This appendix presents more mathematical versions of the sorting models presented in Chapter 3. All of these models assume that individuals do not differ by skill, but only in their strength endowment.

Model A

Here I expand Model A to allow the possibility that the distributions may overlap. Individuals get strength endowments that are random draws from normal distributions. Males and females draw their strength endowments from different distributions, and the male distribution has a higher mean. In general individuals can freely choose among T occupations. In occupation i individual j will produce qij = ai + biSj units of output and will have earnings of Piqij, where pi is the piece-rate. An individual will choose the occupation in which he or she has the highest earnings. Since earnings functions are linear, each occupation will have at most one interval of possible S values over which it is the best occupational choice. (It is possible that an occupation will attract no workers if its earnings are always below those of another occupation.)

What will happen if we take the income functions from Figures 3.2 and 3.5, but allow the strength endowments to overlap? Suppose that females have strength endowments that are normally distributed with a mean of 25 and a standard deviation of 15. Males have strength endowments that are normally distributed with a mean of 75 and a standard deviation of 15.