【上海交通大学-安泰】中级宏观经济学作业答案.docx

1. In the United States, the capital share of GDP is about 50%, the average growth in total output is 3% per year, the depreciate rate is 5% per year, the capital-output ratio is 4. Suppose that the production function is Cobb-Douglas, so that the capital share of output is constant, and the United States has been in a steady state. (10points)a) What must the saving rate be in the initial steady state?Y/Y = 0.03 = n = 0.03sf(k) = (n +)k s = (n +)k/f(k)= (0.03 + 0.05)* 4 = 0.32b) What is the marginal product of capital in the initial steady state?MPK * K/Y = 0.5= MPK = 0.5/(K/Y) = 0.5/4 = 0.125c) What will the capital-output ratio be at the Golden Rule steady state?MPK = (n +) = 0.03 +0.05 = 0.08MPK * K/Y = 0.5= K/Y = 0.5/0.08 = 6.25d) What must the saving rate be to reach the Golden Rate steady state?sf(k) = (n +)k s = (n +)k/f(k) = 0.08 *6.25 = 0.52. Prove each of the following statements about the steady state of the Solow model with population growth and technological progress. (10 points)a) The capital-output ratio is constantIn the steady state, we know that sy = (+ n + g)k. This implies that k/y = s/( + n + g). Since s, , n, and g are constant, this means that the ratio k/y is also constant. Since k/y = K/(L* E)/Y/(L* E) = K/Y, we can conclude that in the steady state, the capital-output ratio is constantb) Capital and labor each earn a constant share of an economys income.We know that capitals share of income =MPK (K/Y). In the steady state, we know from part (a) that the capital-output ratio K/Y is constant. We also know that the MPK is a function of k, which is constant in the steady state; therefore the MPK itself must be constant. Thus, capitals share of income is constant. Labors share of income is 1 - capitals share. Hence, if capitals share is constant, we see that labors share of income is also constantc) Total capital income and total labor income both grow at the rate of population growth plus the rate of technological progress, n + g.We know that in the steady state, total income grows at n + g-the rate of population growth plus the rate of technological change. In part (b) we showed that labors and capitals share of income is constant. If the shares are constant, and total income grows at the rate n + g, then labor income and capital income must also grow at the rate n + g.d) The real rental price of capital is constant, and the real wage grows at the rate of technological progress g. Define the real rental price of capital R as:R = Total Capital Income/Capital Stock= (MPK *K)/K= MPK.We know that in the steady state, the MPK is constant because capital per effective worker k is constant. Therefore, we can conclude that the real rental price of capital is constant in the steady state.Output is divided between capital income and labor income. Therefore, the wage per efficiency unit of labor can be expressed as:w = f(k) MPK k.In the steady state, f(k), MPK and k are all constant numbers because capital per effective worker k is constant. Therefore, the wage per efficiency unit of labor is constant. The wage per unit of labor is related to the wage per efficiency unit of labor by the equationWage per Unit of L = wE.Thus, the growth rate of real wage equals the growth rate of E, g.3. The country of Leverett is a small open economy. Suddenly, a change in world fashions makes the exports of Leverett unpopular. (10 points)a) What happens in Leverett to saving, investment, net exports, the interest rate, and the exchange rate?When Leveretts exports become less popular, its domestic saving Y C G does not change. This is because we assume that Y is determined by the amount of capital and labor, consumption depends only on disposable income, and government spending is a fixed exogenous variable. Investment also does not change, since investment depends on the interest rate, and Leverett is a small open economy that takes the world interest rate as given. Because neither saving nor investment changes, net exports, which equal S I, do not change either. This is shown in the Figure as the unmoving S I curve. The decreased popularity of Leveretts exports leads to a shift inward of the net exports curve, as shown in the Figure. At the new equilibrium, net exports are unchanged but the currency has depreciated.Even though Leveretts exports are less popular, its trade balance has remained the same. The reason for this is that the depreciated currency provides a stimulus to net exports, which overcomes the unpopularity of its exports by making them cheaper.b) The citizens of Leverett like to travel abroad. How will this change in the exchange rate affect them?Leveretts currency now buys less foreign currency, so traveling abroad is more expensive. This is an example of the fact that imports (including foreign travel) have become more expensiveas required to keep net exports unchanged in the face of decreased demand for exports.c) The fiscal policymakers of Leverett want to adjust taxes to maintain the exchange rate at its previous level. What should they do? If they do this, what are the overall effects on saving, investment, net exports, and the interest rate?If the government reduces taxes, then disposable income and consumption rise. Hence, saving falls so that net exports also fall. This fall in net exports puts upward pressure on the exchange rate that offsets the decreased world demand. Investment and the interest rate would be unaffected by this policy since Leverett takes the world interest rate as given.4. Consider an economy with the following Cobb-Douglas production function:Y=K1/3L2/3The economy has 1000 units of capital and a labor force of 1000 workers.a) Derive the equation describing labor demand in this economy as a function of the real wage and the capital stockThe demand for labor is determined by the amount of labor that profit-maximizing firm wants to hire at a given real wage. The profit-maximizing condition is that the firm hire labor until the marginal product of labor equals the real wage, MPL = W/PThe marginal product of labor is found by differentiating the production functionwith respect to laborMPL=dYdL=d(K1/3L2/3)dL=23K1/3L-1/3In order to solve for labor demand, we set the MPL equal to the real wage and solve for L:23K1/3L-1/3=WP=L=827K(WP)-3Notice that this expression has the intuitively desirable feature that increases in the real wage reduce the demand for labor.b) If the real wage can adjust to equilibrate labor supply and labor demand, what is the real wage? In this equilibrium, what are employment, output, and the total amount earned by workers?We assume that the 1,000 units of capital and the 1,000 units of labor are supplied inelastically (i.e., they will work at any price). In this case we know that all 1,000 units of each will be used in equilibrium, so we can substitute them into the above labor demand function and solve for W/P.1000=8271000(WP)-3 WP=23In equilibrium, employment will be 1,000, and multiplying this by 2/3 we find that the workers earn 667 units of output. The total output is given by the production function:Y=K1/3L2/3=10001/310002/3=1000Notice that workers get two-thirds of output, which is consistent with what we know about the CobbDouglas production function from the appendix to Chapter 3.c) Now suppose that Congress, concerned about the welfare of the working class, passes a law requiring firms to pay worker a real wage of 1 unit of output. How does this wage compare to the equilibrium wage?The congressionally mandated wage of 1 unit of output is above the equilibrium wage of 2/3 units of output.d) Congress cannot dictate how many workers firms hire at the mandated wage. Given this fact, what are the effects of this law? Specifically, what happens to employment, output, and the total amount earned by workers?Firms will use their labor demand function to decide how many workers to hire at the given real wage of 1 and capital stock of 1,000: L=(8/27)1000(1)-3=296, so 296 workers will be hired for a total compensation of 296 units of output.e) Will Congress succeed in its goal of helping the working class? Explain.The policy redistributes output from the 704 workers who become involuntarily unemployed to the 296 workers who get paid more than before. The lucky workers benefit less than the losers lose as the total compensation to the working class falls from 667 to 296 units of output.5. Suppose we know that workers tend to receive 70% of all the income earned in Germany, that the German depreciation rate is 10%, and that there is no population growth or technological progress.