Question

A competitive firm’s production function is

            Q = 5 + 20L - .5L² + 40K – K²,

and its demand function is

            P_Q = MR_Q = d = $6.

The input prices of L and K are P_L = $6 and P_K = $12. Use Excel to find the profit-maximizing and cost minimizing amounts of L and K to employ.  

L = _______

K = _____

Find the cost minimizing ratios of marginal product to input prices:

Ratios = _____

Answer 1

Consider the given problem here the profit function of the firm is given below.

=> A = P*Q – PL*L – PK*K.

=> A = P*{5 + 20*L – 0.5*L^2 + 40*K – K^2} – PL*L – PK*K.

The FOC form profit maximization required “dA/dL = dA/dK = 0”.

=> dA/dL = 0, => P*{20 – 0.5*2*L} – PL = 0, => 20 –L = PL /P, where “P=PL = 6”.

=> 20 –L = 1, => L=19.

=> dA/dK = 0, => P*{40 – 2*K} - PK = 0, => 40 – 2*K = PK /P, where “P=6” and “PK=12”.

=> 40 – 2*K = 2, => K = 38/2 = 19, => K = 19.

So, the optimum use of inputs are “L=K=19”.

The marginal productivity to input price ratio are.

=> MPL/PL = (20-0.2*2*L)/PL = (20-L)/6 = 1/6.

=> MPK/PK = (40-2*K)/PK = (40-2*K)/12 = 2/12 = 1/6.