SØK/ECON 535 Imperfect Competition and Strategic Interaction
STATIC OLIGOPOLY MODELS
Lecture notes 10.09.02
Nash Equilibrium
Describes strategic interaction by equilibrium in non-cooperative games.
Strategies: ai , i = 1,2,..., n .
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Payoff (profit) functions: Π ( ai ; a− i ) .
Definition: a strategy combination (a * 1,...,a * n) constitutes a Nash equilibrium
if, for i = 1,...,n,
∀ai : Π i (ai∗ , a−∗ i ) ≥ Π i (ai , a−∗ i )
The Cournot model (1838)
Strategies are quantities (supply/output): qi , i = 1,2,...n .
Firms choose simultaneously (i.e. without knowing their competitors’
strategies).
Duopoly (n = 2)
Homogeneous products.
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Profit functions: Π ( qi , q j ) = P (Q ) qi − Ci ( qi ) ,Q = q1 + q2 , i = 1,2 .
Π i is assumed strictly concave and two times differentiable.
FOC: P ′ (Q ) qi + P (Q ) − Ci′ ( qi ) = 0
Negative external effect on competitor’s profit P ′ (Q ) q j ; therefore producers
choose a larger quantity than that which would maximise total profits:
P (Q ) − Ci′ ( qi ) = −P ′ (Q ) qi < −P ′ (Q ) [q1 + q2 ]
⇒ Qd > Qm
Price is lower and output greater than in monopoly (perfect cartel).
At equilibrium, marginal costs are equal only if firms are symmetric; therefore,
for given aggregate output, total costs are generally not minimised.
From the FOC we obtain the Lerner index:
Li ≡
P − Ci′ α i
=
P
ε
where α i = qi Q is i’s market share and ε = −P ′ (Q ) Q P is the price elasticity
of demand.
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Reaction functions (from FOC): Π i (Ri (q j ),qj ) = 0. 1
Ri′ = −
Π iij
Π iii
<0
if marginal profits decrease as a function of the competitor’s supply (because
Π iij = P ′′qi + P ′ , a sufficient condition for this result is that the inverse demand
function is convex, i.e. P ′′ < 0 ).
If so, quantities are strategic substitutes.
Figure (equilibrium described by reaction functions)
Example: Linear demand and cost functions: Ci ( qi ) = ci qi and P (Q ) = 1 − Q
Ri ( q j ) =
1 − q j − ci
2
Note: a reduction in ci shifts the reaction curve outwards and increases i’s
profit-maximising supply.
Oligopoly (n ≥ 2)
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Profit functions: Π ( qi ;Q− i ) = P (Q ) qi − Ci ( qi ) ,Q− i = ∑ j ≠ì q j
Same FOC as above, and same formula for Lerner-indices.
From the individual Lerner-indices:
L≡
1
We define
P − C′ H
=
P
ε
Π ii = ∂Π i ∂qi , Π iii = ∂ 2 Π i ∂qi
2
and so on.
2
2
where H = ∑ i =1[α i ] is the Hirschmann-Herfindahl concentration index and
n
C ′ = ∑ i =1α i Ci′ the (weighted) average marginal cost.
n
Example: linear demand and cost functions
Symmetric model: ci = c, i = 1,2,..., n :
1
n
1
q=
[1 − c ]
n +1
n
Q=
[1 − c ]
n +1
1
p=c+
[1 − c ] → c as n → ∞
n +1
H=
Concentration indices
Summary statistics for market ‘concentration’, i.e. distribution of market
shares.
Ideally, an index should
treat firms symmetrically;
not decrease in value if part of a firm’s market share is transferred to a
larger firm; and
decrease in value if, in a symmetric industry, the number of firms increase.
Order firms such that α1 ≥ α 2 ≥ ... ≥ α n :
m
m-firm concentration ratio: K = ∑ i =1α i
m
2
Hirschmann-Herfindahl index: H = ∑ i =1[α i ]
n
These indices satisfy the above conditions:
take on values between 0 and 1;
maximised (=1) at monopoly;
for a given number of firms n, the index reaches its lowest value when
firms are equally large.
Hirschmann-Herfindahl index puts more weight on smaller firms.
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Concentration indices may be useful if there is a relationship between market
concentration and competition, or, more specifically, performance/welfare.
This is the case in the Cournot model (under certain conditions, see above).
I other cases, this may not necessarily be true (cf. the Bertrand model).
The Bertrand model (1883)
Strategies: prices pi , i = 1,2,..., n .
Firms choose strategies simultaneously.
Homogenous duopoly (n = 2)
Buyers will shop where the price is lowest (if firms charge same prices we
assume they share the market equally).
Demand:
D ( pi ) if pi < p j

Di ( pi , p j ) =  21 D ( pi ) if pi = p j
0
if pi > p j

where D ( p ) is market demand.
Assume marginal costs are constant and identical (= c).
Nash equilibrium: p1 = p2 = c .
( pi > p j cannot be an equilibrium because j can increase profits by raising
price; pi = p j > c cannot be an equilibrium because either firm would want to
undercut; pi = p j < c cannot be an equilibrium because either firm would
want to raise its price.)
Conclusion: duopoly equals perfect competition!
In the case of asymmetric costs ( ci > c j ), the only equilibrium involves j
obtaining a monopoly at price equal to the cost of i; that is, p j = pi (technical
assumptions to guarantee existence.
The "Bertrand Paradox"
Solutions to the paradox:
product differentiation.
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capacity constraints (Edgeworth);
inter-temporal (dynamic) competition (threat of price wars);
Product differentiation
Demand does not necessarily disappear even if price exceeds those of the
competitors.
(
)
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Profit functions: π ( pi , p j ) = pi Di ( pi , p j ) − Ci Di ( pi , p j ) , i = 1,2.
π i assumed strictly concave and twice differentiable.
(
)
i
Reaction functions (from first-order conditions): π i ri ( p j ) , p j = 0 .
ri′ = −
π iji
>0
π iii
if marginal profit is increasing in competitor’s price.
If so, prices are strategic complements.
Figure (equilibrium described by reaction functions)
Price competition with capacity constraints
Example (costly capacity):
Demand: D ( p ) = 1 − p
Unit capacity costs: c ∈  34 ,1
Variable costs are constant and normalised to 0
Capacity constraints: qi ≤ qi , i = 1, 2 .
Rationing
Assume p1 < p2 and D ( p1 ) > q1 : Who gets to buy from Firm 1?
Efficient rationing: residual demand equals D ( p2 ) − q1 if D ( p1 ) > q1 , 0
otherwise.
Two stage game
1)
Firms choose capacities simultaneously.
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2)
Firms choose prices simultaneously. Trade takes place and profits are
realised.
We are looking for a Subgame Perfect Equilibrium, in which, from each stage
of the game, the strategies constitute a Nash equilibrium of the continuation
game.
Solves the game by backward induction.
STAGE 2:
Gross monopoly profits equal 1/4. Therefore, no firm will ever choose q >
because that would lead to negative profits.
1
3
Given q ≤ 31 , i = 1, 2 , there exists a unique Nash equilibrium in which
p1 = p2 = p* = 1 − q1 − q2 .
Proof of existence:
There is no point in reducing prices because capacity constraints bind.
Profits for firm 1 when p1 > p2 = p * equals p1 [1 − p1 − q2 ] . The marginal
profits is
1 − 2 p1 − q2 < 1 − 2 [1 − q1 − q2 ] − q2 = 2q1 + q2 − 1 ≤ 0.
STAGE 1:
Reduced-form profit functions, given equilibrium play at Stage 2, are
[
]
Π i (q i , q j ) = 1 − q i − q j q i , i = 12
, .
In other words, Cournot profits.
Conclusion: The two-stage game leads to the same equilibrium outcome as
in the Cournot model. The difference is that here prices are set by firms
rather than by some Walrasian auctioneer or other equilibrating ‘mechanism’.
Caveats:
For larger capacities ( qi > 31 ), an equilibrium in pure strategies generally
does not exist and one may have to consider mixed strategies (KrepsSheinkman, 1983).
The equivalence with the Cournot model does not necessarily hold for
other rationing rules (for example, proportional rationing, such that
everyone with willingness to pay above the lowest price has equal
probability of getting to buy at this price).
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Institutional detail – pricing rules and capacity constraints
Market outcomes may depend on the finer details of market institutions; in
particular, how firms are remunerated.
Assume demand D is perfectly inelastic.
There are two (symmetric) firms with capacities k and unit costs c.
Firms set prices simultaneously: pi ∈ c, P  .
A market price is set equal to the price of the marginal supplier; that is,
 min { p1, p2 } if D ≤ k
P=
max { p1, p2 } if D > k
Low-demand periods ( D ≤ k )
There is a unique (Bertrand like) equilibrium in pure strategies in which
p1 = p2 = c .
High-demand periods ( D > k )
There are many equilibria, all having outcomes with one firm pricing at P and
the other pricing so low as to avoid undercutting; that is, pi = P and
p j = b ≥ c, bk ≤ P [D − k ] .
Ref: N-H von der Fehr and D Harbord (1993): Spot market competition in the
UK electricity industry, The Economic Journal, 103, 531-46.
Under the Bertrand assumption (i.e. each firm being paid its own price) we
would have the same equilibrium in low-demand periods, but in high-demand
periods equilibria would be different; in particular, only mixed-strategy
equilibria exists.
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