Uncertainty and the Welfare Economics of Medical Care

VOLUME LIII DECEMBER 1963 NUMBER 5

UNCERTAINTY AND THE WELFARE ECONOMICS OF MEDICAL CARE

By KENNETH J. ARROW*

I. Introduction:Scope and Method This paper is an exploratoryand tentativestudy of the specific differentiaof medical care as the object of normativeeconomics. It is contendedhere, on the basis of comparisonof obviouscharacteris- tics of the medical-careindustry with the normsof welfareeconomics, that the special economicproblems of medicalcare can be explained as adaptationsto the existenceof uncertaintyin the incidenceof disease and in theefficacy of treatment. It shouldbe notedthat the subjectis the medical-careindustry, not health.The causal factorsin health are many,and the provisionof medical care is only one. Particularlyat low levels of income,other commoditiessuch as nutrition,shelter, clothing, and sanitationmay be much more significant.It is the complexof servicesthat center about the physician,private and grouppractice, hospitals, and public health,which I proposeto discuss. The focus of discussionwill be on the way the operationof the medical-careindustry and the efficacywith which it satisfiesthe needs of societydiffer from a norm,if at all. The "norm"that the econo- mistusually uses forthe purposes of such comparisonsis theoperation of a competitivemodel, that is, the flowsof servicesthat would be * The author is professorof economicsat StanfordUniversity. He wishes to expresshis thanks for useful commentsto F. Bator, R. Dorfman,V. Fuchs, Dr. S. Gilson, R. Kessel, S. Mushkin,and C. R. Rorem. This paper was preparedunder the sponsorshipof the Ford Foundation as part of a seriesof papers on the economicsof health,education, and welfare. 942 THE AMERICAN ECONOMIC REVIEW offeredand purchasedand the pricesthat would be paid forthem if each individualin themarket offered or purchasedservices at thegoing prices as if his decisionshad no influenceover them,and the going prices were such that the amountsof serviceswhich were available equalled the total amountswhich other individualswere willingto purchase,with no imposedrestrictions on supplyor demand. The interestin the competitivemodel stemspartly from its pre- sumeddescriptive power and partlyfrom its implicationsfor economic efficiency.In particular,we can state the followingwell-known proposition (First OptimalityTheorem). If a competitiveequilibrium existsat all, and if all commoditiesrelevant to costs or utilitiesare in factpriced in the market,then the equilibriumis necessarilyoptimal in the followingprecise sense (due to V. Pareto): There is no other allocationof resourcesto serviceswhich will make all participantsin themarket better off. Both the conditionsof thisoptimality theorem and the definitionof optimalitycall forcomment. A definitionis just a definition,but when the definiendumis a wordalready in commonuse withhWighly favor- able connotations,it is clear thatwe are reallytrying to be persuasive; we are implicitlyrecommending the achievementof optimalstates.' It is reasonableenough to assertthat a changein allocationwhich makes all participantsbetter off is one thatcertainly should be made; thisis a value judgment,not a descriptiveproposition, but it is a veryweak one. From thisit followsthat it is not desirableto put up witha non- optimalallocation. But it does not followthat if we are at an allocationwhich is optimalin the Paretosense, we shouldnot changeto any other.We cannotindeed make a changethat does not hurtsomeone; but we can still desire to changeto anotherallocation if the change makesenough participants better off and by so muchthat we feelthat the injuryto othersis not enoughto offsetthe benefits.Such inter- personal comparisonsare, of course,value judgments.The change, however,by the previousargument ought to be an optimalstate; of coursethere are manypossible states, each of whichis optimalin the sensehere used. However,a value judgmenton the desirabilityof each possiblenew distributionof benefitsand costs correspondingto each possible reallocationof resourcesis not,in general,necessary. Judgments about thedistribution can be made separately,in one sense,from those about allocationif certainconditions are fulfilled.Before stating the relevant proposition,it is necessaryto remarkthat the competitiveequilibrium achieveddepends in good measureon the initialdistribution of purchasingpower, which consists of ownershipof assets and skills that 'This point has been stressedby I. M. D. Little [19, pp. 71-74]. For the concept of a "persuasivedefinition," see C. L. Stevenson [27, pp. 210-17]. ARROW: UNCERTAINTY AND MEDICAL CARE 943 commanda price on the market.A transferof assets amongindivid- uals will, in general,change the finalsupplies of goods and services and the prices paid for them.Thus, a transferof purchasingpower fromthe well to the ill will increasethe demandfor medical services. This will manifestitself in the shortrun in an increasein theprice of medicalservices and in the long runin an increasein theamount sup- plied. With this in mind,the followingstatement can be made (Second OptimalityTheorem): If thereare no increasingreturns in production, and if certainother minor conditions are satisfied,then every optimal state is a competitiveequilibrium corresponding to some initialdis- tributionof purchasingpower. Operationally, the significanceof this propositionis thatif the conditionsof the twooptimality theorems are satisfied,and if theallocation mechanism in thereal worldsatisfies the conditionsfor a competitivemodel, then social policycan confineitself to steps taken to alter the distributionof purchasingpower. For any given distributionof purchasingpower, the marketwill, under the assumptionsmade, achieve a competitiveequilibrium which is necessarilyoptimal; and any optimalstate is a competitiveequilibrium correspondingto some distributionof purchasingpower, so that any desiredoptimal state can be achieved. The redistributionof purchasingpower among individualsmost simplytakes the formof money:taxes and subsidies.The implications of such a transferfor individualsatisfactions are, in general,not knownin advance. But we can assumethat society can ex post judge the distributionof satisfactionsand, if deemed unsatisfactory,take steps to correctit by subsequenttransfers. Thus, by successiveap- proximations,a mostpreferred social state can be achieved,with resourceallocation being handled by the marketand publicpolicy con- finedto the redistributionof moneyincome.2 If, on the contrary,the actual marketdiffers significantly from the competitivemodel, or if the assumptionsof the two optimalitythe- oremsare not fulfilled,the separationof allocativeand distributional proceduresbecomes, in most cases, impossible.3 The firststep thenin the analysisof the medical-caremarket is the 2The separationbetween allocation and distributioneven under the above assumptions has 4osSed over problemsin the executionof any desiredredistribution policy; in practice, it is virtuallyimpossible to find a set of taxes and subsidies that will not have an ad- verse effecton the achievementof an optimal state. But this discussion would take us even furtherafield than we have already gone. 'The basic theoremsof welfare economics alluded to so brieflyabove have been the subject of voluminous literature,but no thoroughlysatisfactory statement covering both the theoremsthemselves and the significanceof exceptionsto them exists. The positive assertionsof welfareeconomics and theirrelation to the theoryof competitiveequilibrium are admirably covered in Koopmans [181. The best summary of the various ways in which the theoremscan fail to hold is probablyBator's [6]. 944 THE AMERICAN ECONOMIC REVIEW comparisonbetween the actual marketand thecompetitive model. The methodologyof thiscomparison has been a recurrentsubject of con- troversyin economicsfor over a century.Recently, M. Friedman[15] has vigorouslyargued that the competitiveor any othermodel should be testedsolely by its abilityto predict.In the contextof competition, he comesclose to arguingthat prices and quantitiesare the onlyrele- vant data. This point of view is valuable in stressingthat a certain amountof lack of realismin the assumptionsof a model is no argu- mentagainst its value. But theprice-quantity implications of thecom- petitivemodel for pricing are noteasy to derivewithout major--and, in manycases, impossible-econometricefforts. In thispaper, the institutional organization and theobservable mores of the medicalprofession are includedamong the data to be used in assessingthe competitivenessof the medical-caremarket. I shall also examinethe presenceor absence of the preconditionsfor the equiva- lence of competitiveequilibria and optimalstates. The major competitive preconditions,in the sense used here,are three: the existenceof competitiveequilibrium, the marketabilityof all goods and services relevantto costsand utilities,and nonincreasingretiurns. The firsttwo, as we have seen,insure that competitive equilibrium is necessarilyoptimal; the thirdinsures that everyoptimal state is the competitive equilibriumcorresponding to some distributionof income.4The first and thirdconditions are interrelated;indeed, nonincreasing returns plus some additionalconditions not restrictivein a moderneconomy implythe existenceof a competitiveequilibrium, i.e., implythat there will be someset of priceswhich will clear all markets.5 The conceptof marketabilityis somewhatbroader than the tradi- tional divergencebetween private and social costs and benefits.The latterconcept refers to cases in whichthe organizationof the market does not requirean individualto pay for costs that he imposeson othersas the resultof his actions or does not permithim to receive compensationfor benefits he confers.In the medicalfield, the obvious exampleis the spread of communicablediseases. An individualwho failsto be immunizednot onlyrisks his own health,a disutilitywhich presumablyhe has weighedagainst the utilityof avoidingthe proce- dure,but also thatof others.In an ideal pricesystem, there would be a price whichhe would have to pay to anyonewhose healthis endan- gered,a price sufficientlyhigh so that the otherswould feel compensated; or, alternatively,there would be a pricewhich would be paid to him by othersto inducehim to undergothe immunizationprocedure. 'There are furtherminor conditions,for which see Koopmans [18, pp. 50-551. 5 For a more precisestatement of the existenceconditions, see Koopmans [18, pp. 56-60] or Debreu [12, Ch. 5] . ARROW: UNCERTAINTY AND MEDICAL CARE 945 Eilthersystem would lead to an optimalstate, though the distributional implicationswould be different.It is, of course,not hard to see that such price systemscould not,in fact,be practical; to approximatean optimalstate it would be necessaryto have collectiveintervention in the formof subsidyor tax or compulsion. By tlle absence of marketabilityfor an actionwhich is identifiable, technologicallypossible, and capable of influencingsome individual's welfare,for better or forworse, is meanthere the failureof the existing marketto providea means wherebythe servicescan be both offeredand demandedupon paymentof a price.Nonmarketability may be due to intrinsictechnological characteristics of the productwhich preventa suitableprice frombeing enforced,as in the case of communicablediseases, or it may be due to social or historicalcontrols, such as thoseprohibiting an individualfrom selling himself into slav- ery.This distinctionis, in fact,difficult to make precise,though it is obviouslyof importancefor policy; forthe present purposes, it willbe sufficientto identifynonmarketability with the observedabsence of markets. The instanceof nonmarketabilitywith which we shall be mostcon- cernedis thatof risk-bearing.The relevanceof risk-bearingto medical care seemsobvious; illnessis to a considerableextent an unpredictable phenomenon.The abilityto shiftthe risksof illnessto othersis worth a pricewhich many are willingto pay. Because of poolingand of supe- riorwillingness and ability,others are willingto bear therisks. Never- theless,as we shall see in greaterdetail, a greatmany risks are not covered,and indeedthe marketsfor the servicesof risk-coverageare poorlydeveloped or nonexistent.Why thisshould be so is explainedin moredetail in SectionIV.C below; briefly,it is impossibleto drawup insurancepolicies which will sufficientlydistinguish among risks, par- ticularlysince observationof the resultswill be incapableof distin- guishingbetween avoidable and unavoidablerisks, so that incentives to avoidlosses are diluted. The optimalitytheorems discussed above are usually presentedin the literatureas referringonly to conditionsof certainty,but thereis no difficultyin extendingthem to the case of risks,provided the addi- tionalservices of risk-bearingare includedwith other commodities.6 However,the varietyof possiblerisks in theworld is reallystagger- ing. The relevantcommodities include, in effect,bets on all possible occurrencesin theworld which impinge upon utilities. In fact,many of these "commodities,"i.e., desiredprotection against many risks,are

'The theory, in variant forms, seems to have been firstworked out by Allais [2], Arrow [5], and Baudier [7]. For furthergeneralization, see Debreu [11] and [12, Ch. 71. 946 THE AMERICANECONOMIC REVIEW simplynot available. Thus, a wide class of commoditiesis nonmarketable, and a basic competitiveprecondition is not satisfied.7 There is a stillmore subtle consequence of the introductionof risk- bearing considerations.When there is uncertainty,information or knowledgebecomes a commodity.Like othercommodities, it has a cost of productionand a cost of transmission,and so it is naturallynot spread out over the entirepopulation but concentratedamong those who can profitmost from it. (These costsmay be measuredin timeor disutilityas well as money.) But the demandfor information is difficult to discuss in the rationalterms usually employed.The value of informationis frequentlynot knownin any meaningfulsense to the buyer; if, indeed,he knew enoughto measurethe value of information,he would know the informationitself. But information,in the formof skilledcare, is preciselywhat is beingbouight from most physicians, and, indeed,from most professionals. The elusivecharacter of informationas a commoditysuggests that it departsconsiderably from the usual marketabilityassumptions about commodities.8 That riskand uncertaintyare, in fact,significant elements in medical care hardlyneeds argument. I willhold that virtually all thespecial featuresof this industry,in fact,stem fromthe prevalenceof uncertainty. The nonexistenceof marketsfor the bearing of somerisks in thefirst instancereduces welfare for those who wish to transferthose risks to othersfor a certainprice, as well as forthose who wouldfind it profit- able to take on the riskat suchprices. But it also reducesthe desireto renderor consumeservices which have riskyconsequences; in techni- cal language,these commoditiesare complementaryto risk-bearing. Conversely,the productionand consumptionof commoditiesand services withlittle risk attached act as substitutesfor risk-bearing and are encouragedby marketfailure there with respect to risk-bearing.Thus theobserved commodity pattern will be affectedby thenonexistence of othermarkets. ' It should also be remarkedthat in the presenceof uncertainty,indivisibilities that are sufficientlysmall to create little difficultyfor the existenceand viability of competitive equilibriummay neverthelessgive rise to a considerablerange of increasingreturns because of the operationof the law of large numbers.Since most objects of insurance(lives, fire hazards, etc.) have some element of indivisibility,insurance companies have to be above a certainsize. But it is not clear that this effectis sufficientlygreat to create serious obstacles to the existenceand viability of competitiveequilibrium in practice. 8 One form of production of informationis research.Not only does the product have unconventionalaspects as a commodity,but it is also subject to increasingreturns in use, since new ideas, once developed,can be used over and over withoutbeing consumed,and to difficultiesof marketcontrol, since the cost of reproductionis usually much less than that of production.Hence, it is not surprisingthat a free enterpriseeconomy will tend to underinvestin research;see Nelson [211 and Arrow [4]. ARROW: UNCERTAINTY AND MEDICAL CARE 947 The failureof one or moreof the competitivepreconditions has as its most immediateand obvious consequencea reductionin welfare below that obtainablefrom existing resources and technology,in the sense of a failureto reachan optimalstate in thesense of Pareto. But morecan be said. I proposehere the view that,when the marketfails to achievean optimalstate, society will, to someextent at least,recog- nize the gap, and nonmarketsocial institutionswill arise attemptingto bridgeit.9 Certainly this process is not necessarilyconscious; nor is it uniformlysuccessful in approachingmore closely to optimalitywhen the entirerange of consequencesis considered.It has always been a favoriteactivity of economiststo pointout thatactions which on their face achieve a desirablegoal may have less obvious consequences particularlyover time,which more than offset the originalgains. But it is contendedhere that the special structuralcharacteristics of themedical-care market are largelyattempts to overcomethe lack of optimalitydue to the nonmarketabilityof the bearingof suitablerisks and the imperfectmarketability of information.These compensatory institutionalchanges, with some reinforcementfrom usual profitmo- tives, largely explain the observednoncompetitive behavior of the medical-caremarket, behavior which, in itself,interferes with optimality.The social adjustmenttowards optimality thus puts obstacles in its own path. The doctrinethat societywill seek to achieve optimalityby non- marketmeans if it cannotachieve themin the marketis not novel. Certainly,the government, at least in its economicactivities, is usually implicitlyor explicitlyheld to functionas the agencywhich substitutes for the market'sfailure.'0 I am arguinghere that in some circum- stancesother social institutionswill step into the optimalitygap, and that the medical-careindustry, with its varietyof special institutions, some ancient,some modern,exemplifies this tendency. It may be usefulto remarkhere thata good part of thepreference for redistributionexpressed in governmenttaxation and expenditure policies and privatecharity can be reinterpretedas desire forinsurance. It is noteworthythat virtually nowhere is therea systemof subsidies that has as its aim simplyan equalizationof income.The sub- sidiesor othergovernmental help go to thosewho are disadvantagedin life by eventsthe incidenceof whichis popularlyregarded as unpre-

'An important current situation in which normal market relations have had to be greatly modified in the presence of great risks is the production and procurementof modern weapons; see Peck and Scherer [23, pp. 581-82] (I am indebted for this reference to V. Fuchs) and [1, pp. 71-75]. 0For an explicit statementof this view, see Baumol [8]. But I believe this position is implicitin most discussionsof the functionsof government. 948 THE AMERICAN ECONOMIC REVIEW dictable: the blind,dependent children, the medicallyindigent. Thus, optimality,in a contextwhich includesrisk-bearing, includes much that appears to be motivatedby distributionalvalue judgmentswhen lookedat in a narrowercontext." This methodologicalbackground gives rise to the followingplan for this paper. SectionII is a catalogueof stylizedgeneralizations about the medical-caremarket which differentiate it from the usual commodity markets.In SectionIII the behaviorof the marketis compared withthat of the competitivemodel which disregards the factof uncertainty.In SectionIV, the medical-caremarket is compared,both as to behaviorand as to preconditions,with the ideal competitivemarket thattakes accountof uncertainty;an attemptwill be made to demon- stratethat the characteristicsoutlined in SectionII can be explained eitheras the resultof deviationsfrom the competitive preconditions or as attemptsto compensateby otherinstitutions for these failures. The discussionis not designedto be definitive,but provocative.In particular, I have been charyabout drawingpolicy inferences; to a consider- able extent,they depend on furtherresearch, for which the present paper is intendedto providea framework. II. A Surveyof theSpecial Characteristicsof the Medical-CareMarket'2 This sectionwill list selectivelysome characteristics of medicalcare whichdistinguish it fromthe usual commodityof economicstextbooks. The list is not exhaustive,and it is not claimedthat the characteristics listedare individuallyunique to thismarket. But, takentogether, they do establisha special place formedical care in economicanalysis. A. The Nature of Demand The most obvious distinguishingcharacteristics of an individual's demandfor medical services is that it is not steadyin originas, for example,for food or clothing,but irregularand unpredictable.Medi- cal services,apart frompreventive services, afford satisfaction only in the eventof illness,a departurefrom the normalstate of affairs.It is hard, indeed,to thinkof anothercommodity of significancein the average budgetof whichthis is true.A portionof legal services,de- votedto defensein criminaltrials or to lawsuits,might fall in thiscate- gorybut theincidence is surelyvery much lower (and, of course,there

'Since writingthe above, I findthat Buchanan and Tullock [10, Ch. 13] have argued that all redistributioncan be interpretedas "income insurance." 12For an illuminatingsurvey to which I am much indebted,see S. Mushkin [20]. ARROW: UNCERTAINTY AND MEDICAL CARE 949 are, in fact, stronginstitutional similarities between the legal and medical-caremarkets.)'3 In addition,the demandfor medical services is associated,with a considerableprobability, with an assaulton personalintegrity. There is some riskof deathand a moreconsiderable risk of impairmentof full functioning.In particular,there is a majorpotential for loss or reduc- tionof earningability. The risksare notby themselvesunique; foodis also a necessity,but avoidanceof deprivationof foodcan be guaranteed withsufficient income, where the same cannotbe said of avoidanceof illness.Illness is, thus,not onlyrisky but a costlyrisk in itself,apart fromthe cost of medicalcare. B. ExpectedBehavior of thePhysician It is clear fromeveryday observation that the behaviorexpected of sellersof medicalcare is differentfrom that of businessmen in general. These expectationsare relevantbecause medicalcare belongsto the categoryof commoditiesfor which the productand the activityof productionare identical.In all such cases, thecustomer cannot test the productbefore consuming it, and thereis an elementof trustin the relation.' But the ethicallyunderstood restrictions on the activitiesof a physicianare muchmore severe than on thoseof, say, a barber.His behavioris supposedto be governedby a concernfor the customer's welfarewhich would not be expectedof a salesman.In Talcott Par- sons's terms,there is a "collectivity-orientation,"which distinguishes medicineand otherprofessions from business, where self-interest on the part of participantsis the acceptednorm.'5 A fewillustrations will indicatethe degreeof differencebetween the behaviorexpected of physiciansand thatexpected of the typicalbusi- nessman.18(1) Advertisingand overtprice competitionare virtually eliminatedamong physicians.(2) Advice given by physiciansas to furthertreatment by himselfor othersis supposedto be completely "In governmentaldemand, military power is an example of a service used only irregularlyand unpredictably.Here too, special institutionaland professionalrelations have emerged,though the precisesocial structureis differentfor reasons that are not hard to analyze. " Even with material commodities,testing is never so adequate that all elements of implicittrust can be eliminated.Of course,over the long run, experiencewith the quality of product of a given sellerprovides a check on the possibilityof trust. 15See [22, p. 463]. The whole of [22, Ch. 101 is a most illuminatinganalysis of the social role of medical practice; though Parsons' interestlies in differentareas frommine, I must acknowledgehere my indebtednessto his work. 16 I am indebted to Herbert Klarman of Johns Hopkins Universityfor some of the points discussedin this and the followingparagraph. 950 THE AMERICAN ECONOMIC REVIEW divorcedfrom self-interest. (3) It is at least claimedthat treatment is dictatedby the objectiveneeds of thecase and notlimited by financial considerations."7While the ethicalcompulsion is surelynot as absolute in factas it is in theory,we can hardlysuppose that it has no influence over resourceallocation in thisarea. Charitytreatment in one formor anotherdoes existbecause of thistradition about human rights to adequate medicalcare.'8 (4) The physicianis reliedon as an expertin certifyingto theexistence of illnessesand injuriesfor various legal and otherpurposes. It is sociallyexpected that his concernfor the correct conveyingof informationwill, when appropriate,outweigh his desire to please his customers."g Departurefrom the profitmotive is strikinglymanifested by the overwhelmingpredominance of nonprofitover proprietaryhospitals.20 The hospitalper se offersservices not too differentfrom those of a hotel,and it is certainlynot obvious that the profit motive will not lead to a moreefficient supply. The explanationmay lie eitheron thesupply side or on thatof demand.The simplestexplanation is thatpublic and privatesubsidies decrease the cost to thepatient in nonprofithospitals. A second possibilityis that the associationof profit-makingwith the supplyof medical servicesarouses suspicionand antagonismon the part of patientsand referringphysicians, so theydo prefernonprofit institutions.Either explanation implies a preferenceon thepart of some group,whether donors or patients,against the profitmotive in the supplyof hospitalservices.2'

1T The belief that the ethics of medicinedemands treatmentindependent of the patient's ability to pay is stronglyingrained. Such a perceptiveobserver as Rene Dubos has made the remark that the high cost of anticoagulantsrestricts their use and may contradict classical medical ethics, as though this were an unprecedentedphenomenon. See [13, p. 4191. "A time may come when medical ethics will have to be consideredin the harsh light of economics" (emphasis added). Of course, this expectationamounts to ignoring the scarcity of medical resources; one has only to have been poor to realize the error. We may confidentlyassume that price and income do have some consequences for medical expenditures. 18A needed piece of researchis a study of the exact nature of the variationsof medical care received and medical care paid for as income rises. (The relevant income concept also needs study.) For this purpose,some disaggregationis needed; differencesin hospital care which are essentiallymatters of comfortshould, in the above view, be much more responsiveto income than,e.g., drugs. "9This role is enhanced in a socialist society,where the state itselfis activelyconcerned with illnessin relationto work; see Field [14, Ch. 91. ' About 3 per cent of beds were in proprietaryhospitals in 1958, against 30 per cent in voluntary nonprofit,and the remainderin federal, state, and local hospitals; see [26, Chart 4-2, p. 601. " C. R. Rorem has pointed out to me some furtherfactors in this analysis. (1) Given the social intentionof helping all patients without regard to immediateability to pay, economies of scale would dictate a predominanceof community-sponsoredhospitals. (2) ARROW: UNCERTAINTY AND MEDICAL CARE 951 Conformityto collectivity-orientedbehavior is especiallyimportant sinceit is a commonplacethat the physician-patient relation affects the quality of the medicalcare product.A pure cash nexuswould be in- adequate; if nothingelse, the patientexpects that the same physician will normallytreat him on successiveoccasions. This expectationis strongenough to persisteven in the SovietUnion, where medical care is nominallyremoved from the marketplace [14, pp. 194-96]. That purelypsychic interactions between physician and patienthave effects which are objectivelyindistinguishable in kind fromthe effectsof medicationis evidencedby theuse of theplacebo as a controlin medical experimentation;see Shapiro [25]. C. ProductUncertainty Uncertaintyas to thequality of theproduct is perhapsmore intense herethan in any otherimportant commodity. Recovery from disease is as unpredictableas is its incidence.In most commodities,the possibilityof learningfrom one's ownexperience or thatof othersis strong because thereis an adequate numberof trials.In thecase of severeill- ness, thatis, in general,not true; the uncertaintydue to inexperience is added to theintrinsic difficulty of prediction.Further, the amount of uncertainty,measured in termsof utilityvariability, is certainlymuch greaterfor medical care in severecases than for,say, housesor auto- mobiles,even thoughthese are also expendituressufficiently infre- quent so that theremay be considerableresidual uncertainty. Further,there is a special qualityto theuncertainty; it is verydif- ferenton the two sides of the transaction.Because medicalknowledge is so complicated,the information possessed by thephysician as to the consequencesand possibilitiesof treatmentis necessarilyvery much greaterthan that of the patient,or at least so it is believedby both parties.22Further, both parties are aware of thisinformational inequal- ity,and theirrelation is coloredby thisknowledge. To avoid misunderstanding,observe that the differencein information relevanthere is a differencein informationas to the consequence of a purchaseof medicalcare. There is always an inequalityof informationas to productionmethods between the producerand the pur- chaser of any commodity,but in most cases the customermay well

Some proprietaryhospitals will tend to controltotal costs to the patient more closely,including the fees of physicians,who will thereforetend to prefercommunity-sponsored hospitals. 2"Without tryingto assess the presentsituation, it is clear in retrospectthat at some point in the past the actual differentialknowledge possessed by physicians may not have been much. But from the economic point of view, it is the subjective belief of both parties, as manifestedin their market behavior, that is relevant. 952 THE AMERICAN ECONOMIC REVIEW have as good or nearlyas good an understandingof the utilityof the productas the producer. D. SupplyConditions In competitivetheory, the supply of a commodityis governedby the net returnfrom its productioncompared with the returnderivable fromthe use of the same resourceselsewhere. There are severalsig- nificantdepartures from this theory in the case of medicalcare. Most obviouisly,entry to the professionis restrictedby licensing. Licensing,of course,restricts supply and thereforeincreases the cost of medicalcare. It is defendedas guaranteeinga minimumof quality. Restrictionof entryby licensingoccurs in mostprofessions, including barberingand undertaking. A second featureis perhaps even more remarkable.The cost of medicaleducation today is highand, accordingto the usual figures,is borneonly to a minorextent by thestudent. Thus, theprivate benefits to the enteringstudent considerably exceed the costs. (It is, however, possible that researchcosts, not properlychargeable to education, swellthe apparentdifference.) This subsidyshould, in principle,cause a fallin the priceof medicalservices, which, however, is offsetby ra- tioningthrough limited entry to schools and throughelimination of studentsduring the medical-schoolcareer. These restrictionsbasically rendersuperfluous the licensing,except in regardto graduatesof for- eignschools. The special role of educationalinstitutions in simultaneouslysub- sidizingand rationingentry is commonto all professionsrequiring advanced training.23It is a strikingand insufficientlyremarked phe- nomenonthat such an importantpart of resourceallocation should be performedby nonprofit-orientedagencies. Since this last phenomenongoes well beyondthe purelymedical aspect, we will not dwell on it longerhere except to note that the anomalyis moststriking in themedical field. Educational costs tend to be far higherthere than in any otherbranch of professionaltraining. Whiletuition is thesame, or onlyslightly higher, so thatthe subsidy is muchgreater, at the same timethe earningsof physiciansrank high- est amongprofessional groups, so therewould not at firstblush seem to be any necessityfor special inducementsto enterthe profession. Even if we grantthat, for reasons unexamined here, there is a social interestin subsidizedprofessional education, it is not clear why the rate of subsidizationshould differamong professions. One mightex- 23The degree of subsidy in differentbranches of professionaleducation is worthyof a major researcheffort. ARROW: UNCERTAINTY AND MEDICAL CARE 953 pect thatthe tuitionof medicalstudents would be higherthan that of otherstudents. The highcost of medicaleducation in the UnitedStates is itselfa reflectionof the qualitystandards imposed by the AmericanMedical Associationsince the FlexnerReport, and it is, I believe,only since thenthat the subsidyelement in medicaleducation has becomesignifi- cant. Previously,many medical schools paid theirway or evenyielded a profit. Anotherinteresting feature of limitationon entryto subsidizededu- cation is the extentof individualpreferences concerning the social welfare,as manifestedby contributionsto privateuniversities. But whethersupport is public or private,the importantpoint is thatboth the qualityand the quantityof the supplyof medicalcare are being stronglyinfluenced by social nonmarketforces.24'25 One strikingconsequence of the controlof qualityis the restriction on the rangeoffered. If manyqualities of a commodityare possible,it wouldusually happen in a competitivemarket that many qualities will be offeredon the market,at suitablyvarying prices, to appeal to dif- ierenttastes and incomes.Both thelicensing laws and thestandards of medical-schooltraining have limitedthe possibilitiesof alternative qualities of medical care. The decliningratio of physiciansto total employeesin the medical-careindustry shows that substitution of less trainedpersonnel, technicians, and the like, is not preventedcom- pletely,but the centralrole of the highlytrained physician is not af- fectedat all.26 E. PricingPractices The unusualpricing practices and attitudesof the medicalprofes- sionare wellknown: extensive price discrimination by income(with an extremeof zero pricesfor sufficiently indigent patients) and, formerly, a stronginsistence on fee forservices as againstsuch alternativesas prepayment.

'Strictly speaking,there are four variables in the market for physicians: price,quality of enteringstudents, quality of education,and quantity.The basic marketforces, demand for medical services and supply of enteringstudents, determine two relationsamong the four variables. Hence, if the nonmarketforces determine the last two, marketforces will determineprice and quality of entrants. 'The suipplyof Ph.D.'s is similarlygoverned, but there are other conditions in the marketwhich are much different,especially on the demand side. 'Today oinlythe Soviet Union offersan alternativelower level of medical personnel, the feldshers,who practice primarilyin the rural districts (the institutiondates back to the 18th century). Accordingto Field [14, pp. 98-100, 132-33], thereis clear evidence of strain in the relationsbetween physiciansand feldshers,but it is not certain that the feldsherswill gradually disappear as physicians grow in numbers. 954 THE AMERICAN ECONOMIC REVIEW The oppositionto prepaymentis closelyrelated to an even stronger oppositionto closed-panelpractice (contractualarrangements which bindthe patientto a particulargroup of physicians).Again these atti- tudesseem to differentiateprofessions from business. Prepayment and closed-panelplans are virtuallynonexistent in the legal profession.In ordinarybusiness, on the otherhand, thereexists a wide varietyof exclusiveservice contracts involving sharing of risks; it is assumed that competitionwill selectthose which satisfy needs best.27 The problemsof implicitand explicitprice-fixing should also be mentioned.Price competitionis frownedon. Arrangementsof thistype are not uncommonin serviceindustries, and theyhave not been sub- jected to antitrustaction. How importantthis is is hard to assess. It has beenpointed out manytimes that the apparent rigidity of so-called admiriisteredprices considerablyunderstates the actual flexibility. Here, too,if physiciansfind themselves with unoccupied time, rates are likelyto go down,openly or covertly;if thereis insufficienttime for the demand,rates will surelyrise. The "ethics"of price competition may decreasethe flexibilityof priceresponses, but probablythat is all. III. Comparisonswith the CompetitiveModel underCertainty A. NonmarketableCommodities As already noted,the diffusionof communicablediseases provides an obviousexample of nonmarketinteractions. But froma theoretical viewpoint,the issues are well understood,and thereis littlepoint in expandingon this theme.(This shouldnot be interpretedas minimiz- ing the contributionof public healthto welfare;there is everyreason to supposethat it is considerablymore important than all otheraspects of medicalcare.) Beyondthis special area thereis a moregeneral interdepeiidence, the concernof individualsfor the health of others.The economicmanifes- tationsof thistaste are to be foundin individualdonations to hocpitals and to medicaleducation, as well as in the widelyaccepted responsi- bilitiesof governmentin thisarea. The taste forimproving the health of othersappears to be strongerthan forimproving other aspects of theirwelfare.28 In interdependenciesgenerated by concernfor the welfare of others thereis always a theoreticalcase forcollective action if each partici- pant derivessatisfaction from the contributionsof all. ' The law does impose some limits on risk-shiftingin contracts,for example, its general refusalto honorexculpatory clauses. ' There may be an identificationproblem in this observation. If the failure of the market system is, or appears to be, greater in medical care than in, say, food an individual otherwiseequally concernedabout the two aspects of others'welfare may prefer to help in the first. ARROW:UNCERTAINTY AND MEDICALCARE 955

B. IncreasingReturns Problemsassociated with increasing returns play some role in allocation of resourcesin the medical field,particularly in areas of low densityor low income.Hospitals show increasing returns up to a point; specialistsand some medicalequipment constitute significant indivisibilities.In manyparts of the worldthe individualphysician may be a largeunit relative to demand.In such cases it can be sociallydesirable to subsidizethe appropriatemedical-care unit. The appropriatemode of analysisis muchthe same as forwater-resource projects. Increasing returnsare hardlyapt to be a significantproblem in generalpractice in large citiesin the UnitedStates, and improvedtransportation to some extentreduces their importance elsewhere.

C. Entry The moststriking departure from competitive behavior is restriction on entryto the field,as discussedin II.D above. Friedmanand Kuz- nets, in a detailedexamination of the pre-WorldWar II data, have arguedthat the higher income of physicianscould be attributedto this restriction.29 There is some evidencethat the demandfor admission to medical school has dropped (as indicatedby the numberof applicantsper place and the qualityof thoseadmitted), so thatthe number of medical-schoolplaces is not as significanta barrierto entryas in theearly 1950's [28, pp. 14-15]. But it certainlyhas operatedover the past and it is still operatingto a considerableextent today. It has, of course, constituteda directand unsubtlerestriction on the supplyof medical care. There are severalconsiderations that must be added to help evaluate theimportance of entryrestrictions: (1) Additionalentrants would be, in general,of lowerquality; hence,the addition to thesupply of medical care,properly adjusted for quality, is less thanpurely quantitative calculationswould show.30(2) To achievegenuinely competitive conditions,it wouldbe necessarynot only to removenumerical restrictions on entrybut also to removethe subsidy in medicaleducation. Like any otherproducer, the physicianshould bear all the costs of production,

" See [16, pp. 118-37]. The calculations involve many assumptionsand must be re- garded as tenuous; see the commentsby C. Reinold Noyes in [16, pp. 407-10]. 'It might be argued that the existenceof racial discriminationin entrancehas meant that some of the rejected applicants are superior to some accepted. However, there is no necessaryconnection between an increase in the number of entrantsand a reduction in racial discrimination;so long as there is excess demand for entry,discrimination can continue unabated and new entrantswill be inferiorto those previouslyaccepted. 956 THE AMERICAN ECONOMIC REVIEW including,in this case, education.3'It is not so clear thatthis change would not keep even unrestrictedentry down below the presentlevel. (3) To some extent,the effectof makingtuition carry the fullcost of educationwill be to create too few entrants,rather than too many. Giventhe imperfections of thecapital market, loans forthis purpose to thosewho do nothave thecash are difficultto obtain.The lenderreally has no security.The obviousanswer is some formof insuredloans, as has frequentlybeen argued; not too muchingenuity would be needed to createa creditsystem for medical (and otherbranches of higher) education.Under these conditions the cost wouldstill constitute a de- terrent,but one to be comparedwith the high futureincomes to be obtained. If entrywere governedby ideal competitiveconditions, it may be that the quantityon balance would be increased,though this conclu- sion is not obvious. The average quality would probablyfall, even underan ideal creditsystem, since subsidyplus selectedentry draw some highlyqualified individuals who would otherwiseget into other fields.The declinein qualityis not an over-allsocial loss, since it is accompaniedby increasein qualityin otherfields of endeavor;indeed, if demandsaccurately reflected utilities, there would be a net social gain througha switchto competitiveentry.32 There is a second aspect of entryin whichthe contrastwith competitivebehavior is, in manyrespects, even sharper.It is the exclusion of many imperfectsubstitutes for physicians.The licensinglaws, thoughthey do not effectivelylimit the numberof physicians,do exclude all othersfrom engaging in any one of the activitiesknown as medicalpractice. As a result,costly physician time may be employed at speCifictasks for whichonly a small fractionof theirtraining is needed,and whichcould be performedby othersless well trainedand thereforeless expensive.One mightexpect immunization centers, pri- vatelyoperated, but not necessarilyrequiring the servicesof doctors. In the competitivemodel withoutuncertainty, consumers are pre- siumedto be able to distinguishqualities of the commoditiesthey buy. Under this hypothesis,licensing would be, at best, superfluousand exclude those fromwhom consumers would not buy anyway; but it mightexclude too many. D. Pricing The pricingpractices of the medicalindustry (see II.E above) de- ' One problem here is that the tax laws do not permit depreciationof professional education, so that there is a discriminationagainst this form of investment. "2To anticipate later discussion, this condition is not necessarilyfulfilled. When it comes to quality choices, the market may be inaccurate. ARROW: UNCERTAINTY AND MEDICAL CARE 957 part sharplyfrom the competitivenorm. As Kessel [17] has pointed out withgreat vigor, not onlyis pricediscrimination incompatible with the competitivemodel, but its preservationin the face of the large numberof physiciansis equivalentto a collectivemonopoly. In the past, the oppositionto prepaymentplans has takendistinctly coercive forms,certainly transcending market pressures, to say the least. Kessel has arguedthat price discrimination is designedto maximize profitsalong the classic lines of discriminatingmonopoly and that organizedmedical oppositionto prepaymentwas motivatedby the desire to protectthese profits.In principle,prepayment schemes are compatiblewith discrimination,but in practicethey do not usually discriminate.I do not believethe evidencethat the actual scale of discriminationis profit-maximizingis convincing. In particular,note that forany monopoly,discriminating or otherwise,the elasticity of demand in each marketat the pointof maximumprofits is greaterthan one. But it is almostsurely true for medical care thatthe price elasticity of demandfor all incomelevels is less than one. That price discrimination by incomeis not completelyprofit-maximizing is obvious in the extremecase of charity;Kessel arguesthat this represents an appease- mentof publicopinion. But thisalready shows the incompletenessof the model and suggeststhe relevanceand importanceof social and ethicalfactors. Certainlyone importantpart of the oppositionto prepaymentwas its close relationto closed-panelplans. Prepaymentis a formof insurance,and naturallythe individual physician did notwish to assumethe risks.Pooling was intrinsicallyinvolved, and thisstrongly motivates, as we shall discuissfurther in SectionIV below,control over pricesand benefits.The simplestadministrative form is the closed panel; physicians involvedare, in effect,the insuringagent. From this point of view,Blue Cross solvedthe prepayment problem by universalizingthe closedpanel. The case that price discriminationby incomeis a formof profit maximizationwhich was zealouslydefended by oppositionto fees for serviceseems far fromproven. But it remainstrue that this price discrimination,for whatever cause, is a source of nonoptimality.Hypo- thetically,it means everyonewould be betteroff if priceswere made equal forall, and the richcompensated the poor forthe changesin the relativepositions. The importanceof thiswelfare loss dependson the actual amountof discriminationand on the elasticitiesof demandfor medical servicesby the differentincome groups. If the discussionis simplifiedby consideringonly two income levels, rich and poor,and if the elasticityof demandby eitherone is zero, thenno reallocationof medicalservices will take place and theinitial situation is optimal.The 958 THE AMERICAN ECONOMIC REVIEW onlyeffect of a changein pricewill be the redistributionof incomeas betweenthe medicalprofession and the groupwith the zero elasticity of demand.With low elasticitiesof demand,the gain will be small.To illustrate,suppose the price of medicalcare to therich is doublethat to thepoor, the medicalexpenditures by therich are 20 per centof those by the poor,and the elasticityof demandfor both classes is .5; then thenet social gain due to theabolition of discriminationis slightlyover 1 per cent of previousmedical expenditures.33 The issuesinvolved in theopposition to prepayment,the other major anomalyin medicalpricing, are not meaningfulin the worldof cer- taintyand will be discussedbelow. IV. Comparisonwith the Ideal CompetitiveModel underUncertainty A. Introduction In thissection we will comparethe operations of theactual medicalcare marketwith those of an ideal systemin whichnot onlythe usual commoditiesand servicesbut also insurancepolicies against all con- ceivable risksare available.3 Departuresconsist for the mostpart of 3It is assumed that there are two classes, rich and poor; the price of medical services to the rich is twice that to the poor, medical expendituresby the rich are 20 per cent of those by the poor, and the elasticityof demand for medical services is .5 for both classes. Let us choose our quantity and monetaryunits so that the quantity of medical services consumed by the poor and the price they pay are both 1. Then the rich purchase .1 units of medical services at a price of 2. Given the assumption about the elasticitiesof demand, the demand functionof the rich is DR(p) - .14 p--5 and that of the poor is Dp(p) = p`. The supply of miiedicalservices is assumed fixed and therefore must equal 1.1. If price discriminationwere abolished, the equilibrium price, p, must satisfythe relation, DR(P) + Dp(p) 1.1, and thereforep = 1.07. The quantities of medical care purchased by the rich and poor, respectively,would beDR(7p) = .135 and Dp(75) = .965. The inversedemand functions,the priCeto be paid correspondingto any given quantity are djv(q) = .02/q2, and dp(q) = l/q'. Therefore,the consumers' surplus to the rich generatedby the change is: r 135 (1) J (.02/q2)dq- 7(.135 - .1), and similarlythe loss in consumers'surplus by the poor is:

(2) f (l/q2)dq - p(1 - .965) .965 If (2) is subtractedfrom (1), the second terms cancel, and the aggregateincrease in consumers'surplus is .0156, or a little over 1 per cent of the initial expenditures. 'A strikingillustration of the desire for securityin medical care is provided by the expressedpreferences of etmigr&sfrom the Soviet Union as between Soviet medical practice and German or American practice; see Field [14, Ch. 12]. Those in Germany pre- ferredthe German systemto the Soviet, but those in the United States preferred(in a ratio of 3 to 1) the Soviet system. The reasons given boil down to the certaintyof medical care, independentof income or health fluctuations. ARROW: UNCERTAINTY AND MEDICAL CARE 959 insurancepolicies that mightconceivably be written,but are in fact not. Whetherthese potential commoditiesare nonmarketable,or, merelybecause of some imperfectionin the market,are not actually marketed,is a somewhatfine point. To recall what has already been said in SectionI, thereare two kindsof risksinvolved in medicalcare: the risk of becomingill, and the risk of total or incompleteor delayed recovery.The loss due to illnessis onlypartially the cost of medicalcare. It also consistsof dis- comfortand loss of productivetime during illness, and, in mole serious cases, death or prolongeddeprivation of normalfunction. From the pointof view of the welfareeconomics of uncertainty,both losses are risksagainst which individuals would like to insure.The nonexistence of suitableinsurance policies for either risk implies a loss of welfare. B. The Theoryof Ideal Insurancc In this section,the basic principlesof an optimalregime for risk- bearingwill be presented.For illustration,reference will usually be made to the case of insuranceagainst cost in medicalcare. The princi- ples are equally applicable to any of the risks. There is no single sourceto whichthe readercan be easily referred,though I thinkthe principlesare at least reasonablywell understood. As a basis forthe analysis, the assumption is made thateach individual acts so as to maximizethe expectedvalue of a utilityfunction. If we thinkof utilityas attachedto income,then the costs of medical care act as a randomdeduction from this income, and it is theexpected value of theutility of incomeafter medical costs that we are concerned with. (Income aftermnedical costs is the abilityto spend moneyon otherobjects which give satisfaction. We presupposethat illness is not a source of satisfactionin itself; to the extentthat it is a sourceof dissatisfaction,the illness should enterinto the utilityfunction as a separatevariable.) The expected-utilityhypothesis, due originallyto Daniel Bernoulli(1738), is plausibleand is themost analytically man- ageable of all hypothesesthat have been proposedto explainbehavior underuncertainty. In any case, the resultsto followprobably would not be significantlyaffected by movingto anothermode of analysis. It is furtherassumed that individuals are normallyrisk-averters. In utilityterms, this means that they have a diminishingmarginal utility of income.This assumptionmay reasonablybe takento hold formost of the sKgnificantaffairs of life fora majorityof people,but the presence of gamiblingprovides some difficulty in the fullapplication of this view. It followsfrom the assumptionof riskaversion that if an individual is givena choice betweena probabilitydistribution of income, witha givenmean n, and thecertainty of theincome m, he wouldprefer 960 THE AMERICAN ECONOMIC REVIEW the latter.Suppose, therefore, an agency,a large insurancecompany plan,or thegovernment, stands ready to offerinsurance against medical costs on an actuariallyfair basis; thatis, if the costs of medicalcare are a randomvariable with mean n, the companywill chargea premium n, and agree to indemnifythe individualfor all medicalcosts. Underthese circumstances, the individualwill certainlyprefer to take out a policyand will have a welfaregain thereby. Will this be a social gain? Obviouslyyes, if the insuranceagent is sufferingno social loss. Under the assuimptionthat medicalrisks on differentindividuals are basically independent,the poolingof them reducesthe riskinvolved to theinsurer to relativelysmall proportions. In the limit,the welfareloss, even assumingrisk aversion on the part of theinsurer, would vanish and thereis a netsocial gainwhich may be of quite substantialmagnitude. In fact,of course,the pooling of risks does not go to the limit; thereis only a finitenumber of themand theremay be some interdependenceamong the risksdue to epidemics and thelike. But thena premium,perhaps slightly above theactuarial level,would be sufficientto offsetthis welfare loss. From the pointof view of the individual,since he has a strictpreference for the actuari- ally fair policy over assumingthe risks himself,he will still have a preferencefor an actuariallyunfair policy, provided, of course,that it is not too unfair. In additionto a residualdegree of riskaversion by insurers,there are otherreasons for the loadingof the premium(i.e., an excess of premiumover the actuarialvalue). Insuranceinvolves administrative costs.Also, because of theirregularity of paymentsthere is likelyto be a cost of capital tied up. Suppose,to take a simplecase, theinsurance companyis not willingto sell any insurancepolicy that a consumer wants but will chargea fixed-percentageloading above the actuarial value forits premium.Then it can be shownthat the mostpreferred policy fromthe point of view of an individualis a coveragewith a deductibleamount; that is, the insurancepolicy provides 100 per cent coveragefor all medicalcosts in excess of some fixed-dollarlimit. If, however,the insurancecompany has some degreeof riskaversion, its loadingmay also dependon the degreeof uncertaintyof the risk.In that case, the Pareto optimalpolicy will involvesome elementof coinsurance,i.e., the coveragefor costs over the minimumlimit will be some fractionless than 100 per cent (for proofsof thesestatements, see Appendix). These resultscan also be appliedto the hypotheticalconcept of insuranceagainst failureto recoverfrom illness. For simplicity,let us assumethat the cost of failureto recoveris regardedpurely as a money cost, eithersimply productive opportunities foregone or, moregener- ARROW: UNCERTAINTY AND MEDICAL CARE 961 ally, the money equivalentof all dissatisfactions.Suppose further that,given that a personis ill, the expectedvalue of medicalcare is greaterthan its cost; thatis, theexpected money value attributableto recoverywith medical help is greaterthan resources devoted to medical help. However,the recovery,though on the average beneficial,is uncertain;in the absence of insurancea risk-avertermay well prefer not to take a chance on furtherimpoverishment by buyingmedical care. A suitableinsurance policy would, however, mean that he paid nothingif he doesn'tbenefit; since the expectedvalue is greaterthan thecost, there would be a netsocial gain.35 C. Problemsof Insurance 1. The moralhazard. The welfarecase forinsurance policies of all sortsis overwhelming.It followsthat the governmentshould under- take insurancein thosecases wherethis market, for whatever reason, has failed to emerge.Nevertheless, there are a numberof significant practicallimitations on the use of insurance.It is importantto under- stand them,though I do not believe that theyalter the case forthe creationof a muchwider class of insurancepolicies than now exists. One of the limitswhich has been muchstressed in insurancelitera- tureis theeffect of insuranceon incentives.What is desiredin thecase of insuranceis that the eventagainst which insurance is takenbe out of the controlof the individual.Unfortunately, in real lifethis separa- tion can neverbe made perfectly.The outbreakof firein one's house or businessmay be largelyuncontrollable by the individual,but the probabilityof fireis somewhatinfluenced by carelessness,and of course arson is a possibility,if an extremeone. Similarly,in medicalpolicies the cost of medical care is not completelydetermined by the illness sufferedby the individualbut dependson the choiceof a doctorand his willingnessto use medicalservices. It is frequentlyobserved that widespreadmedical insurance increases the demandfor medical care. Coinsuranceprovisions have been introducedinto many major medical policiesto meetthis contingency as well as the riskaversion of thein- surancecompanies. To some extentthe professionalrelationship between physician and patientlimits the normal hazard in variousforms of medicalinsurance. By certifyingto the necessityof giventreatment or the lack thereof, the physicianacts as a controllingagent on behalfof the insurance companies.Needless to say, it is a far fromperfect check; the phy- siciansthemselves are not underany controland it may be convenient forthem or pleasingto theirpatients to prescribemore expensive medi- "It is a popularbelief that the Chinese,at one time,paid theirphysicians when well but not whensick. 962 THE AMERICAN ECONOMIC REVIEW cation,private nurses, more frequenttreatments, and othermarginal variationsof care. It is probablytrue that hospitalization and surgery are moreunder the casual inspectionof othersthan is generalpractice and thereforeless subjectto moralhazard; thismay be one reasonwhy insurancepolicies in thosefields have been morewidespread. 2. Alternativemethods of insurancepayment. It is interestingthat no less thanthree different methods of coverageof thecosts of medical carehave arisen:prepayment, indemnities according to a fixedschedule, and insuranceagainst costs, whateverthey may be. In prepayment plans, insurancein effectis paid in kind-that is, directlyin medical services.The othertwo formsboth involve cash paymentsto thebene- ficiary,but in the one case the amountsto be paid involvinga medical contingencyare fixedin advance,while in the otherthe insurance car- rierpays all the costs,whatever they may be, subject,of course,to provisionslike deductibles and coinsurance. In hypotheticallyperfect markets these three formsof insurance would be equivalent.The indemnitiesstipulated would, in fact,equal the marketprice of the services,so thatvalue to theinsured would be the same if he were to be paid the fixedsum or the marketprice or were giventhe servicesfree. In fact,of course,insurance against full costsand prepaymentplans bothoffer insurance against uncertainty as to the priceof medicalservices, in additionto uncertaintyabout their needs. Further,by theirmode of compensationto the physician,prepaymentplans are inevitablybound up withclosed panels so that the freedomof choiceof thephysician by thepatient is less thanit would be undera schememore strictly confined to theprovision of insurance. These remarksare tentative,and the questionof coexistenceof the differentschemes should be a fruitfulsubject forinvestigation. 3. Third-partycontrol over payments.The moral hazard in physicians' controlnoted in paragraph1 above shows itselfin thosein- suranceschemes where the physician has the greatestcontrol, namely, major medical insurance.Here therehas been a markedrise in ex- pendituresover time.In prepaymentplans, wherethe insuranceand medicalservice are suppliedby the same group,the incentiveto keep medicalcosts to a minimumis strongest.In plans of the Blue Cross group,there has developeda conflictof interestbetween the insurance carrierand the medical-servicesupplier, in this case particularlythe hospital. The need forthird-party control is reinforcedby anotheraspect of the moralhazard. Insuranceremoves the incentiveon the part of individuals,patients, and physiciansto shop aroundfor better prices for hospitalizationand surgicalcare. The marketforces, therefore, tend to be replacedby directinstitutional control. ARROW: UNCERTAINTY AND MEDICAL CARE 963 4. Administrativecosts. The pure theoryof insurancesketched in SectionB above omitsone veryimportant consideration: the costs of operatingan insurancecompany. There are severaltypes of operating costs,but one of the mostimportant categories includes commissions and acquisitioncosts, selling costs in usual economicterminology. Not only does this mean that insurancepolicies must be sold forconsid- erablymore than theiractuarial value, but it also means thereis a greatdifferential among different types of insurance.It is verystriking to observethat among health insurance policies of insurancecompanies in 1958,expenses of one sortor anotherconstitute 51.6 per centof total premiumincome for individual policies, and only9.5 per centfor group policies [26, Table 14-1,p. 272]. This strikingdifferential would seem to implyenormous economies of scale in theprovision of insurance, quite apart fromthe coverageof the risksthemselves. Obviously, this provides a verystrong argument for widespread plans, including,in particular,compulsory ones. 5. Predictabilityand insurance.Clearly, from the risk-aversion point of view,insurance is morevaluable, the greaterthe uncertaintyin the risk being insuredagainst. This is usually used as an argumentfor puttinggreater emphasis on insuranceagainst hospitalization and sur- gerythan otherforms of medicalcare. The empiricalassumption has been challengedby 0. W. Andersonand others[3, pp. 53-54], whoas- sertedthat out-of-hospitalexpenses were equally as unpredictableas in-hospitalcosts. What was in factshown was thatthe probabilityof costs exceeding$200 is about the same forthe twocategories, but this is not,of course,a correctmeasure of predictability, and a quickglance at the supportingevidence shows that in relationto the average cost the variabilityis muchlower for ordinary medical expenses. Thus, for the city of Birmingham,the mean expenditureon surgerywas $7, as opposed to $20 for othermedical expenses,but of those who paid somethingfor surgery the average bill was $99, as against$36 forthose withsome ordinarymedical cost. Eighty-twoper cent of thoseinter- viewedhad no surgery,and only20 per centhad no ordinarymedical expenses[3, Tables A-13,A-18, and A-19 on pp. 72, 77, and 79, respectively]. The issue of predictabilityalso has bearingon the meritsof insurance againstchronic illness or maternity.On a lifetimeinsurance basis, insuranceagainst chronic illness makes sense, since thisis bothhighly unpredictableand highlysignificant in costs. Amongpeople who al- readyhave chronicillness, or symptomswhich reliably indicate it, insurancein thestrict sense is probablypointless. 6. Pooling of unequal risks.Hypothetically, insurance requires for its fullsocial benefita maximumpossible discrimination of risks.Those 964 THE AMERICAN ECONOMIC REVIEW in groupsof higherincidences of illnessshould pay higherpremiums. In fact,however, there is a tendencyto equalize,rather than to differentiate,premniums, especially in the Blue Cross and similarwidespread schemes.This constitutes,in effect,a redistributionof incomefrom thosewith a low propensityto illnessto thosewith a highpropensity. The equalization,of course,could not in factbe carriedthrough if the marketwere genuinelycompetitive. Under thosecircumsances, insurance plans could arise which chargedlower premiumsto preferred risksand draw themoff, leaving the plan whichdoes not discriminate amongrisks with only an adverseselection of them. As we have alreadyseen in the case of incomeredistribution, some of thismay be thoughtof as insurancewith a longertime perspeCtive. If a plan guaranteesto everybodya premiumthat corresponds to total experiencebut not to experienceas it mightbe segregatedby smaller subgroups,everybody is, in effect,insured against a changein his basic stateof healthwhich would lead to a reclassification.This corresponds preciselyto the use of a level premiumin life insuranceinstead of a premiumvarying by age, as would be the case forterm insurance. 7. Gaps and coverage.We maybriefly note that, at any rateto date, insurancesagainst the cost of medicalcare are farfrom universal. Cer- tain groups-the unemployed,the institutionalized,and the aged-are almostcompletely uncovered. Of total expenditures,between one-fifth and one-fourthare coveredby insurance.It shouldbe noted,however, that over half of all hospitalexpenses and about 35 per cent of the medicalpayments of thosewith bills of $1,000 a yearand over,are in- cluded [26, p. 376]. Thus, the coverageon the morevariable parts of medicalexpenditure is somewhatbetter than the over-all figures would indicate,but it mustbe assumedthat the insurancemechanism is still very far fromachieving the full coverageof whichit is capable. D. Uncertaintyof Effectsof Treatment 1. There are reallytwo major aspects of uncertaintyfor an indi- vidualalready suffering from an illness.He is uncertainabout the effec- tivenessof medicaltreatment, and his uncertaintymay be quite different fromthat of his physician,based onithe presumably quite different medicalknowledges. 2. Ideal insurance.This will necessarilyinvolve insurance against a failureto benefitfrom medical care, whether through recovery, relief of pain,or arrestof furtherdeterioration. One formwould be a system in whichthe paymentto thephysician is made in accordancewith the degreeof benefit.Since thiswould involvetransferring the risksfrom the patientto the physician,who mightcertainly have an aversionto bearingthem, there is room forinsurance carriers to pool the risks, ARROW: UNCERTAINTY AND MEDICAL CARE 965 eitherby contractwith physicians or by contractwith the potential patients.Under ideal insurance,medical care willalways be undertaken in any case in whichthe expectedutility, taking account of theprob- abilities,exceeds the expectedmedical cost. This prescriptionwould lead to an economicoptimum. If we thinkof the failureto recover mainlyin termsof lost workingtime, then this policy would, in fact, maxi-mizeeconomic welfare as ordinarilymeasured. 3. The conceptsof trustand delegation.In the absenceof ideal insurance,there arise institutionswhich offersome sort of substitute guarantees.Under ideal insurancethe patientwould actually have no concernwith the informationalinequality between himself and the physician,since he would onlybe payingby resultsanyway, and his utilityposition would in factbe thoroughlyguaranteed. In its absence he waantsto have some guaranteethat at least the physicianis using his knowledgeto the best advantage.This leads to the settingup of a relationshipof trustand confidence,one whichthe physicianhas a social obligationto live up to. Since thepatient does not,at least in his belief,know as muchas the physician,he cannotcompletely enforce standardsof care. In part, he replaces directobservation by gener- alized beliefin theability of thephysician." To put it anotherway, the social obligationfor best practiceis part of the commoditythe physician sells,even thoughit is a part thatis not subjectto thoroughin- spectionby the buyer. One consequenceof such trustrelations is thatthe physician cannot act, or at least appear to act, as if he is maximizinghis incomeat every momentof time.As a signal to the buyerof his intentionsto act as thorouglhlyin the buyer'sbehalf as possible,the physicianavoids the obviousstigmata of profit-maximizing.Purely arms-length bargaining behaviorwould be incompatible,not logically,but surelypsychologi- cally,with the trustrelations. From these special relationscome the variousforms of ethicalbehavior discussed above, and so also, I sug- gest,the relativeun-importance of profit-making in hospitals. The very word,"profit," is a signalthat denies the trust relations. PriCediscrimination and its extreme,free treatment for the indigent, also follow.If the obligationof the physicianis understoodto be first of all to the welfareof the patient,then in particularit takespreced- ence over financialdifficulties. As a second consequenceof informationalinequality between phy- siclan and patientand the lack of insuranceof a suitabletype, the patientmust delegate to the physicianmuch of his freedomof choice.

" Francis Bator points out to me that some protectioncan be achieved, at a price, by securingadditional opinions. 966 THE AMERICAN ECONOMIC REVIEW He does not have the knowledgeto make decisionson treatment,re- ferral,or hospitalization.To justifythis delegation, the physician finds himselfsomewhat limited, just as any agentwould in similarcircum- stances.The safestcourse to take to avoid not beinga trueagent is to give the sociallyprescribed "best" treatmentof the day. Compromise in quality,even forthe purposeof savingthe patient money, is to risk an imputationof failureto liveup to thesocial bond. The special trustrelation of physicians(and allied occuptions,such as priests) extendsto thirdparties so that the certificationsof physiciansas to illnessand injuryare acceptedas especiallyreliable (see Section IJ.B above). The social value to all concernedof such pre- sumptivelyreliable sources of information is obvious. Notice the generalprinciple here. Because thereare barriersto the informationflow and because thereis no marketin whichthe risks involvedcan be insured,coordination of purchaseand sales musttalke place throughconvergent expectations, but theseare greatlyassisted by havingclear and prominentsignals, and these,in turn,force pat- ternsof behaviorwhich are not in themselveslogical necessitiesfor optimality.37 4. Licensingand educationalstandards. Delegation and trustare the social institutionsdesigned to obviatethe problem of informationalinequality.The generaluncertainty about tneprospects of medicaltreat- mentis sociallyhandled by rigidentry requirements. These are designedto reduce the uncertaintyin the mind of the consumeras to thequality of productinsofar as thisis possible.38I thinkthis explanation,which is perhapsthe naive one, is muchmore tenable than any idea of a monopolyseeking to increaseincomes. No doubtrestriction on entryis desirablefrom the point of view of the existingphysicians, but the public pressureneeded to achieve the restrictionmust come fromdeeper causes. The social demandfor guaranteed quality can be metin morethan one way,however. At least threeattitudes can be takenby thestate or othersocial institutionstoward entry into an occupationor toward the productionof commoditiesin general; examplesof all threetypes exist. (1) The occupationcan be licensed,nonqualified entrants being simplyexcluded. The licensingmay be more complexthan it is in medicine;individuals could be licensedfor some, but not all, medical activities,for example. Indeed, the presentall-or-none approach could "The situation is very reminiscentof the crucial role of the focal point in Schelling's theory of tacit games, in which two parties have to fin-da common course of action withoutbeing able to communicate;see [24, esp. pp. 225 ff.]. ' How well they achieve this end is another matter.R. Kessel points out to me that they merely guarantee training,not continued good performanceas medical technology changes. ARROW: UNCERTAINTY AND MEDICAL CARE 967 be criticizedas beinginsufficient with regard to complicatedspecialist treatment,as well as excessivewith regardto minormedical skills. Gradedlicensing may, however, be muchharder to enforce.Controls could be exercisedanalogous to thosefor foods; theycan be excluded as beingdangerous, or theycan be permittedfor animals but not for humans.(2) The state or otheragrency can certifyor label, without compulsoryexclusion. The categoryof CertifiedPsychologist is now underactive discussion;canned goods are graded.Certification can be done by nongovernmentalagencies, as in the medical-boardexamina- tionsfor specialists. (3) Nothingat all may be done; consumersmake theirown choices. The choiceamong these alternatives in any givencase dependson the degreeof difficultyconsumers have in makingthe choiceunaided, and on theconsequences of errorsof judgment.It itsthe general social con- sensus,clearly, that the laisscz-faire solution for medicine is intolerable. The certificationproposal never seems to have beendiscussed seriously. It is beyondthe scope of thispaper to discussthese proposals in detail. I wish simplyto pointout thatthey should be judged in termsof the abilityto relievethe uncertainty of thepatient in regardto thequality of the commodktyhe is purchasing,and thatentry restrictions are the consequencesof an apparentinability to devisea systemin whichthe risks of gaps in medicalknowledge and skill are borneprimarily by thepatient, not the physician.

Postscript I wish to repeat here what has been suggestedabove in several places: that the failureof the marketto insureagainst uncertainties has createdmany social institutionsin whichthe usual assumptionsof the marketare to some extentcontradicted. The medicalprofession is only one example,though in manyrespects an extremeone. All pro- fessionsshare some of the same properties.The eConomicimportance of personaland especiallyfamily relationships, though declining, is by no means trivialin the mostadvanced economies; it is based on non- marketrelations that create guarantees of behaviorwhich would otherwise be afflictedwith excessive uncertainty. Many otherexamples can be given.The logicand limitationsof ideal competitivebehavior under uncertaintyforce us to recognizethe incompletedescription of reality suppliedby theimpersonal price system.

REFERENCES 1. A. A. ALCEIAN, K. J.ARROW, AND WV.M. CAPRON, An EconomicAnalysis of the Marketfor Scientists and Engineers,RAND RM-2190-RC. Santa Monica 1958. 968 THE AMERICAN ECONOMIC REVIEW

2. M. ALLAIS, "Ge'neralisationdes theories de l'6quilibre economique general et du rendementsocial au cas du risque,"in CentreNational de la RechercheScientifique, Econometrie, Paris 1953,pp. 1-20. 3. 0. WV.ANDERSON AND STAFF OF THE NATIONAL OPINION RESEARCH CENTER, VoluntaryHealth Insurancein Two Cities. Cambridge, Mass. 1957. 4. K. J. ARROW, "EIconomic Welfare and the Allocationof Resourcesfor Invention,"in Nat. Bur. Econ. Research,The Role and Direction of InventiveActivity: Economic and Social Factors, Princeton1962, pp. 609-25. 5. , "Les role des valeursboursieres pour la repartitionla meilleure des risques," in CentreNational de la RechercheScientifique, Econo- mietrie,Paris 1953, pp. 41-46. 6. F. M. BATOR, "The Anatomyof Market Failure," Quart. Jour.Econ. Aug. 1958, 72, 351-79. 7. E. BAUDIER, "L'introductiondu tempsdans la theoriede l'equilibregen- eral,"Les CahziersEconomiques, Dec. 1959,9-16. 8. W. J. BAUMOL, WelfareEconomics azd the Theoryof the State. Cam- bridge,MIass. 1952. 9. K. BORCT-I,"The Safety Loading of ReinsurancePremiums," Skandi- naviskAktuariehdskrift, 1960, pp. 163-84. 10. J. M. BUCHANAN AND G. TULLOCK, The Calculus of Consent. Ann Arbor1962. 11. G. DEBREU, "Une economiquede l'incertain,"Economie Appliqutee 1960, 13, 111-16. 12. --, Theoryof Values.New York 1959. 13. R. DUBOS, "Medical Utopias," Daedalus, 1959,88, 410-24. 14. M. G. FIELD, Doctor and Patient in Soviet Russia. Cambridge,Mass. 1957. 15. MILTON FRIEDMAN, "The Methodologyof Positive Economics," in Essays in PositiveEconomics, Chicago 1953, pp. 3-43. 16. AND S. S. KuZNETS, Income from IndependentProfessional Practice.Nat. Bur. Econ. Research,New York 1945. 17. R. A. KESSEL, "Price Discrimination in Medicine," Jour. Law and Econ.., 1958, 1, 20-53. 18. T. C. KOOPMANS, "Allocation of Resources and the Price System," in Three Essays on the State of EconomicScience, New York 1957, pp. 1-120. 19. I. M. D. LITTLE, A Critiqueof WelfareEconomics. Oxford 1950. 20. SELMA MUSHKIN, "Towards a Definitionof Health Economics,"Public Health Reports,1958, 73, 785-93. 21. R. R. NELSON, "The Simple Economicsof Basic ScientificResearch," Jour.Pol. Econt.,June 1959, 67, 297-306. 22. T. PARSONS, The Social System.Glencoe 1951. 23. M. J. PECK AND F. M. SCHERER, The TVeaponzsAcquisition Process: An Economic Analysis. Div. of Research, Graduate School of Business, Harvard University,Boston 1962. ARROW: UNCERTAINTY AND MEDICAL CARE 969

24. T. C. SCHELLING, T he Strategyof Conflict.Cambridge, Mass. 1960. 25. A. K. SHAPIRO, "A Contributionto a Historyof the Placebo Effect," BehavioralScience, 1960, 5, 109-35. 26. H. M. SOMERS AND A. R. SOMERS, Doctors, Patients,and IHealthIn- surance.The BrookingsInstitution, Washington 1961. 27. C. L. STEVENSON, Ethicsand Lan-guage.New Haven 1945. 28. U. S. DEPARTMENT OF HEALTH, EDUCATION AND WELFARE, Physicians fora GrowZOingAmterica, Public Health ServicePublication No. 709, Oct. 1959.

APPENDIX On OptimalInsurance Policies The two propositionsabout the natureof optimalinsurance policies as- sertedin SectionIV.B above willbe provedhere. Proposition1. If an insurancecompany is willingto offerany insurance policyagainst loss desiredby the buyerat a premiumwhich depends only on the policy's actuarial value, then the policy chosen by a risk-averting buyerwill take the formof 100 per cent coverageabove a deductiblemini- mum. Note: The premiumwill, in general, exceed the actuarial value; it is onlyrequired that two policieswith the same actuarialvalue willbe offered by the companyfor the same premium. Proof:Let W be the initialwealth of the individual,X his loss, a random variable, I(X) the amount of insurancepaid if loss X occurs,P the premium,and Y(X) the wealth of the individual afterpaying the premium, incurringthe loss, and receivingthe insurancebenefit. (1) Y(X) =W-P-X + I(X). The individualvalues alternativepolicies by the expectedutility of his finalwealth position, Y(X). Let U(y) be the utilityof finalwealth, y; then his aim is to maximize, (2) Et U[Y(X)]}, wherethe symbol,E, denotesmathematical expectation. An insurancepayment is necessarilynonnegative, so the insurancepolicy mustsatisfy the condition, (3) I(X) > 0 forall X. If a policyis optimal,it must in particularbe betterin the sense of the criterion(2), than any other policy with the same actuarial expectation, E[I(X)I. Considera policy that pays some positiveamount of insuranceat one level of loss, say X,, but whichpermits the finalwealth at some other loss level, say X2, to be lower than that correspondingto X,. Then, it is intuitivelyobvious that a risk-averterwould preferan alternativepolicy withthe same actuarial value whichwould offerslightly less protectionfor losses in the neighborhoodof X, and slightlyhigher protection for those in the neighborhoodof X2, sincerisk aversion implies that the marginalutility 970 THE AMERICAN ECONOMIC REVIEW of Y(X) is greaterwhen Y(X) is smaller:hence, the originalpolicy cannot be optimal. To provethis formally, let 11(X) be the originalpolicy, with 11(X)>0 and Yl(Xl) > Y2(X2),where Y1(X) is definedin termsof 11(X) by (I). Choose a sufficientlysmall so that, (4) 11(X) > 0 for X1 < X < X1 + 3, (5) Y1(X') < Y1(X) for X2 < X' < X2 + 3a X1 < X < X1 + &. (This choice of a is possible if the functions11(X), Y1(X) are continuous; this can be proved to be truefor the optimalpolicy, and thereforewe need only considerthis case.) Let 7r- be the probabilitythat the loss, X, lies in the interval (X1, X1+6), 72 the probabilitythat X lies in the interval(X2, X2+3). From (4) and (5) we can choose E>0 and sufficientlysmall so that, (6) I1(X) -72E > 0 for Xi < X < X1 + a,

(7) Y1(X') + 7riE< Y1(X) - 7r2E for X2< X' < X2+ 3, X1 < X < X1 + S. Now definea new insurancepolicy, 12(X), whichis the same as 11(X) except that it is smallerby 72E in the intervalfrom X1 to X1+8 and largerby 7r,,E in the intervalfrom X2 to X2+8. From (6), 12(X) >0 everywhere,so that (3) is satisfied.We willshow that E[11(X)] = E[I2(X)] and that I2(X) yields the higherexpected utility, so that I1(X) is not optimal. Note that I2(X) - 1(X) equals -7i-2E forX1 < X < X1+6, 7rWEfor X2 < X < X2+6, and 0 elsewhere.Let ?(X) be the densityof the randomvariable X. Then, X1+8 E [I2(X) - 11(X) ] [I2(X) - 1I(X)]cp(X)dX X1 rx2+8 +1 [I2(X) - 11(X)]dX Y2 rXl+8 r X24-S =(-72E) J (X)dX + (WiE) p(X)dX 1 j X2

= - (w2E)7rl + (lriE)r2 = 0, so that the two policieshave the same actuarial value and, by assumption, the same premium. Define Y2(X) in terms of 12(X) by (1). Then Y2(X) - Y1(X) = 12(X) -11(X). From (7), (8) Y1(X') < Y2(X') < Y2(X) < Y1(X) for X2

(9) E{U[Y2(X)] - U[Y1(X)]} J{U[Y2(x)] - u[Y,(x)]}I(x)dx X1 rx2+5 +1 {U [y2(X)] - U[yl(x)])} (X)dX. X2 By the Mean Value Theorem,for any given value of X, (10) U[y2(X)] - U[y1(X)]= U'[Y(X)][y2(X) - Y1(X)] uU' [Y(X) ] [I2(X)- I1(X)]. whereY(X) lies between Y1(X) and Y2(X). From (8), Y(X') < Y(X) for X2 < X' < X2 + 6, X1 < X X1 + a, and, since U'(y) is a diminishingfunction of y fora risk-averter, u' [Y(X')] > U'[Y(X)] or, equivalently,for some numberu, (11) U'[Y(X')]>u for X2 < X' < X2 + 6, U'[Y(X)] < u for X1 < X < X1 + . Now substitute(10) into (9),

E { U [ Y2(X)] U [Y1(X)]} = 7r2E u [Y(X)]O(X)dX 1 rX2+8 + 7rWE U' [Y(X)]O(X)dX X2 From (11), it follows that, E { U [ Y2(X) ]- U [ Y1(x)]} > - 7r2EUrl + rlEU7r2= 0, so that the second policy is preferred. It has thus been shown that a policy cannot be optimal if,for some Xi and X2, I(X1) >0, Y(X1) > Y(X2). This may be put in a differentform: Let Y1,i. be the minimumvalue taken on by Y(X) under the optimal policy; then we must have I(X) = 0 if Y(X)> Ymin.In other words,a minimum final wealth level is set; if the loss would not bring wealth below this level, no benefitis paid, but if it would, then the benefitis sufficientto bringup the finalwealth positionto the stipulatedminimum. This is, of course, preciselya descriptionof 100 per cent coverage for loss above a deductible. We turnto the second proposition.It is now supposedthat the insurance company,as well as the insured,is a risk-averter;however, there are no administrativeor othercosts to be coveredbeyond protection against loss. Proposition2. If the insuredand the insurerare both risk-avertersand thereare no costs otherthan coverageof losses,then any nontrivialPareto- 972 THE AMERICAN ECONOMIC REVIEW optimal policy, I(X), as a function of the loss, X, must have the property, O< dI/dX < 1. That is, any increment in loss will be partly but not wholly compensated by the insurance company; this type of provision is known as coinsurance. Proposition 2 is due to Borch [9, Sec. 2]; we give here a somewhat simpler proof. Proof: Let U(y) be the utility function of the insured, V(z) that of the insurer. Let W0 and W1 be the initial wealths of the two, respectively. In this case, we let I(X) be the insurance benefits less the premium; for the present purpose, this is the only significant magnitude (since the premium is independent of X, this definition does not change the value of dI/dX). The final wealth positions of the insured and insurer are: (12) Y(X) = wo - X + I(X), Z(X) = T - 1(X), respectively. Any given insurance policy then defines expected utilities, u=Et U[Y(X)]} and v=Et V[Z(X)]}, for the insured and insurer, respectively. If we plot all points (u, v) obtained by considering all possible insurance policies, the resulting expected-utility-possibilityset has a boundary that is convex to the northeast. To see this, let 11(X) and I2(X) be any two policies, and let (X1, v1) and (u2, v2) be the corresponding points in the two- dimensional expected-utility-possibility set. Let a third insurance policy, I(X), be defined as the average of the two given ones, I(X) = (21)Il(X) + (1)I2(X), for each X. Then, if Y(X), Y1(X), and Y2(X) are the final wealth positions of the insured, and Z(X), Z1(X), and Z2(X) those of the insurer for each of the three policies, I(X), I1(X), and I2(X), respectively, Y(X) (2)Y1(X) + (2)Y2(X),

Z(X) (2)Z1(X) + (G)Z2(X), and, because both parties have diminishing marginal utility,

U[Y(X)] > (21)U[YI1(X)] + (21)U[Y2(X)], V[Z(X)] > (2)V[Z1(X)] + (21)V[Z2(X)]. Since these statements hold for all X, they also hold when expectations are taken. Hence, there is a point (u, v) in the expected-utility-possibilityset for which u>?(2)U1+(2)U2, v?>(2)v1+(2)v2. Since this statement holds for every pair of points (u,, yi) and (u2, v2) in the expected-utility-possibility set, and in particular for pairs of points on the northeast boundary, it follows that the boundary must be convex to the northeast. From this, in turn, it follows that any given Pareto-optimal point (i.e., any point oIn the northeast boundary) can be obtained by maximizing a linear function, au+fv, with suitably chosen a and ,Bnonnegative and at least one positive, over the expected-utility-possibilityset. In other words, a Pareto-optimal insurance policy, I(X), is one which maximizes, aE{U[Y(X)]} + 0E{V[Z(X)]} -E{laU[Y(X)] + 0V[Z(X)]}, ARROW: UNCERTAINTY AND MEDICAL CARE 973 forsome a?> O, > 0, a > 0 or / > 0. To maximizethis expectation,it is obviously sufficientto maximize: (13) alU[Y(X)] + /3v[Z(X)], with respectto I(X), foreach X. Since, for given X, it followsfrom (12) that, dY(X)/dI(X) -1, dZ(X)/dI(X) - 1, it followsby differentiationof (13) that E(X) is the solutionof th-ieequation, (14) aU'[Y(X)] - OV'[Z(X)] = 0. The cases a 0 or ,B=0 lead to obvious trivialities(olne partysimply hands over all his wealthto the other),so we assumea >O, /3>0. Now differentiate (14) withrespect to X and use the relations,derived from (12), dY/dX - (dI/dX) - 1 dZ/dX= - (dI/dX). a U Y(X) l(dI/dX) - 11+ /3V"[Z (X) j (dI/dX) = 0, or dI/dX = aU" [Y(X) /{aU" [Y(X)j + f3V"[Z(X)]}. Since U"[Y(X)]<0, V"[Z(X)]<0 by the hypothesisthat both parties are risk-averters,Proposition,. 2 follows.