Consider a market with just one firm. The demand in the market is p = 18 – Q and the firm has a linear cost function C(Q) = 2Q

a. How much output will this firm produce. What will be the profit and consumers surplus?
b. Suppose a second firm with the same cost function enters the market and the two firms compete in a Cournot style (simultaneous output choice). What will be the equilibrium price and quantity in the market? What is the total market profit and CS?

a. A monopolist will produce to maximize profits given by:
= (18 – Q)Q - 2Q
The derivative w.r.t Q:
- 2Q - 2 = 0
So Q = 8
The price is then $10. The profit is $10(8 ) - 2(8 ) = 64. The Consumer surplus is (18 - 10 )(8 )/2 = 32.
b. Each firm will maximize profit by:
π = (18 - Q - Qo)Q - 2Q
where Q0 is the output of the other firm. The derivative is:
18 - Qo - 2Q - 2 = 0
The best response is:
Q = 8 - Qo/2
Solving both best response functions simultaneously:
Q = 8 - (8 - Q/2 )/2
Q = 16/3 = 5.33. The total market quantity is then 10.66. The price is 18 - 10.66 = 7.34. The profit for each firm is:
5.33 (7.34 ) - 5.33(2 ) = 28.41
The total market profit is then 56.82
The consumer surplus is (18 - 7.34 )(10.66 )/2 = 56.87