4) Production in the country of StockVille can be characterized by the following

Question

4) Production in the country of StockVille can be characterized by the following function: Y= 2K0.20.8 and that Stockville initially has 200 units of labor and 1000 units of capital. a) b) c) d) Calculate the marginal product of labor and marginal product of capital for StockVille. Show with calculus that the production function exhibits diminishing returns to capital. Calculate the total payments to labor and capital in StockVille. Suppose that in Stock Ville there is a large increase in investment that adds a lot more capital to the economy. What happens to the rental rate of capital and real wage rate in StockVille? Briefly explain 2 e Aain suppose that in StockVille there is a large increase in investment that adds a lot more capital to the economy. What happens to the income shares of capital and labor in this economy? No math is necessary but it may help to see the answer if you pretend that capital increased from 1000 to 1500 while the amount of labor stayed the same. f) In the production function for Stock Ville briefly explain what the "2" represents.

Explanation / Answer

a) Marginal product of labor is the derivative of production function wrt labor and Marginal product of capital is the derivative of production function wrt capital.

MPL = 2x0.8xK^(0.2)xL^(0.8-1) = 1.6(K/L)^0.2. Similarly, MPK = 0.4(L/K)^0.8

b) We see that MPK = 0.4(L/K)^0.8. This implies that for any values of K and L > or = 0, MPK is positive. Now find the derivative of MPK with respect to K.

d(MPK)/dK = 0.4*(-0.8)*(L^0.8)*(K^-1.8). = -0.32(L^0.8)*(K^-1.8)

For any values of K and L > or = 0, MPK is now negative. This shows that as capital rises MPK falls. This is basic postulate of diminshing marginal returns to capital

c) At L = 200 and K = 1000, MPK is 0.4*(200/1000)^0.8 = 0.1104. For 1000 units of capital, payment made to capital is 110.4. At 200 units of labor, MPL is 1.6*(1000/200)^0.2 = 2.2076. For 200 units of labor, payment made to labor is 2.2076*200 = 441.52.

d) At the current rate, MPL > MPK. Adding more capital will reduce the rental price of capital because its supply is increased and capital owners will be willing to accpet a lower rate as well. Real wage will increase as less labor is used and so MPL will increase.

e) Income share of capital and income share of labor both will increase if labor employed is unchanged. This is because rental rate of capital has reduced due to low MPK while K is increased. MPL has K in the denominator so a rise in K will increase MPL. Labor share in income rises essentially. Capital share rises because K rises even though MPK falls

f) 2 is the technology parameter that indicates how much of the production is increased due to technology.

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