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TH1EAMERICAN ECONOMIC REVIEW VOLUME LIII DECEMBER UNCERTAINTY ECONOMICS By 1963 NUMBER AND THE WELFARE OF MEDICAL CARE KENNETH J ARROW* I Introduction: Scope and Method This paper is an exploratory and tentative study of the specific of It differentia medicalcare as the object of normative economics is contended here,on the basis of comparison obviouscharacterisof tics of themedical-care of industry withthe norms welfare economics, that the special economic problems medicalcare can be explained of as adaptations the existence uncertainty the incidence disto of in of ease and in theefficacy treatment of It shouldbe notedthatthe subjectis the medical-care not industry, health.The causal factorsin healthare many,and the provision of medicalcare is only one Particularly low levels of income, other at commodities such as nutrition, shelter, and sanitation clothing, may be much more significant is the complexof servicesthat center It about the physician, privateand grouppractice, and public hospitals, whichI proposeto discuss health, The focus of discussionwill be on the way the operationof the medical-care and industry the efficacy whichit satisfies needs with the of societydiffer if from norm, at all The "norm"that the econoa of mistusuallyuses forthepurposes such comparisons theoperation is of a competitive model,that is, the flowsof servicesthat would be * The author is professor economicsat StanfordUniversity wishes to expresshis of He thanks for usefulcommentsto F Bator, R Dorfman,V Fuchs, Dr S Gilson, R Kessel, S Mushkin,and C R Rorem This paper was preparedunderthe sponsorship the Ford of Foundation as part of a seriesof papers on the economicsof health,education,and welfare 942 THE AMERICAN ECONOMIC REVIEW offered purchasedand the pricesthat wouldbe paid forthemif and each individual themarket in offered purchased or services thegoing at prices as if his decisionshad no influence over them,and the going prices were such that the amountsof serviceswhichwere available equalled the total amountswhich otherindividualswere willingto purchase, withno imposedrestrictions supplyor demand on The interest the competitive in model stemspartlyfromits presumeddescriptive powerand partlyfrom implications economic its for efficiency particular, can state the following In we well-known proposition (First OptimalityTheorem) If a competitive equilibrium existsat all, and if all commodities relevant costsor utilities in to are factpricedin the market, thenthe equilibrium necessarily is optimal in the following precisesense (due to V Pareto): There is no other allocationof resources serviceswhichwill make all participants to in themarket better off Both the conditions thisoptimality of theorem thedefinition and of optimality forcomment definition just a definition, when call A is but the definiendum a wordalreadyin common withhWighly is use favorable connotations, is clearthatwe are reallytrying be persuasive; it to we are implicitly recommending achievement optimal the of states.'It is reasonable enoughto assertthata changein allocation whichmakes all participants better is one thatcertainly off shouldbe made; thisis a value judgment, a descriptive not proposition, it is a veryweak but one Fromthisit follows thatit is notdesirable put up witha nonto optimalallocation.But it does not follow thatif we are at an allocationwhichis optimal theParetosense,we shouldnotchangeto any in other.We cannotindeedmake a changethatdoes not hurtsomeone; but we can still desireto changeto anotherallocationif the change makesenough participants better and by so muchthatwe feelthat off the injuryto othersis not enoughto offset benefits the Such interpersonalcomparisons are, of course,value judgments The change, however, the previousargument by oughtto be an optimalstate; of coursethereare manypossiblestates,each of whichis optimal the in sensehereused a However, value judgment thedesirability each possiblenew on of of distribution benefits and costs corresponding each possiblereto allocationof resources not,in general, is necessary Judgments about thedistribution be made separately, one sense,from can in thoseabout if allocation certain conditions fulfilled are Beforestating relevant the it to proposition, is necessary remark thatthe competitive equilibrium achieveddependsin good measureon the initialdistribution purof of chasingpower,whichconsistsof ownership assets and skills that 'This point has been stressedby I M D Little [19, pp 71-74] For the concept of a "persuasivedefinition," C L Stevenson[27, pp 210-17] see ARROW: UNCERTAINTY AND MEDICAL CARE 943 command price on the market transfer assets amongindivida A of uals will,in general,changethe finalsuppliesof goods and services and the pricespaid forthem.Thus, a transfer purchasing of power from well to the ill will increasethe demandformedicalservices the This will manifest itself the shortrunin an increasein thepriceof in medicalservices and in thelongrunin an increasein theamount supplied With this in mind,the following can statement be made (Second Optimality Theorem): If there no increasing are in returns production, and if certainotherminor conditions satisfied, are theneveryoptimal state is a competitive equilibrium corresponding some initialdisto tribution purchasing of power.Operationally, significance this the of proposition thatif theconditions thetwooptimality is of theorems are and in mechanism thereal worldsatisfies the satisfied, if theallocation conditions a competitive for thensocial policycan confine itself model, to steps takento alter the distribution purchasing of power.For any given distribution purchasing of power,the marketwill, under the assumptions made, achieve a competitive equilibrium whichis necessarilyoptimal;and any optimal stateis a competitive equilibrium corresponding some distribution purchasing to of power, so that any desiredoptimalstatecan be achieved The redistribution purchasing of power among individualsmost takesthe form money:taxesand subsidies simply of The implications of such a transfer for individualsatisfactions are, in general,not knownin advance But we can assumethatsociety can ex post judge the distribution satisfactions of and, if deemedunsatisfactory, take steps to correctit by subsequenttransfers Thus, by successiveapproximations, mostpreferred a social state can be achieved,withresourceallocationbeinghandledby the market and publicpolicyconfined the redistribution money to of income.2 If, on the contrary, actual market the differs significantly the from competitive model,or if the assumptions the two optimality of theoremsare not fulfilled, separation allocativeand distributional the of procedures becomes,in mostcases, impossible.3 The first step thenin theanalysisof themedical-care market the is 2The separationbetween allocation and distribution even under the above assumptions has 4osSed over problemsin the executionof any desiredredistribution policy; in practice, it is virtuallyimpossibleto find a set of taxes and subsidies that will not have an adverse effecton the achievementof an optimal state But this discussionwould take us even further afieldthan we have already gone 'The basic theoremsof welfare economics alluded to so brieflyabove have been the subject of voluminous literature, but no thoroughly satisfactory statementcoveringboth the theoremsthemselvesand the significance exceptionsto them exists The positive of assertionsof welfareeconomicsand theirrelationto the theoryof competitive equilibrium are admirably covered in Koopmans [181 The best summary of the various ways in whichthe theorems can fail to hold is probablyBator's [6] 944 THE AMERICAN ECONOMIC REVIEW comparison between actual market the and thecompetitive model.The of methodology thiscomparison been a recurrent has subjectof conin troversy economics overa century for Recently, Friedman[15] M has vigorously arguedthatthe competitive any othermodelshould or be testedsolelyby its abilityto predict thecontext competition, In of he comesclose to arguing thatpricesand quantities the onlyreleare vant data This point of view is valuable in stressing that a certain amountof lack of realismin the assumptions a modelis no arguof mentagainstits value But theprice-quantity implications thecomof petitive modelforpricing noteasy to derive are without in major and, manycases, impossible-econometric efforts In thispaper,theinstitutional organization theobservable and mores of the medicalprofession includedamongthe data to be used in are assessingthe competitiveness the medical-care of I market shall also examinethe presenceor absence of the preconditions the equivafor lence of competitive equilibria and optimalstates.The majorcompetitivepreconditions, the sense used here,are three:the existence in of competitive equilibrium, marketability all goods and services the of relevant costsand utilities, nonincreasing to and retiurns first The two, as we have seen,insurethatcompetitive equilibrium necessarily is optimal; the thirdinsuresthat everyoptimalstate is the competitive equilibrium corresponding some distribution income.4 to of The first and thirdconditions are interrelated; indeed,nonincreasing returns plus some additionalconditions restrictive a moderneconomy not in implythe existence a competitive of equilibrium, implythatthere i.e., willbe someset of priceswhichwillclearall markets.5 The conceptof marketability somewhat is broaderthan the traditional divergence betweenprivateand social costs and benefits The latterconceptrefers cases in whichthe organization the market to of does not requirean individualto pay forcosts that he imposeson othersas the resultof his actionsor does not permit him to receive compensation benefits confers themedicalfield, obvious for he In the exampleis the spread of communicable diseases An individualwho failsto be immunized onlyriskshis ownhealth, disutility not a which he presumably has weighedagainstthe utility avoidingthe proceof In dure,but also thatof others an ideal pricesystem, there wouldbe a price whichhe would have to pay to anyonewhose healthis endangered,a price sufficiently so thatthe otherswould feelcompenhigh sated; or,alternatively, therewouldbe a pricewhich wouldbe paid to himby othersto inducehimto undergo immunization the procedure 'There are further minor conditions,for which see Koopmans [18, pp 50-551 For a more precisestatement of the existenceconditions, Koopmans [18, pp 56-60] see or Debreu [12, Ch 5] ARROW: UNCERTAINTY AND MEDICAL CARE 945 Eilther system wouldlead to an optimal the state,though distributional implications would be different is, of course,not hard to see that It such pricesystems could not,in fact,be practical;to approximate an optimalstate it would be necessary have collective to intervention in the form subsidyor tax or compulsion of By tlle absenceof marketability an actionwhichis identifiable, for technologically possible,and capable of influencing some individual's for or welfare, better forworse,is meantherethe failure theexistof ing market providea meanswhereby servicescan be bothofto the feredand demanded upon payment a price.Nonmarketability of may be due to intrinsic technological of characteristics the productwhich prevent suitableprice from a beingenforced, in the case of comas municable diseases,or it may be due to social or historical controls, such as thoseprohibiting individual an from sellinghimself intoslavery.This distinction in fact,difficult make precise,though is to it is, of obviously importance policy; forthepresent for it purposes, willbe sufficient identify to nonmarketability the observedabsence of with markets The instanceof nonmarketability whichwe shall be mostconwith cernedis thatof risk-bearing relevance risk-bearing medical The of to care seemsobvious; illnessis to a considerable extent unpredictable an phenomenon abilityto shift risksof illnessto others worth The the is a pricewhich manyare willing pay Because of pooling to and of superiorwillingness ability, and others willing bear therisks.Neverare to theless,as we shall see in greaterdetail,a greatmanyrisksare not and indeedthe markets the servicesof risk-coverage covered, for are or poorlydeveloped nonexistent Whythisshouldbe so is explained in moredetailin SectionIV.C below; briefly, is impossible drawup it to insurance policieswhichwill sufficiently distinguish amongrisks,parsince observation the resultswill be incapableof distinof ticularly betweenavoidable and unavoidablerisks,so that incentives guishing to avoidlossesare diluted The optimality theorems discussedabove are usuallypresented in the literature referring to conditions certainty, there as only of but is no difficulty extending in themto thecase of risks, provided addithe tionalservices risk-bearing included of are withothercommodities.6 the However, variety possiblerisksin theworldis reallystaggerof ing The relevantcommodities include,in effect, bets on all possible in occurrences theworldwhich impinge In uponutilities fact, manyof these "commodities," desiredprotection i.e., againstmanyrisks,are 'The theory,in variant forms,seems to have been firstworked out by Allais [2], Arrow [5], and Baudier [7] For further see generalization, Debreu [11] and [12, Ch 71 946 THE AMERICAN ECONOMIC REVIEW simply available.Thus, a wide class of commodities nonmarketnot is and a basic competitive precondition not satisfied.7 is able, There is a stillmoresubtleconsequence theintroduction riskof of bearing considerations When there is uncertainty, information or knowledge becomes commodity a Like other it commodities, has a cost of production and a cost of transmission, so it is naturally not and spread out over the entirepopulationbut concentrated amongthose who can profit mostfrom (These costsmaybe measured timeor it in as disutility well as money.)But the demandforinformation diffiis cult to discussin the rationaltermsusuallyemployed The value of information frequently knownin any meaningful is not sense to the buyer; if, indeed,he knew enoughto measurethe value of information,he would know the information itself.But information, the in form skilledcare,is precisely of whatis beingbouight from mostphysicians,and, indeed,frommostprofessionals The elusivecharacter of information a commodity as suggests thatit departsconsiderably from the usual marketability assumptions about commodities.8 That riskand uncertainty in fact,significant are, elements mediin cal care hardly needsargument willholdthatvirtually thespecial I all features thisindustry, fact,stemfrom prevalence uncerof in the of tainty The nonexistence markets thebearing somerisksin thefirst of for of instancereduceswelfareforthosewho wishto transfer thoserisksto others a certain for price,as wellas forthosewhowouldfind profitit able to take on theriskat suchprices.But it also reduces desireto the render consume or services whichhave riskyconsequences;in technical language,these commodities complementary risk-bearing are to the Conversely, production consumption commodities servand of and ices withlittleriskattachedact as substitutes risk-bearing are for and encouraged market by failure therewithrespectto risk-bearing Thus theobserved commodity pattern be affected thenonexistence will by of other markets ' It should also be remarkedthat in the presence of uncertainty, indivisibilities that are small to create little difficulty the existenceand viability of competitive sufficiently for equilibriummay nevertheless give rise to a considerablerange of increasingreturnsbecause 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 effect sufficiently is great to createserious obstacles to the existenceand viability of competitiveequilibriumin practice One form of productionof information research.Not only does the product have is unconventional aspects as a commodity, but it is also subject to increasing in returns use, since new ideas, once developed,can be used over and over withoutbeing consumed,and to difficulties marketcontrol,since the cost of reproduction usually much less than of is that of production.Hence, it is not surprising that a free enterprise economy will tend to underinvest research;see Nelson [211 and Arrow [4] in ARROW: UNCERTAINTY AND MEDICAL CARE 947 The failure one or moreof the competitive of has preconditions as its most immediate and obvious consequence reduction welfare in a below that obtainablefromexisting in resources and technology, the senseof a failure reachan optimal to statein thesenseof Pareto.But morecan be said I proposeheretheviewthat,whenthemarket fails to achievean optimal state,society will,to someextent least,recogat nize thegap, and nonmarket social institutions ariseattempting will to this bridgeit.9Certainly processis notnecessarily conscious;noris it in successful approaching uniformly morecloselyto optimality when the entirerangeof consequences considered has always been a is It favorite of to activity economists pointout thatactionswhichon their face achieve a desirablegoal may have less obvious consequences over time,whichmorethanoffset original particularly the gains But it is contended here that the special structural characteristics of themedical-care market largely are to attempts overcome lack of the due of optimality to the nonmarketability thebearing suitablerisks of and the imperfect of These compensatory marketability information institutional changes,withsome reinforcement usual profit from motives, largelyexplain the observednoncompetitive behaviorof the medical-care market, behaviorwhich,in itself,interferes with optiThe socialadjustment towards mality thus in optimality putsobstacles its ownpath The doctrine that societywill seek to achieve optimality nonby marketmeans if it cannotachieve themin the marketis not novel the at Certainly, government, least in its economic is activities, usually or implicitly explicitly held to function theagency as whichsubstitutes I for the market'sfailure.'0 am arguinghere that in some circumstancesothersocial institutions step intothe optimality will gap, and thatthe medical-care withits variety special institutions, of industry, some modern, some ancient, this exemplifies tendency It may be usefulto remark herethata good part of thepreference for redistribution expressedin government taxationand expenditure can be reinterpreted desireforinsurpoliciesand privatecharity as ance It is noteworthy virtually that nowhere therea system subis of sidies thathas as its aim simplyan equalization income.The subof sidiesor other governmental go to thosewhoare disadvantaged help in life by eventsthe incidence whichis popularly of regarded unpreas 'An important current situation in which normal market relations have had to be greatly modifiedin 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 functions government of 948 THE AMERICAN ECONOMIC REVIEW dictable: the blind,dependent the children, medically indigent Thus, in optimality, a contextwhichincludesrisk-bearing, includesmuch thatappears to be motivated distributional by value judgments when lookedat in a narrower context." This methodological background givesriseto the following plan for thispaper SectionII is a catalogueof stylized generalizations about themedical-care market which differentiatefrom usual commodit the ity markets SectionIII the behaviorof the marketis compared In withthatof the competitive modelwhichdisregards factof uncerthe In tainty SectionIV, themedical-care market compared, is bothas to behaviorand as to preconditions, withthe ideal competitive market thattakesaccountof uncertainty; attempt be made to demonan will stratethat the characteristics in outlined SectionII can be explained either theresult deviations as of from competitive the or preconditions as attempts compensate otherinstitutions thesefailures to by for The is discussion not designed be definitive, provocative particuto In but lar, I have been charyabout drawing policyinferences; a considerto able extent, they depend on further for research, whichthe present paper is intended providea framework to II A Surveyof theSpecial Characteristics the of Medical-CareMarket'2 This section willlist selectively somecharacteristics medicalcare of which it distinguish from usual commodity economics the of textbooks The listis notexhaustive, it is notclaimedthatthecharacteristics and listedare individually uniqueto thismarket But,takentogether, they establish specialplace formedicalcare in economic a analysis A The Nature of Demand The most obvious distinguishing characteristics an individual's of demandformedicalservicesis that it is not steadyin originas, for example,forfoodor clothing, irregular but and unpredictable Medical services, apart from preventive services, afford satisfaction onlyin the eventof illness, departure a from normal the stateof affairs is It hard, indeed,to thinkof anothercommodity significance the of in average budgetof whichthis is true.A portion legal services, of devotedto defense criminal in trialsor to lawsuits, might in thiscatefall gorybut theincidence surely is verymuchlower(and, of course, there 'Since writingthe above, I findthat Buchanan and Tullock [10, Ch 13] have argued that all redistribution be interpreted "incomeinsurance." can as 12For an illuminating survey to which I am much indebted,see S Mushkin [20] ARROW: UNCERTAINTY AND MEDICAL CARE 949 are, in fact, stronginstitutional similarities betweenthe legal and medical-care markets.)'3 In addition,the demandformedicalservicesis associated,witha considerable probability, an assaulton personal Thereis with integrity some riskof deathand a moreconsiderable of impairment full of risk functioning particular, In thereis a majorpotential loss or reducfor tionof earning ability The risksare notby themselves unique; foodis also a necessity, avoidanceofdeprivation foodcan be guaranteed but of withsufficient income, wherethe same cannotbe said of avoidanceof illness.Illness is, thus,not onlyriskybut a costlyriskin itself, apart from cost of medicalcare the B ExpectedBehaviorof thePhysician It is clear from everyday observation thatthebehavior expected of sellersof medicalcare is different from thatof businessmenin general These expectations relevant are because medicalcare belongsto the category commodities whichtheproduct of for and the activity of production identical all suchcases,thecustomer are In cannottestthe productbeforeconsuming and thereis an element trustin the it, of relation.' But the ethically understood restrictions the activities on of a physician muchmoreseverethanon thoseof,say, a barber.His are behavioris supposedto be governed a concernforthe customer's by welfarewhichwould not be expectedof a salesman.In Talcott Parsons's terms, thereis a "collectivity-orientation," distinguishes which medicine and otherprofessions frombusiness, whereself-interest on the part of participants the acceptednorm.'5 is A fewillustrations indicatethedegreeof difference will between the behavior expectedof physicians and thatexpected the typicalbusiof nessman.18 Advertising overtprice competition virtually (1) and are eliminated among physicians.(2) Advice givenby physicians to as further treatment himself othersis supposedto be completely or by "In governmentaldemand, military power is an example of a service used only irregularlyand unpredictably.Here too, special institutionaland professionalrelations have emerged, thoughthe precisesocial structure different reasonsthat are not hard is for to analyze " Even with material commodities,testingis never so adequate that all elements of implicittrustcan be eliminated.Of course,over the long run, experience with the quality of productof a given sellerprovidesa check on the possibility trust of 15See [22, p 463] The whole of [22, Ch 101 is a most illuminating analysis of the social role of medical practice; though Parsons' interest lies in different areas frommine, I mustacknowledgehere my indebtedness his work to 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 is divorcedfrom self-interest It is at least claimedthattreatment (3) dictated the objective needsof thecase and notlimited financial by by considerations."7 Whiletheethicalcompulsion surely as absolute is not in factas it is in theory, can hardly we supposethatit has no influence over resource allocationin thisarea Charity treatment one form in or another does existbecause of thistradition abouthumanrights adeto quate medicalcare.'8 (4) The physician reliedon as an expertin is to certifying theexistence illnesses of and injuries variouslegaland for otherpurposes.It is sociallyexpected thathis concern thecorrect for of conveying information whenappropriate, will, outweigh desire his to please his customers."g Departurefromthe profit motiveis strikingly manifested the by overproprietary of overwhelming predominance nonprofit hospitals.20 The hospitalper se offers servicesnot too different fromthose of a and it is certainly obviousthattheprofit not motive notlead will hotel, The explanation to a moreefficient on supply maylie either thesupply side or on thatof demand The simplest is explanation thatpublicand privatesubsidies decreasethecostto thepatient nonprofit in hospitals A secondpossibility that the associationof profit-making the is with on supplyof medical servicesarouses suspicionand antagonism the so part of patientsand referring physicians, theydo prefer nonprofit Eitherexplanation a on institutions implies preference thepartofsome donors or patients, group,whether against the profit motivein the of hospitalservices.2' supply 1T The belief that the ethics of medicinedemands treatment independent the patient's of ability to pay is strongly ingrained.Such a perceptiveobserveras 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 unprecedented phenomenon.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 confidently assume that price and income 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 is also needs study.) For this purpose,some disaggregation needed; differences hospital in care which are essentiallymattersof comfortshould, in the above view, be much more responsive incomethan,e.g., drugs to "9 This role is enhanced in a socialistsociety,where the state itselfis activelyconcerned with illnessin relationto work; see Field [14, Ch 91 ' About per cent of beds were in proprietary hospitalsin 1958, against 30 per cent in and the remainderin federal,state, and local hospitals; see [26, voluntary nonprofit, Chart 4-2, p 601 " C R Rorem has pointed out to me some further factorsin this analysis (1) Given the social intentionof helping all patientswithout regard to immediateability to pay, economies of scale would dictate a predominanceof community-sponsored hospitals (2) ARROW: UNCERTAINTY AND MEDICAL CARE 959 insurance policies that mightconceivably written, are in fact be but not Whetherthese potential commodities are nonmarketable, or, merely because of some imperfection the market, not actually in are marketed, a somewhat is finepoint To recall what has alreadybeen said in SectionI, thereare two kindsof risksinvolvedin medicalcare: the riskof becoming and ill, the risk of total or incomplete delayedrecovery or The loss due to illnessis onlypartially cost of medicalcare It also consists disthe of comfort loss ofproductive and timeduring and,in mole serious illness, cases, death or prolonged From the deprivation normalfunction of pointof view of thewelfare economics uncertainty, losses are of both risksagainstwhichindividuals wouldlike to insure.The nonexistence of suitableinsurance policiesforeither a riskimplies loss of welfare B The Theory Ideal Insuran of cc In this section,the basic principles an optimalregimeforriskof bearingwill be presented For illustration, reference will usually be made to thecase of insurance againstcostin medicalcare The principles are equally applicable to any of the risks There is no single sourceto whichthe readercan be easily referred, I though thinkthe principles at least reasonably are well understood As a basis fortheanalysis, assumption madethateach individthe is ual acts so as to maximize expected the value of a utility If function we thinkof utility attachedto income,thenthe costs of medical as care act as a random deduction from income, it is theexpected this and value of theutility income of after medicalcoststhatwe are concerned with (Income aftermnedical costs is the abilityto spend moneyon otherobjectswhichgivesatisfaction presuppose We thatillnessis not a sourceof satisfaction itself; to the extentthatit is a sourceof in the dissatisfaction, illnessshould enterinto the utilityfunction a as separatevariable.) The expected-utility due to hypothesis, originally Daniel Bernoulli (1738), is plausibleand is themostanalytically manageable of all hypotheses thathave been proposed explainbehavior to underuncertainty any case, the resultsto follow In probablywould not be significantly affected moving another by to modeof analysis It is further assumedthatindividuals normally are risk-averters In thismeansthattheyhave a diminishing utility terms, marginal utility of income This assumption may reasonably takento hold formost be of the sKgnificant affairs lifefora majority people,but thepresof of ence of gamibling provides somedifficulty thefullapplication this in of view It follows from assumption riskaversionthatif an indithe of vidual is givena choicebetween probability a distribution income, of with givenmean n,and thecertainty theincome he wouldprefer a of m, 960 THE AMERICAN ECONOMIC REVIEW the latter.Suppose,therefore, agency,a large insurance an company plan,or thegovernment, standsreadyto offer insurance againstmedical fairbasis; thatis, if the costsof medicalcare costs on an actuarially are a randomvariablewithmean n, the company will chargea premiumn, and agree to indemnify individual all medicalcosts the for Underthesecircumstances, individual the to will certainly prefer take out a policyand willhave a welfare gain thereby Will thisbe a social gain? Obviously yes,if the insurance agentis no suffering social loss Under the assuimption medicalriskson that different individualsare basically independent, poolingof them the reducestheriskinvolved theinsurer relatively to to smallproportions In the limit, welfare the loss,even assuming riskaversion thepart on oftheinsurer, wouldvanishand there a netsocialgainwhich is maybe of quite substantial In magnitude fact,of course,thepoolingof risks does not go to the limit; thereis only a finite number themand of theremay be some interdependence amongthe risksdue to epidemics and thelike But thena premium, perhapsslightly above theactuarial to level,wouldbe sufficient offset welfare this loss Fromthepointof viewof the individual, sincehe has a strict preference theactuarifor ally fairpolicy over assumingthe riskshimself, will still have a he for preference an actuarially unfairpolicy,provided, course,that of it is not too unfair In additionto a residualdegreeof riskaversionby insurers, there are otherreasonsforthe loadingof the premium (i.e., an excess of over the actuarialvalue) Insuranceinvolvesadministrative premium costs.Also,becauseof theirregularity payments of there likely be is to a cost of capitaltiedup Suppose,to take a simplecase, theinsurance companyis not willingto sell any insurance policythata consumer wants but will chargea fixed-percentage loadingabove the actuarial value forits premium Then it can be shownthatthe mostpreferred policy fromthe point of view of an individualis a coveragewitha deductible amount;thatis, theinsurance policyprovides100 per cent coverageforall medicalcosts in excess of some fixed-dollar limit.If, the has some degreeof riskaversion, company its however, insurance loadingmay also dependon the degreeof uncertainty the risk.In of that case, the Pareto optimalpolicywill involvesome element coof insurance, i.e., the coverageforcosts over the minimum limitwill be some fraction less than 100 per cent (for proofsof thesestatements, see Appendix) These results can also be appliedto thehypothetical conceptof insuranceagainst failureto recoverfrom illness.For simplicity, us let assumethatthecostof failure recover regarded to is purely a money as cost,eithersimplyproductive opportunities foregone moregeneror, ARROW: UNCERTAINTY AND MEDICAL CARE 961 ally, the moneyequivalentof all dissatisfactions Suppose further that,giventhata personis ill, the expectedvalue of medicalcare is greater thanits cost; thatis, theexpected money value attributable to recovery withmedicalhelp is greater thanresources devotedto medical help However,the recovery, though the averagebeneficial, on is uncertain;in the absence of insurance risk-averter well prefer a may not to take a chance on further impoverishment buyingmedical by care A suitableinsurance policywould,however, mean thathe paid nothing he doesn'tbenefit; if sincethe expected value is greater than thecost,there wouldbe a netsocialgain.35 C Problemsof Insurance The moralhazard The welfare case forinsurance policiesof all sortsis overwhelming followsthat the government It shouldundertake insurance thosecases wherethismarket, whatever in for reason, has failedto emerge thereare a number significant of Nevertheless, practicallimitations theuse of insurance is important underon It to stand them,thoughI not believethat theyalter the case forthe of creation a muchwiderclass of insurance policiesthannowexists One of the limits whichhas been muchstressed insurance in literatureis theeffect insurance incentives of on Whatis desired thecase in of insurance thatthe eventagainstwhichinsurance takenbe out is is of the control theindividual of Unfortunately, real lifethisseparain tioncan neverbe made perfectly outbreak firein one's house of The or businessmay be largelyuncontrollable the individual, the by but of probability fire somewhat is influenced carelessness, ofcourse by and arson is a possibility, an extreme if one Similarly, medicalpolicies in the cost of medicalcare is not completely determined the illness by suffered the individual by but dependson the choiceof a doctorand his willingness use medicalservices.It is frequently to observed that widespread medicalinsurance increasesthe demandformedicalcare Coinsurance have beenintroduced manymajormedical provisions into policiesto meetthiscontingency wellas theriskaversion theinas of surancecompanies To some extent professional the relationship between and physician patient limits normal the hazardin variousforms medicalinsurance of to By certifying the necessity giventreatment the lack thereof, of or the physicianacts as a controlling agent on behalfof the insurance companies Needless to say, it is a far from perfect check; the physiciansthemselves not underany control are and it maybe convenient or forthem pleasingto their patients prescribe to moreexpensive medi"It is a popular belief thatthe Chinese, one time, at paid their physicians whenwell but not whensick 962 THE AMERICAN ECONOMIC REVIEW cation,privatenurses,more frequent and treatments, othermarginal variations care It is probably of and truethathospitalization surgery are moreunderthe casual inspection others of thanis general practice and therefore subjectto moralhazard; thismaybe one reasonwhy less insurance policiesin thosefields have been morewidespread Alternative methods insurance It that payment is interesting of no less thanthree different methods coverage thecostsofmedical of of carehavearisen:prepayment, to indemnities according a fixed schedule, and insuranceagainst costs,whatever theymay be In prepayment plans, insurance effect paid in kind-that is, directly medical in is in services The other two forms bothinvolvecash payments thebeneto ficiary, in the one case theamounts be paid involving medical but a to contingency fixed advance,whilein theothertheinsurance are in carrierpays all the costs,whatever theymay be, subject,of course,to provisions deductibles coinsurance like and In hypothetically perfectmarketsthese threeformsof insurance wouldbe equivalent The indemnities stipulated would,in fact,equal the market priceof the services, thatvalue to theinsured so wouldbe the same if he wereto be paid the fixedsum or the market priceor weregiventhe servicesfree.In fact,of course,insurance againstfull costsand prepayment plans bothoffer insurance againstuncertainty as to the priceof medicalservices, addition uncertainty in to about their needs Further, theirmode of compensation the physician, by to prepayment plans are inevitably boundup withclosedpanels so thatthe freedom choiceof thephysician thepatient less thanit would of by is be undera scheme morestrictly confined theprovision insurance to of These remarks are tentative, and the questionof coexistence the of different schemesshouldbe a fruitful subjectforinvestigation Third-party controlover payments The moralhazard in physicians' controlnotedin paragraph1 above showsitselfin thoseinsuranceschemes wherethephysician thegreatest has control, namely, major medical insurance Here therehas been a markedrise in exover time.In prepayment penditures plans, wherethe insurance and medicalserviceare suppliedby the same group,the incentive keep to medicalcosts to a minimum strongest plans of the Blue Cross is In group, there has developed conflict interest a of between insurance the carrierand the medical-service in supplier, this case particularly the hospital The need forthird-party control reinforced another is by aspect of the moralhazard Insuranceremoves incentive the part of inthe on and dividuals, patients, physicians shoparoundforbetter to pricesfor hospitalization surgical and care The market forces, therefore, to tend be replaced direct by institutional control ARROW: UNCERTAINTY AND MEDICAL CARE 963 Administrative in costs The pure theory insurance of sketched SectionB above omitsone veryimportant the consideration: costs of operating insurance an company Thereare severaltypesof operating costs,but one of the mostimportant categories includescommissions and acquisition costs,sellingcostsin usual economic Not terminology only does this mean that insurance policiesmustbe sold forconsiderablymore than theiractuarialvalue, but it also means thereis a greatdifferential amongdifferent typesof insurance is verystriking It to observe thatamonghealthinsurance policiesof insurance companies in 1958,expenses one sortor another of constitute percentoftotal 51.6 premium incomeforindividual policies, and only9.5 per centforgroup policies[26, Table 14-1,p 272] This striking differential wouldseem toimply enormous economies scalein theprovision insurance, of of quite apart from coverageof the risksthemselves the Obviously, thisprovides a verystrong argument widespread for plans,including, parin ticular, compulsory ones Predictability insurance and Clearly, from risk-aversion the point of view,insurance morevaluable,the greater uncertainty the is the in risk being insuredagainst.This is usually used as an argument for putting greater emphasis insurance on againsthospitalization surand gerythan otherforms medicalcare The empirical of assumption has been challenged W Anderson others[3, pp 53-54],whoasby and sertedthat out-of-hospital expenseswere equally as unpredictable as in-hospital costs.What was in factshownwas thatthe probability of costsexceeding $200 is aboutthesame forthetwocategories, this but is not,ofcourse, correct a measure predictability, a quickglance of and at the supporting evidenceshowsthatin relation the averagecost to thevariability muchlowerforordinary is medicalexpenses Thus, for the cityof Birmingham, mean expenditure surgery the on was $7, as opposed to $20 for othermedical expenses,but of those who paid for something surgery averagebillwas $99, as against$36 forthose the withsome ordinary medicalcost Eighty-two cent of thoseinterper viewedhad no surgery, only20 per centhad no ordinary and medical expenses[3, Tables A-13,A-18, and A-19 on pp 72, 77, and 79, respectively] The issue of predictability has bearingon the merits insuralso of ance againstchronic illnessor maternity a lifetime On insurance basis, insurance illnessmakessense,sincethisis bothhighly againstchronic in and highlysignificant costs Amongpeople who alunpredictable whichreliably readyhave chronic illness,or symptoms indicateit, inin senseis probably surance thestrict pointless insurance Poolingof unequal risks.Hypothetically, requiresfor a its fullsocial benefit maximum of Those possiblediscrimination risks 964 THE AMERICAN ECONOMIC REVIEW in groupsof higher incidences illnessshouldpay higher of premiums In fact,however, thereis a tendency equalize,rather to thanto differentiate, premniums, especially theBlue Crossand similar in widespread schemes.This constitutes, effect, redistribution incomefrom in a of thosewitha low propensity illnessto thosewitha highpropensity to The equalization, course, of if could not in factbe carriedthrough the market weregenuinely insurcompetitive Underthosecircumsances, ance plans could arise which chargedlower premiums preferred to risksand drawthemoff, leavingtheplan whichdoes notdiscriminate amongrisks with onlyan adverseselection them of As we have alreadyseen in the case of incomeredistribution, some of thismaybe thought as insurance of witha longer timeperspeCtive If a plan guarantees everybody premium to a thatcorresponds total to experience not to experience it might segregated smaller but as be by subgroups, everybody in effect, is, insured againsta changein his basic stateof healthwhich wouldlead to a reclassification corresponds This precisely the use of a level premium life insurance to in insteadof a premium varying age, as wouldbe the case forterminsurance by Gaps and coverage maybriefly We notethat, any rateto date, at insurances againstthecostof medicalcare are farfrom Ceruniversal tain groups-the unemployed, institutionalized, the aged-are the and almostcompletely uncovered totalexpenditures, Of between one-fifth and one-fourth coveredby insurance shouldbe noted, are It however, that over half of all hospitalexpensesand about 35 per cent of the medicalpayments thosewithbillsof $1,000 a yearand over,are inof cluded [26, p 376] Thus, the coverageon themorevariablepartsof medicalexpenditure somewhat is better thantheover-all figures would indicate, it mustbe assumedthattheinsurance but mechanism still is the fullcoverageof whichit is capable veryfar fromachieving D Uncertainty Effects Treatment of of There are reallytwo major aspects of uncertainty an indifor vidualalreadysuffering an illness is uncertain from He abouttheeffecof tiveness medicaltreatment, his uncertainty be quite differand may ent from thatof his physician, based onithepresumably quitedifferent medicalknowledges Ideal insurance.This will necessarily involveinsurance against a failure benefit to from medicalcare,whether through recovery, relief of pain,or arrest further of deterioration form One wouldbe a system in whichthepayment thephysician made in accordance to is withthe Since thiswouldinvolvetransferring risksfrom degreeof benefit the the patientto the physician, who might certainly have an aversionto bearingthem,thereis roomforinsurance carriers pool the risks, to ARROW: UNCERTAINTY AND MEDICAL CARE 965 eitherby contract withphysicians by contract or withthe potential patients Underideal insurance, medical carewillalwaysbe undertaken in any case in whichthe expected utility, takingaccountof theprobabilities,exceeds the expectedmedicalcost This prescription would lead to an economicoptimum we thinkof the failureto recover If mainlyin termsof lost working time,thenthispolicywould,in fact, maxi-mize economic welfare ordinarily as measured The concepts trustand delegation theabsenceof ideal inof In surance,therearise institutions some sort of substitute whichoffer guarantees Underideal insurance patientwouldactuallyhave no the concernwith the informational and the inequalitybetweenhimself physician, since he would onlybe payingby resultsanyway, and his utility position In wouldin factbe thoroughly guaranteed its absence he waants have some guarantee to is thatat least the physician using his knowledge thebest advantage.This leads to thesetting of a to up of relationship trustand confidence, whichthe physicianhas a one social obligation live up to Sincethepatient to does not,at leastin his belief,knowas muchas the physician, cannotcompletely he enforce standardsof care In part, he replacesdirectobservation generby alized beliefin theability thephysician." putit another of To way,the social obligation best practiceis part of the commodity phyfor the siciansells,even though is a partthatis not subjectto thorough it inspection the buyer by One consequence suchtrust of relations thatthephysician is cannot or at leastappearto act,as ifhe is maximizing income every his at act, moment time.As a signalto the buyerof his intentions act as of to in thorouglhly the buyer'sbehalfas possible,the physician avoids the obviousstigmata profit-maximizing of Purelyarms-length bargaining behaviorwould be incompatible, logically, not but surelypsychologically,withthe trustrelations From thesespecial relations come the variousforms ethicalbehavior of discussedabove, and so also, I suggest,the relative of un-importanceprofit-makinghospitals in The very word,"profit," a signalthatdeniesthetrust is relations PriCediscrimination its extreme, treatment theindigent, and free for also follow the obligation thephysician understood be first If of is to of all to the welfare thepatient, of thenin particular takesprecedit ence over financial difficulties As a secondconsequence informational of inequality between physiclan and patientand the lack of insurance a suitabletype,the of muchof his freedom choice patientmustdelegateto thephysician of " Francis Bator points out to me that some protectioncan be achieved,at a price,by securingadditionalopinions 966 THE AMERICAN ECONOMIC REVIEW He does not have the knowledge make decisionson treatment, to referral, hospitalization justify or To thisdelegation, physician the finds himself somewhat limited, just as any agentwouldin similarcircumstances.The safestcourseto take to avoid notbeinga trueagentis to give the sociallyprescribed "best" treatment the day Compromise of in quality, even forthepurposeof savingthepatient money, to risk is an imputation failure liveup tothesocialbond of to The specialtrust relation physicians of (and allied occuptions, such as priests) extendsto thirdpartiesso that the certifications phyof siciansas to illnessand injury acceptedas especially are reliable(see SectionIJ.B above) The social value to all concerned such preof sumptively reliable sources informationobvious of is Notice the generalprinciple here.Because thereare barriers the to information and because thereis no marketin whichthe risks flow involved can be insured, coordination purchase of and sales musttalke place through convergent expectations, theseare greatly but assisted by havingclear and prominent signals,and these,in turn,forcepatternsof behaviorwhichare not in themselves logical necessities for optimality.37 Licensing and educational standards Delegationand trust the are social institutions to designed obviatetheproblem informational of inThe general equality uncertainty abouttneprospects medicaltreatof mentis sociallyhandledby rigidentryrequirements These are designedto reducethe uncertainty the mindof the consumer to in as thequalityof product insofar thisis possible.38 think as I thisexplanation,whichis perhapsthe naive one, is muchmoretenablethanany idea of a monopoly seekingto increaseincomes doubtrestriction No on entry desirablefrom pointof viewof theexisting is the physicians, but the public pressureneeded to achieve the restriction mustcome from deepercauses The social demandforguaranteed qualitycan be metin morethan one way,however least three At attitudes be takenby thestateor can othersocial institutions towardentryinto an occupationor toward the production commodities general;examples all threetypes of in of exist (1) The occupation can be licensed, nonqualified entrants being simplyexcluded.The licensing may be more complexthan it is in medicine;individuals could be licensedforsome,but not all, medical for activities, example.Indeed,thepresent all-or-none approachcould "The situation is very reminiscent the crucial role of the focal point in Schelling's of theory of tacit games, in which two parties have to fin-d common course of action a 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 performance medical technology as changes ARROW: UNCERTAINTY AND MEDICAL CARE 967 be criticized beinginsufficient regardto complicated as with specialist as treatment, well as excessivewith regardto minormedicalskills Gradedlicensing be may,however, muchharderto enforce Controls could be exercised analogousto thoseforfoods; theycan be excluded as beingdangerous, theycan be permitted animalsbut not for or for humans.(2) The state or otheragrency certify label, without or can compulsory is exclusion The category Certified of Psychologist now underactivediscussion;cannedgoodsare graded.Certification be can done by nongovernmental examinaagencies,as in the medical-board tionsforspecialists Nothing all maybe done; consumers (3) at make their ownchoices The choiceamongthesealternatives anygivencase depends the in on degreeof difficulty consumers have in making choiceunaided, the and on theconsequences errors judgment its general of of It the social conthatthelaisscz-faire sensus, solution medicine intolerable clearly, for is The certification proposalneverseemsto have beendiscussed seriously It is beyond scope of thispaperto discusstheseproposals detail the in I wishsimply pointout thattheyshouldbe judgedin terms the to of abilityto relievetheuncertainty thepatient regard thequality of in to of thecommodkty is purchasing, thatentry he and restrictions the are of to consequences an apparentinability devisea system whichthe in risksof gaps in medicalknowledge and skill are borneprimarily by thepatient, thephysician not Postscript I wish to repeat here what has been suggestedabove in several places: that the failureof the marketto insureagainstuncertainties has createdmanysocial institutions whichtheusual assumptions in of the market to someextent are contradicted medicalprofession The is only one example,though manyrespects extreme in an one All professions sharesome of the same properties The eConomic importance of personaland especially family relationships, though declining, by is no meanstrivialin the mostadvancedeconomies;it is based on nonmarket relations thatcreateguarantees behavior of which wouldotherwise be afflicted excessive with uncertainty Many otherexamples can be given.The logicand limitations ideal competitive of behavior under uncertainty forceus to recognize incomplete the description reality of supplied theimpersonal by pricesystem REFERENCES A A ALCEIAN, K J.ARROW, AND WV M CAPRON, An EconomicAnalysis of theMarketforScientists and Engineers, RAND RM-2190-RC.Santa Monica 1958 THE AMERICAN ECONOMIC REVIEW 968 M ALLAIS, "Ge'neralisation des theories de l'6quilibre economique general et du rendement social au cas du risque,"in CentreNationalde la Recherche Scientifique, Econometrie, Paris 1953,pp 1-20 WV ANDERSON AND STAFF OF THE NATIONAL OPINION RESEARCH CENTER, Voluntary Health Insurancein Two Cities Cambridge, Mass 1957 K J ARROW, "EIconomic Welfare and the Allocationof Resourcesfor Invention,"in Nat Bur Econ Research,The Role and Directionof InventiveActivity:Economic and Social Factors, Princeton1962, pp 609-25 , "Les role des valeursboursieres pour la repartition meilleure la des risques,"in CentreNational de la RechercheScientifique, Economietrie, Paris 1953,pp 41-46 F M BATOR, "The Anatomyof Market Failure,"Quart Jour.Econ Aug 1958, 72, 351-79 E BAUDIER, "L'introduction tempsdans la theorie l'equilibre du de general,"Les Cahziers Economiques, Dec 1959,9-16 W J BAUMOL, WelfareEconomicsazd the Theoryof the State Cambridge, MIass.1952 K BORCT-I, "The SafetyLoading of ReinsurancePremiums," SkandinaviskAktuariehdskrift, pp 163-84 1960, 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 Positive Economics, Chicago1953,pp 3-43 16 AND S S KuZNETS, Income from Independent Professional 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 Welfare Oxford 1950 Economics 20 SELMA MUSHKIN, "Towards a Definition Health Economics," of Public Health Reports, 1958,73, 785-93 21 R R NELSON, "The Simple Economicsof Basic Scientific Research," Pol Econt.,June1959,67, 297-306 Jour 22 T PARSONS, The Social System Glencoe1951 23 M J PECK AND F M SCHERER, The TVeaponzs Process: An Acquisition Economic Analysis Div of Research, Graduate School of Business, HarvardUniversity, Boston 1962 ARROW: UNCERTAINTY AND MEDICAL CARE 969 24 T C SCHELLING, T he Strategy Conflict of Cambridge, Mass 1960 25 A K SHAPIRO, "A Contribution a Historyof the Placebo Effect," to BehavioralScience,1960,5, 109-35 26 H M SOMERS AND A R SOMERS, Doctors,Patients,and IHealthInsurance The Brookings Institution, Washington 1961 27 C L STEVENSON, Ethicsand Lan-guage.New Haven 1945 28 U S DEPARTMENT OF HEALTH, EDUCATION AND WELFARE, Physicians fora GrowZOing Public Health ServicePublication No 709, Oct Amterica, 1959 APPENDIX On OptimalInsurancePolicies The two propositions about the natureof optimalinsurancepoliciesassertedin SectionIV.B above willbe provedhere Proposition If an insurancecompanyis willingto offer any insurance policyagainst loss desiredby the buyerat a premium whichdependsonly on the policy's actuarial value, then the policy chosen by a risk-averting buyerwill take the form 100 per cent coverageabove a deductibleminiof mum Note: The premiumwill, in general,exceed the actuarial value; it is onlyrequiredthat two policieswiththesame actuarialvalue willbe offered by the companyforthe same premium Proof:Let W be the initialwealthof the individual, his loss, a random X variable, I(X) the amount of insurancepaid if loss X occurs,P the premium,and Y(X) the wealth of the individualafterpaying the premium, incurring loss,and receiving insurance the the benefit (1) Y(X) =W-P-X + I(X) The individualvalues alternativepolicies by the expectedutilityof his finalwealthposition,Y(X) Let U(y) be the utilityof finalwealth,y; then his aim is to maximize, (2) Et U[Y(X)]}, wherethesymbol, denotesmathematical E, expectation An insurance paymentis necessarily nonnegative, the insurance so policy mustsatisfy condition, the (3) I(X) > forall X If a policyis optimal,it must in particularbe betterin the sense of the criterion (2), than any otherpolicy with the same actuarial expectation, E[I(X)I Considera policythat pays some positiveamountof insuranceat one level of loss, say X,, but whichpermitsthe finalwealthat some other loss level, say X2, to be lower than that corresponding X, Then, it is to intuitively obvious that a risk-averter would prefer alternativepolicy an withthe same actuarialvalue whichwould offer slightly less protection for losses in the neighborhood X, and slightly of higher protection thosein for the neighborhood X2, sinceriskaversionimpliesthatthe marginal of utility 970 THE AMERICAN ECONOMIC REVIEW of Y(X) is greater whenY(X) is smaller:hence,theoriginal policycannotbe optimal To provethisformally, 11(X) be theoriginal let policy,with 11(X)>0 and in Yl(Xl) > Y2(X2),whereY1(X) is defined termsof 11(X) by (I) Choose a sufficiently smallso that, (4) 11(X) > 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; thiscan be provedto be trueforthe optimalpolicy,and therefore need we only considerthis case.) Let 7r- be the probabilitythat the loss, X, lies in the interval (X1, 72 the probability that X lies in the interval(X2, X2+3) From (4) X1+6), and (5) we can choose E>0 and sufficiently small so that, (6) I1(X) -72E > for Xi < X < X1 + a, (7) Y1(X') + 7riE Y1(X) < - 7r2E for X2< X' < X2+ 3, X1 < X < X1 + S Now define newinsurance a policy,12(X), whichis the same as 11(X) except that it is smallerby 72E in the intervalfromX1 to X1+8 and largerby 7r,,E in the intervalfromX2 to X2+8 From (6), 12(X) >0 everywhere, that so (3) is satisfied willshowthat E[11(X)] = E[I2(X)] and thatI2(X) yields We the higherexpectedutility, that I1(X) is not optimal so Note that I2(X) - 1(X) equals -7i-2E forX1< X < X1+6, 7rWE X2 < X for < X2+6, and elsewhere Let ?(X) be the densityof the randomvariable X Then, X 1+8 E [I2(X) - 11(X) ] [I2(X) - 1I(X)]cp(X)dX X1 +1 rx2+8 [I2(X) rXl+8 =(-72E) = - - Y2 J (w2E)7rl + (X)dX + (lriE)r2 = 11(X)]dX (WiE) r X24-S p(X)dX j X2 0, so that the two policieshave the same actuarialvalue 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