Question

Suppose the final goods production function is fixed-proportion, Q = f(E, L) = min{E,L}, where Q is output level, E is energy input and L is the labor in- put. Let m be the marginal cost of energy per unit and w be the price of labor per unit. Suppose the demand function for final good is P = 1 - Q

a). (10) Suppose energy and final good are produced by two different firms. Derive the cost function of final good production.

b). (10) Suppose the final good market is perfect competitive, determine the downstream firms profit maximization problem and the demand function for energy.

c). (10) Suppose the upstream firm is monopoly. Now if two firms merge together, determine the optimal monopoly pricing of the merged firm.

Answer 1

(1) Q = 21L + 9L2 - L3

(a) MPL = dQ / dL = 21 + 18L - 3L2

MPL is maximum when (dMPL / dL) = 0

18 - 6L = 0

6L = 18

L = 3

(b) APL = Q / L = 21 + 9L - L2

APL is maximum when (dAPL / dL) = 0

9 - 2L = 0

2L = 9

L = 4.5

(c) Q is maximum when (dQ / dL) = 0

21 + 18L - 3L2 = 0

7 + 6L - L2 = 0

L2 - 6L - 7 = 0

L2 - 7L + L - 7 = 0

L x (L - 7) + 1 x (L - 7) = 0

(L - 7) (L + 1) = 0

L = 7 or L = - 1

Since L cannot be negative, L = 7

Note: First question is answered.

Thank you...