- #1
Poisonous
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Homework Statement
My Textbook gives the following example problem:
Cournot Duopoly with incomplete information.
The profit functions are given by:
u_i = q_i(θ_¡ - q_i - q_j)
Firm 1 has one type θ_1 = 1, but firm 2 has private information about its type θ_2. Firm 1
believes that θ_2= 3/4 with probability 1/2 and θ _2 = 5/4 with probability 1/2, and this belief is common knowledge.
We will look for a pure strategy equilibrium of this game. Firm 2 of type θ_2’s decision
problem is to
max q_2: q_2(θ_2 - q_1 - q_2)
which is solved at
q_'2(θ_2) = (θ_2-q_1)/2
Firm 1’s decision problem, on the other hand, is
max q_1: 1/2 * q_1 (1 - q_1 - q_'2(3/4)) + 1/2 * q_1(1-q_1-q_'2(5/4))
which is solved at
q_'1 = (2-q_'2 (3/4)-q_'2(5/4))/4
Solving yields,
q_'1 = 1/3
q_'2(3/4) = 11/24
q_'2(5/4) = 5/24
Atempt at a solution
I can't figure out how they got from the equations q_'1 and q_'2 to the answers given in the "solving yields" part. It seems like simultaneous equations, but I'm not sure what to do with the q_1 or really where to begin.