Consider a duopoly market, where two firms sell differentiated prod-ucts, which are imperfect substitutes. The market can be modelledas a static price competition game, similar to a linear city model.The two firms choose prices p1 and p2 simultaneously. The deriveddemand functions for the two firms are: D1 (P1, P2) = + P2P1and D2 (P1, P2) =+ 2, where S > 0 and the parameter t > 0measures the degree of product differentiation. Both firms haveconstant marginal cost c > 0 for production.S2tS(a) Derive the Nash equilibrium of this game, including the prices,outputs and profits of the two firms.Pj-Pi derive(b) From the demand functions, q; = D; (pi, Pj) =the residual inverse demand functions: p; = P:(qi, P¡) (work outP;(qi, P;)). Show that for t > 0, P;(4;, P;) is downward-sloping,aP:(gi-Pj)+2ti.e.,< 0. Argue that, taking p; 2 0 as given, firm iis like a monopolist facing a residual inverse demand, and theoptimal q; (which equates marginal revenue and marginal cost)or p; makes P;(qi, Pi) = 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? Ar-gue that the Bertrand Paradox (i.e., the prediction of the staticBertrand duopoly model, where pj = P = c) holds only in theextreme case of t = 0.

Demand Function for Firm 1Q1 = S2+P2−P12tDemand Function for Firm 2Q1 = S2+P1−P22tBoth Firm Have…

Consider a duopoly market, where two firms sell differentiated prod- ucts, which are imperfect substitutes. The market can be modelled as a static price competition game, similar to a linear city model. The two firms choose prices p1 and p2 simultaneously. The derived demand functions for the two firms are: D1 (P1, P2) = + PI and D2 (P1, P2) =+ ",2, where S> 0 and the parameter t > 0 measures the degree of product differentiation. Both firms have constant marginal cost c> 0 for production. S P2-P1 2t S 2t (a) Derive the Nash equilibrium of this game, including the prices, outputs and profits of the two firms. (b) From the demand functions, q; = D; (p;, P;) =+ ", derive the residual inverse demand functions: p; = P:(q;i, Pj) (work out P;(qi, P;)). Show that for t > 0, P;(9i, P;) is downward-sloping, aP:(qi-Pj) 2t %3D i.e., < 0. Argue that, taking p; > 0 as given, firm i is like a monopolist facing a residual inverse demand, and the optimal q; (which equates marginal revenue and marginal cost) or pi makes P;(qi, P;) = 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;(gi, P;) as t → 0? Is it downward sloping? Ar- gue 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.

14th Edition

ISBN:9781305506381

Author:James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris

Publisher:James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris

Chapter12: Price And Output Determination: Oligopoly

Section: Chapter Questions

Problem 2E