## Best response curves

In a duopoly, the residual demand curve faced by one firm is the market demand curve minus the supply of the rival firm: .

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: and .

This means the inverse residual demand curves for each firm are .

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 .

So the total revenue is .

The marginal revenue is .

The profit maximising quantity of output is where MR = MC, so .

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 so that means .

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