# Cournot Model in a Duopoly Market

## 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).

## The Attempt at a Solution

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

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StoneTemplePython
Gold Member
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...

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
Homework Helper
Gold Member
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.