The Impact of Price Discrimination in Markets with Adverse Selection

75 Pages·2016·2.78 MB·English

The Impact of Price Discrimination in Markets with
Adverse Selection
AndréVeiga*
UniversityofOxford
Thisversion: November25,2016
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Abstract
Would consumer surplus increase if annuity rates were age-neutral? This
papercharacterizesthesociallyoptimalcontractibilityofagivensignalinmar-
ketswithadverseselection. Asignal(e.g.,age)partitionsconsumersintosub-
sets(e.g.,youngandold).Aregulatorrestrictsprice-discriminationonthebasis
ofthesignaliftheconsumersubsetswherethelevelofcostishigherarealsothe
subsetswherethereisgreaterdeadweightlossduetoadverseselection. Such
signalsareempiricallycommon. Toillustratethewelfarebenefitofpricedis-
crimination policy, I use a structural model to estimate its impact on the UK
annuitiesmarket. Themodelisestimatedusingproprietarydatathatinclude
theannuityseller’sestimateofeachindividual’slongevity.Ifindthatrestricting
pricediscriminationcanincreaseconsumersurplusbytheequivalentofabout
£6.5millionperyear.
Keywords:AdverseSelection,PriceDiscrimination,StructuralEstimation
JELClassificationCodes:D82,L52,D41
*IthankSimonCowan,IanCrawford,AndrewDilnot,JamesDuffy,LiranEinav,BasileGrassi,Sukriti
Issar, Ian Jewitt, Michael Keane, Paul Klemperer, Jonathan Levin, Margaret Meyer, Simon Quinn,
HowardSmith,KirstenSmith,JohnVickers,andE.GlenWeyl. SegyeShinprovidedoutstandingre-
searchassistance. DepartmentofEconomicsandSomervilleCollege, UniversityofOxford, Manor
RoadBuilding,OxfordOX13UQ,UK;[email protected]@gmail.com.
1
1 Introduction
Wouldconsumersurplusincreaseifannuityrateswerenotallowedtovarywithage?
Shouldtheseratesvarywithgender? Thereislittleguidanceastowhichindividual
characteristics(e.g.,gender,age)shouldbecontractibleinagivensetting.Theissue
isespeciallyrelevantinmarketslikeannuitiesandinsurance,sincecharacteristics
like age and gender are often correlated with the private information responsible
foradverseselection.
Thedisparatesetofcontractibilityregulationsacrosscountriesandmarketssug-
geststhattheireffectonwelfareisnotwellunderstood. Forinstance, acontrover-
sial 2012 EU ruling mandated that all insurance prices be gender-neutral but age,
placeofresidenceandothercharacteristicsremainfullycontractible. Similarly,the
UnitedStatesAffordableCareActmandatesthatpre-existinghealthconditionsare
notcontractiblebyinsurers, butpricescanvarywiththeindividual’szipcodeand
smoking status. Understanding the welfare effect of such policies is particularly
importantbecausethesepoliciesarepoliticallyexpedienttochangeandhavealow
implementationcost.Conversely,policieslikesubsidiesandmandatesrequiresub-
stantialpublicfundsortheycanbeperceivedasrestrictingconsumerchoice.
Thispapercharacterizesthesociallyoptimalcontractibilityofagivensignal. A
signal (e.g., age) partitions consumers into subsets (e.g., young and old). Restrict-
ing the contractibility of age shrinks the difference in the prices charged to young
andoldindividuals.Ishowhowtheoptimalcontractibilityofasignalisdetermined
by the characteristics of the consumer partition induced by the signal. I then il-
lustrateempiricallythepotentialwelfaregainofsuchapolicyinthecontextofUK
annuities. Theresultssuggestthatoptimalpricediscriminationpolicycanachieve
welfaregainsequivalentto0.09%oftotalannuitizedwealth,or£6.5millionperyear.
Iconsideranindustryaffectedbyselection,wherefirmscompeteinprices(asin
Akerlof(1970),butunlikeRothschildandStiglitz(1976)). Aproductisadverselyse-
lectedifitsinfra-marginalbuyers(thosewiththehighestrelativevaluationsforthat
product)arecostlierthanitsmarginalbuyers.Ifallindividualsmustbechargedthe
sameprice(forinformationorregulatoryreasons),firmswillcoverthehighcostin-
curred on infra-marginal buyers by charging a high price to all individuals. Then,
low-costmarginalindividualswillfaceapricehigherthantheywouldfaceifinfor-
mation were symmetric. Therefore, an adversely selected product exhibits a price
thatisinefficientlyhigh.
2
Often,firmshaveaccesstoasignal(e.g.,age)thatisimperfectlycorrelatedwith
individualcostsandvaluations. Realizationsofthesignalpartitionconsumersinto
multiplesubsets(e.g., youngandold). Aregulatorcan, forinstance, mandatethat
thepricedifferencebetweentheyoungandtheoldcannotexceedagivenamount.
A’fullcommunityrating’or’fullCR’policywouldmandatethepricedifferencebe
zero. A’fullpricediscrimination’or’fullPD’policywouldimposenoconstraint.
The key tension is that CR raises prices for some individuals and lowers it for
others. Iftheindustryiscompetitive,firmsbreakeveninanyscenario. Relativeto
PD,CRneverresultsinaParetoimprovement,butitcanincreaseoverallconsumer
surplus.Thispapercharacterizesthepolicythatmaximizestotalconsumersurplus.
If each consumer subset is homogeneous in cost, full PD is the welfare max-
imizing CR policy. In this case, there is no private information within each con-
sumersubset. Underperfectcompetition,fullPDeliminatesalldeadweightloss,as
inPauly(1970). CRintroducesadverseselectionwhereitpreviouslydidnotexist.
However,thereoftenremainssomeprivateinformationconditionalonthesig-
nal realization. Suppose young consumers possess significant amounts of private
information. Forinstance,ayoungannuitybuyermightbepurchasinganannuity
at an early age because a health condition has forced her to retire, or because she
iswealthyenoughtoretireearly. Thisprivateinformationcancreateadverseselec-
tion withinthe subsetof youngbuyers. Suppose thereis asingle annuity contract
forsale.UnderfullPD,adverseselectioncausestheannuitypriceamongtheyoung
tobeexcessivelyhigh,resultinginadeadweightloss. Ifoldannuitybuyersareho-
mogeneous, the price charged to the old is efficient under full PD. Now suppose a
regulationmarginallyshrinksthepricedifferenceacrossthesetwoconsumersub-
sets. Priceschargedtotheoldwouldrise, butthiswouldhavenofirst-ordereffect
on welfare, because the old were charged the efficient price. The price charged to
theyoungwouldfall,increasingwelfarewithinthatconsumersubset. Thus,overall
welfarewouldincrease.
Inthis(extremebutillustrative)example,CRisbeneficialbecausethehigh-cost
consumersubset(theyoung)isalsothesubsetexperiencingthegreateradversese-
lectiondistortion. Thispatternisempiricallycommoninawiderangeofmarkets,
becauseindividualstendtohaveprivateinformationaboutcostlyoutcomes(Hen-
dren(2013),Brownetal.(2014)). Therefore,someamountofCRisoftendesirable.
Moreover,theintuitionappliestoawiderangeofmarketsfromannuitiestohealth
insurance,lifeinsuranceandcreditmarkets.
3
The intuition described above generalizes in several ways. The model can ac-
commodate signals that partition consumers into more than two subsets. I also
extend the model to a setting where firms offer multiple products, but consumers
mustpurchaseoneofthem,asisthecaseinmyempiricalapplicationtoUKannu-
ities,describedbelow. Inbothextensions,thequalitativeresultsandtheirintuition
aresimilartothosedescribedabove.
IproceedbyillustratingempiricallythepotentialbenefitofCRpolicyinthecon-
text of the £12bn/year UK annuities market. Annuities provide a stream of pay-
ments conditional on the annuitant being alive. In this setting, annuity purchase
is mandatory but individuals can choose their guarantee period. For instance, a
five-yearguaranteeimpliesthebuyer’sestatereceivestheannuitypaymentsforfive
yearseveniftheindividualdieswithinthatperiod. Contractswithshorterguaran-
tees have larger monthly payments. All else being equal, short guarantees are ap-
pealing to individuals with high longevity since they are unlikely to be affected by
guarantees. Because long-lived individuals are costlier buyers of annuities, there
can be adverse selection into contracts with short guarantees. This adverse selec-
tion raises the cost of short-guarantee contracts, lowering the monthly payments
in such contracts. In equilibrium, adverse selection results in too few individuals
choosingcontractswithshortguarantees(butlargemonthlypayments).
I being by estimating the joint distribution of longevities and bequest motives
among UK mandatory annuitants. I use a structural model of annuity contract
choice that makes the following assumptions. First, individual mortality follows a
proportionalhazardGompertzprocess. Second,periodutilitiesfromconsumption
and bequests both exhibit constant relative risk aversion with the same curvature
parameter. Third,retirementtimingisexogenoustothechoiceincontract.
The model is estimated using a proprietary dataset of individual level annuity
purchases from a large UK insurer over a span of two years. The data include all
informationusedinpricingandseveralotherindividual-levelcovariates. Thedata
alsoincludethefirm’sestimateofeachconsumer’slifeexpectancy,usingthefirm’s
proprietary algorithm. This is, to my knowledge, the only dataset to include the
firm’s perceived cost in an annuities setting. This information allows me to take
individual longevity as observed (to the econometrician), although it is not con-
tractiblebythefirm.
The model is identified by variation in individual choices and variation in the
ratesofferedtodifferentindividuals. Ratesvarywiththesizeofpensionfundsbe-
4
causefirmsareallowedtopricediscriminateonthischaracteristic. However,fund
sizedoesnotdirectlyaffectguaranteechoicesundertheCRRAassumption,sothis
variationisexogenousgiventhemodel’sassumptions.Tomaketheestimationmax-
imallyflexible,Iestimatethemodelindependentlyinfourconsumersubsets: men
andwomen,purchasingatages60and65.
Iusetheestimationresultstocomputethecompetitiveequilibriumatacontin-
uum of policies between full PD and full CR. At each policy, I measure consumer
welfareas theincrease innon-annuitized wealththat woulddeliver thesame util-
ity as the individual’s preferred annuity contract. For concreteness, I express this
amountasashareoftotalannuitizedwealth(£12bnin2013). Ifindthatoptimally
restrictingthecontractibilityofgenderincreasesconsumersurplusamong65-year-
olds (men and women) by the equivalent of 0.01% of annuitized wealth, and by
0.09%among60-year-olds. Optimallyrestrictingthecontractibilityofageincreases
welfare by 0.11% among women (60- and 65-year-olds). However, full PD on the
basisofageistheoptimalpolicyformen.
Akeytheoreticalinnovationofthisarticleistocharacterize, inatractableway,
theoptimalcontractibilityofanarbitrarysignal. Levin(2001)findsthat,inmarkets
with adverse selection, revealing private information always increases the proba-
bilityoftrade. Thatarticledoesnotconsiderwelfaredirectly. Moreover, itfocuses
on binary signals that partition individuals into one subset where everyone has a
higher valuation than anyone in the other subset. That result does not hold when
moregeneralsignalsareconsidered.
I focus on a novel benefit of CR: reducing static deadweight loss from adverse
selection. Arrow(1963);Handel,HendelandWhinston(2015);Koch(2014)empha-
sizethatage-basedCRcanincreasewelfarebecauseitallowsindividualstoobtain
insurance against changes in health status over one’s lifetime (i.e., reclassification
risk). MyresultsshowthatCRcanbebeneficialeveninsettingswherereclassifica-
tionriskisabsent,asisthecaseinannuitymarkets.
Theinnovationsofmytheoreticalmodelincludeallowingforresidualasymmet-
ricinformationconditionalonsignalrealizationsandconsideringthecontinuumof
policiesbetweenfullPDandfullCR.Bycontrast,thiskindofheterogeneityhasnot
beenconsideredbytheexistingliteratureonthird-degreepricediscrimination(e.g.,
Aguirre,CowanandVickers(2010),Schmalensee(1981)).ChenandSchwartz(2015)
consideramonopolysellingtotwosubsetsofindividuals,eachwithadifferentlevel
of cost, but each consumer subset is homogenous. Moreover, this literature tends
5
tofocusoncomparingthetwoextremepoliciesoffullPDandfullCR.
My focus is on CR policies which are implementable using only knowledge of
aggregate quantities such as the joint distribution of cost and willingness to pay.
By contrast, Bergemann, Brooks and Morris (2015) shows that any feasible split of
consumer and producer surplus is achievable under an appropriate information
structure. However, implementing these allocations would require a regulator to
knoweachindividual’svaluation.
Crocker and Snow (1986), Hoy (1982) and others find that costless consumer
categorization(i.e.,PD)canexpandtheutilitiespossibilityfrontierandevenleadto
a Pareto improvement. In this context, Finkelstein, Poterba and Rothschild (2009)
considerspecificallytheannuitiesmarket.Thosearticlesconsidersettingswithen-
dogenousquality(likeRothschildandStiglitz(1976))whereasIconsiderfirmsthat
compete only in prices. Competition in prices with otherwise fixed contract char-
acteristics is common in many significantly regulated markets, like annuities and
healthinsurance.
Myempiricalframeworkissimilartothatofotherstudiesofannuitychoicesuch
asKotlikoffandSpivak(1981),Mitchelletal.(1999),Davidoff,BrownandDiamond
(2005). The most closely related article is Einav, Finkelstein and Schrimpf (2010,
henceforthEFS),whoalsostudyUKannuities. Theirdatasetdoesnotincludeindi-
vidualexpectedlongevityoranumberofothercovariatesIobserve. Also,because
itislessrecent,theirdatadoesnotincludevariationinrates. Moreover,theirfocus
isonoptimalmandates,whereasIfocusonCRpolicy.
Tomyknowledge, thisarticleisthefirststudyofCRinthecontextofannuities
markets. Blumberg and Buettgens (2013), Orsini and Tebaldi (2015), and Ericson
and Starc (2015) empirically study the effect of age-based CR in the context of US
health insurance. Orsini and Tebaldi (2015) find CR has little effect on enrollment
butasignificanteffectongovernmentexpenditures. EricsonandStarc(2015)find
thatCRreducesprofitandincreasesoverallconsumersurplus.
Thepaperisorganizedasfollows.Section2containsthetheoreticalresults.Sec-
tion3describesthedataandinstitutionalcontext.Section4describesthestructural
modelofannuitychoice. Section5describestheestimationandresults. Section6
contains the counterfactuals. Section 7 concludes. The appendices following the
main text contain all proofs and additional details regarding the data, estimation
andadditionalcalibrations.
6
2 Theory
21 BaselineModel
Ibeginbydescribingabaselinemodelwherethereisasinglesetofconsumersand
asingleproductbeingsold,asinEFC.1 Iconsideracontinuumofconsumerswith
unitmass. Asinglecontractisoffered(e.g.,anannuityorahealthinsuranceplan).
Consumerwillingnesstopayisu [0,u],withsmoothPDFf(u).2
2
Allfirmsareidentical,andIfocusonallocationswhereallfirmschargethesame
pricep.Pricecompetitioniscommoninregulatedmarketssuchasannuities(Einav,
Finkelstein and Cullen (2010)). A consumer buys the contract if u p. Industry
�
demandis
u
Q = Q(p) = f(u)du
Zp
withslopeQ = Q (p) < 0. Thesemi-elasticityofdemandis
0 0
Q
0
� = �(p) = > 0.
�Q
Individualswithwillingnesstopayuhaveacostc = c(u) 0.3 Iwillrefertoc(p)
�
as the “industry marginal cost” since it is the derivative of industry total cost with
respecttoquantity,atpricep.4 Theindustryaveragecostis
1 u
AC = AC(p) = c(u)f(u)du,
Q(p)
Zp
withslopeAC = �(AC c).Industryprofitis
0
�
⇡(p) = Q(p)(p AC(p)).
�
withslope⇡ = ⇡ (p) = Q (p c)+Q.
0 0 0
�
1RelativetoEFC,Iconsiderdemand,cost,andsoonasfunctionsofprices,notquantities. This
setup is more natural in the context of price discrimination and will result in simpler expressions
fortheresultsbelow. Itisstraightforwardtoexpressallfunctionsintermsofquantities,asinEinav,
FinkelsteinandCullen(2010);MahoneyandWeyl(2014).
2AppendixIexploresexamplesofpossiblemicro-foundationsofu.
3Thatis,c(u)istheexpectedcostamongallindividualswithvaluationu.
4LettingtotalcostbeC = uc(u)f(u)du,then dC = dC 1 = cQ0 =c.
p dQ dp dQ/dp Q0
R
7
Iusethefollowingdefinitionofconsumersurplus:
v¯
U = U (p) = (u p)f(u)du,
�
Zp
with slope U = U (p) = Q(p). Therefore, welfare is W (p) = U (p)+⇡(p), with
0 0
�
slope W = W (p) = Q (p c). Notice that, when a product is priced efficiently
0 0 0
�
(i.e., at marginal cost, so p c = 0), then a marginal increase in the price has no
�
first-ordereffectonwelfare.
Thisdefinitionofconsumersurplusimpliesincomeeffectsarenegligible,sore-
distribution has no intrinsic value. The definition also implies the absence of the
Hirshleifer(1971)effect,inwhichex-postredistributionprovidedarationaleforre-
strictingtheinformationavailabletoinsurers. Therefore, anybenefitofCRinthis
settingmustariseduetoadecreaseinstaticdeadweightloss.5
I assume free entry of firms into this industry. Under the regularity conditions
presentedbelow,auniquepure-strategiesNashequilibriumpricep?existsthatsat-
isfies
⇡(p?) = 0 p? = AC(p?).
)
At the equilibrium price p?, each firm breaks even and so does the industry as a
whole.
However,anywelfare-maximizing(interior)pricep??satisfies
W (p??) = 0 p?? = c(p??).
0
)
Thatis,thesociallyoptimalpricep??isdeterminedbythefixedpointofc(p),whereas
theequilibriumpricep?isdeterminedbythefixedpointofAC(p).Therefore,when-
everc(p) = AC(p),thecompetitivepricewillnotbethewelfare-maximizingprice
6
(p? = p??).
6
AsemphasizedbyEFC,c(p)mightdifferfromAC(p)duetoadverseoradvanta-
geousselection. Theproduct“buy”isadverselyselectedifc = c (u) > 0,sothose
0 0
withhigherwillingnesstopayalsohavehighercost. Adverseselectionimpliesthat,
atagivenp,infra-marginalbuyershaveahighercostthanmarginalbuyers. There-
fore,
E[c u > p] > c AC > c.
| )
5ThisdefinitionofconsumersurplusandwelfareisubiquitousinthePDliteratureandcommon
intheadverseselectionliterature.
8
AC,c
AC(p)
c(p)
p
p** p*
Figure 1: An example of equilibrium and socially optimal price in a market with
adverseselection. Thediagonaldashedlineisthe45°line.
Thiswedgebetweenc(p)andAC(p)generatestheselectiondistortionemphasized
by EFC. If the product is adversely selected, then p? > p??. That is, the product’s
equilibrium price will be higher than is socially optimal, which distorts individual
choices.Relativetoasettingwithsymmetricinformation,fewerindividualsbuythe
product. Figure1illustratestheequilibriumandsociallyoptimalpriceinamarket
wheretheproductisadverselyselected.
It is also possible that the product is advantageously selected, if c (u) < 0, im-
0
plyingAC < candAC < 0. Inthiscase,infra-marginalbuyersarelesscostlythan
0
marginalbuyers,whichdrivesdownthepriceoftheproductresultinginp? < p??.In
thiscase,relativetoasettingwithsymmetricinformation,moreindividualschoose
tobuytheproduct. Iwillassumec(u)ismonotonic,soselectionisgloballysigned.
Because AC = �(AC c), adverse selection also implies AC > 0. The slope
0 0
�
ofAC isproportionaltothewedgebetweenAC andc,andthereforeAC captures
0
theimportanceofadverseselection. Forinstance,withoutselection,theAC curve
isflat(c = 0 AC = 0).6 IfAC islargeandnegative,thenadvantageousselection
0 0 0
)
issignificant.
6Noticealsothatasp u¯,AC c 0,andthereforeAC0 0. Moreover,AC(u)istheaverage
! � ! !
costintheentirepopulation.
9
Imakefourregularityassumptions. First,IassumeQislog-concaveso� (p) >
0
0.Thisassumptionimplieswelfareisresponsivetopriceswhenmarketsexperience
alargepricedistortion,butsmallwelfaregainsresultfromcorrectingpricesinmar-
kets that are already close to their efficient price. Second, I assume c < 1, which
0
impliesadverseselectioncannotbetooextreme. Thisassumptionrequiresthat,as
valuationsuincrease,thesurplusconsumersobtainbeyondsimplyimposingacost
ontheinsurer(u c(u))isincreasing. Third, Iassumeu c(u) > 0overtherele-
� �
vantrangeofprices,whichimpliesconsumersderivesomesurplusbeyondthecost
theyimposeontheinsurer. Thisconditionisvalid,forinstance,ifallconsumerare
sufficientlyriskaverseorifmoralhazardissufficientlysmall. Fourth,Iassumethe
relevant range of prices is always below the monopoly price.7 These assumptions
implythefollowing:
Lemma 1. ⇡ < 0, AC < 1 and p? = AC(p?) is the unique pure-strategies Nash
00 0
equilibrium.
22 PriceDiscrimination
PD becomes relevant if firms observe a signal that partitions the set of consumers
intomultiplesubsets. Supposetwosuchsubsetsexist,indexedbym A,B (say,
2{ }
youngandold). Letthesubscriptmidentifythedemand, price, andsoonineach
set.8 Iassume,forsimplicity,thatthesame(unit)massofconsumersexistsineach
market.9 Notice that I consider an arbitrary partition A,B , in contrast to Levin
{ }
(2001)whoassumedthatthesubsetsA,B arerankedbystrongsetorder.
Icontinuetoassumefreeentryoffirmsintotheindustry. However,ifthereare
constraintsonPD,firmscannotrejectawillingbuyer,becausesuchrejectionwould
amounttoPD.Therefore,eachfirmmustservebothconsumersubsetsand,inequi-
librium,theindustrywillchargepricesp ,p suchthatfirmsbreakevenacrossboth
A B
consumersubsets:
⇡ (p )+⇡ (p ) = 0. (1)
A A B B
Under full PD, prices are p = p? and p = p? , such that ⇡ (p?) = ⇡ (p? ) = 0.
A A B B A A B B
7TheassumptionsinAppendixAimplyAC(p)iseitherdecreasingoracontraction,andtherefore
hasauniquefixedpoint.TheconditionisequivalenttotheassumptioninMahoneyandWeyl(2014)
thataveragecost(asafunctionofquantity)iseverywherelesssteepthaninversedemand.
8Forinstance,Q (p )isdemandinsubsetA;p? istheequilibriumpriceinsubsetBunderfullPD.
A A B
9Itwouldbestraightforwardtoscaleeachmarketupbyadifferentfactorcapturingitssize.