Question

Assume a firm's production is as follows:

    Q= L^0.5K

A. Does this firm exhibit diminsishing returns with respect to labor?

B. Does this firm exhibit dimishing returns with respect to capital?

C. In the long run, does this firm exhibit increasing, constant, or decreasing returns to scale?

D. Construct a short run cost function for this firm, assuming capital is fixed.

E. Identify the TVC and TFC in the short run cost function

F. Construct a MC function

Answer 1

The production is Q= L^0.5K

A. Find the marginal product of labor

MPL = 0.5L^-0.5*K. From the marginal product of labor, we see that as labor units are increased, marginal returns to labor fall (labor is inversely related to MPL). Hence there are diminishing returns to labor

B. Does this firm exhibit dimishing returns with respect to capital?

Find the marginal product of capital

MPK = L^0.5. From the marginal product of capital we see that as capital units are increased, marginal returns to capital remains unchanged. Hence there are constant returns to capital

C. We multiply the inputs by 2.

New Q = (2L^0.5)(2K)

= 2^(0.5 + 1)L^0.5K

= 2^1.5 * Old Q.

Since new output is increased by more than 2, we suggest that in the long run this firm exhibits increasing, returns to scale

D. When K is constant, we have L = (Q/K)^2. Now cost function is C = wL + rK

C = w (Q/K)^2 + r K

E. TVC is the portion that contains/depends on output. Hence TVC is w (Q/K)^2 and so TFC = rk

F. Marginal cost function is the derivative of cost function

MC = dC/dQ

= 2wQ/K^2