collusion with private monitoring and durable goods

8/2/2019 Collusion With Private Monitoring and Durable Goods

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Introduction and MotivationGeneral Model

Consumers Patience: A Simple ExampleReferences

Collusion with Private Monitoring andDurable Goods

Enrique Ide Carvallo2012

Ponticia Universidad Cat olica de ChileInstituto de Economa

March 29, 2012

Enrique Ide Carvallo Collusion with Private Monitoring and Durable Goods
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MotivationLiterature ReviewSketch of my Model

Contents

1 Introduction and MotivationMotivationLiterature ReviewSketch of my Model

2 General ModelAssumptionsCharacterizing the Cooperative PhaseOptimal Deviation and the Cartels Problem

3 Consumers Patience: A Simple ExampleOutlineResults



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Motivation

Two of the favorite hobbies of IO researchers...

How durability affects market power and industry congurationStudy of Collusion. Lately, involving asymmetric information amongrms.

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Motivation

Two of the favorite hobbies of IO researchers...

How durability affects market power and industry congurationStudy of Collusion. Lately, involving asymmetric information amongrms.

Idea: connect those two strands of literature.

Kind of asymmetric info we are studying today? Private Monitoring.


I d i d M i i


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Motivation

So...


I t d ti d M ti ti
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Literature Review - Durable Good Markets

Coase Conjecture (Coase, 1972): Loss Monopoly Power in DurableGood Markets with Patient Consumers


Introduction and Motivation
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Coase Conjecture (Coase, 1972): Loss Monopoly Power in DurableGood Markets with Patient Consumers

However Ausubel and Deneckere (1987); Gul (1987); Dutta et al.(2007) showed that a Duopoly can overcome the conjecture

Presence of a rival, helps credibly commit to a price over time

Moreover, same authors prove that more patient consumers helpscollusion




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Figure 1: Collusion with Durable Goods and Perfect Monitoring - Ausubel andDeneckere (1987); Gul (1987); Dutta et al. (2007)

Enrique Ide Carvallo Collusion with Private Monitoring and Durable Goods Introduction and Motivation M i i


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Literature Review - Collusion with Asymetric Information

Big area of research lately, although it has intrigued researchers atleast since Stigler (1964)


Introduction and Motivation M ti ti


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t oduct o a d ot vat oGeneral Model




Big area of research lately, although it has intrigued researchers atleast since Stigler (1964)

Imperfect Public Monitoring: some variable is public knowledge(either price or quantity)

Green and Porter (1984); Porter (1983a); Abreu et al. (1986);Harrington and Skrzypacz (2007)

Private Monitoring: rms price AND quantities are privateinformationHarrington and Skrzypacz (2011)


Introduction and Motivation Motivation


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General ModelConsumers Patience: A Simple Example

References



Green and Porter (1984): Collusion when price is public knowledge ispossible by periodically switching to a punishment phasePresence of Price Wars in equilibrium path.

Shortcomings...Empirically: we dont usually see price wars in cartelized industriesTheoretically: when monitoring is through sales Harrington andSkrzypacz (2007) showed that collusion is unsustainable withsymmetric punishments


Introduction and Motivation Motivation


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References



Green and Porter (1984): Collusion when price is public knowledge ispossible by periodically switching to a punishment phasePresence of Price Wars in equilibrium path.

Shortcomings...Empirically: we dont usually see price wars in cartelized industriesTheoretically: when monitoring is through sales Harrington andSkrzypacz (2007) showed that collusion is unsustainable withsymmetric punishments

Private Monitoring: Harrington and Skrzypacz (2011) and Chan andZhang (2009) proved that with asymmetric punishments, collusion ispossible even when self-reported sales are used to monitorcompliance.


Introduction and MotivationG l M d l Motivation
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References


Sketch of my Model

Overlapping Generation Model with Durable Goods in the fashion of Dutta et al. (2007), but stochastic environment.

Private Monitoring between rms

Consumers cannot see the effective price charged to anotherconsumer (realistic assumption)

We assume rms use Green and Porter (1984)s trigger strategies


Introduction and MotivationG l M d l Motivation
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ReferencesLiterature ReviewSketch of my Model

Sketch of my Model - Results

Collusion is sustainable

But contrary to the previous ndings, more patient consumers donot unambiguously facilitate collusion

Reason: new counteracting effect of consumers patience: the demand speculation effect

Given price-wars in equilibrium path, more patient consumers wait

until bad realization of demand forces rm to switch phase to buycheaper harms collusive prot difficults collusion.


Introduction and MotivationGeneral Model Motivation


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ReferencesLiterature ReviewSketch of my Model

Sketch of my Model - Results

More generally when consumers may delay purchases, if asymmetric,transfer-style schemes are available, symmetric punishment phasesare an inefficiently costly discipline device.

Transfer-schemes leave the pro-collusive effect of consumerspatience while eliminating the demand speculation effect

Adds up to the theoretical contribution explaining why we do not see

periodical Price Wars in cartelized industries.


Introduction and MotivationGeneral Model Assumptions
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ReferencesCharacterizing the Cooperative PhaseOptimal Deviation and the Cartels Problem

Contents





Introduction and MotivationGeneral Model AssumptionsCh i i h C i Ph
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ReferencesCharacterizing the Cooperative PhaseOptimal Deviation and the Cartels Problem

Assumptions

Assumptions about PopulationContinuum of consumers of mass 1, indexed by k , distributed on[0, 1]. Fraction t isyoungand 1 t are old. t i .i .d . r.v. dist. cdf G : [0, 1] [0, 1], where E [ t ] = (0 , 1).


Introduction and MotivationGeneral Model AssumptionsCh t i i g th C ti Ph
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Characterizing the Cooperative PhaseOptimal Deviation and the Cartels Problem

Assumptions

Assumptions about PopulationContinuum of consumers of mass 1, indexed by k , distributed on[0, 1]. Fraction t isyoungand 1 t are old. t i .i .d . r.v. dist. cdf G : [0, 1] [0, 1], where E [ t ] = (0 , 1).

Assumptions regarding ConsumersThe good is homogeneous and perfectly durableConsumers differ on their valuation v [0, v ] for the good.The valuation of a newly born generation always dist. cdf F : [0, v ] [0, 1]Buyers have rational expectations and disc. future by factor [0, 1]But because they die with chance E [ t ] = effective discountfactor is (1 ) .Resale is not allowed, and in case of a price tie, we simply assumethat the demand splits by halves.


Introduction and MotivationGeneral Model AssumptionsCharacterizing the Cooperative Phase


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Assumptions

Assumptions about FirmsIndustry composed by two identical rms with constant marginal cost= 0 and same discount factor


Introduction and MotivationGeneral Model AssumptionsCharacterizing the Cooperative Phase
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Assumptions

Assumptions about FirmsIndustry composed by two identical rms with constant marginal cost= 0 and same discount factor

Informational AssumptionsFirms cannot observe neither current or past charged prices, norquantities sold by rivalConsumers cannot see the effective price charged to anotherconsumer rms may charged different prices to different buyers.

However consumers personal characteristic are private information.Price offers must be made simultaneously.Neither rms nor consumers are able to observe the currentrealization of t



C P i A Si l E lAssumptionsCharacterizing the Cooperative Phase
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Assumptions

Denition

The players action space consist of all p i , t (k ) functions satisfying p i , t : [0, 1] [0, + )

A contingent strategy is therefore an innite sequence s i = {p i , t (k )}t = t =0 with p i ,0(k ) given, and where s i , t +1 : R

( t +1)+



Consumers Patience: A Simple ExampleAssumptionsCharacterizing the Cooperative Phase
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g pOptimal Deviation and the Cartels Problem

Assumptions

Strategy considered:

Start playing p i ,0(k ) = p k [0, 1] (cooperative phase). Then,If in cooperative phase in period t 1 and

q i , t 1 q then p i , t (k ) = p k [0, 1] and remain in thecooperative phase.q i , t 1 < q then p i , t (k ) = 0 k [0, 1] and go to the punishmentphase

If in the th period of the punishment phase in period t 1 and

< T 1 then p i , t (k ) = 0 k [0, 1] and continue in thepunishment phase T 1 then p i , t (k ) = p k [0, 1] and go to the cooperativephase


Introduction and MotivationGeneral ModelConsumers Patience: A Simple Example

AssumptionsCharacterizing the Cooperative Phase
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Consumer s Patience: A Simple ExampleReferences

g pOptimal Deviation and the Cartels Problem

Characterizing the Cooperative Phase

Suppose P (Price War ) = G () if both rms always cooperateThen E t p t +1 = [1 P (Price War )] p = [1 G ()] p

But

t i .i .d

Et p = [1 G ()] p

>t .



AssumptionsCharacterizing the Cooperative Phase
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Consumer s Patience: A Simple ExampleReferences Optimal Deviation and the Cartels Problem


Suppose P (Price War ) = G () if both rms always cooperateThen E t p t +1 = [1 P (Price War )] p = [1 G ()] p

But

t i .i .d

Et p = [1 G ()] p

>t .

Now, we can separate population in three groups:1 Young consumers2 Old consumers who have not experience a price war

3 Old consumers who have experience a price war



AssumptionsCharacterizing the Cooperative PhaseO i l D i i d h C l P bl
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Consumer s Patience: A Simple ExampleReferences Optimal Deviation and the Cartels Problem


Young Consumers Problem

max U = max {v p , (v E t p t +1 ), 2 (v E t p t +2 ), . . .} (1)

However, because E t p = [1 G ()] p > t and < 1, we cansimplify it to

max U = max {v p , (v E t p )} (2)



AssumptionsCharacterizing the Cooperative PhaseOptimal Deviation and the Cartels Problem
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p pReferences Optimal Deviation and the Cartels Problem


Young Consumers Problem

max U = max {v p , (v E t p t +1 ), 2 (v E t p t +2 ), . . .} (1)

However, because E t p = [1 G ()] p > t and < 1, we cansimplify it to

max U = max {v p , (v E t p )} (2)

Thus he buys today if and only if:

v p (v E

t p ) v v C

(p , q , ) =

p E t p

1 (3)

Therefore total demand emanated from this group is

2

{1 F [v C (p , q , )]} (4)



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References Optimal Deviation and the Cartel s Problem


Old consumers who have not experience a price war

Have only received offersp = p . Hence they bought the good whenyoung if v v C (p , q , ), or have not otherwise.

But, in this period they also received p = p , hence they only buy if v v C (p , q , ). Thus their expected demand is zero.

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R f



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References Optimal Deviation and the Cartel s Problem


Hence

P (q i , t q ) = P t 2

{1 F [v C (p , q , )]} q = G 2 q

1 F [v C (p , q , )]

Which implies that

(p , q , ) =2 q

1 F [v C (p , q , )]

Thus, the expected future price E t p is:

E t p = p [1 G ()] = p 1 G 2 q

1 F [v C (p , q , )]



References

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References p


Which, given rational expectations means that v C must satisfy

v C (p , q , ) =p

1 1 1 G

2 q 1 F [v C (p , q , )]

(5)

Proposition

There exists a continuous increasing functionv C : [0, v ]x [0, 1] x [0, 1 ] [0, v ], convex in p with

1 v C (p = 0 , q , ) = 0 (q , ) [0, 1]x [0, 1 ]2 v C (p , q = 0 , ) = p (p , ) [0, v ]x [0, 1 ]3 v C (p , q , = 0) = p (p , q ) [0, v ]x [0, 1]4 v C (p , q = 1 , ) = p 1 (p , ) [0, v ] x [0, 1 ]

that satises condition ( 5 ).



References

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References


Note that because v C (p , q , ) is increasing in (and therefore in ). Wehave

C = p 2 {1 F [v C (p , q , )]} C = p 2 {1 F [v

C (p , q , )]}

Iff . Moreover, if we assume that v C () is differentiable, it isstraightforward to prove that the price p that maximizes this expression isdecreasing in patience of consumers.

These facts will be the heart of our argument.



References

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References

Optimal Deviations and the Cartels Problem

For the Cartel to be successful, must be sustainable for every possiblehistory of { }

= t 1 =0



References

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References


For the Cartel to be successful, must be sustainable for every possiblehistory of { }

= t 1 =0

The incentives to deviate are increasing in residual demand fromoldconsumers without experience of punishment phase yet.

more stringent history of s: set of old consumers who alreadybought the good equal to zero



References

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DenitionA deviation for rm i consists on region U and a price p (k ) < p k U



References



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Suppose for example U = [0 , ], then

q j , t (p , q ; ) = t [1 ]

2{1 F [v C (p , q , )]}

Thus the expected future price predicted by consumers living in the [0, ]interval will be:

E t p = p 1 G 2q

{1 }{1 F [v C (p , q , )]}= p 1 G

1



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Suppose for example U = [0 , ], then

q j , t (p , q ; ) = t [1 ]

2{1 F [v C (p , q , )]}

Thus the expected future price predicted by consumers living in the [0, ]interval will be:

E t p = p 1 G 2q

{1 }{1 F [v C (p , q , )]}= p 1 G

1

Notice that this price expectation is independent of the actual deviationprice offered p (k )



References

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Now, given this expectations, consumer belonging to the fraction willbuy the good today to rm i only if v v D ( p , , p , q , ). Wherev

D

( p (k ), , p , q , ) = p E t p

1 =

1

1 p (k ) p 1 G

1



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Now, given this expectations, consumer belonging to the fraction willbuy the good today to rm i only if v v D ( p , , p , q , ). Wherev

D

( p (k ), , p , q , ) = p E t p

1 =

1

1 p (k ) p 1 G

1

The optimal deviation price for consumer k [0, ]:

maxp (k ) [0, p ]

p (k ) 1 F v D + (1 ) max 0, F v C F v D

But then is clear that optimal value of p (k ) is independent of k . Thus

p (k ) = p for all k [0, ].Enrique Ide Carvallo Collusion with Private Monitoring and Durable Goods


References

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Then, this period deviation prots are given by D + (1 ) C t , where D comes from the maximization of the previous problem.



References

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Then, this period deviation prots are given by D + (1 ) C t , where D comes from the maximization of the previous problem.

Notice the trade-off the deviator faces:The greater the deviation region, the greater are deviation prots.However, the greater the deviation region, the greater probability of switching to a punishment phase tomorrow

decreases todays demand in the deviation region forcing thedeviator rm to offer lower deviation prices




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We can now obtain our incentive compatibility constraint. DeneV (p , = 0) as the value function of rm i if he does not deviates.




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We can now obtain our incentive compatibility constraint. DeneV (p , = 0) as the value function of rm i if he does not deviates.Then,

V (p , = 0) = C t (p , q ; )+ G 2 q 1 F [v C (p , q , )]T V (p , = 0)

+ 1 G 2 q

1 F [v C (p , q , )]V (p , = 0) (6)

V (p , 0) = C t (p , q ; )

1 + G () { T +1 }(7)




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Now the incentive compatibility constraint requires that

( = 0) arg

max D

(,

) + (1 ) C

t (p , q ; )

+ G

1 T V (p , 0) + 1 G

1

V (p , 0) (8)






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Now the incentive compatibility constraint requires that

(

= 0) arg

max D

(,

) + (1

) C

t (p ,

q ;

)

+ G

1 T V (p , 0) + 1 G

1

V (p , 0) (8)

The cartels problem is therefore maximizes (7) subject to ( 8).




OutlineResults
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Contents







OutlineResults
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Consumers Patience: A Simple ExampleOutline - Assumptions

Things get simpler if t is discrete-binary r.v.

Turns out in this case q independent of p and T

solve problem sequentially: obtain rst q and then for p and T

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OutlineResults


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Consumers Patience: A Simple ExampleOutline - Cooperative Phase

In this case, if no rm ever deviates P (Price War | No Dev ) = 1 E t p = p > t




OutlineResults


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In this case, if no rm ever deviates P (Price War | No Dev ) = 1 E t p = p > t

Thus,

v C (p , ) = p 1 1

(10)

E t q i = 1 F (v C )

2 C =

p 1 F (v C )

2Again, it is self-evident that C is decreasing in and that the price thatmaximizes it, is also decreasing in that parameter.




OutlineResults


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Figure 2: Optimal Prices - Linear Valuations




OutlineResults


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Consumers Patience: A Simple ExampleOutline - Optimal Deviation

Suppose the rival deviates on U = [0 , ] of the market.




OutlineResults


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Suppose the rival deviates on U = [0 , ] of the market.

Because P (Price War | No Dev ), and therefore C , independent of q ,

optimal q should maximizeP

(Price War | Dev )Formally,

q = arg maxq

{P (Price War | Dev ) = (1 )+

P {1

} 1

F (v C )

2 q (11)



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OutlineResults

l l


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Obvious solution:

q = 1 F (v C )

2 (Price War | Dev ) = (1 ) + = 1 (12)

Independent of the deviation fraction .

Therefore, if a rm is ever going to deviate = 1 v D = p D

1 .Then per-unit deviation prots are:

maxp D [0, p ] D

= p D

1 F (v

D

) + 1 max 0, F (v

C

) F (v D

)+ (1 )max 0, F (v C ) F (v D )

(13)




OutlineResults

C P i A Si l E l


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Consumers Patience: A Simple ExampleOutline - Cartels Problem

Dene V as value function on date t given cooperative phase. Then

V = C + (1 ) T V + V V = C

1 { + (1 ) T }(14)



References

OutlineResults

C P i A Si l E l


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V = C + (1 ) T V + V V = C

1 { + (1 ) T }(14)

The incentive-compatibility constraint requires:

D + T +1 V C + (1 ) T V + V

D

C 1 1 T

1 { + (1 ) T } (15)



References

OutlineResults

C P ti A Si l E l
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V = C + (1 ) T V + V V = C

1 { + (1 ) T }(14)

The incentive-compatibility constraint requires:

D + T +1 V C + (1 ) T V + V

D

C 1 1 T

1 { + (1 ) T } (15)

Therefore the Cartels Problem is to maximizes (14) subject to ( 15).



References

OutlineResults

C P ti A Si l E l


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Consumers Patience: A Simple ExampleResults - Simulations

Two different though experiments:1 Assume that collusion is sustainable for a given parameter

conguration and collusive price. How does the length of thepunishment period changes in response to changes in , to keep theagreement sustainable?

2 Given a parameter conguration, how does the cartels value

function changes in response to changes in

, and how does thecollusion outcome evolves?



References

OutlineResults

Consumers Patience: A Simple Example


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Consumers Patience: A Simple ExampleResults: Simulations - Though Experiment 1

Figure 3: = 0 .5 / = 0 .5 Figure 4: = 0 .75 / = 0 .5



General ModelConsumers Patience: A Simple ExampleReferences

OutlineResults



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Consumer s Patience: A Simple ExampleResults: Simulations - Though Experiment 2

Figure 5: = 0 .9 Figure 6: = 0 , 9




OutlineResults



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Consumer s Patience: A Simple ExampleResults: Simulations - Collusive Outcome

Collusive Outcome for: = 0 .9 = 0 , 5 = 0 , 75




OutlineResults



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Consumer s Patience: A Simple ExampleResults - Intuition

More patient consumers do not unambiguously facilitate collusion.

This contrasts with previous results found in the literature.




OutlineResults



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More patient consumers do not unambiguously facilitate collusion.

This contrasts with previous results found in the literature.

Intuition: New, counteracting effect: the demand speculation effect Given price-wars in equilibrium path, more patient consumers waituntil bad realization of demand forces rm to switch phase to buycheaper harms collusive prot difficults collusion.

Thus the overall effect: consumer-induced discipline + the demand speculation effect , can go either way.




OutlineResults



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More generally, when consumers may delay purchases, symmetricpunishment phases are an inefficiently costly discipline mechanism.

Firms would be better-off with asymmetric, transfer-style, schemes. Eliminates the anti-collusive effect of active consumers, leavingthe pro-collusive effect only

Adds up to the theoretical contribution explaining why we do not seeperiodical Price Wars in cartelized industries.




OutlineResults



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More generally, when consumers may delay purchases, symmetricpunishment phases are an inefficiently costly discipline mechanism.

Firms would be better-off with asymmetric, transfer-style, schemes. Eliminates the anti-collusive effect of active consumers, leavingthe pro-collusive effect only

Adds up to the theoretical contribution explaining why we do not see

periodical Price Wars in cartelized industries.




OutlineResults



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Consumer s Patience: A Simple ExampleResults - Robustness

Results do not change if we allow other distributions for t .

In fact, for non-binary distributions, it is more likely that theanti-collusive effect of consumers patience dominates:

Reason: More complex distributions difficult cartels inference

should weaken the pro-collusive effects of active consumers andshould not affect the anti-collusive effect.




OutlineResults



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Consumer s Patience: A Simple ExampleResults - Robustness

Result should be robust to more complex punishment-phasesstrategies (see Abreu et al. (1986))

Results driven by the existence , and not the form of the punishmentphase.




OutlineResults

Some Ideas


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The model is well-suited for analyzing whether the Cartel shouldagree on a price or in a geographical/buyer allocation.

I have not prove it, but intuitively in this setting the Cartel shouldnot use buyer allocation because it looses the discipline coming fromactive consumers,




Abreu, D., D. Pearce, and E. Stacchetti ,Optimal Cartel Equilibriawith Imperfect Monitoring, Journal of Economic Theory, 1986, 39,


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with Imperfect Monitoring, Journal of Economic Theory , 1986, 39 ,251269.

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de la Fuente, Angel , Mathematical Methods and Models for Economist ,Cambridge University Press, 2000.


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Ca b dge U ve s ty ess, 000.Dutta, P., A. Matros, and J. Weibull ,Long-run Price Competition,

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, , ( ),Levenstein, M. and V. Suslow ,What Determines Cartel Success?,

Journal of Economic Literature , 2006, 44 , 4395.Liski, M. and JP. Montero , Forward Trading and Collusion inOligopoly, Journal of Economic Theory , 2006, 131, 212230.

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collusion with private monitoring and durable goods

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