# The Cournet solution for the duopoly

Given the reaction function of duopolist A, QA = (12-QB)/2 (1) and the reaction function of duopolist B, QB = (12-QA)/2 (2), find the Cournot solution, that is to find QA and QB by substituting (1) into (2).

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#### Solution Preview

First, lets find the QA and QB of the equilibrium point. The reaction functions are expressed in term of quantity which maximizes the profit of the firm. For example, QA = (12 - QB) / 2 gives the function that maximizes the profit for firm A. Since the idea is to find the equilibrium point, we want to find QA and QB which maximizes both firm A and firm B's profit to the largest extent. To do this, all we need to do is substitute (1) into (2).

QA = (12 - QB) / 2 (1)

QB = (12 - QA) / 2 (2)

QA = (12 - QB)/2

QA = [12 - (12 - QA)/2]/2

2QA = 12 - 6 + QA/2

4QA = 12 + QA

QA = 12/3

QA = 4

Since the two reaction functions are identical, QB would give the same ...

#### Solution Summary

The Cournet solution for the duopoly is shown.