Collusion and price rigidity

M.I.T.

_UBRARIES

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COLLUSION

AND

PRICE

RIGIDITY

Susan Athey

Kyle Bagwell

Chris Sanchirico

No.

98-23

November,

1998

massachusetts

institute

of

technology

50 memorial

drive

Cambridge,

mass.

02139

WORKING

PAPER

DEPARTMENT

OF

ECONOMICS

COLLUSION

AND

PRICE

RIGIDITY

Susan Athey

Kyle Bagwell

Chris Sanchirico

No.

98-23

November,

1998

MASSACHUSEHS

INSTITUTE

OF

TECHNOLOGY

50

MEMORIAL

DRIVE

CAMBRIDGE,

MASS. 02142

MASSftCh JSETTSINSTITUTE

OF^'ECHMQ_LOGY

DEC

4

₁₉₉₈

Collusion

and

Price Rigidity

Susan

Athey,

Kyle

Bagwell,

and

Chris Sanchirico*

Last Revised:

November

1998.

Abstract

We

consider

an

infinitely-repeatedBertrand

game,

in

which

prices are perfectly ob-served

and each

firm receives a privately-observed, i.i.d. cost

shock

in

each

pciiod.

We

focus

on

symmetric

perfectpublic equilibria

(SPPE),

v^'herein

any "punishments"

are

borne

equally

by

all

firms.

We

identify

a

tradeoff thatis associated with collusive pricing

schemes

in

which

the price to

be

charged

by

each firm

is strictly increasing in its cost level:

such

"fully-sorting"

schemes

offerefficiency benefits, as they ensurethatthelowest-cost

firm

makes

thecurrentsale, but they

also

imply an

informational cost (distorted pricing and/or equilibrium-path price wars), since a higher-cost firm

must

be

deterred

from mimicking

a lower-cost

firm

by

charging

a

lower

price.

A

rigid-pricing

scheme,

where

a

fum's

collusive priceis

independent of

itscurrent costposition,

sac-rifices efficiency benefitsbutalsodiminishestheinformationalcost.

For

a

wide

range

of

settings,

the optimal

symmetric

collusive

scheme

requires (i). the

absence of equiUbrium-path

price wars,

and

(ii).

a

rigidprice. Iffirmsare sufficiently impatient,

however,

the rigid-pricing

scheme

cannot

be

enforced,

and

the collusive price

of

lower-cost firms

may

be

distorted

downward,

in orderto

diminishtheincentiveto cheat.

JEL

Classification

Numbers: C73,

L13, L14.

Keywords:

Collusion,repeated

games,

privateinformation,pricerigidity, auctions.

*M.I.T.and

NBER

(Athey);ColumbiaUniversityEconomics Department and Graduate School ofBusiness,and

NBER

(Bagwell);

andColumbiaUniversityEconomics Department andLaw School(Sanchirico). >\feare grateful toGleimEllison,EricMaskin, DavidPearce, LarsStole,JeanTirole, and seminarparticipants atBarcelona(PompeuFabra), BostonCollege,Columbia,

Har-vard,M.LT,N.YU., Northwestern and Tbulouse O.D.E.L)for helpful discussions. Atheythanks the National ScienceFoundation (SBR-9631760)forgenerousfinancialsupport,andflieCowlesFoundationatYale Universityfor hospitalityandsupport.

1.

Introduction

In thestandard

model

of

collusion,

symmetric

firmsinteractin

an

infinitely-repeatedBertrand

game

in

which

past prices are perfectly observed.

While

the standard

model

deUvers

a

number

of

useful

insights, a limitation

of

this

model

is thatit

presumes

an

unchanging

market

environment.

This

limitation isimportant for

two

reasons. First,the

scope

for testing

a

proposed

theory

of

collusion

is greater, ifthe theory offers predictions

concerning

the

manner

in

which

collusive prices

may

vary with underlying

market

conditions.

Second,

a theory

of

collusion is

of

potential value for

macroeconomists,

ifthe theory

determines

the

response

of

collusive prices to

demand

and

cost

shocks.

These

limitations

have been

partially

addressed

in

two

celebratedextensions

of

thestandard col-lusion

model.

Rotemberg

and

Saloner

(1986) extend

the standard

model

to includethe possibility

of publicly-observed

demand

shocks

that are i.i.d.

over

time.

When

the

demand

shock

is large,

theincentive tocheat (undercutthe collusive price)isacute,

and

collusion

becomes

more

difficult toenforce.

Rather

than foregocollusive activity altogether, firms then

reduce

the collusive price,

therebydiminishingthe incentive tocheat

and

bringingincentives

back

inline. Inthis

way,

Rotem-berg

and

Saloner (1986)offera prediction

of

countercyclical

markups

forcollusive

markets

thatare subject to

demand

shocks.'

Like

thestandard

model,

their

model

does

notpredict that actual "price

wars" occur

on

the equilibriumpath;rather, thesuccess of collusion varies along theequilibrium path

with

the

demand

shocks

thatareencountered.

Following

the seminal

work

of

Stigler (1964),

a

second

literature stresses that a firm

may

be

unable

to perfectly

monitor

the

behavior

of

itsrivals.

Green

and

Porter (1984) explorethis

possi-bilityin

an

infinitely-repeated

Coumot

model.

They

assume

that

a firm

cannot observe

theoutput choices

of

rivalsbutthatallfirms

observe a

publicsignal (the

market

price) thatisinfluenced

both

by

output choices

and

an unobserved

demand

shock.^

A

colluding

firm

thatwitnesses

a

low

mar-ket pricethenfaces

an

inference

problem,

as itisunclear

whether

thelow-price

outcome

aroseas

a

consequence of

a

bad

demand

shock

or a secretoutput

expansion

by

arival.

The

Green-Porter

(1984)

model

thus representscollusion in thecontext

of a

repeated

moral-hazard

(hidden-action)

model,

and a

centralfeature

of

theiranalysisisthat

wars occur along

theequilibriumpath following

bad

demand

shocks.

In this paper,

we

propose a

third extension

of

the standard collusion

model.

We

consider

an

^

_As

_{HaltiwangerandHarrington (1991)andBagwell and}

Staiger (1997)explain,theassociationbetweencollusion

andcountercyclicalmarkups can beoverturned

when

theassumption ofi.i.d.

demand

shocksisrelaxed.

^ _{Whilethe Green-Porter (1984)model}

isdevelopedinthe contextof

Coumot

competition, themaininsightscanbe

c^tured

m

arepeatedBertrandsetting,as Tirole(1988)shows.

The

Green-Porter (1984)modelisfurtherextendedby

infinitely-repeated Bertrand

game,

in

which

each

firmis privately

informed of

its unit cost level in

each

period, thereisa

continuum

ofpossiblecosts,

and

the cost realizationis i.i.d. across firms

and

time. Currentprice selections(butnotcostreahzations)are

pubUcly observed each

period.

We

thus represent collusion inthecontext

of

a repeatedadverse-selection (hidden-information)

model

with perfectly-observedactions (prices).^

Our

model

is well-designed to contribute to a long-standing issue in Industrial Organization

concerning

the relationship

between

cdllusion

and

price rigidity in the presence

of

cost shocks.

Empuical

studies

by

Mills (1927),

Means

(1935)

and

Carlton (1986, 1989)

have concluded

that prices are

more

rigid in concentrated industries. This empirical finding suggests that collusion

is associated with a greater

tendency toward

price rigidity. Further support for this suggestion is

offered

by

Scherer (1980,pp. 184-93),

who

summarizes

a

number

of

studiesthatfind thatcollusion

isoften

implemented with

"rules

of

thumb,"

whereby

the collusive priceisfixedat

some

focallevel,

such

asacertainpercentage

above

a

commonly-observed

wholesaleprice."

When

firmscolludein

this

manner,

the collusiveprice is

independent

of

the privatecost positions

(beyond

the

wholesale

price)

of

the

member

firms.

At

the

same

time, the Industrial Organization Uteraturehas as yet failed to provide a

satisfac-tory theory that

Imks

pricerigiditywithcollusion.

The

best

known

theoryisthe

"kinked

demand

curve" theory offered

by

Sweezy

(1939)

and

Hall

and Hitch

(1939).

As

Scherer (1980)

and

Tu-ole

(1988) discuss,

however,

this theory has important shortcomings.

The

theory

employs

an

ad

hoc

behavioralpostulate,

compresses

a

dynamic

story into astatic

framework, and does

not

determine

the collusiveprice.^

Given

these Umitations,IndustrialOrganization

economists have

gravitated

to-ward

the

more

informal

view

that pricerigidityis

appeaUng

tocollusive firms,

because

arigid-price collusive

scheme

preventsmistrust

and

reducesthe risk

of

aprice

wan

Carlton (1989)explains:*

^ Ourmodeling frameworkisrelated torecentworkthatextendstheGreen-Porter (1984)modeltoallowfor

privately-observeddemandsignals. Compte(1998)and Kandori andMatsushima(1998),forexample,supposethatfirms

pub-liclychoose "messages" afterprivatelyobservingtheirrespective

demand

signals. Inoursetting, bycontrast,firms

are privatelyinformedas totheirrespective costsandthepublicactionisa(payoff-relevant)pricechoice.

^ _{Evidence fromothersourcesalso indicates that rigidpricing}isaubiquitous featureofcollusiveendeavors. For example,inaBusinessVkek (1975)article,anumberofexecutivesinavarietyofindustriesdescribe informal(and

sometimesformal)agreementstomaintainaparticular price. HallandHitch (1939) providesimilaraccounts. Scherer (1980)describesacollusiveschemewhereby GeneralElectricand Westinghouse adopted apricingformulafor turbine generators.Collusionseemsespeciallyprevalentinmarketswithhomogeneousgoods(see, e.g.,

Hay

andKelley (1974)

andScherer (1980,p. 203))andrelatively inelasticdemands(see, e.g.,Eckbo(1976)).

^ Inthekinked-demand-curvetheory,itisassumedthatafirm believesthat rivals will(not)match aprice (increase) decrease. This behavioralpostulateimplies akinkinthe

demand

curve,from whichitfollowsthatpriceandoutput

areunresponsive tosmall maiginal costfluctuations. Maskinand lirole(1988)offer anequilibriummodelofthe

kinked-demandcurvetheory, basedonthehypothesisthatfirms

make

alternatingpricechoicesin aninfinitegame.

Their theorydoesnot includethe possibilityofcostshocks,however.

^ Forasimilarview,seeScherer (1980,p. 180),

who

writes"Mostoligopolies...appearwillingtoforego themodest

gains associatedwithmicro-meterlike adjustnientofprices to fleetingchangesin

demand

andcosts inordertoavoid

5ie riskofmoreserious lossesfrompoorly coordinated pricingpoliciesandpricewarfare." SeealsoScherer (1980,

"Thepropertyofthekinkeddemandcurve theorythatpriceisunresponsivetosomecost f lucDiationsispreservedinmost

discussionsofoUgopolytheory whetherornotbased onthekinkeddemandcurve. Thereasoningis thatinoligopoUes

pricesfluctuate less inresponsetocostchanges(especiallysmall ones) than theywouldotherwiseinorder nottodisturb existing oligopolistic discipline.Anytimeapricechangeoccursinanoligopoly, thereisarisk thatapricewarcould break

out. Hence,firmsare reluctant tochangeprice." (Carlton,1989,pp. 914-15).

Inthepresentpaper,

we

develop a

rigorous evaluation

of

thisinformal reasoning.

Focusing

on

the private cost fluctuations that firms experience, v^^e explore the extent to

which

"mistrust" limits

colluding firms' ability to

respond

to their respective cost positions.

The

costs

of

mistrust are

formalized in

terms

of

the price

wars

and

pricing distortions that are required to dissuade firms

from

misrepresentingtheir privateinformation.

Our

formal

analysisbegins

with

the static

Bertrand

game

with

private cost information.

This

game,

which

has

been

previously

analyzed

by

Spulber

(1995), constitutes the stage

game

of

our

repeated-game model.

Inthe

unique

symmetric

Nash

equiUbrium,

the pricing strategy is strictly increasing in the firm's costlevel.

An

appealingfeature

of

this pricing

scheme

isthat sales in the currentperiodareallocated tothefirm

with

thelowestcost.

This

isthe efficiency benefit

of a

"fully-sorting" (i.e., strictly-increasing) pricing

scheme.

The

pricing function is strictly increasing as a

consequence of

the winner-take-allnature

of

the

game,

together

with

the single-crossingproperty presentinthe firm's

expected

profitfunction.

The

single-crossingproperty arises

because

when

a

firm

has

lower

costs, itvalues

more

theincrease in

expected

sales that a

lower

price implies.

At

the

Nash

equilibrium,

a

higher-cost

firm

isthus

unwiUing

to

"mimic"

the

lower

price thatit

would

charge

were

costslower, despite theincreasein

expected

sales thata

lower

price

would

confer.

We

turn nextto the

repeated-game

model and

explore

whether

firms

can

support

better-than-Nash

profits

when

they interact repeatedly.

We

focus

on

the class

of symmetric

perfect public

equilibria

(SPPE).

An

SPPE

collusive

scheme

ata

given

point in

time

can

be

described

by

(i)

a

price for

each

cost type

and

(ii)

an

associated equilibrium continuation value for

each

vector

of

current prices,

where

thecontinuation valueis

symmetric

acrossfirms. In

an

SPPE,

therefore,

col-luding firms

move

symmetrically

through

any

cooperative orprice-war phases.

By

contrast,in

an

asymmetric

perfect publicequilibrium

(APPE),

firms

keep

track

of

past sales

and

may

coordinate

current price selections in orderto treat

more

favorably

a firm

thathas

experienced

low

sales in therecentpast.

Athey and Bagwell

(1998) study optimal

APPE.

After describing

our

results

about

SPPE,

we

will return to

compare

the

two

kinds

of

equilibria.

We

observe

that

a

collusive

scheme must

satisfy

two

kinds

of

incentive constraints. First, for

every firm

and

costlevel, the short-termgain

from

cheating

with

an

off-schedule deviation (i.e.,

with a

price thatisnot assignedto

any

costtype

and

thatthus represents

a

cleardeviation)

must be

unattractive, in

view

of

the (off-the-equilibrium-path) price

war

that

such a

deviation

would

imply.

As

isusual in

repeated-game

treatments ofcollusion, this constraint issure to

be

met

iffirmsare

sufficiently patient.

Second,

the

proposed conduct

must

also

be such

that

no

firm

iseverattractedto

an

on-scheduledeviation,

whereby

afirmof a givencosttype misrepresentsitsprivateinformation

and

selectsapriceintendedfor adifferentcost type.

To

characterize theoptimal

SPPE,

we

build

on

the

dynamic-programming

techniques putforth

by

Abreu,

Pearce

and

Stacchetti(1986, 1990)

and Fudenberg, Levine and

Maskin

(1994).

We

draw

an

analogy

between

our

repeated hidden-information

game

and

thestatic

mechanism

designliterature,

in

which

theon-scheduleincentive constraintis

analogous

to thestandardincentive-compatibility constraint, the off-schedule incentive constraint serves as a counterpart to the traditional partici-pation constraint,

and

the continuation values of the repeated

game

play the role

of

"transfers."

However,

unlike

a

standard

mechanism

design

problem

in

which

transfersare unrestricted, theset offeasiblecontinuation valuesis limited

and endogenously

determined.

For example,

aprice

war

is

analogous

toatransferthatis

borne

symmetrically

by

allfirms.

We

break

ouranalysis of optimal

SPPE

into

two

parts.

We

suppose

first thatfirms arepatient, sothattheoff-scheduleincentive constraintismet. Inthisanalysis, theon-scheduleincentive

con-straint capturesthe informational costs

of

collusion that confront privately-informedfirms.

The

central

problem

isthat the

scheme must

be

constructedsothat

a

higher-costfirm

does

not

have an

incentive tomisrepresentits costs as lower, thereby securing for itself

a lower

price

and

a higher

expected

market

share. In

an

SPPE,

theinformationalcosts

of

collusion

may

be

manifested

in

two

ways.

First,the prices

of

lower-cost firms

may

be

distorted to

sub-monopoly

levels.

This

is

a

poten-tiallyeffective

means

of

eliciting truthfulcostinformation,since

under

thesingle-crossingproperty higher-costfirmsfind

lower

priceslessappealing.

Second,

followingthe selection

of

lower

prices, the collusive

scheme

may

sometimes

call for

an

equilibrium-pathprice

war

infuture periods.

The

current-periodbenefits

of

a

lower

pricethen

may

be

of

sufficient

magnitude

to

compensate

forthe future cost

of a

pricewar,onlyifthe firmtruly has

lower

costs inthecurrentperiod.

A

richarray

of

collusive

schemes

fitwithinthe

SPPE

category.

At one

extreme, firms

may

incur

theinformationalcosts

of

collusionpurelyinterms

of

distorted pricing.

An

example

is the

Nash-pricing

scheme,

in

which

firms repeatedly play the

Nash

equilibrium

of

the static

game.

At

the otherextreme, firms

may

adopt

a monopoly-pricing scheme,

where

each firm

observesitscostfor theperiod

and

thenselectsits

monopoly

price. If

demand

slopes

down

(so thatthe

monopoly

price varies

with

cost), this

scheme

satisfiestheon-scheduleincentive constraint

only

ifequilibrium-path

wars sometimes

follow the selection

of

lower-cost

monopoly

prices.

More

generally, collusive pricing

schemes

may

involve

both

distorted pricing

and

equilibrium-path wars.

While

all

SPPE

coUusive

pricing

schemes

callfor prices thatare(weakly) increasingincosts,

some

schemes

allow

schemes

are fully sorting.

One

scheme

of

particular interest is the rigid-pricing

scheme.

Under

this

scheme,

each firm

selectsthe

same

price in

each

period,regardless

of

itscurrent cost position,

and

sotheinformational

costs ofcollusion are neutralized:

a

high-costfirm has

no

incentive to misrepresentits cost, since

the

same

price is required

under

any

realization.

Under

a

rigid-pricing

scheme,

therefore,

any

deviation is oflf-schedule, representing

an

unambiguous

violation

of

the agreement.

The

threat

of

a

price

war

may

thus

be

limited to off-equilibrium path events.

The

downside

of

the

rigid-pricmg scheme, however,

is that it

does

not offer efficiency benefits:

one firm

may

have lower

coststhanitsrivals,

and

yetthefirms sharethemarket.

We

thus seethe outline

of

a

broad

tradeoff:

iffirmsseek acollusive

scheme

that

emphasizes

efficiency benefits (e.g.,afiilly-sorting

scheme),

then the informational costs

of

collusion

may

be

large; while if firms

seek

a

collusive

scheme

that

moderates

theinformationalcosts

of

collusion (e.g., the rigid-pricing

scheme),

then they

may

sacrificeefficiency benefits.

Our

first

main

finding is that firms fare poorly

under any

SPPE

collusive

scheme

that insists

upon

fullsorting. Infact,considering the entire setoffully-sorting

SPPE,

we

findthat in thebest

such

equilibrium firms

employ

the

Nash

pricing

scheme:

itisbetter to

have

low

(Nash)

prices

and

no

actualprice

wars

thanhigher(e.g.,

monopoly)

prices

and

actualpricewars.

Thus,

ifcolluding firms are to

improve

upon

Nash

profits,the collusive pricing

scheme must

exhibit

some

rigidity.

We

next considerthefullclass

of

SPPE

collusion

schemes

and

reporta

second

main

finding: if

firmsare sufficiently patient

and

demand

is inelastic, then

an

optimal

SPPE

collusive

scheme

can

be

achieved without recoursetoequilibrium-pathpricewars.

We

also

provide

sufficientconditions

under

which

theoptimal

SPPE

does

notrely

on

equilibrium-pathpricewars,for patientfirms

when

demand

is

downward

sloping;

however,

we

identify

a

limited set

of

circumstances

where

they

might

be

used.

The

finding that optimal collusion generally

does

not require price

wars

stands

insharp contrast tothe analysis presented

by Green and

Porter (1984), suggesting

a fundamental

difference

between

repeated

games

with

hidden

actions

and

those

with

hidden

information

and

publicly-observedactions.

Armed

withthese findings,

we

next

add

some

additional structure

and

provide

a

characterization

of

theoptimal

SPPE

collusive

scheme.

Specifically,

when

firms are patient

and

the distribution

of

costtypesis logconcave,

we

establish

a

third

main

finding: if

demand

isinelastic,optimal

SPPE

collusionischaracterized

by

a

rigid-pricing

scheme,

in

which

firms selectthe

same

price(namely,

the reservation price

of

consumers)

in

each

period,

whatever

theircostlevels.

We

provide

as well sufficientconditions for rigid-pricing

behavior

inthe optimal

SPPE when

demand

is

downward

and

collusion described above.

We

then turn to the

second

part

of our

analysis,considering impatientfirms.

We

begin

by

show-ingthat impatience

on

the part of firms creates

an

additional disadvantage to usingprice wars: a

scheme

with highprices today sustained

by

wars

in the future

makes

a deviation especially

prof-itabletoday, while simultaneouslyreducingthevalue

of

cooperationinthefuture.

Our

second

(no-wars) findingthereforecontinuesto hold,

even

when

firmsareimpatient. Next,

we

observe

thatthe ofif-scheduleincentive constraintis particularly

demanding

forlower-costtypes. Intuitively,

when

a firm

draws

a lower-costtype, the temptation to undercutthe assigned priceis severe, sincethe resulting market-share gainis thenespecially appealing.

For

impatientfirms, acollusive

scheme

thus

must

ensurethatlower-cost types receive sufficient

market

share

and

select sufficiently

low

prices in equilibrium,sothatthegains

from

cheating arenot toogreat.

This

logic is reminiscent

of

the

argument

made

by Rotemberg

and

Saloner (1986), although here it isprivatecost

shocks

(as

opposed

to public

demand

shocks) that necessitate modification

of

thecollusive

scheme.

We

confirm

thislogic

with

our

fourth

main

finding: iffirms arenot

suffi-cientlypatienttoenforcethe rigid-pricing

scheme,

they

may

stillsupportapartially-rigidcollusive

scheme,

in

which

the price

of

lower-cost typesisdepressed

below

that

of

higher-costtypesinorder

to mitigate the former's incentive to cheat. This finding suggeststhat

symmetric

collusion

between

impatient firms

may

be

marked by

occasional(and

perhaps

substantial)pricereductions

by

individ-ual firms.

These

departures

occur

when

a firm

receives

a

favorable costshock,

and

they represent a permitted "escape clause" (i.e.,

an

opportunity tocut prices

and

increase

market

share

without

triggering retaliation)withinthe collusive

scheme.

We

further

show

thattheoptimalpricing

problem

forimpatientfirms

can

also

be

studiedusing

the toolsofstatic

mechanism

design,althoughthe

problem

is

more

difficultthanthe standard case

where

theoutside option,

which

affectsthe participation constraint, is

a

constant. Inthe repeated

game

context, the value

of

the outside option (that is,

openly

cheating

on

the

agreement)

varies

both with

the firm's cost type

and

the collusive

agreement

itself,

through

the price specified for

each

costtypeinthe present, aswellas

through

the futurevalue

of

cooperation.

Finally,

we

considerthe

consequences of

relaxing

our

restrictionto

symmetric schemes.

Asym-metric

schemes

allow

one firm

to

enjoy a

more

profitable continuationvalue than another,

and

in

this

way

such

schemes

facilitatetransfers

^m

one firm

to another.

While

we

do

not

propose a

full treatmentof

APPE

(see

Athey

and

Bagwell

(1998)),

we

do

provide a simple

example

thatcaptures

the qualitative benefits offered

by

asymmetric schemes.

The

scheme

we

study allows

only

two

prices

on

theequilibriumpath.

The

low

price

may

be

arbitrarilyclose to the

high

one; but,

over

some

efficiency benefits in the first period

of each

cycle, since

a

high-cost firm will hesitate to

expend

itsonly opportunitytochargethe

low

price

when

it

knows

that

tomorrow

its

expected

costs

wiU

be

lower. In contrast, low-cost firms will

wish

to increase their

expected market

share inthe

firstperiod, since

on

average,

winning

will

be

lessvaluablein the

second

period.

Although

APPE

can

provide

such

efficiency benefits,

we

focus here

on

SPPE

for three

main

reasons. First,

under

many

conditions the optimal

SPPE

is stationary.

While

we

do

not

propose

a

theory

of

how

firms learn

and

coordinate

on

aparticularequilibrium, the rigid-price

SPPE

(and

more

generally, stationary

symmetric

equilibria) are appealingly simple.

By

contrast,

a

necessary condition for

APPE

to

improve

upon

SPPE

is that the

asymmetric

equilibrium is nonstationary.

Such schemes

are

more

sophisticated,

and

may

be

most

plausible

when

a small

number

of firms

interact frequently

and

communicate

explicitly.

Second,

most

of

theinformalliterature

about

col-lusion (discussed above) highlights firms' fear

of

industry-wide

breakdowns

in collusion,

and

it

is thus interesting to analyze

SPPE, which

isolate this consideration. Finally,

SPPE

are the

only

alternative iffirms

can observe

the prices offeredin themarket, but they

cannot observe

(orinfer)

theidentities

of

thefirms

who

offertheseprices.

This

situation arisesin

procurement

auctions

with

more

than

two

bidders,ifthe

winning

bid-but notthe

name

of

the

winning

supplier-is

announced.

Our

findingsare related tothose

developed

by

McAfee

and

McMillan

(1992),intheiranalysis

of

bidding

rings. Inastatic

model,

they

show

thata fixedprice(i.e., "identicalbidding")istheoptimal

strategy forbiddingcartelsin first-priceauctions for

a

single object,

when

thecartels are

"weak"

(i.e.,firmsareunableto

make

transfers).

The

analysis that

we

presentfortheinelastic-demand case

may

be

understood

interms

of

a

procurement

auction. Infact,in

our

analysis

of

theoptimal

SPPE,

we

generalize the weak-cartel

model,

since the static

mechanism

we

analyze is directly derived

from

a

repeated

game

and

allows for

a

restricted class

of

transfers (corresponding to

symmetric

price wars).'

Our

rigid-pricingfindingthusprovidesadditional theoreticalsupportforthe

common

practice

of

identicalbidding.*

We

also

extend

theprevious analysis inotherdirections, including

downward-sloping

demand

and

impatientfirms.

The

paper

proceedsas follows. InSections

2

and

3,

we

describe thestatic

and

repeated

games,

respectively.

We

present

our

findings for

SPPE

among

patientfirmsinSection4. Impatient firms

are considered in Section 5.

We

explore

simple

asymmetric

schemes

in Section 6.

Concluding

thoughts arepresentedinSection7.

"^

_McAfee

andMcMillan(1992)alsoconsider "strong"cartels,inwhichtransfersare unrestricted. (SeealsoCramton

andPalfrey (1990)and Kihlstrom and Vives(1992).)Incomparison, ouranalysisof

SPPE

restrictstransfers toa range ofsymmetricvalues (namely,

SPPE

continuationvalues). Likewise,in

APPE,

firmshaveavailableanexpanded(but

stillrestricted) classoftransfers(namely,

APPE

continuationvalues).

* _{See,forexample,}

_Comanor

_{andSchankerman(1976),}

_{who show}

_{thatfixed-pricerules(orcloselyrelatedsimple}

rotationschemes) have been usedbyasubstantialfractionofthedocumentedcasesofbidcUngringsinprocurement

2.

The

Static

Game

We

begin

by

considering

a

static

game

of

Bertrand competition

when

firmspossessprivate informa-tion.

This

game

illustratesthe

immediate

tradeoffsthatconfront firms indetermining theirpricing

policies. In

subsequent

sections,

we

examine

a

dynamic game,

sothat

we may

consideraswellthe

long-term

impUcations of

differentpricingpoUcies.

The

static

game

serves as the stage

game

for the

dynamic

analysis.

2.1

The Model

We

suppose

thatthereare

n

ex

ante identicalfirmsthat

engage

in

Bertrand

competitionfor salesina

homogenous-good

market.

Following Spulber

(1995),

we

modify

thestandard Bertrand

model

with

the

assumption

that

each

firmisprivately

informed

as toitsunitcostlevel. Specifically,

we

assume

thatfirmi's "type" Qiis

drawn

in

an

i.i.d. fashion

from

thesupport[^,Q\accordingtothe

commonly

known

distribution function FiQ),

where

the

corresponding

density j{&)

=

F'{Q) is

everywhere

positive. Afterthe firms learntheirrespective cost types, they simultaneously

choose

prices.

Let

Pi

G

J?+ denotethe price

chosen

by

firmi.

The

associatedprice profileisthen represented

by

the vector,

_p

=

_(pi, ...,p„).

On

the

demand

side

of

themarket,

we

assume

a

unit

mass

of

identical

consumers,

each

of

whom

possesses a

demand

function

D

(p),

where p

is the price at

which

the

consumer

buys.

We

allow that the

demand

function is either (i) inelastic, or (ii) positive, twice continuously differentiable

and

decreasing (i.e.,

"downward

sloping"). In the

former

case,

we

suppose

that

D{p)

—

\

up

to

some

reservation pricer,

where

r

>

6?

A

price strategy for

firm

iis a functionpi{6i)

mapping from

theset

of

cost types, [9,6], tothe

set

of

possibleprices,

R^.

The

functionpiis

assumed

continuouslydifferentiable, except

perhaps

atafinite

number

ofpoints (so as to allow for schedules

with

jumps).'"

A

price strategy profile

is thus a vector

p{0)

=

(pi (9i),

p_i

{0-i)),

where

6

=

{$i, 0-i) isthe vector

of

cost types

and

p_i

(O-i) isthe profile

of

rivalprice strategies.

Each

firm

chooses

itsprice strategy

with

the goal

of

maximizing

its

expected

profit,

given

its cost type.

To

represent

a

firm's

expected

profit,

we

require

two

further definitions. First,

we

define it_{p,6)

=

[p

—

6]D

(p) as the profit that

a

firm

receives

when

itsetsthe price

_p

and

hascosttype

6

and "wins"

the entire unit

mass

of consumers.

We

assume

that tt_(p,6) has a

unique maximizer,

p"^

(9),

and

is strictly quasi-concave."

Second,

we

specify

a

Bertrand market-share-allocationfunction,rui _{(p), that indicates}firmi's

market

share

® _{Inthecaseofinelasticdemand,}

_we

_may

_{therefore interpret thestaticmodelasamodelofa procurement}

auction.

>\feassumethatr

>

flfor simplicity.

^° _{Thisrestriction}

includesawiderangeofpossible pricingstrategies,anditservestominimizetechnical

complica-tions thatarenotcentral toourdiscussion.

^^ _{Underinelasticdemand,p'"(6)}

₌

_r.

_When

_{demandslopesdown,p"^{0)}

isincreasingin9.

when

thevectorofrealizedpricesisp.

This

functionallocates

consumers evenly

among

firmsthat

tie for the lowest price in the market; therefore, rrii _(p)

=

1 ifp,

<

mirij^ipj, rrii _(p)

=

if

Pi

>

min.^^Pj,

and

mi(p)

=

1/k

ifthere are /c

—

1 other firmsthattiefirmi forthe lowestprice.

We

may now

define firm z's

ex

post

profit as ttj _(p,6i)

=

it_{(p^, di)}

mi

_(p).

This

is the profit thatfirmi enjoys after allprices are posted,

when

it

wins

a share

m^

_(p)

of consumers.

It is also

convenient

torepresent

firm

i'sinterim profit,

which

isthe

expected

profitforfirmi

when

ithascost

type6i, selects the pricep^

and

anticipates thatrivalprices will

be determined

by

therival pricing strategy profile,

_{p_t}

_{0-i).

Firm

i's interimprofit function is thus defined as ni(pj,p_i,^j)

=

E[Ki{pi,

p_t

(^-t) ,Oi)],

where

theexpectationistaken

over

0_j. Finally,

we

may

alsodefine

firm

i's

ex

ante profit.

This

isthe profit that

firm

i expects before

any

firm (includingiitself)learns its

cost type,

when

itisanticipated that firms' eventual prices are

determined

by

thepricingstrategy

profile,

_p

(0).

Firm

i's

ex

anteprofit is

given

by

ni(p)

=

E[U.i{p(6) ,_6i)],

where

the

remaining

expectationistaken

over firm

i's

own

costlevel.

2.2

Equilibrium

Pricing

and

Profit

We

return

now

to the interim profit function defined

above

and

describe the essential tradeoff that confronts a firm

when

it sets its price. Interim profit

may

be

written as Ili{Pi,P-i,9i)

=

K{Pi,

9i)Emi(pi,p-i{6

_i)), indicating that

firm

i's price p^ affects its profit

both

by

altering its

"profit-if-win" (i.e.,7r{pi,9i))

and

itsprobability

of

winning

(i.e.,

_{Emi{pi, p_j(0_J)). Thus,}

when

afirm

of a

given type considers

whether

to

lower

itsprice

from

some

status

quo

level, it

must weigh

the effect

of

an

increaseinthe

chance of winning

against the direct effect

of

the pricereduction

on

profit-if-win.

An

important feature

of

the

model

is that different types feel differently

about

this tradeoff.

In particular, theinterimprofit fiinction satisfies

a

single-crossingproperty:

lower

types find the

expected-market-share increasethat

accompanies a

pricereduction

more

appecding than

do

higher

types. Inthefirstplace,since

lower

types

have lower

totalcosts,they

have

higherprofit-if-win(i.e., 7rfl(p,6)

<

0). Therefore, theincreased

chance of winning

(i.e, theincrease in

_{Emi{p, p_t(0_J)}

from

lowering

priceis

of

greaterbenefitto

a

lower-costfirm. Secondly, the reductionin

profit-if-win from

lowering

priceislessseverefor

lower

types(i.e.,tt^(p,6)

>0):

revenue

changes

arethe

same

acrossalltypes,but since

lower

types also

have lower

marginalcost,their costsrise

by

less

inresponse to increased

demand.

This

second

effectisstrict

when

demand

slopes

down,

as then

the

lower

price

does

raise

demand, and

itisneutral inthe case

of

inelastic

demand.

As

isstandard, the single crossing property impliesthat higher-cost firms

must

selecthigher prices(i.e,_Pi{6i) is

Employing

techniques

from

the auction literature

(Maskin and

Riley, 1984),

Spulber

(1995) analyses this stage

game

in detail

and

finds in part:

Proposition

1 (Spulber (1995))

The

static

game

has

a

unique

Nash

equilibrium,

which

1) is

symmetric: _Pi

=

p^, Vi; 2)iscontinuouslydifferentiable

and

strictlyincreasing

over

6

€

(^,6); 3)

is

below

the

monopoly

price: p^_{6)

_<

p^{9),

"iO

<

6 and;

4) yields positive interimprofit

for

all

typesbutthe highest6,

who

never

wins

and

whose

price p^{6)

=6

would

yieldzeroprofit

even

ifit

did.

Notice that the

symmetric

equilibrium pricing strategy p^ is continuous

and

strictly increasing.

These

properties arise asa

consequence of

the "winner-take-all" nature

of

thestatic

game.

If, for

example,

theschedule

were

to

jump

up

discontinuouslyat9,thena

firm

with

type

6

would do

better increasingits price intothe

"gap"

(provided_p^ifi)

_<

p^{d)); whileifthe priceschedule

were

to

have

a flat section that included type 6, then

a

firm ofthis type

would

gain

by

shading

its price

downward

aslight

amount

(converting "ties" into "wins").

These

restrictions are standard inthe

literature,but, as

we

shall see,theydisappear

when

we move

totherepeated

game.

3.

The

Repeated

Game

Inthis section,

we

definetherepeated

game

thatisthe focus

of our

study.

We

alsopresenta

"Fac-tored

Program"

and

establish

a

relationship

between

solutions tothis

program and

optimal

SPPE.

We

usethisrelationship in

subsequent

sections,

where

we

characterize optimal

SPPE.

3.1

The Model

Imagine

thatfirms

meet

periodafterperiodtoplaythestage

game

describedintheprevioussection,

each with

theobjective

of

maximizing

its

expected

discounted

stream

of

profit.

Assume

furtherthat,

upon

entering a period

of

play,

a firm

observes

only

the historyof: (i)its

own

costdraws, (ii) its

own

pricingschedules,

and

(iii)the realized prices

of

itsrivals.

Thus,

we

assume

that

a firm does

notobserverivaltypesorrivalprice schedules.

Formally,

we

describe therepeated

game

inthe followingterms.

A

fullpath

of

playis

an

infi-nite

sequence

{0*,p*},

with

a given

pair in the

sequence

representing

a

vector

of

types

and

price

schedules atdatet.

The

infinite

sequence

implies

a

publichistory

of

realized pricevectors {p*},

and pathwise

payoffsfor firmi

may

be

thusdefinedas

where

6\ is the cost level for

firm

i atdate

t,6

£

[0,1) isthediscountfactor,

and

7ri(p*, 6\) is

ex

which

may

be

written as hi

—

{Ol,pl,p*Li}J=i-

(The

null history is the firm's information set at

the

beginning of

thefirstperiod.)

A

(pure) strategy for

firm

iassociates apriceschedule Si{hi){9i)

with

each

informationsethi.

Each

strategy profile s

=

(si, ...,s^)induces aprobability distribution

over

play paths {0',

p*}

in the usual

manner.

The

expected

discounted

payoff

from

s is thus the

expectationVi{s)

=

E[vi{{6^,p*})]taken

with

respecttothis

measure on

playpaths.

We

would hke

our

dynamic

game

to

have a

recursive structure, sothat

dynamic-programming

techniques (as

developed

by

Abreu, Pearce and

Stacchetti (1986, 1990)

[APS])

may

be

applied.

We

do

this at

two

levels. First,

we

assume

that

firm

types arei.i.d. across

time

(andfirms), sothat the

game

is "structurally recursive."

Second,

we

follow

Fudenberg,

Levine

and

Maskin

(1994)

[FLM]

and

restrictattentiontoarecursive solutionconcept;namely,

we

focus

upon

thosesequential equilibria in

which

players condition

only

on

the history

of

realized prices

and

not

on

their

own

private history

of

types

and

priceschedules.

Such

strategies arecalledpublicstrategies

and

such

sequential equilibria are calledperfect public equilibria (PPE).

Note

that

PPE

are tested against deviations that

do

condition

on

private histories; that is, they are true sequential equilibria in the

game. Note

also that

a

public strategy

may

be

abbreviated as

a

map

from

finite public histories

{p*}^i

to price schedules.

Using

familiar

arguments

(see,e.g.,

FLM),

these

two

restrictionsimply:

Lemma

2

(Recursion

of

Perfect

Public

Equilibria)

The

continuation

of

any

PPE

after

any

history

isitself

a

PPE

inthefull

game and

yieldspayoffsinthecontinuation

equal

to

what would

have

been

obtained

had

thestrategy profile

been used

from

thestart.

Before

moving on

to aformal analysis

of

the perfectpublicequilibria fortherepeated

game,

we

highlight

some

ofthe

novel

features

of

this

game.

First,

a

PPE

ina repeated

game

with hidden

information

must be

immune

to

two

kinds

of

current-perioddeviations.

A

firmdeviates "off-schedule"

when

it

chooses a

pricenotspecified for

any

costrealization: i.e.,apricenotinthe

range of

_Pt.

When

a firm

prices inthis

manner,

it

has

unam-biguouslydeviated;consequently,ifthecollusive

scheme

prescribes

a

punishment

(i.e.,

a

reduction

incontinuationvalues) following

such

adeviation,

and

iftheprospect

of

such a

punishment

deters

a

firm

from

undertakingtheoff-scheduledeviation(as will

be

trueiffirms aresufficientlypatient), thenthat

punishment

will

never

actually

occur

(i.e., it isoff theequilibriumpath).

By

contrast,

a

firm

deviates "on-schedule"

when

it

chooses a

price that is assigned to

some

cost level,

but

not

its

own.

For example, a

firm

may

be

tempted

to

choose

the

lower

price assignedto

a

lower

cost

realization inordertoincreaseits

chances of

winning

themarket. Importantly,

an on-schedule

de-viationis notperfectlyobservable,as

a

deviation, torivalplayers:

a

rival

can

not

be

surethatthe deviating

firm

was

nottruly

of

the cost type thatit is imitating.

Thus,

firms

may

be

tempted

to

constructacollusive

scheme

that

imposes

aharsh

punishment

when

low

prices are chosen,asthis

would

servetopreventon-scheduledeviations

by

higher-costtypes.

But

such a

punishment

would

be

costly, sinceit

would

occur

alongtheequilibriumpath

of

play,

whenever

firmsactually realized

low

costs.

Secondly,

we

notethattherepeated

game

admits aricherrange

of

equilibriumpricingschedules than didthe static

game.

As

discussedinSection 2, theequilibriumpricing schedule forthe static

game

can have

neitherflat

segments nor jumps. This

is

no

longertrue intherepeated

game. For

example,

we

may

specify thatundercutting

a

flatprice leads to

a

harsh

punishment,

and

this

may

be

sufficient to deterthe (off-schedule) deviation.

More

generally, the repeated

game

adds

athird ingredient to the tradeoffthat afirm confronts

when

it considers reducing price.

Now

lowering

price

means

notjust: 1)

an

increased

chance

ofwinning,

and

2)

lower

profit-if-win, butalso 3) a

differentcontinuationvalue.

Finally,

we

observe thatthe

mixture

of

on-

and

off-scheduleincentive

problems

is

endogenous

to the firms' choice

of

price schedules.

A

completely

flat schedule

means

a

relatively acute off-schedule incentive

problem,

but

no

on-schedule

problem.

A

price schedule that spans the

fuU

relevant

range

ofpriceshas

no

off-schedule

problem,

but possibly

an

acuteon-schedule

problem.

With

thechoiceof apriceschedule,therefore, firms

endogenously determine

the support

of

actual pricesthat

may

be

observed.

3.2

Optimal

Symmetric

Collusion:

The Program

Ina

symmetric

perfect public equilibrium

(SPPE),

followingevery publichistory,firms

adopt

sym-metric price schedules: Si{p^,...,

p^)

=

Sj{p^, ...,p^), Vz,j,T,

p\

...,

p^.

Symmetry

means

that

allfirms suffer future

punishments

and rewards

together.

For example,

it

might be

thatfirm 1 runs

the risk

of

triggering

an

industry-wide

breakdown

incollusion

should

itrepeatedlychaige

implau-sibly

low

prices.

Symmetric

equilibriathus capturethe "fear

of

breakdown"

thatcollusive firms

may

experience,asdiscussedinthe Introduction.'^

Our

goal is to formulate the

problem

of

characterizing the optimal

SPPE

in

such a

way

that

we

can

employ

the tools

of

(static)

mechanism

design.

We

begin

by

constructing

a

programming

problem

(The

"Factored

Program")

interms

of

current-periodprice schedules

and

"continuation

'^ _Despite

itsclear restrictiveness,aricharrayofstrategy profilesstillfitunderthesymmetricheading.Forinstance,

anyamountofhistorydependenceis allowed. Thus,pricewars canbetriggeredbyanynumberofperiodsof"bad behaviot" Thismeansinparticular thatthelaw oflargenumbers

may

beutilized: sincetypesaredrawni.i.d. across timeandfirms,an accuratemeasure oftheaverage typeofeachfirmbecomesavailable, andthis can becompared

totherealizedaverageannouncedtype(i.e.,the typecorrespondingto theaimouncedprice)todeterminewhether a

given firm hascheated (See,e.g.,Radno*(1981)andRubinsteinand"Ykari(1983)foradiscussionof

how

thelaw of

largenumbers

may

be usedtoimplement such "reviewstrategies.")Moreover,gradationsofpunishmentareavailable: pricewars canvaryinseverityandlengthandso

may

betailored tothe"fitthe crime."

payofffunctions,"

and

claim

from

APS

logic that

any

solution to this

problem

corresponds

to an optimal

SPPE. To

this end,

we

first

make some

notational adjustments that reflectthe

symmetric

setting. Let Tl{p)

denote

the

ex

ante profit to

a

firm

when

all firms use the

symmetric

pricing strategyp,

and

letU.{p',p),representthe

ex

anteprofit toafirm

when

itadoptsthepricing schedule p'whileallotherfirms

employ

the

symmetric

pricing strategy

p

.

(The

notation for thecontinuation

payoff

function

developed

below

is interpreted similarly.) Let V^

G

$ft

denote

the set

of

SPPE

continuation values

and

write

_^

=

infV^

and

v

=

sup

Vg. Note, initially, that

with

a

continuum

of

possiblepricing strategies thereis

no

a

priori basis

from which

to argue that either

v

E

Vs

o-veVs.

Any

symmetric

public strategy profile s

=

(s,...,s)

can he

factored into a first-period price

schedule

_p

and a

continuation

payoff

function v: 5R"

—

> 5?.

The

continuation

payoff

function describes the

repeated-game payoff v{p) enjoyed

by

all firms

from

the perspective

of

period

2

onward

after

each

first-periodprice realization

_p

=

_{(p^, ...p„)}

G

SR".

Each

firm's

payoff

from

s

can

then

be

writtenasll{p)

+

6Ev{p).

Now

considerthefollowing

problem

in

which

we

choose

factorizations _{p,v) ratherthan

choos-ing strategiesdirectly:

The

Factored

Program:

Choose

price schedule

p

and

continuation

payoff

function

v

to

maxi-mize

n(p)

+

6Ev{p)

subjectto:

Vp

e

3?^,

v{p)

e

V^,

and

Vp',

n(p)

+

8Ev{p)

>

n{p',p)

+

6Ev{pf,p).

An

important ingredientin

our

argument

iscontainedinthefollowingresult:

Lemma

3

Ifthe

symmetric

public strategyprofile s*

=

(s*,...,s*)factorsinto(p*,

v*),and

{p*,v*)

solves the

Factored

Program,

thens* is

an

optimal

SPPE.

The

lemma

follows

from

the fact that (p,v) is the factorization

of

some

SPPE

if

and

only

ifit

satisfiesthe

Factored

Problem's

two

constraints.

To

seethisequivalence,notefirstthat,

by

Lemma

2, every

SPPE

must

satisfy

v{p)

G

Vg.

Every

SPPE

also satisfies the

second

constraint,

because

an

SPPE

must

be

immune

todeviationsinthefirstperiod. Conversely,

one

shows

that

any

(p,v)

satisfyingthe constraintsisthefactorization

of

an

SPPE

by

appending

to

_p

the

SPPE

continuation

strategy profilesunderlying v{p),

and

thenarguing

from

theone-stage-deviation-principlethatthe

constructedstrategy profileisitself

an

SPPE.

We

referthe readerto

APS

forfulldetails.

To

analyzetheFactored

Program

further,

we

rewriteitinequivalentinterim-profitform, parsing

the constraints into on-

and

off-schedule conditions.

Using

abbreviated notation in light

of

the

symmetric

analysis:

The

Interim

Program: Choose

priceschedule

p

and

continuationpayoff function

v

to

maximize

Ei[Yi{p{ei),p,ei)

+

5E_iv{p{ei),p)\

subjectto:

Off-Schedule Constraints: Vp'

_^

p{[9,0]),

(IC-ofifl)

Vp_i,

v{p',p_,)eV,

(ic-off2) yOi, u(p{ei),p,Oi)

+

8E.iv{p{ei),p)

>

n(p',p,Oi)

+

8E.iv{p',p)

On-Schedule

Constraints: \/9i,

(IC-onl)

Vp_i,

v{p{ei),p_i)eVs

(ic-on2) v^i, n{p{ei),p,Oi)

+

6E_iv(p{ei),p)

>

n(p{ei),p,Oi)

+

6E_iv{p{9i),p).

Notice

how

the on-schedule constraints are written in "direct" form: for

given

p

and

v, the on-scheduleconstraintrequires thata

firm

with typedi

does

better

by

"announcing"

thatitstypeis _6i

than

by

announcing

some

othertype, 9i. This observation suggests thatthe

on-schedule

constraint

can be understood

as corresponding to a "truth-telling" constraint in

an

appropriate

mechanism

design formulation.

4.

Optimal

Symmetric

Collusion

Among

Patient

Firms

We

break our

analysis

of

symmetric

collusion into

two

parts. Inthe current section,

we

suppose

that firms arepatient, so thatthe ofif-schedule incentive constraints aresatisfied.

The

assumption

ofpatientfirmsisthen relaxedinSection5.

Our

strategyisto

show

that

when

firmsarepatient,theInterim

Program

can

be

furtherrelaxedto yield

what

we

callthe

Mechanism

Design Program.

We

thenuse standardtools

from

the

mechanism

designliteraturetocharacterizeoptimal

SPPE,

soas to offerpredictionsregarding

observed

pricing

patterns incollusivesettings.

4.1

The

Mechanism

Design

Program

To

begin,

we

define

a

new

function that represents interim profit in direct form:

H(9,

_9;p)

=

The

Mechanism

Design

Program:

Choose

priceschedule

_p

and

function

T

: [d,'9\

—

> 3? to

maximize

E[H{9,9;p)-T{e)]

subject to : V^,

T{e)

>

(ic-onM)

v^,e,

H{e,

_e]p)

-

T{e)

>

h{9,

e-.p)

-

T{e).

To

relate the Interim

and

Mechanism

Design Programs,

we

first

observe

that,

while

the

Mech-anism Design

Program

uses different notation, the objective function is the

same.

We

thus

fo-cus

on

comparing

the constraints.

Observe

thatfor

any

(p^,v^)

which

satisfies the constraints

of

the Interim

Program,

we

can

find a

corresponding

T^

such

that (p^,

T^)

satisfies the constraints

of

the

Mechanism

Design Program.

To

see this, recall that

v

=

supK,

and

define T^{9i)

=

6{v

—

E-iV^{p^(9i),p^)).

Then,

the restriction

v

e

Vg implies that

v{p)

<v

for all p,

and

thus

T\9)

>

0. Further, {p^,v^) satisfies (IC-on2) in theInterim

Program

if

and

onlyif (p^,

T^)

satis-fies

(IC-onM).

Now

consider a

(p^,T^) which

satisfies the constraints

of

the

Mechanism

Design Program.

As

we

discuss below, the ofif-schedule constraints

of

the Interim

Program

can

be

satisfied

when

firms are sufficientlypatient.

Turning

to the

on-schedule

constraints,

we

can

define

v^

such

that

T^{9i)

=

5{v

-

E_iV^{p^(9i),p^)),

whereby (IC-onM)

and

(IC-on2) are equivalent.

But

(IC-onl)

requires further that

v^{p^{9i),p_i)

^

^a,

and

so it

remains

toverify that

such a

v^

can

be

found.

There

are

two

cases: (a) the

"no

wars"

case,

where

T^

=

0;

and

(b)thecase

where

T^

is

sometimes

positive.

Consider

firstcase(a),

whereby

our

construction requires

v^{p^{9),p^))

=v

forall 0. If

K

is

closed, so that

v

E

Vs, this also satisfies (IC-onl).

We

proceed

by showing

that

v

can indeed

be

obtainedas

an

SPPE

ifthe solution tothe

Mechanism

Design

Program

is (p*,

T*

=

0).

Thus,

suppose

forthe

moment

that

{p*,T*

=

0).If

we

relaxtheInterim

Program by

replacingthe

(IC-onl) constraint,v{p{9), p_j)

G

K,

with

the constraintv{p{9),

_{p_^}

e

K

U

v,

two

conclusions

follow. First, (p*,v*(p*{9),p*))

=

v) solvestherelaxed Interim

Program. Second,

any

solution to therelaxed Interim

Program

yields profitsatleastaslargeas the

supremum

of

the profits available in the(unrelaxed) Interim

Program,

v.Formally,

EH{9,

9;

p*)+6v

>

v,or

EH(9,

9;p*)/(l

-6)>v.

But, since(p*,

T*

=

0) satisfies

(IC-onM), p*

must

satisfy

(IC-0n2)

for

any

continuation

value

functionthat

does

not

vary

with

the

announced

costtype.

Thus,

the collusive

scheme where

p*

is

used

inevery periodis

an

SPPE

so long

as theoff-scheduleconstraints are satisfied. Ifp* yields

per-periodprofitsgreaterthanthestatic

Nash

equilibrium(as

we

will establish inSections4.3.4

and

4.4.3),theoff-scheduleconstraints

can

be

satisfied

when

firmsare sufficiently patient(supported

by

the threatof foreverreverting to thestatic

Nash

equilibrium). Therefore, since

EH{6,

_6;p*)/(l

—

6)

G

Vg, the definition

of

v

implies

v

>

EH{d,6;p*)/{l

—

6).

As

the reverse inequality

was

established above,

we

thus

conclude

that thebest

SPPE

is the stationary equilibrium

where

each

firm uses p* in everyperiod.

Summarizing:

Proposition

4

(Staticizing)

Suppose

thatthe

Mechanism

Design

Program

issolved

by

(p*,T*),

where

T*

=

0.

Then

36*

_e

_{(0, 1)}

such

that, if

6

>

6*,

(i)p*

maximizes

EH

{9,e;p) subjecttoyd,6,

H{9,

9;p)

>

H{9,

9;p).

(ii)In

an

optimal

SPPE,

firms adopt

p*afterallequilibrium-pathhistories. (Hi)

V

=

EH

_{(9, 9;}p*)/{I

-

6)

e

Vs.

This resultestablishesconditions

under

which

the

problem

of

finding

an

optimal

SPPE

can be

reduced

to asimple

and

familiar(static)

mechanism

designproblem. In the

remainder

of Section

4

we

show

that,

under

many

conditions, the solutionto the

Mechanism

Design

Program

(p*,

T*)

in fact entails

T*

=

0.

However,

we

also describe a limited set

of

circumstances

under

which

price

wars

may

be used

(case(b)

from

above). Itispossibletogeneralize part(iii)

of

the staticizing

propositiontothecase

of

nonstationaryequilibria,

showing

that

when

firmsare sufficientlypatient,

E[H{9,

9;p*)

—

T*{9)]/{1

—

6)

=

v.

However,

such

aresultrequires that

K

is

convex,

sothatall

positive, finite

punishments

are available

when

firms are sufficiently patient."

Because

most

of

our

results

concern

thecase

where

T*

=

0,

we

willnot

pursue

thatextensionformallyhere.

4.2

Consequences

of Incentive

Compatibility

Now

that the

problem

of

finding

an

optimal

SPPE

has

been

reduced

to

a

standard

programming

problem,

we

proceed

toanalyzethe

program.

Our

firststep inthisprocessisto characterizethe

im-plicationsoftheon-scheduleconstraint thatis

found

inthe

Mechanism

Design

Program

(IC-onM).

We

do

thisin thefollowing

lemma

(where

we

use

thenotation

He{9,9]p)

=

^H{9,

9;p)Ig^g):

Lemma

5

(Constraint

Reduction)

(p,

T)

satisfies

(IC-onM)

if

and

onlyif(p,

T)

alsosatisfies:

(i). _p{9) is

weakly

increasing,

and

(ii).

H{9,

_9;p)

-

T{9)

=

H{9,9;p)

-

T{9)-

_J

_He{9,

9;p)d9.

Thisresultisstandardinthe

mechanism

designliterature,"

and

itfollows

from

the single-crossing

^^

_As

_{dicussed above,}

_we

_{will establish that there}

isalwaysvariation inthesetV^,which can be

made

arbitrarilylarge

bytakingScloseto1.

One way

toguaranteeconvexitywouldbetoextendthemodeltoallow public randomization.

For example,foreachprice vector,

we

couldspecifyaprobability thatthe firmsrevert tothestaticNashequilibrium.

^^ _{For astandard}