Joint-Search Theory
Bulent Guler1 Fatih Guvenen 2 Gianluca Violante3
1Indiana University 2University of Minnesota
3New York University
Goal of the Chapter
Theoretical characterizationof thejointjob search problem of a household
(i.e., a couple)
Starting point: McCall (1970)-Mortensen (1970), and Burdett (1978)
We study two environments where joint decision leads todifferent outcome
from single-agent:
1 Couple hasconcave utilityover pooled income
2 Couple receives job offers frommultiple locations, and faces acost of living apart
Goal of the Chapter
Theoretical characterizationof thejointjob search problem of a household
(i.e., a couple)
Starting point: McCall (1970)-Mortensen (1970), and Burdett (1978)
We study two environments where joint decision leads todifferent outcome
from single-agent:
1 Couple hasconcave utilityover pooled income
2 Couple receives job offers frommultiple locations, and faces acost of living apart
Goal of the Chapter
Theoretical characterizationof thejointjob search problem of a household
(i.e., a couple)
Starting point: McCall (1970)-Mortensen (1970), and Burdett (1978)
We study two environments where joint decision leads todifferent outcome
from single-agent:
1 Couple hasconcave utilityover pooled income
2 Couple receives job offers frommultiple locations, and faces acost of living apart
Joint-Search Problem
Decision unit⇒couple: a pair of infinitely livedsymmetric spousesindexed
byi={1, 2}
Discount rater, income flowsyi ∈ {wi,b}
Household intra-period utility: u(y1+y2)
Couplepools incomeand there is no storage(relaxed later)
Searchonlyduring unemployment (relaxed later)
At rateα, unemployed draws offer from exogenous distributionF(w)
Wageconstantduring employment spell
Joint-Search Problem
Decision unit⇒couple: a pair of infinitely livedsymmetric spousesindexed
byi={1, 2}
Discount rater, income flowsyi ∈ {wi,b}
Household intra-period utility: u(y1+y2)
Couplepools incomeand there is no storage(relaxed later)
Searchonlyduring unemployment (relaxed later)
At rateα, unemployed draws offer from exogenous distributionF(w)
Wageconstantduring employment spell
Joint-Search Problem
Decision unit⇒couple: a pair of infinitely livedsymmetric spousesindexed
byi={1, 2}
Discount rater, income flowsyi ∈ {wi,b}
Household intra-period utility: u(y1+y2)
Couplepools incomeand there is no storage(relaxed later)
Searchonlyduring unemployment (relaxed later)
Value Functions
Flow value fordual-worker couple:
rT(w1,w2) =u(w1+w2)
Flow value forworker-searcher couple:
rΩ(w1) =u(w1+b) +α Z
max{T(w1,w2)−Ω(w1),Ω(w2)−Ω(w1), 0}dF(w2)
Flow value fordual-searcher couple:
rU =u(2b) +2α Z
Value Functions
Flow value fordual-worker couple:
rT(w1,w2) =u(w1+w2)
Flow value forworker-searcher couple:
rΩ(w1) =u(w1+b) +α Z
max{T(w1,w2)−Ω(w1),Ω(w2)−Ω(w1), 0}dF(w2)
Flow value fordual-searcher couple:
rU =u(2b) +2α Z
Value Functions
Flow value fordual-worker couple:
rT(w1,w2) =u(w1+w2)
Flow value forworker-searcher couple:
rΩ(w1) =u(w1+b) +α Z
max{T(w1,w2)−Ω(w1),Ω(w2)−Ω(w1), 0}dF(w2)
Flow value fordual-searcher couple:
rU =u(2b) +2α Z
Reservation Wage Functions
Dual-searcher couple:
I Accept iffwi≥w∗∗such thatΩ(w∗∗) =U
Worker-searcher couple(spouse 1 employed):
I w1≥ψ(w2)such thatT(ψ(w2),w2) =Ω(w2): 1 does not quit
F 2 accepts offer iffw2≥φ(w1)such thatT(w1,φ(w1)) =Ω(w1)
I w1<ψ(w2)such thatT(ψ(w2),w2) =Ω(w2): 1 quits
F 2 accepts offer iffw2≥φ(w1)such thatΩ(φ(w1)) =Ω(w1)
Reservation Wage Functions
Dual-searcher couple:
I Accept iffwi≥w∗∗such thatΩ(w∗∗) =U
Worker-searcher couple(spouse 1 employed):
I w1≥ψ(w2)such thatT(ψ(w2),w2) =Ω(w2): 1 does not quit
F 2 accepts offer iffw2≥φ(w1)such thatT(w1,φ(w1)) =Ω(w1)
I w1<ψ(w2)such thatT(ψ(w2),w2) =Ω(w2): 1 quits
F 2 accepts offer iffw2≥φ(w1)such thatΩ(φ(w1)) =Ω(w1)
Reservation Wage Functions
Dual-searcher couple:
I Accept iffwi≥w∗∗such thatΩ(w∗∗) =U
Worker-searcher couple(spouse 1 employed):
I w1≥ψ(w2)such thatT(ψ(w2),w2) =Ω(w2): 1 does not quit
F 2 accepts offer iffw2≥φ(w1)such thatT(w1,φ(w1)) =Ω(w1)
I w1<ψ(w2)such thatT(ψ(w2),w2) =Ω(w2): 1 quits
F 2 accepts offer iffw2≥φ(w1)such thatΩ(φ(w1)) =Ω(w1)
Reservation Wage Functions
Dual-searcher couple:
I Accept iffwi≥w∗∗such thatΩ(w∗∗) =U
Worker-searcher couple(spouse 1 employed):
I w1≥ψ(w2)such thatT(ψ(w2),w2) =Ω(w2): 1 does not quit
F 2 accepts offer iffw2≥φ(w1)such thatT(w1,φ(w1)) =Ω(w1)
I w1<ψ(w2)such thatT(ψ(w2),w2) =Ω(w2): 1 quits
F 2 accepts offer iffw2≥φ(w1)such thatΩ(φ(w1)) =Ω(w1)
CARA case: Results
w∗∗<w∗
Intuition: Income maximizationversusconsumption smoothing
φ(wi) =
w1 if wi <w∗
w∗ if wi ≥w∗
CARA case: Results
w∗∗<w∗
Intuition: Income maximizationversusconsumption smoothing
φ(wi) =
w1 if wi <w∗
w∗ if wi ≥w∗
Breadwinner Cycle
0 20 40 60 80 100 120 140 160 180 200 0.4 0.6 0.8 1 1.2 Time (weeks) Wage
General Characterization for HARA family:
Dual-searchercouple isless choosythan thesingle-searcher: w∗∗<w∗
∃wˆ >w∗∗ such that ∀wi ∈(w∗∗,wˆ):
φ(wi) =wi ⇒Breadwinner Cycle always exists
∀wi ≥wˆ:
φ0(wi) =
>0 if DARA
=0 if CARA
<0 if IARA
General Characterization for HARA family:
Dual-searchercouple isless choosythan thesingle-searcher: w∗∗<w∗
∃wˆ >w∗∗ such that ∀wi ∈(w∗∗,wˆ):
φ(wi) =wi ⇒Breadwinner Cycle always exists
∀wi ≥wˆ:
φ0(wi) =
>0 if DARA
=0 if CARA
<0 if IARA
General Characterization for HARA family:
Dual-searchercouple isless choosythan thesingle-searcher: w∗∗<w∗
∃wˆ >w∗∗ such that ∀wi ∈(w∗∗,wˆ):
φ(wi) =wi ⇒Breadwinner Cycle always exists
∀wi ≥wˆ:
φ0(wi) =
>0 if DARA
=0 if CARA
<0 if IARA
Exogenous separation w/ CARA and DARA utility
w∗∗<w∗: Breadwinner cycle still exists...
The reservation wage function,φ(wi), is strictly increasingeverywhere
From the definition ofφ(wi):
u(w1+φ(w1))−u(w1+b) +δ[Ω(φ(w1))−Ω(w1)] +
α
r+2δ
R
φ(w1)[u(w1,w2)−u(w1,φ(w1))]dF(w2) +δ[U−Ω(w1)] +O(αδ)
Exogenous separation w/ CARA and DARA utility
w∗∗<w∗: Breadwinner cycle still exists...
The reservation wage function,φ(wi), is strictly increasingeverywhere
From the definition ofφ(wi):
u(w1+φ(w1))−u(w1+b) +δ[Ω(φ(w1))−Ω(w1)] +
α
r+2δ
R
φ(w1)[u(w1,w2)−u(w1,φ(w1))]dF(w2) +δ[U−Ω(w1)] +O(αδ)
Equivalence Results
Search strategies of joint-search problem and single-search problem are
identicalunder:
1 Risk-neutrality
2 On the job search with same offer arrival rates during unemployment and employment: αe=αu
Equivalence Results
Search strategies of joint-search problem and single-search problem are
identicalunder:
1 Risk-neutrality
2 On the job search with same offer arrival rates during unemployment and employment: αe=αu
Numerical Example: single vs couple
Model period: one weekand interest rater =0.001 (annual 5.3%)
Preferences: CRRA(DARA) with risk aversion coef γ∈ {0, 2}
Exogenous separation rateδ=0.0054 (annual 0.25)
Wage offer distributionlognormalwithE[logw]=0 andSD[logw]=0.1
Offer arrival rate,α, matches annual unemployment rate5.5%
Unemployment income flow,b=0.4
Numerical Example: single vs couple
Model period: one weekand interest rater =0.001 (annual 5.3%)
Preferences: CRRA(DARA) with risk aversion coef γ∈ {0, 2}
Exogenous separation rateδ=0.0054 (annual 0.25)
Wage offer distributionlognormalwithE[logw]=0 andSD[logw]=0.1
Offer arrival rate,α, matches annual unemployment rate5.5%
Unemployment income flow,b=0.4
Single vs Couple: Comparison
ρ=0 ρ=2 ρ=4
Single Joint Single Joint Single Joint Res. wagew∗/w∗∗ 1.02 1.02 0.98 0.75 0.81 0.58
Res. wageφ(1) − n/a − 1.03 − 0.941
Double ind. wˆ − 1.02 − 1.02 − 0.94
Mean wage 1.06 1.06 1.07 1.10 1.01 1.05
Mm ratio 1.04 1.04 1.09 1.47 1.23 1.81
Unemp. rate 5.5% 5.5% 5.4% 7.6% 5.4% 7.7%
Unemp. duration 9.9 9.9 9.7 12.6 9.8 13.3
Dual-searcher − 6 − 4.7 − 7.7
Worker-searcher − 9.8 − 14.2 − 13.6
Job quit rate − 0% − 11.1% − 5.55%
Model’s Predictions (DARA case)
Lowest wageaccepted by couplessmallerthan for singles
Breadwinner cycles: W-S ⇒S-W andwage↑
Unemployment durationfor dual searcher couplelowerthan for single
searcher
Unemployment durationfor single searcherlowerthan for worker-searcher
couple
Model’s Predictions (DARA case)
Lowest wageaccepted by couplessmallerthan for singles
Breadwinner cycles: W-S ⇒S-W andwage↑
Unemployment durationfor dual searcher couplelowerthan for single
searcher
Unemployment durationfor single searcherlowerthan for worker-searcher
couple
Model with multiple locations
Risk-neutrality
Inside location (i) and outside location (o)
Offer arrive at rateαi andαo, drawn from the same distribution F
Fixedcost of living apartκ (in consumption units) for the couple
No cost of migration across locations
Three reservation wages/functions to characterize:
I dual-searcher couple: w∗∗
Model with multiple locations
Risk-neutrality
Inside location (i) and outside location (o)
Offer arrive at rateαi andαo, drawn from the same distribution F
Fixedcost of living apartκ (in consumption units) for the couple
No cost of migration across locations
Three reservation wages/functions to characterize:
Numerical Example: Single vs Couple
κ=0 κ =0.1 κ=0.3
Single Joint Joint Joint
Mean wage 1.058 1.058 1.06 1.045
Mm ratio 1.04 1.04 1.09 1.11
Unemployment rate 5.5% 5.5% 6.9% 13.7%
Unemployment duration 9.9 9.9 9.8 13.0
Dual-searcher − 6.5 3.3 3.0
Worker-searcher − 9.3 12.9 28.0
Movers (% of population) 0.52% 0.52% 0.74% 1.26%
Stayers (% of population) 1.12% 1.12% 1.53% 3.4%
Tied-movers/Movers − 0% 29% 56%
Tied-stayer/Stayers − 0% 11% 23%
Job quit rate − 0% 23% 50%
