2500+ words Essay

Intra-household bargaining

SESS 0042

21st October 2019

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What we know so far…

 People do not match randomly  They choose their partners with a view

to some overall gain from relationship  The gain depends on each partner’s

traits  The gain also depends on the

operation of the household  Next question: how do they split it up?

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Transferable utility = endogenous division of spoils

Abby Barbar a

Carrie Dana

Jake 15 11 12 3

Kyle 10 14 2 4

Len 8 11 9 10

Martin 12 8 10 13

Are these matches stable with transferable utility?

Kyle: Hey B, wt r u getting out of that r-ship? Barbara: x utils. Kyle: U kidding? So Len gets 11-x? Barbara: Yes. U? Kyle: Abby keeps 10-y utils, I get y. … Kyle: U know wt Im thinking? Barbara: Wt? Kyle: Right now, the 2 of us get x+y; u & me 2gethr, we’d have 14>x+y. Barbara: Wt r u saying? Kyle: Dump him 4 me. He can have Abby. Barbara: OK. I luv u.

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Formal analysis  Barbara + Len = x + 11-x = 11  Kyle + Abby = y + 10-y = 10  x+y 21-x-y 21  Either: x + y < 14 => Len + Abby = 21 – x –y

> 7  Then Kyle & Barbara will get together &

have 14  14 is enough to increase both people’s

utility  Or: x + y > 14 => Len + Abby = 21 – x –y <

7  Then Len & Abby will get together & have 8  8>7, so again, both can be made better off

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Transferable utility = endogenous division of spoils

Abby Barbar a

Carrie Dana

Jake 15 11 12 3

Kyle 10 14 2 4

Len 8 11 9 10

Martin 12 8 10 13

These matches are NOT stable: break-ups will follow.

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Limits on splits  W/o further assumptions, the split of

resources is not unique  Yet, both sides need to pay attention:

 To what their partners may get elsewhere  To what they themselves could get

elsewhere  Assume:

 One side proposes, the other just accepts/rejects

 Value of singlehood is zero  Everybody is a maximizer & out for

him/herself

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Even if one side has all the (legal) power, the other side will not get just zero (why? competition)

Abby Barbar a

Carrie Dana

Jake 15 11 12 3

Kyle 10 14 2 4

Len 8 11 9 10

Martin 12 8 10 13 12:1

12:2

9:0

15:0 3:12

0:14

0:9

1:12

14:0

13:0

0:15

0:13

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Men’s alternatives deteriorate: (some) women benefit

Abby Barbar a

Carrie Dana

Jake 15 11 11 3

Kyle 10 14 1 4

Len 8 11 8 10

Martin 12 8 9 13 11:2

11:3

8:0

14:1 3:12

0:14

0:8

1:12

15:0

14:0

13:0

0:15

0:13

15:0 12:2 9:0 12:13:12 0:14 0:9 1:12Initial equilibrium:

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Women’s outside options improved: (some) women benefit

Abby Barbar a

Carrie Dana

Jake 15 11 12 3

Kyle 10 14 2 4

Len 8 11 9 10

Martin 12 8 10 13 9:4

9:5

6:3

12:3 3:12

0:14

0:9

1:12

U0=3

15:0 12:2 9:0 12:13:12 0:14 0:9 1:12

15:0

14:0

13:0

0:15

0:13

Initial equilibrium:

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Abby Barbar a

Carrie Dana

Jake 15 11 12 3

Kyle 10 14 2 4

Len 8 11 9 10

Martin 12 8 10 13 10:3

10:4

9:0

3:12

0:14

0:9

1:12

U0=5

15:0 12:2 9:0 12:13:12 0:14 0:9 1:12

10:5

14:0

13:0

0:15

0:13

7:2

Initial equilibrium:

Women’s outside options improved: (some) women benefit

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Abby Barbar a

Carrie Dana

Jake 15 0 12 3

Kyle 10 0 2 4

Len 8 0 9 10

Martin 12 0 10 13 5:8

9:0

3:12

0:9

1:12

withdraws

15:0 12:2 9:0 12:13:12 0:14 0:9 1:12

15:0

13:0

0:15

0:13

2:7

Initial equilibrium:

Sex ratio favors women: (some) women benefit

5:10

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Bargaining power…  …responds to market characteristics…

 The reservation utility (U0) of participants  This depends on value of outside options

 The sex ratio (affects intensity of competition)  Level of gains from marriage produced in all

marriages (affects what competitors can offer)  …to legal and social constraints…

 It matters who proposes and who responds  …and that with predictable consequences.

 High U0, own scarcity, high productivity within relationships and high own productivity increase one’s take

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More formally  Denote U0 the utility from remaining alone  Denote Z the total resources to be split  Denote Zij the resources when man i marries woman j  Denote mij and fij the man’s a woman’s share, so that mij +

fij = Zij  Assume PAM for simplicity  Then:

),(max),(max

),(max),(max

00

00

ikkiikikii

ikkiikikii

fUfmUZ

mUmfUZ





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What’s love got to do with it?

 Same would apply to labor market:  Outside option: how much do I value being

penniless but with lots of free time?  Relative demand/supply: what is the

unemployment rate & job vacancy rate?  Productivity: how efficient are people in

other firms and I relative to them?  What is the labor law?  What are the in/formal rules of salary

bargaining?

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Introducing love  Love = caring about the other person’s utility  Assume total relationship gain Z = m + f  Some limits on the split m:f are determined

by the market  If man cares…

 Fully: her consumption gives him as much utility as his own, so MUM(m) = MUM(f) > 0

 Somewhat: MUM(m) > MUM(f) > 0  Not at all (market allocation follows): MUM(m) > MUM(f) = 0

 In other words: UM = U(m, U(f)); UF = U(f)

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Caring and utility

m/f

MUm

m = f f < m

MUm(m)

MUm(f)

Clearly, MUm(m) > MUm(f) and the distance can capture how much less he cares for her than for himself.

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How love affects the split

m

f

Z

Z

45° = equal sharing

Market- determined split if men propose

Market-determined split if women propose

Jake’s voluntary transfer to Abby

1 )(

)( 

fMU

mMU MRS

m

m

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If caring is present

 The market bounds need not be binding  Actual sharing will be independent of

market determinants, such as the sex ratio

 There is a clear shift closer to equal sharing

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How does caring affect the marriage market?

 Assume each man loves exactly one woman  Assume optimal matches are along the

diagonal  Assume MUm(m) = 20 – m  Assume MUm(f) = 0.8*(20 – f) = 16 – 0.8f  Equilibrium split equalizes the MU’s:

Ex. Jake loves Abby and ZJake+Abby = 15. Jake offers m = (4+0.8*15)/1.8 = 8.88; f = 15 - 8.88 = 6.11

)()(8.01620)( fMUmZmmMU mm 

mmmZ 8.18.08.01620 

m Z

 

8.1

8.04

(How) will his altruism affect her behavior?

 So far, Jake cares about Abby’s utility, but Abby only cares about her own consumption

 Jake has clear incentives to increase Z:  He gets direct utility from his own portion of Z  He gets indirect (altruistic) utility from Abby’s portion

 Abby’s incentives:  She only gets direct utility from her portion of Z  Split of Z is determined by Jake, however

 Would Abby care about increasing Z?  Assume Jake is a chess-lover  Abby has opportunity to costlessly learn to play chess  This would increase Z through extra joy of Jake 20

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Will Abby care about raising Z?

m

f

Z

Z

45° = equal sharing

Market- determined split if men propose

Market-determined split if women propose

Clearly, even though only Jake enjoys chess, Abby’s take also increased because Jake adjusted his transfer up. Abby gets a share of the extra Z

she generated.

Rotten kid theorem  First formulated by Becker (1974) in the context of

parent-child altruism (hence: RTK)  Generalizable to all altruist-beneficiary relationships  RTK: The self-interested person (here: Abby) will seek

to maximize Z, even though only Jake is the altruist (as long as transfers are positive)

 It’s because his altruism provides a feedback loop  Once Jake’s transfers to her fall to zero, Abby will stop

maximizing Z  Implications:

 All intra-household externalities (= impact of one’s actions on the other’s gains) will be internalized

 This means spouses may even “take a hit” for each other and then adjust through the amount of transfer

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What if Abby’s extra effort also increases her overall marriage market attractiveness?

Abby Barbar a

Carrie Dana

Jake 15 11 12 3

Kyle 10 14 2 4

Len 8 11 9 10

Martin 12 8 10 13 12:1

12:2

9:0

15:0

15:5

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Abby learns to bake a mean apple pie!

Abby Barbar a

Carrie Dana

Jake 20 11 12 3

Kyle 15 14 2 4

Len 13 11 9 10

Martin 17 8 10 13 12:1

12:2

9:0

20:0

14:0

13:0

15:0 12:2 9:0 12:1Initial equilibrium:

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Abby’s actions raise Z and her outside next best offer

m

f

Z

Z

45° = equal sharing

Market- determined split if men propose

Market-determined split if women propose

Again, Abby gets a share of the extra Z she generated but Jake’s voluntary transfer is smaller, as larger portion of Abby’s take is due to her increased

desirability on market.

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If men care each about his woman…

Abby Barbar a

Carrie Dana

Jake 15 11 12 3

Kyle 10 14 2 4

Len 8 11 9 10

Martin 12 8 10 13

6.2:2.8

15:0 12:2 9:0 12:1

8.9:6.1

8.4:5.6

8:5

Initial equilibrium:Abby’s U0=5

equil.:

10:5 10:4 7:2 10:3

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Jake’s utility

m/f

MUm

MUm(m)

MUm(f)

m=8. 9

f=6.1

Abby Barbar a

Carrie Dana

ZJake+j 15 11 12 3

UJake+j NA

6.2:2.88.4:5.6 8:58.9:6.1

+

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Effect of love/caring

 Loved person’s received offer improves relative to no-love situation

 It turns the loved person’s consumption into a semi-public good in the eyes of the lover

 Increases one’s utility from one particular match relative to others (weaker incentive to search on)

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What if Abby loves Jake back? UM=U(m, UF(f)); UF=U(f, UM(m))

m

f

Z

Z

45° = equal sharing

Note: Even if Jake and Abby love each other, they can still clash over the split of Z because they have different

optimal allocations.

Abby’s IC before she loved Jake

Once she loves him, her utility reflects his consumption, too.

How Abby would split Z, if she had the power to

propose.

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What if the women love their men back?

Abby Barbar a

Carrie Dana

Jake 15 11 12 3

Kyle 10 14 2 4

Len 8 11 9 10

Martin 12 8 10 13

6.2:2.8

15:0 12:2 9:0 12:1

8.9:6.1

8.4:5.6

8:5

Initial equilibrium:Abby’s U0=5

equil.:

10:5 10:4 7:2 10:3

It would make luring the women away even harder: they get U out of his consumption.

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Why are some men ambivalent about women’s empowerment?

m

f

Z

Z

45° = equal sharing

Note: How these various outcomes rank against each other will depend on the shape and position of the curves and on the movement of the

threat points. Still, overall: Abby unambiguously gains; Jake – it depends.

This is where Jake

optimally wants to be.

This is where Jake minimally needs to

be, if he wants Abby to

stay.

This is where Abby optimally wants to be and where Jake

actually ends up once Abby starts calling the shots.

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Conclusions: theory

 Market forces matter  Love matters also  Love can be a great stabilizer (by

creating public goods out of private) … …as long as it is itself stable

 Empirical predictions: in-demand people get a sweeter deal; better alternatives (e.g. U0) and being loved begets that also

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Empirical evidence

 Public goods are enjoyed by both without need to bargain over split (only over quantity)

 Many private HH goods are hard to observe

 Therefore: focus on best observables: market consumption and sex

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Arcidiacono, Beauchamp & McElroy (2010)

 Study of high-school dating  Look at preferences for sexual activity  Male and female students differ

 in preference for sex  as well as the dating market situation

 H0: favorable sex ratio on one’s dating market will allow him/her to secure more of what he/she likes

 Control for age, sex, school chars etc.

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Arcidiacono, Beauchamp & McElroy (2010)

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Arcidiacono, Beauchamp & McElroy (2010)

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Arcidiacono, Beauchamp & McElroy (2010)

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Arcidiacono, Beauchamp & McElroy (2010)

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Arcidiacono, Beauchamp & McElroy (2010)

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Moehling (2005)  A study of intra-household bargaining  Looks at spending power of working children

vs non-working children  Why not look at wives? There is variation among

children within HHold (but only one wife)  Take a sample of working families from 1917-

1919 and look at each member’s consumption

 Not all consumption is “assignable”, so she looks at clothing

 Empirical prediction: working children will have greater say in the family

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What historical changes have affected the division?

 Marriage and divorce legislation  Ease of divorce, custody, alimony rules have moved in favor of

women  Contraception and in-door plumbing

 Has made sex “less expensive” and “less icky”  Marriage market boundaries

 Spread of inter-racial/-ethnic/-religious marriage  Affects the sex ratio

 Outside options  Women work more than before  Female singlehood not as stigmatized as before  Single-person HH more manageable  Single-parenthood easier than before

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Conclusions

 Relationship is a positive-sum game… …but some division needs to be

made  Marriage/dating market is like any other

 Intensity of competition among suppliers benefits the buyer

 Strong outside option strengthens one’s bargaining position