Strategic
commitment
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
Strategic
commitment
Relaxing
competition
Strategic commitment to affect future competition
Another topic:
Repeated interaction
Product differentiation
Capacity constraints (Cournot oligopoly model)
Are two firms are sufficient for a perfectly competitive outcome?
Competition can be relaxed by
Pricing behavior by two (or more) firm with identical and
constant unit costs (Bertrand oligopoly model)
Pricing behavior by a monopoly firm
To be covered under
Market power & product differentiation
ECON5200
Fall 2009
Department of Economics, University of Oslo
Geir B. Asheim
Applications of game theory 1
Market power & product differentiation
Strategic
commitment
Relaxing
competition
Monopoly
Duopoly
Pricing
behavior
Geir B.
Asheim
Market
power
Strategic
commitment
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Geir B.
Asheim
Pricing
behavior
Market
power
Market
power
with equality if q m > 0
Other distortions; investment in cost reductions & quality choice.
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.
p (q m )q m + p(q m ) ≤ c (q m )
First-order condition:
q
max p(q)q − c(q)
p(·) = x −1 (·): inverse demand function; with p(0) = p̄.
p
max px(p) − c(x(p))
x(·): continuous and strictly decreasing demand function for all p
with x(p) > 0; x(p) = 0 for p ≥ p̄
Pricing behavior by a profit-maximizing monopolist
4 November
Seminar.
27 October
Principal-agent problems.
20 October
Adverse selection, signaling and screening.
13 October
Market power & product differentiation.
3 lectures where game theory is applied:
Introduction
Strategic
commitment
Capacity
competition
Product
differentiation
Repeated
interaction
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
Strategic
commitment
Relaxing
competition
Monopoly
Duopoly
Not a static game, but
repeated interaction
Not homogeneous products, but
product differentiation
Not no capacity contraints, but
capacity constraints (Cournot oligopoly model)
Competition can be relaxed by
Firms set prices. But the conclusion of the Bertrand model —
that two firms are sufficient for perfectly competitive outcomes –
is unrealistic.
Relaxing competition
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.
Proposition
Strategic
commitment
Capacity
competition
Product
differentiation
Repeated
interaction
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
Strategic
commitment
Relaxing
competition
Monopoly
Duopoly
Pricing
behavior
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 ) =
if pj = pk
2 x(pj )
⎪
⎩
0
if pj > pk
Geir B.
Asheim
Pricing
behavior
Market
power
Structure: (1) Prices set (2) Demand determines quantities
Market
power
Pricing behavior by profit-maximizing duopolists
Bertrand model with homogeneous products (1)
with equality if qj > 0
For each q̄k , let bj (q̄k ) denote j’s best response.
p (qj + q̄k )qj + p(qj + q̄k ) ≤ c
First-order condition:
qj
max p(qj + q̄k )qj − cqj
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:
Structure: (1) Quantities set (2) Demand determines price
Quantity choices by profit-maximizing duopolists
Cournot duopoly model (1)
Conclusion: If J firms have constant unit cost c,
then all firms set price equal to c, provided J ≥ 2.
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.
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.
Proof.
Pricing behavior by profit-maximizing duopolists
Bertrand model with homogeneous products (2)
Strategic
commitment
Capacity
competition
Product
differentiation
Repeated
interaction
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
Strategic
commitment
Capacity
competition
Product
differentiation
Repeated
interaction
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
+
+
q2∗ )q1∗
q2∗ )q2∗
+
+
p(q1∗
p(q1∗
+
+
q2∗ )
q2∗ )
with equality if
with equality if
≤c
≤c
q1∗
q2∗
>0
>0
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.
With efficient rationing, the game considered by Kreps &
Scheinkman (1983) leads to the Cournot outcome.
Proposition
2
1
How are consumers rationed?
In the subgames starting at stage (3), all possible profiles of
capacities and prices must be considered.
Structure: (1) Capacities set
(2) Prices set
(3) Demand determines quantities
Capacity choices followed by price competition
Kreps & Scheinkman (1983)
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 ).
Proposition
If p(0) > c, then (q1∗ , q2∗ ) 0, implying that
∗ ∗
q +q
p (q1∗ + q2∗ ) 1 2 2 + p(q1∗ + q2∗ ) = c
p (q1∗
p (q1∗
(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
Quantity choices by profit-maximizing duopolists
Cournot duopoly model (2)
Strategic
commitment
Capacity
competition
Product
differentiation
Repeated
interaction
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
Strategic
commitment
Capacity
competition
Product
differentiation
Repeated
interaction
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
Not covered
How do firms product differentiate to relax price
competition?
Bertrand model where demanded quantity is a continuous
function of the price profile
Horizontal differentiation
How does product differentiation affect price competition?
Two issues
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.
Part 1: p(q1∗ + q2∗ ) > c in a Nash equilibrium.
Follows from the FOC and the assumption that p (q) < 0.
Proof.
Quantity choices by profit-maximizing duopolists
Cournot duopoly model (3)
Strategic
commitment
Capacity
competition
Product
differentiation
Repeated
interaction
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
Strategic
commitment
Capacity
competition
Product
differentiation
Repeated
interaction
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
⎪
⎩
and
0
(t+pk −pj )M
2t
⎧
⎪
⎨M
t + p2 − p1
2t
xj (pj , pk ) =
ẑ =
t + p1 − p2
2t
if pj ∈ (pk − t, pk + t)
if pj ≥ pk + t
if pj ≤ pk − t
1 − ẑ =
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
Horizontal product differentiation
Linear city with linear transportation costs (1)
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).
Proposition
pj
max(pj − c)x(pj , pk )
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.
Structure: (1) Prices set (2) Demand determines quantities
Pricing behavior by profit-maximizing duopolists
Bertrand model with differentiated products (1)
Strategic
commitment
Capacity
competition
Product
differentiation
Repeated
interaction
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
Strategic
commitment
Capacity
competition
Product
differentiation
Repeated
interaction
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
(t + pk − pj )M
2t
What product differentiation maximizes welfare?
Do firms maximize product differentiate to relax price
competition? Problematic to discuss choice of product
differentiation with linear transportation costs. Why?
Unique Nash equilibrium, (p1∗ , p2∗ ) satisfies:
p1∗ = p2∗ = c + t. Hence, price exceeds c.
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
(pj − c)
pj ∈[pk −t,pk +t]
max
Horizontal product differentiation
Linear city with linear transportation costs (2)
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.
Case 1: min{p1 , p2 } < c. At least one firm j has negative profit.
Non-negative profit by setting pj ≥ 0.
To be shown: (p1 , p2 ) is not a Nash equilibrium if
min{p1 , p2 } ≤ c.
Proof.
Pricing behavior by profit-maximizing duopolists
Bertrand model with differentiated products (2)
Strategic
commitment
Relaxing
competition
Pricing
behavior
Assume sufficient differentiability.
Assume unique Nash equilibrium: (s1∗ (k), s2∗ (k)).
Best response functions: b1 (s2 , k) and b2 (s1 ).
Payoffs: π1 (s1 , s2 , k) and π2 (s1 , s2 )
Actions are “aggressive”:
∂π1 (s1 , s2 , k)/∂s2 < 0 and ∂π2 (s1 , s2 )/∂s1 < 0.
Strategic
commitment
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Structure:
(1) Firm 1 makes strategic investment k. Observable.
(2) Firms 1 and 2 play some oligopoly game, choosing s1 and s2
Geir B.
Asheim
Strategic
commitment
Capacity
competition
Product
differentiation
Repeated
interaction
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
Market
power
Strategic commitment
to affect future competition (1)
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 .
Proposition
(p(2)t ) = (c, c), (c, c), . . .
(p(1)t ) = (c, c), (c, c), . . .
(p(0) ) = (p, p), (p, p), . . .
t
Consider the following paths, where p ∈ (c, p m ]:
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).
Market
power
Strategic
commitment
Capacity
competition
Product
differentiation
Repeated
interaction
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
Repeated Bertrand duopoly (1)
1
2
1
(p − c)x(p)
Payoff without deviation
2
or, equivalently, δ ≥ 12 .
≤
db2
=
ds1
1
db2 ∂b1
dk
ds1 ∂k
∂b1
∂k
2 ∂b1
− db
ds1 ∂s2
ds2∗ =
∂b1 ∗ ∂b1
dk
ds +
∂s2 2
∂k
db2
ds2∗
=
dk
ds1
db2 ∂b1
1−
ds1 ∂s2
ds2∗
Identity: s2∗ = b2 (b1 (s2∗ , k))
Strategic commitment
to affect future competition (2)
What if there are more than 2 firms?
What if prices are not perfectly observable?
Is a unilateral one-period deviation from (c, c) profitable?
Payoff with deviation is non-positive.
Payoff without deviation equals zero.
Not profitable if 1 − δ ≤
Supremum of payoff with deviation
(1 − δ)(p − c)x(p)
Is a unilateral one-period deviation from (p, p) profitable?
Enough to consider unilateral one-period deviations. Why?
Proof.
Repeated Bertrand duopoly (2)
Strategic
commitment
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
Strategic
commitment
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
=
<0
∂b1 (·)
∂k
∂b1
∂k
db2 ∂b1
1− ds ∂s
1
2
>0
∂b1 (·)
∂k
db2
ds1
<0
>0
ds2∗ (k)
dk
>0
<0
∂b1 (·)
∂k
∂b1 (·)
∂k
date Entry
db2 (·)
ds1
>0
> 0 Δk < 0
Puppy Dog
ds2∗ (k)
dk
db2 (·)
ds1
> 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
ds2∗ (k)
dk
big & strong to look small & weak to look
tough & aggressive soft & inoffensive
< 0 Δk > 0
Top Dog
<0
Strategic
substitutes:
ds2∗ (k)
dk
To Accommo-
<0
>0
Strategic
complements:
ds2∗ (k)
dk
ds2∗ (k)
dk
>0
db2 (·)
ds1
db2 (·)
ds1
<0
Strategic
complements:
Strategic
substitutes:
ds2∗ (k)
dk
Strategic commitment
to affect future competition (5)
ds2∗
dk
Strategic commitment
to affect future competition (3)
Strategic
commitment
Relaxing
competition
Pricing
behavior
Geir B.
Asheim
Market
power
∂b1 (·)
∂k
∂b1 (·)
∂k
Entry
<0
>0
To Deter
<0
>0
> 0 Δk > 0
Top Dog
ds2∗ (k)
dk
db2 (·)
ds1
Strategic
complements:
> 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
ds2∗ (k)
dk
big & strong to look big & strong to look
tough & aggressive tough & aggressive
< 0 Δk > 0
Top Dog
ds2∗ (k)
dk
db2 (·)
ds1
Strategic
substitutes:
Strategic commitment
to affect future competition (4)