Consider a market with just one firm. The demand in the market is p =

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?

ANSWER

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