## Best response curves

September 29, 2011

In a duopoly, the residual demand curve faced by one firm is the market demand curve minus the supply of the rival firm: $D^r (P) = D (P) - S^o (P)$.

In the simple model I’m using for these examples, the market demand is Q = 500 – P and the firm (both firms in this duopoly case) have no fixed costs and a constant marginal cost of 150. So if the firms are firm A and firm B, then the residual demand curves for each firm are: $Q_A = 500 - P - Q_B$ and $Q_B = 500 - P - Q_A$.

This means the inverse residual demand curves for each firm are $P= 500 - Q_A - Q_B$.

Given that we know each firm will want to produce at the profit maximising point, where MR = MC, we can use these equations to find out an expression for the profit maximising output for each term in terms of the quantity produced by the other – a best response function.

For firm A, the inverse demand function is $P= 500 - Q_A - Q_B$.

So the total revenue is $TR_A = PQ_A = 500Q_A - {Q_A}^2 - {Q_B}{Q_A}$.

The marginal revenue is $MR_A = \frac{d(TR_A)}{dQ} = 500 - 2Q_A - Q_B$.

The profit maximising quantity of output is where MR = MC, so $500 - 2Q_A - Q_B = 150 \Rightarrow 2 Q_A = 350 - Q_B \Rightarrow Q_A = 175 - 0.5 Q_B$.

This is the best response function for firm A.

This graph has the quantity produced by firm A on the horizontal axis and the quantity produced by firm B on the vertical axis. From this curve you can see what the optimum amount for firm A to produce is, when the amount produced by firm B is at a particular level.

For instance if firm B produces 200, what would be the optimum for firm A to produce?

The best response function is $Q_A = 175 - 0.5 Q_B$ so that means $Q_A = 175 - 0.5 (200) = 75$.

If firm B produces output of 200, then firm A’s best response is to produce output of 75.