In: Economics
Two firms compete with quantities as in Cournot. Each firm has a marginal cost of $12. The industry demand is P=48-2Q. How much output will each firm produce individually?
Given that MC1 = MC2 = $12 and demand function is P=48-2Q where Q = Q1 + Q2
The demand curve is P = 48 - 2(Q1 + Q2) = 48 - 2Q1 - 2Q2
Profit function of firm 1 is given by
π1 = TR1 - TC1
Here, TR1 = P * Q1 = (48 - 2Q1 - 2Q2) * Q1 = 48Q1 - 2Q12 - 2Q1Q2
TC1 = MC1* Q1 = 12Q1
So, π1 = TR1 - TC1 = 48Q1 - 2Q12 - 2Q1Q2 - 12Q1
The best response function of firm 1 is given by
∂π1 / ∂Q1 = 0
48 - 4Q1 - 2Q2 - 12 = 0
4Q1 + 2Q2 = 36
or 2Q1 + Q2 = 18 ......(1)
Profit function of firm 2 is given by
π2 = TR2 - TC2
Here, TR2 = P * Q2= (48 - 2Q1 - 2Q2) * Q2= 48Q2 - 2Q1Q2 - 2Q22
TC2 = MC2* Q2 = 12Q2
So, π2 = TR2 - TC2 = 48Q2 - 2Q1Q2 - 2Q22 - 12Q2
The best response function of firm 2 is given by
∂π2 / ∂Q2 = 0
48 - 2Q1 - 4Q2 - 12 = 0
2Q1 + 4Q2 = 36
or Q1 + 2Q2 = 18 .......(2)
Solving equations (1) and (2) simultaneously, we get
Q1 = 6 units and Q2 = 6 units
Thus, each firm will produce 6 units of output individually.