I answered questions a-e, but need help with f and g.

1. Consider the 2x2x2 Heckscher-Ohlin model (two goods, 7 = 1,2, two factors and two countries, j = 1,2). The production technologies are as follows; Y; = KgLi (1) Y2 = KfQL;z_ 4 (2) where Y); is output of good 7 in country j, Kj; is capital used in the production of good i in country 7, and L;; is labor used in the production of good in country j. 0 a. The endowments of capital (K, j = 1,2) and labor (Lj, 7 =1,2) in the two countries are such that ';{:11 > K=j Consumers in both countries have identical, homothetic preferences. All markets are perfectly competitive. a) Write down the profit maximization problem for firms operating in sector 7 in coun- try j. (2 marks) b) From the first order conditions to firms' profit maximization problem in each sector, solve for the algebraic equilibrium relationship between the factor intensity and relative factor prices. (5 marks) c) Based on your answer to part (b), what can you say about the relative factor intensities of the two sectors? Why? (3 marks) d) Further manipulate the first order conditions in order to find the algebraic equi- librium relationship between relative goods prices and relative factor prices when both goods are produced. (5 marks) e) Suppose that the two countries open up to free trade. What will the move from autarky to free trade imply for the relative price of good 1 in country 27 What will the pattern of trade be? NB. You do not need to derive anything in this subquestion. (5 marks) g) Illustrate your findings to part (f) using a Lerner diagram. Show how the relative price change of the commodities affects the nominal returns to the two factors and the relative factor intensities of the two sectors. Make sure to carefully label the graph and explain the changes you find. (5 marks) 1a) The profit for each sector is then the total revenue minus the total costs: max n. = PY. - WL - rK ji Given the production technologies from the Heckscher-Ohlin model, the profit functions become: Sector 1 1-a TI. = P K. L 1 j1 j1 WL. - rK. j j1 j1 Sector 2 It, = pKI-B 12 12 - WL. - rK j 121b) First-order conditions require that the partial derivatives of the profit function with respect to capital and labour are equal to zero: Sector 1 j1 dK =pak -1 1-a *j1 j1 j1 -r = 0-r =pak -1, 1-a j1 /1 dL =p (1 - a)K L i1 j1 j1 -W= 0 - w = p (1 - a)K L 1 j1 j1 W p (1-a) K L = j1 j1 1-a 1 r papa-1 1-a K j1 1 j1 Sector 2 dit. 12 dK = p,BK -1/ 1-B j2 12 12 0 -r, = p,BKB-121-B dit.. 12 12 12 dL =p, (1 - B)KB L-B 12 j2 - W. = 0 - W, =P, (1 - B)KB L-B W P, (1-B)KB L -B 12 j2 12 j2 r 1-B jz P, BRB-1, 1-B B K 12 12 j2 Equilibrium: 1-a Lj1 1-B 412 a K j1 B K 121c) Good 1 is produced in a sector that's labour-intensive (since a is smaller), i.e. the sector prefers using more labour compared to capital. Meanwhile, good 2 is produced in a sector that's capital-intensive (since B is larger), i.e. it leans towards using more capital than (1-p) Lz Bi-w) K, and we know that L workers. Furthermore, we can manipulate the equation: K'l = j1 L. L, B(1 a) > a(1 B), which means -~ > =, i.e. the capital-to-labour ratio is lower in K;l sz sector 2 than in sector 1. The intuition comes from the fact that in a competitive equilibrium, the return to each factor of production must equalize across sectors. Firms in each sector choose their capital and labour inputs based on the factor prices such that the ratio of the marginal products of the factors equals the ratio of the factor prices. Given that the elasticity of output with respect to capital (B) is higher in sector 2, the sector 2 firms find it more profitable to use more capital relative to labour. Alternatively, with a lower a, the sector 1 firms find it more profitable to use more labour relative to capital. 1d) First-order conditions with respect to capital and labour for each sector: Sector 1 a1_1-a r.= ;r;ilcu'(}_1 L}_1 =p(1- K L" Y; pl( 0K, Ly Sector 1 g-1,1-8 = pzsz sz _ BB Ratio of goods prices to factor prices: P (1-a)KL i1 j1 W P, (1-B)KB L-B W j2 j2 P (1-a) KPI -B P2(1-B) = 12 j2 K L -a j1 j1 Ratio of the rental rate: Pak"-1, 1-a j1 /1 r P,BRB-1, 1-B = r = 1 j2 P, a KB-1, 1-B E P,B Ka-1, 1-a j1 j1 Solve for the relative goods prices: P1 (1-B)KB L-B P2 (1-a)K L j1 j1 P1 BRB-1, 1-B 12 12 of"-1, 1-a (1-B)KB 12 j2 BRB-1, 1-B = 12 j2 (1-a) K L -a j1 j1 axa-1, 1-a j1 j1 (1-a) (1-B)KB L-B 12 12 (1-B)KB-1, 1-B OK" I- j1 /1 By"-1, 1-a j1 /1 (1-a) Li1 (1-B) Liz K j1 B K 1-a W a = ( -, " ) 1-B BTherefore, the equilibrium relationship between relative goods prices and relative factor prices is that the ratio of goods prices adjusted for the capital share of income in each sector must equal the wage-rental ratio adjusted for the labour share of income in each sector. 1e) When country 2 opens to free trade, the relative price of good 1 is likely to fall due to increased supply aimed at the global market. Assuming country 2 is rich in the factor used intensively in producing good 1, it will export this good, which depresses its local price relative to good 2. Under the Heckscher-Ohlin model, country 2 will export goods that use its abundant factor and import those that require its scarce factor, following its comparative advantage and influencing the trade pattern accordingly. 1f) 19)