A TheoryofMonopolyPricingSchemes
withDemandUncertainty
By MILTON HARRIS AND ARTUR RAVIV*
A few types of monopolistic pricing conceivable mechanism is feasible. We
schemesare commonlyused in themarketing therebyseek to providean explanationof the
of manydifferent
products.The most com- use of the typesof schemesdiscussed above
monlyused schemeis thesimplesingleprice
and to identifythe conditionsunder which
strategyin which a sellerposts a price and
each particularschemewill be used. For exoffersto sell to anyone wishingto purchase ample, when would one expect to observea
at thisprice.Anotherwidelyused marketing single-pricedstrategyas opposed to some
techniqueis some formof auction,forexam- formof auction.'
ple, Treasurybills and bonds, some corpoPreviousworkin this area has proceeded
ratefinancialsecurities,
artobjects,oil leases,
by imposing a given pricing scheme and
government
contracts,etc. There are several analyzingoptimalvalues for the parameters
typesof auctionproceduresclassifiedby the
of thisscheme.David Baron (1971), Duncan
way in whichbids are solicited(sealed bid or
Holthausen, and Hayne Leland considered
the
open) and by themethodfordetermining
thesinglepriceschemeand characterizedthe
finalallocationas a functionof thebids (for
optimal single price allocation. Holthausen
example,competitiveor "second price" aucalso considereda schemein whichquantity
tions,discriminating
auction, etc.). A third is set priorto therealizationof demandwith
scheme is one in which various prices are
price determinedby ex post marketclearing.
chargedand buyerspayinghigherpricesare
Othershave extendedthistypeof analysisto
assigned higher priorityin receiving the
allow differentprices to be set ex ante in
product. We call this technique "priority different
periods dependingon expecteddepricing."For example,naturalgas and elecmand in those periods.This typeof pricing
tric power are sold to some industrialconscheme,generallyreferredto as peak load
sumersusinga priority
pricingscheme(users
pricing,has been analyzed by Roger Sherpayinglowerprices are cut offbeforethose
man and Michael Visscher,M. A. Crew and
paying higherprices in times of shortage). P. R. Kleindorfer,Gardner Brown and M.
Another example is the close-out sale in
Bruce Johnson,Robert Meyer, and others.
which the price is reduced over time and
Most of these studies characterizeoptimal
buyerswillingto accept lower probabilities peak load pricesand capacity,giventhepeak
of obtaining the product may be able to
load pricing scheme. Joseph Stiglitz conpurchaseit at lowerprices.A thirdexample
sideredthepossibilityof pricediscrimination
is thewayin whichmail deliveryis pricedby
based on quantitypurchased,that is, nonmost post offices; lower-pricedthird-class linearpricing.He characterizesoptimalpaymail is handled only afterall first-class
mail
ment schedules which give total cost to a
has been serviced.
consumer as a function of the quantity
The purpose of this paper is to derive purchased.
the formof an optimalmarketendogenously
Anotherpricing scheme analyzed in the
ing schemein a contextin whichalmostany
literatureis similarto what we have called
priority pricing. Maurice Marchand and
*Associateprofessor,Graduate School of Industrial
Carnegie-MellonUniversity;and assoAdministration,
ciate professor,Graduate School of IndustrialAdministrationand Faculty of Management,Tel Aviv University,respectively.Commentsof an anonymousreferee
and George Borts are gratefullyacknowledgedalong
with the supportof the National Science Foundation,
grantSOC-7826262.
347
'Obviously, pricingschemesotherthan the ones discussed above are also observed, namely, price discriminationbased on directlyobservablecharacteristics
of the buyers.We do not seek to explain theseschemes
but insteadfocus on situationsin whichthe buyersare
identicalin termsof theirdirectlyobservablecharacteristics.
348
THE AMERICAN ECONOMIC REVIEW
John Tschirhartand Frank Jen consider a
scheme in which an interruptible
serviceis
priced accordingto its reliability.Given this
pricingscheme,thesestudiescharacterizean
optimalprice-reliability
schedule.
A separate literatureanalyzes auctions as
a methodof marketinggoods. Most studies
in this area analyze the propertiesof given
auctions (see, for example, Baron, 1972,
Charles Holt, Steven Matthews,John Riley
and WilliamSamuelson,and RobertWilson).
The object of thesestudieswas generallyto
compare two differenttypes of auctions
based on some criterionsuch as seller's
expectedrevenue.A somewhatbroaderview
was taken by William Vickreyand Roger
Myerson(1981) who searchedforan optimal
auctiondesignamong a largerclass of feasible auctions.In ourearlierpaper,we analyzed
optimalauctiondesign,but also provedthat
in a certainenvironment
an auctionis indeed
an optimalallocationmechanism.
In the presentpaper, we provide an explanation of the use of various observed
monopolypricingschemes.Our approach is
based on the presumptionthatthe observed
marketscheme is chosen optimallyby the
sellerfroina largeclass of feasibleallocation
mechanisms.This class contains the three
pricingschemesdiscussedabove (singleprice,
prioritypricing,and auctions) as well as
most other conceivable schemes (for example, non-linearpricing).Thus, in contrastto
previousstudies,we do not imposea particular marketingtechnique,but instead derive
an optimalschemeendogenously.
This deeper
approach allows us to explain observed
marketingschemes.
Our model consistsof a single,monopolistic sellerand N potentialbuyers.The seller
produces a homogeneousproductwith constant marginal production cost up to a
capacity limit. The capacity limit may or
may not be binding.Each buyeris assumed
to demand up to one unit of the productat
any price at or below his reservationprice.
Buyers are identical except for reservation
price. A centralassumptionof the model is
that thereis asymmetric
information
among
the agents.In particular,each buyerknows
only his own reservationprice and not that
of any otherbuyer.The sellerdoes not ob-
JUNE 1981
serve any buyer's reservationprice. Each
agent has Bayesian priorsregardingreservation priceshe cannot observe.Our approach
to derivingoptimal marketingschemes is
based on the methodologysuggested by
Harris and Robert Townsend, and is describedbrieflyin Section II.
Our resultsconsistof characterizing
optimal marketingschemes for the above environment.It turns out that the optimal
marketing
techniquedependscruciallyon the
assumptionsregardingthe capacitylimit.In
particular,we show that
(i) When potential demand exceeds
capacity, the prioritypricing scheme discussed above is optimal for the seller. We
show explicitlyhow the optimal priority
prices depend on the exogenous aspects of
the model, namely,marginalcost, capacity,
and thepriorsof theagents.Anotherscheme
whichis optimalin some environments
is a
modifiedversion of the (Vickrey)competitive auction. In this auction, buyers may
submitsealed bids above a minimum
acceptable bid and a uniformpriceis chargedto all
accepted bids. The minimumacceptable bid
is equal to the lowest priorityprice. Again
we show how the equilibriumprice depends
on thebids and theexogenousaspectsof the
model.
(ii) When capacityexceeds potentialdemand, the single price scheme is optimal.
The optimalpriceis shownto be determined
by the usual "marginalrevenueequals marginal cost" conditionmodifiedto accountfor
the specific type of uncertaindemand assumed in the model.
(iii) Whencapacitycan be chosenby the
low cost of capacity,it
seller,forsufficiently
is optimalto choose capacityequal to maximum potentialdemand and charge a single
price.
The explanationof theuse of variouspricing schemes provided by these results appears to be consistentwiththe observations
describedat thebeginningof thissection.In
particular,we seem to observeprioritypricing or auctionsmainlywhencapacitylimitations are important.On the otherhand, we
observea singlemonopolypricemainlywhen
capacity constraintsare not binding.Obviously oil leases, securities,rare art objects,
VOL. 71 NO. 3
349
HARRIS AND RA VI V: MONOPOL Y PRICING SCHEMES
governmentcontractsare sold by monopolistsand potentialdemandfortheseproducts
exceedscapacity.As mentionedabove, these
products are generallysold at auction as
predictedby our analysis.Similarlydiscontinuedproductsor styles,naturalgas, electric
power, mail delivery,are also examples of
productsin whichcapacityis limitedrelative
to potential demand. These products are
oftensold by prioritypricing.Furthermore
mostproductssold by monopolistsat a single
priceare productsforwhichcapacitylimitations are unimportant
or increasesin capacityare relativelycheap. Other,morespecific
positiveimplicationsof the analysisare discussed below.
1U)
0
0
z
(9
c
I. TheEconomic
Model
A. ProductionTechnology
and
Consumers'Preferences
We considera simple"partial-equilibrium"
environmentpopulated by a monopolistic
seller and N potentialcustomers.The monopolist is described by his marginalcost
function,his preferences,and the information he possesses regardingdemand for his
product. In order to keep the analysis as
simpleas possible,we suppose thatmarginal
cost is a constant,c, foroutputbelow some
capacity limit,Q (see Figure 1).2 It is not
possible to produce more than Q units of
output.Initially(in Section IV) we assume
thatthecapacitylimitis exogenouslyor historicallyfixed.Later (in SectionV) we relax
thisassumptionto considerthecase in which
capacity can be chosen by the monopolist.
We also assume that outputcan be chosen
afteracquiringinformationabout demand.
This informationis acquired throughthe
mechanismused to marketthe product.The
objectiveof themonopolistis assumedto be
expected profitmaximization.Because the
monopolistdoes not have perfectinformation about the demand forhis productuntil
afterchoosinga pricingschemeand actually
2This assumptionhas been used extensivelyin the
literatureon imperfectcompetitionunder demand uncertainty(see, for example,Brown and Johnson,Crew
and Kleindorfer,
and Meyer).
Q
OUTPUT
FIGURE 1. THE MONOPOLIST'S COST STRUCTURE
sellingthe product,he mustconsiderprofits
to be randomwhen he makes his decisions.
The exact natureof thisrandomnesswill be
discussed below in connectionwith the description of the monopolist's information
about demand.
Each of the N potentialcustomersin our
model is characterizedby his marginalwillingnessto pay forthemonopolist'sproduct,
his endowmentof money,and his informationconcerningthedemandsof otherbuyers
and the cost structureof the monopolist.
Again in order to simplifythe analysis,we
assume thata typicalbuyer'sdemandforthe
productis givenby a unitarydemand curve.
That is, each potential customer i(i =
1,..., N) has a fixed reservationprice (or
marginalwillingnessto pay) denotedRi, and
is willing to buy at most one unit of the
productat any pricep-Ri and no units at
prices above Ri (see Figure 2).3 Note that
the reservationprices may differ across
customers.In what follows,it will be con3This assumptionis standardin the auctions literature (see, for example,Vickrey,Wilson, Matthews).It
has recentlycome to our attentionthatEric Maskin and
Riley have been workingon an extensionof thepresent
model to the case of lineardownward-sloping
demand.
350
THE AMERICAN ECONOMIC REVIEW
JUNE 1981
B. TheInformation
Structure:
KnowsWhatand When
W1ho
XRj
QUANTITY OF PRODUCT
FIGURE 2. DEMAND CURVE OF A TYPICAL CUSTOMER
venientto have a representation
of the individual consumer demands in the form of
utilityfunctions.This utilityfor buyer i,
U(di, qi, Ri), is a functionof the amountof
moneyhe holds,di, thequantityof the good
he consumes,qi, and his reservation
priceRi.
The form of the functionU which corresponds to the unitarydemand functiondescribedabove is
(1)
U(di,
qi, Rj)
-di + R imin(qi,
1)
Thus, forqi < 1, buyeri has a fixedmarginal
rate of substitution
Ri betweenthegood and
money,that is, his marginalwillingnessto
pay for the good Ri is independentof the
quantityhe consumesforquantitiesless than
1. Moreover, buyer i has no utility for
amountsof the good in excess of one unit,
that is, his marginalwillingnessto pay for
quantitiesin excessof one unitis zero. In all
respects other than reservationprice, the
buyersare assumedto be identical.In particular,each customerstartsoffwithd unitsof
money(dollars) to spend on thisgood.
To completethe descriptionof the model,
we mustspecifythe information
available to
each agent.If thesellerkneweach consumer's
true reservationprice with certainty,it is
clear thathis optimalstrategywould be simply to sell one unit to each of the Q highest
reservation
pricebuyersand chargeeach such
buyerhis truereservationprice.Thus a crucial assumption of the model is that the
monopolistcannot tellbuyersapart.That is,
the monopolistdoes not know the true reservationprice of any consumerunless that
price is revealed through the consumer's
purchasingbehavioror voluntarilyrevealed
by the consumer. In order to model the
monopolist'simperfectinformationregarding the reservationprices of the buyers,we
assume that the monopolisthas some prior
beliefsabout theseprices.The simplestsuch
assumptionis thatthemonopolistviewseach
Ri as havingequal probabilityof being any
one of the numbersX,,...,Xk.
That is, in
makinghis decisionsabout how to price his
product,the monopolistacts as if the Ri's
are drawn at random independentlyfrom
a uniform distributionon the set X=
{xI, .. ., Xk}. (All of the major resultswould
continueto hold witha nonuniform
distributionon X.) It is in thissensethatthedemand
faced by the monopolistis random,that is,
no individualconsumer'sdemand is subject
to random shocks,but the monopolist,because of his ignorance,is forced to view
demand as uncertain.4Otherthan the actual
reservation
pricesof thebuyers,themonopolist knowseverything
else about the environment,namelyhis marginalcost c, his capacityQ, thenumberof potentialbuyersN, and
themonetaryresourcesof each buyerd.
Justas the monopolistis assumed not to
know the true reservationprices of the
4Obviously, we are takinga Bayesian view of decision makingunderuncertainty.
We believe thisview to
be appropriate.Again, it is standard in the auctions
literature.Note thatbecause of the monopolist'suncertaintyregardingindividualdemand, aggregatedemand
is also uncertainfrom his point of view, unless the
numberof buyersN is verylarge.
HARRIS AND RA VI V: MONOPOL Y PRICING SCHEMES
VOL. 71 NO. 3
351
II. TheMonopolist's
Problem
Having set out the model, we must now
specifyexactly the problem which the monopolistsolves in order to discoveran optimal scheme or mechanismfor pricingand
FIGuRE 3. THE SET OF POSSIBLE RESERVATIONPRICES, X
sellinghis product.In formulating
thisproblem, we imaginethat the monopolisthas at
his disposal a large class of alternativepricbuyers, each buyer is assumed to be ining schemes.This class of schemesincludes
formedonly about his ownreservationprice.
both simpleone-stagemechanismsas well as
Thus buyeri knowsRi but doesn't know R1
more complicated,sequentialones. The mofor some other buyerj. Again we model
nopolist then searches over this class of
buyeri's uncertainty
about R1 by assuming schemesuntil he findsone which resultsin
that buyer i believes that Ri has an equal
maximalexpectedprofits.
In orderto make our optimalityresultsas
chance of being any of the numbers
x1,..., Xk. Finally, each buyer is informed strongas possible,we includein the class of
about all otheraspects of the environment, available pricingschemesalmostany mechanamely,c, Q, N, and d.
nism whichone can imagineforpricingand
Finally, thereare two more assumptions sellinga homogeneousproduct.5Perhapsthe
whichwill simplifythe analysisbut will not
simplestimaginableschemeis to announcea
affectthe qualitativeresultsregardingwhen
price and sell Q units to the firstQ buyers
it is optimal to use which kind of pricing who are willingto pay that price. We call
mechanism.The firstof these is that the
this the "singleprice" scheme,and we shall
possible reservationprices xi are equally
see thatin some cases, thisschemeis optimal
spaced on an interval[a,/3] as in Figure 3.
for the monopolist.Anothersimple type of
The distance between any two successive schemeis a sealed-bidauction.In thismechpointsis denoted S>0. Later (in SectionV)
anism,each potentialbuyeris asked to write
we will investigatethe resultsas k -* oc and
down a bid for one unit, that is, to write
8 -O0, i.e.,as X approachestheinterval[a, /3]. down a numberwhich is interpreted
as his
Second, we assume that each buyer has
declared reservationprice. Afterall buyers
enoughmoneyto purchaseone unit,even at
have submittedtheir bids, the number of
thehighestreservation
units allocated to each buyerand the price
price,i.e.,d -xk.
The remainderof the paper is devoted to
each mustpay is determinedfollowingsome
pre-announcedallocation rule as a function
characterizing
marketingschemeswhich are
of thesubmittedbids. Varioustypesof sealed
optimal for the seller. For example, would
we expectthe monopolistto sell his product
5Formally,
wedefinea mechanism
as a multistage
(or
by settinga monopolisticprice on a take-itoftwoobjects.Firstis a
gamewhichconsists
or-leave-itbasis (i.e., the approach usually sequential)
finite
sequenceofsets(or messagespaces),one foreach
assumedin textbooks),or by adoptingsome
player,
whichdefinethesetoffeasible
playsormessages
more complicatedscheme in order to disforeachplayerat eachstage.The setof feasibleplays
criminateamong buyers?Obviouslyany atforan agentat stages coulddependon thehistory
of
stages- 1. Secondis a function,
temptat discrimination
is limitedby thefact thegameup through
ruleof themechanism,
whichdethatthe sellerdoes not know the reservation calledtheallocation
of theentire
termines
thefinalallocationas a function
prices of individual buyers. Nevertheless, history
ofthesequenceof
ofthegame(i.e.,as a function
some discrimination
is possible; forexample, actualplaysor messages).
A mechanism
is feasibleifit
resultsin a feasibleallocationof goodsand moneyfor
usingauctions.We will see below thatunder
certainconditions,simplemonopolypricing any sequenceof playsby the agents.The class over
themonopolist
searchesis simplytheset of all
is optimal,whereasunder otherconditions, which
feasible
Thesedefinitions
areadaptedfrom
mechanisms.
more complicatedpricingschemesprove suA morepreciseand formal
HarrisandTownsend.
treata = xI
perior.
x2
X3
.
.
Xk l
xk-
mentmaybe foundthere.
352
THE AMERICAN ECONOMIC REVIEW
bid auctionscan be representedby different
allocation rules. For example, the U.S.
Treasuryoftenuses a discriminating
sealed
bid auction forpricingand sellingTreasury
bills. In this auction,the Q highestbidders
each receive a unit and each pays his bid
price.This is an exampleof a nonsequential
schemesince all buyerssubmitbids simultaneously (or equivalently,no bid is announceduntilall bids are in). An exampleof
a sequential scheme is the "open English"
auction. In this scheme, buyers announce
bids openly.Aftera bid is announced,any
otherbuyermay raise the bid. When no one
is willingto raise a bid, the highestbidder
gets the unit and pays his bid price. All of
the above schemesare includedin the class
of mechanismswhich are available to the
monopolist.In addition any other scheme,
sequential or nonsequential,as long as it
alwaysresultsin a feasibleallocationofgoods
and money (no matterhow the buyersbehave duringthescheme),is allowed.
The problem of the monopolist as describedthus farappears to be horriblycomplicated.Firstof all, theclass of objectsover
which he must search consists of possibly
highlycomplexschemes(or games). Second,
the monopolist,in orderto evaluate a given
scheme,musttake into account the strategic
behaviorof thebuyerswho participatein the
scheme. That is, the monopolistmust take
into account the fact that,given a scheme,
each buyerwill act in his own interestsgiven
his expectationsabout the behaviorof other
the probbuyers.In view of thiscomplexity,
lem seems completelyunapproachable.Fortunately,however,both of theabove complicationscan be vastlysimplifiedby appealing
to the followingresult.
THEOREM 1: RevelationPrinciple.6For any
schemethereis an equivalentschemewhichis
bothdirectand truthful.
61t is well beyond the scope of this paper to prove
Theorem1. This theoremis essentiallya combinationof
Theorems I and 2 of Harris and Townsend. A result
similar to our Theorem I may be found in Myerson
(1979). The terminology
"RevelationPrinciple"is also
due to Myerson.
JUNE 1981
In the theorem,
we use the termsdirect
and truthful
schemeswhichwe mustnow
define.A directschemeis a verysimple
schemein whicheach buyerwritesdowna
bid or declaredreservation
price,ri(without
knowing
whattheotherbuyersare writing).
The declaredreservation
pricemustbe one
of thepossiblereservation
prices,thatis, a
member
ofthesetX. Oncethesedeclarations
havebeensubmitted,
an allocation
rulegives
theactualallocationofmoneydi, and productqi, as a function
ofthedeclarations.
The
allocationruleforsucha directmechanism
is
denotedby
[di(r), qi(r)]
i= 1,.,N
where r= (r1, . ..., rN). The seller is assumed
to producethe total quantityallocatedto
buyers,:EN= qi(r), and to receivepayments
fromthebuyerstotaling
2iN 1[d-di(r)]. Dif-
ferenttypesof directschemescan be modeled by different
allocationrules,that is,
different
d 's and qi's. A truthful
schemeis a
directschemein whichtheoptimalstrategy
foranybuyeri is to declarehistruereservationprice,thatis,ri=R , giventhatall other
buyersare declaringtheirtruereservation
prices.7'8
7In thelanguage
ofgametheory,
a truthful
schemeis
a directgamein whichtruth
is a Nash(or,more
telling
precisely,
Bayes)equilibrium.
Notall directschemes
are
A somewhat
truthful.
ridiculous
exampleis thedirect
schemein whichthe Q unitsare allocatedto the Q
lowestbiddersat theirbid prices.In thisscheme,
each
buyerwoulddeclarethelowestpossiblereservation
price
ofhistruereservation
price.
xi regardless
8Someroughintuition
forTheorem1 maybe gained
as follows.Firstsupposewe have a givenpricingscheme
whichis notdirect.Supposealso thattheequilibrium
of thebuyersin thisschemeas functions
strategies
of
thetruereservation
We can construct
pricesareknown.
an equivalenttruthful
schemeas follows.Insteadof
havingeachbuyercomputeand reporthisequilibrium
strategy
(as a function
ofhistruereservation
price),we
simplyask each buyerto reporthis reservation
price
whileinforming
himthatthisreported
value willbe
used to calculatethe exactsame strategies
he would
havechosenhimself.
Thatis, thereported
reservation
priceis pluggedintothebuyer'sequilibrium
strategy
function.
The resulting
strategies
willthenbe used to
determine
theallocation
just as in theoriginalprocedure.Sincethebuyers
hadnoreasontolie tothemselves
abouttheirrespective
reservation
priceswhentheywere
VOL. 71 NO. 3
HARRIS AND RAVIV: MONOPOLY PRICING SCHEMES
353
In viewof Theorem1, the searchfora
declare x (as opposed to any otherdeclaraprofit-maximizing
pricingschememay be
tiony) giventhatall otherbuyersare declarIn the mathematicalexpresconfinedto directpricingschemeswithout ing truthfully.
any loss of generality.Thus, we can ignore
complicated
sequentialschemessuchas the
open Englishauction,without
anyfearthat
in so doingwe have thrownout schemes
whichmight
resultinhigher
expected
profits.
recallthatall directschemesare
Moreover,
verysimilar.They all requirethe buyers
to submitdeclaredreservasimultaneously
tionprices.Directschemesdiffer
onlywith
respectto theirallocationrules,thatis, with
respectto how the goods and moneyare
of the declaredreallocatedas a function
inorderto search
servation
prices.Therefore,
over directschemes,we need only search
overfeasibleallocationrules[di(r),qi(r)].
We can, however,
usingTheorem1 simplifytheproblemstillfurther.
Namely,we
can, also withoutloss of generality,restrict
sion, the symbol R _i refersto the vector
of true reservationprices of buyers other
than i. The expression di(x, R _) means
dj(R1, ... Ri-1l x, Rj+1i ... , RN), and theexpressionsqi(x, R _j), di(y, R _j), qi(y, R _0
are to be interpretedsimilarly.The expectation Ei refersto the expectationover the
unknown(to buyer i) reservationprices of
thebuyersotherthani usingthepriorbeliefs
describedin Section II. The utilityfunction
U is the one given in equation (1) of Section I. In order to consider only truthful
schemes, we will consider only allocation
ruleswhichsatisfy(SS).
In additionto (SS), thereare severalother
which we must place on the alrestrictions
location rules which the monopolist may
choose. First of all, we do not allow the
monopolistto forcebuyersto participatein
any pricingscheme.In orderto be eligible,a
truthfulscheme must provide each buyer
with at least as much expectedutilityas he
would obtainin autarky(i.e., by retaininghis
endowmentof d dollars and not acquiring
any goods).9 The mathematicalversion of
thiscondition,whichwe referto as individual
rationality
(IR) is
attention
to truthful
schemes.
Now,ifwe are
our attentionto truthful
goingto restrict
we musthave somewayof telling
schemes,
thesefromotherdirectschemessimplyby
inspecting
allocationrules.Recall thatin a
eachbuyer'sexpected
truthful
scheme,
utility
is higherif he declareshis truereservation
pricethanif he declaresanyotherpossible
reservation
price,giventhateveryone
elseis
theallocaalso tellingthetruth.Therefore,
andonlythose (IR)
tionrulesoftruthful
schemes,
EjU[dj(R), qi(R), Rj]
allocationrules,satisfythe following
selfforeveryi and Ri.
selection(SS) conditions:forany buyeri
Finally,we requirethatthe allocationrule
and anytwopossiblereservation
pricesx, y
be feasiblein thesense thatno morethanNd
in X,
(SS)
EjU[dj(x, R_j), qj(x,R_j), x]
Z,EiU[dj(y,
R _j), qi(y, R _i), X]
is simplya mathematical
verThiscondition
thatif buyeri's true
sion of thestatement
reservation
priceis x, thenhe is betteroffto
computingtheirown strategies,
theywill have no incentive in the new scheme to lie about theirreservation
prices. Consequently,the new scheme will result in
exactly the same equilibriumoutcome as the original
one, and moreover,in the new schemethe equilibrium
declarationswill be truthful.
dollars and no more than Q units of the
good may be allocated to thebuyersand the
allocationsmay not be negative.These conditionsmustbe fulfilledno matterwhat the
true reservationprices are. Mathematically,
we require
N
(T.1)
dji(R)<Nd
9This does not preventthe sellerfromexcludingany
particularbuyer, say i. He can still exclude buyer i
simplyby settingdj(R)=d, qi(R)-O. If this restriction werenot present,it is clear thatan optimalscheme
would be simplyto requireeach buyerto hand over his
d dollarswithoutreceivinganythingin return.
354
THE AMERICAN ECONOMIC REVIEW
foreveryR,
(T.2)
qi(R)--Q
foreveryR,
(T.3)
di(R) _>0, qi (R)~> 0
foreveryi and R.
The problemwhich faces the monopolist
can now be statedas a simplemathematical
programming
problem,namelyto choose an
allocation rule [di(R), qi(R)] which maximizes expectedprofitssubject to the conditions(SS), (IR), and (T).'? In mathematical
form,we assume the monopolistsolves
[PI
max E
N
2
[d-di(R)-cqi(R)]
subjectto (SS), (IR), and (T). Here E refers
to the expectationoverR.
The remainderof the paper is devoted to
characterizingsolutionsto problem[P] under various assumptionsregardingcapacity
Q,and itsrelationto thenumberof potential
buyersN.
III. A MethodforSolvingtheProblem
Our generalapproach to findingan optimal pricingscheme,thatis, solvingproblem
[P], consists of two stages. First,we show
that [P] is equivalent to a much simpler
problem.Second,we ignoresome of thecontraintsof [P], solve the reduced problem
[P'], and thenshow thatthe solutionof [P']
satisfiesthe ignored constraintsand thereforesolves [P].
Showingthat[P] can be simplified
without
loss of generalityinvolvesthreesteps. First
recall fromequation (1) that the expected
utilityappearingin the (SS) and (IR) con'0Note that we now writethe allocation rule as a
functionof the trueratherthanthedeclaredreservation
prices.This is allowed since we are restricting
attention
to truthful
schemesso thatdeclaredand actual reservation pricesare identical.
JUNE 1981
straintsof problem[PI involvesa non-linear
expressioncontainingmin(qi,1). It is easy to
see thatany solutionof [P] will have qi(R)
1 since productionis costlyand qi(R)> 1
results in the same value of the expected
utilityexpressionsof thebuyersas qi( R) 1.
Thereforewe can replace theutilityfunction
in problem [P] by the linear expression
d1(R)+Rjqj(R) and add the additionalconstraintsqi(R)< 1, forall i and R.
The second simplificationis achieved by
noticingthat,since all agents are the same
except for reservationprice, it should be
possible to find a solution to [P] in which
any two agentswho draw the same reservation price get the same bundle. Although
theircommonbundle depends on the other
agents' reservationprices,it should only depend on the numberof agents havingeach
possible reservationprice (i.e., each xi) and
not explicitlyon which agents have which
reservationprices.More formally,
we state
LEMMA: Problem[ P] has a symmetric
solution,thatis, thereis a solution[d q7] of [ P]
such that d*(R1,.*,Ri,
R
d*(R')
for any i=
q*(R.,
RN)=q*(R'),
Rj,
R' of theele1,..., N and any rearrangement
mentsof R in whichRi appears in thefirst
place.
n
PROOF:
First we simplynote that since [P] is a
linear programany convex combinationof
solutions of [P] is itselfa solution. Next,
given any solution,we can constructN! other solutionswhich are the same as the
firstsolutionexcept that the agents are renamed and elementsof R are rearranged.
Finally these solutionsmay be averaged to
obtain a new solution with the symmetry
propertyclaimed.
From now on we will onlybe interestedin
symmetric
solutions,thatis, thosethatsatisfy
theconclusionof the Lemma. Now fromthe
Lemma,it followsthat
(2a)
)=
Eidi(fx,R i__ Eld,(x, R_ l)
VOL. 71 NO. 3
HARRIS AND RAVIV: MONOPOLY PRICING SCHEMES
355
N
foranyxE X. Usingtheseequalities,it folforanyR,
lowsthatthe(SS) and (IR) constraints
for (T.2)
qi(R)<Q
buyersotherthanbuyer1 areredundant.
involvesshowing
The thirdsimplification
qi (R) >Odi(R) 0, I1~thatthe(IR) constraints
ex- (T.3)
are redundant
ceptforj= 1. The proofconsistsofrewriting
then for any i and R, and [di, qi] satisfythe
(SS) and (IR) usingtheaboveresults,
of theLemma.
conditions
it followsfrom symmetry
notingthat since xj >xj
Thiscompletes
thederivation
of a simpler
(SS) that
whichis equivalent
to[P]. Hereafter
problem
problem.
equivalent
[P] refers
to thissimpler
E,[d,(xj R 1)+xjqj(xj, R-1)]
however.InWe do not solve[P] directly,
to ignoresomeof the
steadit is convenient
-,>El[d,(xj-, R - ) +xj- ql(xj- , R - )]
of [P]. This leads to an even
constraints
set,
simpler
problemwitha largerconstraint
whichwe call problem[P']. If we can solve
Thus if (IR) holds forj = 1, it holds for all
thatthesolutionalso satisfies
j> 1 as well.Therefore
loss [P'] andverify
we can,without
of [P], thenwe will
of generality,
ex- thedroppedconstraints
drop all (IR) constraints
havefounda solutionof[P]. Theconstraints
cepttheone forj= 1.
Finally,by summingequations(2a) and whichare ignoredare all the (SS) con(2b) overi= 1,..., N and takingexpectations straintsin whichm=/=j-1 and the constraints
withrespectto x, we obtain
di(R)>O. It is easyto showthatany
optimalsolutionof [P'] will satisfyall the
N
and the (IR)
(remaining)
(SS) constraints
E 2 di(R)=NEd1(R),
as equalities.
Theseequalitiesand
constraint
i=lI
x1-x11 =8 can thenbe used to derivethe
following
N
E 2 qi(R)=NEq1(R)
(3)
i= I
Eldl(x1, R-1)
Consequently,
usingall of theaboveresults,
to
problem[P] is equivalent
[PI
max
[di(R), qi(R)]
where
Eir
=NE[d-d1(R)-cq,(R)]
subjectto
(SS)
E1[d1(xj,R_j)+xjq1(xj,R_j)]
aEl[dl(xm, R - ) +xjql(xm, R - )]
forj#&m,j,m= 1,..., k
(IR)
(T.1)
El d,(x1, R _)+x1q1(x1,
N
i= I
i-I
2 Zm'-Zj[Xt +( j-l)8]
m=1
=d+S
di(R)
Nd
R_1)] kd
foranyR,
(4)
zj =E1q,(xj,R_1)
forj
,...,k
Substituting
(3) into the objectivefunction
yields,aftersomemanipulation
k
(S) E= NT 2 [x1+ (2 j-k-l)8-c]zj
j=1
wherep 1/k. In orderto solve [P'], we
side of (5) with
maximizethe right-hand
respectto thezi's subjectto theconstraints
[P']. To do this,wemust
impliedbyproblem
theconceptof an allocation
firstintroduce
rankby rank.Supposewe havea particular
couldinvolve
ingof thebuyers(theranking
ties,i.e.,morethanone buyercouldreceive
356
THE AMERICAN ECONOMIC REVIEW
the same rank). An allocationby rank is the
allocationin whichunitsof the productare
assigned to buyersaccordingto theirrank,
fromhighestto lowest. That is, each buyer
withthe highestnumberedrank is allocated
one unit(or his proportionalshare of the Q
unitsif morethan Q buyershave thehighest
rank).The remainingunitsare thenallocated
in the same way to buyers of the second
highestrank,etc. This continuesuntil all Q
units are allocated. An allocation by rank
withcut-off
rank p is an allocation by rank
with the additional condition that buyers
with rank strictlybelow p are not assigned
any units even if less than Q units are assignedto buyerswithrankp and above.
In order to maximizethe right-handside
of (5), note that the expressionis linear in
the zJ's and the coefficientsof the zJ's increase withj. Thereforelet T be the smallest
j forwhich the coefficientof Zj is nonnegative,thatis, T is such that
(6)
forj2 T
xl+6(2j-k-l)?'c
<c
forj< T
Consequently to maximize the right-hand
sideof(5) we choosezI, Z2,... 9 ZT-1 as small
as possible,thenchoose Zk as large as possible, zk- I as largeas possiblegivenour choice
of zk, etc., down to ZT. Because of the nonnegativityconstraints,we must have zj 0
for each j. Therefore,we choose zr=
To make Zk as large as
Z2*=...=zT_l=0.
possible,one should choose ql(xk, R - )= 1
for all R -1 (since qi(R) <I for all R), that
is, givebuyer1 one unitwhenhis reservation
price is the highestone possible. Because of
the symmetryconstraint,however,this requires giving any buyer whose reservation
priceis xk one unit.If Q<N, thenumberof
buyerswith reservationprice xk mightexceed Q. In thiscase, thelargestZk is obtained
by sharingtheQ unitsamongall buyerswith
reservationprice Zk- Once Z* is determined,
the maximizationof Zk-I is achieved by allocating the remainingunits in the same
fashion. This continues until ZT is determined.It shouldbe clear fromthisdiscussion thatmaximizingthe right-handside of
(5) is achievedby choosingqI(x1, R - 1) to be
the quantitieswhichare assignedto agent 1
JUNE 1981
in an allocationby rankwithcut-offrank T
where the rankingof agents is by true reservationprice, that is, assign each agent
with reservationprice x; rankj. The maximizingvalues zT,..., zk are then the values
which resultsfromchoosingq,(x1,Rl) in
this way. Explicitexpressionsfor the values
Zj* are needed only in the formalproofsand
are therefore
givenonlyin theAppendix.
Most of the work of solving [P'] has
now been done. The maximizingchoices
ZT--4
Z,
however,
dependon whether
Q<
N or Q2N. Since N is the maximumpossible demand, we referto these two cases as
"potential demand exceeds capacity (Q<
N )" and "capacityexceedspotentialdemand
(Q2N)". We analyze these two cases separatelyin the followingsection.
IV. OptimalPricing
Schemeswith
Exogenous
Capacity
In thissectionwe assumethatthecapacity
of the seller, Q, is exogenouslyfixed. The
analysis of optimal pricingschemes under
thisassumptionis importantfortworeasons.
First, in many observed situationsthis assumptiondoes, in fact,characterizethe situation. For example, the quantity of
Rembrandt originals is historicallyfixed.
Other examples of this type come easily to
mind (rare coins, stamps,etc.). Anotherexample is the fixedcapacityof a givenairline
flight.Also manymanufacturing
facilitiesare
characterizedby fixed capacity,at least in
the shortrun (utilities,etc.). Anotherexample occurs at the end of a season or model
year whenno further
productiontakesplace
and existingstocks are simplydisposed of.
As will be seen, our analysis explains the
procedures often observed in these fixedcapacity environments.
Second, the analysis
of the fixed-capacity
case is a necessaryfirst
step in examiningthe case of endogenous
capacitywhichwe pursuein SectionV.
A. PotentialDemandExceeds Capacity
(Q<N)
Clearly,if the seller had full information
about the reservationprice of each buyer,it
would be optimal for him to sell one unit
each to buyerswith the highestreservation
VOL. 71 NO. 3
HARRIS AND RA VIV: MONOPOLY PRICING SCHEMES
prices and to charge each buyer his true
reservation
price,thatis, to discriminate
perfectlyamongbuyers.In the absence of such
fullinformation,
it is not possible to achieve
such perfectdiscrimination.It is possible,
however,to discriminateto some extentby
chargingsome buyers higherprices in exchangeforhigherpriorityaccess to theproduct. Higherreservation-price
buyerswill be
willingto pay somewhathigherprices,relative to low reservation-price
buyers,in exchange forhigherpriority.This fact can be
used by the selleras a basis for discrimination. Our resultsfor this subsectionexhibit
two optimalpricingschemeseach of which
exploitsthisidea. One of thesetwo schemes
is optimalfor any values of the parameters
of themodel(i.e., N, Q, k, d, etc.). The other
is optimal only for some values of these
parameters.
The firstschemewe analyze is a "priority
pricingscheme."A priority
pricingschemeis
one in whichthesellerannouncesa schedule
Pm< ... <pn of priorityprices. Each buyer
choosesa priority
pricewhichhe is willingto
pay. Buyersare thenrankedby the priority
prices theychoose, that is, buyerschoosing
higher priorityprices are assigned higher
rank. Buyersnot wishingto buy at any of
thesepricessimplydo not announcea choice
of price and are assigned rank 0< m. The
product is then allocated by this ranking
with cut-offrank m. Each buyer pays his
chosen priorityprice times the quantityhe
receives.We showin thenexttheoremthata
prioritypricingschemeis an optimalmechanism provided that the priorityprices are
properlychosen. A sketchof the proof of
thistheorem,and all subsequenttheorems,
is
givenin theAppendix.
357
B. Anybuyerwithtruereservation
price xi
choosespriority
pricep* whenfaced withthe
above scheme,for i = T,..., k. Buyers with
reservation
priceless thanXT willchoosenotto
buy.
By adoptingthe scheme describedin Theorem2, thesellerin effectconvertsa homogeneous productinto a heterogenousone. This
is accomplishedby attachingto the product
various probabilities of being allowed to
purchaseit accordingto thepriorityscheme.
The seller is thus enabled to discriminate
acrossbuyers:buyerswithhigherreservation
priceswillchoose to pay higherpriority
prices
in exchangeforhigherreliability
(probability
of obtainingthe product).Notice, however,
that the seller cannot capture all the rents
possible under full informationsince each
prioritypricep* is strictly
less thanits correspondingreservationpricexi, exceptforthe
lowest pricepT (see equation (7)). This loss
of rentsreflectsthefactthatthesellercannot
directlyobservebuyer'spreferences."
Under the optimalschemeof Theorem2,
it may happen that not all Q units will be
sold. This occurswhen fewerthan Q buyers
have reservationpricesat or above p*T= XT.
In thiscase, the sellercould have sold more
unitsifhe werewillingto offerpriority
prices
below p*T.Moreover,therecould be prices
below p* whichare stillabove marginalcost
(see equation (6)). The question then arises
whynot sell moreunitsat theselowerprices.
Suppose thatthe sellerdecided to reducethe
lowest priorityprice in order to sell more
units.Obviously,in orderto attractany additional buyers, the lowest price must be
reduced at least to xT I (fromXT). In this
case, however,buyerswithreservationprice
XT would find it advantageous to choose
THEOREM 2:
priorityprice xT 1 instead of XT. Similarly
A. WhenQ<N, an optimalschemeis the buyerswithreservation
pricesabove XT would
prioritypricing scheme withpriorityprices findit in theirinterestto choose lowerpriorrank,is defined itypricesthanbefore.The consequentloss of
P*T9.. Pk whereT, thecut-off
by(6), k is thenumberofpossiblereservation revenuemorethancompensatesforthereveprices,and
i- I
(7)
Pi*=x-(/
fori=T+1,...,k,p
(z,-...,
),Z,
j=T
=xT
z are thenumbers
definedabove).
"The readermightwonderwhyit is not optimalto
charge priorityprices XT,... Xk instead of P k-.
since fori> T, x; >p7. The reason is thatif the prices
were XT,...,Xk,
any buyer with reservationprice xi
would not choose to pay priorityprice xi but would
choose to pay pricexj whereXj is the reservationprice
just belowp7*.
358
THE AMERICAN ECONOMIC REVIEW
nue gainedfromany additionalsales. This is
why it may be optimal to set the lowest
prioritypricesuch thatforsome realizations
of demand,less than Q unitsare sold.
The prioritypricingscheme shown to be
optimal in Theorem 2 is in fact often observedin reality.For example,end of model
year or end of season sales can be viewedas
prioritypricingschemes.In such sales,prices
are graduallyreducedand some buyerspay
lowerpricesthanotherswhiletakinggreater
riskof not being able to purchasethe product. Our model predictsthat in these'sales,
the scheduleof pricereductionswill be independent of remainingquantities.This appears to providea meansof testingthemodel.
Anotherexample of prioritypricingoccurs
in the provisionof natural gas. Here some
industrialusers of natural gas pay higher
pricesin exchangeforbeinglast to be cut off
in case of shortage.Standbyticketsat lower
prices offeredby airlines provide another
example.A similarexampleinvolvesthe sale
of ticketsforBroadwayplays in New York:
at 4:00 P.M. remainingtickets(if any) for
thatevening'sperformance
go on sale at half
price.
Anotherschemewhichmay also be optimal is a common price auction with minimumacceptablebid. A common
priceauction
in thisenvironment
is a sealed bid auctionin
whicheach buyersubmitsa bid forone unit
of theproduct.A bid can be any reservation
price, that is, any elementof X. The bids
determinethe common price paid by every
"successful"bidderand the numberof units
allocated to each bid. Differentcommon
price auctionsare characterizedby different
methodsof determining
thisprice as a functionof thebids. A common
priceauctionwith
minimum
acceptablebid b is a commonprice
auction withthe additionalfeaturethatany
bid below b is rejectedautomaticallyregardless of whetheror not thereare Q bids at or
above b.
It turnsout that for some values of the
parametersN, Q, and d, one can constructa
commonpriceauctionwhichyieldsthesame
expectedprofitas thepriority
pricingscheme
just discussed. In this auction, the seller
allocatesthe Q unitsby rankwheretherank
correspondsto thebid. The selleralso sets a
JUNE 1981
minimumacceptablebid givenby XT, thatis,
thecut-offrankis T as in thepriority
pricing
scheme(see equation(6)). The commonprice
paid by all successfulbiddersis determined
as follows.Suppose that the lowest bid at
which any units are awarded in the allocation by rank (the lowest accepted bid) is xi
and the highestbid below xi (the highest
rejectedbid) is xj. If all bidderswho bid xi
receiveone uniteach, thenthecommonprice
paid by all successfulbidders(thosewho bid
xi and above) is x1+A16 where A1>O is
givenexplicitlyin Theorem3. If bidderswho
bid xi each receiveless than one unit (i.e.,
theyshare),thenthe commonprice is xi. If
thelowestacceptedbid is XT (i.e.,Xi =XT),
thenall successfulbidderspay XT. This auction is a modificationof the competitive
auctionproposedby Vickrey.In the Vickrey
auction, the common price is the highest
rejectedbid. The optimal auction just presented differsfromthe Vickreyauction in
two ways. First it includes a minimumacceptable bid.'2 Second we exploit the assumed discretenessof possible reservation
prices by sometimessettingthe price above
the highestrejectedbid. In the theorembelow we referto this auction as the Modified
Auction.'3
Vickrey
THEOREM 3:
A. WhenQ <N, an optimalpricingscheme
is theModifiedVickrey
Auctionprovidedthat
(8)
fori= T,..., k-1
xi +Ai16Ad
where Ai=1 I-
a.
-a
a
fori=l,...,k-1
and theexpressions
fora1, ai, and bi are given
in theAppendix.
B. Anybuyerwithtruereservation
price at
or above xT willbid his truereservation
price
whenfaced with the above auction. Buyers
withreservation
prices below XT will choose
nottoparticipate.
'2Riley and Samuelsonconsiderauctionsand derive
an optimalminimumbid. Their expression,when specialized to our model,is the same as ours.
13Theorem3 may be viewed as a generalizationof
Theorem2 of our earlierpaper to morethantwoagents.
VOL. 71 NO. 3
HARRIS AND RAVIV: MONOPOLY PRICING SCHEMES
The interpretation
of condition
(8) in Theorem3 is thattheModifiedVickrey
Auction
mayrequirea successful
bidderto paymore
forsomerealizations
of
thanhisendowment
thereservation
prices.Thatis forsomevalues of N, Q, and d, the ModifiedVickrey
Auctionis not feasible,althoughis optimal
whenever
feasible.It can be shownthatthe
ModifiedVickrey
Auctionis alwaysfeasible
if thereis onlyone unitavailable(Q= 1). It
can also be shownthatthisauctionis feasible if the possiblereservation
prices(the
are
sufficiently
close
together
(i.e., for
xj's)
sufficiently
largek andsmallS). '
In caseswheretheModifiedVickrey
Auctionis feasible(and therefore
optimal),
Theorems2 and 3 implythatone mightexpect
to observeeitherone of two marketing
in whichpotenschemesin an environment
tialdemandexceedscapacity.The theory
is
not richenoughto predictwhichof these
twoschemeswouldbe used.Note,however,
thatthetwoschemesarequitedifferent.
The
prioritypricingschemedoes not require
one
simultaneous
participation.
Forexample,
waytoimplement
thisschemeis first
tooffer
the productforsale at Pk, thengradually
359
pricesbeforethemechanism
ownreservation
is chosen).For a buyerwithhighreservation
price,theauctionpricewill,on average,be
pricehe wouldchoosein
belowthepriority
pricingscheme.On the other
the priority
price,
hand,fora buyerwithlowreservation
theauctionprice
ifhe purchases
theproduct,
price,
willon averagebe belowhisreservation
price.
butabovehispriority
At thispointone mightwonderwhether
the usual singleprice schemeis also optimal.
to observethistypeof schemein close out
sales.'5 On the other hand, the auction
schemerequiresthatall bids be submitted
beforetheallocationof theproductand the
Whichschemeone
price are determined.
mightexpectto observemaybe determined
as therelative
costof operaby suchfactors
tionwhicharenotincludedinouranalysis.
Finally,
notethatthetwoschemes
arealso
fromthepointofviewofthebuyers.
different
Buyerswithhighreservation
priceswillprefertheauctionschemewhilethosewithlow
reservation
priceswillprefer
priority
pricing
(recallthatbuyersareassumedtoknowtheir
In thisscheme,thesellersetsa pricebefore
of demand,thensellsto each
therealization
at thatprice,
buyerwhodemandstheproduct
up to the limitof his capacity.The single
priceapproachhasbeenanalyzedextensively
In orderforthismechain theliterature.'6
nismto be optimalin thepresentenvironment,it mustyieldthesameexpectedprofit
as thepriority
pricingscheme.It turnsout,
demandexceeds
thatwhenpotential
however,
a singlepriceyieldsstrictly
capacity,setting
pricing.
thanpriority
lowerexpectedprofits
To see this,supposethatthebestsingleprice
is P. Now considerthe prioritypricing
schemewithlowestpriority
priceequal to P.
Since Q<N, thereare,withpositiveprobability,buyerswillingto pay morethanP in
in theallocation
orderto gethigherpriority
of thegood (namelythosewithreservation
pricingscheme
pricesaboveP). The priority
allowsthesellerto exploitthisfactbychargingthesebuyershigherprices.The quantity
sold undereitherthe singleprice scheme
pricingwithlowest
withpriceP or priority
priceP willbe thesameforeachrealization
the priority
of demand.Therefore,
pricing
This
schemewithlowestpriceP is superior.
shows that when Q<N, the singleprice
schemeis strictly
suboptimalsinceit does
not exploitthepotentialfordiscrimination
oftheproduct.'7
offered
bythescarcity
'4Somepreliminary
resultssuggestthatwhenthe
either
infinity,
buyersapproaches
numberof potential
or no
to singlepricing
all optimalschemesdegenerate
common
priceauctionis optimal.
sale,ifthebuyersareallowed
15Insucha sequential
to observethequantity
leftat eachpointin time,then
scheme.Conpricing
themechanism
is notthepriority
fromthe
theoutcomemightbe suboptimal
sequently,
thatthesellermight
seller'spointofview.Thissuggests
stocks.
endeavor
toconcealremaining
16See,forexample,Baron(1971),Crewand Kleinand
andVisscher,
Holthausen,
Leland,Sherman
dorfer,
andJen.
Tschirhart
(2 and 3) that,
'7NotefrompartB of each theorem
aftereither
optimalschemehasbeenplayedout,buyers
pricesof otherbuyersif
willlearnthetruereservation
pricespaid or
theyare allowedto observethepriority
in this
thebidsof otherbuyers.One mightconjecture,
by the
wouldbe thwarted
case, thatdiscrimination
of buyerswithlow reservation
pricesto resell
attempt
lowertheprice to Pk-
etc. In factwe seem
360
THE AMERICAN ECONOMIC REVIEW
TABLE I-NUMERICAL
JUNE 1981
EXAMPLE
PriceReceived
nI
n2
n3
3
2
2
1
1
I
0
0
0
0
0
1
0
2
0
I
3
2
1
0
0
0
1
0
2
1
0
1
2
3
Probability
PP
M VA
1/27
3/27
3/27
3/27
3/27
6/27
1/27
3/27
3/27
1/27
3
3-6/7
4-11/19
3-6/7
4-11/19
4-11/19
3-6/7
4-11/19
4-11/19
4-11/19
4-1/3
3
3-2/3
3-2/3
4
5
4-5/9
4
4-5/9
5
5
4-1/3
E?T
A numericalexamplemighthelp to clarify
thepointsmade above. For thisexample,we
take capacityto be Q= 1 unit and marginal
productioncosts to be c=O. Suppose there
are N= 3 potential buyers each of whom
may have a reservation
priceof $3, $4, or $5
per unit(i.e., a =3, /B=5,k=3, 6=1). In this
case, from(6), the cut-offrank in eitherthe
prioritypricing scheme or the Modified
VickreyAuctionis T= 1. Thus even a buyer
with a reservationprice of x =3 has some
probabilityof receivingsome of thegood.
We firstcompute the values of zj*. Although these can be computed using the
formulaein the Appendix,here we compute
themdirectlyas an aid to intuition.Recall
thatzr is theexpectednumberof unitsthata
buyerwith lowest reservationprice ($3) receives in an allocationby rank.Since Q = 1,
theonlyway a buyerwitha reservation
price
of 3 can obtain any units is if both of the
otherbuyersalso have reservationprices of
3. This event has probability(1/3)(1/3)=
1/9 and if it occurs,each buyerwill obtain
1/3 unit. Thereforezr =(1/9)(1/3) = 1/27,
Z 7/27, z3 = 19/27,whereZ2 and Z3 were
calculated similarly.Using these values and
theformulagivenin Theorem2, we calculate
the priorityprices as p =3, p = 3-6/7,
theirunitsto buyerswithhigherreservation
prices.This
cannot happen, however,since in either scheme, the
equilibriumallocationis by rankwithrank determined
by true reservationprice. Thus no buyer receivesany
unitsunlessall buyerswithhigherreservationpricesare
fullysatisfied.
= 4- 11/19. Therefore,for example, if a
buyer is willingto pay p3*, he will get one
unit if neitherof the other two buyersare
willingto pay p*, he will get one-halfunitif
exactlyone of the otherbuyersis willingto
pay p*, and he will get one-thirdunitif both
of theotherbuyersare willingto pay p*. His
expected allocation of the good is just Z.
Consequently,if a buyerwhose reservation
priceis x3 = 5 statesthathe willpayp3*,then
hisexpected
utility
is (x3 -p3*)Z3* = 8/27.On
the otherhand, if this buyerstated that he
was only willing to pay p*, his expected
utility would be (x3 -p*)Z2 = 8/27. This
demonstratesthat any buyerwhose reservationpriceis x3 = 5 is just willingto payp3* as
opposed to p*. The otherself-selection
propertiesfollowsimilarly.
Withregardto the ModifiedVickreyAuction,usingTheorem3, we computeA1 = 2/3,
A2 = 5/9. This means that if exactly one
person bids 5 and at least one otherperson
bids 4, thepricewillbe 4-5/9. If one person
bids 5 or 4 and theothertwobid 3, theprice
is 3-2/3. The othercases are shownin Table
1. This table comparesthepricesreceivedby
the sellerunderprioritypricingand the auction schemesas a functionof the patternof
true reservationprices of the buyers. The
firstcolumn gives the assumed patternof
buyers'reservationprices (n 1,n2,n3) where
n is the numberof buyerswithreservation
pricexj. The second columngivestheprobabilitywhichthe sellerattachesto thepattern
givenin the firstcolumn.The thirdcolumn
gives the price receivedby the sellerin the
p3*
VOL. 71 NO. 3
HARRIS AND RAVIV: MONOPOLY PRICING SCHEMES
priority
pricingscheme(PP) whilethefourth
columngives the price receivedin the ModifiedVickreyAuction(MVA). At thebottom
of columns three and four is shown the
expectedprofitforeach scheme.
Note thatexpectedprofitin both schemes
is 4-1/3. In contrast,if themonopolistwere
to chargea singleprice of $4, his expected
profitswould be 4 X (26/27) whichis smaller
than 4-1/3. Any other single price would
yieldeven smallerexpectedprofits.
361
thansay Q <N units.In thiscase, theamount
sold will be lower and the price received
higher,on average,than in the single price
set up. As the theoremshows,the tradeoffis
not in favor of artificiallyrestricting
maximum output.'8
Note also thatthemonopolypriceXT will,
in general,be such thatless thanN unitsare
sold with positiveprobability.In particular
this will occur wheneversome buyershave
reservationprices strictlybelow XT. ObviouslyifXT = a theprobabilityof thiseventis
B. CapacityExceedsPotentialDemand
zero. In general,however,xT> a. The rela(Q>N)
tionshipwilldepend on thecost and demand
parameters,as can be seen from(6). Having
At the end of the previoussubsection,we
a monopolypriceabove a can also be viewed
argued that settinga single price is strictly as restricting
output.This typeof restriction
suboptimal.It is clear fromthe discussion is differentfrom the one discussed in the
thatthe drivingassumptionwas that Q<N.
previousparagraph.There,in orderto make
It mightbe conjectured,therefore,
thatwhen
discriminationfeasible, the seller must reQ ,>N, a singleprice is optimal.In thissubstrictoutput below N independentlyof the
section,we provethisconjecture.
realizationof demand. Under a singleprice
scheme the seller stands ready to sell all N
THEOREM 4:
units if N buyershave reservationprices at
A. When Q>>N, an optimal marketing or above XT. This lattertype of output reschemeis to set a singleprice equal to XT,
strictionsimplyreflectsthe usual "marginal
whereT is definedby (6). Each buyerwilling revenue= marginal cost" condition, as is
topay XT (namelythosewithreservation
prices
shownin part B of Theorem4.
at or above xT) receivesone unitof theprodFinally the theoremcan be interpreted
as
uctand pays XT.
showingthatin monopolyenvironments
with
B. The monopoly
price, XT, is thesmallest stochasticdemand and no bindingcapacity
reservation
price such thatmarginalexpected restriction,
an ex ante price-setting
strategy
revenueis greaterthanor equal to marginal is superior to an ex ante quantity-setting
cost.
strategy.'9Most previous studies analyzed
the problem of monopoly behavior under
To gain some insightinto thisresult,supuncertaindemand by assumingthatthe mopose Q?>N and thesellerannouncedpriority nopolist chooses a price ex ante. See, for
prices and offeredhigherpriorityto higher example, Baron (1971), Crew and Kleinpayingcustomers.If the sellerstands ready dorfer,Leland, Meyer, and Sherman and
to sell N unitsif all N buyersare willingto
Visscher.Our analysisthusjustifiesthis appay at least the lowest priorityprice, then proach provided capacity exceeds potential
higherpriorityaccess is meaningless.In this demand.Previousstudieshave,however,emcase, clearlyall buyerswill choose to pay the
ployed the price-setting
assumptionalso in
lowestpriorityprice since theygain nothing
frompaying a higherprice (in eithercase,
18Ourmodel assumes that Q is knownby everyone.
theygetone unitforsure).In otherwords,in
If Q were,say, randomwithsome probabilityof being
the absence of scarcityrelativeto demand, below N, thisargumentwould not go through,and some
the sellercannot discriminateacross buyers form of prioritypricing would be optimal. We are
gratefulto George Bortsforpointingthisout to us.
based on the probabilityof receivingthe
'9In fact,Theorem4 impliesonlythatpricesettingis
product.Therefore,in orderto discriminate, weakly
superiorto quantitysetting.The strictsuperiorthesellermustcreatescarcityby statingthat, ity (in a limitingcase) followsfromthe resultof Secunder no circumstances,will he sell more
tionV.
362
THE AMERICAN ECONOMIC REVIEW
cases whererationingmay be required.Our
resultsindicate that in such situationsone
would expect a monopolist not to use a
simpleex ante price-setting
strategy,
but instead use eitherprioritypricingor an auction.
V. Optimal
Pricing
Schemeswith
Endogenous
Capacity
In this section we suppose that the monopolist can choose his capacity. Capacity
must be chosen before demand is known.
From theresultsof SectionIV, we knowthat
if capacityis chosen to be less thenpotential
demand, then a prioritypricingscheme is
optimalwhereasif capacityis chosen to exceed potentialdemand, then a single price
scheme is optimal. Thus the problem of
choosing capacity can be reduced to the
question of whetherto choose Q = N (obviously Q > N is pointless) or Q <N and if
Q<N is optimalwhatis theoptimalQ.
Clearly,if increasingcapacityis free,then
the case of endogenous capacity choice is
equivalentto havingan exogenouscapacity
limitwhichis neverbinding.This is precisely
thecase analyzedin SectionIVB. It therefore
followsimmediatelythatif increasingcapacityis free,thenchoosingQ = N and usingthe
single price schemeis optimal. If, however,
capacityis at all costly,a necessarycondition
fora singlepricewithQ= N to remainoptimal,is thatexpectedprofitsare strictly
higher
under monopoly pricing with Q = N than
under prioritypricingwith Q <N for any
Q<N. In thenexttheoremwe showthatthis
conditionis in fact satisfiedfor the limiting
case in whichthe distribution
of reservation
prices approaches the continuous uniform
distribution
on the interval[a, /3].The proof
proceedsby takinglimitsof expectedprofits
as k, the numberof equally spaced reservation pricesin the interval[a,/3], approaches
infinity.
THEOREM 5: If capacitycan be costlessly
chosen,thenfor the limitingcase in whichk
approachesinfinity,
the singleprice XT with
Q =N is strictlysuperiorto any mechanism
withQ<N, forany Q<N.
JUNE 1981
Since we showstrictsuperiority
of a single
price scheme with Q=N when capacity is
costlessly
chosen,it followsthateven if there
is a cost of choosingcapacity,a singleprice
is still optimal provided that this cost is
sufficiently
small.Clearly,ifcapacityis sufficientlycostly,prioritypricingwill be superior withsome Q<N.
VI. Conclusions
In this paper, we have endogenouslyderivedoptimalmonopolypricingschemes.We
found that a crucial aspect of the environment for determiningan optimal pricing
schemeis capacitylimitations.In particular,
the policy of charginga single price on a
take-it-or-leave-it
basis is optimalif and only
if capacity restrictions
are unimportant.On
the other hand, when capacity restrictions
are importanta more complicated scheme
which we call "prioritypricing"is optimal.
We also find that for some environments
with potentiallybindingcapacity,a certain
common price auction is also optimal. It is
argued in the paper that these results are
consistentwithcasual observations.
APPENDIX: PROOFS OF THEOREMS
2, 3, 4, 5
For the sake of brevity,these proofsare
only sketchedhere.More detailedproofsare
available fromus on request.
Derivationof zT,.. Zkfor Q<N:
The appropriatechoice of z*,..., zk was
discussedin the textfollowingequation (6).
It can be shown that this procedure for
choosing Zj* results in the followingformulae:
(Al)
z7 `
i-1
m=I
(am+bm)+ai fori=T,...,k
where
(A2)
ai =
(k-i)'(i-
PN
1)N-j-
I
N-1
Q-2N
2 (
1
fori= 1,... , k-I
VOL. 71 NO. 3
HARRIS AND RA VIV: MONOPOLY PRICING SCHEMES
N-I
Q-1
(k-i)'(i-I)N-j-1
(A4)
N-1
j/
fori=1,..., k
Q-1/
bi=(PN-1
-(i-
[iNQ
NAT1
(k-i)Q
1
fori= 1,..., k-I
1)NQ]
Evaluatingtheright-hand
side of (5) using
(A1) and recallingthat zr =z =...=Z*
=0, we obtain the followingexpressionfor
the case Q<N:
k
i-1
i=T
m=l
It is not difficultto verifyusing(A8) that
thisallocationis feasiblefor[P] and thatthe
objective functionevaluated at this allocation equals the right-hand
side of (A5). This
shows that[d*, q*] solves [P]. Part B of the
theoremis clear since [d*, q*] satisfies(SS).
Proofof Theorem3:
The proposed solution for q* is given in
(A7) and now
d* (xi, R_ )d-
Proofof Theorem2:
Considertheallocation
(A6)
d*(xi, R-,)=d-pi*q*(xi
(A7)
q*(xi, R - )
R-1)
k
Oif 2
m=i+
=
min
1
nm(R)>Qori<T
k
~~Q- 2
[R(Q) ifR(Q+ 1) R(Q) >XT
R(Q+ 1)+
8Am[R(Q+Il]
if R(Q) >R(Q+ 1) XT
XT if R(Q+ 1) <XT
tth highestvalue among the R1, i=
R
I,...,N, fort=Q,Q+1 and foranyxnEX,
m(xn)=n. The expressionsfor the An are
givenin the statementof the theorem.It can
be shownthatAn>O forn=1,...,k-1.
The expected utilityof agent 1 for this
allocation when R, =xi is, again using (A8)
and also the definitions
of a, an and bn,
(A9)
d-E w(xi, R -1)q1 (xi, R 1)+xiz~
nm(R)
Q Mim(R
w(R) q*(Xi, R_1
where
(AS) Ev=Np E E Zi[(am+bm)+ai]
where Z =x1 +(2i-k-1)S-c.
Note that
(AS) givesthe value of any solutionof [P'].
363
i-l
d+xiZ*z-
) otherwise
a x
n=T
i-l
where nm(R)=number of elements of R
which equal xm for m= 1,..., k, and where
Pi* foriz T is givenin the statementof the
theorem and, for notational convenience,
p 0=0 for i<T. Let d and qj forj>l be
determinedfrom the symmetryconditions
givenin thestatementof Lemma 1. It can be
shownthat
(A8)
i-1
Elq*(Xi,R-)=
z (am +bm)+afj=zi
m=1
_
2
-aixiT- I
-XT
bn(xn+AnS)
n=T
2
n= 1
( an +bn) ifiz,>T
dif i<T
The remainderof the proof follows the
same strategyas that of Theorem 3 using
(A8) and (A9).
Proofof Theorem4:
When Q >N, it is clear that maximizing
the expressionon the right-handside of (5)
364
THE AMERICAN ECONOMIC REVIEW
involves choosing ql(xi,R_1)=l for iZ T,
i.e., choosingzT = ...
= 1. Therefore
the
objectivefunctionis
k
(AIO)
Ev=N99 :E [xl +8(2i-ki=T
=N(k-
I)-c]
T+ 1)(xT -c)
R
(Xi,9 -1)
R-,)=d-XTql
Il(iR_
=
0
xi
fori < T
i I(Xk +c)
But as mentionedabove, XTiS the smallestxi
such that(Al 1) holds. This provespart B of
the theorem.
Proofof Theorem5:
Clearlyit sufficesto show thatin thelimit
as k -x o, expected profitsusing a single
price xT with Q=N is strictlygreaterthan
(AS) forany Q<N.
From (6), it can be shownthat
(A12)
= lim (T/k) =maxro,
t 00
0 2a+3
k-_o
0I
2(13-a)
(A13)
lim qrM(k)=
lim
k-3oo
k-3oo
N
k (k-t+1)(XT-c)
IN(a - c)
ifc<2a-,B
N(
ifc>2a-f3
_)2
a)
/3#Next, denote by v(Q, k) the value of expected profitsfor any Q<N givenin (A5).
Thus
The remainderof theproofof partA follows
the same patternas in Theorems2 and 3.
To prove part B, the change in expected
revenue AERi when the monopoly price
changesfromxi to xi+I is givenby AER1=
(N/k)[(k -i + lI)xi-(k -i)xi+ I] = (Nlk)
(2xi-xk). The change in expectedquantity
sold is (N/k)[(k-i + 1)-(k- i)] = N/k.
Thereforethe marginalexpectedrevenueis
2xi -xk which is greaterthan or equal to
marginalcost c if and onlyif
(Al 1)
Q =N. From theproofof Theorem4
=
The proposed solutionforthis theoremis
d* (xi,
JUNE 1981
<< x
Now denote by vrM(k)the value of expected profitsunder a singleprice XT when
v(Q, k)=Nqpl(xl-c)
k
:E z*
i=T
k
- (k + )8 :E z*+2
i=T
k
2 E iz*t
i=T
It can be shown,usingthe above expression
thatlimk Toov(Q,
k) <limk , vM(k).
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