Consider two Cournot competitors selling complementary goods with demand curves given by:

p1 = 100 - q1 + .5q2
p2 = 100 - q2 + .5q1
Suppose each firm has a marginal and average cost of $10.
a. What about the demand equations indicate that these goods are complements? How do they differ from the standard Cournot model?
b. Find the equilibrium prices and quantities.
c. Suppose the two firms merge. By doing so, the newly merged firm will act to maximize the joint profits ((q1,q2 ) = 1(q1,q2 ) + 2(q1,q2 )). Find the joint-profit maximizing price and quantities.
d. Are the combined profits greater or smaller from merging? That is, is merging profitable for the firms?
e. Are consumers better or worse off with the firms merging? How does this compare to the mergers of Cournot competitors selling substitutes? What does this imply about antitrust policy towards mergers of firms selling complementary goods (such as airplanes and engines, computers and processors, cars and tire companies, etc).

a. The output of the rival firm actually increases the own firm's demand curve. Thus, these goods are complements rather than substitutes as in the typical model
b. The objective function of a firm is
= (100 - q1 + .5q2 )q1 - 10q1
The derivative is
100 - 2q1 + .5q2 - 10 = 0
q1 = (90 + .5q2 )/2
The firm's will produce where outputs are the same (symmetrical)
2q = 90 + .5q q = 60. Both firms produce 60 units. The price of each good is 70
c. The combined profit is:
= (100 - q1 + .5q2 )q1 - 10q1 + (100 - q2 + .5q1 )q2 - 10q2
The derivative w.r.t. q1 is
100 - 2q1 + .5q2 - 10 + .5q2 = 0
The outputs will be the same because the derivatives are symmetrical.
90 - q1 = 0
q1 = 90
So 90 units is produced by each firm ("plant") for a total of 180 units. The price in each market is 55.
d. Greater
e. Consumers are better off because the price is lower and the quantity is higher in each market.