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Market Structures
Jan Zouhar Games and Decisions
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
Collusive vs. Non-Collusive Oligopolies
Jan Zouhar Games and Decisions
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
Oligopoly – Model Specification
Jan Zouhar Games and Decisions
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).
Bertrand Duopoly
Jan Zouhar Games and Decisions
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), q2 = 0
p1 > p2 → q1 = 0, q2 = D(p2)
p1 = 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)
Bertrand Duopoly (cont’d)
Jan Zouhar Games and Decisions
6
NE depends on the MC of the players:
c1 = c2 → p1* = p2* = c1 = c2 → zero economic profit for both
c1 < c2 → p1* = c2 – ε → player 1 wins all, positive profit
c1 > c2 → p2* = c1 – ε → player 2 wins all, positive profit
more precisely: graphical best-response analysis – reaction curves:
p1
p2
c1
c2
45°
p2(p1)
p1(p2)
monopoly price
(no player 1) 2Mp
1Mp
p2 just below p1
(curve just below 45°)
NE
Bertrand Duopoly (cont’d)
Jan Zouhar Games and Decisions
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
Cournot Oligopoly – Formal Treatment
Jan Zouhar Games and Decisions
8
model type – normal-form game with the following elements:
list of firms: 1,2,…,N
strategy spaces: X1, X2,…,XN
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: (x1,x2,…,xN)
(where xi ∈ Xi )
cost functions: c1(x1), c2(x2),…, cN(xN)
total cost as the function of output level
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: 1 1( ,..., ) ( ... ) ( )i N i i i N i ix x TR TC x f x x c x
Nash Equilibrium in Cournot Oligopoly
Jan Zouhar Games and Decisions
9
mathematical definition:
A strategy profile (x1*, x2*,…, xN*) is a NE if for all i = 1,…,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:
NE: for i = 1,…,N
1 2 1 2( *, *,..., ,..., *) ( *, *,..., *,..., *)i i N i i Nx x x x x x x x
1 1
1 2 1 2( ,... ) argmax ( , ,..., )N i Nx X
r x x x x x
1 1 1* ( *,..., *, *,..., *)i i i i Nx r x x x x
Example 1: Cournot Duopoly
Jan Zouhar Games and Decisions
10
price function:
other characteristics:
profit functions:
best response of player 1 to x2: profit-maximizing (π1-maximizing) value
of x1 for the given x2
1 2 1 2( ) 100 ( )p f x x x x
1 1 2 1 1 2 1 1
1 1 2 1
21 1 1 2
21 1 1 2
22 1 2 2 2 1 2
( , ) ( ) ( )
100 ( ) 150 12
100 150 12 1
88 150
( , ) 100 2
x x x f x x c x
x x x x
x x x x x
x x x x
x x x x x x
1 1 1 1
22 2 2 2
0, ( ) 150 12
0, ( )
X c x x
X c x x
0 20 40 60 80-2500
-2000
-1500
-1000
-500
0
500
1000
1500
x1
1
x2 = 10
x2 = 25
x2 = 50
profit of player 1 for three different levels of x2:
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
Example 1: Cournot Duopoly (cont’d)
Jan Zouhar Games and Decisions
11
!1 1 2
1 21
( , )88 2 0
x xx x
x
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 1 2 3 4 5 6 7 8 9 10-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Jan Zouhar Games and Decisions
Note: first order conditions are generally not sufficient for a maximum, only
necessary conditions for extreme (but: concave function → global maximum)
f(x)
x
( )0
f x
x
Example 1: Cournot Duopoly (cont’d)
Jan Zouhar Games and Decisions
13
we can also write the result in terms of the reaction function r1:
similarly, for player 2, we obtain:
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
1 2 1 1 2 1
1 2 2 2 1 2
88 2 * * 0 * ( *) * 36or
100 * 4 * 0 * ( *) * 16
x x x r x x
x x x r x x
1!
2 1 21 2 2 2 1 4
2
( , )100 4 0 ( ) 25
xx xx x x r x
x
2!
1 1 21 2 1 1 2 2
1
( , )88 2 0 ( ) 44
xx xx x x r x
x
Question:
What are the equilibrium profits and price?
Collusive Oligopoly
Jan Zouhar Games and Decisions
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” {∅}
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
Question:
How many different coalitions can be formed with N firms?
Collusive Oligopoly (cont’d)
Jan Zouhar Games and Decisions
15
characteristic function for grand coalition:
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:
1
1( ,..., ) 1
( ) max ( ,..., )N
N
i Nx x i
v Q x x
1( )
( ) max ( ,..., )i i K
i Ni Kx
v K x x
Example 2: Collusive Duopoly
Jan Zouhar Games and Decisions
16
consider the same duopoly as in example 1, only with capacity
constraints:
price function:
other characteristics:
profit functions:
first, we’ll find the equilibrium characteristic function:
we already know the NE values:
immediately, we have:
1 2 1 2( ) 100 ( )p f x x x x
1 1 1 1
22 2 2 2
0,40 ( ) 150 12
0,20 ( )
X c x x
X c x x
21 1 2 1 1 1 2
22 1 2 2 2 1 2
( , ) 88 150
( , ) 100 2
x x x x x x
x x x x x x
1 1
2 2
* 36 * 1146
* 16 * 512
x
x
1 1
2 2
1 1 1
2 2 2
(1) max ( ,16) (36,16) 1146
(2) max (36, ) (36,16) 512
x X
x X
v x
v x
Example 2: Collusive Duopoly (cont’d)
Jan Zouhar Games and Decisions
17
for grand coalition Q = {1,2}, we obtain:
function of two variables now, but still concave (see next slide)
→ first-order conditions (both partial derivatives equal zero)
v(Q) = v(1,2) = 1822
1 2
1 2
1 1 2 2 1 2( , )
2 21 2 1 2 1 2
( , )
(1,2) max ( , ) ( , )
max 88 100 2 2 150
x x
x x
v x x x x
x x x x x x
!1,2 1 2
1 2 opt1 opt1
1,2opt!1,2 1 2 2
1 22
( , )88 2 2 0
381822
( , ) 6100 2 4 0
x xx x
xx
x x xx x
x
Example 2: Collusive Duopoly (cont’d)
Jan Zouhar Games and Decisions
-100
-50
0
50
100
-100
-50
0
50
100
-8
-6
-4
-2
0
2
x 104
x1
x2
y
-100 -50 0 50 100-100
-80
-60
-40
-20
0
20
40
60
80
100
x1
x2
π1,2
Jan Zouhar Games and Decisions
-2-1.5
-1-0.5
00.5
11.5
2
-2
-1
0
1
2
-0.5
0
0.5
x1
x2
y
first-order conditions: necessary conditions for local
extremes (not sufficient, not for maxima only!)
Example 2: Collusive Duopoly (cont’d)
Jan Zouhar Games and Decisions
20
the complete equilibrium characteristic function is as follows:
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:
( ) 0
(1) 1146
(2) 512
(1,2) 1822
v
v
v
v
1 1 1 1
2 2 2 2
21 1 1 1
22 2 2 2 2
(1) max ( ,20) max 68 1 150 (34,20) 1006
(2) max (40, ) max 60 2 (40,15) 450
x X x X
x X x X
v x x x
v x x x
Example 2: Collusive Duopoly (cont’d)
Jan Zouhar Games and Decisions
21
a comparison of the two characteristic functions:
core of the oligopoly: a division of payoffs a1,a2 such that
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
1 2 1 2
1 1
2 2
1822, 1822,
1146, or 1006,
512, 450.
a a a a
a a
a a
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