Technology. Technology is the same in both the Home and Foreign countries. In both countries, skilled workers are self-employed. They rent capital at the rental rate r and produce computers according to the production function: QC = H1/2K 1/2 C (1) Their income is given by RC = PCQC - rKC, where KC is the capital employed to produce computers. Unskilled workers are also self-employed. They rent capital and use it to produce desks according to the production function: QD = L 1/2K 1/2 D (2) Their income is given by RD = PDQD - rKD, where KC is the capital employed to produce computers. [NOTE: I updated the production function for desk, so that you will have a beautiful “closed-form” solution to capital rental rate in (c).] Endowments. Home country is endowed with the following factor amounts: H = 9, K =20, L = 4. The Foreign country is endowed with the following factors: H* = 1, K* = 20, L* = 4. Preference. Preferences are identical in the two countries and are described by the following utility function: U(XC, XD) = X 1/3 C X 2/3 D . (3) In the economy all the factors are always fully employed. Assume for now that the two countries are in autarky. (a) In this model, what are the specific factors to produce each product? (b) Given the production functions, write down the PPF for the Home country using the idea that capital is employed in both sectors. Plot it on a graph with QD as the horizontal axis and QC as the vertical axis.
Homework 2 Format: type version is encouraged. Handwriting-and-scan version is acceptable. Remark 1 This problem set helps you review the Specific Factor model and (part of) Heckscher- Ohlin Model of international trade. You may discuss in groups, with each group up to 5 students. But you need to write down your homework by yourself. 1. (Specific Factors Model: Autarky Economy) In this economy two countries, Home and Foreign, produce two goods, Computers and Desks, using three factors of production, skilled labor (H), unskilled labor (L) and capital (K). Technology. Technology is the same in both the Home and Foreign countries. In both countries, skilled workers are self-employed. They rent capital at the rental rate r and produce computers according to the production function: QC = H 1/2K 1/2 C (1) Their income is given by RC = PCQC − rKC , where KC is the capital employed to produce computers. Unskilled workers are also self-employed. They rent capital and use it to produce desks according to the production function: QD = L 1/2K 1/2 D (2) Their income is given by RD = PDQD − rKD, where KC is the capital employed to produce computers. [NOTE: I updated the production function for desk, so that you will have a beautiful “closed-form” solution to capital rental rate in (c).] Endowments. Home country is endowed with the following factor amounts: H = 9, K =20, L = 4. The Foreign country is endowed with the following factors: H* = 1, K* = 20, L* = 4. Preference. Preferences are identical in the two countries and are described by the following utility function: U(XC , XD) = X 1/3 C X 2/3 D . (3) In the economy all the factors are always fully employed. Assume for now that the two countries are in autarky. (a) In this model, what are the specific factors to produce each product? (b) Given the production functions, write down the PPF for the Home country using the idea that capital is employed in both sectors. Plot it on a graph with QD as the horizontal axis and QC as the vertical axis. 1 (c) Suppose prices of Computers and Desks, PC and PD. Consider the Home country and find the autarky equilibrium return to capital r using the graph showing the Value of Marginal Product of Capital P ∗MPK for both goods, following the steps: i. Draw the capital demand for these two industries in one graph. (Hint: capital demand in each industry is determined by P ∗MPK = r.) ii. What is equilibrium capital rental rate r and the allocation of Capital to the production of the two goods (as a function of PC and PD)? iii. What are the corresponding outputs of each product (as a function of PC and PD)? In particular, what is the relative supply function QC/QD as a function of PC/PD? (d) Suppose the output prices are PC and PD, and given that the capital rental rate r is a function of PC and PD, solve for the demand for computers and desks by solving the consumer utility maximization problem for skilled workers and unskilled workers at Home country. In particular, what is the relative demand function XC/XD as a function of PC/PD? (e) Using relative demand and relative supply, find the equilibrium autarky relative price for the Home country. Draw the relative supply and relative demand function for Home in one figure and mark the equilibrium point as A. [NOTE: because you can only find the relative price PC/PD, you could use Desk as the numeraire good by setting its price PD to one.] 2. (Specific Factors Model: trade) In the above question, now suppose the two countries are allowed to trade with each other freely. (a) Find the world relative supply function. [Hint: solve for the foreign supply function as you did for Home. Combine Home and Foreign to get the result.] (b) What is the world relative demand function? (c) Draw the relative supply and relative demand function for the world in one figure, and find the equilibrium world relative prices PC/PD. Mark the world equilibrium point as B. i. Is it higher or lower than the Autarky equilibrium relative price at Home? ii. Compare it with the Autarky equilibrium point A as you find in Question 3 (e), show which factors gain and which lose from allowing free trade between the two countries. (d) What is the effect of this change in relative price on the income of the skilled (RS) and unskilled workers (RU). 3. (Heckscher-Ohlin Model: Autarky) Consider a economy, Mexico, with two goods (footballs F and soccer balls S), and two factors of production (high skilled labor H and low skilled labor L). Suppose that footballs and soccer balls are produced with production function: QF = H φ FL (1−φ) F (4) QS = H σ SL (1−σ) S (5) where φ = 1/2, σ = 1/4. The total endowment of high and low skilled workers in the country is HF +HS = H, LH + LS = L. (6) 2 (a) Which product is more skill-intensive? (b) The optimal production decision in these two industries implies that the ratio technical rate of substitution is the same in the two industries, i.e. MPHF MPLF = MPHS MPLS . (7) Where MPHF is the marginal product of high-skill worker in the football industry, and other variables are similarly defined. i. Write down the explicit form of Equation (7), given the demand function. ii. Given endowment K and L, derive the equilibrium production possibility frontier (PPF) for this economy, using the equation derived in (i) together with equation (6) and the two production functions. (c) Suppose the output price ratio is fixed at PFPS , find the equilibrium output for F and S in the graph with PPF. (d) If the output prices increases to 2PFPS . How does this affect the purchasing power of high-skilled and low-skilled workers? Please explain this based our discussion in class. (e) If Mexico introduces many high-skill workers from the United States and the total amount of high-skilled workers is doubled to 2H. i. How will this change the shape of PPF? Please explain clearly using either words as we did in class, or using the mathematical formula you derived above. ii. How will this affect the outputs of each product? Please show it in the figure. 3