Question

Suppose a business estimates his production function to be ?? = ?? ^0.25?? ^0.75 where Q is the output, K amount of capital and L is amount of labor. Price of labor (wage rate) is $10 and price of capital is $15.

(a) Calculate the slope of isoquant curve.

(b) Calculate the slope of isocost curve.

(c) Suppose the firm wants to produce 100 units of output. Find the optimal combination of labor and capital.

Answer 1

a)

Isoquant = ?? = ?? ^0.25?? ^0.75

Marginal product of Labor = dQ/dL = 0.25?? ^-0.75?? ^0.75

Marginal product of Capital = dQ/dK = 0.75?? ^0.25?? ^-0.25

Slope of isoquant = Marginal Product of labor/Marginal product of capital = 0.25?? ^-0.75?? ^0.75 / 0.75?? ^0.25?? ^-0.25 = 0.25K/0.75L = K/3L

b)

Slope of isocost = Price of L/ Price of K = 10/15 = 2/3

c)

At equilibrium, the slope of isocost equals the slope of isoquant, hence

K/3L = 2/3

K = 2L

Substituting the value of K = 2L and Q = 100 in the production function we get,

100 = ?? ^0.25(2L) ^0.75

100 = L*2^0.75

L = 59.46

K = 2L = 118.93