Prefs: Both workers and managers are risk neutral and discount the future
at rate
β.
•
Prod Tech:
—
Managers own a technology
y
t
=
z
t
h
t
where
h
t
∈
{
0
,
1
}
is worker
input.
—
If a manager is matched (hires) with a worker, the pair generates
output
z
t
but if the manager does not hire a worker, they generate
zero output. Output is nonstochastic since we will assume
z
t
=
z
.
—
If a manager is not matched with a worker, it costs
c
to post a vacancy
(job opening).
—
If a worker is not matched with a manager (i.e. if the worker is unem
ployed), he receives the utility
b<z
(this could be home production
or the “replacement rate”).
—
Once a manager and a worker are matched, there is an exogenous
probability
δ
that the technology breaks down and the match is de
stroyed.
—
There is free entry by managers.
•
Match Tech:
—
The measure of successful matches in a period is given by
M
(
u
t
,v
t
)
where
u
t
and
v
t
are aggregate unemployment and vacancies, respec
tively. Since there is a unit measure of workers,
1=
u
t
+
n
t
where
n
t
is aggregate per capita employment.
1
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We will take it to be a CobbDouglas technology
M
(
u
t
,v
t
)=
u
α
t
v
1
−
α
t
,
which is increasing in both arguments, homogeneous of degree 1. The
parameter
α
measures the elasticity of the matching function with
respect to unemployment.
1
—
The assumption of constant returns to scale is consistent with em
pirical
f
ndings by Blanchard and Diamond (1989) “The Beveridge
Curve”, Brookings Papers on Economic Activity, p. 160. They
found evidence for CRS by regressing data for log(new hires), a mea
sure of matches, on data for log unemployment and log vacancies and
testing that the coe
ﬃ
cients were
α
and
(1
−
α
)
respectively.
—
The probability of
f
lling a vacancy is denoted
q
(
θ
t
M
(
u
t
t
)
v
t
=
θ
−
α
t
(1)
where
θ
t
=
v
t
u
t
is a measure of “labor market tightness”. By this
de
f
nition, if unemployment is low or vacancies (job openings) is high,
the labor market it tight (it is hard for “
f
rms” to
f
nd workers). Since
q
0
(
θ
t
−
αθ
−
(
α
+1)
t
<
0
,
we know that if the labor market gets tighter, the probability of
f
lling
a job falls.
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 Spring '07
 CORBAE
 Steady State, Unemployment, Beveridge curve

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