Market
power
Geir B.
Asheim
Pricing
behavior
Market power & product differentiation
Applications of game theory 1
Relaxing
competition
Strategic
commitment
Geir B. Asheim
Department of Economics, University of Oslo
ECON5200
Fall 2009
Introduction
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Strategic
commitment
3 lectures where game theory is applied:
13 October
Market power & product differentiation.
20 October
Adverse selection, signaling and screening.
27 October
Principal-agent problems.
4 November
Seminar.
To be covered under
Market power & product differentiation
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Strategic
commitment
Pricing behavior by a monopoly firm
Pricing behavior by two (or more) firm with identical and
constant unit costs (Bertrand oligopoly model)
Are two firms are sufficient for a perfectly competitive outcome?
Competition can be relaxed by
Capacity constraints (Cournot oligopoly model)
Product differentiation
Repeated interaction
Another topic:
Strategic commitment to affect future competition
Pricing behavior by a profit-maximizing monopolist
Market
power
Geir B.
Asheim
x(·): continuous and strictly decreasing demand function for all p
with x(p) > 0; x(p) = 0 for p ≥ p̄
max px(p) − c(x(p))
Pricing
behavior
Monopoly
Duopoly
p
p(·) = x −1 (·): inverse demand function; with p(0) = p̄.
Relaxing
competition
max p(q)q − c(q)
Strategic
commitment
q
First-order condition:
p (q m )q m + p(q m ) ≤ c (q m )
with equality if q m > 0
If q m > 0, then p(q m ) > c (q m ): Price under monopoly exceeds
marginal cost; this leads to quantity distortion and welfare loss.
Other distortions; investment in cost reductions & quality choice.
Pricing behavior by profit-maximizing duopolists
Bertrand model with homogeneous products (1)
Market
power
Structure: (1) Prices set (2) Demand determines quantities
Geir B.
Asheim
x(·): continuous and strictly decreasing demand function for all
p with x(p) > 0; x(p) = 0 for p ≥ p̄. Firms 1 and 2 have
constant unit cost c and set prices p1 and p2 simultaneously.
Sales are given by:
⎧
⎪
if pj < pk
⎨ x(pj )
1
xj (pj , pk ) =
x(pj ) if pj = pk
2
⎪
⎩
0
if pj > pk
Pricing
behavior
Monopoly
Duopoly
Relaxing
competition
Strategic
commitment
Proposition
There is a unique Nash equilibrium (p1∗ , p2∗ ) in the Bertrand
duopoly model. In this equilibrium, both firms set their prices
equal to cost: p1∗ = p2∗ = c.
Pricing behavior by profit-maximizing duopolists
Bertrand model with homogeneous products (2)
Market
power
Geir B.
Asheim
Pricing
behavior
Monopoly
Duopoly
Relaxing
competition
Strategic
commitment
Proof.
Part 1: (p1∗ , p2∗ ) = (c, c) is a Nash equilibrium. Assume that
pk = c. If firm j chooses pj = c, then profit equals 0. If firm j
chooses pj < c, then profit is negative. If firm j chooses pj > c,
then profit is 0. Therefore, pj = c is best response to pk = c.
Part 2: (p1 , p2 ) is not a Nash equil. if (p1 , p2 ) = (c, c).
Case 1: min{p1 , p2 } < c. At least one firm j has negative profit.
Zero profit by setting pj ≥ 0.
Case 2: pk > pj = c. Firm j can increase its profit by setting
pj = 12 (pk + c).
Case 3: pk ≥ pj > c. Firm k can increase its profit by setting
pk = pj − ε, for ε > 0 sufficiently small.
Conclusion: If J firms have constant unit cost c,
then all firms set price equal to c, provided J ≥ 2.
Relaxing competition
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Capacity
competition
Product
differentiation
Repeated
interaction
Strategic
commitment
Firms set prices. But the conclusion of the Bertrand model —
that two firms are sufficient for perfectly competitive outcomes –
is unrealistic.
Competition can be relaxed by
Not no capacity contraints, but
capacity constraints (Cournot oligopoly model)
Not homogeneous products, but
product differentiation
Not a static game, but
repeated interaction
Quantity choices by profit-maximizing duopolists
Cournot duopoly model (1)
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Capacity
competition
Product
differentiation
Repeated
interaction
Structure: (1) Quantities set (2) Demand determines price
Assume firms first choose quantities q1 and q2 and that these are
sold at price p(q1 + q2 ) in the market. Each firm j’s problem:
max p(qj + q̄k )qj − cqj
qj
First-order condition:
Strategic
commitment
p (qj + q̄k )qj + p(qj + q̄k ) ≤ c
with equality if qj > 0
For each q̄k , let bj (q̄k ) denote j’s best response.
Quantity choices by profit-maximizing duopolists
Cournot duopoly model (2)
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Capacity
competition
Product
differentiation
Repeated
interaction
Strategic
commitment
(q1∗ , q2∗ ) is a Nash equilibrium if and only if for each j,
qj∗ ∈ bj (qk∗ ). Hence, if (q1∗ , q2∗ ) is a Nash equilibrium, then
p (q1∗ + q2∗ )q1∗ + p(q1∗ + q2∗ ) ≤ c
with equality if q1∗ > 0
p (q1∗ + q2∗ )q2∗ + p(q1∗ + q2∗ ) ≤ c
with equality if q2∗ > 0
If p(0) > c, then (q1∗ , q2∗ ) 0, implying that
∗ ∗
q +q
∗
∗
p (q1 + q2 ) 1 2 2 + p(q1∗ + q2∗ ) = c
Proposition
In a Nash equilibrium of the Cournot duopoly model with
p(0) > c > 0 and p (q) < 0 for all q > 0, the market price is
greater than c (the competitive price) and smaller than p(q m ).
Quantity choices by profit-maximizing duopolists
Cournot duopoly model (3)
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Capacity
competition
Product
differentiation
Repeated
interaction
Strategic
commitment
Proof.
Part 1: p(q1∗ + q2∗ ) > c in a Nash equilibrium.
Follows from the FOC and the assumption that p (q) < 0.
Part 2: p(q1∗ + q2∗ ) < p(q m ) in a Nash equilibrium.
Since p (q) < 0 for all q > 0, this is equivalent to q1∗ + q2∗ > q m .
Subpart (i): q1∗ + q2∗ ≥ q m .
Suppose q1∗ + q2∗ < q m . If one firm j increases quantity to
q m − qk∗ , then total profit is increased and the profit of the other
firm is decreased. Hence, firm j can do better.
Subpart (ii): q1∗ + q2∗ = q m .
Suppose q1∗ + q2∗ = q m . The FOCs for monopoly and Cournot
duopoly cannot both be satisfied.
Capacity choices followed by price competition
Kreps & Scheinkman (1983)
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Capacity
competition
Product
differentiation
Repeated
interaction
Strategic
commitment
Structure: (1) Capacities set
(2) Prices set
(3) Demand determines quantities
In the subgames starting at stage (3), all possible profiles of
capacities and prices must be considered.
How are consumers rationed?
1
2
Proportional rationing: Any consumer willing to pay a good
offered at p has an equal change of buying at p.
Efficient rationing: A consumer with the highest
willingness-to-pay is served first.
Proposition
With efficient rationing, the game considered by Kreps &
Scheinkman (1983) leads to the Cournot outcome.
Two issues
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Capacity
competition
Product
differentiation
Repeated
interaction
Strategic
commitment
How does product differentiation affect price competition?
Bertrand model where demanded quantity is a continuous
function of the price profile
Horizontal differentiation
How do firms product differentiate to relax price
competition?
Not covered
Pricing behavior by profit-maximizing duopolists
Bertrand model with differentiated products (1)
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Structure: (1) Prices set (2) Demand determines quantities
Firms 1 and 2 have constant unit cost c and set prices p1 and p2
simultaneously. Sales are given by xj (pj , pk ), which is the
continuous demand function for firm j.
Capacity
competition
Product
differentiation
Repeated
interaction
Strategic
commitment
max(pj − c)x(pj , pk )
pj
Proposition
If x1 (p1 , p2 ) > 0 when p1 ≤ min{c, p2 } and x2 (p2 , p1 ) > 0 when
p2 ≤ min{c, p1 }, then in any Nash equilibrium (p1∗ , p2∗ ) we have
that (p1∗ , p2∗ ) (c, c).
Pricing behavior by profit-maximizing duopolists
Bertrand model with differentiated products (2)
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Capacity
competition
Product
differentiation
Repeated
interaction
Strategic
commitment
Proof.
To be shown: (p1 , p2 ) is not a Nash equilibrium if
min{p1 , p2 } ≤ c.
Case 1: min{p1 , p2 } < c. At least one firm j has negative profit.
Non-negative profit by setting pj ≥ 0.
Case 2: min{p1 , p2 } = c. If pj = c and pk ≥ c, then by
assumption xj (pj , pk ) > 0, and firm j’s profit is zero.
Since xj (·, ·) is continuous, firm j can attain positive profit by
setting pj = c + ε for ε > 0 sufficiently small.
Horizontal product differentiation
Linear city with linear transportation costs (1)
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Capacity
competition
Product
differentiation
Repeated
interaction
Strategic
commitment
Assume that M consumers are distributed uniformly along the
unit inverval: [0, 1]. Also, assume that firm 1 is located at 0 and
firm 2 is located at 1. For a consumer located at z (∈ [0, 1]) the
cost to buy from firm 1 is p1 + tz and the cost to buy from firm
2 is p2 + t(1 − z). Finally, assume that every consumer buys one
(and only one) unit from a firm with lowest cost.
Indifferent consumer ẑ: p1 + tẑ = p2 + t(1 − ẑ), which implies
ẑ =
t + p2 − p1
2t
xj (pj , pk ) =
and
1 − ẑ =
⎧
⎪
⎨M
⎪
⎩
(t+pk −pj )M
2t
0
t + p1 − p2
2t
if pj ≤ pk − t
if pj ∈ (pk − t, pk + t)
if pj ≥ pk + t
Horizontal product differentiation
Linear city with linear transportation costs (2)
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Capacity
competition
Product
differentiation
Repeated
interaction
Strategic
commitment
max
(pj − c)
pj ∈[pk −t,pk +t]
(t + pk − pj )M
2t
FOC if pj ∈ (pk − t, pk + t): t + pk + c − 2pj = 0
⎧
⎪
if pk ≥ c + 3t
⎨ p̄k − t
t+pk +c
bj (p̄k ) =
if p̄k ∈ (c − t, c + 3t)
2
⎪
⎩
p̄k + t
if p̄k ≤ c − t
Unique Nash equilibrium, (p1∗ , p2∗ ) satisfies:
p1∗ = p2∗ = c + t. Hence, price exceeds c.
Do firms maximize product differentiate to relax price
competition? Problematic to discuss choice of product
differentiation with linear transportation costs. Why?
What product differentiation maximizes welfare?
Repeated Bertrand duopoly (1)
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Capacity
competition
Product
differentiation
Repeated
interaction
Firms 1 and 2 have constant unit cost c. In each stage, firms
choose prices simultaneously. The chosen prices determine
quantities in this stage and becomes observable for both firms
before the next stage. Infinite number of stages. Payoff is
profits discounted at constant discount factor δ ∈ (0, 1).
Consider the following paths, where p ∈ (c, p m ]:
(p(0)t ) = (p, p), (p, p), . . .
Strategic
commitment
(p(1)t ) = (c, c), (c, c), . . .
(p(2)t ) = (c, c), (c, c), . . .
Proposition
The simple strategy profile σ((p(0)t ), (p(1)t ), (p(2)t )) is a
subgame-perfect equilibrium if and only if δ ≥ 12 .
Repeated Bertrand duopoly (2)
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Capacity
competition
Product
differentiation
Repeated
interaction
Strategic
commitment
Proof.
Enough to consider unilateral one-period deviations. Why?
Is a unilateral one-period deviation from (p, p) profitable?
(1 − δ)(p − c)x(p)
Supremum of payoff with deviation
Not profitable if 1 − δ ≤
1
2
≤
1
2 (p
− c)x(p)
Payoff without deviation
or, equivalently, δ ≥ 12 .
Is a unilateral one-period deviation from (c, c) profitable?
Payoff with deviation is non-positive.
Payoff without deviation equals zero.
What if there are more than 2 firms?
What if prices are not perfectly observable?
Strategic commitment
to affect future competition (1)
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Strategic
commitment
Structure:
(1) Firm 1 makes strategic investment k. Observable.
(2) Firms 1 and 2 play some oligopoly game, choosing s1 and s2
Payoffs: π1 (s1 , s2 , k) and π2 (s1 , s2 )
Actions are “aggressive”:
∂π1 (s1 , s2 , k)/∂s2 < 0 and ∂π2 (s1 , s2 )/∂s1 < 0.
Best response functions: b1 (s2 , k) and b2 (s1 ).
Assume unique Nash equilibrium: (s1∗ (k), s2∗ (k)).
Assume sufficient differentiability.
Strategic commitment
to affect future competition (2)
Market
power
Geir B.
Asheim
Pricing
behavior
Relaxing
competition
Strategic
commitment
Identity: s2∗ = b2 (b1 (s2∗ , k))
db2
ds2∗ =
ds1
db2 ∂b1
1−
ds1 ∂s2
ds2∗
dk
=
∂b1 ∗ ∂b1
ds +
dk
∂s2 2
∂k
db2
ds1
ds2∗ =
1
db2 ∂b1
dk
ds1 ∂k
∂b1
∂k
2 ∂b1
− db
ds1 ∂s2
Strategic commitment
to affect future competition (3)
Market
power
Geir B.
Asheim
Pricing
behavior
ds2∗
dk
=
db2
ds1
∂b1
∂k
db ∂b
1− ds 2 ∂s 1
1
2
Relaxing
competition
Strategic
commitment
∂b1 (·)
∂k
∂b1 (·)
∂k
>0
<0
Strategic
substitutes:
Strategic
complements:
db2 (·)
ds1
db2 (·)
ds1
ds2∗ (k)
dk
ds2∗ (k)
dk
<0
>0
<0
ds2∗ (k)
dk
>0
>0
ds2∗ (k)
dk
<0
Strategic commitment
to affect future competition (4)
Market
power
Geir B.
Asheim
To Deter
Pricing
behavior
Entry
Relaxing
competition
Strategic
commitment
∂b1 (·)
∂k
∂b1 (·)
∂k
>0
Strategic
substitutes:
Strategic
complements:
db2 (·)
ds1
db2 (·)
ds1
<0
ds2∗ (k)
dk
< 0 Δk > 0
Top Dog
>0
ds2∗ (k)
dk
> 0 Δk > 0
Top Dog
big & strong to look big & strong to look
tough & aggressive tough & aggressive
<0
ds2∗ (k)
dk
> 0 Δk < 0
ds2∗ (k)
dk
< 0 Δk < 0
Mean & Hungry Look Mean & Hungry Look
small & firm to look small & firm to look
tough & aggressive tough & aggressive
Strategic commitment
to affect future competition (5)
Market
power
Geir B.
Asheim
To AccommoPricing
behavior
date Entry
Relaxing
competition
Strategic
commitment
∂b1 (·)
∂k
∂b1 (·)
∂k
>0
Strategic
substitutes:
Strategic
complements:
db2 (·)
ds1
db2 (·)
ds1
<0
ds2∗ (k)
dk
< 0 Δk > 0
Top Dog
>0
ds2∗ (k)
dk
> 0 Δk < 0
Puppy Dog
big & strong to look small & weak to look
tough & aggressive soft & inoffensive
<0
ds2∗ (k)
dk
> 0 Δk < 0
ds2∗ (k)
dk
< 0 Δk > 0
Fat Cat
Mean & Hungry Look
small & firm to look fat & mellow to look
tough & aggressive soft & inoffensive