【上海交通大学-安泰】中级宏观经济学作业答案

1、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 marg

2、inal 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

3、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

4、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 i

5、n 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 = Tota

6、l 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 co

7、nstant. 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?

8、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

9、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 t

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