1. (Maple worksheet)

2. 1. The two labor demand curves are W^c=400-L^c and W^a=400-L^a. Subtracting the second from the first yields equation 1, W^c-W^a=L^a-L^c. The distribution of equalizing differences implies equation 2, W^c-W^a=L^c/10. Setting 1 and 2 equal implies 3, L^a-L^c=L^c/10. Defining equation 4 to be L^a+L^c=420, we can solve 3 and 4 simultaneously to get L^c=200 and L^a=220. Substituting these into equation 1 yields W^c-W^a=$20/day.
   2. Set equation 1 from Part 1 equal to 21, Fred's equalizing differential. Substituting equation 4 from above into this, and solving for L^c, yields L^c=399/2. Substituting this into the labor demand equation yields W^c=401/2 > $200/day. Further substitutions yield W^a=359/2 < $180/day. In the diverse economy, Fred is an auto worker because his equalizing difference is $21 and the wage differential is only $20. In this economy he is paid a wage of $180/day. In the clone economy, in which everyone shares Fred's preferences, he is paid a wage of 359/2, which is less than $180/day. So he gains from diversity.
   3. All women will work in auto repair, plus some men. L^cm and L^am represent the number of men in coal mining and auto working, respectively. From the distribution of equalizing differences we know that L^cm=294(W^c-W^a)/42=7(W^c-W^a). We can rewrite this as L^cm=7[400-L^cm-274+L^am] where the term in [ ] is simply the difference of the two demand equations. Simplifying this equation yields 7L^am+882=8L^cm. But since we know the total number of men in the economy, we can write down another equation in these same two variables, namely L^cm+L^am=294. Solving these two linear equations simultaneously gives L^am=98 and L^cm=196. Substituting these back into the demand equations gives W^c=$204/day and W^a=$176/day. So the equilibrium wage differential, W^c-W^a=$28/day.
      
      The average wage for women will be $176/day, since all of the women are in auto working. For men, the average wage will be (196/294)*204 + (98/294)*176 @ $194.67/day. That is, the fraction of men in coal mining times the coal mining wage, plus the fraction of men in auto working times the auto working wage. All women are worse off as a result of the discrimination. Women who were auto workers before were paid $180 and now they are only paid $176, so all of these women lose $4/day. Losses for the women who originally worked in mining depend on their individual equalizing differences. A female miner with a differential of $0 would be equally happy earning $200 in auto working. But instead she is forced to go into auto working and get paid $176, a $24/day loss. Similarly, a female miner with a differential of $1/day would be equally happy earning $199 in auto working or $200 in mining. So when she is forced into auto working at a wage of $176 she loses $199 – $176 = $23/day.
      
      More generally, if W^c0 is the pre-discrimination coal mining wage, W^a1 is the post-discrimination auto working wage, and k is the equalizing difference for a particular female coal miner, then her losses as a result of discrimination are given by W^c0 – k – W^a1. For the "marginal" female coal miner, whose equalizing difference is $20/day, the same as the difference in market wages W^c-W^a, she will lose $4/day.
      
      Since we have measured the losses for each female in the economy, we have answered part 4 of this question. Damages from a class action discrimination suit awarded to a particular woman would ideally equal her daily losses multiplied by the number of days that she was banned from coal mining. We say "ideally" because in practice it is extremely difficult to know a person's true equalizing difference, since the person has an incentive to lie about it. [Note: For the class action suit we compare the results when there is exclusion to those that would hold without exclusion. For individual litigation, one should only take into account the difference of the utility a woman would get as a miner (given that everything else stays as it is) with that which she obtains as an auto worker. ]
      
      For men the calculation of losses is a bit tricky since men are free to switch jobs. The 140 men who were miners originally will remain miners after discrimination, because the market wage differential is even higher ($28/day as opposed to $20/day). These men will each gain $4/day as a result of discrimination, since their wage increases from $200/day to $204/day. The 98 men who remain in auto work will lose $4/day, since their wage has decreased from $180/day to $176/day. Some men who were previously auto workers (the 56 men who have equalizing differences between $20 and $28) will become miners. Of these men, those with equalizing differences from $20 to $24 will gain, and those with equalizing differences greater than $24 will lose. To see this, note that at their original wage in the auto industry, they would have required a wage of W^c0+(k-20) in coal mining to be indifferent between the two occupations. The difference between this "counterfactual" wage and the actual wage W^c1 that they actually get when they switch to coal mining is a measure of their welfare change. If it is negative (W^c0+k-20-W^c1<0) then they benefit, and if it is positive (W^c0+k-20-W^c1>0) then they lose. Hence, the people who gain from discrimination are of two types: the male miners who remain miners, and the male auto workers who switch into mining and have equalizing differences less than $24/day. Everyone else in the economy loses.

3. 1. Nothing happens in the North in the short run. Labor supply in the South increases by 5 percent, so the wage rate falls by 10 percent.
   2. (Maple worksheet)
   3. In the long run (if there is no change in the capital stock), wages must be the same in the two regions, and since employment rises by 2% and the elasticity is 1/2, the wage must fall by 4%, to $14.40.