Oligopoly: Cournot, Bertrand, Stackelberg - Yale CampusPress [PDF]

Oligopoly: Cournot, Bertrand, Stackelberg

Oligopoly market power many …rms: i = 1; 2; ::; I with cost function ci (qi ) ; ci0 (qi )

0; ci00 (qi )

0

each …rm i produces quantity qi aggregate quantity is q=

I X

qi ;

q

i =1

i

=

X

qi

j 6=i

inverse demand associates price to quantity o¤ered: p( ) = q

1

( );

p 0 (q) < 0

Oligopolist’s Choice

Antoine Augustin Cournot (28/8/1801 – 31/3/1877): the supply of spring water teacher of August Walras, father of Leon Walras, …rst to draw supply and demand curves, 30 years before Marshall the problem of the oligpolist is to choose quantity qi : max fp (qi ; q i ) q qi

c (qi )g

solving the oligpolist’s problem p 0 (qi ; q i ) qi + p (qi ; q i )

ci0 (qi ) = 0

oligopolist is more aggressive than monopolist

Linear Demand and Constant Marginal Cost consider inverse demand p (q) = a

bq; a; b > 0

consider cost function (constant marginal cost) ci (qi ) = c qi ; c > 0 the problem for the oligopolist is 80 1 < X max @a b qj A qi q : j

cqi

9 = ;

the …rst order condition lead to the best response: 0 1 P X a b j 6=i qj @a b A qj 2bqi c = 0 , qi = 2b j 6=i

c

we have the best response: 1 0 X a @a b qj 2bqi A c = 0 , qi = j 6=i

b

P

j 6=i

qj

c

2b

or

qi = BRi (q i ) the action are strategic substitutes dBR (qj ) dqi = = dqj dqj

1 ; for all I > 1. I +1 2

a c b(I + 1)

Stackelberg Game

…rm a is the …rst mover, …rm b is the second mover leadership advantage value of commitment

Bertrand Pricing

…rms compete in prices rather than quantities …rm with lowest price receives the entire demand demand function is q (min fp1 ; :::; pI g) all …rms have same marginal cost: c (qi ) = cqi

Di¤erentiated Products

consider q i = ai with b 2 (0; 1)

pi + bpj

a higher price of a competitor leads to an increase of price j strategic complements

Hotelling Pricing Game

linear city horizontally di¤erentiated products marginal cost equal to zero