Two firms compete as duopolists, producing identical goods in quantities q1 and q2,
respectively. Assume the inverse market demand function for their product is
P = 504 – (q1 + q2).
If both companies have the identical cost function TC = 0.5q2 (i.e., MC1 = q1 and
MC2 = q2), find the following solutions:
(i)
Total profit-maximizing industry output Q (= q1+q2) under the Cartel solution,
where P = 504 – Q and TC = 0.5Q2.
(ii)
a. the profit-maximizing output for each individual firm under the Bertrand
solution.
b. the profit-maximizing output for each individual firm under the Competitive
Fringe solution, where firm #1 represents the dominant firm in the market.
(iii)
a. the profit-maximizing output for each individual firm under the Cournot
solution.
b. the profit-maximizing output for each individual firm under the Stackelberg
solution, where firm #1 represents the market leader.
ANS:
(i) Cartel solution: Q = 168. [Since MR = 504 - 2Q and MC = Q]
(ii) Bertrand solution: (q1, q2) = (168, 168). [Since both CV1 & CV2 = -1, the
equilibrium outputs solve P = MC1 and P = MC2].
Competitive Fringe solution: The non-dominant firm [#2] has an (initial) CV2 of -1;
that is used to form its profit-max equation P = MC2 [after rearranging, q2 = 252 –
0.5q1.] But the equation-based CV2 = q2/q1 = -0.5, and the dominant firm [#1] is
credited with (correctly) recognizing this, so its profit-max equation is ultimately
504 – 2.5q1 – q2 = 0, or q1 = 201.6 - 0.4q2. The solution to this equation pair is
(q1, q2) = (126, 189).
** Note that firm #2 actually has a bigger market share than firm #1! This is not an
illogical result here, since firm #2 in this context is an amalgamation of a large number of
small, perfectly competitive firms.
(iii) Cournot solution: (q1, q2) = (126, 126). [Since both CV1 & CV2 = 0, the
equilibrium outputs solve MR1 = MC1 and MR2 = MC2].
Stackelberg solution: The non-leader firm [#2] has an (initial) CV2 of 0; that is used
to form its profit-max equation MR2 = MC2 [after rearranging, q2 =168 – 1/3q1.]
But the equation-based CV2 = q2/q1 = -1/3, and the leader firm [#1] is credited
(as in the Competitive Fringe solution) with correctly recognizing this, so its profitmax equation is ultimately 504 – 8/3q1 – q2 = 0, or q1 = 189 – 3/8q2. The solution to
this equation pair is (q1, q2) = (144, 120).