In this assignment, you will solve a version of the specic factors model to see how trade can create both winners and losers through its effects on real factor incomes. There is a single country, Home, and two sectors, 1 and 2. Home is endowed with E units of labor, K1 units of capital specic to sector 1, and I22 units of capital specic to sector 2. Household preferences in Home are Cobb-Douglas with potentially asymmetric weights: U (01. C2) : Cngl's where C1 and 02 denote the household's consumption of goods 1 and 2 respectively. The parameter S E (0, 1) measures how much the household likes good 1 relative to good 2. The production technology available is the same in each sector, and is given by the following Cobb-Douglas production function: X7 = Bil-Am where Xi, Ki. and Li denote the output, capital input, and labor input of sectori respectively. The parameter /\\ E (U, 1) measures how important labor is for production relative to capital. In what follows, assume that factor endowments are: L=1 13:1 132:1 2 and that the weight on labor in the production function is A = 7. We will examine how different values for the parameter S affect the welfare gains from trade. First, we will solve for demand and supply assuming that the goods prices 301 and p2 (and hence the relative price p, E plfpg) are known. Let m, n, and 7-2 denote the wage, rental rate of sector 1 capital, and rental rate of sector 2 capital respectively. Demand The utility maximization problem for a household with income I is: IrraxCiq021'g (71.02 s.t. pl Y1+p202 : I (a) What is equilibrium] relative consumption, 0,. E Cl/CQ, given goods prices? Supply The prot maximization problem for a rm in sector 1' is: , , zlA A __ z , . it??? {pihi L, - \"In - {UL} (b) Write down the rst-order conditions for this problem). Now note that capital market clearing requires: K, : K, for i E {1.2}. (c) What is the equilibrium relative allocation of labor across sectors, L,r E ill/Lg, given goods prices? (d) What is equilibrium] relative output, XT E Xl/X'z, given goods prices? Autarky Now suppose that Home is in autarky. Goods market clearing therefore requires that consumption equals production: Ci = Xi for i e {1, 2}. (e) What is the equilibrium relative goods price p, in autarky? How does the relative price change with S? Why? Now note that labor market clearing also requires: L1 + L2 = L. (f) What is the allocation of labor in each sector, L, and L2? (g) What are real factor incomes in terms of goods 1 and 2, i.e. 2, w, ri, " "2, and 12? p2 Trade Now suppose that Home can trade with the rest of the world at some prices {pl , pl } and suppose that the world relative price is pr = p, /p2 = 1. (h) Repeat your calculations in parts (f)-(g) to determine real factor incomes under trade. (i) How does trade affect real factor incomes for (i) workers, (ii) owners of sector 1 capital, and (iii) owners of sector 2 capital? (Hint: your answer will depend on whether S ? ). (j) Explain the intuition for your findings in part (i)