On the Economics of Polygyny

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On the Economics of Polygyny
Ted Bergstrom, UCSB

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Poly Families

• PolygamyMultiple 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

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Murdochs 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.

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Bridewealth vs Dowry

• Bridewealth is payment from the grooms family
  to the brides male relatives.
• Dowry is not the reverse of bridewealth. Dowry is
  usually a payment from the brides family to the
  newly married couplea kind of pre-mortem
  inheritanceJack Goody.
• Indirect dowry A payment from the grooms family
  to the newly wed couple.

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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?
  

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Economics of African Polygyny

• Some comparative statics of bride markets
• General equilibrium analysis
• Welfare analysis

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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.

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Bridewealth the Currency

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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 cruciallyto 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)

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Two questions

• Why is polygyny common and polyandry rare?
• Why is bride price in Africa positive and not
  negative?

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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.

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A formal model

• Mens 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,

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Two Reasons for Demand
Child Production
Labor Services
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Bride prices, gross and net

• Value of woman js labor wj
• Bride price of woman j Bj
• Assume all women equally fertile, in equilibrium
  all have same net cost, pBj-wj.
• Net cost of buying a woman and giving her r units
  of goods is pr.

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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

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Expected Fertility Function
fr

r
r
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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 rp the cost of buying and
  supporting an additional wife.
• Condition is df(r)/dr f(r)/(rp)

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Optimal resources per wife

fr
r
r
-P
r

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The efficient way to spend Z on raising children

• Solve equation df (r )/drf(r)(rp) for r
• Let nZ/(rp)
• Buy n wives. Give each wife r units of
  resources for herself and her children.
• You will then get nf(r)Z f(r)/(rp)
  children.
• Note constant returns per dollar. Price of a kid
  is f(r)/(rp).

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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 dont care who
  daughter marries. (In contrast to case of dowry.)
• See Borger-Mulderhoff for empirical work on
• Bride prices among Kipsigis.

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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.

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Comparative Statics

fr
-P
-P
r
r
r
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Equilibrium Analysis

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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?

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Dry run simple g.e. model

• Assume men have identical preferences and
  endowments.
• Each man has one sister.
• Each mans labor produces m units and each
  womans labor produces w units of resources.
• Each man receives his sisters bride price as an
  inheritance.

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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 PB-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..

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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
• xrmw
• Let x and r solve this problem. These are
  the equilibrium consumptions.

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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 rr, this
  condition determines P as the solution to
• Pr f(r) /f(r)

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Geometry of Solution

fr
r
r
-P
r

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Comparative statics result

• If children are a normal good. (Income effect
  positive) then bride price and resources per wife
  are increasing functions of mw.
• Proof See diagram.

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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
• wisiBi wipmi
• 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.

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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

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Case of equal productivity

• Suppose that men all have productivity m and
  women all have productivity w.
• Average wealth of men is
• Bmpwm
• Wealth of man i is
• Bsim
• Therefore man i has ni wives where
• ni Bsi/ (Bm) m/(Bm)

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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 mgt0, variance of number of wives is smaller
  than variance of sex ratio.

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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 brides male
  relatives by groom. Dowry is paid to the newly
  wed couple by brides relatives.

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Which was scarce?

Women?
Land?
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Threshold productivity
fr
r0

r
r
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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 femaletypically did not marry or have
  children.
• Late marriage in Norway

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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.

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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,
  domesticslater would move to cities or emigrate.

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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.

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Evolution of Sex ratios Natures 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.)
• DarwinFisher equilibrium theory.

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