2. Consider an economy with J countries producing J different country-specific goods, as in the Armington economy of lecture note 2. Consumers everywhere have the prefer- ences shown in question 1, with N = J. In country i, there are H; > 0 consumers, each endowed with one unit of labor. Everyone supplies labor inelastically. There is a linear labor-only technology for producing country-specific good i, and the productivity of this technology is > 0. Write y; for the amount of good i supplied by country i. Countries can freely trade all goods. a. Explain why the relative prices 1/e Pi Pj for all i and j are the equilibrium prices for this economy. You can appeal to what you know about homothetic utility functions without going through the full construction of the competitive equilibrium. b. (i) Determine the ratio of per-capita income for any pair of countries i and j. (ii) Show that per-capita income in country i declines relative to per-capita income in country j if the population of country i increases relative to that of country j. C. (i) If > 1, what happens to relative per-capita incomes when one country becomes more productive relative to the other? (ii) Answer the same question for the case = (0,1). E d. Real income is nominal income divided by the price index P(p). (i) Use the relative prices obtained in a to show that Pjy,/H; P(p) (1/c)/(1-1/2) *(EA))" (ii) Show that real per-capita income in country j falls if its population increases. (iii) Now assume that > 1. Show that real per-capita income in country j rises when this country becomes more productive. (One can show that it may fall if < 1, but you are not asked to show that here.)