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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.

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
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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 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 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, qi= D¡ (p; , p¡ )= SG+P), derive the residual inverse demand functions: p; = P; (qi , p¡) (work out: P; (qi , P;)). Show that for t > 0, P;(qi , P;) is downward 2t aPi (qi ,Pj) . sloping, i.e., < 0. Argue that, taking p; 20 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 , 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? Is it downward sloping? Argue that the Bertrand Paradox (i.e., the prediction of the static Bertrand duopoly model, where p *1 =p *2 = c) holds only in the extreme case of t = 0
Transcribed Image Text: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 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 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, qi= D¡ (p; , p¡ )= SG+P), derive the residual inverse demand functions: p; = P; (qi , p¡) (work out: P; (qi , P;)). Show that for t > 0, P;(qi , P;) is downward 2t aPi (qi ,Pj) . sloping, i.e., < 0. Argue that, taking p; 20 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 , 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? Is it downward sloping? Argue that the Bertrand Paradox (i.e., the prediction of the static Bertrand duopoly model, where p *1 =p *2 = c) holds only in the extreme case of t = 0
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Why is (P2-P1/2T) and (P1-P2/2T) not multiplied by S for the q1 and q2 equations in the first line?

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Follow-up Question

I cannot figure out how the equation goes from q1 = D1 (p1; p2) = S(1/2 + (p2-p1/2t)) to q1 = S/2 +(p2-p1/2t)

why is S not in the numerator of both terms? i.e. q1 = S/2 + S(p2-p1/2t)

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Follow-up Question
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 p2₂ simultaneously. The derived demand functions for the two firms are: D₁ (P₁, P2) = S(+ P2²-P₁) and D₂ (P₁, P2)= S(+²₁=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. 2t (b) From the demand functions, qi= Di (Pi, Pj )= S(+ PP¹), derive the residual inverse demand functions: P₁ = Pi (qi, Pj) (work out: Pi (qi, Pj)). Show that for t > 0, Pi(qi, Pj) is downward OPi (qi Pj) sloping, 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 P₁ = (qi, Pj) = pi > c, i.e., firm i has market power. əqi O ⠀ (c) Calculate the limits of the equilibrium prices and profit as t → 0 ? What is Pi (qi, Pj) as t → 0? Is it downward sloping? Argue that the Bertrand Paradox (i.e., the prediction of the static Bertrand duopoly model, where p *₁ =p *2 = c) holds only in the extreme case of t = 0
Transcribed Image Text: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 p2₂ simultaneously. The derived demand functions for the two firms are: D₁ (P₁, P2) = S(+ P2²-P₁) and D₂ (P₁, P2)= S(+²₁=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. 2t (b) From the demand functions, qi= Di (Pi, Pj )= S(+ PP¹), derive the residual inverse demand functions: P₁ = Pi (qi, Pj) (work out: Pi (qi, Pj)). Show that for t > 0, Pi(qi, Pj) is downward OPi (qi Pj) sloping, 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 P₁ = (qi, Pj) = pi > c, i.e., firm i has market power. əqi O ⠀ (c) Calculate the limits of the equilibrium prices and profit as t → 0 ? What is Pi (qi, Pj) as t → 0? Is it downward sloping? Argue that the Bertrand Paradox (i.e., the prediction of the static Bertrand duopoly model, where p *₁ =p *2 = c) holds only in the extreme case of t = 0
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