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.