A cost minimizing firm has the following short run production function: Q=f (L,K

Question

A cost minimizing firm has the following short run production function:
Q=f (L,K)=72L+5L2-0.2L3

a) Briefly explain why this is a short run production function.

b) At what level of employment would diminishing returns set in for the variable input?

c) Fin the levels of employment that define stage two in the production process.

d)If the market determined real wage rate is $20 determine the amount of labor a cost minimizing firm would hire in order to minimize the total cost of production.

Explanation / Answer

a) It is a short run production function since capital (K) does not appear in the function b) First find the derivative of the function w.r.t L f'(L) = 72 + 10L - 0.6L^2. Now find the maximum of this function f'(L) = 72 + 10L - 0.6L^2 = 0, L > 0 This occurs approximately at L = 22. Above this point, the marginal product of labor would begin to decline c) In stage two, output increases at a decreasing rate. This point occurs at the maximum of the second derivative, so first find that: f''(L) = 10 - 1.2L f"(L) = 10 - 1.2L = 0 10 = 1.2L 10/1.2 = L 8.33 = L This is the point at which output begins increasing at a decreasing rate. This stage continues until output begins to decline rather than increase, which is where diminishing returns set in, which we found above as L = 22

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