Please help me with the following:
5. Suppose that inverse demand is given by
D(Q) = 56 − 2Q, Q = q1 + q2
and the cost function is
TC(qi ) = 20qi + f
Find the Stackelberg equilibrium and compare it to the Cournot equilibrium.
6. Demand and costs are as given in the preceding question.
(a) Find the limit output for fixed costs ( f ) equal to 50, 32, 18, and 2.
(b) What is the SPNE for the entry game with the following timing: in the first-stage firm 1 can commit to its output; in the second stage firm 2 can enter and choose its output for fixed costs equal to 50, 32, 18, and 2?
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For simplicity, I will use x for the output of firm 1 (i.e q1) and y for the output of firm 2 (i.e. q2).
The inverse demand function, in terms of x and y, is P = 56 - 2x - 2y
The cost functions for each firm is C1 = 20x + f and C2 = 20y + f
The profit of firm 1 = x(56 - 2x - 2y) - 20x - f and the profit of firm 2 = y(56 - 2x - 2y) - 20y - f.
Expanding the brackets gives profit 1 = 56x - 2x^2 - 2xy - 20x - f and profit 2 = 56y - 2xy - 2y^2 - 20y - f.
Differentiating and setting to zero gives
56 - 4x - 2y - 20 = 0
56 - 4y ...
The stackelberg equilibrium and SPNE are provided. Cost functions for inverse demands are analyzed.