Quantitative Comparative Statics
for a Multimarket Paradox
Philipp von Falkenhausen
Technische Universität Berlin
July 11, 2013
Joint work with Tobias Harks
Agenda
1
Comparative Statics
2
Quantitative Comparative Statics
Agenda
1
Comparative Statics
2
Quantitative Comparative Statics
Comparative Statics
System at
equilibrium
marginal
−−−−−−−−−−→
parameter change
Marginal change
of equilibrium
Comparative Statics
System at
equilibrium
marginal
−−−−−−−−−−→
parameter change
Marginal change
of equilibrium
Examples
Introduction of export taxes/subsidies (Brander and Spencer
1985, Eaton and Grossman 1986)
Demand or cost shift (Quirmbach 1988, Février and Linnemer
2004)
Forced reduction of produced quantity (Gaudet and Salant 1991)
Multimarket Cournot Oligopoly
Oligopoly Finite number of competing firms i ∈ N producing some
good. Firm i has cost ci (qi ) for producing quantity qi .
Multimarket markets m ∈ M served with the good.
Cournot Each firm i produces
P quantity qi,m in market m, the
market price is pm ( i qi,m )
Multimarket Cournot Oligopoly
Oligopoly Finite number of competing firms i ∈ N producing some
good. Firm i has cost ci (qi ) for producing quantity qi .
Multimarket markets m ∈ M served with the good.
Cournot Each firm i produces
P quantity qi,m in market m, the
market price is pm ( i qi,m )
P
P
revenue of firm i: m∈M pm ( j qj,m )qi,m
profit of firm i: revenue - cost
marginal revenue of firm i on market m:
0
πi,m (qi,m , q−i,m ) := pm (qi,m + q−i,m ) + pm
(qi,m + q−i,m )qi,m
Multimarket Cournot Oligopoly
Oligopoly Finite number of competing firms i ∈ N producing some
good. Firm i has cost ci (qi ) for producing quantity qi .
Multimarket markets m ∈ M served with the good.
Cournot Each firm i produces
P quantity qi,m in market m, the
market price is pm ( i qi,m )
P
P
revenue of firm i: m∈M pm ( j qj,m )qi,m
profit of firm i: revenue - cost
marginal revenue of firm i on market m:
0
πi,m (qi,m , q−i,m ) := pm (qi,m + q−i,m ) + pm
(qi,m + q−i,m )qi,m
Definition (Cournot Equilibrium)
Given choices q−i of other firms, firm i produces qi such that
marginal revenue on market m = marginal cost
X
πi,m (qi,m , q−i,m ) = ci0 (
qi,m )
m
for all m ∈ M
Paradox: Price increase reduces profit of monopolist
Example by Bulow, Geanakoplos, Klemperer (1985)
Market 1
Market 2
Firm a
Price: p1 (q1 ) = 50
Firm a, firm b
Price: p2 (q2 ) = 200 − q2
Both firms have cost: ci (qi ) = 21 qi2
Paradox: Price increase reduces profit of monopolist
Example by Bulow, Geanakoplos, Klemperer (1985)
Market 1
Market 2
Firm a
Price: p1 (q1 ) = 50
Firm a, firm b
Price: p2 (q2 ) = 200 − q2
Both firms have cost: ci (qi ) = 21 qi2
When the price on market 1 increases by 10%, the profit of firm a
decreases by 0.76%.
Paradox: Price increase reduces profit of monopolist
Example by Bulow, Geanakoplos, Klemperer (1985)
Market 1
Market 2
Firm a
Price: p1 (q1 ) = 50
Firm a, firm b
Price: p2 (q2 ) = 200 − q2
Both firms have cost: ci (qi ) = 21 qi2
When the price on market 1 increases by 10%, the profit of firm a
decreases by 0.76%.
Definition (Strategic Substitutes, Bulow et al. 1985)
"Less ‘aggressive’ play (e.g., [...] lower quantity) by
one firm raises competing firms’ marginal profitabilities."
Profit loss of 0.76% - so what?!
Comparative statics studies marginal changes of a parameter.
Open questions
1
significance: are changes in a given parameter worth considering?
2
robustness: how sensitive is the game to changes of a parameter?
⇒ Quantitative approach
Agenda
1
Comparative Statics
2
Quantitative Comparative Statics
Model
Assumptions
Market 2
Market 1
Firm a
Firms i 6= a
Firm a
Price is affine, decreasing function of quantity
Cost is convex, differentiable function of quantity
Objective
What is max. impact of price shock δ on market 1 on profit of firm a?
γ :=
eq. profit after shock
eq. profit before shock
Main result
Theorem (Main result)
For an instance with n firms and a positive price shock
γ≥
3
(3n − 1)(n + 1)
≥ .
4
4n2
The profit loss is at most 25%.
Main result
Theorem (Main result)
For an instance with n firms and a positive price shock
γ≥
3
(3n − 1)(n + 1)
≥ .
4
4n2
The profit loss is at most 25%.
Corollary (Dual result)
For an instance with n firms and a negative price shock
γ≤
4n2
4
≤ .
(3n − 1)(n + 1)
3
Main result
Theorem (Main result)
For an instance with n firms and a positive price shock
γ≥
3
(3n − 1)(n + 1)
≥ .
4
4n2
The profit loss is at most 25%.
Corollary (Dual result)
For an instance with n firms and a negative price shock
γ≤
4n2
4
≤ .
(3n − 1)(n + 1)
3
Complementing Bound
Large class of instances where 25% profit loss is attained.
Proof Overview
1
Establish basics
I
I
2
Series of simplifications
I
3
Cournot equilibrium unique
Price shock triggers strategic substitution
Given instance G, construct simplified G̃ with γ̃ ≤ γ
Proof theorem for simplified game
Proof Overview
1
Establish basics
I
I
2
Series of simplifications
I
3
Cournot equilibrium unique
Price shock triggers strategic substitution
Given instance G, construct simplified G̃ with γ̃ ≤ γ
Proof theorem for simplified game
Paradox: Explained
Firm a
Firm b
cb0 (qb )
ca0 (qa )
qa
Initial equilibrium x
→
Price shock δ
qb
→
New equilibrium y
Paradox: Explained
Firm a
Firm b
π
a,
2(
a,
cb0 (qb )
ca0 (qa )
q
2,
xb
π
,2 )
b,
2(
q
b,
Initial equilibrium x
→
2,
xa
,2 )
qa
xb,2
Price shock δ
→
qb
New equilibrium y
Paradox: Explained
Firm a
Firm b
π
a,
2(
a,
cb0 (qb )
ca0 (qa )
q
2,
xb
π
,2 )
b,
2(
b,
p1
Initial equilibrium x
→
q
2,
xa
,2 )
qa
xb,2
Price shock δ
→
qb
New equilibrium y
Paradox: Explained
Firm a
Firm b
π
a,
2(
a,
cb0 (qb )
ca0 (qa )
q
2,
xb
π
,2 )
b,
2(
b,
p1
xa,2
xa,2 + xa,1
Initial equilibrium x
→
q
2,
xa
,2 )
qa
xb,2
Price shock δ
→
qb
New equilibrium y
Paradox: Explained
Firm a
Firm b
π
a,
2(
a,
cb0 (qb )
ca0 (qa )
q
2,
xb
π
,2 )
p1 + δ
p1
xa,2
xa,2 + xa,1
Initial equilibrium x
→
b,
2(
q
b,
2,
xa
,2 )
qa
xb,2
Price shock δ
→
qb
New equilibrium y
Paradox: Explained
Firm a
Firm b
π
a,
2(
a,
cb0 (qb )
ca0 (qa )
q
2,
xb
π
,2 )
p1 + δ
p1
Initial equilibrium x
→
b,
2(
q
b,
2,
xa
,2 )
qa
xb,2
Price shock δ
→
qb
New equilibrium y
Paradox: Explained
Firm a
Firm b
π
π
a,
2(
ca0 (qa )
q
a,
2,
xb
b,
2(
,2 )
cb0 (qb )
q
b,
2,
ya
,2 )
p1 + δ
qa
Initial equilibrium x
→
Price shock δ
yb,2
→
qb
New equilibrium y
Paradox: Explained
Firm a
Firm b
π
π
a,
2(
ca0 (qa )
q
a,
2,
xb
b,
2(
,2 )
cb0 (qb )
q
b,
2,
ya
,2 )
p1 + δ
qa
Initial equilibrium x
→
Price shock δ
yb,2
→
qb
New equilibrium y
Paradox: Explained
Firm a
Firm b
π
ca0 (qa )
π
a,
2(
2(
cb0 (qb )
q
b,
2,
ya
,2 )
p1 + δ
q
a,
b,
2,
yb
,2 )
Initial equilibrium x
qa
→
Price shock δ
yb,2
→
qb
New equilibrium y
Paradox: Explained
Firm a
Firm b
π
ca0 (qa )
π
a,
2(
2(
cb0 (qb )
q
b,
2,
ya
,2 )
p1 + δ
q
a,
b,
2,
ya,2
yb
,2 )
Initial equilibrium x
ya,2 + ya,1
→
qa
Price shock δ
yb,2
→
qb
New equilibrium y
Paradox: Explained
Firm a
Firm b
π
ca0 (qa )
π
a,
2(
2(
cb0 (qb )
q
b,
2,
ya
,2 )
p1 + δ
q
a,
b,
2,
ya,2
yb
,2 )
Initial equilibrium x
ya,2 + ya,1
→
qa
Price shock δ
yb,2
→
qb
New equilibrium y
Proof Overview
1
Establish basics
I
I
2
Series of simplifications
I
3
Cournot equilibrium unique
Price shock triggers strategic substitution
Given instance G, construct simplified G̃ with γ̃ ≤ γ
Proof theorem for simplified game
Proof Overview
1
Establish basics
I
I
2
Series of simplifications
I
3
Cournot equilibrium unique
Price shock triggers strategic substitution
Given instance G, construct simplified G̃ with γ̃ ≤ γ
Proof theorem for simplified game
Competitors Are Most Aggressive With Linear Cost
Lemma
For given G, let G̃ be similiar to G except that c̃i (qi ) = ci0 (xi )qi for all
firms i 6= a. Then, γ̃ ≤ γ.
Competitors Are Most Aggressive With Linear Cost
Lemma
For given G, let G̃ be similiar to G except that c̃i (qi ) = ci0 (xi )qi for all
firms i 6= a. Then, γ̃ ≤ γ.
Proof (by picture).
linear cost
π i,
(q
i,
y
i,
(q
2
2
π i,
strictly convex cost
ci0 (qi )
y−
i)
i)
π i,
−
π i,
x
i,
(q
2
(q
2
i,
i)
i)
−
x−
xi,2 yi,2
qi
More strategic substitution: yi,2 < ỹi,2 .
xi,2
ỹi,2
c̃i0 (x)
qi
Strategic Substitution Independent of Cost Function
* Assume linear cost for competitors
Lemma
For given G, let G̃ be similiar to G except that all firms i 6= a have cost
ci (qi ) = c̃qi . Then, there is c̃ such that γ̃ = γ.
Strategic Substitution Independent of Cost Function
* Assume linear cost for competitors
Lemma
For given G, let G̃ be similiar to G except that all firms i 6= a have cost
ci (qi ) = c̃qi . Then, there is c̃ such that γ̃ = γ.
Proof (by picture).
π i,
(q
2
i,
i)
y−
c3
c2
c1
qi
Strategic Substitution Independent of Cost Function
* Assume linear cost for competitors
Lemma
For given G, let G̃ be similiar to G except that all firms i 6= a have cost
ci (qi ) = c̃qi . Then, there is c̃ such that γ̃ = γ.
Proof (by picture).
π i,
Lemma
(q
2
For any instance with n firms, when a
firm produces 1 unit less, its
competitors produce n−1
n units more.
i,
i)
y−
c3
c2
c1
X
qi
i6=a
(yi,2 − xi,2 ) =
n−1
(xa,2 − ya,2 )
n
Elastic Demand in Market 1
* Assume identical, linear cost for competitors
Lemma
For given G, let G̃ be similiar to G except that the price of market 1 is
fixed at p̃1 = πa,1 (x). Then, γ̃ ≤ γ.
Proof.
More elastic demand enhances effect of price shock, but does not
affect initial equilibrium.
Steep Cost -> Fixed Capacity
* Assume identical, linear cost for competitors; fixed price on market 1
Lemma
For given G, let G̃ be similar to G except for c̃a . If c̃a (q) = ca (q) for
q ≤ xa,2 and c̃a0 (q) > ca0 (q) for q > xa,2 , then γ̃ ≤ γ.
Steep Cost -> Fixed Capacity
* Assume identical, linear cost for competitors; fixed price on market 1
Lemma
For given G, let G̃ be similar to G except for c̃a . If c̃a (q) = ca (q) for
q ≤ xa,2 and c̃a0 (q) > ca0 (q) for q > xa,2 , then γ̃ ≤ γ.
Proof (by picture).
π
a,
π
2(
q
a,
2,
ca0 (qa )
x−
a,
2)
p1 + δ
p1
a,
2(
q
a,
2,
c̃a0 (qa )
x−
a,
2)
p1 + δ
p1
qa
qa
Steeper cost has no effect on market 2, but limits additional production
on market 1 after price shock.
Final simplification
* Assume identical, linear cost for competitors; fixed price on market 1; steep cost for
firm a at quantities greater than xa,2 .
Lemma
Let G̃ be similiar to G except that p̃1 = 0, p̃2 (q2 ) = p2 (q2 ) − p1 ,
c̃ = c − p1 and c̃a (qa ) = 0 for qa ≤ xa,2 . Then, γ̃ ≤ γ.
Proof Overview
1
Establish basics
I
I
2
Series of simplifications
I
3
Cournot equilibrium unique
Price shock triggers strategic substitution
Given instance G, construct simplified G̃ with γ̃ ≤ γ
Proof theorem for simplified game
Proof Overview
1
Establish basics
I
I
2
Series of simplifications
I
3
Cournot equilibrium unique
Price shock triggers strategic substitution
Given instance G, construct simplified G̃ with γ̃ ≤ γ
Proof theorem for simplified game
Proof of Main Theorem for Simplified Instances
* Assume identical, linear cost for competitors; p1 ≡ 0; firm a has 0 cost for quantities
less than xa,2 and prohibitively high for more.
Theorem (Main result)
For an instance with n firms and a positive price shock
γ≥
(3n − 1)(n + 1)
3
≥ .
2
4
4n
Proof of Main Theorem for Simplified Instances
* Assume identical, linear cost for competitors; p1 ≡ 0; firm a has 0 cost for quantities
less than xa,2 and prohibitively high for more.
Theorem (Main result)
For an instance with n firms and a positive price shock
γ≥
(3n − 1)(n + 1)
3
≥ .
2
4
4n
Proof.
c̃a0 (qa )
πa
a
(q
,2
,x
,2
a,
−
)
2
qa
Proof of Main Theorem for Simplified Instances
* Assume identical, linear cost for competitors; p1 ≡ 0; firm a has 0 cost for quantities
less than xa,2 and prohibitively high for more.
Theorem (Main result)
For an instance with n firms and a positive price shock
γ≥
(3n − 1)(n + 1)
3
≥ .
2
4
4n
Proof.
c̃a0 (qa )
πa
(q
,2
a,
δ
,y
2
−
a,
)
2
qa
Proof of Main Theorem for Simplified Instances
* Assume identical, linear cost for competitors; p1 ≡ 0; firm a has 0 cost for quantities
less than xa,2 and prohibitively high for more.
Theorem (Main result)
For an instance with n firms and a positive price shock
γ≥
(3n − 1)(n + 1)
3
≥ .
2
4
4n
Proof.
Easy to calculate γ in simplified
instances.
c̃a0 (qa )
πa
(q
,2
a,
δ
,y
2
−
a,
)
2
qa
Proof of Main Theorem for Simplified Instances
* Assume identical, linear cost for competitors; p1 ≡ 0; firm a has 0 cost for quantities
less than xa,2 and prohibitively high for more.
Theorem (Main result)
For an instance with n firms and a positive price shock
γ≥
(3n − 1)(n + 1)
3
≥ .
2
4
4n
Proof.
Easy to calculate γ in simplified
instances.
c̃a0 (qa )
πa
γ is quadratic function of δ
(q
,2
worst case δ is function of xa,2
a,
δ
,y
2
maximum xa,2 minimizes γ.
−
a,
)
2
qa
Instances where 25% profit loss is attained
Any instance, where
firm a’s competitors have linear cost
firm a can produce some quantity xa,2 (not more) at cost 0 and
sells this quantity the initial equilibrium on market 2
market 1 has constant price, initially 0
can have a price shock that leads to a 25% profit loss for firm a.
Specifically, this is independent of the price function on market 2 and
the competitors’ cost functions.
Summary
Comparative Statics studies marginal parameter changes
Benefits of quantifying such results: Significance, Robustness
Application here: Paradox in Multimarket Cournot Oligopoly
Main result Positive price shock in monopoly market can lead to profit
loss of at most 25%.
Dual result Profit gain from price decrease at most 33%.
Side result Exact quantification of strategic substitution when
competitors have linear cost (which is worst case).