Equilibrium Unemployment
Joao Gomes, Jeremy Greenwood and Sergio Rebelo
FACTS
In the US economy the following variables move
countercyclically:
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Unemployment rate.
FACTS
In the US economy the following variables move
countercyclically:
I
I
Unemployment rate.
Average duration of unemployment.
FACTS
In the US economy the following variables move
countercyclically:
I
I
I
Unemployment rate.
Average duration of unemployment.
Flows into and out of unemployment.
GOAL
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To construct a general equilibrium model in which
individual job opportunities are affected by both
aggregate and idiosyncratic shocks and that is consistent
with these key facts about unemployment.
GOAL
I
To construct a general equilibrium model in which
individual job opportunities are affected by both
aggregate and idiosyncratic shocks and that is consistent
with these key facts about unemployment.
Why?
GOAL
I
To construct a general equilibrium model in which
individual job opportunities are affected by both
aggregate and idiosyncratic shocks and that is consistent
with these key facts about unemployment.
Why?
I
Such a model would be an ideal laboratory to examine
such questions as the impact of unemployment insurance
and the cost of business cycles fluctuations.
The Model (1)
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Heterogeneous agent economy.
The Model (1)
I
I
Heterogeneous agent economy.
Agents maximize expected value of lifetime utility:
E0
∞
X
t=0
β t U (˜
ct − D (lt ))
The Model (1)
I
I
Heterogeneous agent economy.
Agents maximize expected value of lifetime utility:
E0
∞
X
β t U (˜
ct − D (lt ))
t=0
I
Employed agents pay taxes
The Model (1)
I
I
Heterogeneous agent economy.
Agents maximize expected value of lifetime utility:
E0
∞
X
β t U (˜
ct − D (lt ))
t=0
I
I
Employed agents pay taxes
Unemployed agents receive unemployment insurance and
do not pay taxes
The Model (1)
I
I
Heterogeneous agent economy.
Agents maximize expected value of lifetime utility:
E0
∞
X
β t U (˜
ct − D (lt ))
t=0
I
I
I
Employed agents pay taxes
Unemployed agents receive unemployment insurance and
do not pay taxes
Incomplete markets (there is only a riskfree bond)
The Model (1)
I
I
Heterogeneous agent economy.
Agents maximize expected value of lifetime utility:
E0
∞
X
β t U (˜
ct − D (lt ))
t=0
I
I
I
I
Employed agents pay taxes
Unemployed agents receive unemployment insurance and
do not pay taxes
Incomplete markets (there is only a riskfree bond)
exogenous borrowing constraint
The Model (2)
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Every period an agent receives a job opportunity
O (k, l; ε, λ)
The Model (2)
I
I
Every period an agent receives a job opportunity
O (k, l; ε, λ)
If he accepts it he earns labor income
O (k, l; ε, λ) − (r + δ) k and pays taxes τ . Also ε0 is
drawn from G (ε0 |ε)
The Model (2)
I
I
I
Every period an agent receives a job opportunity
O (k, l; ε, λ)
If he accepts it he earns labor income
O (k, l; ε, λ) − (r + δ) k and pays taxes τ . Also ε0 is
drawn from G (ε0 |ε)
If he rejects it he gets UI benefits µ and his next ε0 is
drawn from H (ε0 )
The Problem of a Worker
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A worker first chooses k and l as follows
Y (ε, λ; Z ) = max [O (k, l; ε, λ) − (R(λ; Z ) + δ)k − d(l)]
k,l
The Problem of a Worker
I
A worker first chooses k and l as follows
Y (ε, λ; Z ) = max [O (k, l; ε, λ) − (R(λ; Z ) + δ)k − d(l)]
k,l
The Choice problem of a worker is
Z
W (a, ε, λ; Z )
s.t c + a0
a0
Z0
=
max
{U (c) + β
0
c,a
max [W (a0 , ε0 , λ0 ; Z 0 ) , S (a0 , λ0 ; Z 0 )]
×dG (ε0 |ε) dF (λ0 |λ) dε0 dλ0 }
= Y (ε, λ; Z ) + [1 + R(λ; Z )] a − T (λ; Z ) ,
≥ ¯a
= TZ
The Problem of a Searcher
The Choice problem of a searcher is :
Z
max W a0 , ε0 , λ0 ; Z 0 , S a0 , ε0 ; Z 0
c,a
×dH (ε) dF λ0 |λ dε0 dλ0 }
S (a, λ; Z ) = max0 {U (c) + β
s.t c + a0 = [1 + R(λ : Z )] a + µ,
a0 ≥ ¯a
Z 0 = TZ
The decision rule governing whether someone works or not is:
Ω (a, ε, λ; Z ) =
1 if W (a, ε, λ; Z ) ≥ S (a, ε; Z )
0 otherwise.
The decision rule governing whether someone works or not is:
Ω (a, ε, λ; Z ) =
1 if W (a, ε, λ; Z ) ≥ S (a, ε; Z )
0 otherwise.
The government maintains a balanced budget each period:
Z
µ
Z
[1 − Ω (a, ε, λ; Z )] dZ (a, ε) dadε = τ
Ω (a, ε, λ; Z ) dZ (a, ε) dadε
The decision rule governing whether someone works or not is:
Ω (a, ε, λ; Z ) =
1 if W (a, ε, λ; Z ) ≥ S (a, ε; Z )
0 otherwise.
The government maintains a balanced budget each period:
Z
Z
[1 − Ω (a, ε, λ; Z )] dZ (a, ε) dadε = τ
µ
Ω (a, ε, λ; Z ) dZ (a, ε) dadε
The capital market clears
Z
Z
K (ε, λ; Z ) Ω (a, ε, λ; Z ) dZ (a, ε) dadε =
adZ (a, ε) dadε
The law of motion for the economy-wide distribution of
wealth, or Z’=TZ, is described by:
0
0
0
Z (a , ε ) =
Z
{I (A (a, ε, λ; Z ) − a0 ) [Ω (a, ε, λ; Z ) G (ε0 |ε)
+ (1 − Ω (a, ε, λ; Z )) dH (ε0 )]dZ (a, ε) dadε},
where I (x)=1 if x ≤ 0 and I (x)=0 otherwise.
Calibration
The instantaneous utility function is:
1−σ
c˜ − l 1+θ / (1 + θ)
−1
U (˜
c − D (l)) =
, θ > 0, σ > 0
1−σ
The production function is:
O (k, l; ε, λ) = exp (λ + ε) k α l 1−α
Income Process Calibration
Aggregate shocks.
λ0 = ρλ λ + ξ,
ξ ∼ N 0, σλ2
Income Process Calibration
Aggregate shocks.
λ0 = ρλ λ + ξ,
ξ ∼ N 0, σλ2
Idiosyncratic shocks.
The worker shock ε evolves according to
ε0 = ρε ε + η,
η ∼ N 0, σε2
and the searcher draws a value of ε with
ε = υ,
υ ∼ N 0, συ2
Income Process Calibration
Aggregate shocks.
λ0 = ρλ λ + ξ,
ξ ∼ N 0, σλ2
Idiosyncratic shocks.
The worker shock ε evolves according to
ε0 = ρε ε + η,
η ∼ N 0, σε2
and the searcher draws a value of ε with
ε = υ,
υ ∼ N 0, συ2
Finally unemployment compensation is set to µ = ηy ∗
Calibration parameters
Income dynamics
Heaton and Lucas(1996)
ln (yit /yit−1 ) = υ0 + υ1 ln (yit−1 /yit−2 ) + υ2 ln (yt /yt−1 ) + µit
Hubbard et al’s (1995)
ln (yit ) = υ1 ln (yit−1 ) + µit
Model υ1 = 0.5 and σµi =0.19 vs Data υ1 = 0.95 and σµi =0.14
Comparative Statics
Impulse response, positive shock
Impulse response, positive shock
Impulse response, positive shock
Welfare Cost of business cycle fluctuations
"
Eb
∞
X
t=0
#
β t U (ct ) = En
"
∞
X
t=0
#
β t U ($ct )
Welfare Cost of business cycle fluctuations
"
Eb
∞
X
#
β t U (ct ) = En
t=0
∞
X
#
β t U ($ct )
t=0
(
$=
"
P∞ t 1−σ )1/(1−σ)
Eb
t=0 β ct
P∞
t 1−σ
En
t=0 β ct
Welfare Cost of business cycle fluctuations
"
Eb
∞
X
#
β t U (ct ) = En
t=0
∞
X
#
β t U ($ct )
t=0
(
$=
"
P∞ t 1−σ )1/(1−σ)
Eb
t=0 β ct
P∞
t 1−σ
En
t=0 β ct
For the model economy $ − 1 = 0.0056.
This implies that the agent prefers to live in an economy with
aggregate shocks.
Some Remarks
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The model does not distinguish between unemployment
due to quits and layoffs
Some Remarks
I
I
The model does not distinguish between unemployment
due to quits and layoffs
Flows between employment and nonparticipation are as
large as flows between employment and unemployment.
Some Remarks
I
I
I
The model does not distinguish between unemployment
due to quits and layoffs
Flows between employment and nonparticipation are as
large as flows between employment and unemployment.
What about the long term unemployed?
Some Remarks
I
I
I
I
The model does not distinguish between unemployment
due to quits and layoffs
Flows between employment and nonparticipation are as
large as flows between employment and unemployment.
What about the long term unemployed?
The model abstracts from vacancies, so that the number
of new job openings always equals the number of
unemployed workers