A reaction curve for Firm 1 is a function Q1*() that takes as input the
quantity produced by Firm 2 and returns the optimal output for Firm 1 given Firm
2's production decisions. In other words, Q1*(Q2) is Firm 1's best
response to Firm 2's choice of Q2. Likewise, Q2*(Q1) is Firm 2's best
response to Firm 1's choice of Q1.
Let's assume the two firms face a single market demand curve as follows:
Q = 100 - P
where
P is the single market price and
Q is the total quantity of output in
the market. For simplicity's sake, let's assume that both firms face cost
structures as follows:
MC_1 = 10
MC_2 = 12
Given this market demand curve and cost structure, we want to find the reaction
curve for Firm 1. In the Cournot model, we assume Q2 is fixed and proceed.
Firm 1's reaction curve will satisfy its profit maximizing condition, MR = MC.
In order to find Firm 1's marginal
revenue, we first determine its
total revenue, which can be described as follows
Total Revenue = P * Q1 = (100 - Q) * Q1
= (100 - (Q1 + Q2)) * Q1
= 100Q1 - Q1 ^ 2 - Q2 * Q1
The marginal revenue is simply the first derivative of the total revenue with
respect to Q1 (recall that we assume Q2 is fixed). The marginal revenue
for Firm 1 is thus:
MR1 = 100 - 2 * Q1 - Q2\
Imposing the profit maximizing condition of MR = MC, we conclude that Firm 1's
reaction curve is:
100 - 2 * Q1* - Q2 = 10 => Q1* = 45 - Q2/2
That is, for every choice of Q2, Q1* is Firm 1's optimal choice of
output. We can perform analogous analysis for Firm 2 (which differs only in
that its marginal costs are 12 rather than 10) to determine its reaction curve,
but we leave the process as a simple exercise for the reader. We find Firm 2's
reaction curve to be:
Q2* = 44 - Q1/2
Ad-free experience
Study Guides for 1,000+ titles
Full Text content for 250+ titles
No Fear Translations of Shakespeare’s plays (along with audio!) and other classic works
Flashcards
Mastery Quizzes
Infographics
Graphic Novels
AP® Practice & Lessons
Note-taking
Bookmarking
Dashboard
