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10

C

h

a

p Nash equilibrium

e

r • What if there are no dominated or dominant strategies?

9 • Then we need to use the Nash equilibrium concept.

S • Change the airline game to a pricing game:

– 60 potential passengers with a reservation price of $500

a – 120 additional passengers with a reservation price of $220

i

c – price discrimination is not possible (perhaps for regulatory

G reasons or because the airlines don’t know the passenger types)

a – costs are $200 per passenger no matter when the plane leaves

m – airlines must choose between a price of $500 and a price of

e $220

a – if equal prices are charged the passengers are evenly shared

n – the low-price airline gets all the passengers

d

C • The pay-off matrix is now:

o

u

r

11

C

h

a

p The example

If Delta prices high

e The Pay-Off Matrix

If both price high

r and American low

9 then both get 30 then American gets

passengers.

If Delta prices

Profit

low

all 180 passengers.

American

S and

per American

passenger

high

is If both price low

Profit per passenger

a then $300

Delta gets they each get 90

is $20

i PH = $500 PL = $220

all 180 passengers. passengers.

c Profit per passenger

Profit per passenger

G

a is $20 is $20

PH = $500 ($9000,$9000) ($0, $3600)

m

e Delta

a PL = $220 ($3600, $0) ($1800, $1800)

n

d

C

o

u

r

12

C

h

a (PH, PH) is a Nash

p Nash equilibrium

e (PH, PL) cannot be

There is no simple

equilibrium. The Pay-Off Matrix

There are two Nash

r (PL, PL) is a Nash

a Nash equilibrium.

way to choose between

If both are pricing

equilibria to this version

9 equilibrium.

If American prices

Custom and familiarity these equilibria

high then neither wants

of the game American

If both are pricing

S low then Delta should

might lead both to

to change

“Regret” might

(PL, PH) cannot be low then neither wants

also price low

price high

a cause both to

a Nash equilibrium. to change

i PH = $500 PL = $220

price low

If American prices

c

G high then Delta should

a also price high

PH = $500 ($9000,$9000) ($0, $3600)

($9000, $9000)

m

e Delta

a PL = $220 ($3600, $0) ($1800, $1800)

n

d

C

o

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13

C

h

a

p Oligopoly models

e

r • There are three dominant oligopoly models

9 – Cournot

S – Bertrand

– Stackelberg

a

i • They are distinguished by

c – the decision variable that firms choose

G – the timing of the underlying game

a

m • Concentrate on the Cournot model in this section

e

a

n

d

C

o

u

r

14

C

h

a

p The Cournot model

e

r • Start with a duopoly

9 • Two firms making an identical product (Cournot

S supposed this was spring water)

• Demand for this product is

a

i P = A - BQ = A - B(q1 + q2)

c

G where q1 is output of firm 1 and q2 is output of firm 2

a

m • Marginal cost for each firm is constant at c per unit

e • To get the demand curve for one of the firms we treat

the output of the other firm as constant

a

n • So for firm 2, demand is P = (A - Bq1) - Bq2

d

C

o

u

r

15

C

h

a

p The Cournot model 2 If the output of

e $ firm 1 is increased

r P = (A - Bq1) - Bq2

9 the demand curve

The profit-maximizing for firm 2 moves

A -

S choice of output by Bq1 to the left

firm 2 depends upon

a

i the output of firm 1 A - Bq’1

c Marginal revenue for

G Demand

Solve this

firm 2 is

a c MC

m for output MR2

MR2 = (A - Bq1) - 2Bq2

e q2 q*2

MR2 = MC

a Quantity

n ∴

d A - Bq1 - 2Bq2 = q*2 = (A - c)/2B - q1/2

C c

o

u

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16

C

h

a

p The Cournot model 3

e

r q*2 = (A - c)/2B - q1/2

9 This is the reaction function for firm 2

S It gives firm 2’s profit-maximizing choice of output

for any choice of output by firm 1

a

i There is also a reaction function for firm 1

c

G By exactly the same argument it can be written:

a

m q*1 = (A - c)/2B - q2/2

e Cournot-Nash equilibrium requires that both firms be on

a their reaction functions.

n

d

C

o

u

r

17

C

h

a

p Cournot-Nash equilibrium

e q2

r If firm 2 produces The reaction function

The Cournot-Nash

9 (A-c)/B then firm for firm 1 is

(A-c)/B equilibrium is at

1 will choose to q*1 = (A-c)/2B - q2/2

S the intersection

Firm 1’s reaction function

produce no output

of the reaction

If firm 2 produces

a The reaction function

i functions

nothing then firm

c for firm 2 is

(A-c)/2B 1 will produce the

G q*2 = (A-c)/2B - q1/2

C monopoly output

a qC

m (A-c)/2B

2

e Firm 2’s reaction function

q1

a (A-c)/2B (A-c)/B

qC

n 1

d

C

o

u

r

18

C

h

a

p Cournot-Nash equilibrium 2

e

r q*1 = (A - c)/2B -

q2

9 q*2/2

q*2 = (A - c)/2B -

S (A-c)/B q*1/2

∴ q*2 = (A - c)/2B - (A - c)/4B

a Firm 1’s reaction function + q*2/4

i

c ∴ 3q*2/4 = (A - c)/4B

G (A-c)/2B ∴ q*2 = (A - c)/3B

a C

m (A-c)/3B

e ∴ q*1 = (A - c)/3B

Firm 2’s reaction function

a q1

n (A-c)/2B (A-c)/B

d (A-c)/3B

C

o

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19

C

h

a

p Cournot-Nash equilibrium 3

e

r • In equilibrium each firm produces qC1 = qC2 = (A -

9 c)/3B

S • Total output is, therefore, Q* = 2(A - c)/3B

• Recall that demand is P = A - BQ

a • So the equilibrium price is P* = A - 2(A - c)/3 = (A + 2c)/3

i

c • Profit of firm 1 is (P* - c)qC1 = (A - c)2/9

G • Profit of firm 2 is the same

a

m • A monopolist would produce QM = (A - c)/2B

e • Competition between the firms causes them to

overproduce. Price is lower than the monopoly price

a

n • But output is less than the competitive output (A - c)/B

d where price equals marginal cost

C

o

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r

20

C

h

a

p Cournot-Nash equilibrium: many firms

e

r • What if there are more than two firms?

9 • Much the same approach.

S • Say that there are N identical firms producing identical

a products

i This denotes output

• Total output Q = q1 + q2 + … + qN

c of every firm other

G • Demand is P = A - BQ = A - B(q1 + q2 + … + qN)

a than firm 1

• Consider firm 1. It’s demand curve can be written:

m

e P = A - B(q2 + … + qN) - Bq1

• Use a simplifying notation: Q-1 = q2 + q3 + … + qN

a

n • So demand for firm 1 is P = (A - BQ-1) - Bq1

d

C

o

u

r

21

C

h

a If the output of

p The Cournot model: many firms 2

e the other firms

$

P = (A - BQ-1) - Bq1

r is increased

9 The profit-maximizing the demand curve

choice of output by firm A - BQ-1

S for firm 1 moves

1 depends upon the to the left

a output of the other firms

i A - BQ’-1

c Marginal revenue for

G Solve this

firm 1 is Demand

a c MC

for output

m MR

MR1 = (A - BQ-1) - 2Bq1

q1

e 1

q*1

MR1 = MC

a Quantity

n ∴

A - BQ-1 - 2Bq1 = c q*1 = (A - c)/2B - Q-1/2

d

C

o

u

r


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PESO

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AUTORE

Atreyu

PUBBLICATO

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DESCRIZIONE DISPENSA

Materiale didattico per il corso di Impresa, management e mercati della prof.ssa Ornella Tarola. Trattasi di slides aventi ad oggetto il tema dei regimi di monopolio ed oligopolio. In particolare, sono affrontati i seguenti argomenti: la teoria dei giochi, l'equilibrio di Nash, modelli di oligopolio, il modello di Cournot, l'equilibrio di Cournot - Nash, il rapporto tra concentrazione e profitto.


DETTAGLI
Corso di laurea: Corso di laurea magistrale in scienze delle pubbliche amministrazioni
SSD:
A.A.: 2011-2012

I contenuti di questa pagina costituiscono rielaborazioni personali del Publisher Atreyu di informazioni apprese con la frequenza delle lezioni di Impresa management e mercati e studio autonomo di eventuali libri di riferimento in preparazione dell'esame finale o della tesi. Non devono intendersi come materiale ufficiale dell'università La Sapienza - Uniroma1 o del prof Tarola Ornella.

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