Question

In: Economics

Two firms compete with quantities as in Cournot. Each firm has a marginal cost of $12....

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?

Solutions

Expert Solution

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.


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