Purchasing health care services from providers with unknown altruism

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JournalofHealthEconomics24(2005)73–93
Purchasing health care services from providers
(cid:1)
with unknown altruism
∗
William Jack
DepartmentofEconomics,GeorgetownUniversity,RoomICC580,
Washington,DC20057,USA
Received16September2003;receivedinrevisedform26May2004;accepted4June2004
Availableonline5November2004
Abstract
Cost-sharingrulesforpayingphysicianshavebeenadvancedasawayofgeneratingincentives
for the provision of quality care, while recognizing their potential negative effects on production
efficiency.However,theoptimalsharingratetypicallydependsonthedegreetowhichthephysician
actsintheinterestofthepatient,whatweidentifyasthephysician’saltruism.Sincethedegreeof
altruismislikelytovaryacrossphysicians,andtobeprivateinformation,thestandardrulesforsetting
thecost-sharingrateareunlikelytobeoptimal.Thispaperderivesconditionsfortheoptimalnon-
linearcost-sharingmechanisminthepresenceofasymmetricinformationaboutaltruism,andshows
howitcansometimesbeimplementedthroughamenuoflinearcost-sharingschemes.Themodelcan
beusedtorationalizethedesignofthefund-holdersystemforgeneralpractictionersthatoperatedin
the1990sintheUnitedKingdom.
©2004ElsevierB.V.Allrightsreserved.
JELclassification: D8;H4;I1
Keywords: Physicianincentives;Cost-sharing;Altruism
(cid:1)
ThispaperwaspartlywrittenwhiletheauthorwasavisitingscholarattheCenterforHealthEconomics
ResearchandEvaluationattheUniversityofTechnology,Sydney,NewSouthWales.
∗ Tel.:[phone];fax:[phone].
E-mailaddress:[email protected].
URL:www.georgetown.edu/faculty/wgj.
0167-6296/$–seefrontmatter©2004ElsevierB.V.Allrightsreserved.
doi:10.1016/j.jhealeco.2004.06.001
74 W.Jack/JournalofHealthEconomics24(2005)73–93
1. Introduction
It is now widely recognized among health economists that in budgeting resources for
healthcareservices,itmattersasmuchhowresourceflowsareimplementedashowmuch
of a resource envelope is available. That is, research has focused increasingly on how to
purchaseservices,inadditiontothestandardquestionofhowmuchhealthcareshouldbe
provided.Thelatterkindofquestioncanbeaddressedwithtoolsofcost-benefitandcost-
effectivenessanalysis.Thispapercontributestotheliteratureonthefirstquestion:howto
purchasemedicalcare.
Muchoftheexistingliteraturepertainingtothisquestionhasrecentlybeenthoroughly
reviewed by Newhouse (2002). A recurring theme in the design of physician and
hospital payment mechanisms is the need to provide incentives for the provision
of quality care without cream-skimming, while also maintaining incentives for cost-
control. Cost-reimbursement is good for the former, while fixed budgets are good
for the latter, but it has proven perennially difficult to achieve both objectives si-
multaneously, except when consumers themselves are good judges of quality and can
induce the delivery of high quality by threatening to switch providers (Ma, 1994;
GlazerandMcGuire,1994).
Animportantelementofanumberofmodelsofphysicianpaymentmechanismsisthe
extenttowhichprovidersvaluethebenefitsofcarethatotherwiseaccruetopatients.For
example,EllisandMcGuire(1986)showedthatifamedicalcareprovidercaresaboutnet
revenueandpatientbenefitsinthesamewaythatasocialplannerdoes(sothephysician
isa“perfectagent”),thenfullyprospectivepaymentisoptimal.Thereisnoproblemwith
under-provision of quality simply because it is assumed that providers care about it as
muchasthesocialplannerdoes.If,ontheotherhand,providersplaceasmallerweighton
patientbenefits,thenunderprospectivepaymenttheywillleantowardsunder-provisionof
quality.Thisdistortioncanbecorrectedbyreimbursingsomeoftheincurredcosts:since
thereisaone-to-onerelationshipbetweencostsandquality,subsidizingcostsisthesameas
subsidizingquality,andthesocialoptimumcanbeimplemented.Thus,EllisandMcGuire’s
contributionhastheflavorofcorrectivePigouviantaxation—qualityprovisionimposesa
positiveexternalityonconsumers(becausetheirbenefitsareunder-valuedbytheprovider)
andsoissubsidized.
Chalkley and Malcomson (1998) introduced the idea that there may not be a one-to-
one relationship between costs and quality, so that if the physician places less than full
weight on consumer well-being, the corrective subsidy inherent in Ellis and McGuire’s
cost-sharing rule would not yield the full optimum. In particular, costs depend on both
quality(intheusualway)andeffortexertedbythephysician.Thefulloptimumcouldbe
implemented by subsidizing quality directly, but only if this is contractible. Subsidizing
costs, however, is not a perfect substitute for subsidizing quality, as it reduces incentives
toexerteffort.Thesizeofthesubsidyoptimallyemployeddependsontheextenttowhich
theprovidervaluesconsumerbenefits.Ifthephysicianplacesanyvalueonpatientbenefits,
thenaprospectivepaymentwillinduceoptimalcost-reducingeffort,andsomepositive,but
sub-optimal,levelofquality.Asmallshifttocost-reimbursementiswelfareimproving,as
thepositiveimpactofincreasedqualityisoffirstorder,whilethenegativeimpactthrough
higher costs is second order. Intuitively, as the physician values patient well-being more,
W.Jack/JournalofHealthEconomics24(2005)73–93 75
qualitychoicesunderprospectivepaymentmovetowardsthesocialoptimum,andthesize
of the corrective cost-reimbursement rate falls.1 Just as the Ellis and McGuire model is
reminiscentofPigou’s(1947)analysisofexternalities,ChalkleyandMalcomson’saccount
canbeseenasbeinginthespiritoftheGreenwaldandStiglitz(1986)examinationofthe
secondbest.
In both papers, the optimal cost-sharing rate (which achieves the first best in Ellis
and McGuire, and the second best in Chalkley and Malcomson) depends on the ex-
tent of agency – that is, on the extent to which the provider incorporates patient bene-
fits when making decisions about quality and effort. It is arguable, however, that such
preference parameters are at least as difficult to discern as the quality of services pro-
vided, if not more so. Indeed, it is very likely that providers vary widely in their altru-
ismtowardspatients,soitwouldbenecessarytopaydifferentprovidersdifferently—the
more benevolent having a smaller share of incurred costs reimbursed. There is no
guarantee, however that faced with a menu of cost-sharing schemes designed for the
range of provider types, each type would choose the cost-reimbursement scheme meant
forhim.
We employ standard techniques from the optimal regulation literature (Laffont and
Tirole, 1993) to examine the way a purchaser should optimally contract with providers
withheterogeneousandunknowndegreesofaltruism.Thesetechniqueshavebeenusedin
thehealtheconomicsliteraturepreviously,butwhenheterogeneityiswithrespecttoproduc-
tivityorcost.Forexample,ChalkleyandMalcomson(2002)usedthetechniquestostudythe
roleofcostreimbursementwhenpatientswithagivenadministrativelydefinedcondition
(i.e.,agivenDRG)differintheseverityoftheirillnesses.Intheirmodel,providerswere
identical,buttherewasdemand-sideheterogeneityinthesensethattheseverityofpatient
conditionswithinagivendiagnosticrelatedgroup,coulddifferacrossproviders.Similarly,
Jack(2002)assumedprovidersdiffereddirectlyaccordingtoameasureofproductivityina
modeloftheinstitutionaldesignofahealthsystemwhendecisionsabouthospitalclosures
areincludedincontractdesign.
The next section sets up the structure of the model, including the technological
relationship between physician effort, the quality of care, and costs, as well as a de-
scription of the objectives of the purchaser. In Section 3 the first best outcome is
illustrated, when provider characteristics and actions are contractible. Section 4 ex-
amines the second best policy when just provider characteristics, but not quality, are
observable, similar to Chalkley and Malcomson’s (1998) analysis. Section 5 presents
what we call the third best—the optimal payment mechanism when neither provider
characteristics nor quality of care are observable. Section 6 presents an illustration of
the use of the model in rationalizing the design of the UK’s fund-holder experiment,
wherein general practices were given the option of adopting fund-holder (high cost-
sharing) status and continuing with the status quo (low cost-sharing). Section 7 briefly
concludes.
1 GlaeserandShleifer(2001)presentamodelthatprovidesamotivationforwhyfirmsmightchoosenon-profit
status.TheirmodeliscloselyrelatedtoChalkleyandMalcomson’sinthatreducingthefinancialrewardtocost
reduction(i.e.,havingsomedegreeofcost-sharing)reducesincentivestostintonquality.
76 W.Jack/JournalofHealthEconomics24(2005)73–93
2. Modelset-up
Inthissection,wefirstdescribethetechnologicalenvironmentrelatingphysicianeffort,
qualityofservices,andfinancialcosts,andthenspecifysocialobjectivesthatarepursued
bythepurchaser.
2.1. Effort,quality,andcosts
A health care purchaser (for example, the government) contracts with a population of
medicalcareproviders.Providerscanchoosethequality,q>0,ofcaredelivered,andcan
affect the financial cost, C, by exerting effort, e>0. We assume C(q, e)>0 satisfies the
convexityassumptionsC >0,C >0,C <0,C >0,andC <0,wheresubscriptsdenote
1 11 2 22 12
partial derivatives. It is helpful to think of C as the cost the provider incurs when buying
inputs (including the labor services of other staff), so that higher quality services cost
more,buteffortreducesthesecosts.Noticethattheprovidercan“reducecosts”bysimply
providingalowerquality.ThereisnoheterogeneityinC(·,·)acrossphysicians,sotheyall
have the same case mix and labor productivity (or else differences in these factors have
beenaccountedfor).2
As well as the financial costs of provision, the provider suffers some extra disutility
associatedwithherchoiceofqualityandeffort.Usually,wethinkofthisasjustthepersonal
costsofeffortthatdonotshowupinpurchasedinputs,whatLaffontandTirole(1993)for
examplehavedenotedψ(e),whereψ(·)ispositive,increasing,andstrictlyconvex.However,
inthispaper,weadmitthepossibilitythatproviderscareaboutthequalityoftheservices
theyprovide.Itissuggestedthattheyderivesomedirectbenefitθϕ(q)fromprovidingquality
q.θ∈[(cid:1) ,(cid:1) ]⊂R+ parameterizesthestrengthofaprovider’spreferenceforquality,and
0 1
ϕ(·)ispositive,increasing,andweaklyconcave.Thus,thenetdisutilityofprovidingquality
qandexertingefforteis:
γ(q,e,θ)=ψ(e)−θφ(q). (1)
Although,weshallsometimesrefertoθasameasureofbenevolenceoraltruism,itshould
benotedthatconsumerwell-beingdoesnotfullyentertheprovider’sutilityfunction.Thisis
fortworeasons:first,althoughtheprovidervaluesquality(thedirectbenefitsofwhichaccrue
topatients),shemightnotvalueitinthesamewayasconsumers.Thatis,ϕ(·)mightnot
necessarilymeasurethebenefitofcaretoconsumers.Thesecond,andmoreconsequential
reasonisthat,whiletheprovidervaluesquality,shedoesnotvaluecostsavingsinasimilar
way.Because,thefinancialcostsofservicedeliveryarebornebyconsumersthroughtaxes,
cost savings benefit patients, but the physician does not recognize these benefits when
choosingthelevelandqualityofmedicalcare.Thisaspectoftheformulationisconsistent
withmuchoftheliteratureonphysicianobjectives,assurveyedforexamplebyMcGuire
(2000),althoughwerecognizethat,asMcGuirenotes(page521–22),thereissomeempirical
2 Othermodelsintheliterature(MaandMcGuire,1997;McGuire,2000)formulatethequalityeffortbywriting
qualityasafunctionofeffortandotherinputsz,e.g.,q(z,e).Ifthepricesofotherinputsarefixed,thisfunction
canbeinvertedtoyieldC(q,e)asabove.
W.Jack/JournalofHealthEconomics24(2005)73–93 77
evidencesuggestingthatphysiciansrespondalsotothefinancialcostsbornebyconsumers
(Eisenberg,1986).3
2.2. Socialobjectives
Thepurchasercaresaboutconsumerandproviderutility(i.e.,consumerandproducer
surplus).Netconsumerutilityisgrossbenefitsfromquality,S(q),increasingandconcave,
lessthefinancialcostsofpayingforservices.Followingthestandardliterature,weassume
theproviderisreimburseditsfinancialcostsc,andlettbethetransfer(ifany)totheprovider
in excess of these costs. Thus, the gross transfer received by the provider is t+c. The
consumer,however,mustforgoanamount(1+λ)(t+c),whereλ>0isthemarginalexcess
burdenoftaxation.Thus,netconsumerutility,orconsumersurplus,isS(q)−(1+λ)(t+c).
Theprovider’snetutilityisu=t−γ,whichisdecomposedintonetprofitplusbenefits
fromquality,
u=t−γ =t−ψ+θφ≡π+θφ.
Utilityisdecomposedinthisfashionbecause,followingHammond(1987)andChalkley
andMalcomson(1998),wedonotwishtoincludethedirectbenefitsderivedbytheprovider
insocialwelfare.Thatis,thepurchaser’sobjectivefunctionis:
W =S(q)−(1+λ)(t+c)+t−ψ =S(q)−(1+λ)(ψ+c)−λπ. (2)
Leaving profits with the provider is costly, simply because it costs money to transfer
resourcestothemfromconsumers.Aprovideroftypeθ willagreetosignacontractonly
ifhernetutilityisnon-negative,u≥0.Thisinturnmeansthatthepurchasermustensure
thatπ≥−θφ.4
Itisinstructivetonotetheconsequencesofincludingthealtruisticcomponentofprovider
utilityinsocialwelfare.Thewelfarefunctionthenbecomes:
W+ =S(q)−(1+λ)(t+c)+t−γ =S(q)−(1+λ)(γ +c)−λu.
Leavingutilityrentstoprovidersisagaincostly.Ifinbothcasesproviderscanbekept
totheirreservationutilitylevels,thentheonlydifferencebetweentheformulationsisthat
the quality of care optimally purchased under W (and the health care budget allocated
+
toproviders)willbegreaterthanthatpurchasedunderW.Inbothcases,theparticipation
constraint is u≥0, and even in the third best, when this cannot be made to bind for all
typesofprovider(i.e.,forallθ),thequalitativefeaturesoftheoptimalcontract,suchasthe
3 Wenote,howeverthatinthemodelofthispaper,allcostsfinancedbyconsumersarechannelledthroughthe
taxsystem,soitisarguablethatproviderswouldbemuchlessresponsivetoincreasesinthemthaniftheywere
paiddirectlybyconsumersatthepointofservice.
4 Itmightbenecessaryinpracticetoaddtheconstraintthatt>0,since,otherwisetheproviderwouldbe
requiredtopaythepurchaserfortheprivilegeofworking.
78 W.Jack/JournalofHealthEconomics24(2005)73–93
directionofdistortionstobudgetsandquality,areunaffectedbytheinclusionofaltruistic
benefitsinthewelfarefunction.
3. Thefirstbest
Ifapurchasercouldwriteacontractbasedonqualityandcost,andifitknewthetype
of provider it was purchasing from, it would set t=ψ−θϕ (i.e., π=−θφ) and solve the
followingmaximizationproblem:
maxS(q)−(1+λ)(ψ(e)+C(q,e,θ))+λθφ(q),
q,e
withassociatedfirstorderconditionsforaninteriorsolution:
S(cid:10)(q)−[(1 +λ)Cq−λθφ(cid:10)(q)]=0, (3)
and
−(ψ(cid:10)(e)+Ce)=0. (4)
Denote the optimal values by q*(θ) and e*(θ). Quality should be expanded until the
marginalbenefittoconsumersisequaltothemarginalcosttheybear(Eq.(3)).Themarginal
cost of quality borne by consumers is composed of a gross marginal payment to cover
additional input expenses (1+λ)C , offset by the marginal saving on distortionary costs
q
becauseprovidersareeffectivelypartiallypaidinkind,λθϕ(cid:10) (q).Providereffortshouldbe
exertedtoequatethemarginaldisutility,ψ(cid:10) (e),withthemarginalcostsavings,−C (Eq.
e
(4)).
To examine how the first best solution varies with θ, write Eqs. (3) and (4) as g(q, e,
θ)=0 and h(q, e)=0, respectively. The second order condition for a maximum, which is
assumedtobesatisfied,requiresinpartthat∆=g h −g h >0.Optimalqualityandeffort
1 2 2 1
choicesvaryaccordingto:
(cid:1) (cid:2) (cid:1) (cid:2)
q∗(cid:10)(θ) λφ(cid:10) −h
= 2 .
e∗(cid:10)(θ) ∆ h1
Ash ispositiveandh isnegative,qualityandeffortbothincreasewithθatthefirstbest.
1 2
Thesetwoeffectsmeanthatattheoptimum,thevariationofinputexpendituresc*(θ)and
theoptimaltransfertotheprovidert*(θ)withθisnotdeterminedwithoutfurtherstructural
assumptions.Thenetutilityearnedisidenticallyzeroforallprovidertypes,u*(θ)=0.
4. Thesecondbest:non-contractiblequalityandeffort
Supposethepurchaserattemptstoimplementthefirstbestoptimumbygivingaprovider
oftypeθ abudgetc* (θ)withwhichtoprovideservices,andatransferorsalaryt*(θ),to
W.Jack/JournalofHealthEconomics24(2005)73–93 79
compensateherforhereffortcosts.5Ifqualityandeffortchoicesarenotdirectlyobservable,
then this transfer will in general not yield the first best outcome. In particular, given any
transferc,aslongassheiswillingtoparticipate,theproviderwillchoosequalityandeffort
tosolvethefollowingproblem:
minγ(q,e,θ) s.t. C(q,e)=c. (5)
q,e
Letqˆ(c,θ)andeˆ(c,θ)betheprovider’soptimalchoices(assumedunique),whichsatisfy:
−θφ(cid:10)+µCq =0 (6)
and
ψ(cid:10)+µCe =0 (7)
for some non-negative Lagrange multiplier µ(c, θ). The convexity assumptions on C, ϕ,
andψ implythatqˆ andeˆ areincreasinginθ.Wefurthermakethefollowingassumptions
regardingthedependenceofqˆ andeˆ onc:
Assumption1.
dqˆ deˆ
>0 <0,
dc dc
d2qˆ d2eˆ
<0 >0,
dc2 dc2
and
d2qˆ d2eˆ
>0 <0.
dθc2 dθdc
Theseassumptionsensurethatindifferencecurvesin(c,t)-spacearedownwardsloping
andconvex.Thefirstpairofassumptionssaythatahigherbudgetleadstohigherquality,
butcrowdsoutsomeeffort.Thesecondpairrepresentakindofdecreasingreturnsassump-
tion.Furtherincreasesinabudgetleadtodiminishingqualityimprovements,andsmaller
crowdingouteffectsonthesupplyofeffort.Finally,thethirdpairsaythatthehigherisθ,
thelargertheeffectonqualityandeffortofincrementalbudgetresources.Thefirstorder
effectsofprovidertypeandbudgetsareillustratedinFig.1.
Eqs.(6)and(7)arethesameasthefirstorderconditionsforthefirstbestoptimumifand
onlyifthebudgetistherightsize(sothatµ(c,θ)=1),andiftheprovidervaluesqualitythe
samewayasthepurchaser,uptoanadditiveconstant:thatis,onlyifθϕ(q)=S(q)+k,for
someconstantk.Toseethis,notefrom(6)thatifcissettoensureµ=1,theproviderchooses
qualitytosatisfyC =θϕ(cid:10)(q).Substitutingthisinto(3),wefindthatthesocialoptimumis
q
5 Theseresourcescould,ofcourse,betransferredjointly,forinstanceasatotalbudgetb=t+c.Weassume
thatthepurchasercouldthenmonitorexpendituresoninputsanddeclarethecontractvoidiftheseexpenditures
didnotmatchc.
80 W.Jack/JournalofHealthEconomics24(2005)73–93
Fig.1. Boldlinesrepresenteffort-qualitypairsyieldingconstantfinancialcosts,andthinlinesrepresentconstant
disutilityofprovisionlines.(i)Morealtruisticprovidersputinmoreeffortandproducehigherqualityforthesame
budget.(ii)Alargerbudgetleadstohigherqualityandlowereffort.
attainedonlyifθϕ(q)=S(q)+k.EllisandMcGuire(1986)referredtoaproviderwithsuch
acoincidenceofpreferencesasa“perfectagent”.
Iftheproviderisnotaperfectagentforthepurchaser(theinterestingcase),whatisthe
bestthepurchasercando?ThisisessentiallythequestionaddressedbyEllisandMcGuire
(1986)andChalkleyandMalcomson(1998).Thepurchaser’sdecisionvariablesarenow
thebudgetcandtransfertmadetotheprovider.Weassumeinthissectionthatthepurchaser
knowstheprovider’stype,θ.
Foragivenbudgetc,denotetheminimizeddisutilityin(5)byγˆ(c,θ),andlet:
γˆ(c,θ)=ψˆ(c,θ)−θφˆ(c,θ).
Thatis,ψˆ (c,θ)isthecostofeffortoptimallyincurredbytheproviderwhengivenabudgetc,
andθφˆ(c,θ)isthedirectbenefitfromqualityoptimallyderived.Thenetdisutilityfunction,
γˆ,hasthefollowingproperties,whichareimpliedbyourunderlyingassumptions:
γˆ =γ =−φ(qˆ(c,θ)<0
2 3 .
γˆ =−µ(c,θ)<0
1
Inaddition,wemakethefollowingsinglecrossingassumption:6
Assumption2.
γˆ <0 (8)
12
6 Thesinglecrossingpropertydoesnotquitefollowfromourpreviousassumptions.Toseethis,notethat:
(cid:3) (cid:4) (cid:3) (cid:4)
γˆ12= ddθ ddγcˆ = ddθ ψ(cid:10)ddceˆ −θφ(cid:10)ddqcˆ =ψ(cid:10)ddc2deˆθ +ψ(cid:10)(cid:10)ddθeˆ ddceˆ −φ(cid:10)ddqcˆ −θφ(cid:10)ddc2dqˆθ −θφ(cid:10)(cid:10)ddqθˆ ddqcˆ
Underourmaintainedassumptions,allthetermsinthisexpressionarenegative,exceptthelast.Notethatifϕis
linear,thenthesinglecrossingpropertyholdsautomatically.
W.Jack/JournalofHealthEconomics24(2005)73–93 81
Similarly,definethesocialbenefitassociatedwithaθ-typeproviderprovidingserviceswith
abudgetcas:
Sˆ(c,θ)≡S(qˆ(c,θ)).
Because, it knows θ and can thereby infer the provider’s quality and effort choices, the
purchasercanalsoinferdisutilitycosts,γˆ (c,θ).Andbecauseleavingrentstoprovidersis
inefficient,thepurchasersetst=γˆ(c,θ).Redefinewelfarefrom(2)intermsofcandθas:
Wˆ (c,θ)=Sˆ(c,θ)−(1+λ)(ψˆ(c,θ)+c)+λθφˆ(c,θ). (9)
From our assumptions on providers’ supply of effort and quality, we are guaranteed that
Wˆ is concave in c.7 In addition, for later reference, we make a final assumption that the
marginalwelfareimpactofbudgetresourcesdeliveredtomorealtruisticphysiciansislarger.
Thatis,
Assumption3.
Wˆ >0.
12
Since the level of effort (and hence, its marginal cost) is higher when θ is larger, this
assumptionisconsistentwithextrabudgetresourcesleadingtoalargerreductionineffort
costsamongstmorebenevolentphysicians.8
Theoptimalbudgettogivetheproviderthensatisfiesthefirstordercondition:
Wˆ (c,θ)=Sˆ (c,θ)−(1+λ)(ψˆ (c,θ)+1)+λθφˆ (c,θ)=0. (10)
1 1 1 1
7 ConcavityofWˆ isassured,since:
(cid:3) (cid:4) (cid:1) (cid:2)
Wˆ11 =Sˆ11+λθφˆ11−(1+λ)ψˆ11=(S(cid:10)(cid:10)+λθφ(cid:10)(cid:10)) ddqcˆ 2+(S(cid:10)+λθφ(cid:10)) dd2cq2ˆ
(cid:1) (cid:3) (cid:4) (cid:1) (cid:2)(cid:2)
−(1+λ) ψ(cid:10)(cid:10) dqˆ 2+ψ(cid:10) d2eˆ <0.
dc dc2
8 Formally,asufficientconditionforWˆ12>0isSˆ12>0andψˆ12<0.Differentiating(butomittingfunction
arguments):
Sˆ =S(cid:10)(cid:10)dqˆ dqˆ +S(cid:10) d2qˆ
12 dθ dc dθdc
and
ψˆ =ψ(cid:10)(cid:10)deˆ deˆ +ψ(cid:10) d2eˆ
12 dθdc dθdc
Givenourassumptionsonthefunctionsqˆ andeˆ,ψˆ12<0asrequired.Ontheotherhand,thefirstterminthe
expressionforSˆ12isnegative,whilethesecondispositive.Assumption3issatisfiedaslongasthesecondterm
dominatesthefirst.
82 W.Jack/JournalofHealthEconomics24(2005)73–93
Denotethesolutiontothisconditionbycˆ(θ).UnderAssumption3,theoptimalbudgetis
increasinginθ.Ontheotherhand,theadditionaltransferorsalary,ˆt(θ)fallswithθsince,
dγˆ
ˆt(θ)= =γˆ cˆ(cid:10)(θ)+γˆ <0.
dθ 1 2
Thus,amorebenevolentproviderisgivenalargerbudgetbecauseshemakesitgofurther,
butsheisgivenasmallersalaryasthevalueofthepaymentshereceivesinkind(derived
fromquality)islarger.
4.1. Implementationthroughcost-sharing
Thesecondbestoptimumisabudgetcˆ(θ)andtransferˆt(θ).Theproviderisnotthought
ofchoosingcinthisinterpretation—shejusttakescasfixed,anddecideshowmucheffort
toexert.However,thepurchasercanimplementthesameoutcomeusingacost-sharingrule.
Tothisend,supposetheproviderisabletochoosethesizeofherbudget,andthatwhenshe
choosescshereceivesatransferTˆ(c).Letuswritetheprovider’snetutilityas:
uˆ =(c,θ)=Tˆ(c)−γˆ (c,θ).
1
Shesimplychoosesctomaximizethisexpression,thereby,choosingctosatisfy:
Tˆ(cid:10)(c)=γˆ (c,θ),
1
whichisnegative.Evaluatetherighthandsideofthisexpressionatthesecondbestoptimum
budget,cˆ(θ),anddenoteitsabsolutevalueby:
τˆ(θ)=−γˆ (cˆ(θ),θ).
1
Iftheproviderisreimbursedcostsc,andgivenanadditionaltransfer:
Tˆ(c)=αˆ(θ)−τˆ(θ)c
forsomeconstantαˆ(θ),shewillindeedchoosethedesiredbudget,cˆ(θ).Nowjustchoose
αˆ(θ)tosatisfy,
αˆ(θ)−τˆ(θ)cˆ(θ)=γˆ(cˆ(θ),θ),
andtheproviderwillearnzeronetutilityfromthecontract.Finally,thebudgetreimburse-
mentandextratransfercomponentscanbecombinedtoyieldagrosspaymenttotheprovider
of:
Pˆ(c)=Tˆ(c)+c=α(θ)+(1−τˆ(θ))c≡α(θ)+σˆ(θ)c.
Thatis,thegrosstransfertotheproviderisafixedsumplusashare,σˆ(θ)=1−τˆ(θ),of
realizedcosts.Theparametersofthesesharingschemesvaryacrossproviders.Inparticular,
σˆ(cid:10)(θ)=−τˆ(cid:10)(θ)=γˆ cˆ(cid:10)(θ)+γˆ <0, (11)
11 12
so a more altruistic provider has a smaller share of realized costs reimbursed. This is
consistent with the Chalkley and Malcomson result alluded to in the introduction—the