LECTURE 4: JOINT-SEARCH THEORY
September 17, 2012
Lecture 4: Joint-Search Theory September 17, 2012 1 / 32
Introduction
RISE OF DUAL-CAREER COUPLES
Female labor force participation rate stands at 60% compared to 75% formales.
% of households in which wife provides majority of household income hastripled since 1970.
Now 1/3 of US households have two main breadwinners.
Bottom Line: For many households, job search is increasingly becominga joint decision process.
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Introduction
RISE OF DUAL-CAREER COUPLES
FIGURE : Fraction of Household Income Provided by Each Spouse
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Introduction
RISE OF DUAL-CAREER COUPLES
Female labor force participation rate stands at 60% compared to 75% formales.
% of households in which wife provides majority of household income hastripled since 1970.
Now 1/3 of US households have two main breadwinners.
Bottom Line: For many households, job search is increasingly becominga joint decision process.
Lecture 4: Joint-Search Theory September 17, 2012 4 / 32
Introduction
AIM OF THE PAPER
Theoretical characterization of the joint job search problem of ahousehold (i.e., a couple)
Same economic environment as in McCall (1970)-Mortensen (1970), andBurdett (1978)
We study two cases where joint decision leads to different outcomes fromsingle-agent:
1 Couple has concave utility over pooled income
2 Couple receives job offers from multiple locations, and faces a cost of livingapart
Systematic comparison to single-agent search problem
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Introduction
PREVIEW OF RESULTS
1 Risk-aversion (HARA): New opportunities
Joint-search similar to on-the-job search
reservation wage of unemployed couple is lower than reservation wage ofsingle-agent
endogenous quits into unemployment ) couple climbs up wage ladder(“breadwinner-cycle”)
Extension to nonparticipation
2 Multiple locations: New frictions (even with risk neutrality)
Joint-search process generates “tied-movers” and “tied-stayers” (Mincer,1978)
Couple misses job opportunities grabbed by single-agent facing sameenvironment
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Introduction
RELATED LITERATURE
Shimer-Rogerson-Wright (2005)’s survey: no mention of householdsearch.
Burdett-Mortensen (1977): pioneering effort to characterize joint-searchproblem
Danforth (1979): search with risk-free saving
Mincer (1978): facts on family migration decisions
Dey-Flinn (2007) and Gemici (2007): quantitative search models withtwo-person households
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Joint-Search Model
JOINT-SEARCH MODEL
Decision unit: couple, i.e. a pair of infinitely lived symmetric spousesindexed by i = 1,2
Discount rate r , income flows: y
i
2 {b,wi
}
Couple pools income (i.e., “unitary household”) and there is no storage(relaxed later)
Household intra-period utility: u(y1 + y2)
Search only during unemployment (relaxed later)
At rate a unemployed draw offer from F(w), exogenous
Wage constant during employment spell
No exogenous separation into unemployment (relaxed later)
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Joint-Search Model
VALUE FUNCTIONS
Flow value for dual-worker couple:
rT (w1,w2) = u(w1 +w2)
Flow value for worker-searcher couple:
r⌦(w1) = u(w1 +b)+aˆ
max [T (w1,w2)�⌦(w1) ,⌦(w2)�⌦(w1) ,0]dF (w2)
Flow value for dual-searcher couple:
rU = u (2b)+2aˆ
max [⌦(w)�U,0]dF (w)
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Joint-Search Model
RESERVATION WAGE FUNCTIONS
Dual-searcher couple:
Accept offer iff w1 > w
⇤⇤ s.t. ⌦(w⇤⇤) = U
Worker-searcher couple (spouse 1 employed):
T (w1,w2)� ⌦(w2):
2 accepts offer iff w2 > f(w1) s.t. T (w1,f(w1)) = ⌦(w1)1 does not quit.
T (w1,w2)< ⌦(w2):
2 accepts offer iff w2 > f(w1) s.t. ⌦(f(w1)) = ⌦(w1)1 quits upon acceptance
Quit decision:
1 quits job iff w1 < y(w2) where y is defined by T (y(w2),w2) = ⌦(w2)
) y(.) = f(.)
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Joint-Search Model
1. RISK NEUTRALITY: JOINT SEARCH = SINGLE SEARCH
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Joint-Search Model
2. CARA
Result I: w
⇤⇤ < w
⇤
Trade-off: consumption smoothing vs income maximization
Result II:
f (w1) =
⇢w1 if w1 < w
⇤ (quit)w
⇤ if w1 � w
⇤ (no quit)
Because of CARA, the reservation wage of the unemployed spouse isindependent of the wage of the employed spouse w1.
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Joint-Search Model
2. CARA
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Joint-Search Model
DYNAMICS WITH BREADWINNER CYCLE
0 20 40 60 80 100 120 140 160 180 200
0.4
0.6
0.8
1
1.2
Time (weeks)
Wag
e
0 20 40 60 80 100 120 140 160 180 200
0.4
0.6
0.8
1
1.2
Time (weeks)
Wag
e
Single 2
Spouse 2
Single 1Spouse 1
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Joint-Search Model
3. GENERAL CHARACTERIZATION FOR HARA:
Unemployed couple less choosy than single: w
⇤⇤ < w
⇤
9bw � w
⇤ s.t. for w1 2 (w⇤, bw):
f (w1) = w1 (i.e., 450 line)
Breadwinner cycle always exists!
For w1 � bw :
f 0 (w1)
8<
:
> 0 if DARA
= 0 if CARA
< 0 if IARA
But Shape/type of region changes depending on DARA, CARA or IARA.
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Joint-Search Model
DARA:
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Joint-Search Model
CARA
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Joint-Search Model
IARA
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Joint-Search Model
AN ISOMORPHISM: MODEL OF MULTIPLE JOB HOLDING
Suppose unit of active time can be divided into two subperiods: e.g.,day-shift and night-shift
Single worker can be either:
1 Searching for first job, while producing 2b at home
2 Holding two jobs with wages (w1,w2)
3 Working on one job at wage w1, while searching for the second job andproducing only b at home
If wage offer on the second job is high enough, worker quits her first job andsearches for a better secondary job
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Joint-Search Model
THREE EXTENSIONS
Symmetric on-the-job search (ae
= au
)
Joint-searching reduces to single-agent search
CARA with saving and “loose enough” borrowing limit
Joint-searching reduces to single-agent search
Exogenous separation
In the CARA and DARA cases, f (w) is strictly increasing. Strengthensbreadwinner cycle.
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Numerical Example
ILLUSTRATIVE SIMULATION: SINGLES VS COUPLES
Period: one week [discount rate r = 0.001]
CRRA (DARA) with risk aversion coefficient r 2 {0,2,4,8}
Yearly exogenous separation rate 25%
Wage distribution: Log-normal, with mean wage normalized to zero andSD(logw) = 0.1
Offer arrival rates au
set to match u = 5.5%
Value of leisure b = 0.40
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Numerical Example
COMPARISON OF SINGLE VERSUS JOINT SEARCH
r = 0 r = 2Single Joint Single Joint
Mean wage 1.06 1.06 1.07 1.10Mm ratio 1.04 1.04 1.09 1.47Unemp. rate 5.5% 5.5% 5.4% 7.6%Unemp. duration 9.9 9.9 9.7 12.6
Dual-searcher � 6 � 4.7Worker-searcher � 9.8 � 14.2
Quits/separations � 0% � 11.1%EQVAR- income � 0% � 1.1%
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Numerical Example
COMPARISON OF SINGLE VERSUS JOINT SEARCH
r = 0 r = 2Single Joint Single Joint
Mean wage 1.06 1.06 1.07 1.10Mm ratio 1.04 1.04 1.09 1.47Unemp. rate 5.5% 5.5% 5.4% 7.6%Unemp. duration 9.9 9.9 9.7 12.6
Dual-searcher � 6 � 4.7Worker-searcher � 9.8 � 14.2
Quits/separations � 0% � 11.1%EQVAR- income � 0% � 1.1%
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Numerical Example
COMPARISON OF SINGLE VERSUS JOINT SEARCH
r = 0 r = 2Single Joint Single Joint
Mean wage 1.06 1.06 1.07 1.10Mm ratio 1.04 1.04 1.09 1.47Unemp. rate 5.5% 5.5% 5.4% 7.6%Unemp. duration 9.9 9.9 9.7 12.6
Dual-searcher � 6 � 4.7Worker-searcher � 9.8 � 14.2
Quits/separations � 0% � 11.1%EQVAR- income � 0% � 1.1%
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Numerical Example
COMPARISON OF SINGLE VERSUS JOINT SEARCH
r = 0 r = 2Single Joint Single Joint
Mean wage 1.06 1.06 1.07 1.10Mm ratio 1.04 1.04 1.09 1.47Unemp. rate 5.5% 5.5% 5.4% 7.6%Unemp. duration 9.9 9.9 9.7 12.6
Dual-searcher � 6 � 4.7Worker-searcher � 9.8 � 14.2
Quits/separations � 0% � 11.1%EQVAR- income � 0% � 1.1%
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Numerical Example
NON-MONOTONIC BEHAVIOR IN RRA
r = 2 r = 4 r = 8Single Joint Single Joint Single Joint
w
⇤/w
⇤⇤ 0.98 0.75 0.81 0.58 0.60 0.48E (w) 1.07 1.10 1.01 1.05 1.001 1.01Mm 1.09 1.47 1.23 1.81 1.67 2.10U rate 5.4% 7.6% 5.4% 7.7% 5.3% 5.6%U duration 9.7 12.6 9.8 13.3 9.6 10.0
D-S � 4.7 � 7.7 � 7.1W-S � 14.2 � 13.6 � 9.6
Quits � 11.1% � 5.55% � 0.74%EQ-cons. � 4.5% � 14% � 26%E- income � 1.1% � 2.8% � 0.7%
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Two-Location Model
TWO-LOCATION MODEL
Risk-neutrality
Inside location (i) and outside location (o)
Offers arrive at rate a and ao
, drawn from the same distribution F
Fixed cost of living apart k (in consumption units) for the couple
No cost of migration across locations
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Two-Location Model
VALUE FUNCTIONS
Dual-worker and Separate dual-worker couple:
rT (w1,w2) = w1 +w2
rS(w1,w2) = w1 +w2 �k
Worker-searcher couple
r⌦(w1)=w1+b+ai
ˆmax [T (w1,w2)�⌦(w1) ,⌦(w2)�⌦(w1) ,0]dF (w2)
+ao
ˆmax [S(w1,w2)�⌦(w1) ,⌦(w2)�⌦(w1) ,0]dF (w2)
Dual-searcher couple
rU = 2b+2(ai
+ao
)
ˆmax [⌦(w)�U,0]dF (w)
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Two-Location Model
RESERVATION WAGES: INSIDE AND OUTSIDE OFFERS
Outside Offers Inside Offers
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Two-Location Model
TIED-MOVERS AND TIED-STAYERS
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Two-Location Model
ILLUSTRATIVE SIMULATION: 9 LOCATIONS
k = 0 k = 0.3Single Joint Joint
Reservation wage w
⇤/w
⇤⇤ 1.02 1.02 0.94Mean wage 1.058 1.058 1.045Unemployment rate 5.5% 5.5% 13.7%Unemployment duration 9.9 9.9 13.0
Dual-searcher � 6.5 3.0Worker-searcher � 9.3 28.0
Movers/Population 0.52% 0.52% 1.16%Stayers/Population 1.12% 1.12% 3.4%Tied-movers/Movers � 0% 60%Tied-stayer/Stayers � 0% 23%Job quit rate � 0% 50%EQVAR-cons � 0% �6.5%
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Two-Location Model
FUTURE WORK
Theoretical: Explore
asymmetries between spouses
asymmetries in locations (i.e., size of cities or labor markets)
Quantitative:
We plan to use Danish data (IDA) for a structural estimation of a richermodel based on this framework.
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