Lecture 1: Introduction

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LECTURE 6: OLIGOPOLY Jan Zouhar

Games and Decisions

Market Structures 2 

the list of basic market structure types (seller-side types only): Number of sellers

Seller entry barriers

Deadweight loss

Perfect competition

Many

No

None

Monopolistic competition

Many

No

None

Oligopoly

Few

Yes

Medium

Monopoly

One

Yes

High

Games and Decisions

Jan Zouhar

Collusive vs. Non-Collusive Oligopolies 3 

note: oligopoly differs from monopoly (allocation-wise) only if there’s no collusion 

 



collusion: a largely illegal form of cooperation amongst the sellers that includes price fixing, market division, total industry output control, profit division, etc. controlled by competition/anti-trust laws well-known collusion cases: OPEC, telecommunication, drugs, sports, chip dumping (poker)

game-theoretical models: 



cooperative setting (collusive oligopoly) → coalition theory  games in the characteristic-function form non-cooperative setting (competitive, non-collusive oligopoly) → normal form game analysis  NE’s etc.; however, matrices can’t typically be used for payoffs

Games and Decisions

Jan Zouhar

Oligopoly – Model Specification 4 

to make the analysis simple, we’ll make several assumptions:

1. single-product model: oligopolists produce a single type of homogenous

product 2. one strategic variable: firms decide about prices or output levels 3. static model: single-period analysis only 

in dynamic models, there are more diverse strategic options: elimination of competitors even with contemporary losses etc.

4. single objective: all firms maximize their individual profit

Three basic non-cooperative oligopoly models: 

Bertrand oligopoly – firms simultaneously choose prices



Cournot oligopoly – firms simultaneously choose quantities



Stackelberg oligopoly – firms choose quantities sequentially 

note: sequential-move games are typically not modelled as normal-form games. Instead, we use the extensive-form approach (not this lecture).

Games and Decisions

Jan Zouhar

Bertrand Duopoly 5 





Bertrand duopoly (2 oligopolists only) – model notation: 

market demand function:

q = D(p)



prices charged by the players:

p1, p2



resulting quantities:

q1, q2



unit costs:

c1, c2

(for simplicity: AC = MC = c)

homogenous product → lower price attracts all the consumers 

p1 < p2 →

q1 = D(p1),



p1 > p2 → q1 = 0,



p1 = p2 →

q2 = 0

q2 = D(p2)

equal market share, q1 = q2 = ½ D(p1) = ½ D(p2)

as long as the prices are higher than c1 and c2, both oligopolists tend to push prices down (below the other player’s price) 



imagine the prices are equal and above c1; by lowering the price just slightly, player 1 can gain the whole market (if p2 stays the same) best response of player 1 to p2 is to choose p1 = p2 – ε (“just below” p2) (until the prices reach c1 → player 1 suffers a loss below)

Games and Decisions

Jan Zouhar

Bertrand Duopoly

(cont’d)

6 



NE

depends on the MC of the players:



c1 = c2 → p1* = p2* = c1 = c2



c1 < c2 → p1* = c2 – ε

→ player 1 wins all, positive profit



c1 > c2 → p2* = c1 – ε

→ player 2 wins all, positive profit

→ zero economic profit for both

more precisely: graphical best-response analysis – reaction curves: p1(p2)

p2 monopoly price (no player 1)

45°

p2(p1)

p2M NE

p2 just below p1 (curve just below 45°)

c2

c1 Games and Decisions

p1M

p1 Jan Zouhar

Bertrand Duopoly

(cont’d)

7

→ price competition leads to fairly efficient allocation Critique of the Bertrand model (or, when Bertrand model fails to work) 

capacity constraints of production 



e.g., consider the c1 < c2 situation: if player 1 can’t supply enough for the whole market, player 2 can still charge p2 above c2 and attract some customers (and achieve a positive profit) if c1 = c2 and neither player can supply to all customers, either player can raise the output price above c



lack of product homogeneity (homogeneity disputable in most cases)



transaction/transportation costs: 

may differ for the specific customer–firm interactions  e.g., shops at both ends of a street: people tend to pick the closer one  if transportation costs are accounted for, the consumer expenditures vary even if prices are equal

Games and Decisions

Jan Zouhar

Cournot Oligopoly – Formal Treatment 8 

model type – normal-form game with the following elements:  







list of firms:

1,2,…,N

strategy spaces:  potential quantities: typically intervals like [0,1000] → infinite alternatives!  the output level produced by ith player (the strategy adopted) is denoted xi  a strategy profile is an N-tuple: (where xi ∈ Xi )

X1, X2,…,XN

cost functions:  total cost as the function of output level

c1(x1), c2(x2),…, cN(xN)

(x1,x2,…,xN)

price function (or inverse demand function): p = f(x1 + x2 + … + xN)  i.e., market price is the function of total industry output

profit of ith firm:

Games and Decisions

i ( x1 ,..., x N )  TRi  TCi  xi  f ( x1  ...  x N )  ci ( xi ) Jan Zouhar

Nash Equilibrium in Cournot Oligopoly 9 

mathematical definition: A strategy profile (x1*, x2*,…, xN*) is a NE if for all i = 1,…,N

i ( x1*, x2*,..., xi ,..., x N *)  i ( x1*, x2*,..., xi*,..., x N *) holds for all xi ∈ Xi . 

finding the NE: best-response approach (again)  NE: 

the strategies have to be the best responses to one another

best-response functions: 

for player 1: r1(x2,…,xN) is the best-response x1 chosen by player 1, given that the other player’s quantities are x2,…,xN



mathematically: r1 ( x2 ,...x N )  arg max  i ( x1 , x2 ,..., x N ) x1 X1

 NE:

xi*  ri ( x1*,..., xi 1*, xi 1*,..., x N *) for i = 1,…,N

Games and Decisions

Jan Zouhar

Example 1: Cournot Duopoly 10 

price function: p  f ( x1  x2 )  100  ( x1  x2 )



other characteristics: X1  0,   X 2  0,  



c1 ( x1 )  150  12x1 c2 ( x2 )  x22

profit functions: 1 ( x1 , x2 )  x1  f ( x1  x2 )  c1 ( x1 )   x1  100  ( x1  x2 )   150  12x1  

 100x1  x12  x1x2  150  12x1   88x1  x12  x1x2  150

 2 ( x1 , x2 )  

 100x2  2x22  x1x2

best response of player 1 to x2: profit-maximizing (π1-maximizing) value of x1 for the given x2

Games and Decisions

Jan Zouhar

Example 1: Cournot Duopoly

(cont’d)

11 

profit of player 1 for three different levels of x2: 1500 1000 500

1

best response to x2 = 10 is x1 ≈ 40, best response to x2 = 25 is x1 ≈ 33, best response to x2 = 50 is x1 ≈ 20,

0

-500 -1000 -1500

x 2 = 10 x 2 = 25

-2000

x 2 = 50 -2500

0

20

40

60

80

x1 

best response for an arbitrary level of x2: as the function π1(x1,x2) is strictly concave in x1 for any x2, we can use the first-order condition for a local extreme ! 1 ( x1 , x2 )  88  2x1  x2  0 x1

Games and Decisions

Jan Zouhar

Note: first order conditions are generally not sufficient for a maximum, only necessary conditions for extreme (but: concave function → global maximum)

f ( x ) 0 x 3 2.5 2 1.5

f(x)

1 0.5 0 -0.5 -1 -1.5

0

1

2

3

4

5

6

7

8

9

10

x Games and Decisions

Jan Zouhar

Example 1: Cournot Duopoly

(cont’d)

13 

we can also write the result in terms of the reaction function r1: ! 1 ( x1 , x2 )  88  2x1  x2  0 x1



x

x1  r1 ( x2 )  44  2 2

similarly, for player 2, we obtain: !  2 ( x1 , x2 )  100  x1  4x2  0 x2







x

x2  r2 ( x1 )  25  1 4

altogether, we have 2 linear equations; for NE strategies, both have to hold at the same time → in order to find the NE, we just need to solve 88  2x1*  x2*  0 100  x1*  4x2*  0

or

x1*  r1( x2*) x2*  r 2( x1*)



x1*  36 x2*  16

Question: What are the equilibrium profits and price? Games and Decisions

Jan Zouhar

Collusive Oligopoly 14 





model framework: as in case of Cournot oligopoly, only that players can form coalitions coalition: a group of firms that coordinate output levels and redistribute profits grand coalition: the coalition of all oligopolists, Q = {1,2,…,N }  

other coalitions are denoted by K,L,… a “single-firm coalition” is still called a coalition, e.g. K = {2}, and so is the “empty coalition” {∅}

Question: How many different coalitions can be formed with N firms? 

characteristic function (of the oligopoly): a function v(K) that assigns to any coalition K the maximum attainable total profit of K 

payoff function: single player, individual payoff, for a given strategy profile



characteristic function: coalition, sum of members’ profits, max. attainable

Games and Decisions

Jan Zouhar

Collusive Oligopoly

(cont’d)

15 

characteristic function for grand coalition:

v(Q )  

max

N

  i ( x1 ,..., x N )

( x1 ,...,x N ) i 1

characteristic function for other coalitions: profit of coalition members depends on the quantity chosen by non-members

→ what will the other players do? (Generally, difficult to say.) 1. minimax characteristic function: assume non-members supply as

much as they can (up to their capacity constraints) 2. equilibrium characteristic function: assume the other players

choose the NE quantities 

characteristic function for an arbitrary coalition:

v( K )  max

( xi )iK

Games and Decisions

iK  i ( x1 ,..., x N ) Jan Zouhar

Example 2: Collusive Duopoly 16 

consider the same duopoly as in example 1, only with capacity constraints: p  f ( x1  x2 )  100  ( x1  x2 )  price function: 



other characteristics:

profit functions:

X1  0,40 

c1 ( x1 )  150  12x1

X 2  0,20 

c2 ( x2 )  x22

1 ( x1 , x2 )  88x1  x12  x1x2  150  2 ( x1 , x2 )  100x2  2x22  x1x2



first, we’ll find the equilibrium characteristic function:

1*  1146  2*  512



we already know the NE values:



immediately, we have: v(1)  max 1 ( x1 ,16)  1 (36,16)  1146

x1*  36 x2*  16

x1 X1

v(2)  max  2 (36, x2 )   2 (36,16)  512 Games and Decisions

x2 X 2

Jan Zouhar

Example 2: Collusive Duopoly

(cont’d)

17



for grand coalition Q = {1,2}, we obtain: v(1,2)  max 1 ( x1 , x2 )   2 ( x1 , x2 )  ( x1 ,x2 )

 max 88x1  100x2  x12  2x22  2x1x2  150 ( x1 ,x2 )



function of two variables now, but still concave (see next slide)

→ first-order conditions (both partial derivatives equal zero)

1,2 ( x1 , x2 ) x1 1,2 ( x1 , x2 ) x2 

!

 88  2x1  2x2  0 !

 100  2x1  4 x2  0



x1opt  38  opt  1,2  1822 opt x2  6 

v(Q) = v(1,2) = 1822

Games and Decisions

Jan Zouhar

Example 2: Collusive Duopoly

(cont’d)

100 4

x 10

80 2 60 0

πy1,2

40

-2

20

x2

-4

0 -20

-6 -40 -8 100

-60 50

100 50

0

x2

0

-50

-50 -100

Games and Decisions

-100

x1

-80 -100 -100

-50

0

x1

50

Jan Zouhar

100

first-order conditions: necessary conditions for local extremes (not sufficient, not for maxima only!)

0.5

y

0

-0.5 2 1 0

x2

Games and Decisions

-1 -2

-2

-1.5

-1

-0.5

0.5

0

1

1.5

2

x1 Jan Zouhar

Example 2: Collusive Duopoly

(cont’d)

20 

the complete equilibrium characteristic function is as follows: v( )  0 v(1)  1146 v(2)  512 v(1,2)  1822



minimax characteristic function: 



v(∅) and v(1,2) are the same as in the equilibrium char. function

for v(1) and v(2), we calculate the players’ profits under the condition that the other player produces up to his/her capacity constraint: v(1)  max 1 ( x1 ,20)  max 68x1  x12  150  1 (34,20)  1006 x1 X1

x1 X1

v(2)  max  2 (40, x2 )  max 60x2  2x22   2 (40,15)  450 x2 X 2

Games and Decisions

x2 X 2

Jan Zouhar

Example 2: Collusive Duopoly

(cont’d)

21 



a comparison of the two characteristic functions: equilibrium

minimax

v(∅)

0

0

v(1)

1146

1006

v(2)

512

450

v(1,2)

1822

1822

v(1,2) – v(1) – v(2)

164

366

core of the oligopoly: a division of payoffs a1,a2 such that a1  a2  1822, a1  1146, a2  512,

Games and Decisions

or

a1  a2  1822, a1  1006, a2  450. Jan Zouhar

LECTURE 6: OLIGOPOLY Jan Zouhar

Games and Decisions

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Lecture 1: Introduction

LECTURE 6: OLIGOPOLY Jan Zouhar Games and Decisions Market Structures 2  the list of basic market structure types (seller-side types only): Numbe...

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