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. 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; (pi, p;) = § + , derive the residual inverse demand functions: p; = P;(qi, P;) (work out P;(qi, P;)). Show that for t > 0, P;(qi, P3) is downward-sloping, OP (41-P;) < 0. Argue that, taking p; 2 0 as given, firm i i.e., is like a monopolist facing a residual inverse demand, and the optimal q: (which equates marginal revenue and marginal cost) or p; makes P:(q;, P;) = På > 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 static Bertrand duopoly model, where p¡ = p = c) holds only in the extreme case of t = 0.
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. 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; (pi, p;) = § + , derive the residual inverse demand functions: p; = P;(qi, P;) (work out P;(qi, P;)). Show that for t > 0, P;(qi, P3) is downward-sloping, OP (41-P;) < 0. Argue that, taking p; 2 0 as given, firm i i.e., is like a monopolist facing a residual inverse demand, and the optimal q: (which equates marginal revenue and marginal cost) or p; makes P:(q;, P;) = På > 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 static Bertrand duopoly model, where p¡ = p = c) holds only in the extreme case of t = 0.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Hello,
I had a question on this question, can you pls let me know if my answers are wrong or right?
a) I had equilibrium
p1 = p2 = st + c
q1 = q2 = s/2
Profit = s2t/2
b) I managed to show the dpi/dqi <0. I got Pi = 1/2(st + Pj +c) . Is the Pi larger than c?
I need help on b, pls..
Transcribed Image Text: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.
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; (pi, P;) = § +", derive
the residual inverse demand functions: Pi = P:(q;, P3) (work out
P:(qi, P3)). Show that for t > 0, P:(4i, P;) is downward-sloping,
aP,(q;-P;)
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:(q;is P3) = 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:(q;, 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 pj = p; = c) holds only in the
extreme case of t = 0.
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