Question

Suppose that the production function for iPods is Q = 20K0.5L0.5. The marginal product of labor is 10(K/L)0.5, and the marginal product of capital is 10(L/K)0.5. Suppose that labor can be hired for $6, and capital can be rented for $9. If the firm has exactly $300 to spend on producing iPods, what is the maximum number of iPods it can produce?

Answer 1

Q = 20K^0.5*L^0.5, MPL = 10(K/L)^0.5, MPK = 10(L/K)^0.5.
PL = $6, PK = $9.
In microeconomic theory, the Marginal Rate of Technical Substitution (MRTS) is the amount by which the quantity of one input has to be reduced when one extra unit of another input is used, so that output remains constant.
The firm's marginal rate of technical substitution will be:
MRTS = MPL/MPK = (10(K/L)^0.5) / ((10(L/K)^0.5) = (K/L)^0.5 / (L/K)^0.5 = (K/L)^0.5*(K/L)^0.5 = K/L
In this case the highest amount of output is produced, when K = L, so 6L + 9K = 300,
6L + 9L = 300,
15L = 300
L = K = 20 units.
Q = 20*20^0.5*20^0.5 = 400 iPods.