The following model comes from Ted Begstrom's 'On the economics of polygyny', and asks you to think about bride prices (dowries, etc). Question 1-5 below ask you to derive results of this model. This paper is referenced in chapter 12 of the textbook. Suppose that men care about two things when choosing a wife; expected number of children a wife can provide and material resources required. Let f(r) be the expected number of children that a woman with r units of material resources produces. Assume that f(0)=0, that f is bounded from above (there exists some finite number f^ such that f(r)f^ for all r ), that f(r)0 and f(r)0 for r0, and that there is some r^ such that f(r) 0 for 0rr^ and f(r)0 for rr^. Furthermore, while each woman shares the same function f, women may have different abilities to produce their own material resources. We say that a woman who can on her own produce material resources wj has bride price bj. This means the total cost of purchasing bride j and supplying her with r units of material resources is bjwj+r. We can also write pj=bjwj, that is p describes the net cost of purchasing a bride. Since we have assumed that all women share the same f, and differ only in their material resource productivity, it must be that in competitive equilibrium there isa uniform net bride price p such p=bj+wj for all women j. So the cost of purchasing bride j and supplying her with r units of material resources is p+r. Since a woman is capable of producing f(r) children in expectation, this means that the cost per expected child is p+r(r). That is, in a polygynous society a man faces a tradeoff between (expected) number of wives and the material resources supplied to each wife.5 Question 5 An ordinary good is defined to be one where when the price of it increases while income and other prices remain the same, then the demand will decrease (that is, an ordinary good obeys the law of ordinary demand). Show that if expected children are an ordinary good, then brides are also an ordinary good..

1. The following model comes from Ted Begstrom's 'On the economics of polygyny', and asks you to
think about bride prices (dowries, etc). Question 1-5 below ask you to derive results of this model.
This paper is referenced in chapter 12 of the textbook. Suppose that men care about two things
when choosing a wife; expected number of children a wife can provide and material resources
required. Let f(r) be the expected number of children that a woman with r units of material
resources produces. Assume that f(0)=0, that f is bounded from above (there exists some finite
number f^ such that f(r)f^ for all r ), that f(r)0 and f(r)0 for r0, and that there is some r^ such that f(r)
0 for 0rr^ and f(r)0 for rr^. Furthermore, while each woman shares the same function f, women
may have different abilities to produce their own material resources. We say that a woman who
can on her own produce material resources wj has bride price bj. This means the total cost of
purchasing bride j and supplying her with r units of material resources is bjwj+r. We can also write
pj=bjwj, that is p describes the net cost of purchasing a bride. Since we have assumed that all
women share the same f, and differ only in their material resource productivity, it must be that in
competitive equilibrium there isa uniform net bride price p such p=bj+wj for all women j. So the
cost of purchasing bride j and supplying her with r units of material resources is p+r. Since a
woman is capable of producing f(r) children in expectation, this means that the cost per expected
child is p+r(r). That is, in a polygynous society a man faces a tradeoff between (expected) number
of wives and the material resources supplied to each wife.5 Question 5 An ordinary good is
defined to be one where when the price of it increases while income and other prices remain the
same, then the demand will decrease (that is, an ordinary good obeys the law of ordinary
demand). Show that if expected children are an ordinary good, then brides are also an ordinary
good.