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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

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