P2-P1 2t 1. Consider a duopoly market, where two firms sell differentiated products, which are imperfect...

 

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P2-P1 2t 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) = +21 and D2 (P1, P2) +12, where S > 0, and the parameter t > 0 measures the degree of product differentiation. Both firms have constant marginal cost c> 0 of production. = 2t (a) Derive the Nash equilibrium of this game, including the prices, out- puts and profits of the two firms. == Pj-Pi 2t (b) From the demand functions, q = Di (Pi, Pj) = +, derive the residual inverse demand functions: Pi = Pi(i,P) (work out Pi(P)). Show that for t > 0, Pi(qi,P) is downward-sloping, OP(qPj) i.e., <0. Argue that, taking p; 0 as given, firm i is 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. qi = (c) Calculate the limits of the equilibrium prices and profits as t 0. What is Pi(qi, Pi) as t0? Is it downward sloping? Argue that the Bertrand Paradox (i.e., the prediction of the static Bertrand duopoly model, where p = p2 = c) holds only in the extreme case of t = 0. P2-P1 2t 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) = +21 and D2 (P1, P2) +12, where S > 0, and the parameter t > 0 measures the degree of product differentiation. Both firms have constant marginal cost c> 0 of production. = 2t (a) Derive the Nash equilibrium of this game, including the prices, out- puts and profits of the two firms. == Pj-Pi 2t (b) From the demand functions, q = Di (Pi, Pj) = +, derive the residual inverse demand functions: Pi = Pi(i,P) (work out Pi(P)). Show that for t > 0, Pi(qi,P) is downward-sloping, OP(qPj) i.e., <0. Argue that, taking p; 0 as given, firm i is 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. qi = (c) Calculate the limits of the equilibrium prices and profits as t 0. What is Pi(qi, Pi) as t0? Is it downward sloping? Argue that the Bertrand Paradox (i.e., the prediction of the static Bertrand duopoly model, where p = p2 = c) holds only in the extreme case of t = 0.