1. Consider a duopoly market, where two firms sell differentiated products, which are imperfect substitutes. The market can be modelled as a static price competition game. The two firms choose prices P1 and p2 simultane- ously. The demand functions for the two firms are: D: (P1, P2) = { + P2771 and D2 (P1, P2) S + 21,722, where S > 0, and the parameter t > 0 measures the degree of product differentiation. Both firms have constant marginal cost c> 0 of production. 2 2t 2 (a) Derive the Nash equilibrium of this game, including the prices, out- puts and profits of the two firms. (b) From the demand functions, qi = Di (Pi, P;) + Pj Pi derive the residual inverse demand functions: Pi = Pi(qi, Pj) (work out Pi(qi,P;)). Show that for t > 0, P(qi,P;) is downward-sloping, OP (qi;P) i.e., 0 as given, firm i is aqi like a monopolist facing a residual inverse demand, and the optimal qi (which equates marginal revenue and marginal cost) or pi makes Pi(qi, Pj) = pi > c, i.e., firm i has market power. (c) Calculate the limits of the equilibrium prices and profits as t 0. What is P;(qi, P;) as t ? 0? Is it downward sloping? Argue that the Bertrand Paradox (i.e., the prediction of the static Bertrand duopoly model, where pi= p = c) holds only in the extreme case of t = 0.

1. Consider a duopoly market, where two firms sell differentiated products, which are imperfect substitutes. The market can be modelled as a static price competition game. The two firms choose prices p1 and p2 simultane- ously. The demand functions for the two firms are: D1 (p1, p2) = 5 + 22-P1 and D2 (p1, p2) = 2 + 2. P1-P2, where S > 0, and the parameter t > 0 measures the degree of product differentiation. Both firms have constant marginal cost c > 0 of production. (a) Derive the Nash equilibrium of this game, including the prices, out- puts and profits of the two firms. (b) From the demand functions, qi = Di (Pi, pj) = 5 + Pi Pi 2t , derive the residual inverse demand functions: p; = P.(qi, p; ) (work out Pi(qi, p;)). Show that for t > 0, Pi(qi,p; ) is downward-sloping, i.e., aqi P: (q p)) c, i.e., firm i has market power. (c) Calculate the limits of the equilibrium prices and profits as t - 0. What is Pi(qi, p;) as t - 0? Is it downward sloping? Argue that the Bertrand Paradox (i.e., the prediction of the static Bertrand duopoly model, where pf = p? = c) holds only in the extreme case of t = 0