Industrial Organization - Nonlinear pricing and exclusion

Introduction Contracts as a barrier to entry Nonlinear pricing and exclusion BO T (qE , qI ) T (qI ) Comparing

Industrial OrganizationNonlinear pricing and exclusion

Laurent Linnemer

CREST (LEI)

2014/15

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IntroductionDominant firm and exclusion

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Landmark antitrust cases

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Theories of harm?

Predation?

• Hard to convince judges (and economists)• Likely to be rare in practice

In this class: Static scenario of exclusion

• link the shape of price-quantity schedules to demandbehavior and exclusionary strategy

• relate the nonlinearities to uncertainty about rivalcharacteristics

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Two types of models on exclusion

Naked exclusion (complete information)

• Rasmusen, Ramseyer, and Wiley (1991)• Segal and Whinston (2000)• DeGraba (2013)

Rent shifting (incomplete information)

• Aghion and Bolton (1987)• Marx and Shaffer (1999) (complete info)• Choné and Linnemer (2014a)• Choné and Linnemer (2014b)

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http://www.jstor.org/stable/2006909


http://dx.doi.org/10.1016/j.ijindorg.2013.09.001



https://ideas.repec.org/p/crs/wpaper/2014-16.html



Citations forAghion and Bolton (1987)“Contracts as a barrier to entry” 1987-November 2014: 238 citations

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Contracts as a barrier to entryAghion and Bolton (1987)

Three players: one buyer, two sellers

• Buyer: needs 1 unit, valued at 1• Dominant firm: cost 0 < cI < 1• Entrant: cost cE ∈ [0,1] distribution F (.), f ().

Welfare

• 1− cI if I is the seller• 1− cE if E is the seller

Timing

• B and I sign a contract (commitment) before cE is known• B and E learn cE and bargain

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Contracts as a barrier to entryAghion and Bolton (1987)

The dominant firm makes a take it or leave it offerT =

{P if qI = 1P0 if qI = 0

• P regular price• P0 stipulated damage for breach of contract

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Aghion and Bolton (1987)Equilibrium

Choice of B and E for given P, P0, and cE

• Surplus for B − E if B buys from E : 1− cE − P0

• Surplus for B − E if B buys from I: 1− P• B buys from E iff 1− cE − P0 > 1− P• B buys from E iff P − P0 > cE

• If E has all the bargaining power: PE = P − P0

By choosing ∆ = P − P0 the pair B − I controls entryChoice of ∆ to maximize B − I profits?

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Aghion and Bolton (1987)Equilibrium

Choice of ∆ to maximize B − I profitsF (∆) (1−∆) + (1− F (∆)) (1− cI)f.o.c. is:cI −∆ = F (∆)/f (∆)

Assumptions

• F/f is↗• F (0)/f (0) = 0

A unique solution denoted ∆∗

• Main property: 0 < ∆∗ < cI inefficient exclusion• Example: cE uniformly distributed over [0,1]

• Then ∆∗ = cI/216 / 41



Aghion and Bolton (1987)Split of the profit between I and B

Individual profits

• ΠI = F (∆)P0+(1− F (∆)) (P − cI) = P−cI−F (∆) (∆− cI)

• ΠB = F (∆) (1−∆− P0) + (1− F (∆)) (1− P) = 1− P• constraint: ∆ = ∆∗

P is used to split the profit between B and Iand P0 is such that P − P0 = ∆∗

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DiscussionIn Aghion and Bolton the focus is on the stipulateddamages but

T =

{P if qI = 1 (and qE = 0)P0 if qI = 0 (and qE = 1)

equivalent to

T = P0 + (P − P0)qI

Three interpretations

• stipulated damages• two-part tariff (with price below marginal cost)• exclusivity offer

Also not clear if the price schedule is of the formT (qI) or T (qE ,qI)

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Nonlinear pricing and exclusionChoné and Linnemer (2014a)

A general model

• Same timing and uncertainty as Aghion Bolton• Continuous quantity choice• Differentiated goods• Disposal costs

Three types of price schedule

• Conditional: T (qE ,qI)

• Non conditional: T (qI)

• Exclusivity: (T (qI) and T x(qI))

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Nonlinear pricing and exclusionThe model

Utility of B: V (qE ,qI) =

maxxE≤qE ,xI≤qI

vExE + vIxI − h(xE , xI)− γ(qE − xE )− γ(qI − xI).

where h is a convex function, ∂h/∂xE (0,0) = ∂h/∂xI(0,0) = 0

∂2h/∂xE∂xI > 0 ; exple: h(xE , xI) = x2E/2 + x2

I /2 + σxExI ,0 ≤ σ < 1

Total Welfare when x = q (no disposal)W (qE ,qI) = V (qE ,qI ; vE )− cEqE − cIqI

Key parameters:

• Surplus: ωE = vE − cE and ωI = vI − cI

• Conditionally efficient quantities: q∗E (qI ;ωE ) and q∗I (qE ;ωI)

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Preferences of the Buyer

W = cst

No disposal

Some units ofgood I are

disposed of

Some units ofeach good are

disposed of

Some units ofgood E are

disposed of

qI = vI + γ − σqE

qE = vE + γ − σqIq∗∗I

q∗∗E

A

vI + γ

vE + γ

qI

qE

q∗I(qE ;ωI)

ωI

q∗E(qI ;ωE)

ωE

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Last stages (complete information)Choices of qI and qE by B and E

Given T = T (qE ,qI) or T (qI)

SBE = maxqE ,qI

V (qE ,qI)− T − cEqE ,

Key observations:

• If T = T (qI), then qE = q∗E (qI ;ωE )

• If T = T (qE ,qI) = P(qE ) + cIqI , then qI = q∗I (qE ;ωI)

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Last stages (complete information)Split of the surplus between B and E

Outside options0 and U0

B = maxqI≥0

V (0,qI)− T (0,qI)

Surplus∆SBE = SBE − U0

B

SplitΠE = β ∆SBEΠB = (1− β) ∆SBE + U0

B

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Negotiation between B and IAt this stage: incomplete information

B and I max their joint expected profitEcE ,vE ΠBI = EcE ,vE {W (qE ,qI ; cE , vE )− ΠE}

Two inefficiencies

1. Ex post inefficiencyqI 6= q∗I (qE ; θE )

2. Inefficient foreclosure

• Complete when qE = 0 < q∗E• Partial when 0 < qE < q∗E

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Buyer OpportunismTake the rebates and run!

Why would the Buyer pay more for less units?25 / 41


Buyer OpportunismPricing should take disposal costs into account

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Buyer OpportunismOptimality result

PropositionAt the second-best optimum, the buyer consumes all units (ofboth goods) she has purchased.• Good E: BI have nothing to gain from pushing B to

purchase too much• Good I: A matter of credibility⇒ ∂T/∂qI ≥ −γ

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Conditional price-quantity schedule T (qE ,qI)Rewriting the program of B and I who can control both qE and qI

Notations

• F distribution of ωE on [ωE , ωE ]

• f continuous density function• The hazard rate f/(1− F ) is↗ in ωE .

Surplus created by B and E∆SBE (ωE ) = max

qE ,qI≥0ωEqE + vIqI − h(qE ,qI)− T (qE ,qI)− U0

B

⇒ ∂∆SBE∂ωE

= qE (ωE ) then using ΠE = β∆SBE ⇒∫ ωEωE

ΠE (ωE )f (ωE ) dωE = ΠE (ωE ) + β∫ ωEωE

qE (ωE )[1− F (ωE )] dωE

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Conditional price-quantity schedule T (qE ,qI)Rewriting the program of B and I who can control both qE and qI

Virtual surplusEωE ΠBI = EωE S v(qE ,qI ;ωE )− ΠE (ωE ) with

S v(qE ,qI ;ωE ) = ωEqE + ωIqI − h(qE ,qI)− βqE1− F (ωE )

f (ωE )

Pointwise Maximization of S v(qE ,qI ;ωE )

• ωE − ∂h∂qE

(qcE ,q

cI ) ≤ β 1−F (ωE )

f (ωE )

• ∂h∂qI

(qcE ,q

cI ) = ωI

• ⇒ qI is conditionally efficient, qI = q∗I (qE )

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W = cst

qcI

qcE

C

q∗E

(qI ;ωE − β 1−F

f

)

ωE − β 1−Ff

Sv = cst

No disposal

Some units ofgood I are

disposed of

Some units ofeach good are

disposed of

Some units ofgood E are

disposed of

qI = vI + γ − σqE

qE = vE + γ − σqIq∗∗I

q∗∗E

A

vE + γ

qI

qE

q∗I(qE ;ωI)

ωI

q∗E(qI ;ωE)

ωE


Conditional schedule T (qE ,qI)

Proposition

1. B and I agree on T = P(qE ) + cIqI with P ′′ < 0 and

P ′(qE ) = β(1− F (ωE ))/f (ωE )

2. ∀ωE , qcE ≤ q∗∗E . The quantity purchased from the

incumbent firm, qcI = q∗I (qc

E ;ωE ), is efficient given qcE and

distorted upwards relative to q∗∗I .3. The magnitude of the disposal costs, γ, does not affect the

buyer’s supply policy or the price quantity scheduleT (qE ,qI).

Remark: No market share rebates (suboptimal)

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Unconditional price-quantity schedule T (qI)Rewriting the program of B and I

The key: qE is conditionally efficient

• BI can control qE only through qI

• ⇒ qE = q∗E (qI ;ωE )

Surplus created by B and E∆SBE (ωE ) =maxqI≥0

ωEq∗E (qI ;ωE ) + vIqI − h(q∗E (qI ;ωE ),qI)− T (qI)− U0B

⇒ ∂∆SBE∂ωE

= q∗E (qI ;ωE ) then using ΠE = β∆SBE ⇒∫ ωEωE

ΠE (ωE )f (ωE ) dωE =

ΠE (ωE ) + β∫ ωEωE

q∗E (qI ;ωE )[1− F (ωE )] dωE

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Unconditional schedule T (qI)Rewriting the program of B and I

Virtual surplusS̃ v(qI ;ωE ) = S v(q∗E (qI ;ωE ),qI ;ωE ) =

ωEq∗E (qI ;ωE ) + ωIqI − h(q∗E (qI ;ωE ),qI)− βq∗E (qI ;ωE )1−F (ωE )f (ωE )

Pointwise Maximization of S v(q∗E (qI ;ωE ),qI ;ωE )

Technical requirements for equilibria à la Martimort-Stole 2009• Assumption: quasi-concavity of S̃ v(qI ;ωE ) in qI

• Assumption: ∂2S v/∂qI∂ωE < 0

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W = cst

ωI/σ

qcI = quI

qcE quE

C U

q∗E

(qI ;ωE − β 1−F

f

)

ωE − β 1−Ff

Sv = cst

q∗∗I

q∗∗E

A

qI

qE

q∗I(qE ;ωI)

ωI

q∗E(qI ;ωE)

ωE


Unconditional schedule T (qI)

Proposition

• quE = q∗E (qu

I ;ωE ) ≤ q∗∗E and quI ≥ q∗∗I

• When γ is large, the possibility of buyer opportunism doesnot constrain pricing policy and (qu

E ,quI ) such that

• ωI − ∂h∂qI

(quE ,q

uI ) = β 1−F (ωE )

f (ωE )∂q∗

E∂qI

• −γ < T ′ ≤ cI

• When γ is small, T ′ = −γ• (qu

E ,quI ) is efficient if goods are independent (or β = 0)

• (quE ,q

uI )→ the efficient allocation as γ → −cI

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Unconditional schedule T (qI)

Proposition

• T (qI) concave if q∗E not too convex in qI (true in thequadratic case)

• T ′(quI (ωE )) = max

(cI + 1−F (ωE )

f (ωE )

∂q∗E

∂qI, −γ

)< cI

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Comparing T (qE ,qI), T (qI), and (T ,T x)Equilibrium quantities of good E under each type of price schedule

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Comparing T (qE ,qI), T (qI), and (T ,T x)Equilibrium quantities of good I under each type of price schedule

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Comparing T (qE ,qI), T (qI), and (T ,T x)Buyer opportunism in regimes u and x : qI is above q∗

I (qE ;ωI)

0 q∗∗E (ωE)

q∗∗I (ωE)

q∗∗I (ωE)

qI

qI

qE

(qE = q∗∗E (ωE), qI = q∗I(ωE))

(qE = qcE(ωE), qI = qcI(ωE))

(qE = quE(ωE), qI = quI (ωE))

quE(ω̃uE)

(qE = qxE(ωE), qI = qxI (ωE))

quE(ωxE)

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Comparing T (qE ,qI), T (qI), and (T ,T x)Price-quantity schedules in the three regimes

quI (ωE)

$

qI

cIqI + T (0) T (qI)

qI

T (qI)

+X

T x(qI)

q∗I(0;ωI)qxI (ωxE)

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Comparing T (qE ,qI), T (qI), and (T ,T x)Welfare at the first-best and second-best allocations

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Industrial Organization - Nonlinear pricing and exclusion