model with 2 firms and constant marginal cost.
p1 = firm 1 price, p2 = firm 2 price
q1 = firm 1 quantity, q2 = firm 2 quantity
c = marginal cost, identical for both firms
Equilibrium prices will be:
p1 = p2 = P(q1 + q2)
This implies that firm 1’s profit is given by Π1 = q1(P(q1 + q2) − c)
Calculate firm 1’s residual demand: Suppose firm 1 believes firm 2 is producing quantity q2. What is firm 1's optimal quantity? Consider the diagram 1. If firm 1 decides not to produce anything, then price is given by P(0 + q2) = P(q2). If firm 1 produces q1' then price is given by P(q1' + q2). More generally, for each quantity that firm 1 might decide to set, price is given by the curve d1(q2). The curve d1(q2) is called firm 1’s residual demand; it gives all possible combinations of firm 1’s quantity and price for a given value of q2.