Question #73965

Guwahati Tyres, a small producer of automobile typres, has the following production function:
Q=100K0.5L0.5.
During the last production period, the firm operated efficiently and used input rates of 100 and 25 for capital and labor respectively.
a.) What is the marginal product of capital and marginal product of labor based on input rates specified?
b.) If the price of capital was Rs.20 per unit what was the wage rate?
C.)for the next production,the price per unit of capital is expected to increase to Rs.25, while the wage rate and the labor input will remain unchanged under the terms of the labor contract with the labor union. if the firm maintains efficient production, what input rate of capital will be used?

Expert's answer

Q=100K^0.5L^0.5.

K = 100, L = 25.

a.) The marginal product of capital and marginal product of labor are:

MPK = Q'(K) = 50L^0.5/K^0.5 = 25.

MPL = Q'(L) = 50K^0.5/L^0.5 = 100.

b.) If the price of capital was r = Rs.20 per unit, then the wage rate w is:

MPK/r = MPL/w,

25/20 = 100/w,

w = Rs.80.

C.) If r = Rs.25, then the input rate of capital K will be:

MPK/r = MPL/w,

MPK/25 = 100/80,

MPK = 31.25,

MPK = 50L^0.5/K^0.5 = 31.25,

50*5/K^0.5 = 31.25,

K = (250/31.25)^2 = 64 units.

K = 100, L = 25.

a.) The marginal product of capital and marginal product of labor are:

MPK = Q'(K) = 50L^0.5/K^0.5 = 25.

MPL = Q'(L) = 50K^0.5/L^0.5 = 100.

b.) If the price of capital was r = Rs.20 per unit, then the wage rate w is:

MPK/r = MPL/w,

25/20 = 100/w,

w = Rs.80.

C.) If r = Rs.25, then the input rate of capital K will be:

MPK/r = MPL/w,

MPK/25 = 100/80,

MPK = 31.25,

MPK = 50L^0.5/K^0.5 = 31.25,

50*5/K^0.5 = 31.25,

K = (250/31.25)^2 = 64 units.

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