Job Search Models
Jesús Fernández-Villaverde
University of Pennsylvania
February 12, 2016
Jesús Fernández-Villaverde (PENN)
Job Search
February 12, 2016
1 / 27
Introduction
Motivation
We want to have dynamic models of the job market.
Examples of questions we are interested in:
1
Why is there unemployment?
2
Why does unemployment ‡uctuate over the business cycle?
3
Why does unemployment ‡uctuate in the lower frequencies?
4
Why are unemployment rates di¤erent across countries?
5
Is the unemployment level e¢ cient?
6
What are the e¤ects of labor market regulation?
7
What are the e¤ects of UI?
8
What determines the distribution of jobs and wages?
Equilibrium models of unemployment.
Labor market frictions.
Jesús Fernández-Villaverde (PENN)
Job Search
February 12, 2016
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Introduction
Search Models
We will begin with a simple model of job search.
Matching is costly. Think about getting a date.
We can bring our intuition to the job market. Why?
Useful to illustrate many ideas and for policy analysis.
Contributions of:
1
Stigler (1961).
2
McCall (1970).
Static problem versus sequential.
Jesús Fernández-Villaverde (PENN)
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Stigler’s Model
Stigler’s Model
Risk-neutral agent.
Easier to think as an agent asking for bids.
Samples o¤ers i.i.d. from F (w ) .
Decide ex-ante how many o¤ers n she is going to ask for.
Each o¤er has a cost c.
Jesús Fernández-Villaverde (PENN)
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Stigler’s Model
Optimal Number of O¤ers
Remember that:
Mn = E min (w1, w2 , ..., wn ) =
Then, gain of additional o¤er is:
Gn
= Mn
=
Z ∞
1
(1
Mn
F (w ))
n 1
dw
0
=
=
Z ∞
Z0 ∞
Z ∞
Z ∞
(1
F (w ))n dw
0
(1
F (w ))n dw
(1
F (w ))n
0
(1
F (w ))
n 1
dw
Z ∞
1
(1
F (w )) dw
0
(1
F (w ))n
1
F (w ) dw
0
Then Gn is a decreasing function with limn !∞ Gn = 0.
Optimal rule: set n such that Gn c > Gn +1 .
Basic problem of static decisions: What if I get the lowest possible
price in my …rst o¤er?
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McCall’s Model
McCall’s Model
An agent searches for a job, taking market conditions as given.
Preferences:
∞
E ∑ βt xt
t =0
where
xt =
= w if employed
= z if unemployed
Interpretation of w and z.
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McCall’s Model
Job O¤ers
An unemployed agent gets every period one o¤er i.i.d. from
distribution F (w ) .
O¤er can be rejected (unemployed next period) or accepted (wage
posting by …rms).
No recall of o¤ers (no restrictive because of stationarity of the
problem).
Job last forever (neither quitting nor …ring).
Undirected search (alternative: directed search).
Jesús Fernández-Villaverde (PENN)
Job Search
February 12, 2016
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McCall’s Model
Bellman Equations
Value function of employed agent:
W (w ) = w + βW (w )
Clearly: W (w ) =
w
1 β.
Value function of unemployed agent:
U =z+β
Z ∞
max fU, W (w )g dF (w )
U =z+β
Z ∞
max U,
Then:
0
0
w
1
β
dF (w )
Lebesgue integral: discrete and continuous components.
Jesús Fernández-Villaverde (PENN)
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McCall’s Model
Reservation Wage Property
There exist a reservation wage wR
W (wR ) = U =
such that if w
otherwise.
wR
1
β
wR the worker should accept the o¤er and reject
Then:
wR = T (wR ) = (1
β) z + β
Z ∞
0
max fwR , w g dF (w )
that is a contraction (that is, limN !∞ T N (w0 ) = wR and wR is
unique).
Jesús Fernández-Villaverde (PENN)
Job Search
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McCall’s Model
Characterizing Strategy I
Note:
wR
1
β
=z+β
Z wR
0
=z+β
wR
Adding wR
wR
1
Z wR
0
Z wR
Z ∞
wR
w
,
0
1
dF (w ) +
Z ∞
wR
β 1
β
dF (w ) )
dF (w ) =
β
β
wR 1
Z ∞
wR
w
dF (w ) + β
dF (w ) )
1 β
β
wR 1
z=β
dF (w )
0
R∞
wR
max
Z ∞
βw
wR
1
wR
dF (w )
β
dF (w ) to both sides:
wR
Jesús Fernández-Villaverde (PENN)
z=
β
1
β
Z ∞
wR
Job Search
(w
wR ) dF (w )
February 12, 2016
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McCall’s Model
Characterizing Strategy II
Interpretation
=
wR z
| {z }
Cost of Search one more time
β
1
|
β
Z ∞
wR
(w wR ) dF (w )
{z
}
Expected Gain of one more search
Sequential nature of the problem.
Note that
g (wR ) =
β
1
g 0 (wR ) =
g 00 (wR ) =
Jesús Fernández-Villaverde (PENN)
β
β
1
β
1
Z ∞
β
β
(w
wR
(1
f (wR )
Job Search
wR ) dF (w )
F (wR )) < 0
0
February 12, 2016
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McCall’s Model
Characterizing Strategy III
Integrating by parts
Z ∞
(w
wR
wR ) dF (w ) =
Then:
wR
Jesús Fernández-Villaverde (PENN)
z=
β
1
β
Z ∞
Z ∞
(1
F (w )) dw
wR
(1
F (w )) dw
wR
Job Search
February 12, 2016
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McCall’s Model
Characterizing Strategy IV
Note that
wR
z
=
=
=
=
β
1
β
β
1
β
1
β
β
β
1
β
Z
Z wR
0
∞
(w
(w
Jesús Fernández-Villaverde (PENN)
β
1
β
β
β
1
β
wdF (w )
1
0
Ew
1
Z wR
(w
wR ) dF (w )
Z wR
(w
wR ) dF (w )
0
wR ) dF (w )
wR ) dF (w )
0
Z ∞
β
wR ) dF (w ) +
(w
wR
β
β
1
Z ∞
wR
wR
β
β
1
Job Search
β
Z wR
0
β
Z
0
wR
0
(w
(w
wR ) dF (w )
wR ) dF (w )
February 12, 2016
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McCall’s Model
Characterizing Strategy V
Now:
wR
(1
z=
β
1
β
1
z ) = βEw
β) (wR
wR
Ew
β
z = β (Ew
β
β
wR
1
βwR
z)
β
β
β
Z
Z wR
0
wR
Z wR 0
(w
0
(w
(w
wR ) dF (w ) )
wR ) dF (w ) )
wR ) dF (w )
Rw
R wR
Integrating by parts 0 R (w wR ) dF (w ) =
F (w ) dw and
0
then:
Z wR
wR z = β (Ew z ) + β
F (w ) dw
0
Jesús Fernández-Villaverde (PENN)
Job Search
February 12, 2016
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McCall’s Model
Comparative Statics
Factors that a¤ect search strategy:
1
2
3
Value of unemployment z. Unemployment insurance: length and
generosity of unemployment insurance vary greatly across countries.
US replacement rate is 34%. Germany, France, and Italy the
replacement rate is about 67%, with duration well beyond the …rst
year of unemployment.
Distribution ofR o¤ers. Let Fe (wR) be a mean-preserving spread of
w
w
F (w ). Then 0 R Fe (w ) dw > 0 R F (w ) dw for all wR and
w
eR > wR .
Minimum Wages: If the minimum wage is so high that it makes
certain jobs unpro…table, less jobs are o¤ered and job …nding rates
decline.
Jesús Fernández-Villaverde (PENN)
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McCall’s Model
Problems
Rothschild (1973): Where does the distribution F (w ) come from?
Diamond (1971): Why is the distribution not degenerate?.
Intuition:
1
2
3
4
5
In a model such as the previous one, workers follow a reservation
wage strategy.
Hence, …rms do not gain anything out of posting any w > wR .
At the same time, …rms will never hire anyone if they post w < wR .
Therefore, F (w ) will have a unit mass at wR . (Rothschild’s Paradox).
Moreover (Diamond’s Paradox):
wR
z = β (Ew
wR
z) + β
z = β (wR
wR = z
Jesús Fernández-Villaverde (PENN)
Job Search
Z wR
0
F (w ) dw )
z) )
February 12, 2016
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McCall’s Model
Answers
1
Exogenously given: di¤erent productivity opportunities.
2
Endogenous:
1
Lucas and Prescott model of islands economy.
2
Bargaining.
3
Directed search.
Jesús Fernández-Villaverde (PENN)
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Islands Models
Lucas and Prescott (1974)
Continuum of workers.
Workers are risk neutral.
A large number of separated labor markets (islands).
There is a …rm in each island subject to productivity shocks.
Wage is determined competitively in each island.
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Islands Models
Firms
Each island has an aggregate production function:
θf (n )
where θ is a productivity shock, n is labor, and f has decreasing
returns to scale.
θ evolve according to kernel π θ, θ 0 .
There is a stationary distribution of θ.
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Islands Models
Worker
At the beginning of the period, worker observes:
1
Productivity θ.
2
Amount of worker on the island x.
3
Distribution of islands in the economy Ψ (θ, x ) .
They decide whether or not to move:
1
If it stays, workers will get wage w (θ, x ) .
2
If it moves, it does not work this period and picks which island to move
to.
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Islands Models
Equilibrium within the Island
Firms maximize:
w (θ, x ) = θf 0 (n (θ, x ))
Markets clear:
n (θ, x )
Jesús Fernández-Villaverde (PENN)
x + arrivals
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Islands Models
Value Function for the Worker
The Bellman equation for the worker is given by:
v (θ, x ) = max βvu , w (θ, x ) + β
Z
v θ0, x 0 d θ
where vu is the value of search.
Three cases:
1
v (θ, x ) = βvu : some workers are leaving the market.
2
v (θ, x ) > βvu : no worker is leaving the market. Some may or may not
arrive.
3
v (θ, x ) < βvu : cannot happen.
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Islands Models
Case 2
No worker is leaving but some workers are arriving:
vu =
Z
v θ0, x 0 d θ
Thus:
v (θ, x ) = θf 0 (n (θ, x )) + βvu .
No worker is leaving and no workers are arriving:
v (θ, x ) = θf 0 (n (θ, x )) + β
Jesús Fernández-Villaverde (PENN)
Z
v θ0, x 0 d θ
Job Search
θf 0 (n (θ, x )) + βvu
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Islands Models
A New Expression
Putting all these cases together:
v (θ, x ) = max βvu , θf 0 (n (θ, x )) + min βvu , β
Z
v θ0, x 0 d θ
Functional equation on v (θ, x ) .
Unique solution.
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Islands Models
Evolution of the Labor Force
Some agents leave the market. Then x 0 = n (θ, x ) solves:
0
θf (n (θ, x )) + β
Z
v θ 0 , x 0 d θ = βvu
No worker is leaving but some will arrive next period. Then x 0 solves:
Z
v θ 0 , x 0 d θ = vu
No worker is leaving and no workers will arrive next period. Then:
x0 = x
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Islands Models
Stationary Distribution
The evolution of (θ, x ) is then governed by a function Γ θ 0 , x 0 θ, x
that embodies the equations above.
Then the stationary distribution solves:
Ψ (θ, x ) =
Z
Γ θ 0 , x 0 θ, x Ψ (θ, x ) d θ
From the stationary distribution we can …nd vu .
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Islands Models
Álvarez and Veracierto (1999)
Now, instead of going to their favorite island, unemployed workers
search for a new job randomly.
Every period they …nd one island from distribution Ψ (θ, x ) .
They decide whether to accept it or reject it.
Endogenous distribution of wage o¤ers.
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