Joint-Search Theory Bulent Guler 1 Fatih Guvenen 2 Gianluca Violante - - PowerPoint PPT Presentation

▶
joint search theory
SMART_READER_LITE
LIVE PREVIEW

Joint-Search Theory Bulent Guler 1 Fatih Guvenen 2 Gianluca Violante - - PowerPoint PPT Presentation

Apr 09, 2024 116 likes •485 views

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

slide-1
SLIDE 1

Joint-Search Theory

Bulent Guler 1 Fatih Guvenen 2 Gianluca Violante 3

1Indiana University 2University of Minnesota 3New York University

Indiana University

GGV (UT-Austin, NYU) Joint-Search Theory IUB 1 / 19

slide-2
SLIDE 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

slide-3
SLIDE 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

slide-4
SLIDE 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

slide-5
SLIDE 5

Joint-Search Problem

Decision unit ⇒ couple: a pair of infinitely lived symmetric spouses indexed by i = {1, 2} Discount rate r, income flows yi ∈ {wi, b} Household intra-period utility: u(y1 + y2)

GGV (UT-Austin, NYU) Joint-Search Theory IUB 3 / 19

slide-6
SLIDE 6

Joint-Search Problem

Decision unit ⇒ couple: a pair of infinitely lived symmetric spouses indexed by i = {1, 2} Discount rate r, income flows yi ∈ {wi, b} Household intra-period utility: u(y1 + y2) 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

slide-7
SLIDE 7

Joint-Search Problem

Decision unit ⇒ couple: a pair of infinitely lived symmetric spouses indexed by i = {1, 2} Discount rate r, income flows yi ∈ {wi, b} Household intra-period utility: u(y1 + y2) 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

slide-8
SLIDE 8

Value Functions

Flow value for dual-worker couple: rT (w1, w2) = u (w1 + w2)

GGV (UT-Austin, NYU) Joint-Search Theory IUB 4 / 19

slide-9
SLIDE 9

Value Functions

Flow value for dual-worker couple: rT (w1, w2) = u (w1 + w2) Flow value for worker-searcher couple:

rΩ (w1) = u (w1 + b) + α

  • max {T (w1, w2) − Ω (w1) , Ω (w2) − Ω (w1), 0} dF (w2)

GGV (UT-Austin, NYU) Joint-Search Theory IUB 4 / 19

slide-10
SLIDE 10

Value Functions

Flow value for dual-worker couple: rT (w1, w2) = u (w1 + w2) Flow value for worker-searcher couple:

rΩ (w1) = u (w1 + b) + α

  • max {T (w1, w2) − Ω (w1) , Ω (w2) − Ω (w1), 0} dF (w2)

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

slide-11
SLIDE 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

slide-12
SLIDE 12

Reservation Wage Functions

Dual-searcher couple:

◮ Accept iff wi ≥ w∗∗ such that Ω (w∗∗) = U

Worker-searcher couple (spouse 1 employed):

◮ w1 ≥ ψ (w2) such that T (ψ (w2) , w2) = Ω (w2): 1 does not quit ⋆ 2 accepts offer iff w2 ≥ φ (w1) such that T (w1, φ (w1)) = Ω (w1) GGV (UT-Austin, NYU) Joint-Search Theory IUB 5 / 19

slide-13
SLIDE 13

Reservation Wage Functions

Dual-searcher couple:

◮ Accept iff wi ≥ w∗∗ such that Ω (w∗∗) = U

Worker-searcher couple (spouse 1 employed):

◮ w1 ≥ ψ (w2) such that T (ψ (w2) , w2) = Ω (w2): 1 does not quit ⋆ 2 accepts offer iff w2 ≥ φ (w1) such that T (w1, φ (w1)) = Ω (w1) ◮ w1 < ψ (w2) such that T (ψ (w2) , w2) = Ω (w2): 1 quits ⋆ 2 accepts offer iff w2 ≥ φ (w1) such that Ω (φ (w1)) = Ω (w1) GGV (UT-Austin, NYU) Joint-Search Theory IUB 5 / 19

slide-14
SLIDE 14

Reservation Wage Functions

Dual-searcher couple:

◮ Accept iff wi ≥ w∗∗ such that Ω (w∗∗) = U

Worker-searcher couple (spouse 1 employed):

◮ w1 ≥ ψ (w2) such that T (ψ (w2) , w2) = Ω (w2): 1 does not quit ⋆ 2 accepts offer iff w2 ≥ φ (w1) such that T (w1, φ (w1)) = Ω (w1) ◮ w1 < ψ (w2) such that T (ψ (w2) , w2) = Ω (w2): 1 quits ⋆ 2 accepts offer iff w2 ≥ φ (w1) such that Ω (φ (w1)) = Ω (w1) ◮ Note that φ (.) = ψ (.) GGV (UT-Austin, NYU) Joint-Search Theory IUB 5 / 19

slide-15
SLIDE 15

CARA case: Results

w∗∗ < w∗ Intuition: Income maximization versus consumption smoothing

GGV (UT-Austin, NYU) Joint-Search Theory IUB 6 / 19

slide-16
SLIDE 16

CARA case: Results

w∗∗ < w∗ Intuition: Income maximization versus consumption smoothing φ (wi) =

  • w1

if wi < w∗ w∗ if wi ≥ w∗

  • Quit might be optimal ⇒ “Breadwinner Cycle”

GGV (UT-Austin, NYU) Joint-Search Theory IUB 6 / 19

slide-17
SLIDE 17

CARA Case: Graphical Representation

GGV (UT-Austin, NYU) Joint-Search Theory IUB 7 / 19

slide-18
SLIDE 18

Breadwinner Cycle

20 40 60 80 100 120 140 160 180 200 0.4 0.6 0.8 1 1.2 Time (weeks) Wage 20 40 60 80 100 120 140 160 180 200 0.4 0.6 0.8 1 1.2 Time (weeks) Wage

Single 2 Spouse 2 Single 1 Spouse 1 GGV (UT-Austin, NYU) Joint-Search Theory IUB 8 / 19

slide-19
SLIDE 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

slide-20
SLIDE 20

General Characterization for HARA family:

Dual-searcher couple is less choosy than the single-searcher: w∗∗ < w∗ ∃ ˆ w > w∗∗ such that ∀wi ∈ (w∗∗, ˆ w): φ (wi) = wi ⇒ Breadwinner Cycle always exists

GGV (UT-Austin, NYU) Joint-Search Theory IUB 9 / 19

slide-21
SLIDE 21

General Characterization for HARA family:

Dual-searcher couple is less choosy than the single-searcher: w∗∗ < w∗ ∃ ˆ w > w∗∗ such that ∀wi ∈ (w∗∗, ˆ w): φ (wi) = wi ⇒ Breadwinner Cycle always exists ∀wi ≥ ˆ w: φ′ (wi) =    > 0 if DARA = 0 if CARA < 0 if IARA   

GGV (UT-Austin, NYU) Joint-Search Theory IUB 9 / 19

slide-22
SLIDE 22

DARA Case: Graphical Representation

GGV (UT-Austin, NYU) Joint-Search Theory IUB 10 / 19

slide-23
SLIDE 23

Exogenous separation w/ CARA and DARA utility

w∗∗ < w∗: Breadwinner cycle still exists... The reservation wage function, φ (wi), is strictly increasing everywhere

GGV (UT-Austin, NYU) Joint-Search Theory IUB 11 / 19

slide-24
SLIDE 24

Exogenous separation w/ CARA and DARA utility

w∗∗ < w∗: Breadwinner cycle still exists... The reservation wage function, φ (wi), is strictly increasing everywhere From the definition of φ (wi): u (w1 + φ (w1)) − u (w1 + b) + δ [Ω (φ (w1)) − Ω (w1)] +

α r+2δ

  • φ(w1) [u (w1, w2) − u (w1, φ (w1))] dF (w2) + δ [U − Ω (w1)] + O (αδ)

Intuition: Downfall in consumption in case of separation increases with w1

GGV (UT-Austin, NYU) Joint-Search Theory IUB 11 / 19

slide-25
SLIDE 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

slide-26
SLIDE 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: αe = αu

3

CARA with saving and “loose enough” borrowing limit

GGV (UT-Austin, NYU) Joint-Search Theory IUB 12 / 19

slide-27
SLIDE 27

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

slide-28
SLIDE 28

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

slide-29
SLIDE 29

Single vs Couple: Comparison

ρ = 0 ρ = 2 ρ = 4 Single Joint Single Joint Single Joint

  • Res. wage w∗/w∗∗

1.02 1.02 0.98 0.75 0.81 0.58

  • Res. wage φ (1)

− n/a − 1.03 − 0.941 Double ind. ˆ w − 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% EQVAR- cons. − 0% − 4.5% − 14% EQVAR- income − 0% − 1.1% − 2.8%

GGV (UT-Austin, NYU) Joint-Search Theory IUB 14 / 19

slide-30
SLIDE 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

slide-31
SLIDE 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

slide-32
SLIDE 32

Model with multiple locations

Risk-neutrality Inside location (i) and outside location (o) Offer arrive at rate αi and αo, drawn from the same distribution F

GGV (UT-Austin, NYU) Joint-Search Theory IUB 16 / 19

slide-33
SLIDE 33

Model with multiple locations

Risk-neutrality Inside location (i) and outside location (o) Offer arrive at rate αi and αo, 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), φo (w)} for inside and outside offers

(respectively)

GGV (UT-Austin, NYU) Joint-Search Theory IUB 16 / 19

slide-34
SLIDE 34

Outside Offers: Graphical Representation

GGV (UT-Austin, NYU) Joint-Search Theory IUB 17 / 19

slide-35
SLIDE 35

Outside Offers: Graphical Representation

GGV (UT-Austin, NYU) Joint-Search Theory IUB 18 / 19

slide-36
SLIDE 36

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% EQVAR-cons − 0% −0.8% −6.5%

GGV (UT-Austin, NYU) Joint-Search Theory IUB 19 / 19