Chapter 17: Duopoly Reaction Curves
Nicola Giocoli BASIC CONCEPTS A reaction curve (RC), also called reaction function or best-reply function, is the locus of optimal, that is, profit-maximizing, actions Ri (aj) that a firm i may undertake for any given action aj chosen by a rival firm j. The RC diagram is the standard tool for the graphical analysis of duopoly. In the diagram, the market equilibrium is at the intersection of the RCs, one for each firm. The commonest case of a RC diagram is that of the Cournot duopoly model. Consider two firms 1 and 2 producing a homogeneous product with output levels q1 and q2 and aggregate output Q 5 q1 1 q2. Provided invertibility conditions are met, the inverse demand function gives the market price associated with aggregate output p (Q) 5 p (q1 1 q2) . Assume each firm has a cost function ci (qi) , i 5 1, 2, and take the strategic variable for both firms to be the output level, so that firm 1’s maximization problem is: max q p1 (q1, q2) 5 p (Q) q1 2 c1 (q1) . Given that firm 1’s profit also depends on firm’s 2 output, firm 1’s optimal choice must also take into account firm 2’s choice. A similar problem can be formulated for firm 2. Following the so-called Cournot assumption, we model each firm as taking as given the rival’s quantity. The first order condition (FOC) for each firm is therefore: 1 0pi (q1, q2) 5 p (Q) 1 pr (Q) qi 2...
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