（这个题要求运用微积分)考虑有三种投入的柯布—道格拉斯生产函数。K是资本（机器数量)，L是劳动（工人人数)，H是

(这个题要求运用微积分)考虑有三种投入的柯布—道格拉斯生产函数。K是资本(机器数量)，L是劳动(工人人数)，H是人力资本(工人中持有大学学历的人数)。生产函数是：Y=K^1/3L^1/3H^1/3

a．推导出劳动的边际产量。人力资本量的增加怎样影响劳动的边际产量？

b．推导出人力资本的边际产量的表达式。人力资本量的增加怎样影响人力资本的边际产量？

c．支付给劳动的收入份额是多少？支付给人力资本的收入份额是多少？在这个经济的国民收入核算中，你认为工人会得到多大份额的总收入？(提示：考虑人力资本的收益体现在哪里。)

d．一个无技术工人赚到劳动的边际产量，而一个技术工人赚到劳动的边际产量加人力资本的边际产量。用(a)与(b)的答案，找出技术工人与无技术工人工资的比率。人力资本量增加如何影响这个比率？请解释。

e．一些人倡议把政府融资的大学奖学金作为创造一个更加平等的社会的途径。另一些人认为，大学奖学金只能帮助那些能上大学的人。你能用前面问题的答案来讨论这种争论吗？

(This problem requires the use of calculus) Consider a Cobb-Douglas production function with three inputs. K is capital (the number of machines), L is labor (the number of workers), and H is human capital (the number of college degrees among the workers). The production function is Y=K^1/3L^1/3H^1/3

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 (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. Do your answers to the preceding questions shed light on this debate？