Cournot Model in a Duopoly Market

[samchan5167]

Homework Statement

Two retailers compete on price in a market. Firm 1’s demand depends on both its own price and firm 2’s price as follows: q1 = b – p1 + ap2. Similarly, firm 2’s demand depends on its own price and firm 1’s price: q2 = b – p2 + ap1. Their marginal costs of producing one unit of product are both c.
a. Find the expression of firm 1’s equilibrium price (as a function of a, b and c).
b. Find the expression of firm 1’s equilibrium profit (as a function of a, b and c).

Homework Equations

The Attempt at a Solution

I can't seem to figure this out as normally the price function would be given...

 

[StoneTemplePython]

Can you help us out with some Relevant Equations here? I have vaguely fond memories of the Cournot model from game theory, but emphasis on vague at the moment...

 

[samchan5167]

Two firms producing the same kind of product in quantities of q1 and q2, respectively:
Market clearing price p = a - b (q1 + q2)

Profit for firm i:
πi = (p – c) qi = [a - b (q1 + q2) – c] qi ,where c is the unit production cost.

Define B = (a – c)/b,
πi = b (B – q1 – q2) qi

Objective: choose qi to maximize profit,
maxqi b(B – q1 – q2) qi

Normally, after you find q1 and q2, you sub them into the price equation and then you can get the equilibrium price. However, in this question, there’s no value given for a or b or c and price and profit are to be found rather than finding q1 or q2. Thanks for replying.

 

[rude man]

However, in this question, there’s no value given for a or b or c

Why are you concerned? You're supposed to solve in terms of a, b & c, not some set of actual numbers.