3. 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) 2 2t = 2t and D2 (P1, P2) = + P₁-P2, where S > 0 and the parameter t > 0 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. S (b) From the demand functions, qi = Di (Pi, Pj) = {{+ the residual inverse demand functions: pi Pi(qi,Pj) Əqi = Pj-Pi derive ' 2t Pi(qi, Pj) (work out Pi(qi, Pj)). Show that for t > 0, Pi(qi, Pj) is downward-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 pį 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,Pj) as t → 0? Is it downward sloping? Ar- gue 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.
3. 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) 2 2t = 2t and D2 (P1, P2) = + P₁-P2, where S > 0 and the parameter t > 0 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. S (b) From the demand functions, qi = Di (Pi, Pj) = {{+ the residual inverse demand functions: pi Pi(qi,Pj) Əqi = Pj-Pi derive ' 2t Pi(qi, Pj) (work out Pi(qi, Pj)). Show that for t > 0, Pi(qi, Pj) is downward-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 pį 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,Pj) as t → 0? Is it downward sloping? Ar- gue 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.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:3. 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) = ;+"", where S > 0 and the parameter t > 0
measures the degree of product differentiation. Both firms have
constant marginal cost c > 0 for production.
P2=P1
2t
S
(a) Derive the Nash equilibrium of this game, including the prices,
outputs and profits of the two firms.
S
Pj¬Pi derive
(b) From the demand functions, q¡ = D¡ (pi, Pj)
the residual inverse demand functions: p; = P;(qi, Pj) (work out
P:(qi, p;)). Show that for t > 0, P;(qi, P;) is downward-sloping,
OP:(4i-Pj)
2t
i.e.,
< 0. Argue that, taking p; 2 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 p; makes P;(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, Pj) as t
gue that the Bertrand Paradox (i.e., the prediction of the static
Bertrand duopoly model, where pi = p = c) holds only in the
0? Is it downward sloping? Ar-
extreme case of t = 0.
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