Uncertainty and the Welfare Economics of Medical Care
Kenneth J. Arrow
The American Economic Review, Vol. 53, No. 5. (Dec., 1963), pp. 941-973.
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THE AMERICAN
ECONOM C REVIE
VOLUME LIII DECEMBER 1963 NUMBER 5
UNCERTAINTY AND THE WELFARE 

ECONOMICS OF MEDICAL CARE 

I. Introdz~ction:Scope and ~Wethod
This paper is an exploratory and tentative study of the specific
differentia of medical care as the object of normative economics. It
is contended here, on the basis of comparison of obvious characteristics
of the medical-care industry with the norms of welfare economics,
that the special economic problems of medical care can be explained
as adaptations to the existence of uncertainty in the incidence of disease
and in the efficacy of treatment.
It should be noted that the subject is the medical-care industry, not
health. The causal factors in health are many, and the provision of
medical care is only one. Particularly at low levels of income, other
commodities such as nutrition, shelter, clothing, and sanitation may
be much more significant. It is the complex of services that center
about the physician, private and group practice, hospitals, and public
health, which I propose to discuss.
The focus of discussion will be on the way the operation of the
medical-care industry and the efficacy with which it satisfies the needs
of society differ from a norm, if at all. The "norm" that the economist
usually uses for the purposes of such comparisons is the operation
of a competitive model, that is, the flows of services that would be
*The author is professor of economics at Stanford University. I-Ie wishes to express his
thanks for useful comments to F. Eatoi, R. Dorfman, V. Fuchs, Dr. S. Gilson, R. Kessel,
S. Mushkin, and C. R. Rorem. This paper was prepared under the sponsorship of the Ford
Foundation as part of a series of papers on the economics of health, education, and welfare.
942 THE AMERICAN ECONOMIC REVIEW
offered and purchased and the prices that would be paid for them if
each individual in the marliet offered or purchased services at the going
prices as if his decisions had no influence over them, and the going
prices were szch that the amounts of services which were available
equalled the total amounts which other individuals were willing to
purchase, with no imposed restrictions on supply or demand.
The interest in the competitive model stems partly from its presumed
descriptive power and partly from its implications for econoinic
efficiency. In pzrticular, we can state the following well-known proposition
(First Optimality Theorem). If a conipztitive equilibrium
exists at all, and if all commodities relevant to costs or utilities are in
fact priced in the market, then the equilibrium is necessarily optimal
in the folloxx-;vingprecise sense (due to V. Pareto): There is no other
a1lo:atian of resources to services which will inake all participants in
the ~ilarketbetter off.
Both the co:~ditionsof this optimality theorem and the definition of
optimality call for comment. A definition is just a definition. but when
the dcfiniendunz is a word already in common use with highly favorable
connotations, it is clear that me are really trying to be pcrsvasive;
we are implicitly reconlmending the achievement of optimal states.' I t
is reasoilable enough to assert that a change in allocation which makes
all participants better off is one that certajnly should be nade; this is
a value judgment, not a descriptive proposition, but it is a very weak
one. From thjs it follows that it is not desirable to put up with a nonoptimal
allocation. But it does not folloxv that if vx are at an ailocation
which is optimal in the Pareto sense, we should not change to any
other. We cacnot indeed make a change that does not hurt someone;
but we can still desire to change to another allocation if the change
makes enough participants better off and by so much that xT;e feel that
the injury to others is not enough to offset the benefits. Such interpersonal
comparisons are, of course, value judgments. The change,
however, by the previous argument ought to be an optimal state; of
course there are many possible states, each of which is optinlal in the
sense here used.
However, a value judgment on the desirability of each possible new
distribution of benefits and costs corresponding to each possible reallocation
of resources is not. in general, necessary. Judgments about
the distribution can be made separately, in one sense, from those about
allocation if certain conditions are fulfilled. Eefore stating the relevant
proposition, it is necessary to remark that the competitive equilibrium
achieved depends in good measure on the initial distribution of purchasing
power, which consists of ownership of assets and skills that
'This point has been stressed by I. M. D. Little [19, pp. 71-74]. For the concept of a
"persuasive definition," see C. L. Stevenson [27, pp. 210-171.
943ARROW: UNCERTAINTY AND MEDICAL CARE
command a price on the market. A transfer of assets among individuals
will, in general, change the final supplies of goods and services
and the prices paid for them. Thus, a transfer of purchasing power
from the well to the ill will increase the demand for medical services.
This will manifest itself in the short run in an increase in the price of
medical services and in the long run in an increase in the amount sup-
plied.
With this in mind, the following statement can be made (Second
Optimality Theorem) : If there are no increasing returns in production,
and if certain other minor conditions are satisfied, then every optimal
state is a competitive equilibrium corresponding to some initial distribution
of purchasing power. Operationally, the significance of this
proposition is that if the conditions of the two optimality theorens are
satisfied, and if the allocation mechanism in the real world satisfies the
conditions for a competitive model, then social policy can confi~eitself
to steps taken to alter the distribution of purchasing power. For any
given distribution of purchasing power, the market will, under the
assumptions made, achieve a competitive equilibrium which is necessarily
optimal; and any optimal state is a competitive equilibrium corresponding
to some distribution of purchasing power, so that any
desired optimal state can be achieved.
The redistribution of purchasing power among individuals most
simply takes the form of money: taxes and subsidies. The implications
of such a transfer for individual satisfactions are, in general, not
known in advance. But we can assume that society can ex post judge
the distribution of satisfactions and, if deemed unsatisfactory, take
steps to correct it by subsequent transfers. Thus, by successive approximations,
a most preferred social state can be achieved, with resource
allocation being handled by the market and public policy confined
to the redistribution of money income.'
If, on the conti-ary, the actual market differs significantly from the
competitive model, or if the assumptions of the two optimality theorems
are not fulfilled, the separation of allocative and distributional
procedures becomes, in most cases, impo~sible.~
The first step then in the analysis of the medical-care market is the
a The srnaration between allocation and distribution even under the above assumptions
has :,.los;:d os.er prob1e:n; in the e:;ecutior, of 3r1y desired redistribution policy; in przctice,
it is virtually impossible to find a set of taxes and subsidies that will not have an adverse
effect on the achievement of an optimal state. But this discussion would take us
even further afield than we have already gone.
'The basic theorems of welfare economics alluded to so briefly above have been the
subject of voluminous literature, but no thoroughly satisfactory statement covering both
the theorems themselves and the significance of exceptions to them exists. The pcsitive
assertions of welfare economics and their relation to the theory of competitive equilibrium
are admirably covered in ICoopmans [IS]. The best summary of the various ways in
which the theorems can fail to hold is probably Bator's [61.
9.44 THE AMERICAN ECONOMIC REVIEW
comparison between the actual market and the competitive model. The
methodology of this comparison has been a recurrent subject of controversy
in economics for over a century. Recently, M. Friedman [IS]
has vigorously argued that the competitive or ar,y other model should
be tested solely by its ability to predict. In the context of competition,
he comes close to arguing that prices and quantities are the only relevant
data. This point of view is valuable in stressing that a certain
amount of lack of realism in the acsumptions of a model is no argument
against its value. But the price-quantity irnplicai-ions of the competitive
model for pricing are not easy to derive without major--and, in
many cases, impossible-econometric efiorts.
In this paper, the institutional organization and the observable mores
of the medical profession are included among the data to be used in
awessing the coinpetitiveness of tlie medical-care market. I shall also
examine the presence or absence of the preconditions for the equivalence
of competitive equilibria and optimal states. The major competitive
preconditions, in the sense used here, are three: the existence of
competitive equilibrium, the ma~ketnhilityof all goods and services
relevant to cocts and utilities, and noflincreasing ret;rn.ns. The first two,
as we have seen, insure that competitive equilibrium is necessarily optimal;
the third insures that every optimal state is the competitive
equilibrium corresponding to some distribution of income.Yhe first
and third conditions are interrelated; indeed, nonincreasing returns
plus some additional conditions not restrictive in a modern economy
imply the existence of a competitive equilibrium, i.e., imply that there
will be some set of prices which will clear all markets."
The concept of marketability is somewhat broader than the traditional
divergence between private and social costs and benefits. The
latter concept refers to cases in which the organization of the market
does nct reqaire an individual to pay for coqts that he imposes on
others as the result of his actions or does not permit him to receive
compensation for benefits he confers. In the medical field, the obvious
example is the spread of communicable diseases. An individual who
fails to be immunized not only risks his own health, a disutility which
presumably he has weighed against the utility of avoiding the procedure,
but also that of others. In an ideal price system, there would be a
price which he would have to pay to anyone whose health is endangered,
a price sufficiently high so tllat the others would feel compensated;
or, alternatively, there would be a price which would be paid to
him by others to induce him to undergo the immunization procedure.
*There are further minor conditions, for which see Koopmans [IS, pp. 50-551.
For a more precise statement of the existence conditions, see Koopmans 118, pp. 56-60]
or Debreu [12, Ch. 51.
ARROW: UNCERTAINTY AND MEDICAL CARE 945
Either system would lead to ail optimal state, though the distributional
implications would be different. It is, of course, not hard to see that
such price systems could not, in fact, be practical; to approximate an
optima! state it would be necessary to have collective intervention in
the form of subsidy or tax or compulsion.
By the absence of marketability for an action which is identifiable,
technologically possible, and capable of influencing some individuaľs
welfare, for better or for worse, is meant here the failure of the existing
market to provide a means whereby the services can be both offered
and demanded upon payment of a price. Nonniarketability may
be due to intrinsic techiiological characteristics of the product which
prevent a suitable price from being enforced, as in the case of communicable
diseases, or it may be due to social or historical controls,
such as those prohibiting an individual from selling himself into slavery.
This distinction is, in fact, difficult to make precise, though it is
obviously of i~nportancefor policy; for the present purposes, it will be
sufficient to identify non~narketabilitywith the observed absence of
markets.
The instance of nonmarketability with which x7e shall be most concerned
is that of risk-bearing. The relevance of risk-bearing to medical
care seems obvious; illness is to a considerable extent an unpredictable
phenomenon. The ability to shift the risks of illness to others is worth
a price which many are willing to pay. Because of pooling and of superior
?villingness and ability, others are willing to bear the risks. Nevertheless,
as we shall see in greater detail, a great many risks are not
covered, and indeed the markets for the services of risk-coverage are
poorly developed or nonexistent. Why this should be so is explained in
more detail in Section 1V.C below; briefly, it is impossible to draw up
insurance policies which will sufficiently distinguish among risks, particularly
since observation of the results will be incapable of distinguishing
between avoidable and unavoidable risks, so that incentives
to avoid losses are diluted.
The optimality theorems discussed above are usually presented in
the literature as referring only to conditions of certainty, but there is
no difficulty in extending them to the case of risks, provided the additional
services of risk-bearing are included with other c~niniodities.~
However, the variety of possible risks in the w ~ r l dis really staggering.
The relevant commodities include, in effect, bets on all possible
occurrences in the world which impinge upon utilities. In fact, many of
these "commodities," i.e., desired protection against many risks, are
6 T l ~ etheory, in vxiant forms, seems to have teen first worked out by hllais 123,
Arrow [ 5 ] , and BauCier [ I ] . For further generalization, see Debreu [ll] and [I.?, Ch. 71.
946 THE AMERICAN ECONOMIC REVIEJV
simply not available. Thus, a wide class of commodities is nonmarketable,
and a basic competitive precondition is not sati~fied.~
There is a still more subtle consequence of the introduction of riskbearing
considerations. When there is uncertainty, information or
knowledge becomes a commodity. Like other commodities, it has a cost
of production and a cost of transmission, and so it is naturally not
spread out over the entire population but concentrated among those
who can profit most from it. (These costs may be measured in time or
disutility as well as money.) But the demand for information is difficult
to discuss in the rational terms usually employed. The value of
information is frequently not known in any meaningful sense to the
buyer; if, indeed, he knew enough to measure the value of information,
he would know the information itself. But information, in the
form of skilled care, is precisely what is being bought from most physicians,
and, indeed, from most professionals. The elusive character of
information as a commodity suggests that it departs considerably from
the usual marketability assumptions about commodities.'
That risk and i~ncertaintyare, in fact, significant elements in medical
care hardly needs argument. I will hold that virtually all the special
features of this industry, in fact, stem from the prevalence of uncer-
tainty.
The nonexistence of markets for the bearing of some risks in the first
instance reduces welfare for those who wish to transfer those risks to
others for a certain price, as well as for those who would find it profitable
to take on the risk at such prices. But it also reduces the desire to
render or consume services which have risky consequences; in technical
language, these commodities are complementary to risk-bearing.
Conversely, the production and consumption of commodities and services
with little risk attached act as substitutes for risk-bearing and are
encouraged by market failure there with respect to risk-bearing. Thus
the observed commodity pattern will be affected by the nonexistence of
other markets.
' I t should also be remarked that in the presence of uncertainty, indivisibiiities that are
sufficiently small to create little difficulty for the evistence and viability of competitive
equilibrium may nevertheless give rise to a considerable range of increasing returns because
of the operation of the law of large numbers. Since most objects of insurance (lives,
fire hazards, etc.) hare some element of indivisibility, insurance companies have to be
above a certain si7e. Gut it is not clezr that this effect is sufficiently great to create serious
obstacles to the existence and viability of competitive equilibrium in practice.
'One form of production of information is research. Not only does the product have
unconventionsl aspects as a commodity, but it is also subject to increasing returns in use,
since new ideas, once developed, can be used over and over without being consumed, and
to difiiculties of market control, since the cost of reproduction is usually much less than
that of production. Hence, it is not surprising that a free enterprise economy will tend
to underinvest in research; see Nelson [211 and Arrow [4].
947ARROW: UNCERTAIYTY AND MEDICAI, CARE
The failure of one or more of the competitive preconditions has as
its most immediate and obvious consequence a reduction in welfare
below that obtainable from existing resources and technology, in the
sense of a failure to reach an optimal state in the sense of Pareto. But
more can be said. I propose here the view that, when the market fails
to achieve an optimal state, society will, to some extent at least, recognize
the gap, and nonmarket social institutions will arise attempting to
bridge it.Tertainly this process is not necessarily conscious; nor is it
uniformly successful in approaching rnore closely to optinlality when
the entire range of consequences is considered. It has a l ~ ~ a y sbeen a
favorite activity of econonlists to point out that actions which on their
face achieve a desirable goal may have less obvious consequences,
particularly over time, which more than offset the original gains.
But it is contended here that the special structural characteristics
of the medical-care market are largely attempts to overcome the lack of
optimality due to the nonmarketability of the bearing of suitable risks
and the imperfect marketability of icforination. These compensatory
institutional changes, with some reinforcement from usual profit motives,
largely explain the observed noncompetitive behavior of the
medical-care market, behavior ~vhich,in itself, interferes with optimality.
The social adjustment towards optimality thus puts obstacles in
its own path.
The doctrine that society will seek to achieve optimality by nonmarket
means if it cannot achieve them in the market is not novel.
Certainly, the government, at least in its economic activities, is usually
implicitly or explicitly held to function as the agency which substitutes
for the markeťs failure.'' I am arguing here that in some circumstances
other social institutions will step into the optimality gap, and
that the medical-care industry, with its variety of special institutions,
some ancient, some modern, exemplifies this tendency.
It may be useful to remark here that a good part of the preference
for redistribution expressed in government taxation and expenditure
policies and private charity can be reinterpreted as desire for insurance.
It is noteworthy that virtually nowhere is there a system of subsidies
that has as its aim simply an equalization of income. The subsidies
or other governmental help go to those who are disadvantaged in
life by events the incidence of which is popularly regarded 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 procurement of
modern weapons; see Peck and Scherer [23, pp. 581-821 (I am indebted for this reference
to V. Fuchs) and [I, pp. 71-75],
10
For an explicit statement of this view, see Baumol [81. But I bciieve this position
is implicit in most discussions of the functions of government.
948 THE AMERICAN ECONOMIC REVIEW
dictable: the blind, dependent children, the medically indigent. Thus,
optimality, in a context which includes risk-bearing, includes much
that appears to be motivated by distributional value judgments when
looked at in a narrower context.''
This methodological background gives rise to the following plan for
this paper. Section I1 is a catalogue of stylized generalizations about
the medical-care market which differentiate it from the usual cornmodity
markets. In Section I11 the behavior of the market is compared
with that of the competitive model which disregards the fact of uncertainty.
In Section IV, the medical-care market is compared, both as to
behavior and as to preconditions, with the ideal competitive market
that takes account of uncertainty; an attempt will be made to demonstrate
that the characteristics outlined in Section I1 can be explained
either as the result of deviations from the competitive preconditions or
as attempts to compensate by other institutions for these failures. The
discussion is not designed to be definitive, but provocative. In particular,
I have been chary about drawing policy inferences; to a considerable
extent, they depend on further research, for which the present
paper is intended to provide a framework.
11. A Survey of the Special Characteristics of the
Medical-Care Market12
This section will list selectively some characteristics of medical care
which distinguish it from the usual commodity of economics textbooks.
The list is not exhaustive, and it is not claimed that the characteristics
listed are individually unique to this market. But, taken together, they
do establish a special place for medical care in economic analysis.
A. The Nature of Demand
The most obvious distinguishing characteristics of an individuaľs
demand for medical services is that it is not steady in origin as, for
example, for food or clothing, but irregular and unpredictable. Medical
services, apart from preventive services, afford satisfaction only in
the event of illness, a departure from the normal state of affairs. It is
hard, indeed, to think of another commodity of significance in the
average budget of which this is true. A portion of legal services, devoted
to defense in criminal trials or to lawsuits, might fall in this category
but the incidence is surely very much lower (and, of course, there
11 Since writing the above, I find that Buchanan and Tullock [lo, Ch. 131 have argued
that all redistribution can be interpreted as "income insurance."
"For an illuminating survey to which I am much indebted, see S. Mushkin [201.
949ARROW: UNCERTAINTY AND MEDICAL CARE
are, in fact, strong institutional similarities between the legal and
medical-care markets.)13
In addition, the demand for medical services is associated, with a
considerable probability, with an assault on personal integrity. There is
some risk of death and a more considerable risk of impairment of full
functioning. In particular, there is a major potential for loss or reduction
of earning ability. The risks are not by themselves unique; food is
also a necessity, but avoidance of deprivation of food can be guaranteed
with sufficient income, where the same cannot be said of avoidance of
illness. Illness is, thus, not only risky but a costly risk in itself, apart
from the cost of medical care.
B. Expected Behavior of the Physician
I t is clear from everyday observation that the behavior expected of
sellers of medical care is different from that of business men in general.
These e:ipectations are relevant because medical care belongs to
the category of commodities for which the product and the activity of
production are identical. In all such cases, the customer cannot test the
product before consuming it, and there is an element of trust in the
relation.'" But the ethically understood restrictions on the activities of
a physician are much more severe than on those of, say, a barber. His
behavior is supposed to be governed by a concern for the customer's
welfare which would not be expected of a salesman. In Talcott Parsons's
terms, there is a "collectivity-orientation,') which distinguishes
medicine and other professions from business, where self-interest on
the part of participants is the accepted norm.15
A few illustrations will indicate the degree of difference between the
behavior expected of physicians and that expected of the typical businessman.''
(1) Advertising and overt price competition are virtually
eliminated among physicians. ( 2 ) Advice given by physicians as to
further treatment by himself or others is supposed to be completely
"In governmental demand, military power is an example of a service used only
irregularly and unpredictably. Here too, special institutional and professional relations
have emerged, though the precise social structure is different for reasons that are not hard
t3 analyze.
"Even with material commodities, testing is never so adequate that all elements of
implicit trust can be eliminated. Of course, over the long run, experience with the quality
of product of a given seller provides a check on the possibility of trust.
See 122, p. 4631. The whole of 122, Ch. 101 is a most illuminating analysis of the
social role of medical practice; though Parsons' interest lies in different areas from mine,
I must acknowledge here my indebtedness to his work.
"I am indebted to Herbert Klarman of Johns Hopkins University for some of the
points discussed in this and the following paragraph.
95'0 THE AMERICAN ECONOMIC REVIEW
divorced from self-interest. (3) It is at least claimed that treatment is
dictated by the objective needs of the case and not limited by financial
considerations." While the ethical compulsion is surely not as absolute
in fact as it is in theory, we can hardly suppose that it has no influence
over resource allocation in this area. Charity treatment in one form or
another does exist because of this tradition about human rights to adequate
medical care.ls (4) The physician is relied on as an expert in
certifying to the existence of illnesses and injuries for various legal and
other purposes. I t is socially expected that his concern for the correct
conveying of information will, when appropriate, outweigh his desire
to please his customers.'"
Departure from the profit motive is strikingly manifested by the
overwhelming predominance of nonprofit over proprietary hospitals.'O
The hospital per se offers services not too different from those of a
hotel, and it is certainly not obvious that the profit motive will not lead
to a more efficient supply. The explanation may lie either on the supply
side or on that of demand. The simplest explanation is that public and
private subsidies decrease the cost to the patient in nonprofit hospitals.
A second possibility is that the association of profit-making with the
supply of medical services arouses suspicion and antagonism on the
part of patients and referring physicians, so they do prefer nonprofit
institutions. Either explanation implies a preference on the part of some
group, whether donors or patients, against the profit motive in the
supply of hospital services."
"The belief that the ethics of medicine demands treatment independent of the patienťs
ability to pay is strongly ingrained. Such a perceptive observer as RenC Dubos has made
the remark that the high cost of anticoagulants restricts 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 considered in the harsh
light of economics" (emphasis added). Of course, this expectation amounts to ignoring
the scarcity of medical resources; one has only to have heen poor to realize the error.
We may confidently assume that price and income do have some consequences for
medical expenditures.
needed piece of research is a study of the exact nature of the variations of medical
care received and medical care paid for as income rises. (The relevant income concept
also needs study.) For this purpose, some disaggregation is needed; differences in hospital
care which are essentially matters of comfort should, in the above view, be much more
responsive to income than, e.g., drugs.
I9 This role is enhanced in a socialist society, where the state itself is actively concerned
with illness in relation to work; see Field [14, Ch. 91.
mAbcut 3 per cent of beds were in proprietary hospitals in 1955, against 30 per cent in
voluntary nonprofit, and the remainder in federal, state, and local hospitals; see [26,
Chart 4-2, p. 601.
"C. R. Rorem has pointed out to me some further factors in this analysis. (1) Given
the social intention of helping all patients without regard to immediate ability to pay,
economies of scale would dictate a predcminance of con~n~unity-sponsoredhospitals. (2)
ARROW: UNCERTAINTY AND MEDICAL CARE 951
Conformity to collectivity-oriented behavior is especially important
since it is a commonplace that the physician-patient relation affectsthe
quality of the medical care product. A pure cash nexus would be inadequate;
if nothing else, the patient expects that the same physician
will normally treat him on successive occasions. This expectation is
strong enough to persist even in the Soviet Union, where medical care
is nominally removed from the market place [14, pp. 194-961. That
purely psychic interactions between physician and patient have e2ects
which are objectively indistinguishable in kind from the effects of
medication is evidenced by the use of the placebo as a control in inedical
experimentation; see Shapiro [2 51.
C. P ~ ~ d l i ~ iUncertainty
Uncertainty as to the quality of the product is perhaps more inten.:c
here than in any other important commodity. Recovery from disease is
as unpredictable as is its incidence. In most commodities, the possibility
of learning from one's own experience or that of others is strong
because there is an adequate number of trials. In the case of severe illness,
that is, in general, not true; the uncertainty due to igexperience
is added to the intrinsic difficulty of prediction. Further, the amount of
uncertainty, measured in terms of utility variability, is certainly much
greater for medical care in severe cases than for, say, houses or automobiles,
even though these are also expenditures sufficiently infrequent
so that there may be co~siderableresidual uncertainty.
Further, there is a special quality to the uncertainty; it is very different
on the two sides of the transaction. Because medical knowledge
is so complicated, the information possessed by the physician as to the
consequences and possibilities of treatment is necessarily very much
greater than that of the patient, or at least so it is believed by both
parties.22Further, both parties are aware of this informational icequality,
and their relation is colored by this knowledge.
To avoid misunderstanding, observe that the difference in information
relevant here is a difference in information as to the consequence
of a purchase of medical care. There is always an inequality of information
as to prodxiion methods between the producer and the purchaser
of any cornrnodity, but in most cases the customer may well
Some proprietary hospitals will tend to control total costs to the patient more closely, ircluding
the fees of physicians, who will therefore tend to prefer community-sponsored
hospitals.
22iVithouttrying to assess the present situation, it is clear in retrospect that at some
point in the past the actual differential knowledge possessed by p11ysic:ans m . y not
have been much. But from the economic point of view, it is the subjective belief of
both parties, as manifested in their market behavior, that is relevant.
952 THE AMERICAN ECONOMIC REVIEW
have as good or nearly as good an understanding of the utility of the
product as the producer.
D. Supply Conditions
In competitive theory, the supply of a commodity is governed by the
net return from its production compared with the return derivable
from the use of the same resources elsewhere. There are several significant
departures from this theory in the case of medical care.
Most obviously, entry to the profession is restricted by licensing.
Licensing, of course, restricts supply and therefore increases the cost of
medical care. It is defended as guaranteeing a minimum of quality.
Restriction of entry by licensing occurs in most professions, including
barbering and undertaking.
A second feature is perhaps even more remarkable. The cost of
medical education today is high and, according to the usual figures, is
borne only to a minor extent by the student. Thus, the private benefits
to the entering student considerably exceed the costs. (It is, however,
possible that research costs, not properly chargeable to education,
swell the apparent difference.) This subsidy should, in principle, cause
a fall in the price of medical services, which, however, is offset by rationing
through limited entry to schools and through elimination of
students during the medical-school career. These restrictions basically
render superfluous the licensing, except in regard to graduates of foreign
schools.
The special role of educational institutions in simultaneously subsidizing
and rationing entry is common to all professions requiring
advanced training.23It is a striking and insufficiently remarked phenomenon
that such an important part of resource allocation should be
performed by nonprofit-oriented agencies.
Since this last phenomenon goes well beyond the purely medical
aspect, we will not dwell on it longer here except to note that the
anomaly is most striking in the medical field. Educational costs tend to
be far higher there than in any other branch of professional training.
While tuition is the same, or only slightly higher, so that the subsidy is
much greater, at the same time the earnings of physicians rank highest
among professional groups, so there would not at first blush seem
to be any necessity for special inducements to enter the profession.
Even if we grant that, for reasons unexamined here, there is a social
interest in subsidized professional education, it is not clear why the
rate of subsidization should differ among professions. One might ex23T11edegree
of subsidy in different branches of professional education is worthy of a
major research effort.
953ARROW: UNCERTAINTY AND MEDICAL CARE
pect that the tuition of medical students would be higher than that of
other students.
The high cost of medical education in the United States is itself a
reflection of the quality standards imposed by the American Medical
Association since the Flexner Report, and it is, I believe, only since
then that the subsidy element in medical education has become significant.
Previcusly, mal:y medical schools paid their way or even yielded
a profit.
Xncther interesting feature of limitatioil on entry to subsidized education
is the extent of individual preferences concerning the social
welfare, as manifested by contributions to private universities. But
whether support is public or private, the important point is that both
the quality and the quantity of the supply of medical care are being
strongly influenced by social nonmarket
One striking consequence of the control of quality is the restriction
on the range offered. If many qualities oi a commodity are possible, it
ivould usually happen in a competitive market that many qualities will
be offered on the market, at suitably varying prices, to appeal to dif:@renttastes
and incomes. Both the licensing laws and the standards of
xedical-school training have limited the possibilities of alternative
qualities of medical care. The declining ratio of physicians to total
employees ill the medical-care industry shows that substitution of less
trained personnel, technicians, and the like, is not prevented completely,
but the central role of the highly trained physician is not affected
at all.26
E. Pricing Practices
The unusual pricing practices and attitudes of the n~edicalprofession
are well known: extensive price discrimination by income (with an
extreme of zero prices for sufficiently indigent patients) and, formerly,
a strong insistence on fee for services as against such alternatives as
prepayment.
*A
Strictly speaking, there are four vari:rbles in the market for physicians: price, quality
of entering students, quality of education, and quantity. The basic market forces, demand
for medical services and supply of entering students, determine two relations among the
four variables. Hence, if the nonmarket forces determine the last two, market forces will
determine price ar.d quality of entrants.
"The sapply of Ph.D.'s is similarly governed, hut there are other conditions in the
market which are much difiereot, especially on the dernand side.
28
Today oi~lythe Soviet Union offers an alternntive lower level of medical personnel,
the feldshers, who practice primarily in the rural districts (the institution dates back
to the 18th century). According to Fie!d [11, pp. 98-100, 132-331, there is clear evidence
of strain in the relations between physicians and feldshers, but it is not certain that the
feldihers will gradually disappear as physicians grow in numbers.
954 THE AMERICAN ECONOMIC REVIEW
The opposition to prepayment is closely related to an even stronger
opposition to closed-panel practice (contractual arrangements which
bind the patient to a particular group of physicians). Again these attitudes
seem to differentiate professions from business. Prepayment and
closed-panel plans are virtually nonexistent in the legal profession. In
ordinary business, on the other hand, there exists a wide variety of
exclusive service contracts involving sharing of risks; it is assumed
that competition will select those which satisfy needs best."
The problems of implicit and explicit price-fixing should also be
mentioned. Price competition is frowned on. Arrangements of this type
are not uncommon in service industries, and they have not been subjected
to antitrust action. How important this is is hard to assess. I t
has been pointed out many times that the apparent rigidity of so-called
admiiiistered prices considerably understates the actual flexibility.
Here, too, if physicians find themselves with unoccupied time, rates are
likely to go down, openly or covertly; if there is insufiicient time for
the demand, rates will surely rise. The "ethics" of price competition
may decrease the flexibility of price responses, but probably that is all.
111. Comparisons with the Competitive Model under Certainty
As already noted, the diffusion of conlmunicable diseases provides
an obvious example of nonmarket interactions. But from a thearetical
viewpoint, the issues are well understood, and there is little point in
expanding on this theme. (This should not be interpreted as minirnizing
the contribution of public health to welfare; there is every reason
to suppose that it is considerably more important than all other aspects
of medical care.)
Beyond this special area there is a more general interdepe~~dence.the
concern of individuals for the health of others. The economic manifestations
of this taste are to be found in individual donations to hospitals
and to medical education, as well as in the widely accepted responsibilities
of government in this area. The taste for improving the health
of others appears to be stronger than for improving other aspects of
their elf are.'^
I11 interdependencies generated by concern for the welfare of others
there is always a theoretical case for collective action if each participant
derives satisfaction from the contributions of all.
nThe law does impose some limits on risk-shifting in contracts, for example, its general
refusal to honor e~culpatoryclauses.
=There may be an identification problem 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
otherwise equally concerned about the two aspects of others' welfare ma:; prefer
to help in the first.
ARROW: UNCERTAINTY AND MEDICAL CARE 955
B. Incrcasirtg Returns
Problen~sassociated with increasing returns play some role in allocation
of resources in the medical field, particularlv in areas of low
density or low income. Hospitals show increasing returns up to a point;
specialists and some medical equipment constitute significant indivisibilities.
In many parts of the world the individual physician may be a
large unit relative to demand. In such cases it can be socially desirable
to subsidize the appropriate medical-care unit. The appropriate mode
of analysis is much the same as for water-resource projects. Increasing
returns are hardly apt to be a significant problem in general practice in
large cities in the United States, and improved transportation to some
extent reduces their importance elsewhere.
C. Entry
The most striking departure from competitive behavior is restriction
on entry to the field, as discussed in 1I.D above. Friedman and Kuznets,
in a detailed examination of the pre-World War I1 data, have
argued that the higher income of physicians could be attributed to this
re~triction.~'
There is some evidence that the demand for admission to medical
school has dropped (as indicated by the number of applicants per
place and the quality of those admitted), so that the number of medical-school
places is not as significant a barrier to entry as in the early
1950's [25, pp. 14-15]. But it certainly has operated over the past and
it is still operating to a considerable extent today. It has, of course,
constituted a direct and unsubtle restriction on the supply of medical
care.
There are several considerations that must be added to help evaluate
the importance of entry restrictions: (1) Additional entrants would be,
in general, of lower quality; hence, the addition to the supply of medical
care, properly adjusted for quality, is less than purely quantitative
calculations would show.30(2) To achieve genuinely competitive conditions,
it would be necessary not only to remove numerical restrictions
on entry but also to remove the subsidy in medical education. Like any
other producer, the physician should bear all the costs of production,
See [16, pp. 118-371. The calculations involve many assumptions and must be regarded
as tenuous; see the comments by C. Reinold Noyes in [16, pp. 407-101.
301t might be argued that the existence of racial discrimination in entrance has meant
that some of the rejected applicants are superior to some accepted. However, there is
no necessary connection between an increase in the number of entrants and a reduction
in racial discrimination; so long as there is excess demand for entry, discrimination can
continue unabated and new entrants will be inferior to those previously accepted.
956 THE AMERICAN ECONOMIC REVIEW
including, in this case, e d ~ c a t i o n . ~ ~It is not so clear that this change
would not keep even unrestricted entry down below the present level.
(3) To some extent, the effect of making tuition carry the full cost of
education will be to create too few entrants, rather than too many.
Given the imperfections of the capital market, loans for this purpose to
those who do not have the cash are difficult to obtain. The lender really
has no security. The obvious answer is some form of insured loans, as
has frequently been argued; not too much ingenuity would be needed
to create a credit system for medical (and other branches of higher)
education. Under these conditions the cost would still constitute a deterrent,
but one to be compared with the high future incomes to be
obtained.
If entry were governed by ideal competitive conditions, it may be
that the quantity on balance would be increased, though this conclusion
is not obvious. The average quality would probably fall, even
under an ideal credit system, since subsidy plus selected entry draw
some highly qualified individuals who would otherwise get into other
fields. The decline in quality is not an over-all social loss, since it is
accompanied by increase in quality in other fields of endeavor; indeed,
if demands accurately reflected utilities, there would be a net social
gain through a switch to competitive entry."
There is a second aspect of entry in which the contrast with competitive
behavior is, in many respects, even sharper. It is the exclusion
of many imperfect substitutes for physicians. The licensing laws,
though they do not effectively limit the number of physicians, do exclude
all others from engaging in any one of the activities known as
medical practice. As a result, costly physician time may be employed
at specific tasks for which only a small fraction of their training is
needed, and which could be performed by others less xvell trained and
therefore less expensive. One might expect immunization centers, privately
operated, but not necessarily requiring the services of doctors.
In the competitive model without uncertainty, consumers are presumed
to be able to distinguish qualities of the commodities they buy.
Under this hypothesis, licensing would be, at best, superfluous and
exclude those from whom consumers would not buy anyway; but it
might exclude too many.
D. Pricing
The pricing practices of the medical industry (see 1I.E above) deproblem
here is that the tax laws do not permit depreciation of professional
education, so that there is a discrimination against this form of investment.
''To anticipate later discussion, this condition is not necessarily fulfilled. When it
comes to quality choices, the market may be inaccurate.
957ARROW : UNCERTAINTY AND ?rfEDICAL CARE
part sharply from the coxpetitive norm. As Xessel 1171 has pointed
out with great vigor, not only is price discrimination incompatible with
the competitive model, but its preservation in the face of the large
number of physicians is equivalent to a collective monopoly. In the
past, the opposition to prepayment plans has taken distinctly coercive
forms, certainly transcending market pressures, to say the least.
Kessel has argued that price discrimination is designed to maximize
profits along the classic lines of discriminating monopoly and that
organized medical opposition to prepayment was motivated by the
desire to protect these profits. In principle, prepayment schemes are
compatible with discrimination, but in practice they do not usually
discriminate. I do not believe the evidence that the actual scale of discrimination
is profit-maximizing is convincing. In particular, note that
for any monopoly, discriminating or otherwise, the elasticity of demand
in each market at the point of maximum profits is greater than one.
But it is almost surely true for medical care that the price elasticity of
demand for all income levels is less than one. That price discrimination
by income is not completely profit-maximizing is obvio:zs in the
extreme case of charity; Kessel argues that this represents an appeasement
of public opinion. But this already shows the incompleteness of
the model and suggests the relevance and importance of social and
ethical factors.
Certainly one important part of the opposition to prepaym--nt was
its close relation to closed-panel plans. Prepayment is a form of insurance,
and naturally the individual physician did not wish to assume the
risks. Pooling was intrinsically involved, and this strongly motivates, as
we shall discuss further in Section IV below, control over prices and
benefits. The simplest administrative form is the closed panel; physiciacs
involved are, in effect, the insuring agent. From this point of
view, Blue Cross solved the prepayment problem by universalizing the
closed panel.
The case that price discrimination by income is a form of profit
maximization which was zealously defended by opposition to fees for
service seems far from proven. But it remains true that this price discrimination,
for whatever cause, is a source of nonoptimality. Rypothetically,
it means everyone would be better off if prices were made
equal for all, and the rich compensated the poor for the changes in the
relative positions. The importance of this welfare loss depends on the
actual amount of discrimination and on the elasticities of demand for
medical services by the different income groups. If the discussion is
simplified by considering only two income levels, rich and poor, and if
the elasticity of demand by either one is zero, then no reallocation of
medical services will take place and the initial situation is optimal. The
958 THE AMERICAN ECONOMIC REVIEW
only effect of a change in price will be the redistribution of income as
between the medical profession and the group with the zero elasticity
of demand. With low elasticities of demand, the gain will be small. To
illustrate, suppose the price of medical care to the rich is double that to
the poor, the medical expenditures by the rich are 20 per cent of those
by the poor, and the elasticity of demand for both classes is .5; then
the net social gain due to the abolition of discrimination is slightly over
1 per cent of previous medical expenditure^.^^
The issues involved in the opposition to prepayment, the other major
anomaly in medical pricing, are not meaningful in the world of certainty
and will be discussed below.
IV. Comparison with the Idcal Competitive Model under Uncertainty
A. Introduction
In this section we will compare the operations of the actual medicalcare
market with those of an ideal system in which not only the usual
commodities and services but also insurance policies against all conceivable
risks are a~ailable.~'Departures consist for the most part of
83
I t is assumed that there are t v o classes, rich and poor; the price of medical services
to the rich is twice that to the poor, medical expenditures by the rich are 20 per cent
of those by the poor, and the elasticity of demand for medical services is .5 for both
classes. Let us choose our quantity and monetary units 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
elasticities of demand, the demand function of the rich is Dn(p) = .llp-" and that of
the poor is Dr(P) =p-". The supply of medical services is assumed fixed and therefore
must equal 1.1. If price discrimination were abolished, the equilibrium price, 3, rnust
satisfy the relation,
and therefore = 1.07. The quantities of medical care purchased by the rich and poor,
respectively, would be DR(?) = .I35and Dp(3) =,965.
The inverse demand functio:ls, tile price to be paid corresponding to any given quantity
are drr(q)= .02/q2, and d r ( q ) = l / q 2 . Therefore, the consumers' surplus to the rich
generated by the change is :
,136 

(1)  (.02/q2)dq - ?(.I35 - .I),
and similarly the loss in consumers' surplus by the poor is:
If ( 2 ) is subtracted from ( I ) , the second terms cancel, and the aggregate increase in
consumers' surplus is ,0156, or a little over 1 per cent of the initial expenditures.
"X striking illustration of the desire for secn~ityin medical care is provided by the
expressed preferences of dmigrds from the Soviet Union as between Soviet medical practice
and German or American practice; see Field [11,Ch. 121. Those in Germany preferred
the German system to 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 certainty of
medical care, independent of income or health fluctuations.
959ARROV4: UNCERTAINTY AND MEDICAL CARE
insurance policies that might conceivably be written, but are in fact
not. Whether these potential com~oditiesare nonmarketable, or,
merely because of some imperfection in the market, are not actually
marketed, is a somewhat fine point.
To recall what has already been said in Section I, there are two
kinds of risks involved in n~edicalcare: the risk of becoming ill, and
the risk of total or incomplete or delayed recovery. The loss due to
illness is only partially the cost of medical care. It also consists of discomfort
and loss of productive time during illness, and, in more serious
cases, death or prolonged deprivation of normal function. From the
point of view of the welfare economics of uncertainty, both losses are
risks against which individuals would like to insure. The nonexistence
of suitable insurance policies for either risk implies a loss of welfare.
16. The Theory oj idcat Insurnncc
In this section, the basic principles of an optimal regime for riskbearing
ill be presented. For illustration, reference will usually be
made to the case of insurance against cost in medical care. The principles
are equally applicable to any of the risks. There is no single
source to which the reader can be easily referred, though I think the
prisciples are at least reasonably well understood.
As a basis far the analysis, the assumptior, is made that each individual
acts so as to maximize the expected value of a utility function. If
we think of utility as attached to income, then the costs of medical
care act as a random deduction from this income, and it is the expected
value of the utility of income after medical costs that we are concerned
with. (Income after medical costs is the ability to spend money on
other objzcts which give satisfaction. We presuppose that illness is not
a source of satisfaction in itself; to the extent that it is a source of
dissatisfaction, the illness should enter into the utility function as a
separate variable.) The expected-utility hypothesis, due originally to
Daniel Bernoulli (1738), is plausible and is the most analytically manageable
of all hypotheses that have been proposed to explain behavior
under uncertainty. In any case, the results to follow probably would
not be significa~tlyaffected by moving to another mode of analysis.
It is further assumed that individuals are normally risk-averters. In
utility terms, this means that they have a diminishing marginal utility
of income. This assunlption may reasonably be taken to hold for most
of the signZficantaffairs of life for a majority of people, but the presence
of gaixbling provides some difficulty in the full application of this
view. It foIlows from the assumption of risk aversion that if an individual
is given a choice between a probability distribution of income,
with a given mean nt, and the certainty of the income nz, he would prefer
960 THE AMERICAN ECONOMIC REVIEW
the latter. Suppose, therefore, an agency, a large insurance company
plan, or the government, stands ready to offer insurance against medical
costs on an actuarially fair basis; that is, if the costs of medical care
are a random variable with mean fn, the company will charge a premium
m, and agree to indemnify the individual for all ~nedicalcosts.
Under these circumstances, the individual will certainly prefer to take
out a policy and will have a welfare gain thereby.
Will this be a social gain? Obviously yes, if the insurance agent is
suffering no sccial loss. Under the assumption that medical risks on
different individuals are basically independent, the pooling of them
reduces the risk involved to the insurer to relatively small proportions.
In the limit, the wzlfare loss, even assuming risk aversion on the part
of the insurer, would vanish and there is a net social gain which may be
of quite substantial magnitude. In fact, of course, the pooling of risks
does not go to the limit; there is only a finite number of them and
there may be some interdependence among the risks due to epidemics
and the like. Eut then a premium, perhaps slightly above the actuarial
level, would be sufficient to offset this welfare loss. From the point of
view of the individual, since he has a strict preference for the actuarially
fair policy over assuming the risks himself, he will still have a
preference for an actuarially unfair policy, provided, of course, that
it is not too unfair.
In addition to a residual degree of risk aversion by insilrers, there
are other reasons for the loading of the premium (i.e., an excess of
premium over the actuarial value). Insurance involves administrative
costs. Also, because of the irregularity of payments there is likely to be
a cost of capital tied up. Suppose, to take a simple case, the insurance
company is not willing to sell any insurance policy that a consumer
wants but will charge a fixed-percentage loading above the actuarial
value for its premium. Then it can be shown that the most preferred
policy from the point of view of an individual is a coverage with a
deductible amount; that is, the insurance policy provides 100 per cent
coverage for all medical costs in excess of come fixed-dollar limit. If,
however, the insurance company has some degree of risk aversion, its
loading may also depend on the degree of uncertainty of the risk. In
that case, the Pareto optimal policy will involve some element of coinsurance,
i.e., the coverage for costs over the minimum limit will be
some fraction less than 100 per cent (for proofs of these statements,
see Appendix).
These results can also be applied to the hypothetical concept of insurance
agrzinst failure to recover from illness. For simplicity, let us
assume thai the cost of failure to recover is regarded purely as a money
cost, either simply productive opportunities foregone or, more gener-
ARROW: UKCERTAINTE' ASD hlEDICAL CARE 961
ally, the money equivalent of all dis~atisfactions.Suppose further
that, given that a person is ill, the expected value of nledical care is
greater than its cost; that is, the expected money value attributable to
recovery with medical help is greater than resources devoted to medical
help. However, the recovery, though on the average beneficial, is
uncertain; in the absence of insurance a risk-averter may well prefer
not to take a chance on further impoverishment by buying medical
care. A suitable insurance policy vrould, however, mean that he paid
nothing if he doesn't benefit; since the expected value is greater than
the cost, there would be a net social gain.35
C. Problenzs Gf Insurance
1. The ~noralhazard. The welfare case for insurance pclicies of all
sorts is over\\-helming. It follows that the government should undertake
insurance in those cases where this market, for whatever reason,
has failed to emerge. Nevertheless, there are a number of significant
practical limitations on the use of insurance. It is important to understand
them, though I do not believe that they alter the case for the
creation of a much wider class of insurance policies than now exists.
One of the limits which has been much stressed in insurance literature
is the effect of insurance on incentives. What is desired in the case
of insurance is that the event against which insurance is taken be out
of the control of the individual. Unfortunately, in real life this separation
can never be made perfectly. The outbreak of fire in one's house
or business may be largely uncontrollable by the individual, but the
probability of fire is somewhat influenced by carelessness, and of course
arson is a possibility, if an extreme one. Similarly, in nledical policies
the cost of medical care is not completely determined by the illness
suffered by the individual but depends on the choice of a doctor and
his willingness to use medical services. It is frequently observed that
widespread medical insurance increases the demand for medical care.
Coinsurance provisions have been introduced into many major medical
policies to meet this contingency as \.;ell as the risk aversion of the insurance
companies.
To some extent the professional relationship between physician and
patient limits the normal hazard in various forms of medical insurance.
By certifying to the necessity of given treatment or the lack thereof,
the physician acts as a controlling agent on behalf of the insurance
companies. Needless to say, it is a far from perfect check; the physicians
themselves are not under any control and it may be convenient
for them or pleasing to their patients to prescribe more expensive mediIt
is 3 popular belief that the Chinese, at one tine, paid their physicisns when .sell
but not when sick
THE AMERICAN ECONOMIC REVIEW
cation, private nurses, more frequent treatments, and other marginal
variations of care. It is probably true that hospitalization and surgery
are more under the casual inspection of others than is general practice
and therefore less subject to moral hazard; this may be one reason why
insurance policies in those fields have been more widespread.
2. Alternative nzethods of insurance payment. It is interesting that
no less than three different methods of coverage of the costs of medical
care have arisen: prepayment, indemnities according to a fixed schedule,
and insurance against costs, whatever they may be. In prepayment
plans, insurance in effect is paid in kind-that is, directly in medical
services. The other two forms both involve cash payments to the beneficiary,
but in the one case the amounts to be paid involving a medical
contingency are fixed in advance, while in the other the insurance carrier
pays all the costs, whatever they may be, subject, of course, to
provisions like deductibles and coinsurance.
In hypothetically perfect markets these three forms of insurance
would be equivalent. The indemnities stipulated would, in fact, equal
the market price of the services, so that value to the insured would be
the same if he were to be paid the fixed sum or the market price or
were given the services free. In fact, of course, insurance against full
costs and prepayment plans both offer insurance against uncertainty as
to the price of medical services, in addition to uncertainty about their
needs. Further, by their mode of compensation to the physician, prepayment
plans are inevitably bound up with closed panels so that the
freedom of choice of the physician by the patient is less than it would
be under a scheme more strictly confined to the provision of insurance.
These remarks are tentative, and the question of coexistence of the
different schemes should be a fruitful subject for investigation.
3. Third-party contvol over payments. The moral hazard in physicians'
control noted in paragraph 1 above shows itself in those insurance
schemes where the physician has the greatest control, namely,
major medical insurance. Here there has been a marked rise in expenditures
over time. In prepayment plans, where the insurance and
medical service are supplied by the same group, the incentive to keep
medical costs to a minimum is strongest. In plans of the Blue Cross
group, there has developed a conflict of interest between the insurance
carrier and the medical-service supplier, in this case particularly the
hospital.
The need for third-party control is reinforced by another aspect of
the moral hazard. Insurance removes the incentive on the part of jndividuals,
patients, and physicians to shop around for better prices for
hospitalization and surgical care. The market forces, therefore, tend to
be replaced by direct institutional control.
963A.RROItr: UNCERTAINTY AND MEDICAL CARE
4. Adtninistrative costs. The pure theory of insurance sketched in
Section B above omits one very important consideration: the costs of
operating an insurance company. There are several types of operating
costs, but one of the most important categories includes commissions
and acquisition costs, selling costs in usual economic terminology. Not
only does this mean that insurance policies must be sold for considerably
more than their actuarial value, but it also means there is a
great differential among different types of insurance. It is very striking
to observe that among health insurance policies of insurance companies
in 1958, expenses of one sort or another constitute 51.6 per cent of total
premium income for individual policies, and only 9.5 per cent for group
policies [26, Table 14-1, p. 2721. This striking differential would seem
to imply enormous econcmies of scale in the provision of insurance, quite
apart from the coverage of the risks themselves. Obviously, this provides
a very strong argument for widespread plans, including, in particular,
compulsory ones.
5. P~edictabilityand insurance. Clearly, from the risk-aversion point
of view, insurance is more valuable, the greater the uncertainty in the
risk being insured against. This is usually used as an argument for
putting greater emphasis on insurance against hospitalization and surgery
than other forms of medical care. The empirical assumption has
been challenged by 0 . W. Anderson and others [3, pp. 53-54], who asserted
that out-of-hospital expenses were equally as unpredictable as
in-hospital costs. What was in fact shown was that the probability of
costs exceeding $200 is about the same for the two categories, but this
is not, of course, a correct measure of predictability, and a quick glance
at the supporting evidence shows that in relation to the average cost
the variability is much lower for ordinary medical expenses. Thus, for
the city of Birmingham, the mean expendip~reon surgery was $7, as
opposed to $20 for other medical expenses, but of those who paid
something for surgery the average bill was $99, as against $36 for those
with some ordinary medical cost. Eighty-two per cent of those interviewed
had no surgxy, and only 20 per cent had no ordinary medical
expenses [3, Tables A-13, A-18, and A-19 on pp. 72, 77, and 79, re-
spectively].
The issue of predictability also has bearhg on the merits of insnrance
against chronic illness or maternity. On a lifetime insurance basis,
insurance against chronic illness makes sense, since this is both highly
unpredictable and highly significant in costs. Among people who already
have chronic illness, or symptoms which reliably indicate it, insurance
in the strict sense is probably pointless.
6. Pooling o j urzeqz~alrisks. Hypothetically, insurance requires for
its full social benefit a maximum possible discrimination of risks. Those
964 THE AMERICAN ECONOhlIC REVIEW
in groups of higher incidences of illness should pay higher premiums.
In fact, however, there is a tendency to equalize, rather than to differentiate,
premiums, especially in the Blue Cross and similar widespread
schemes. This constitutes, in eifect, a redistribution of income from
those with a low propensity to illness to those with a high propensity.
The equalization, of course, could not in fact be carried through if the
market were genuinely competitive. Under those circumsances, insurance
plans could arise which charged lower premiums to preferred
risks and draw them off, leaving the plan which does not discriminate
among risks with only an adverse selection of them.
As we have already seen in the case of income redistribution, some
of this may be thought of as insurance with a longer time perspective.
If a plan guarantees to everybody a premium that corresponds to total
experience but not to experience as it might be segregated by smaller
subgroups, everybody is, in effect, insured against a change in his basic
state of health which tvould lead to a reclassification. This corresponds
precisely to the use of a level premium in life insurance instead of a
premium varying by age, as would be the case for term insurance.
7. Gaps avd coverage. We may briefly note that, at any rate to date,
insurances against the cost of medical care are far from universal. Certain
groups-the unemployed, the institutionalized, and the aged-are
almost completely uncovered. Of total expenditures, between one-fifth
and one-fourth are covered by insurance. I t should be noted, however,
that over half of all hospital expenses and about 35 per cent of the
medical payments of those with bills of $1:000a year and over, are included
[26, p. 3761. Thus, the coverage on the more variable parts of
medical expenditure is somewhat better than the over-all figures would
indicate, but it must be assumed that the insurance mechanism is still
very far from achieving the full coverage of which it is capable.
1. There are really two niajor aspects of uncertainty for an individual
already suffering from an illness. He is uncertain about the effectiveness
of medical treatment, and his uncertainty may be quite different
from that of his physician, based on the presumably quite different
medical knowledges.
2. Ideal inszira~zce.This will necessarily involve insurance against
a failure to benefit from medical care, whether through recovery, relief
of pain, or arrest of further deterioration. One form would be a system
in which the payment to the physician is nlade in accordance 17;ith the
degree of benefit. Since this would involve transferring the risks from
the patient to the physician, who might certainly have an aversion to
bearing them, there is room for insurance carriers to pool the risks,
965ARROW: UNCERTAINTY AND MEDICAL CARE
either by contract with physicians or by contract with the potential
patients. Under ideal insurance, medical care will always be undertaken
in any case in which the expected utility, taking account of the probabilities,
exceeds the expected medical cost. This prescription would
lead to an economic optimum. If we think of the failure to recover
mainly in terms of lost working time, then this policy would, in fact,
maximize cconon~icwelfare as ordinarily measured.
3. The concepts of trztst and d~lcgation.In the absence of ideal insurance,
there arise institut;ons which offer some sort of substitute
guarantees. Under ideal insurance the patient would actually have no
concern with the informational inequality between himself and the
physician, since he would only be paying by results anyway, and his
utility position would in fact be thoroughly guaranteed. In its absence
he wants to have some guarantee that at least the physician is using
his knowledge to the best advantage. This leads to the setting up of a
relationship of trust and confidence, one which the physician has a
socyal obligation to live up to. Since the patient does not, at least in his
belief, know as much as the physician, he canr,ot completely enforce
standards of care. In part, he replaces direct observation by generalized
belief in the ability of the physician." To put it another way, the
social obligation for best practice is part of the commodity the physician
sells, even thcugh it is a part that is not sxbject to thorough inspection
by the buyer.
One consequence of ~ u c htrust re!a!ior.s is that the physician cannot
act, or at least appear to act, as if he is maximizing his income at every
moment of time. As a signal to the buyer of his intentions to act as
thorougliiy in the buyer's behalf as possible, the physician avoids the
obvious stigma!a of profit-maximizing. Purely arms-length bargainirig
behav:or would be incompatible. ngt logically, but surely psychologically,
with the trust relations. From these special relations come the
vari~usforms of ethical behavior discussed above, and so also, I suggest,
the relative unimportance of profit-making in hospitals. The very
~vord,"profit," is a signal that denies the trust relations.
Price discrimination and its extreme, free treatment for the indigent,
also follow. If the obligation of the physician is understood to be first
of all to the welfare of the patient, then in particular it talies precedence
over financial difficulties.
As a second consequence of informational inequality between physician
and patient and the lack of insurance of a suitable type, the
patient must delegate to the physician much of his freedom of choice.
36F:.an~iiBator points out to me that some protection can be achieved, at a price, by
securing additional opinions.
966 THE AMERICAN ECONOMIC REVIEW
He does not have the knowledge to make decisions on treatment, referral,
or hospitalization. To justify this delegation, the physician finds
himself somewhat limited, just as any agent would in similar circumstances.
The safest course to take to avoid not being a true agent is to
give the socially prescribed "best" treatment of the day. Compromise
in quality, even for the purpose of saving the patient money, is to risk
an imputation of failure to live up to the social bond.
The special trust relation of physicians (and allied occuptions, such
as priests) extends to third parties so that the certifications of physicians
as to illness and injury are accepted as especially reliable (see
Section 1I.B above). The social value to all concerned of such presumptively
reliable sources of information is obvious.
Notice the general principle here. Because there are barriers to the
information flow and because there is no market in wh;ch the ri3k.;
involved can be insured, coordination of purchase and sales must ta!ie
place through convergent expectations, but these are greatly assisted
by having clear and prominent signals, and these, in turn, force patterns
of behavior which are not in themselves logical necessities for
optin~ality.~~
4. Licensing and educational sta~zdards.Delegation and trust are the
social institutions designed to obviate the problem of informational inequality.
The general uncertainty about the prospects of medical treatment
is socially handled by rigid entry requirements. These are designed
to reduce the uncertainty in the micd of the consumer as to
the quality of product insofar as this is possible." 1think this esplanation,
which is perhaps the naive one, is much more tenable than any
idea of a monopoly seeking to increase incomes. No doubt restriction
on entry is desirable from the point of view of the existing physicians,
but the public pressure needed to achieve the restriction must come
from deeper causes.
The social demand for guaranteed quality can be met in more than
one way, however. At least three attitudes can be taken by the state or
other social institutions toward entry into an occupation or toward
the production of commodities in general; examples of all three types
exist. (1) The occupation can be licensed, nonqualified entrants being
simply excluded. The licensing may be inore complex than it is in
medicine; individuals could be licensed for some, but not all, medical
activities, for example. Indeed, the present all-or-none approach could
37Thesituation is very reminiscent of the crucial role of the focal point in Scheliing's
theory of tacit games, in which two parties have to f i ~ da common course of action
without being able to comn~unicate;sce [21, csp. pp. 225 ti.].
"How well they achieve this end is another matter. R. Resscl points out to me that
they merely guarantee training, not continued goo:] performance as medical technology
chaagcs.
4RROTT: UNCERT.4INTY AXD MEDICAL CARE 967
be criticized as being insufficient with regard to complicated specialist
treatment, as well as excessive with regard to nlinor medical skills.
Graded licensing may, however, be much harder to enforce. Controls
cculd be exercised analogous to those for foods; they can be excluded
as being dangerous, or they can be permitted for animals but not for
humans. (2) The state or other agency can certify or label, v~ithcr!t
ccjmpulsory exclusion. The category of Certified Psychologist is now
under active discussion; canned goods are gradcd. Certification can be
done by nongovernmental agencies, as in the medical-board examnations
for specialists. (3) Nothing at all may be done; consumers niake
their own choices.
The choice among these alternatives in any given case depends on the
degree of difficulty consumers have in making the choice unaided, and
on the conscqucnces of errors of judgment. It is the general social coasensus:
clearly, that the lniss~z-fai~csolution for medicine is intolerable.
The certification proposal never seems to have been discussed seriously.
I t is beyond the scope of this paper to discuss these proposals in detail.
I wish simply to point out that they should be judged in terms of the
ability to relieve the uncertainty of the patient in regard to the quality
of the commodrty he is purcl~asing,and that entry restrictions are the
consequences of an apparent inability to devise a system in which the
risks of gaps in medical lrnowlcdge and skill are borne primarily by
the patient, not the physician.
Postscript
I wish to repeat h ~ r ewhzt has been suggested a5ove in several
places: that the failure of the market to insure against uncertainties
has created many social institutions in which the usual assumptions of
the market are to some extent contradicted. The medical profession is
only one example, though in many respects an extreme one. All professions
share some of the same properties. The economic importance
of personal and especial!^ family relationships, though declining, is by
no means trivial in the most advanced economies; it is based on nonmarket
relations that create gtlarantees of behavior which ivould otherwise
be afflicted with excessive uncertainty. hiany other examples can
be given. The logic and limitations of ideal competitive behav!or under
uncertainty force us to recognize the incomplete description of reality
supplied by the impersonal price system.
RZFEREECES
1. A. 4. ALCIIIAN,K. J. ARROW, An Ecor~onzicAnalysisAND Iv. h/I. CAPRO~~,
of t?ze Aia:.kst for Scie~:tistscnd ErzgZ~zc.e~s,RASD RkZ-2190-RC. Santa
Monica 1958.
958 THE AMERICAN ECONOMIC REVIEW
2. XI. .L\LLAIS, "Generalisation des thCories de llCquilibre Cconomique
gknkral et du rendement social au cas du risque," in Centre National de
la Recherche Scientifique, Ecofzonzetrie, Paris 1953, pp. 1-20.
3. 0. \IT. AND TI-IE NATIONALOPINIONRESEARCHANDERSON STAFFOF
CENTER,Voluntary Health I~zsura~zcein Two Ciiies. Cambridge, Mas;.
1957.
4. I<. J. ARROW,'(Econoinic Welfare and the Allocation of Resources for 

Invention," in Nat. Bur. Econ. Research, Tlie Role a d Directiorz of 

I?zaenti.~eBctiiiity: Econonzic and Social Factors, Princeton 1962, pp. 

6G9-25. 

5. -----, "Les rBle des valeurs boursi6res pour la rkpartition la meilleure
des risques," in Centre National de la Recherche ScientiEqxe, Econovzetrie,
Paris 1953, pp. 41-46.
6. F. M. BATOR,"The Anatomy of Market Failure," Quart. JOLLY.Eco:~.
4ug. 1958, 72, 351-79.
7. E. B,~UDIER, du temps dans la thkorie de ľkquilibre g6n-"Ľintroduction 

Cral," Les Cahiers Econonziqzies, Dec. 1959, 9-16. 

8. W. J. B n u ~ o ~ ,  the State. Cam-I4"elfare Econonzics and the Theory of
bridge. Mass. 1952.
9. I<. BORCII,"The Safety Loading of Reinsurance Premiums," Ska~zdinat~iskAktuariehdslzrift,
1960, pp. 163-84.
10. J. M. BUCHANAN  The Calculus ofATCD G. TULLOCIC, Colzscnt. Ann
Arbor 1962.
11. G. DEBREU,"Une economique de ľincertain," Econonzie Appliquc'e,
1960, 13, 111-16.
12. --, Theory of Values. Xew York 1959.
13. R. DUBOS,"Siedical Utopias," Daedalzu, 1959, 82, 410-21.
14. M. G. FIELD,Doctor mzd Patient in Sotliet Russia. Cambridge, Mass.
1957.
15. MILTON FRIEDRIAN,"The ~Iethodology of Positive Economics," in 

Essays in Positive Econonzics, Chicago 1953, pp. 3-43. 

16. --- ASD S. S. I ~ Z N E T S , fronz Irzdependerzt Professio~zalInconze 

Practice. Xat. Bur. Econ. Research, New York 1945. 

17. R. A. RESSCL,"Price Discrimination in Medicine," Jour. Law and Ecofz.,
1958, 1, 20-53.
18. T. C. I<OOPT.LANS, ofLL.Allo~ation Resources and the Price System," in
Threc Essays orz tlze State of Economic Science, New York 1957, pp.
1-120.
19. I. ).I. D. LITTLE,A C~ifiqueof Tlielfare Eco~zomics.Oxford 1950.
20. SEL~TA~IUSIIICIN,"Towards a Definition of Health Economics," Publlc
Healtlz Rcports, 1958, 73, 755-93.
21. R. R. NELSON,('The Simple Economics of Basic Scientific Research,"
Jour. Pol. Ecorz., June 1959, 67, 297-306.
22. T. PARSONS,Tlze Social Systcnz. Glencoe 1951.
23. M. J. PECICAND F. M. SCHERER,T l ~ eTYeapo7zs Acquisition Process: A7z
Eco7zomic A7zalysis. Div. of Research, Graduate School of Business,
Harvard University, Coston 1962.
969ARROW: UNCERTAINTY AND MEDICAL CARE
23. T. C. SCHELLIXG,Ťlze Stvategy oj Co~zflict.Cambridge, Mass. 1960.
25. A. I<. SIIAPIR~,"A Contribution to a I-Iistory of the Placebo Effect,"
Beliaz,ioral Science, 1960, 5, 109-35.
26. H. hl. S O ~ E R SAND A. R. SOMCRS,Doctors, Patie~zts,alzd Hcaltlz inszcrarzce.
The Brooking. Institution, Washington 1961.
27. C. L. STEVENSON,Ethics n~zdLarzguage. Kew I-Iaven 1945.
28. U. S. DEPARTMENTOF HEALTH,EDUCATION LI-ELFARE,AXD PIzysiciarls
for a Gvo:aing A~nerEca,Public Ilealth Service Publication KO. 709, Oct.
1959.
Oiz Optimal Insuralzce Policies
The two propositions about the nature of optimal insurance policies asserted
in Section IIT.Babove will be proved here.
Propositio:r 1. If an insurance company is ~i~illingto offer any insurance
policy against loss desired by the buyer at a premium which depends only
on the policy's actuarial value, then the policy chosen by a risk-averting
buyer will take the form of 100 per cent coverage above a deductible mini-
mum.
Note: The premium will, in general, exceed the actuarial value; it is
only required that two policies with the same actuarial value will be offered
by the company for the same premium.
Proof: Lct Tt- be the initial wealth of the individual, X his loss, a random
variable, I ( X ) the amount of insurance paid if loss X occurs, P the premium,
and Y ( X ) the wealth of the individual after paying the premium,
incurring the loss, and receiving the insurance benefit.
(1)  Y ( X ) = I V - P - X + I ( X ) .
The individual values alternative policies by the expected utility of his
final wealth position, Y ( X ) .Let U(y)be the utility of final wealth, y; then
his aim is to maximize,
( 2 )  C[Y(X)I I,
where the symbol, E, denotes mathematical expectation.
An insurance payment is necessarily nonnegative, so the insurance policy
must satisfy the condition,
(3)  I ( X ) 2 0 for all X.
If a policy is optimll, it must in particular be better in the sense of the
criterion (2), than any other policy with the same actuarial expectation,
E [ I ( X ) ] .Conqider a policy that pays some positive amount of insurance at
one level of loss, say X I , but which perinits the final wealth at some other
loss level, say X z , to be lower than that corresponding to XI. Then, it is
intuitively obvious that a risk-averter would prefer an alternative policy
1i7ith the same actuarial value which ~irouldoffer slightly less protection for
losses in the neighborhood of X I and slightly higher protection for those in
the neighborhood of Xz, since risk aversion implies that the marginal utility
970 THE ARIERICAN ECONOMIC REVIEI?'
of Y(X) is greater when Y(X) is smaller: hence, the original policy cannot be
optimal.
To prove this formally, let Il(X) be the origiilal policy, with I1(X)> O and
Yl(X1)>Y?(X2),where YI(X) is defined in terms of Il(X) by (I). Choose 6
sufficiently small so that,
(4) Il(X) > 0 for XI _< X < X1 +6,
(5) Yl(X') < YI(X) for X-2 _< X' I Xp +6, XI < X < X1 +6.
(This choice of 6 is possible if the f:lnctions I1(X), Yl(X) are continuous;
this can be proved to be true for the optimal policy, and therefore we need
only consider this case.)
Let .irl be the probability that the loss, X I lies in the interval (XI,
X1+6), a2the probability that X lies in the interval (X2,X2+6). From (4)
and (5) we can choose E > O and sufficiently small so that,
(6) I1(X) - 7r2~2 0 for X1 < X I XI +6,
for Xz < X' _< Xz +6, XI < X 5 XI +6.
Now define a ne~vinsurance policy, Iz(X),which is the same as II(X) except
that it is smaller by .ir2e in the interval from X1 to X1+6 and larger by a l e
in the interval from X2 to X2+6. From (6), Iz(X) > O every~vhere,so that
(3) is satisfied. LVe will sho1i7 that E[Il(X)]=E[IZ(X)] and that Is(X) yields
the higher expected utility, so that I1(X) is not optimal.
Note that Iz(X)-II(X) equals -T.ZE for XI<_ X_< X1-t-6, TIE for X 2 5 X
<X2+6, and O else~vhere.Let +(X) be the density of the random variable
X. Then,
so that the two policies have the sa~neactuarial value and, by asauniptio~~,
the same premium.
Define Yz(X) in terms of 12(X) by (1). Then Y2(X)- Yl(S) =12(X)
-Il(X). From (71,
(8) Yi(X1) < Yz(X') < Y2(X) < Yi(X)
for X2 5 X' < X2 +6, XI < X 5 X I + 6.
Since PI(X)- Y2(X)=O outside the intervals (XI, ,rrTl+G), (X2, X2+6), we
ARROIIT: UNCERTAINTY AND MEDICAL CARE 971
can write,
By the Mean Value Theorem, for any given value of X ,
(10) U [ Y ~ ( X ) ] u [ Y ~ ( x ) ]  - Y l ( X ) I- = u ' [ 1 7 ( x ) ][ Y ~ ( x )
= C 1 [ Y ( x ) ] - IJ[ I Z ( X ) I ~ ( x )
where Y ( X )lies between Y l ( X )and Yz(X). From (8))
Y ( X 1 )< Y ( X ) for Xz 5 X' < Xz +6, X I I X 5 X I +6,
and, since U1(y)is a diminishing function of y for a risk-averter,
~ ; ' [ Y ( x ' ) ]> u'[Y(x)]
or, equivalcnt1~-, for some number u,
u'[Y(x')]> u for X Z I X' I X z +6,
(11)
U 1 [ Y ( X ) ]< u for XI < X 5 X1 +6.
Korv substitute (10) into (9).
E ( u [ I ~ ~ ( x ) ]u [ Y ~ ( x ) ] )= - a2e+
J X ~ U 1 [ 1 7 ( X ) ] B ( ~ ) d ~ .
From ( l l ) ,it follows that,
E ( II'[E\(x)]- ) > - TzEUTl +TlEUTz = 0,u[ Y ~ ( x ) ]
so that the second policy is preferred.
I t has thus been shorvn that a policy cannot be optimal if, for some XI
and XZ,I ( X l )>0, Y(X1)>Y(X2).This may be put in a different form: Let
Y,;, be the minimum value taken on by Y ( X ) under the optimal policy;
then we must have I ( X )= O if Y ( X )>I\;,. In other words, a minimum
final wealth level is set; if the loss ~vouldnot bring wealth below this
level, no benefit is paid, but if it would, then the benefit is sufficient to
bring up the final wealth position to the stip~llatedminimum. This is, of
coLzrse, precisely a description of 100 per cent coverage for loss above a
deductible.
Lye turn to the second proposition. I t is now supposed that the insurance
company, as ~vellas the insured, is a risk-averter; hoxever, there are no
administrative or other costs to be covered beyond protection against loss.
Proposition 2. If thc insured and the insurer are both risk-averters and
there are no costs other than coverage of losses, then any nontrivial Pareto-
972 THE AMERICAN ECONOMIC REVIEW
optimal policy, I(X), as a function of the loss, X, must have the property,
O<dI/rEX<l.
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. 21; we give here a somewhat simpler
proof.
Proof: Let C(y) be the utility function of the insured, V(z) that of the
insurer. Let W oand TIT1 be the initial wealths of the two, respectively. I n
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 find wcalth po~itionsof the insured and insurer are:
Y(X) = Iťo - X +I(X),
Z ( 9 ) = TV1 - I(X),
re~pectively.Any givcn insurance policy then defines expected utilities,
16 =E ( C[Y(X)]] and .L =b{V[Z(X)]J , for the insured and insurer, respectively.
If we plot all points (u,a) obtained by considering all possii~leinsurance
policics, the resulting expected-utility-possibility set has a boundary
that is convex to ihe northeast. To see this, let II(X) and Iz(X) be any two
policies, arid let (241, 31) and (uz, z'z) be the corresponding points in the twodimeniional
expected-utility-possif~ilityset. Let a third insurance policy,
I(X), be defined as the avercageof the two given oues,
I(X) = (+)Il(X) +ca)Iz(x),
for tach X. ?hen, if Y(X), Yl(X), and Y2(X) are the final wealth positions
of the insured, and Z(X), Zl(X), and Zn(X) those of the insurer for each of
tlie three policies, I(X), Il(X\, and 1219),respectively,
Y(X) = (3Yl(X) + ($1 Y.(X),
Z(X) = (+)Zl(X) + (+)Zz(X),
aacl, because both parties have diminishing marginal utility,
0-[IT(X)] 2 (3)C[I'~(X)I + (3)z - [ ~ ~ z ( ~ ) l ,
v[z(x) ] 2 (3)v[Zl(X)I + ($1 [ZdX)I.
Since these statements hold for all X, they also hold when expectations are
tnlien. Hence, there is a point (zt, a) in the expected-utility-possibility set
for which u> (+)u1+(3)uZ,z 2 (3)~1~+($)21~.Since this statement holds for
every pair of points ( 1 ~ ~ ,yl) and (ti2, nz) in the expected-utility-possibility
set, and in particular for pairs of points on the ilortheast boundary, it follor\-s
that the boundary must be convex to the northeast.
From this, in turn, it follo\vs that any given Pareto-optimzl point (i.e.,
any point on the northenst boumdary) can be obtained by maximi~inga
linear function, cru+pa, with suitably chohen cr and /3 nonnegative and at
least one poiitive, over the expected-utility-possibility set. In other words,
a Pareto-optimal insurance policy, I(X), is one which maximizes,
.E{ u [ I ' ( ~ ) ] ) +PE(v[z(x)]) = E { ~ L [ Y ( x ) ]+BV[Z(X)]),
973ARROlll;: UKCERTAIIVTV AND MEDICAL CAKE
for soine a>O, P 2 0 , a > O or 8>0. To maximize this expectation, it is obviously
sufficient to maximize:
(13) f f W ~ ( ~ ) l+P W - ( X ) I ,
with respect to I ( X ) , for each X. Since, for given X, it follows fronx (12)
that,
dY(X)/dI(X) = 1, dZ(X)/dI(X) = - 1,
it follows by differentiation of (13) that I!X) is the solution of the equation,
(14) a u 1 [ y ( x ) ] - ~ V ' [ % ( X ) ]= 0.
The cases a=O or /3=O lead to obvious trivialities (oi~cparty simply hands
over all his wealth to the other), so we assume a>O, p>O. NOTVdifferentiate
(14) with respect to S and use the relations, derived from (12),
dY/dX = (ctI/dX) - 1, dZ/dX = - (dI/dX).
0r
dI/dX = a ~ " PV" [z(x) ] ).[I'(x) ] I { al;" [Y(x) ] -/Si11ce
U1'[Y(X)] <0, V1'[Z(X)]<O by the hypothesis that bcth parties are
risl.:-a\ erters, Propoiilio:: 2 fullox s.