Joint-Search Theory Bulent Guler 1 Fatih Guvenen 2 Gianluca Violante 3 1 Indiana University 2 University of Minnesota 3 New York University Indiana University GGV (UT-Austin, NYU) Joint-Search Theory IUB 1 / 19
Goal of the Chapter Theoretical characterization of the joint job search problem of a household (i.e., a couple) Starting point: McCall (1970)-Mortensen (1970), and Burdett (1978) GGV (UT-Austin, NYU) Joint-Search Theory IUB 2 / 19
Goal of the Chapter Theoretical characterization of the joint job 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 to different outcome from single-agent: Couple has concave utility over pooled income 1 Couple receives job offers from multiple locations, and faces a cost of living 2 apart GGV (UT-Austin, NYU) Joint-Search Theory IUB 2 / 19
Goal of the Chapter Theoretical characterization of the joint job 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 to different outcome from single-agent: Couple has concave utility over pooled income 1 Couple receives job offers from multiple locations, and faces a cost of living 2 apart Comparison with canonical (single-agent) job-search model GGV (UT-Austin, NYU) Joint-Search Theory IUB 2 / 19
Joint-Search Problem Decision unit ⇒ couple: a pair of infinitely lived symmetric spouses indexed by i = { 1, 2 } Discount rate r , income flows y i ∈ { wi , b } Household intra-period utility: u ( y 1 + y 2 ) GGV (UT-Austin, NYU) Joint-Search Theory IUB 3 / 19
Joint-Search Problem Decision unit ⇒ couple: a pair of infinitely lived symmetric spouses indexed by i = { 1, 2 } Discount rate r , income flows y i ∈ { wi , b } Household intra-period utility: u ( y 1 + y 2 ) Couple pools income and there is no storage (relaxed later) Search only during unemployment (relaxed later) At rate α , unemployed draws offer from exogenous distribution F ( w ) GGV (UT-Austin, NYU) Joint-Search Theory IUB 3 / 19
Joint-Search Problem Decision unit ⇒ couple: a pair of infinitely lived symmetric spouses indexed by i = { 1, 2 } Discount rate r , income flows y i ∈ { wi , b } Household intra-period utility: u ( y 1 + y 2 ) Couple pools income and there is no storage (relaxed later) Search only during unemployment (relaxed later) At rate α , unemployed draws offer from exogenous distribution F ( w ) Wage constant during employment spell No exogenous separation into unemployment (relaxed later) GGV (UT-Austin, NYU) Joint-Search Theory IUB 3 / 19
Value Functions Flow value for dual-worker couple: rT ( w 1 , w 2 ) = u ( w 1 + w 2 ) GGV (UT-Austin, NYU) Joint-Search Theory IUB 4 / 19
Value Functions Flow value for dual-worker couple: rT ( w 1 , w 2 ) = u ( w 1 + w 2 ) Flow value for worker-searcher couple: � r Ω ( w 1 ) = u ( w 1 + b ) + α max { T ( w 1 , w 2 ) − Ω ( w 1 ) , Ω ( w 2 ) − Ω ( w 1 ) , 0 } dF ( w 2 ) GGV (UT-Austin, NYU) Joint-Search Theory IUB 4 / 19
Value Functions Flow value for dual-worker couple: rT ( w 1 , w 2 ) = u ( w 1 + w 2 ) Flow value for worker-searcher couple: � r Ω ( w 1 ) = u ( w 1 + b ) + α max { T ( w 1 , w 2 ) − Ω ( w 1 ) , Ω ( w 2 ) − Ω ( w 1 ) , 0 } dF ( w 2 ) Flow value for dual-searcher couple: � rU = u ( 2 b ) + 2 α max { Ω ( w ) − U , 0 } dF ( w ) GGV (UT-Austin, NYU) Joint-Search Theory IUB 4 / 19
Reservation Wage Functions Dual-searcher couple: ◮ Accept iff w i ≥ w ∗∗ such that Ω ( w ∗∗ ) = U GGV (UT-Austin, NYU) Joint-Search Theory IUB 5 / 19
Reservation Wage Functions Dual-searcher couple: ◮ Accept iff w i ≥ w ∗∗ such that Ω ( w ∗∗ ) = U Worker-searcher couple (spouse 1 employed): ◮ w 1 ≥ ψ ( w 2 ) such that T ( ψ ( w 2 ) , w 2 ) = Ω ( w 2 ) : 1 does not quit ⋆ 2 accepts offer iff w 2 ≥ φ ( w 1 ) such that T ( w 1 , φ ( w 1 )) = Ω ( w 1 ) GGV (UT-Austin, NYU) Joint-Search Theory IUB 5 / 19
Reservation Wage Functions Dual-searcher couple: ◮ Accept iff w i ≥ w ∗∗ such that Ω ( w ∗∗ ) = U Worker-searcher couple (spouse 1 employed): ◮ w 1 ≥ ψ ( w 2 ) such that T ( ψ ( w 2 ) , w 2 ) = Ω ( w 2 ) : 1 does not quit ⋆ 2 accepts offer iff w 2 ≥ φ ( w 1 ) such that T ( w 1 , φ ( w 1 )) = Ω ( w 1 ) ◮ w 1 < ψ ( w 2 ) such that T ( ψ ( w 2 ) , w 2 ) = Ω ( w 2 ) : 1 quits ⋆ 2 accepts offer iff w 2 ≥ φ ( w 1 ) such that Ω ( φ ( w 1 )) = Ω ( w 1 ) GGV (UT-Austin, NYU) Joint-Search Theory IUB 5 / 19
Reservation Wage Functions Dual-searcher couple: ◮ Accept iff w i ≥ w ∗∗ such that Ω ( w ∗∗ ) = U Worker-searcher couple (spouse 1 employed): ◮ w 1 ≥ ψ ( w 2 ) such that T ( ψ ( w 2 ) , w 2 ) = Ω ( w 2 ) : 1 does not quit ⋆ 2 accepts offer iff w 2 ≥ φ ( w 1 ) such that T ( w 1 , φ ( w 1 )) = Ω ( w 1 ) ◮ w 1 < ψ ( w 2 ) such that T ( ψ ( w 2 ) , w 2 ) = Ω ( w 2 ) : 1 quits ⋆ 2 accepts offer iff w 2 ≥ φ ( w 1 ) such that Ω ( φ ( w 1 )) = Ω ( w 1 ) ◮ Note that φ ( . ) = ψ ( . ) GGV (UT-Austin, NYU) Joint-Search Theory IUB 5 / 19
CARA case: Results w ∗∗ < w ∗ Intuition: Income maximization versus consumption smoothing GGV (UT-Austin, NYU) Joint-Search Theory IUB 6 / 19
CARA case: Results w ∗∗ < w ∗ Intuition: Income maximization versus consumption smoothing w i < w ∗ � � w 1 if φ ( w i ) = w ∗ w i ≥ w ∗ if Quit might be optimal ⇒ “Breadwinner Cycle” GGV (UT-Austin, NYU) Joint-Search Theory IUB 6 / 19
CARA Case: Graphical Representation GGV (UT-Austin, NYU) Joint-Search Theory IUB 7 / 19
Breadwinner Cycle 1.2 1 Wage 0.8 0.6 Single 1 0.4 Spouse 1 0 20 40 60 80 100 120 140 160 180 200 Time (weeks) 1.2 1 Wage 0.8 0.6 Single 2 0.4 Spouse 2 0 20 40 60 80 100 120 140 160 180 200 Time (weeks) GGV (UT-Austin, NYU) Joint-Search Theory IUB 8 / 19
General Characterization for HARA family: Dual-searcher couple is less choosy than the single-searcher: w ∗∗ < w ∗ GGV (UT-Austin, NYU) Joint-Search Theory IUB 9 / 19
General Characterization for HARA family: Dual-searcher couple is less choosy than the single-searcher: w ∗∗ < w ∗ w > w ∗∗ such that ∀ w i ∈ ( w ∗∗ , ˆ ∃ ˆ w ) : φ ( w i ) = w i ⇒ Breadwinner Cycle always exists GGV (UT-Austin, NYU) Joint-Search Theory IUB 9 / 19
General Characterization for HARA family: Dual-searcher couple is less choosy than the single-searcher: w ∗∗ < w ∗ w > w ∗∗ such that ∀ w i ∈ ( w ∗∗ , ˆ ∃ ˆ w ) : φ ( w i ) = w i ⇒ Breadwinner Cycle always exists ∀ w i ≥ ˆ w : > 0 if DARA φ ′ ( w i ) = = 0 if CARA < 0 if IARA GGV (UT-Austin, NYU) Joint-Search Theory IUB 9 / 19
DARA Case: Graphical Representation GGV (UT-Austin, NYU) Joint-Search Theory IUB 10 / 19
Exogenous separation w/ CARA and DARA utility w ∗∗ < w ∗ : Breadwinner cycle still exists... The reservation wage function, φ ( w i ) , is strictly increasing everywhere GGV (UT-Austin, NYU) Joint-Search Theory IUB 11 / 19
Exogenous separation w/ CARA and DARA utility w ∗∗ < w ∗ : Breadwinner cycle still exists... The reservation wage function, φ ( w i ) , is strictly increasing everywhere From the definition of φ ( w i ) : u ( w 1 + φ ( w 1 )) − u ( w 1 + b ) + δ [ Ω ( φ ( w 1 )) − Ω ( w 1 )] + α � φ ( w 1 ) [ u ( w 1 , w 2 ) − u ( w 1 , φ ( w 1 ))] dF ( w 2 ) + δ [ U − Ω ( w 1 )] + O ( αδ ) r + 2 δ Intuition: Downfall in consumption in case of separation increases with w 1 GGV (UT-Austin, NYU) Joint-Search Theory IUB 11 / 19
Equivalence Results Search strategies of joint-search problem and single-search problem are identical under: GGV (UT-Austin, NYU) Joint-Search Theory IUB 12 / 19
Equivalence Results Search strategies of joint-search problem and single-search problem are identical under: Risk-neutrality 1 On the job search with same offer arrival rates during unemployment and 2 employment: α e = α u CARA with saving and “loose enough” borrowing limit 3 GGV (UT-Austin, NYU) Joint-Search Theory IUB 12 / 19
Numerical Example: single vs couple Model period: one week and interest rate r = 0.001 (annual 5.3%) Preferences: CRRA (DARA) with risk aversion coef γ ∈ { 0, 2 } Exogenous separation rate δ = 0.0054 (annual 0.25) Wage offer distribution lognormal with E [logw]=0 and SD [logw]=0.1 GGV (UT-Austin, NYU) Joint-Search Theory IUB 13 / 19
Numerical Example: single vs couple Model period: one week and interest rate r = 0.001 (annual 5.3%) Preferences: CRRA (DARA) with risk aversion coef γ ∈ { 0, 2 } Exogenous separation rate δ = 0.0054 (annual 0.25) Wage offer distribution lognormal with E [logw]=0 and SD [logw]=0.1 Offer arrival rate, α , matches annual unemployment rate 5.5% Unemployment income flow, b = 0.4 Symmetric couple members GGV (UT-Austin, NYU) Joint-Search Theory IUB 13 / 19
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