# ECON 2005 Nash Equilibrium Questionnaire

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Econ2005 Assignment 2
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This assessed coursework accounts for 5% of the total mark of this module.
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1. Consider a duopoly market, where two rms sell di¤erentiated products,
which are imperfect substitutes. The market can be modelled as a static
price competition game. The two rms choose prices p1 and p2 simultaneously. The demand functions for the two rms are: D1 (p1 ; p2 ) = S2 + p22tp1
and D2 (p1 ; p2 ) = S2 + p12tp2 , where S > 0, and the parameter t > 0
measures the degree of product di¤erentiation. Both rms have constant
marginal cost c > 0 of production.
(a) Derive the Nash equilibrium of this game, including the prices, outputs and prots of the two rms.
p
p
(b) From the demand functions, qi = Di (pi ; pj ) = S2 + j2t i , derive
the residual inverse demand functions: pi = Pi (qi ; pj ) (work out
Pi (qi ; pj )). Show that for t > 0, Pi (qi ; pj ) is downward-sloping,
@Pi (qi ;pj )
i.e.,
< 0. Argue that, taking pj 0 as given, rm i is @qi like a monopolist facing a residual inverse demand, and the optimal qi (which equates marginal revenue and marginal cost) or pi makes Pi (qi ; pj ) = pi > c, i.e., rm i has market power.
(c) Calculate the limits of the equilibrium prices and prots 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 p1 = p2 = c) holds only in the extreme case of t = 0.
2. Consider the static game represented by the following payo¤ matrix:
c
r
C 2; 2 0; 3
R 3; 0 1; 1
(a) Find the Nash equilibrium of the above game.
(b) If above static game is repeated T < 1 times without discounting of payo¤s, what will be its subgame perfect Nash equilibrium? (c) If the above static game is repeated an innite number of times with payo¤ discount factor being = 1+1 , what will be its subgame perfect Nash equilibria? Interpret the economic meaning of parameter . Explain the intuition how parameter can a¤ect the equilibrium outcome of the game. 1 Purchase answer to see full attachment Tags: nash Equilibrium discounting Static Game User generated content is uploaded by users for the purposes of learning and should be used following FENTYESSAYS.COM ESSAY's honor code & terms of service.