Question

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Question 1:

FOR THE REST OF THE TEST Assume Y = A*K.5*L.5

In equation form- what is the marginal product of capital ?
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59) In equation form what is the marginal product of labor ?
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60) How do we know that each is diminishing.
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61) Prove that if you double capital and double labor Y doubles.
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62) What is the characteristic described in d called in words ?
63) Write out the Solow Equation for the growth rate of capital per worker as a function of y/k, s, n, and
depreciation.

Answer 1

Y = AK^0.5 L^0.5

58) MPK = derivative of output with respect to capital = 0.5*A*K^-0.5L^0.5.

59) MPL = derivative of output with respect to labor = 0.5*A*K^0.5L^-0.5.

60) Note that second order derivative of output with respect to capital, d^2Y/dK^2 = -0.25K^-1.5L^0.5. Since it is negative, it shows that MPK is diminishing. Similarly, second order derivative of output with respect to labor, d^2Y/dL^2 = -0.25K^1.5L^-1.5. Since it is negative, it shows that MPL is diminishing

61) Double K and L

Y = A(2K)^0.5 (2L)^0.5

Y = A(2^0.5)K^0.5 * (2^0.5)L^0.5

Y = A(2^1)K^0.5 L^0.5

Y = 2A*K^0.5L^0.5.

Hence output is doubled

62) This is called constant returns to scale

63) Steady state capital per worker

k/y = s/(d + n)

The growth rate is given by gy = sy/k - (d + n).