Question

Please show all your work.

Your firm has total costs equal to 40Q₁. The demand you face is P=4000-5Q.

Now your firm has a rival firm, with total costs equal to 40Q₂.

Now there is a third firm, with total costs equal to 40Q₃.

All three firms play Cournot. What is your output and profits? What is the deadweight loss?

Answer 1

In Cournot competition, firms compete on the basis of the quantity to be produced.

Suppose, the firms produce quantity Q1, Q2, Q3

So, for firm 1, he has to choose Q1 given Q2 and Q3

Thus, his profit function is = Revenue - Cost = (4000-5(Q1+Q2+Q3))xQ1-40Q1=3960Q1-5Q1x(Q1+Q2+Q3)

By symmetry, Q1, Q2 and Q3 should be the same. So, 3960Q1-5Q1x(Q1+Q2+Q3)=3960Q1-15Q1^2

Deriving wrt Q1, we get 3960-30Q1

So, Q1=3960/30=132

Thus, each firm produces 132 output totalling at 396 and profit for each firm is 132*(3960-15*132)=261360 for each firm.

Price will be 4000-5x396=2020

Marginal cost is 40, so in perfect competition, Q=(4000-40)/5=792 and price is 40

The deadweight loss is 0.5x(792-396)*(2020-40)= 392040