Quantitative Comparative Statics for a Multimarket Paradox Philipp - - PowerPoint PPT Presentation

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

• f equilibrium

Comparative Statics

System at equilibrium

marginal

− − − − − − − − − − →

parameter change

Marginal change

• f 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 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 quantity qi,m in market m, the market price is pm(

i qi,m)

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: πi,m(qi,m, q−i,m) := pm(qi,m + q−i,m) + p′

m(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 quantity qi,m in market m, the market price is pm(

i qi,m)

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: πi,m(qi,m, q−i,m) := pm(qi,m + q−i,m) + p′

m(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 for all m ∈ M πi,m(qi,m, q−i,m) = c′

i(

• m

qi,m)

Paradox: Price increase reduces profit of monopolist

Example by Bulow, Geanakoplos, Klemperer (1985) Market 1

Firm a Price: p1(q1) = 50

Market 2

Firm a, firm b Price: p2(q2) = 200 − q2 Both firms have cost: ci(qi) = 1

2q2 i

Paradox: Price increase reduces profit of monopolist

Example by Bulow, Geanakoplos, Klemperer (1985) Market 1

Firm a Price: p1(q1) = 50

Market 2

Firm a, firm b Price: p2(q2) = 200 − q2 Both firms have cost: ci(qi) = 1

2q2 i

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

Firm a Price: p1(q1) = 50

Market 2

Firm a, firm b Price: p2(q2) = 200 − q2 Both firms have cost: ci(qi) = 1

2q2 i

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

• ne 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 1

Firm a

Market 2

Firm a Firms i = 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 ≥ (3n − 1)(n + 1) 4n2 ≥ 3 4. The profit loss is at most 25%.

Main result

Theorem (Main result)

For an instance with n firms and a positive price shock γ ≥ (3n − 1)(n + 1) 4n2 ≥ 3 4. The profit loss is at most 25%.

Corollary (Dual result)

For an instance with n firms and a negative price shock γ ≤ 4n2 (3n − 1)(n + 1) ≤ 4 3.

Main result

Theorem (Main result)

For an instance with n firms and a positive price shock γ ≥ (3n − 1)(n + 1) 4n2 ≥ 3 4. The profit loss is at most 25%.

Corollary (Dual result)

For an instance with n firms and a negative price shock γ ≤ 4n2 (3n − 1)(n + 1) ≤ 4 3.

Complementing Bound

Large class of instances where 25% profit loss is attained.

Proof Overview

1

Establish basics

◮ Cournot equilibrium unique ◮ Price shock triggers strategic substitution 2

Series of simplifications

◮ Given instance G, construct simplified ˜

G with ˜ γ ≤ γ

3

Proof theorem for simplified game

Proof Overview

1

Establish basics

◮ Cournot equilibrium unique ◮ Price shock triggers strategic substitution 2

Series of simplifications

◮ Given instance G, construct simplified ˜

G with ˜ γ ≤ γ

3

Proof theorem for simplified game

Paradox: Explained

Firm a qa c′

a(qa)

Firm b qb c′

b(qb)

Initial equilibrium x → Price shock δ → New equilibrium y

Paradox: Explained

Firm a qa πa,2(qa,2, xb,2) c′

a(qa)

Firm b qb c′

b(qb)

π

b , 2

( q

b , 2

, x

a , 2

) xb,2 Initial equilibrium x → Price shock δ → New equilibrium y

Paradox: Explained

Firm a qa πa,2(qa,2, xb,2) p1 c′

a(qa)

Firm b qb c′

b(qb)

π

b , 2

( q

b , 2

, x

a , 2

) xb,2 Initial equilibrium x → Price shock δ → New equilibrium y

Paradox: Explained

Firm a qa πa,2(qa,2, xb,2) p1 xa,2 xa,2 + xa,1 c′

a(qa)

Firm b qb c′

b(qb)

π

b , 2

( q

b , 2

, x

a , 2

) xb,2 Initial equilibrium x → Price shock δ → New equilibrium y

Paradox: Explained

Firm a qa πa,2(qa,2, xb,2) p1 p1 + δ xa,2 xa,2 + xa,1 c′

a(qa)

Firm b qb c′

b(qb)

π

b , 2

( q

b , 2

, x

a , 2

) xb,2 Initial equilibrium x → Price shock δ → New equilibrium y

Paradox: Explained

Firm a qa πa,2(qa,2, xb,2) p1 p1 + δ c′

a(qa)

Firm b qb c′

b(qb)

π

b , 2

( q

b , 2

, x

a , 2

) xb,2 Initial equilibrium x → Price shock δ → New equilibrium y

Paradox: Explained

Firm a qa πa,2(qa,2, xb,2) p1 + δ c′

a(qa)

Firm b qb c′

b(qb)

πb,2(qb,2, ya,2) yb,2 Initial equilibrium x → Price shock δ → New equilibrium y

Paradox: Explained

Firm a qa πa,2(qa,2, xb,2) p1 + δ c′

a(qa)

Firm b qb c′

b(qb)

πb,2(qb,2, ya,2) yb,2 Initial equilibrium x → Price shock δ → New equilibrium y

Paradox: Explained

Firm a qa p1 + δ c′

a(qa)

πa,2(qa,2, yb,2) Firm b qb c′

b(qb)

πb,2(qb,2, ya,2) yb,2 Initial equilibrium x → Price shock δ → New equilibrium y

Paradox: Explained

Firm a qa p1 + δ c′

a(qa)

πa,2(qa,2, yb,2) ya,2 + ya,1 ya,2 Firm b qb c′

b(qb)

πb,2(qb,2, ya,2) yb,2 Initial equilibrium x → Price shock δ → New equilibrium y

Paradox: Explained

Firm a qa p1 + δ c′

a(qa)

πa,2(qa,2, yb,2) ya,2 + ya,1 ya,2 Firm b qb c′

b(qb)

πb,2(qb,2, ya,2) yb,2 Initial equilibrium x → Price shock δ → New equilibrium y

Proof Overview

1

Establish basics

◮ Cournot equilibrium unique ◮ Price shock triggers strategic substitution 2

Series of simplifications

◮ Given instance G, construct simplified ˜

G with ˜ γ ≤ γ

3

Proof theorem for simplified game

Proof Overview

1

Establish basics

◮ Cournot equilibrium unique ◮ Price shock triggers strategic substitution 2

Series of simplifications

◮ Given instance G, construct simplified ˜

G with ˜ γ ≤ γ

3

Proof theorem for simplified game

Competitors Are Most Aggressive With Linear Cost

Lemma

For given G, let ˜ G be similiar to G except that ˜ ci(qi) = c′

i(xi)qi for all

firms i = a. Then, ˜ γ ≤ γ.

Competitors Are Most Aggressive With Linear Cost

Lemma

For given G, let ˜ G be similiar to G except that ˜ ci(qi) = c′

i(xi)qi for all

firms i = a. Then, ˜ γ ≤ γ.

Proof (by picture).

qi π

i , 2

( q

i

, x

− i

) π

i , 2

( q

i

, y

− i

) xi,2 yi,2 c′

i(qi)

strictly convex cost qi π

i , 2

( q

i

, x

− i

) π

i , 2

( q

i

, y

− i

) ˜ c′

i(x)

xi,2 ˜ yi,2 linear cost More strategic substitution: yi,2 < ˜ yi,2.

Strategic Substitution Independent of Cost Function

* Assume linear cost for competitors

Lemma

For given G, let ˜ G be similiar to G except that all firms i = a have cost ci(qi) = ˜

• cqi. Then, there is ˜

c such that ˜ γ = γ.

Strategic Substitution Independent of Cost Function

* Assume linear cost for competitors

Lemma

For given G, let ˜ G be similiar to G except that all firms i = a have cost ci(qi) = ˜

• cqi. Then, there is ˜

c such that ˜ γ = γ.

Proof (by picture).

qi π

i , 2

( q

i

, y

− i

) c1 c2 c3

Strategic Substitution Independent of Cost Function

* Assume linear cost for competitors

Lemma

For given G, let ˜ G be similiar to G except that all firms i = a have cost ci(qi) = ˜

• cqi. Then, there is ˜

c such that ˜ γ = γ.

Proof (by picture).

qi π

i , 2

( q

i

, y

− i

) c1 c2 c3

Lemma

For any instance with n firms, when a firm produces 1 unit less, its competitors produce n−1

n

units more.

• i=a

(yi,2 − xi,2) = n − 1 n (xa,2 − ya,2)

Elastic Demand in Market 1

* Assume identical, linear cost for competitors

Lemma

For given G, let ˜ G be similiar to G except that the price of market 1 is fixed at ˜ p1 = πa,1(x). Then, ˜ γ ≤ 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 ˜ G be similar to G except for ˜

• ca. If ˜

ca(q) = ca(q) for q ≤ xa,2 and ˜ c′

a(q) > c′ a(q) for q > xa,2, then ˜

γ ≤ γ.

Steep Cost -> Fixed Capacity

* Assume identical, linear cost for competitors; fixed price on market 1

Lemma

For given G, let ˜ G be similar to G except for ˜

• ca. If ˜

ca(q) = ca(q) for q ≤ xa,2 and ˜ c′

a(q) > c′ a(q) for q > xa,2, then ˜

γ ≤ γ.

Proof (by picture).

qa p1 π

a , 2

( q

a , 2

, x

− a , 2

) p1 + δ c′

a(qa)

qa p1 ˜ c′

a(qa)

π

a , 2

( q

a , 2

, x

− a , 2

) p1 + δ Steeper cost has no effect on market 2, but limits additional production

• n 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 ˜ G be similiar to G except that ˜ p1 = 0, ˜ p2(q2) = p2(q2) − p1, ˜ c = c − p1 and ˜ ca(qa) = 0 for qa ≤ xa,2. Then, ˜ γ ≤ γ.

Proof Overview

1

Establish basics

◮ Cournot equilibrium unique ◮ Price shock triggers strategic substitution 2

Series of simplifications

◮ Given instance G, construct simplified ˜

G with ˜ γ ≤ γ

3

Proof theorem for simplified game

Proof Overview

1

Establish basics

◮ Cournot equilibrium unique ◮ Price shock triggers strategic substitution 2

Series of simplifications

◮ Given instance G, construct simplified ˜

G with ˜ γ ≤ γ

3

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) 4n2 ≥ 3 4.

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) 4n2 ≥ 3 4.

Proof.

qa ˜ c′

a(qa)

π

a , 2

( q

a , 2

, x

− a , 2

)

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) 4n2 ≥ 3 4.

Proof.

qa ˜ c′

a(qa)

δ π

a , 2

( q

a , 2

, y

− a , 2

)

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) 4n2 ≥ 3 4.

Proof.

qa ˜ c′

a(qa)

δ π

a , 2

( q

a , 2

, y

− a , 2

) Easy to calculate γ in simplified instances.

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) 4n2 ≥ 3 4.

Proof.

qa ˜ c′

a(qa)

δ π

a , 2

( q

a , 2

, y

− a , 2

) Easy to calculate γ in simplified instances. γ is quadratic function of δ worst case δ is function of xa,2 maximum xa,2 minimizes γ.

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).