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Econ2005 Assignment 2

Answer the following questions.

Word-process your coursework to generate a pdf document for submission.

Submit the document electronically to the Blackboard as instructed before the

deadline.

This assessed coursework accounts for 5% of the total mark of this module.

Warning: This assignment is individual work. The TurnItIn similarity score

will be used to ensure Academic Integrity.

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
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nash Equilibrium
discounting
Static Game
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