Two firms 1 and 2 are using both capital (K) and labour (L) to produce identical products. They have different technologies in their productions, which are described as the following production functions:

f1(K1,L1) = aK1 +bL1; f2(K2,L2) = min{aK2,bL2},

where a,b are two positive constants and the subscripts 1 and 2 represent different firms, respectively. The market prices for capital and labour are r and w.

(a) What are the cost-minimizing values of K1 and L1 if Firm 1 produces a given number q1? Show your work and illustrate your answer using an appropriate diagram with necessary details. Note that both explanations and a diagram are required.

(b) What are the cost-minimizing values of K2 and L2 if Firm 2 produces a given amount q2? Show your work and illustrate your answer using an appropriate diagram with necessary details. Note that both explanations and a diagram are required.

MAE201: Competition and Industry Trimester 1 2020 Individual Assignment

(c) What are the minimal total costs TC1 and TC2 for Firms 1 and 2 when they produce q1 and q2 amounts, respectively? What are the average costs AC1 and AC2, respectively?

(d) Use suitable analytical tools (either algebraic or graphic) to show that, if the minimal total costs are the same for both firms (i.e., TC1 = TC2), then Firm 1's output level q1 is at least twice of Firm 2's output level q2, (i.e., q1 2q2).