Who Marries Whom?Optimal matching and assortative marriage

Economic DemographyDemog/Econ C175Prof. Ryan Edwards

April 7, 202014/7/20 11:39 AM demog/econ c175

Agenda

• Schedule• Learning objectives: past and present• Markets and marriage• Household production• Optimal sorting• Positive and negative sorting

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Remaining course requirements

• Labs 9, 10, 11, 13 (there is no Lab 12)

• Final exam will be fully electronic, probably on bCourses, probably 100% multiple choice– Timed, with extra time as usual for DSP allowances– Open book, open note, open everything– On Honor Code that it’s you and you’re working on

your own

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Learning objectives• Economics of Fertility

– Female wages, the Value of Time, the total cost of children

– Income and substitution effects

– The quality-quantity tradeoff, which produces substitution effects on caffeine, toward education

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Learning objectives• Economics of Marriage and Divorce

– Classical (Becker) view: gains from specialization, from partnership-specific goods and investments

– New views

• Divorce: it’s complicated. Good economists are humble about what we can say because of selection

• Similarity in tastes and enjoyment of life might be pretty important for understanding marriage

– Assortative or Assortive Mating: marriage and fertility are choices, and it is interesting to compare how traits typically vary within couples

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Why do people divorce?• There must be gains to divorce if people do it, just like our

argument about marriage

• When the future benefits of a divorce minus the cost of divorce exceed the future gains to this marriage, cut it loose!

• The state of the marriage or partnership market that you face once you divorce is a key input; if the prospects of repartnering are poor, you might stay married

• Sometimes even the best-laid plans can seem inadequate

– Careers can develop in unexpected ways or be lost

– Psychological changes can create new incompatibilities, maybe consumption complementarities fade away

– People may not have been the most honest when they sold themselves!7

Is divorce bad? Is marriage good? Should we design policies to increase marriage?

• Stevenson and Wolfers (2007) provide an economist’s perspective: Marriage and divorce are almost never random treatments

• Many unobservable characteristics determine marriage & divorce

• Wealthy, happy, healthy people are surely more likely to be married, according to our theories. But this says little about whether marriage is good for the marginal couple

• “Gardner and Oswald (2006) find that while people are less happy the year that they separate, a year after the divorce they are happier than they were while married.”

• “Dahl (2005) uses variation in state laws restricting the minimum age of marriage to show that those who are prevented from marrying while young are substantially less likely to end up living in poverty later in their lives.”

8

Assortative matingMatchmaker, matchmaker,Make me a match!Find me a find!Catch me a catch!

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A motivating example:to marry, or not to marry

Susanne Susanne David David(alone) (with D) (with S) (alone)

Production: 10 8 6 2

Does it make sense for Susanne and David to marry?1. If they each keep what they produce (no trade, bargaining,

exchange)2. If they can bargain (and "sell" their services)

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Lessons from example

• For marriage to make sense, no one is worse off

• If trade and bargaining, then arrangement with highest output should be the choice (because spoils can be divided so that both will win)

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The household production function

Zi j= h( Ai , Aj ) [Becker writes Z( )]

Zi j = Total household production of person i with attribute Ai and person j with attribute Aj

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What is "production"

• Not just food, clothing and $$• Also happiness, enjoyment of kids, a lot of

things that aren’t in GDP

824 JOURNAL OF POLITICAL ECONOMY

sorting is in the "core" since no "coalition" outside the core could make

any of its members better off without making some worse off.

Persons entering noncore marriages could not produce more together

than the sum of their incomes in the core. For, if they could, and if any

division of output between mates were feasible, they could find a division

of their output that would make each better off, a contradiction of the

definition of the core. If the sorting along the diagonal were in the core,

this condition states that

mii +fjj ? Z1u alliandj. (16) Conditions (15) and (16) immediately rule out any sorting that does not

maximize the total output of commodities over all marriages, for at

least one Al and one F would then be better off with each other than with

their mates.23 Moreover, the theory of optimal assignments, which has

the same mathematical structure as the sorting of persons by marriage,

implies the existence of a set of incomes that satisfy conditions (15) and

(16) for sortings that maximize total output.24

The solution can be illustrated with the following 2 x 2 matrix of

payoffs:

F1 F2

Ml [8 41 (17) M2 L9 7]

Although the maximum output in any marriage is between M2 and

F1, the optimal sorting is M1 to F1 and M2 to F2. For, if mi11 = 3,

f, = 5, M22 = 5, andf22 = 2, M2 and F1 have no incentive to marry since M22 + 111 = 10 > 9, and neither do M1 and F2 since m1 1 +

1'22 = 5 > 4. In other words, the marriage market chooses not the maximum household commodity output of any single marriage but the

maximum sum of the outputs over all marriages, just as competitive

product markets maximize the sum of the outputs over all firms. Let

us stress again that the commodity output maximized by all households

is not to be identified with national output as usually measured, but in-

cludes conversation, the quantity and quality of children, and other

outputs that never enter or enter only imperfectly into the usual measures.

Put still differently, the marriage market acts as if it maximizes not the

23 If Mi married Fj and Fj married M, in an optimal sorting that did not maximize total output, condition (16) requires that maj + fPj > Z11, all id, pi, or, by summation,

n

ZP = E mij + Lid > sE zoo = Z*. all ij, pi i

Since Z* is the maximum total output, it must exceed Z, by assumption less than the maximum. Hence, a contradiction, and a proof that the optimal sorting cannot produce

less than the maximum total output. 24 For a proof, see Koopmans and Beckman (1957).

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Optimal sorting

• Let’s assume full information, no surprises about household production (ha ha sure)

• Match is made if no alternative is better

• A better match occurs when shifting to a different partner gets you a better outcome

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NOTE

• This is hard• Gary Becker was a Nobel Laureate for a

reason• There are many layers of this onion• There appears to have been an error in the

slides from Spring 2019!• Something this hard WON’T be on the

exam154/7/20 11:39 AM demog/econ c175

Becker's example

Just have two males and two females.Payoff matrix Z is

F1 F2M1 8 4M2 9 7

Who should pair with whom? 164/7/20 11:39 AM demog/econ c175

Becker's example

Just have two males and two females.Payoff matrix Z is

F1 F2M1 8 4M2 9 7

Who should pair with whom? 174/7/20 11:39 AM demog/econ c175

Becker's example

Just have two males and two females.Payoff matrix Z is

F1 F2M1 8 4M2 9 7

Who should pair with whom? 18

In the green squares, partners won’t decide to switch

But there’s some major hosing, right?

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Becker's example

Just have two males and two females.Payoff matrix Z is

F1 F2M1 3+5 4M2 9 5+2

Who should pair with whom? 19

But in the red circles when returns are divvied up M+F as shown,

M2 won’t marry F1 because 9 < 5 + 5

M1 won’t marry F2 because 4 < 3 + 2

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Becker's example

Just have two males and two females.Payoff matrix Z is

F1 F2M1 8 4M2 9 7

Who should pair with whom? 20

Turns out that the red circles also maximize total output

Becker’s point was that private contracts could get us there4/7/20 11:40 AM demog/econ c175

What drives a good match?

• Let’s think about traits across partners

• Some traits are probably complementary

– High education: enjoy talking politics and science

– Low education: don’t enjoy that, enjoy other things

• Some traits are substitutes

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Wages?

• Becker argues that wage levels are substitutes!

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if Im) the time spent at work, is greater than zero. Similarly,

zf = T - S Cf = 1fC- > ?. (25)

Positive or negative assertive mating by wage rates is optimal as

a2z ____w__ - _Zfm 0. (26)

Differentiate Zf with respect to wm to get

zfm - _C-2Cmjf + C - (27) aWm

The first term on the right is clearly negative, so Zfm will be negative if

the second term, alf laWm < 0, is nonpositive, that is, if tm and tf are not gross complements, as these terms are usually defined. 30 Consequently,

a perfectly negative rank correlation between wm and Wf would maximize

total commodity output if the time of M and F were not such gross

complements as to swamp the first term in (27). Considerable empirical

evidence supports the conclusion that tm and tf are not gross complements (Ofek 1970; Smith 1972).

A negative correlation between Wm and Wf maximizes total output because the gain from the division of labor is maximized. Low-wage F

should spend more time in household production than high-wage F

because the foregone value of the time of low-wage F is lower; similarly,

low-wage M should spend more time in household production than high-

wage M. By mating low-wage F with high-wage M and low-wage M

with high-wage F, the cheaper time of both M and F is used more ex- tensively in household production, and the expensive time of both is

used more extensively in market production.

All persons have been assumed to participate in the labor force. During

any year, however, most married women in the United States do not

participate, and a significant number never really participate throughout

their married life. Our analysis does predict that many women would

have only a weak attachment to the labor force since low-wage women

would be discouraged from participation both by their low wage and by

the high wage of their husbands. 31

30 This definition is different from the one given earlier in terms of the sign of the cross- derivative of profit or production functions. The definition in equation (28) is preferable, at least as a predictor of responses to changes in input prices. By "gross" rather than "net" complements is meant in the usual way that the income effect is included along with the

substitution effect. Even if tm and tf were net complements they could still be gross substitutes since the income effect of an increase in wm would tend to increase tf.

31 Low-wage men also would be encouraged to work less both because of their low wage and the relatively high wage of their wives. They would not leave the labor force in large numbers, however, partly because average wage rates of men are so much higher than

those of women and partly because the nonmarket productivity of women is higher than

that of men.

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The real world• Our Lab 10 looks at the 2015 ACS

• American Community Survey (ACS), a continuous micro-census of about 1% of the population

• We look just at California

• In lab, you'll also look at same-sex marriages (a few hundred in sample)

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Summary

• Households, like whole pop, can have a production function

• "free market" of choosing partners and within household à optimal sorting, which maximizes individual welfare (and aggregate)

• A framework for understanding sorting, but not predictive

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Real world example: Edwards and Roff (2010)

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Biological clock for males?

• Over time and exposure, there can be mutations to sperm

• (Famously, Hugh Hefner was unable to produce children toward the end of his life)

• Sample sizes are small, but there appears to be some evidence that schizophrenia may be associated with older dads

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How would you test for this?

• Can you randomly assign older dads to moms and kids?

• Nope

• What if you just look at kids in kindergarten who have older dads?

• Why do they have older dads? Is it random?

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We found two big reasons

• Kids had older dads in the data because:

– Kids had many more older siblings

• What do we suspect about children’s “quality” on average when quantity of children is higher?

– Moms had less education

• This could be associated with family size

• Or it could reflect assortative mating

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Omitted variable bias• Suppose an outcome, like children’s

neurocognitive outcomes (Y) is associated with paternal age (X) and mother’s education (Z)

• If you model ! = # + % & + '

without Z, then your estimate of % will be biased if X and Z are correlated4/7/20 12:07 PM demog/econ c175 29

A very real-world example

• COVID-19 and treatment drugs

• BBC: “Chloroquine, and a related derivative, hydroxychloroquine, have gained attention -despite the World Health Organization (WHO) saying there is no definitive evidence they work.”

• “There are … risks of serious side effects, including renal and liver damage.”

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Omitted variable bias• Suppose an outcome, like surviving

COVID-19 (Y) is associated with chloroquine (X) and something else (Z)

• If you model ! = # + % & + '

without Z, then your estimate of % will be biased if X and Z are correlated

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Randomized controlled trial

• If you randomly select treatment and control groups who are otherwise identical

• And treat the treated with the treatment

• Then the RCT evidence is the difference in outcomes, and that’s pretty legitimate

• But you’d have to deny the treatment to the control group. Nobody’s yet done that much

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