Foundations of Economics for International Business Supplementary Exercises 2

Transcription

1 Foundations of Economics for International Business Supplementary Exercises 2 INSTRUCTOR: XIN TANG Department of World Economics Economics and Management School Wuhan University Fall 205 These tests are only for your practice, you do NOT have to hand them in. However, all materials in the supplementary tests are also subject to be tested in exams. Check the answers carefully! MULTIPLE CHOICES. If Y = AK 0.5 L 0.5 and A,K and L are all 00, the marginal product of capital is: (A) 50. (B) 00 (C) 200 (D), If the production function describing an economy is Y = 00K 0.25 L 0.75, then the share of output going to labor (A) is 25 percent. (B) is 75 percent. (C) depends on the quantities of labor and capital. (D) depends on the state of technology. 3. If the consumption function is given by C = (Y T ), and Y is 6,000 and T is given by T = Y, then C equals: (A) 2,500. (B) 2,800.

2 (C) 3,500. (D) 4, The equation Y = C (Y T ) + I (r ) + G may be solved for the equilibrium level of: (A) income. (B) consumption. (C) government purchases. (D) the interest rate. Figure : The Capital-Labor Ratio 5. Refer to Figure, starting from capital-labor ratio k, the capital-labor ratio will: (A) decrease. (B) remain constant. (C) increase. (D) first decrease and then remain constant. 2 NUMERICAL PROBLEMS AND SHORT ANSWERS. (0%) Consider a Cobb-Douglas production function with three factors. K (number of machines) and L (number of workers) are as usual aggregate capital and labor, H is human capital (you could think of it as the number of college degrees among the workers). The production function is given by Y = K /3 L /3 H /3 2

3 (a) Derive an expression for the marginal product of labor. How does an increase in the amount of human capital affect the marginal product of labor? (b) Derive an expression for the marginal product of human capital. How does an increase in the amount of human capital affect the marginal product of human capital? (c) What is the income share paid to labor? What is the income share paid to human capital? In the national income accounts of this economy, what share of total income do you think workers would appear to receive? (Hint: Consider where the return to human capital shows up.) (d) An unskilled worker earns the marginal product of labor, whereas a skilled worker earns the marginal product of labor plus the marginal product of human capital. Using your answers to parts (a) and (b), find the ratio of the skilled wage to the unskilled wage. How does an increase in the amount of human capital affect this ratio? Explain. (e) Some people advocate government funding of college scholarships as a way of creating a more egalitarian society. Others argue that scholarships help only those who are able to go to college. Based on your answers to part (d), explain your attitude towards this issue. 2. (6%) When the government subsidizes investment, such as with an investment tax credit, the subsidy often applies to only some types of investment. This question asks you to consider the effect of such a change. Suppose there are two types of investment in the economy: business investment and residential investment. And suppose that the government institutes an investment tax credit only for business investment. (a) How does this policy affect the demand curve for business investment? The demand curve for residential investment? (b) Draw the economy s supply and demand for loanable funds. How does this policy affect the supply and demand for loanable funds? What happens to the equilibrium interest rate? (c) Compare the old and the new equilibria. How does this policy affect the total quantity of investment? The quantity of business investment? The quantity of residential investment? 3. (9%) (This question helps your to grasp the proof of Euler Theorem step by step, you may wish to turn to your notes of my lecture and the recitation.) Prove that if function Y = F (K, L) exhibits constant returns to scale, then Y = F K K + F L L. F K and F L represent the partial derivatives to the according argument in the subscripts. (a) z > 0, what property does F (zk, zl) holds if it is constant returns to scale? (b) Use the definition of derivatives by limits to show the derivative with respect to z for the function zy? (c) Use the definition of derivatives by limits to show the derivative with respect to z for the function F (zk, zl)? (Hint: Apply the trick of first adding up and then substracting a specific term.) (d) Based on your answers to question (b) and (c), how do you reach the final conclusion of Euler Theorem? 3

4 (e) Guess the Euler Equation for production function with n arguments where n is any finite integer. 4. (0%) (This final question demonstrates to you the basic logic of Solow-Swan model.) Consider a neoclassical growth model, the production function Y = F (K, L) satisfies Inada conditions and constant returns to scale. Let y = f (k) be the per capita production function. (a) How do you derive y = f (k) from Y = F (K, L) using constant returns to scale? Demonstrate in details. (b) What are the four items in Inada conditions? Explain each term. (c) Suppose that the saving rate s, what are the consumption and investment in terms of capital per capita in this economy? (d) Suppose that the depreciation rate δ is exogenous, what is the equation that characterizes the law of motion of capital per capita k in terms of k. (e) What is the steady-state condition that pins down the level of capital per capita? (f) Let k represent the steady state level of capital per capita, explain graphically (better if mathematically) what happens if the initial capital per capita k 0 is less than k? What if k 0 > k? 3 SOLUTIONS Multiple Choices: A B B D C Short Answers:. (a) The marginal product of labor MPL is found by differentiating the production function with respect to labor: M P L = dy d L = 3 K/3 H /3 L 2/3 An increase in human capital will increase the marginal product of labor because more human capital makes all the existing labor more productive. (b) The marginal product of human capital MPH is found by differentiating the production function with respect to human capital: M P H = dy d H = 3 K/3 L /3 H 2/3 An increase in human capital will decrease the marginal product of human capital because there are diminishing returns. 4

5 (c) The labor share of output is the proportion of output that goes to labor. The total amount of output that goes to labor is the real wage (which, under perfect competition, equals the marginal product of labor) times the quantity of labor. This quantity is divided by the total amount of output to compute the labor share: Labor Share = 3 (K/3 H /3 L 2/3 )L K /3 H /3 L /3 = 3 We can use the same logic to find the human capital share: Human Capital Share = 3 (K/3 L /3 H 2/3 )H K /3 H /3 L /3 = 3 so labor gets one-third of the output, and human capital gets one-third of the output. Since workers own their human capital (we hope!), it will appear that labor gets two-thirds of output. (d) The ratio of the skilled wage to the unskilled wage is: W skilled = M P L + M P H W unskilled M P L = 3 K/3 H /3 L 2/3 + 3 K/3 L /3 H 2/3 = + L H 3 K/3 H /3 L 2/3 Notice that the ratio is always greater than because skilled workers get paid more than unskilled workers. Also, when H increases this ratio falls because the diminishing returns to human capital lower its return, while at the same time increasing the marginal product of unskilled workers. (e) If more college scholarships increase H, then it does lead to a more egalitarian society. The policy lowers the returns to education, decreasing the gap between the wages of more and less educated workers. More importantly, the policy even raises the absolute wage of unskilled workers because their marginal product rises when the number of skilled workers rises. 2. (a) The demand curve for business investment shifts out to the right because the subsidy increases the number of profitable investment opportunities for any given interest rate. The demand curve for residential investment remains unchanged. (b) The total demand curve for investment in the economy shifts out to the right since it represents the sum of business investment, which shifts out to the right, and residential investment, which is unchanged. As a result the real interest rate rises as in Figure 2. The total quantity of investment does not change because it is constrained by the inelastic supply of savings. The investment tax credit leads to a rise in business investment, but an offsetting fall in residential investment. That is, the higher interest rate means that residential investment falls (a movement along the curve), whereas the rightward shift of the business investment curve leads business investment to rise by an equal amount. Figure 3 shows this change. Note that I B + I R = I B 2 + I R 2 = S 5

6 Figure 2: Loan Market Figure 3: Investment Market 6

7 3. (a) F (zk, zl) = zf (K, L). (b) The derivative of zy with respect to z is simply Y. (c) The derivative of F (zk, zl) with respect to z is F K K + F L L. The easiest way to get this answer is to use chain rule which can be found in any calculus textbook. However, itąŕs still possible to get this using limits, for which you should go back to your classnotes. (d) Since we have F (zk, zl) = zf (K, L) = zy, take derivative with respect to z on both sides will give you the answer. (e) Suppose we have a constant return to scale production function with n inputs, that is F (z x, z x 2,, z x n ) = zf (x, x 2,, x n ) for any positive z, then the Euler theorem in this case is Y = F (x, x 2,, x n ) = F x + F 2 x F i x i + + F n x n in which F i denotes the derivative of F with respect to its ith input. 4. (a) Since F exhibits constant return to scale, we have per capita output y = Y /L = F (K, L)/L = F (K/L,), note we have the per capita capital stock k = K/L, therefore we get y = f (k) = F (k,). (b) The four items of Inada condition concern with the behavior of the (per capita) production at both 0 and infinity: f (k) 0 as k 0 f (k) as k f (k) as k 0 f (k) 0 as k. (c) Consumption equals output net of saving, i.e. c = y s y = ( s)f (k). Investment equals saving, i.e. i = s f (k). (d) If we let k denote the next period (year) capital stock (per capita), then we have k = i + ( δ)k = s f (k) + ( δ)k (e) At steady state, we have k = k, so from last equation we have which can be simplified as δk = s f (k). k = k = s f (k) + ( δ)k, (f) If you start from a k 0 which is less than k the steady state value of (per capita) capital stock, then you can use the equation in (d) to compute k, k 2,,etc. This sequence of capital stock will converge to k. The key point can be found by looking how this sequence evolves starting from a k 0 < k, by drawing a figure of s f (k) + ( δ)k against k, which is similar to Figure. 7