On the Economics of PolygynyTed Bergstrom, UCSB

Poly Families

• Polygamy—Multiple mates of unspecified sex and number

• Polygyny---One man with many wives

• Polyandry---One woman with many husbands

• Polygynandry---More than one person of each sex

Murdoch’s Ethnographic Atlas

• 1170 recorded societies.

• 850 are polygynous.

• Polyandry is rare, but is found in Himalayan regions.

• In African countries, percentage of women living in polygynous households ranges from 25% to 55%.

Bridewealth vs Dowry

• Bridewealth is payment from the groom’s family to the bride’s male relatives.

• Dowry is not the reverse of bridewealth. Dowry is usually a payment from the bride’s family to the newly married couple—a kind of pre-mortem inheritance—Jack Goody.

• Indirect dowry– A payment from the groom’s family to the newly wed couple.

Economic Explanation of Marriage Institutions?

• When and why is bride price positive?

• Is polygyny “good” or “bad” for women?

• When are there dowries instead of bride prices?

Economics of African Polygyny

• Some comparative statics of bride markets

• General equilibrium analysis

• Welfare analysis

African polygyny and bride price

• In cattle-raising societies of Africa, polygyny is the norm.

• Most wealth is cattle. Men trade cattle for wives.

• Price is usually high, 20 or more cattle. Very significant fraction of individual wealth.

Bridewealth: the Currency

Wives for Cattle

• A house was constituted by a major wife and her children. If a man had three wives, he would have three separate estates. A house had the right to the products of its gardens, the calves and milk of its cows, the earnings of the wife and minor children, and crucially…to the bridewealth received by its daughters.

• Cattle received in bridewealth for a daughter of the house normally used for sons of the house to acquire wives.

(Adam Kuper Wives for Cattle)

Two questions:

• Why is polygyny common and polyandry rare?

• Why is bride price in Africa positive and not negative?

A Sociobiological Answer

• Polygyny is more common than polyandry because of reproductive technology.– People value descendants.– Sperm is abundant, wombs are scarce. – A woman with a shared husband is about as

fertile as with one or more husbands. – A man’ s fertility increases sharply with

number of wives.

A formal model

• Men’s utility U(x,k) : k is number of surviving children and x is own consumption.

• Fertility function of a wife f( r) where r is consumption goods given to wife and her children.

• Production function for children nf(r),where n is the number of wives,

Two Reasons for Demand

Labor Services Child Production

Bride prices, gross and net

• Value of woman j’s labor: wj• Bride price of woman j: Bj • Assume all women equally fertile, in

equilibrium all have same net cost, p=Bj-wj.• Net cost of buying a woman and giving her r

units of goods is p+r.

The integer problem

• Wives come in indivisible units. Possible ways of dealing with this nuisance.

• Lottery solution. – To buy 1/10 of an expected wife, bet p/10 at odds 10 to

1. If you win, you get p and buy a wife.

• Time-sharing– Polyandry

– Urban Underclass model. W.J. Wilson, R. Willis

r

f[r]

r̂

Expected Fertility Function

Optimal Allocation of Funds

• Tradeoff : number of wives vs resources per wife.

• Marginal condition: Expected fertility gain from dollar spent on resources for one wife equals expected gain from gambling the dollar with winning prize = r*+p the cost of buying and supporting an additional wife.

• Condition is df(r*)/dr = f(r*)/(r*+p)

r

f[r]

r̂

-P r*

Optimal resources per wife

The efficient way to spend $Z on raising children

• Solve equation df (r *)/dr=f(r*)(r*+p) for r*• Let n*=Z/(r*+p)• Buy n* wives. Give each wife r* units of

resources for herself and her children.• You will then get n*f(r*)=Z f(r*)/(r*+p)

children.• Note constant returns per dollar. Price of a

kid is f(r*)/(r*+p).

More about the solution

• In equilibrium, all wives are treated the same, whether they marry rich or poor man, and whether they are more or less productive as workers.This simplifies pricing, parents don’t care who daughter

marries. (In contrast to case of dowry.)

See Borger-Mulderhoff for empirical work on

Bride prices among Kipsigis.

Comparative statics

• Rogers’ Law –The more you have to pay for a bride, the better you will treat her.

• This is a comparison across equilibria. For proof, see diagram.

• Normal goods theorem- Demand curve for wives slopes down if demand curve for kids slopes down.

r

f[r]

-P’ r*

Comparative Statics

-P r*’

Equilibrium Analysis

General Equilibrium Questions

• What determines bride price?

• Why is distribution of income persistently unequal?

• Where does wealth to pay for brides come from?

• Does need to purchase brides reduce other productive investment?

Dry run simple g.e. model

• Assume men have identical preferences and endowments.

• Each man has one sister. • Each man’s labor produces m units and each

woman’s labor produces w units of resources.• Each man receives his sister’s bride price as an

inheritance.

Equilibrium setup

• Let goods be the numeraire and B be bride price. • In equilibrium, each family sells its daughter for B

and gives proceeds to its son. • Men keep their wives’ earnings. Therefore the net

cost of a wife is P=B-w.• In equilibrium, each man will buy one wife and will

allocate his income between consumption x for himself and r for his wife and her children..

Allocation of consumption

• In equilibrium, each man buys one wife and receives the bride price of one sister. He also controls earnings of his wife. Then each

chooses x units of consumption for himself and r for his wife and children to as to maximize

Max U(x, f(r)) subject to

x+r=m+w• Let x* and r* solve this problem. These are the equilibrium

consumptions.

Equilibrium prices

• The equilibrium bride price has to be such that each individual chooses to buy exactly one bride at these prices.

• This will be the case if and only if a marginal dollar spent on a (fractional) extra wife produces as much expected fertility as a marginal dollar spent on supporting resources r.

• Given that we already have solved for r=r*, this condition determines P as the solution to

P+r*= f’(r*) /f(r*)

r

f[r]

r̂

-P r*

Geometry of Solution

Comparative statics result

• If children are a normal good. (Income effect positive) then bride price and resources per wife are increasing functions of m+w.

• Proof: See diagram.

Distribution of wealth

• Suppose that sisters’ bride price revenue is shared equally among brothers.

• Wealth varies with sex ratios across families. • If man has productivity wi and the ratio of girls to

boys in his family is si, then his wealth is

wi+siBi= wi+p+mi where Bi is the average brideprice of his sisters, p is

the net market price of brides and mi is the average productivity of his sisters.

Distribution of wives and fertility

• With polygyny, men produce children at constant marginal cost.

• In equilibrium, all women consume equal amounts• With homothetic preferences, expected number of

children and of wives is proportional to wealth. • Mean number of wives is 1. Therefore number of

wives is ratio of own wealth to average wealth

Case of equal productivity

• Suppose that men all have productivity m and women all have productivity w.

• Average wealth of men is

B+m=p+w+m• Wealth of man i is

Bsi+m • Therefore man i has ni wives where

ni= Bsi/ (B+m) +m/(B+m)

Source of Inequality

• In this simple model, the source of persistent inequality of wealth is differences in the distribution of sex ratios of children.

• Note that the higher the bride price relative to the productivity of men, the greater the variance of income and of the number of wives.

• In special case where men do no work, variance of number of wives equals variance of sex ratio.

• If m>0, variance of number of wives is smaller than variance of sex ratio.

And what about dowry?

• 1) Is a dowry a negative bride price?

• 2) Why do many agricultural societies have dowries?

• Answer 1: No, bride price is paid to bride’s male relatives by groom. Dowry is paid to the newly wed couple by bride’s relatives.

Which was scarce?

Women? Land?

r

f[r]

r̂

Threshold productivity

r0

19th Century Europe Agriculture

• Farms divided into small units able to sustain only one family.

• Farm owners had single wife with many children.

• Large number of landless agricultural labors, male and female—typically did not marry or have children.– Late marriage in Norway

Why monogamy?

• European farms were too small to support more than one wife and her children.

• Even at zero bride price, landowners could not improve their fertility by acquiring an extra wife and her children.

• At zero price, there was excess supply of marriageable women.

Dowries for Farmers

• Landowners would expect a dowry on marriage. • Oldest son would inherit the farm. Parents would

save money or assets to use as dowry for a daughter.

• Oldest son would receive a dowry at marriage.• Other children would work as farm laborers,

domestics—later would move to cities or emigrate.

Why pay a dowry?

• One way to obtain grandchildren is to have your daughter marry a landowner, so that she can reproduce.

• There are more women than farms. • How to persuade a landowner to marry your

daughter? • Offer a dowry.• Equilibrium price would equalize productivity of

money paid to have daughter married, money paid to increase sons’ reproductive prospects.

Evolution of Sex ratios: Nature’s choice and efficient markets

• Why do most mammals produce about equal numbers of males as females.

• Wasteful Nash equilibrium, since one male can mate many females. (Waste less if males help raise offspring.)

• Darwin—Fisher equilibrium theory.

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