Quiz 6

Q:  Demand is given by P=90-Q.
1)  If there is a single monopolist in the market with TC=4q2, what will be the market price?
2)  If there are two firms competing according tot he Cournot model each with TC=4q2, what will be the market price?

A:  1)  The monopolist sets MR=MC.  MC=8q.  TR=P*Q=(90-Q)q so MR=90-2Q.  (Notice that we are using Q for the market quantity and q for the firm's quantity. Since a monopolist is the only firm Q=q.)  The optimal quantity for the firm is where 8q=90-2Q or q=q.  Hence P=90-9=81.

2) Each firm wants to set MR=MC.  Let q1 denote the quantity for firm 1 and q2  the quantity for firm 2.  MC for firm 1 is 8q1 and the MC for firm 2 is 8q2.  TR for firm 1 is P*q1= (90-q1-q2)q1.  So MR for firm 1 is 90-2q1 -q2.  Thus the firm 1's best choice is  where 8q1 = 90-2q1 -q2.  Firm 2's problem is similar and it results in 8q2 = 90-2q2 -q1. By symmetry q2 =q1.  Hence the solution is where 8q1 = 90-2q1 -q1 or 11q1=90 or q1=90/11.  Therefore q2=90/11, Q=180/11, and P=90-180/11.