Joint-Search Theory. Bulent Guler 1 Fatih Guvenen 2 Gianluca Violante 3. Indiana University

Transcription

1 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

2 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

3 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: 1 Couple has concave utility over pooled income 2 Couple receives job offers from multiple locations, and faces a cost of living apart GGV (UT-Austin, NYU) Joint-Search Theory IUB 2 / 19

4 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: 1 Couple has concave utility over pooled income 2 Couple receives job offers from multiple locations, and faces a cost of living apart Comparison with canonical (single-agent) job-search model GGV (UT-Austin, NYU) Joint-Search Theory IUB 2 / 19

5 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

6 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

7 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

8 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

9 Value Functions Flow value for dual-worker couple: rt (w 1, w 2 ) = u (w 1 + w 2 ) Flow value for worker-searcher couple: rω (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

10 Value Functions Flow value for dual-worker couple: rt (w 1, w 2 ) = u (w 1 + w 2 ) Flow value for worker-searcher couple: rω (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 (2b) + 2α max {Ω (w) U, 0} df (w) GGV (UT-Austin, NYU) Joint-Search Theory IUB 4 / 19

11 Reservation Wage Functions Dual-searcher couple: Accept iff wi w such that Ω (w ) = U GGV (UT-Austin, NYU) Joint-Search Theory IUB 5 / 19

12 Reservation Wage Functions Dual-searcher couple: Accept iff wi w such that Ω (w ) = U Worker-searcher couple (spouse 1 employed): w1 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

13 Reservation Wage Functions Dual-searcher couple: Accept iff wi w such that Ω (w ) = U Worker-searcher couple (spouse 1 employed): w1 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 ) w1 < 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

14 Reservation Wage Functions Dual-searcher couple: Accept iff wi w such that Ω (w ) = U Worker-searcher couple (spouse 1 employed): w1 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 ) w1 < 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

15 CARA case: Results w < w Intuition: Income maximization versus consumption smoothing GGV (UT-Austin, NYU) Joint-Search Theory IUB 6 / 19

16 CARA case: Results w < w Intuition: Income maximization versus consumption smoothing { w1 if w φ (w i ) = i < w } w if w i w Quit might be optimal Breadwinner Cycle GGV (UT-Austin, NYU) Joint-Search Theory IUB 6 / 19

17 CARA Case: Graphical Representation GGV (UT-Austin, NYU) Joint-Search Theory IUB 7 / 19

18 Breadwinner Cycle Wage Single 1 Spouse Time (weeks) Wage Single 2 Spouse Time (weeks) GGV (UT-Austin, NYU) Joint-Search Theory IUB 8 / 19

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

20 General Characterization for HARA family: Dual-searcher couple is less choosy than the single-searcher: w < w ŵ > w such that w i (w, ŵ): φ (w i ) = w i Breadwinner Cycle always exists GGV (UT-Austin, NYU) Joint-Search Theory IUB 9 / 19

21 General Characterization for HARA family: Dual-searcher couple is less choosy than the single-searcher: w < w ŵ > w such that w i (w, ŵ): φ (w i ) = w i Breadwinner Cycle always exists w i ŵ: > 0 if DARA φ (w i ) = = 0 if CARA < 0 if IARA GGV (UT-Austin, NYU) Joint-Search Theory IUB 9 / 19

22 DARA Case: Graphical Representation GGV (UT-Austin, NYU) Joint-Search Theory IUB 10 / 19

23 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

24 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 r+2δ φ(w 1 ) [u (w 1, w 2 ) u (w 1, φ (w 1 ))] df (w 2 ) + δ [U Ω (w 1 )] + O (αδ) Intuition: Downfall in consumption in case of separation increases with w 1 GGV (UT-Austin, NYU) Joint-Search Theory IUB 11 / 19

25 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

26 Equivalence Results Search strategies of joint-search problem and single-search problem are identical under: 1 Risk-neutrality 2 On the job search with same offer arrival rates during unemployment and employment: α e = α u 3 CARA with saving and loose enough borrowing limit GGV (UT-Austin, NYU) Joint-Search Theory IUB 12 / 19

27 Numerical Example: single vs couple Model period: one week and interest rate r = (annual 5.3%) Preferences: CRRA (DARA) with risk aversion coef γ {0, 2} Exogenous separation rate δ = (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

28 Numerical Example: single vs couple Model period: one week and interest rate r = (annual 5.3%) Preferences: CRRA (DARA) with risk aversion coef γ {0, 2} Exogenous separation rate δ = (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

29 Single vs Couple: Comparison ρ = 0 ρ = 2 ρ = 4 Single Joint Single Joint Single Joint Res. wage w /w Res. wage φ (1) n/a Double ind. ŵ Mean wage Mm ratio Unemp. rate 5.5% 5.5% 5.4% 7.6% 5.4% 7.7% Unemp. duration Dual-searcher Worker-searcher Job quit rate 0% 11.1% 5.55% EQVAR- cons. 0% 4.5% 14% EQVAR- income 0% 1.1% 2.8% GGV (UT-Austin, NYU) Joint-Search Theory IUB 14 / 19

30 Model s Predictions (DARA case) Lowest wage accepted by couples smaller than for singles Breadwinner cycles: W-S S-W and wage GGV (UT-Austin, NYU) Joint-Search Theory IUB 15 / 19

31 Model s Predictions (DARA case) Lowest wage accepted by couples smaller than for singles Breadwinner cycles: W-S S-W and wage Unemployment duration for dual searcher couple lower than for single searcher Unemployment duration for single searcher lower than for worker-searcher couple Separation rate for married couples higher (because of endogenous quits) GGV (UT-Austin, NYU) Joint-Search Theory IUB 15 / 19

32 Model with multiple locations Risk-neutrality Inside location (i) and outside location (o) Offer arrive at rate α i and α o, drawn from the same distribution F GGV (UT-Austin, NYU) Joint-Search Theory IUB 16 / 19

33 Model with multiple locations Risk-neutrality Inside location (i) and outside location (o) Offer arrive at rate α i and α o, drawn from the same distribution F Fixed cost of living apart κ (in consumption units) for the couple No cost of migration across locations Three reservation wages/functions to characterize: dual-searcher couple: w worker-searcher couples: {φi (w), φ o (w)} for inside and outside offers (respectively) GGV (UT-Austin, NYU) Joint-Search Theory IUB 16 / 19

34 Outside Offers: Graphical Representation GGV (UT-Austin, NYU) Joint-Search Theory IUB 17 / 19

35 Outside Offers: Graphical Representation GGV (UT-Austin, NYU) Joint-Search Theory IUB 18 / 19

36 Numerical Example: Single vs Couple κ = 0 κ = 0.1 κ = 0.3 Single Joint Joint Joint Mean wage Mm ratio Unemployment rate 5.5% 5.5% 6.9% 13.7% Unemployment duration Dual-searcher Worker-searcher 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% EQVAR-cons 0% 0.8% 6.5% GGV (UT-Austin, NYU) Joint-Search Theory IUB 19 / 19