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, similar to a linear city model. The twofirms choose prices p, and p2 simultaneously. The derived demand functions for the two firms are: D1(P1, P2) = SG+P1)and D2 (P1, P2)= S(;+-P2), where S > 0 and the parameter t > 0 measures the2t2tdegree of product differentiation. Both firms have constant marginal cost c> 0 for production.(a) Derive the Nash equilibrium of this game, including the prices, outputs and profit of the two firms.(b) From the demand functions, qi= D¡ (p; , p¡ )= SG+P), derive the residual inverse demandfunctions: p; = P; (qi , p¡) (work out: P; (qi , P;)). Show that for t > 0, P;(qi , P;) is downward2taPi (qi ,Pj) .sloping, i.e.,< 0. Argue that, taking p; 20 as given, firm "i" is like a monopolist facing aresidual inverse demand, and the optimal q; (which equates marginal revenue and marginal cost)or pi makes P; = (qi , Pi) = Pi > c, i.e., firm į has market power.%3D(c) Calculate the limits of the equilibrium prices and profit as t → 0 ? What is P; (qi , p;) as t → 0? Isit downward sloping? Argue that the Bertrand Paradox (i.e., the prediction of the static Bertrandduopoly model, where p *1 =p *2 = c) holds only in the extreme case of t = 0

A duopoly is what is going on where two organizations together own everything, or virtually all, of…

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, 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) = S(;+2P1)and D2 (P1, P2)= S(;+P2), where S > 0 and the parameter t> 0 measures the 2t 2t degree of product differentiation. Both firms have constant marginal cost c> 0 for production. (a) Derive the Nash equilibrium of this game, including the prices, outputs and profit of the two firms. (b) From the demand functions, qq= D¿ (pi , p¡ )= S(÷+Pi), derive the residual inverse demand functions: Pi = P; (qi , p¡) (work out: Pt (q; , P¡)). Show that for t > 0, P¿(q; , P¡) is downward 2t aP: (qt »Pj) . sloping, i.e., < 0. Argue that, taking p; 2 0 as given, firm "i" is like a monopolist facing a 'be residual inverse demand, and the optimal q; (which equates marginal revenue and marginal cost) or pi makes Pį = (qi , Pj) = Pi > c, i.e., firm į has market power.

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, 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) = S(;+2P1)and D2 (P1, P2)= S(;+P2), where S > 0 and the parameter t> 0 measures the 2t 2t degree of product differentiation. Both firms have constant marginal cost c> 0 for production. (a) Derive the Nash equilibrium of this game, including the prices, outputs and profit of the two firms. (b) From the demand functions, qq= D¿ (pi , p¡ )= S(÷+Pi), derive the residual inverse demand functions: Pi = P; (qi , p¡) (work out: Pt (q; , P¡)). Show that for t > 0, P¿(q; , P¡) is downward 2t aP: (qt »Pj) . sloping, i.e., < 0. Argue that, taking p; 2 0 as given, firm "i" is like a monopolist facing a 'be residual inverse demand, and the optimal q; (which equates marginal revenue and marginal cost) or pi makes Pį = (qi , Pj) = Pi > c, i.e., firm į has market power.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter15: Imperfect Competition

Section: Chapter Questions

Problem 15.5P

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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) = ; + 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 (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, Pi) (work out P:(qi, Pi)). Show that for t > 0, P:(q;, P;) is downward-sloping, aP:(gi-Pj) + 2t i.e., 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¡) =…
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) = + P2P1 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 2t S (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 out P;(qi, P;)). Show that for t > 0, P;(4;, P;) is downward-sloping, aP:(gi-Pj) + 2t i.e., 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…
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 pi and p2 simultaneously. The derived demand functions for the two firms are: D1 (p1, P2) and D2 (p1, P2) = + 2, where S > 0 and the parameter t > 0 을 + S + P2-P1 2t 2t measures the degree of product differentiation. Both firms have constant marginal cost c > 0 for production. (a) Derive the Nash equilibrium of this game, including the prices, outputs and profits of the two firms.
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