Question

Player 1 and Player 2 (independently and simultaneously) choose quantities, q1 and q2. The cost of producing qi units is c(qi) = q_i² ; and the demand curve is given by P(Q) = 5 − Q: (i.e., if Player 1 produces q1 and Player 2 produces q2; each sells all his units at price 5 − q1 − q2. Find the profit.

Answer 1

Demand function: P= 5-q1-q2

Cost of firm 1= q1²

Cost of firm 2= q2²

Profit of firm 1= Pr1= P x q1 - Cost of firm 1

Pr1= 5q1-q1² -q1q2-q1²

Differentiate it with respect to q1:

dPr1/dq1= 5-2q1-q2-2q1 =0

4q1+q2= 5 Equation 1

q1= (5-q2)/4 Reaction curve of firm 1

Profit of firm 2= Pr2= P x q2 - Cost of firm 2

Pr2= 5q2-q2² -q1q2-q2²

Differentiate it with respect to q2:

dPr1/dq1= 5-2q2-q1-2q2 =0

q1+4q2= 5 Equation 2

q2= (5-q1)/4 Reaction curve of firm 2

Solve equation 1 and 2: Multiply equation 1 by 4 and then subtract it from equation 2

q1+4q2-16q1-4q2= 5-20

-15q1 = -15

q1= 1 Optimal quantity by firm 1

Use q1= 1 in reaction curve of firm 2:

q2= (5-1)/4= 1 Optimal quantity by firm 2

P= 5-q1-q2= 5-2= 3 Optimal price

Cost of firm 1= 1

Cost of firm 2= 1

Profit of firm 1= P x q1-cost of firm 1= 3-1= 2

Profit of firm 2= P x q2-cost of firm 2= 3-1= 2