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Working Paper Series

Our Money or Your Life: Indemnities vs.
Deductibles in Health Insurance

WP 00-04

This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/

Robert F. Graboyes
Federal Reserve Bank of Richmond

Our Money or Your Life: Indemnities vs.
Deductibles in Health Insurance∗
Robert F. Graboyes**
Federal Reserve Bank of Richmond Working Paper No. 00-4
August 2000
JEL Nos. I11, G22 D81
Keywords: health insurance, heterogeneity, indemnities, deductibles

Abstract
When the value of a medical treatment differs across individuals, it may be socially
beneficial to treat some, but not all, patients. If individuals are ignorant of their health
status ex ante, they should be willing to purchase insurance fully covering treatments for
high-benefit patients (Hs) and denying treatment for low-benefit patients (Ls). But if
prognoses are observable but not verifiable, insurers may have trouble denying care to
Ls. Deductibles force Ls to reveal their status by imposing a marginal cost on treatment,
but at a price of incomplete risk-sharing. Lump-sum indemnities can similarly induce Ls
to forgo treatment but are rare in health insurance markets. They were once more
common and remain so in non-health markets. This paper reviews the potential for
health insurance indemnities. We model an insurance market for a single illness and
derive conditions determining the relative efficiency of indemnities and deductibles.
We define a disease that strikes randomly, where there is no private information, and
where benefits are measured as cure rates. These and other assumptions yield several
rules of thumb: It is never socially or (ex ante) privately beneficial to offer an indemnity
larger than the cost of treatment. The optimal indemnity is always larger than the optimal
deductible. If Ls outnumber Hs, the best deductible contract always yields higher welfare
than the best indemnity contract. As the Ls' cure rate approaches 0, the choice of
indemnity or deductible depends entirely upon the relative numbers of Hs and Ls.

∗

This paper does not necessarily represent the views of the Federal Reserve System or the Federal Reserve
Bank of Richmond. This paper comprised a portion of a doctoral dissertation at Columbia University. The
author thanks the following for their assistance in writing this and two companion articles: Sherry Glied,
Ken Leonard, Dan O’Flaherty, Abigail Tay, and Josh Zivin and also Mark Babunovic, Dolores Clement,
Doug Hadley, Bob Hurley, Jeff Lacker, Roice Luke, Lou Rossiter, Rich Schieken.
**
Federal Reserve Bank of Richmond, Robert.Graboyes@rich.frb.org and Virginia Commonwealth
University/Medical College of Virginia rgraboyes@hsc.vcu.edu

1
1. Indemnities vs. Deductibles in Health Insurance
For some insured patients, the social costs of medical treatment exceed the private
benefits which, in turn, exceed the private costs of treatment. In a first-best world, people
would willingly buy health insurance with a pre-commitment to deny treatment in such
cases. In reality, it is difficult to bind people to such pre-commitments, and the result is
spending beyond the social optimum on at least some aspects of health care.
Given this market failure, the United States employs a number of roundabout
methods to reduce costly expenditures on low-marginal benefit care. Insurers deny care
by fiat (e.g., utilization review, coverage exclusions). They deter usage by imposing outof-pocket marginal costs (e.g., high deductibles). 1 One can argue that advance medical
directives accomplish this goal by moral suasion, regardless of whether that is the
intention of the signatories.
In many non-health markets, insurance offsets damages by means of cash
indemnities; individuals can either use the cash to reverse the damage (e.g., to have dents
removed from a car), or they may use the cash for other purposes which they value more
highly (e.g., a vacation). What people do with their indemnities reveals the value they
place on reversing the damage. Thus, indemnities help to limit resource malallocation.
Cash indemnities are seldom used in health insurance. 2 Instead, benefits usually help
allay the costs of medical services rendered. This restriction can induce people to seek
costly medical services which, for them, have minimal but positive benefits. For many

1

We use “deductibles” and “copayments” interchangeably.
Traditional fee-for-service policies are also referred to as “indemnification” policies. To avoid
confusion, this paper will use “indemnities” to mean payouts unrelated to services rendered, and we will
avoid using “indemnification” as a synonym for “fee-for-service.”
2

2
illnesses, there are obvious reasons why cash indemnities are impractical. 3 For certain
illnesses, though, indemnities might be a reasonable means of restraining health care
spending and making patients happier as well. Here, we model such an illness and search
for conditions under which indemnities are more efficient than deductibles at deterring
low-benefit patients from seeking expensive treatments.
The paper is organized as follows: Section 1.1 defines “ex post” moral hazard and
introduces the demand-side and supply-side approaches to moderating its effects. Section
1.2 illustrates the differences between indemnity and deductible contracts by means of a
simple thought problem. Section 1.3 outlines the history and rarity of indemnities in
health insurance. Section 1.4 looks at the recent history of the search for optimal
demand- and supply-side health insurance incentives. Section 2 constructs a model for
comparing the efficiency of indemnity and deductible contracts in a restrictive setting.
Section 3 is the conclusion and lists directions for future research. Most mathematical
proofs are contained in the Appendix, in order to avoid interrupting the article's flow.

1.1 Ex post Moral Hazard, Deductibles, and Managed Care
The economics of health insurance revolves largely around two problems, adverse
selection and moral hazard, that place economy and cost-sharing at loggerheads. Until the
early 1980s, health economists and health insurers both searched largely for demand-side
solutions. Typical was the search for the optimal deductible, which would minimize the
combined utility loss from moral hazard and risk to personal wealth. Deductibles

3

Indemnities would seem a poor settlement mechanism for an ailment such as lower back pain, the
presence of which is not easily observed, even by a specialist. Here, willingness to undergo treatment
provides evidence that patient actually suffers from the ailment and is not simply gaming the insurer.

3
persuade low-benefit patients to self-select away from treatment by imposing a private
marginal cost; such economy, though, comes at a cost of reduced cost-sharing. From the
early 1980s till the present, academicians, insurers, and other interested parties (notably
employers) shifted their attention to the supply-side. The tools known generically as
“managed care” seek economy and cost-sharing, but require high administrative expenses
(e.g., utilization review, capital expenditures on information management and protocol
development). Managed care also induces the consumer dissatisfaction and costly
gaming typically associated with nonprice rationing.
This brings us to the central question of this paper—whether indemnities can
efficiently persuade high- and low-benefit patients to reveal the values they place on
medical services. This paper returns to the search for efficient demand-side solutions.
To do so, it assumes away adverse selection. It also assumes away the type of moral
hazard we will label “ex ante”—where the promise of future insurance coverage induces
people to take excessive health risks, thereby increasing the incidence of illness.
Ignoring these two monumental problems allows us to focus solely on ex post moral
hazard, where people who are already ill purchase excessive medical services because
third parties bear the costs. Such assumptions are damningly unrealistic for many
illnesses. Other diseases, though, strike more or less randomly and their presence is
easily observed by outside observers. Such illnesses would accord reasonably well with
our assumptions. (Importantly for purposes of the model we build, the diagnosis is
assumed costlessly verifiable, but its prognosis is not.) The analysis modeled here posits

4
such an illness and carveout, or single-benefit insurance policy, to financially protect its
sufferers. 4
In other words, we are restricting this analysis to a class of medical conditions where
we do not face the classic principal-agent problem of health care—private information
held by either the patient or provider. Here, all agents have perfect information, but
providers cannot fully act on this information.

1.2 Indemnities and Deductibles: Pricing Life
Indemnities are so alien to health insurance that it might be helpful to illustrate
indemnities and deductibles through a simple thought problem composed of two cases:
Case 1 (Indemnity): You have been diagnosed with an illness that is 100%
terminal if untreated. The treatment provides a 2% chance of survival but costs
$500,000. Would you prefer full coverage of this treatment or $350,000 in cash?
Our evidence on how people would, in fact, react to such offers is paltry, but
introspection at least suggests the possibility of accepting the cash option. 5 It seems
logical that an individual’s response would depend, among other things, upon age, family
status, the cost of the treatment, level of financial wealth, and—crucially—the survival
rate posited for those treated. If the survival rate, for instance, were 50%, far more

4

“Single-benefit” plans commonly cover mental health, dental, and optical expenses (Weiner and de
Lissovoy 1993, p. 100).
5
There are health insurance mechanisms that require patients to make such Hobson's choices. For
example, to receive care under Medicare Hospice Benefit, a patient must surrender coverage for medical
treatments for the terminal illness. While the patient has the option later of switching back to medical
coverage (and surrendering hospice coverage), the program demands an either/or choice. Perhaps the best
evidence would come from examining the behavior of people who are wealthy enough to pay enormous
sums for experimental treatments not covered by health insurance or who are self-insured. Conceptually,
the simple indemnity policy described here takes a person of modest means, makes them wealthy, and
allows them to decide for or against from that vantage point.

5
people might opt for treatment rather than for cash. In contrast, consider the thought
problem restated in terms of a deductible contract:
Case 2 (Deductible): You have been diagnosed with an illness that is 100%
terminal if untreated. The available treatment will give you a 2% chance of
survival, but it costs $500,000. Your insurance will pay for the treatment, except
for a $3,000 deductible. Would you prefer the treatment or will you opt out of
treatment and keep your $3,000?
The deductible is unlikely to deter many from treatments and a 2% chance. If one
were to choose the cash option in Case 1 and the treatment option in Case 2, then we
would have strong evidence, at least for the person so choosing, that life has a price and
that ex post moral hazard exists. A population composed entirely of such people ought to
willingly bind themselves to forgo treatment in Case 1—if they know that others will be
similarly bound. (The premium will reflect the fealty of subscribers to this accord.)

1.3 Rarity of Health Insurance Indemnities
Rationing-via-indemnity is rare in health insurance. It is common in other types of
insurance—automobile and homeowner policies, for example. The owner of a damaged
car typically obtains several repair estimates and shares them with the insurer, who then
pays a cash settlement in an amount determined by a formula whose arguments include
the estimates and the policy’s provisions. The purpose of the settlement is to restore the
owner’s utility, not to restore his car (though restoring the car may be the end result).
Often, the owner is free to pocket the money, and his utility can wind up higher than,
lower than, or equal to its pre-collision level, depending on his preferences and the

6
amount on the check. Importantly, the owner's post-indemnity choices reveal the value
he places on fixing the car vis-à-vis other goods.
Indemnity insurance was common in health insurance in the days prior to efficacious
interventions. 6,7 A person diagnosed with a serious, perhaps fatal, illness might receive
an indemnity to cover palliatives, home nursing care, and burial, but discouraging
expensive treatments does not appear to have been a primary motive. Today, indemnities
primarily live on in odd corners of health insurance—in policies we don’t even tend to
classify as health insurance. Personal accident insurance often pays a fixed indemnity for
the loss of a limb or eye; as with auto insurance, the insurer doesn’t care what the
recipient does with the payout. Some indemnity provisions also survive in dental
policies. 8 “Dread disease” policies sometimes pay indemnities, though the triggering
event may be the purchase of a medical service (e.g., hospital stay) rather than mere
diagnosis. 9
The disappearance of indemnities began in earnest with the founding of Blue Cross in
the 1930s, which reimbursed beneficiaries for services rendered by hospitals and other

6

“the [insurance] industry focused primarily on lump-sum payment policies to underwrite the expense of
terminal illness and subsequent burial … policy proceeds were fixed and paid independent of whether ill
workers actually purchased medical services. In fact, only about 1 percent of the $97 million in sickness
benefits paid in 1914 went directly for medical care, a reflection of the primitive state of medicine in
reducing morbidity and mortality … By 1930 medical expenses generally outstripped wage losses due to
sickness, yet only 10 percent of the benefits paid under then-existing health insurance plans went to
treatment costs.” Feigenbaum (1997).
7
“Fifty years ago, physicians were little more than diagnosticians, their activities being essentially
‘limited to identification of … illness, the prediction of the likely outcome, and then the guidance of the
patient and his family while the illness ran its full, natural course.” Weisbrod (1991, p. 526), quoting the
Report of the President’s Biomedical Research Panel 1976.
8
Pauly (1986, p. 642).
9
Dread disease policies generally provide only supplemental coverage above and beyond standard health
insurance benefits. American Family Life Insurance Assurance Company markets cancer policies that pay
upon diagnosis; these products are especially popular in Japan. Mutual of Omaha offers a variety of
products, including one that pays both an up-front benefit and reimbursement for treatment. Canada Life
Assurance Co. offers lump-sum benefits on diagnosis of any of 12 diseases. These and other indemnitylike products are discussed in Panko (1999).

7
providers. 10 Arrow (1963) mentioned indemnities as one of three means of paying health
insurance benefits. 11 Shortly thereafter, though, health insurance indemnities largely
disappeared from the market and from the academic literature. Some came to view
service-related benefits as the defining characteristic of health insurance. In the past thirty
years, a small number of academic papers have suggested the use of indemnities in health
insurance. 12 Pauly considered the absence of indemnities to be “puzzling.”13 Indemnities
have reappeared in a few health insurance policies. As an experiment, one insurer
recently offered certain cancer sufferers the ability to collect cash in lieu of treatments
deemed to be futile. The indemnity was equal in value to the expected value of
treatments. 14 This policy was inspired by the viatical contracts recently offered by some
life insurers. A viatical allows a terminally ill person to collect in advance of his death
the discounted value of a life insurance death benefit. 15

10

Blue Cross's explicit goal was to shift expenditures toward hospitals, rather than to improve the health
status of its subscribers (Feigenbaum 1997).
11
Arrow (1963, p. 962).
12
Pauly 1971; Pauly 1986; Gianfrancesco 1983; Feigenbaum 1997.
13
“One puzzle is why indemnities are not more common. Only in dental insurance are benefits typically
conditional on the submission and approval of a diagnosis and plan of treatment, which in a sense defines
the ‘illness’ state. Perhaps the explanation is that for medical conditions, such pretreatment determination
of the illness state may be more costly, or more likely to be in serious error; such procedures are quite
uncommon in medical insurance. Other explanations are possible, of course—the medical profession may
have been more successful in resisting such limits …, or moral hazard in unconstrained dental insurance
may be that much worse.” Pauly (1986, p. 642)
14
This experiment was detailed in Trigon (1995). Non-small cell lung cancer patients were offered the
option of aggressive treatments with a negligible probability of success or a cash indemnity equal to the
expected cost of the treatments—an either-or proposition. Following is a statement from the study report:
“Physicians also report discomfort in discussing treatment options, including the option of no treatment,
with [patients]. Patients and their families seek hope and guidance from physicians who often respond in
terms of statistical opportunity related to various treatment modalities. … As cost of medical care continues
to be an issue of strong debate and discussion in American society, the value of clinical interventions for
members of this patient population must be assessed. Although this can become an emotional and difficult
issue to discuss, if clinical interventions produce no appreciable extension of life and no perceived benefit
in terms of quality of life for these patients, could the money directed toward clinical interventions for these
patients be better utilized in other ways by the patients themselves and, ultimately, by the providers?” For
regulatory and other reasons, the peer group offered this option was small and produced fewer results than
had been hoped for. The study was also reported in Journal of the National Cancer Institute (1996).
15
Crites-Leoni and Chen (1997) and Kosiewicz (1998) discuss medical and legal aspects of viaticals.

8
It may be tempting to dismiss indemnity-based policies altogether in health insurance,
where adverse selection is a huge problem, as are both flavors of moral hazard. The
emotions associated with health care are also problematic; health care providers and
insurers are hard-pressed to deny medical treatment and coverage to patients, even when
those patients have pre-committed to abstain from service in a given circumstance. Many
authors cite these and other characteristics to argue that health care markets behave
differently from other markets. Certainly the problems in health care differ in degree, if
not in quality, from those of other markets.
To some scholars, though, indemnities in homeowner and auto collision policies
suggest some possible applications to health insurance. The typical homeowner, for
instance, would not likely wish his insurer to help select his new wardrobe after a fire or
even to require that he purchase a new wardrobe—perhaps he now prefers a boat. Yet
this micro-management of spending behavior is analogous to the insurer’s role in
monitoring health care. However, we know very little about how or whether indemnity
policies might work in health care markets. Yesterday’s sickness/burial policies and
today’s isolated quasi-health indemnity policies (e.g., personal accident insurance) are
too rare and limited to shed much light on the subject. The model in this paper attempts a
very modest step in the direction of understanding how and whether indemnity policies
might help improve health care and when they wouldn’t.

9
1.4 Deductibles and Managed Care
Evidence shows that people consider cost when making their health care treatment
decisions. 16 To an economist, the ideal in medicine is not to give treatment up to the
point where the marginal benefit to the patient is zero; rather, the ideal is to equate
marginal cost and benefit, a point that seems to run counter to some views expressed in
the medical literature. 17 In many ways, our health care system recognizes that we cannot
afford to provide medical care to all patients who might benefit. Experimental
procedures may not be covered by insurance. Posters in restaurant windows ask passersby to contribute to potential organ recipients. Battlefield triage is perhaps the most vivid
indicator that survival rates guide the choice of who ought to receive medical resources.
One could argue that the disinterest in indemnities is part of a more general pattern of
disinterest in demand-side incentives in health insurance. Until the 1980s, the health
insurance literature focused on the search for ideal deductibles and copayments. 18 With
the advent of widespread private and governmental health insurance, health care
expenditures grew more rapidly than many wished; simultaneously the theoretical
literature cast doubts on the whole notion of optimizing via consumer incentives. 19
Between the 1960s and 1980s, health care spending rose dramatically as a percentage
of GDP and had become a major item on the public policy agenda. Weisbrod (1991)

16

See Manning (1987).
Glied and (1998) and Callahan (1990) survey differences between the views of medical and economics
professionals on what constitutes appropriate objective functions.
18
Ellis and McGuire (1993, p. 1356) said that the bulk of health economics research in the 1970s was in
this area. Feldstein (1972) estimated the ideal deductible for minimizing the combined welfare losses from
inadequate risk-pooling and from excessive insurance-induced health care expenditures. Manning et al.
(1987) reported the results of the RAND Health Insurance Experiment, which strongly supported the
existence and exploitability of downward-sloping demand curves in health care.
19
Rothschild and Stiglitz (1976) and others raised concerns over whether stable equilibria in such
markets would even exist. Glied (1998, p. 38) notes that one problem with demand-side incentives was
that there were few dimensions on which plans could actually compete.
17

10
summarized the connections between overspending and insurance. 20 The role of moral
hazard was mentioned by Glied (1994). 21
Given the limited success of researchers in devising demand-side mechanisms for
controlling costs, the focus in both the academic literature and in public policy shifted
toward supply-side inducements to economize. Some looked toward a centralized model
evident in Medicare and Medicaid; DRGs sorted medical procedures into a finite number
of discrete categories, thereby setting the stage for de facto price controls. 22
The other principal direction was managed care—a decentralized version of cost and
expenditure controls intended to emulate a competitive equilibrium in health care and
insurance. Managed care produced a wide range of supply-side inducements for cost
containment, 23 generally involving inducements for providers to permanently enforce a
situation of excess supply on the market. 24 Specific mechanisms for doing so include
selection and organization of providers, methods for paying providers, and methods for
monitoring service utilization (Glied 1998, pp. 4-5). Often, there are also some demandside mechanisms, such as price differentials for in- and out-of-network providers.

20

“The expansion of health insurance has paid for the development of cost-increasing technologies, and
… the new technologies have expanded demand for insurance.” Weisbrod 1991, p. 524. Other papers
exploring this interplay between insurance and technology include Cutler and Sheiner (1997); Ellis and
McGuire (1993); Goddeeris (1984).
21
In theory, moral hazard may also increase the rate of growth in health spending through its effects on
the development and diffusion of new health care technologies. By reducing the costs to a consumer of
using a new, costly technology, moral hazard expands the market for such innovations. At the same time,
moral hazard may limit the market for innovations that promise to reduce costs. Glied (1994, p. 13)
22
An important area of research toward this end was the development of resource-based relative value
scales, which defined the value of medical care in terms of the quantity and quality of inputs. This
literature was described in great detail in Hsiao et al. (1988) and 11 other articles contained in a special
issue of JAMA.
23
Managed care plans differ across many dimensions. A number of authors have devised taxonomies, of
which Weiner and de Lissovoy (1993) is an example.
24
Ellis and McGuire (1990) wrote that, “[Q]uantity demanded regularly exceeds quantity supplied in
health maintenance organizations (HMOs), where on the margin, consumers often face no cost sharing.”

11
After two decades of managed care, its weaknesses, too, are now apparent. There has
been somewhat of a retreat from the idea that optimal insurance plans are likely to
emerge solely from supply-side mechanisms. While seeking theoretical justifications for
departing from a purely cost-based reimbursement (retrospective payment system), Ellis
and McGuire drew from both the demand-side and supply-side literatures in investigating
the theoretical advantages of combining mechanisms from both sides. 25
In the policy arena, interest in demand-side mechanisms included the development of
the Point of Service plans and Medical Savings Accounts. Ferrara (1995 and 1996)
discusses MSAs in detail and, in the second paper, sketches the key characteristics of a
number of such plans. In one typical plan, policyholders face a $3,000 deductible, so that
all amounts less than that are 100% covered by patients on an out-of-pocket basis. The
plan includes a catastrophic coverage provision so that expenses beyond the $3,000
deductible are completely covered. By imposing the full cost of small-ticket items on
consumers, it discourages overuse of those services and, perhaps more importantly, the
high administrative costs associated with small and frequent claims. On the negative
side, the plan eliminates risk-pooling for these small-ticket items. The plan allows for
complete risk-pooling for big-ticket items; but once again, consumers bear little or no
marginal cost for big-ticket items and, thus, have incentive to overspend.

25

“The problem of designing an optimal health care payment system must be recast when it is recognized
that payment instruments on both the demand and the supply side can be used to attain the social goals of
efficient utilization and minimization of patient financial risk. Neither the literature on optimal insurance
nor the literature on optimal provider reimbursement takes account of the other side of the policy coin.
Empirical evidence from the health sector leaves no doubt that both demand and supply-side payment
practices influence utilization.” Ellis and McGuire (1990), p. 393. Their argument is expanded in Ellis and
McGuire (1993), where they argue that such a combined supply/demand mechanism is better than either
separately at dealing with the pooling/moral hazard tradeoff.

12
2. The Model
We have introduced the notion of indemnity-based health insurance. Now, we
construct a model that asks how such a contract would stack up against other insurance
regimes. This model compares the welfare efficiency of two health insurance contracts,
both of which induce some patients to seek treatment and others to opt out. And we can
compare those two contracts with a regime of full insurance for all patients and one of
zero insurance for anyone. In one contract, deductibles require patients to bear some outof-pocket costs for treatment. In the other, cash indemnities compensate patients for
abstaining from treatment.
Section 2.1 lays out the assumptions and notation for the model. There are well
people (Ws) and sick people (Ss); Some Ss have a high cure rate (Hs) and others a low
cure rate (Ls). We assume away adverse selection and ex ante moral hazard. This means
that a person cannot hide the disease or his susceptibility to it; and he can neither prevent
nor induce the disease, nor even affect his probability of contracting it. We also assume
costless diagnosis.
Section 2.2 presents a criterion for determining whether indemnities or deductibles
are more efficient at deterring Ls from treatment. By using a more specific utility
function, Section 2.3 refines this criterion to an explicit function of the primitive
parameters. By sacrificing some generality, this section also allows us to derive some
rules of thumb—namely that the relative size of the L and H populations is a primary
determinant of the relative efficiency of indemnities and deductibles. Once we have
decided which is the more efficient mechanism for deterring Ls from treatment, Section
2.4 asks whether we wish to deter Ls; in some cases, we may instead wish to cover both

13
Hs and Ls or to cover neither. Finally, Section 2.5 discusses why the particular utility
function used here was chosen.

2.1 Setup
Indemnities would be problematic for illnesses whose likelihood is partly in the
control of the patient or whose presence is undetectable by the insurer. There are,
however, illnesses that appear to strike randomly, that are observable to both patient and
physician, and are expensive to treat.
These assumptions are restrictive, but not prohibitively so for a number of illnesses.
An example might be a cardiovascular accident (CVA) or stroke. 26 The incidence of
CVA is somewhat affected by long-term lifestyle and heredity, but the assumption of
randomness isn’t overly unrealistic. At least for a broad percentage of the population, the
CVA comes with little warning—before its onset, no one knows if the victim is a W or an
S. Once the CVA hits, the severity—one’s status as an H or as an L—is objectively
observable. Diagnosis and monitoring are not costless, but costs are small enough
compared with treatment that we can ignore them. For an insurer, though, it is difficult to
differentiate legally between Hs and Ls—a promise of coverage is valid for either. Some
people may fall clearly into one category or the other, but defining the boundary is legally
problematic. Our model assumes that treatment costs are identical for Hs and Ls; while
this is unrealistic, it sets the bar higher for the model by saying “even if Ls are no more
expensive to treat than Hs, it is still optimal to treat Hs and not Ls.” Were we to assume
higher treatment costs for Ls, such a finding would be more likely.

26

This example was suggested by my colleague, Dr. Richard Schieken, Chairman of the Department of
Pediatric Cardiology at the Medical Campus of Virginia Commonwealth University.

14
Again, CVAs may be an appropriate illness for our model because adverse selection,
ex ante moral hazard, and diagnosis and monitoring costs (relative to treatment costs) are
all small enough for us to ignore. Similar conditions arguably hold for various mental
illnesses (e.g., schizophrenia) and certain cancers (e.g., some leukemias).
The analysis presented here is limited to such illnesses. The model developed below
defines criteria for judging the relative efficiency of deductibles and indemnities as
demand-side rationing tools. We assume away many of the most important issues in
health insurance—adverse selection, costly monitoring, etc. This simplifies the analysis
but limits the findings to a limited range of insurable events where these assumptions are
realistic. By sacrificing generality, we reduce the questions to a tractable set of problems
and provide a foundation on which to build more general models in later papers. We
assume the following:
[I]

[II]
[III]
[IV]
[V]
[VI]

[VII]

Ex post utility is a state-dependent Von Neumann-Morgenstern function where
U(y; w)=U(y; s)+k, with Uy >0 and Uyy <0. y is ex post monetary wealth, w and s
are the two values of a binary variable representing well and sick states, and k is a
constant denoting the difference in utility between the two states for any y. This
functional form means that utility is state-dependent, but marginal utility is not.
The insurance policy protects against a single illness. It is a carveout—similar to a
dread disease policy, although dread disease policies' benefits are often contingent
upon a hospital stay or other medical service.
Adverse selection is not an issue. All agents are equally likely to contract the
illness. That probability is known both to subscribers and insurers.
There is no ex ante moral hazard; the presence or lack of insurance does not
influence the behavior of insured parties before they contract the illness or, hence,
the incidence of disease.
Diagnosis is binary and unambiguous and requires no costly monitoring.
Sick people are classified as Hs or Ls, based on their probability of cure if treated.
An individual's likelihood of cure, κH or κL, is costlessly observable by both the
patient and the insurer. However, the prognosis is not legally verifiable, so
patients can act on the information, but insurers cannot. The insurer cannot, for
example, promise to pay for chemotherapy if the probability of cure is 5%, but not
if it is 1%, though the patient may accept or decline treatment on the same basis.
This is because patients cannot bind themselves to forgo treatment if they are Ls.
There are no loading costs or other fixed costs.

15
[VIII] The cost of treatment is large enough that no one can purchase it without
insurance. In other words, there is no borrowing or capital market.
The driving force behind the model is that one's status as an H or an L is not legally
verifiable. Insurers cannot mandate differential benefit payouts for the two groups. So,
they must impose marginal costs large enough to deter the Ls, but not so high that it
deters Hs. Forcing patients to reveal their preferences, though, prevents us from equating
marginal utility of wealth across individuals, thus reducing welfare.
Notation falls into three general categories—primitive assumptions, contract
parameters and ex post wealth, and welfare under different modes—the word used here to
denote insurance regimes.
Initial conditions: These parameters define the state of the world:
πW
πH
πL
πS
κH
κL
y0
x
k

percent of subscribers who are well
percent of subscribers who are sick and will experience a high cure rate if treated
percent of subscribers who are sick and will experience a low cure rate if treated
percent of subscribers who are sick: π H+ π L
the cure rate for Hs
the cure rate for Ls
initial wealth of all agents
the cost of treatment
the welfare loss of having the disease; it is completely reversed if cured
Contract parameters (indemnities, deductibles, premiums) and ex post wealth:

i
i*
d
d*
pi*
pd*
pf
y

A cash indemnity large enough to deter Ls from seeking treatment
The minimum cash indemnity large enough to deter Ls from seeking treatment
A deductible large enough to deter Ls from seeking treatment
The minimum deductible large enough to deter Ls from seeking treatment
The insurance premium paid by all subscribers under the indemnity contract
The insurance premium paid by all subscribers under the deductible contract
The insurance premium paid by all subscribers under the full-insurance contract
ex post wealth; y0 minus premiums and deductibles paid or indemnities received
Welfare under different modes: Mode H is infeasible because insurers cannot be

legally bound to refuse treatment if they are found to be Ls. I, D, Z, and F are feasible:

16
Ûh
Mode H: Hs 100% covered, Ls not treated; this mode is infeasible.
Ûi
Suboptimal indemnity; deters Ls, but not Hs, from seeking treatment.
Ûi*
Mode I: Optimal indemnity; deters Ls, but not Hs, from seeking treatment.
Ûd
Suboptimal deductible; deters Ls, but not Hs, from seeking treatment.
Ûd*
Mode D: Optimal deductible; deters Ls, but not Hs, from seeking treatment.
Ûz
Mode Z: Zero insurance; neither Hs and Ls are treated
Ûf
Mode F: Full insurance; treatment for Hs and Ls 100% covered
Û
MAX[Ûi* , Ûd* , Ûz, Ûf]; the optimal policy across all modes
U(-;w) State-dependent utility function in well state
U(-;s) State-dependent utility function in sick state

Importantly, these utility functions can be read as either social welfare functions or
individual utility functions. Because no one knows his or her status (W, H, or L) ahead
of time, the socially preferred contract is also the individual optimum. Insurers can offer
zero insurance, full insurance, the optimal indemnity contract, a suboptimal indemnity
contract, the optimal deductible contract, a suboptimal deductible contract; individuals
will always find the socially optimal contract to be privately optimal, as well.

2.2 D/I Boundary Condition: General Case
Based on the above assumptions, Theorem 1 defines the boundary condition signaling
whether the deductible mode D or the indemnity mode I is the more efficient means of
inducing Ls, but not Hs, to opt out of medical treatment. Expected welfare differs under
these two modes solely because of differences in the ex post wealth distribution; for all
individuals, health is identical under these two modes. In both cases, Ws are well, all Hs
are sick and treated (and a fraction κH cured), and all Ls are sick and untreated.
>
Theorem 1: The assumptions imply the D/I boundary condition Ûd* = Ûi* iff
<
>
U(y0 -π Hx+πHd*;w)-U(y0 -π Hx-π Li*;w) = π SκLk, whose sign determines the relative
<

17
efficiency of the optimal deductible contract and the optimal indemnity contract in
deterring Ls from seeking medical treatment.
Theorem 1 is built on Propositions 1-5, shown below and proven in the Appendix.
(P.1) states the level of welfare if mode I is adopted and the indemnity is set at the
minimum deterrent indemnity i*. (P.2) states that the minimum deterrent indemnity is
the optimal deterrent indemnity. (P.3) states the level of welfare if mode D is adopted
and the deductible is set at the minimum deterrent deductible d*. (P.4) states that the
minimum deterrent deductible is the optimal deterrent deductible. (P.5) derives the D/I
boundary condition by combining (P.1) and (P.3) and is the key result of the paper.
(P.1)

ˆ
U i* = U ( y 0 − pi* ; w) − (π S − π H κ H − π Lκ L )k , where p i* = π H x + π L i *

(P.2)

Ûi* >Ûi ∀ i>i*

(P.3)

ˆ
U d * = U ( y 0 − p d* ; w) − (π S − π H κ H + π H κ L )k , where p d* = π H x − π H d *

(P.4)

Ûd* >Ûd ∀ d>d*

(P.5)

>
>
Ûd* = Ûi* iff U(y0 -πHx+π Hd*;w)-U(y0 -π Hx-πLi*;w) = π SκLk
<
<

Theorem 1 tells the insurer whether to offer an indemnity or deductible contract.

2.3 D/I Boundary Condition: Logarithmic Specification
Theorem 2 obtains stronger results by restricting the utility function to a logarithmic
specification, where U(y; w)=ln(y) and U(y; s)=ln(y)-k. This yields a more explicit
version of the D/I boundary condition obtained in Theorem 1, as well as enabling us to
derive some rules of thumb that shed intuition on the D/I boundary condition.

18
Theorem 2: If utility is a logarithmic function U(y;w)=ln(y), the D/I boundary
>
>
1 + (φ −1)π L
ˆ
ˆ
condition becomes U d * = U i* iff
=1.
[φ − (φ −1)π H ]φ π H + π L −1 <
<
Theorem 2 is proven by Propositions 6-8, shown here and proven in the Appendix.
(P.6) and (P.7) derive explicit terms for i* and d*, respectively. Using i* and d*, (P.8)
derives the explicit D/I boundary condition.

(φ − 1)( y 0 − π H x )
, where φ=exp(κLk)
1 + (φ −1)π L

(P.6)

i* =

(P.7)

d* =

(P.8)

>
>
1 + (φ −1)π L
ˆ
ˆ
U d * = U i* iff
=1
[φ − (φ −1)π H ]φ π H + π L −1 <
<

(φ − 1)( y0 − π H x )
φ − (φ − 1)π H

where φ=exp(κLk)

This boundary is a restricted version of the condition shown in (P.5) and again marks
the point at which insurance subscribers are indifferent between the optimal indemnity
policy and the optimal deductible policy. Later, we will use numerical example and
graphics to illustrate the significance and behavior of this criterion.
Theorems 3, 4, and 5 derive some rules of thumb that emanate from Theorem 2.
Theorem 3 shows that we can sometimes ascertain the relative efficiency of D and I
merely by comparing the shares of Ls and Hs in the population; if Ls outnumber Hs, we
will always prefer D to I. Theorem 4 shows that as the cure rate for Ls approaches zero,
the D/I boundary condition collapses to a simple rule of thumb: if there are more Ls than
Hs, use a deductible; if there are more Hs than Ls, use an indemnity (Theorem 3 only
demonstrates the first half of Theorem 4.). Theorem 5 shows that the optimal indemnity
i* is always larger than the optimal deductible d*, a fact that helps explain the behavior in

19
Theorems 3 and 4. Theorem 6 shows that the relative desirability of modes F and I
depend solely on the relative size of the treatment cost x and the indemnity i*.
Theorem 3: If π L≥π H, the optimal deductible policy is always more efficient than
the optimal indemnity policy. This theorem consists of three propositions. (P.9) proves
the equality, (P.10) the inequality, and (P.11) simply combines (P.9) and (P.10).
(P.9)

If π L=πH, then U d * > U i*

(P.10) If π L>πH, then U d * > U i*
(P.11) If π L≥πH, then U d * > U i*
Theorem 3, however, says nothing about how to decide between D and I if π L<π H.
We will find that D is preferred if π L is close to π H, that I is preferred if π L is not close to
πH. The meaning of “close” is defined formally by the D/I boundary condition but does
not translate into any neat rules of thumb. However Theorem 4 shows that as the Ls' cure
rate κL becomes small, so does “close.”
Theorem 4/Proposition 12: In the limit, as the Ls’ cure rate κL approaches 0, the
D/I boundary condition collapses to Ûi* >Ûd* iff π H>π L.
>
>
(P.12) Ûi* = Ûd* iff π H = πL
<
<
So, choosing between the deductible and indemnity contracts becomes simple if Ls
are essentially hopeless cases. The insurer needs only to compare the population shares.
Theorem 5: The optimal indemnity is always larger than the optimal deductible.
(P.13) i*>d*
The intuition behind this is that an individual with a deductible policy must choose
whether or not to seek treatment, he does so from a wealth level of y0 . If an individual

20
with an indemnity policy must choose whether or not to seek treatment, she does so from
a wealth level of y0 +i*. Thus, other things being equal, the person with the indemnity
policy has a greater willingness to pay for treatment than does the person with the
deductible policy. Thus, the reward for forgoing treatment must be larger for the one
with the greater willingness to pay. 27 (Figure 4 in the Appendix offers an intuitive
representation of this theorem.)
So, if the L and H populations are equal, indemnities will yield a less desirable wealth
distribution than will deductibles. With even larger numbers of Ls, the problem becomes
worse. Indemnities will only be desirable if Ls are few, relative to Hs.

2.4 I and D versus Z and F
The preceding theorems have demonstrated some of the intricacies of comparing the
relative efficacy of deductible and indemnity contracts. But we have not yet addressed
when we would wish to use either. The preferred contract is that one whose expected
welfare is higher than all other feasible contracts, or:
(P.13) Û=MAX(Ûi* , Ûd* , Ûz, Ûf)
We now know from the boundary condition how to choose between I and D. Now,
welfare under the more efficient of those two contracts must be compared with welfare
under modes Z and F. Propositions 14 and 15 give the formula for welfare under those

27

This finding calls to mind the Willingness to Pay-Willingness to Accept literature from environmental
economics [See Hanemann (1991) and Boyce et al. (1992). The WTP-WTA literature asks why people
seem to value losses more highly than gains. (Here, the indemnity would pay for the loss of treatment,
while the deductible would pay for the gain of a treatment.) In much of that literature, the mystery comes
from the fact that small losses and small gains seem to differ so widely. Here, though, the losses and gains
are so huge relative to wealth that there is little mystery in the outcome of this proposition. Simply put, the
person accepting an indemnity is, upon diagnosis, far wealthier than the person considering the deductible,
so the disparity in valuation is not surprising.

21
two modes. The derivations of these formulae should be obvious from inspection, so no
proofs are provided.

ˆ
(P.14) U z = U(y 0 ; w ) − πS k
$
(P.15) U f = U( y 0 − p f ; w) − ( π S − π H κ H − π L κ L )k , where p f = πS x
At this point, we can derive one final rule of thumb.
Theorem 6: The relative efficiency of modes F and I is determined solely by the
relative size of the optimal indemnity i* and the cost of treatment x. More formally,
>
>
(P.16) i* = x iff Ûf = Ûi*
<
<
You never want pay a patient more than the cost of treatment to forgo treatment. This
is true even if you can prevent them from then purchasing treatment and pocketing the
difference. The opposite inequality yields a somewhat less intuitive result. If Ls can be
bribed into forgoing treatment for less than the cost of treatment, an indemnity contract
will always be preferable to full insurance. The proof of this is in the Appendix.

2.5 Digression: Medical Insurance without Medicine
The model developed in this paper assumes that for a given level of wealth, utility
depends on the state of health, but marginal utility of wealth (MUW) does not
(assumption [1]). While this assumption is restrictive, it is not an unusual formulation in
health economics papers. The assumption is employed here for several reasons,
including simple practicality—the assumption lends itself to tractable solutions.
A more important reason is to focus solely on incentives for Ls to accept or forgo
medical treatment; with certain utility functions (where assumption [1] does not hold),

22
there is incentive to redistribute wealth between well and sick, even if medical treatment
is not a consideration. We can best explain this distinction by positing a world in which
there are no medical treatments, but there is medical insurance. In fact, “sickness”
policies in the early twentieth century paid sick or dying individuals cash indemnities to
cover burial costs and palliative care or to compensate the sick for their suffering. If
assumption [1] holds, there can be no such flows with fair insurance, as we shall see.
Figures 1, 2, and 3 represent three different ways that one might postulate the statedependent utility function in such a world. Figure 1 accords with assumption [1]; MUW
is always equal in both states. In Figure 2, for any given level of wealth, the MUW is
higher for sick people than for well people. Figure 3 shows the opposite case, where the
MUW is higher for well people than for sick people.

23

UW
US

Figure 1
Marginal utility of
wealth equal in well and
sick states

yW
y0
yS

UW
US

yW

y0

Figure 2
Marginal utility of
wealth higher in sick
state than in well state

yS

UW

US
yS y0

yW

Figure 3
Marginal utility of
wealth higher in well
state than in sick state

24
In each of the three figures, y0 is an individual’s ex ante wealth, yW is his ex post wealth
if he turns out to be well, and yS is his ex post wealth if he turns out to be sick. We
assume this to be a fair insurance policy; all premiums paid in are later paid out as
benefits (indemnities); so, y0 -yW =π S(yS- yW ) in Figure 2 and y0 -yS=π S(yW -yS) in Figure 3.
In Figure 1, all three wealth levels are the same, meaning that premiums and payouts
are both zero. This occurs because MUW in the two states is already equal, so once the
subscribers know their health states, we are already at a Pareto optimum.
In Figure 2, MUW is always higher for Ss than for Ws. So, ex ante, all subscribers
pay a premium of y0 -yW . Ex post, those subscribers who are sick receive an indemnity of
yS–yW thereby equalizing MUW between those in the two states. We can think of this as
compensation for pain and suffering.
Figure 3 shows a more counterintuitive case. Here, the MUW is always higher for
well people than for sick people with the same wealth. All subscribers pay a premium of
y0 -yS. Once the diagnoses are made, however, the insurance plan pays well people an
indemnity of yW –yS. This indemnity seems a little odd in that it compensates well people
for lack of pain and suffering.
Figures 2 and 3 may be more understandable in cases of intrafamily wealth
redistribution. Suppose a person becomes untreatably ill and his family contributes to a
fund to pay for pain-killers, hospice care, and airplane tickets for visiting family
members. This is the Figure 2 case, and it presumes that a few extra dollars are more
precious to the sick person than to the well relatives. Figure 3 is the opposite case: a
person becomes a permanent invalid and turns some bank accounts over to her children
and grandchildren on the logic that they are better able than she to enjoy the wealth.

25
Perhaps the Figure 3 case occurs within the families because of interpersonal utility
functions or because the family structure creates implicit contracts which are enforceable
because one's health condition and possessions are easily observable by other family
members. Whatever the underlying logic and motivations, the Figures 2 and 3 worlds
would confound the central issue of this paper—when do we want to exclude people from
medical treatment, and how can we get them to self-select out of treatment?

3. Conclusion
We have modeled the relative advantages of indemnities and deductibles in
discouraging the use of costly medical services by patients who receive small benefits
from treatments. Under moderately restrictive conditions, several rules of thumb emerge.
First, it will never be optimal for the insurer to offer deterrent indemnities that are larger
than the cost of treatment. Second, if there are more Ls than Hs, insurers should use
deductibles rather than indemnities. Third, if Hs substantially outnumber Ls, indemnities
are more efficient than deductibles. Fourth, as the cure rate for Ls drops toward 0, the
choice of indemnity policy or deductible policy approaches a simple matter of asking
whether there are more Hs than Ls. Fifth, we can derive additional rules that determine
whether the optimal deterrent mechanism improves on full insurance or zero insurance.
In future papers, we will want to expand this research by generalizing the model in
several ways: (1) An important assumption here is that the utility cost of illness is
invariant with respect to wealth. As presented in Section 2.5, other utility functions can
make it harder or easier to justify indemnities. (2) We will want to examine how these
findings hold up when there is a range of cure rates, rather than only two. We will also

26
want to ask whether, in fact, prognoses tend to be massed at a few discrete points or
spread across a continuum. The former could be the case if, for example, different
prognoses result largely from discrete factors, such as the presence or absence of a
comorbid condition. (3) Since adverse selection and ex ante moral hazard are prevalent
with many illnesses, future research might quantify the amount of each that is consistent
with an offer of indemnities. (4) Technological change is an important factor in health
insurance markets, and we will want to examine how changes in the efficacy of
treatments affect such markets.
It will also be important to do empirical research based on this model and variations
on the model. Given the paucity of actual indemnities, such research poses some
practical challenges. The likeliest source of data may come from examining how
individuals of different wealth levels behave with respect to costly, risky treatments.
Such information could either come from the self-insured or from patients whose
illnesses are excluded from coverage (e.g., experimental treatments). Referring to Figure
4 in the Appendix, we could estimate a minimum deterrent deductible d* for people with
income of y0 . Then, we could estimate the minimum deterrent deductible d** for people
whose initial wealth endowment equals y0 +i*. If our model is correct, d** should
coincide with i*. 28
Perhaps the most important area of research would be to ask why it is that indemnities
are so rare today in health insurance. If an indemnity contract is optimal for disease A
and a deductible contract is optimal for disease B, why do we not see health insurance

28

This would conflict with the social psychology literature mentioned by Fuchs and Zeckhauser (1987,
p.265). That literature, he notes, suggests “that individuals are risk averse with respect to gains but risk
preferring with respect to losses” a condition that would drive a wedge between d* and i*.

27
unbundled into separate A and B policies? Several possible reasons come to mind. (1)
The problem may be political. Paying a patient to forgo treatment is analytically
equivalent to refusing to treat a patient unable or unwilling to pay the deductible.
However, if these are perceived by the political system to be ethically different, then that
perception may be sufficient reason to avoid indemnities. (2) Indemnity contracts may be
more unstable dynamically in the wake of changes in technology, wealth, treatment costs,
etc. (3) The time-inconsistency problem may be worse with indemnities than with
deductibles. There may be a temptation for patients to seek treatment after receiving (and
perhaps after spending) an indemnity. Fighting such cases in the legal system may be
difficult for insurers from a public relations standpoint. (4) Perhaps adverse selection and
ex ante moral hazard are simply too overwhelming and prevalent to permit indemnities.
(5) Perhaps the boundaries between different diseases are also observable but not
verifiable so that insurers and patients would engage in costly arguments over diagnosis.

28
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32
APPENDIX

>
Theorem 1: The assumptions imply the D/I boundary condition Ûd* = Ûi* iff
<
>
U(y0 -π Hx+πHd*;w)-U(y0 -π Hx-π Li*;w) = π SκLk, whose sign determines the relative
<
efficiency of the optimal deductible contract and the optimal indemnity contract in
deterring Ls from seeking medical treatment. Proof: Propositions 1-5.
Proposition 1: The minimum deterrent indemnity contract yields welfare of:
(P.1)

ˆ
U i* = U ( y 0 − pi* ; w) − (π S − π H κ H − π Lκ L )k , where p i* = π H x + π L i *

Equation (1.1) defines an indemnity i* where Ls who have already been diagnosed
are indifferent between the indemnity and fully covered medical treatment:
(1.1)

U( y 0 − p i* + i*; s) = κ L U( y 0 − p i* ; w ) + (1 − κ L ) U( y 0 − p i* ; s)

(1.1) is really a limit, since the left-hand side should be at least an epsilon larger than
the right-hand side to assure that Ls refuse treatment. 29 By assumption [1] (1.1) becomes:
(1.2)

U( y 0 − p i* + i*; w) − k = κ L U( y 0 − p i* ; w) + (1 − κ L ) U( y 0 − p i* ; s) − (1 − κ L )k , so

(1.3)

U( y 0 − p i* + i*; w) = U( y 0 − p i* ; w) + κ L k

Since Ls are deterred from seeking treatment, expected utility for all agents becomes:
(1.4)

Ûi *=(π W+π H κ H)U(y0-pi*;w)+πH(1-κ H)[U(y0-pi*;w)-k]+π L[U(y 0-pi*+i*;w)-k]
=(π W+π H)U(y 0-pi*;w)+πLU(y0-pi*+i;w)-[πS-π H κ H]k

(πW +π HκH)U(y0 -pi* ;w) is the contribution to expected utility of Ws and of Hs who are
cured; π H(1-κH)[U(y0 -pi* ;w)-k] represents the utility contribution of those Hs who are

33
treated but are not cured; πL [U(y0 -pi* +i*;w)-k] represents the Ls, who are paid i* to forgo
treatment and thus remain sick. Substituting (1.3) into (1.4),
(1.5)

Ûi* =(πW +π H)U(y0 -pi* ;w)+πL[U(y0 -pi* ;w)+κLk]-[π S-π HκH]k
=U(y0 -pi* ;w)-[π S-πHκH-π LκL]k [Q.E.D.]

Proposition 2: The minimum deterrent indemnity is the optimal indemnity, or:
(P.2)

Ûi* >Ûi ∀ i>i*

Replacing i* with an arbitrary i≥i* in (1.4) results in:
(2.1)

Ûi=(π W +πH)U(y0 -pi;w)+π LU(y0 -pi+i;w)-[π S-πHκH]k,

(2.2)

$
$
$
∂U i ( y 0 − p i ; w)
∂Ui ( y 0 − p i + i ; s)
∂U i
= (π W + π H )
+ πL
∂i
∂i
∂i

and

Remember assumption [1] and note that ∂pi/∂i=-π L and πW +πH=(1-πL); (2.2) becomes:

(2.3)

$
$
$
∂U i ( y 0 − p i ; w ) ∂( y 0 − p i )
∂U i ( y 0 − p i + i ; w ) ∂( y 0 − p i + i )
∂U i
= (1 − π L )
+ πL
∂i
∂i
∂i
∂( y 0 − p i )
∂( y 0 − p i + i )
= −π L (1 − π L )

$
∂U i ( y 0 − p i ; w )
∂( y 0 − p i )

+ π L (1 − π L )

$
∂U i ( y 0 − p i + i; w )
∂( y 0 − p i + i )

This expression is negative iff:

(2.4)

$
∂U i ( y 0 − p i + i ; w )
∂( y 0 − p i + i)

<

$
∂U i ( y 0 − p i ; w )
∂( y 0 − p i )

which is true, since agents experience diminishing marginal utility of wealth. [Q.E.D.]
Proposition 3: The minimum deterrent deductible contract yields welfare of:
(P.3)

29

ˆ
U d * = U ( y 0 − p d* ; w) − (π S − π H κ H + π H κ L )k , where p d* = π H x − π H d *

(1.1) is depicted graphically in Figure 4 below.

34
The minimum deterrent deductible d* makes Ls indifferent between purchasing and
not purchasing medical treatment. This yields (3.1) 30 and, equivalently, (3.2) and (3.3):
(3.1)

U( y0 − p d* ; s) = κ L U(y 0 − pd * − d*; w ) + (1 − κ L )U(y 0 − pd * − d*; s)

(3.2)

U( y 0 − p d* ; w) − k = κ L U( y 0 − p d* − d*; w ) + (1 − κ L ) U( y 0 − p d* − d*; w ) − k

(3.3)

U(y 0 − p d* − d*; w) = U( y0 − pd * ; w) − κ L k

[

]

Expected utility under the minimal deterrent deductible contract is:

$
U d*
(3.4)

= π W U( y 0 − p d* ; w) + π H κ H U( y 0 − p d* − d*; w)

+ π H (1 − κ H ) U( y 0 − p d* − d*; s) + π L U( y 0 − p d* ; s)

[

]

= ( π W + π L ) U( y 0 − p d* ; w) + π H U( y 0 − p d* − d*; w) − π L + π H (1 − κ H ) k
Substituting (3.3) into (3.4) and consolidating:
(3.5)

ˆ
U d * = U ( y 0 − p d* ; w) − (π S − π H κ H + π H κ L )k , with p d* = π H x − π H d * [Q.E.D.]

Proposition 4: The minimum deterrent deductible is the optimal deductible, or:
(P.4)

Ûd* >Ûd ∀ d>d*

In (3.4), replace d* with d≥d*. πW +π L=1-π H and assumption [1] holds, so:

(4.1)

$
$
$
∂U( y 0 − p d ; w ) ∂( y 0 − p d )
∂U( y 0 − p d − d; w ) ∂( y 0 − p d − d)
∂U d
= (1 − π H )
+ πH
∂d
∂d
∂d
∂( y 0 − p d )
∂( y 0 − p d − d )

(4.2)

$
$
$
∂U( y 0 − p d ; w)
∂U( y 0 − p d − d ; w)
∂U d
= π H (1 − π H )
− π H (1 − π H )
∂d
∂( y 0 − p d )
∂( y 0 − p d − d)

The derivative at right is larger than the derivative at left, so (4.1) is negative. [Q.E.D.]
Proposition 5: The deductible/indemnity boundary condition is:

30

(3.1) is depicted graphically in Figure 4 below.

35

(P.5)

>
>
Ûd* = Ûi* iff U(y0 -pd* ;w)-U(y0 - pi* ;w) = π SκLk
<
<

The proof can be seen by inspection; it is merely a comparison of the expressions in
(P.1) and (P.3). (P.5) completes Theorem 1 and is the key result of the paper. [Q.E.D.]
Theorem 2: If utility is a logarithmic function U(y;w)=ln(y), the D/I boundary
>
>
1 + (φ −1)π L
ˆ
ˆ
condition becomes U d * = U i* iff
= 1 . Proof: Propositions 6-8.
[φ − (φ −1)π H ]φ π H + π L −1 <
<
Proposition 6: With a logarithmic utility function, the optimal indemnity is
(P.6)

i* =

(φ − 1)( y 0 − π H x )
, where φ=exp(κLk).
1 + (φ −1)π L

Ls are indifferent between medical treatment and the minimal indemnity in (1.3)
which, restated logarithmically, becomes:
(6.1)

ln( y 0 − p i* + i *) = ln( y 0 − p i* ) + κ L k

Note that pi* =π Hx+πLi*, and take the natural antilogarithm of each side:
(6.2)

( y0 − π H x − π L i * +i *) = φ ( y0 − π H x − π L i *) , where φ = e κ k

(6.3)

( y0 − π H x ) + (1 − π L )i* = φ ( y 0 − π H x ) − φπ L i *

(6.4)

(φ − 1)( y 0 − π H x ) = [1 + (φ − 1)π L ]i *

(6.5)

i* =

L

, and successively:

(φ − 1)( y0 − π H x )
, [Q.E.D.]
1 + (φ − 1)π L

Proposition 7: With a logarithmic utility function, the optimal deductible is:
(P.7)

d* =

(φ − 1)( y0 − π H x )
φ − (φ − 1)π H

36
Ls are indifferent between medical treatment (with an out-of-pocket deductible) or
forgoing treatment in (3.1) which, restated logarithmically, is:
(7.1)

ln( y 0 − p d* − d *) = ln( y 0 − p d* ) − κ L k

Note that pd* =π Hx-π Ld*, and take the natural antilogarithm of each side:
(7.2)

( y0 − π H x + π H d *) = ( y 0 − π H x + π H d * −d *)φ , where again, φ = e κ k

(7.3)

( y0 − π H x ) + π H d * = φ ( y 0 − π H x ) + φ (π H − 1)d *

(7.4)

d* =

L

, then:

(φ − 1)( y0 − π H x )
. [Q.E.D.]
φ − (φ − 1)π H

Proposition 8: With a logarithmic utility function, the D/I boundary condition is:

(P.8)

>
>
1 + (φ −1)π L
ˆ
ˆ
U d * = U i* iff
=1
[φ − (φ −1)π H ]φ π H + π L −1 <
<

Restate the D/I boundary condition from Theorem 1 (Proposition 5) logarithmically:

(8.1)

>
>
ˆ
ˆ
U d * = U i* iff ln ( y0 − π H x + π H d *) − ln ( y 0 − π H x − π L i *) = π S κ L k
<
<

Taking the exponential of the right-hand side equation leaves the sign unchanged, so
(8.2)

>
y − π H x + π H d * > π S κ Lk
ˆ
ˆ
U d * = U i* iff 0
=e
y0 − π H x − π Li * <
<

With φ = e κ L k , substitute in i* from (6.5) and d* from (7.4). Then, (8.2) becomes:

(8.3)

( y 0 − π H x ) + π H ( y 0 − π H x )(φ − 1) >
>
φ − (φ − 1)π H
πS
ˆ
ˆ
U d * = U i* iff
( y0 − π H x )(φ − 1) =φ
<
<
(y0 − π H x) − π L
1 + (φ − 1)π L

and, successively:

37



(8.4)

π H (φ − 1)
φ − (φ − 1)π H


>
>

 =φ π S
ˆ = U iff
ˆ
Ud*
i*

π (φ − 1)  <
<
( y 0 − π H x )1 − L

 1 + (φ − 1)π L 

( y 0 − π H x )1 +

( y 0 − π H x )φ − (φ − 1)π H + π H (φ − 1)  >
φ − (φ − 1)π H

 =φ π
1 + (φ − 1)π L − π L (φ − 1)  <
( y 0 − π H x )

1 + (φ − 1)π L




(8.5)

>
ˆ
ˆ
U d * = U i* iff
<

(8.6)



φ

>
>
ˆ = U iff  φ − (φ − 1)π H  = φ π S
ˆ
Ud*
i*

 <
<
1
 (

1 + φ − 1)π L 

(8.7)



>
>
1 + (φ −1)π L
ˆ
ˆ
U d * = U i* iff
=1
π H + π L −1
[φ − (φ −1)π H ]φ
<
<

S

The boundary condition is a function only of π H, π L, κL, and k. [Q.E.D.]
Theorem 3: If π L≥π H, the optimal deductible policy is always more efficient than
the optimal indemnity policy. Proof: Propositions 9-11.
(P.9)

If π H=π L, then U d * > U i*

If we set π H=π L, we can rearrange and (8.7) to be:
(9.1)

>
>
ˆ
ˆ
U d* = U i* iff φ[1 + (φ − 1)π L ] = [φ − (φ − 1)π L ]φ 2πL ,
<
<

For Proposition 9 only, we adopt several notational conventions. We use a to
represent κLk, so a=κLk and φ=ea. And we use λ and ρ, respectively, to represent the lefthand side and right-hand side of (9.1). (9.1) becomes:
(9.2)

>
>
ˆ
ˆ
U d* = U i* iff λ = ρ ,
<
<

>
or equivalently iff e a 1 + e a − 1 π L = e a − e a − 1 π L e a 2π L
<

[ (

) ][

(

) ]

38
From this, we will prove (P.9). The proof is lengthy, so Figure 4 provides a roadmap.
To prove (P.9), we will show that: [1] π L∈(0,.5); [2] if π L=0, λ=ρ=φ; [3] if π L=.5,
λ=ρ=.5φ(φ+1) [4] λ is increasing and linear in π L; [5] ρ is increasing in π L and strictly

ˆ
ˆ
convex over the relevant range; [6] Therefore, λ>ρ over the relevant range, so U d* > U i* .
[1] through [4] can all be demonstrated by inspection. [1] π L∈(0,.5). If π L=π H=0,
there are no sick people. If π L=πH=.5, there are only sick people. Therefore, the actual
population shares must lie between the two. [2] and [3] can be shown by plugging the
values π L=π H=0 and πL=π H=.5 into λ and ρ. Thus, the two curves coincide at the extreme
values of π L. [4] Since ∂?/∂pL=λ'=φ(φ-1) and ∂2 ?/∂pL2 =λ?=0, λ is linear in π L.
[Throughout this section, superscripts of ', '', and ''' will indicate the 1st , 2nd, and 3rd
derivatives with respect to π L.]
[5] requires us to prove two lemmae. Lemma 9.1 shows that ρ is upward-sloping.
Lemma 9.2 shows that ρ is strictly convex to the axis.
Lemma 9.1: ρ is an increasing function of π L.
According to our notational conventions:

Figure 4: Graphic Representation of Proposition 9
.5φ(φ+1)

λ = φ [1 + (φ − 1)π L ]

φ
πL=0

ρ = [φ − (φ − 1)π L ]φ 2 πL

πL=.5

39
(9.3)

[

]

ρ = [φ − (φ − 1)π L ]φ 2 πL = e a − π L e a + π L e a 2π L

We further define the bracketed term as a function Q(π L,a), so:
(9.4)

ρ = Qe a 2 πL

>
>
so, ρ = 0 iff Q = 0 , which is true by inspection. From (9.3), we can derive:
<
<
(9.5)

ρ′ = Q2ae a 2π L + Q′e a 2 πL , and dividing the right-hand side by e a 2π L :

(9.6)

ρ′ > 0 iff Q 2a + Q ′ > 0 .

From (9.3) we can derive:
(9.7)

Q′ = 1 − e a < 0 , since a>0.

So for π L∈[0,.5], Q is at its minimum at π L=.5. Substituting this value into (9.3) yields:
(9.8)

MIN[Q]=Q(.5,a)=.5(ea+1)>0

From (9.5), ρ' is an increasing function of Q and is at its minimum when Q(π L=.5).
Substituting (9.7) and (9.8) into (9.6)
(9.9)

Q2a + 1 − e a =

ea + 1
2a + 1 − e a = e a + 1 a + 1 − e a = e a a + a + 1 − e a
2

(

)

If (9.9) is positive for all a>0 and for all π L, then ρ is upward-sloping. If a=0, (9.9)
equals 0. And to find the sign of (9.9) for a>0:
(9.10)

[

]

∂ a
e a + a + 1 − e a = e a a + e a + 1 − e a = e a a + 1 > 0 for a≥0.
∂a

So from (9.10) we know that (9.9) is always positive for relevant a and π L. If (9.9) is
positive, then we know from (9.6) that ρ′ > 0 , thus proving Lemma 9.1. [Q.E.D]
Lemma 9.2: ρ is strictly convex to the axis.
Strict convexity requires ρ′′ > 0 for all π L∈(0,.5). From (9.5) we derive:

40

[

]

(9.11) ρ′′ = 2a Q2ae a 2πL + Q′e a 2π L + Q′2ae a 2 πL + Q′′e a 2π L
From (9.7), we see that:
(9.12) Q ′′ = 0 . Substituting this into (9.12):

[

]

(9.13) ρ′′ = 2a Q2ae a 2π L + Q′e a 2π L + Q′2ae a 2πL , so

[

]

(9.14) ρ′′ > 0 iff 2a Q2ae a 2 πL + Q′e a 2π L + Q′2ae a 2 πL > 0
Consolidating terms and dividing through by 4aea 2 π L :
(9.15) ρ′′ > 0 iff Qa + Q ′ > 0 , and since Q"=0
(9.16) ρ′′′ = aQ ′ < 0
From (9.7) and (9.16), Qa+Q' is at a minimum when π L=.5. Substituting (9.8) into (9.16):

(

) (

)

(9.17) ρ′′ > 0 iff .5 e a + 1 a + 1 − e a > 0 , or, consolidating and multiplying by 2:
(9.18) e a a + a + 2 − 2e a > 0 ⇔ ρ′′ > 0 ,

And,

(9.19) a=0⇒ e a a + a + 2 − 2e a = 0 . Taking the derivative,
(9.20)

∂ a
e a + a + 2 − 2e a = e a a + e a + 1 − 2e a = e a a + 1 − e a ,
∂a

so from (9.19) and (9.20)
(9.21) e a a + 1 − e a > 0 ⇒ e a a + a + 2 − 2e a > 0
(9.22) a=0⇒ e a a + 1 − e a = 0 .
(9.23)

And

∂ a
e a + 1 − e a = e aa + e a − ea = ea a > 0
∂a

so from (9.22) and (9.23),
(9.24) a>0⇒ e a a + 1 − e a > 0
Now, through a chain of logic including (9.24), (9.21), and (9.18):

41
(9.25) a>0⇒ e a a + 1 − e a > 0 ⇒ e a a + a + 2 − 2e a > 0 ⇒ ρ′′ > 0
This proves Lemma (9.2), so ρ is strictly convex to the x-axis. Thus, for all
πL=π H∈(0,.5), ρ lies strictly below λ. π L, πL=πH thus implies that a deductible policy is
strictly preferred to an indemnity policy, proving Proposition 9. [Q.E.D.]
Proposition 10 uses Proposition 9 to demonstrate a similar result in inequality form:
(P.10) If π L>πH, then U d * > U i*
If π L>πH, then (8.8) becomes
(10.1)

1 + (φ − 1)(π L* + ξ )
> φ 2π L* −1
φ − (φ − 1)(π L* − ξ )

where π L=π L*+ξ and π H=πL*-ξ and ξ is some constant. Now, we can compare the
expected utilities of (π L,πH)=(πL*,πL*) and (π L,π H)=(π L*+ξ ,πL*-ξ). RHS can be written as
φ π L* +ξ +π L* −ξ −1 , and this does not change as ξ changes. The LHS is
(10.2)

1 + (φ − 1)(π L* + ξ )
, and taking the derivative:
φ − (φ − 1)(π L* − ξ )

(10.3)

∂ 1 + (φ − 1)(π L* + ξ) (φ − 1)[φ − (φ − 1)(πL* − ξ )] − (φ − 1)[1 + (φ − 1)(πL* + ξ)]
=
∂ξ φ − (φ − 1)(πL* − ξ)
[φ − (φ − 1)(πL* − ξ)]2

The sign of this depends on the sign of the numerator, which we can divide by φ-1:
(10.4) [φ-(φ-1)(π L*-ξ)]-[1+(φ-1)(π L*+ξ)]
which reduces to
(10.5) φ-(φ-1)(π L*-ξ)-1-(φ-1)(πL*+ξ)] = (φ-1)(1-2πL*)
Since φ>1>2πL, (10.5) is positive. As π L increases from π L*, the RHS expression also
increases. Since a deductible is preferred if π L=π L, therefore, a deductible is preferred at
(π L,πH)=(πL*,π L*) or at any (π L,π H)=(π L*+ξ,πL*-ξ), [Q.E.D.]

42
Proposition 11 combines Propositions 9 and 10 to prove Theorem 3:
(P.11) If π L≥πH, then U d * > U i* [Q.E.D.]
Theorem 4/Proposition 12: In the limit, as the Ls’ cure rate κL approaches 0, the
D/I boundary condition collapses to Ûi* >Ûd* iff π H>π L. Proof: Proposition 12.
If we are indifferent between the optimal deductible and indemnity contracts, then
(8.8) says that:

(12.1)

[

] =1
−π ]

φ − π L 1 + φπ L − π L

[

φ π H 1 + φ −1 π H

H

In this case, it is also true that:

(12.2)

m( φ)
n( φ)

=

[

] =1
− π ]−1

φ − π L 1 + φπ L − π L − 1
φ

πH

[1 + φ

−1

πH

H

If this function holds, then φ, π H, and π L are such that we are indifferent between I and D.
l'Hôpital's Rule states that:
m(φ )
m′(φ )
= lim
if lim m( φ ) = lim n( φ) = 0
φ →1 n(φ )
φ →1 n ′(φ )
φ→1
φ→1

(12.3) lim

So take the derivatives of m and n with respect to φ and divide m′ by n′:

(12.4)

[φ ][ π ] + [− π φ ][1 + φπ − π ]
=
n ′( φ )
[φ ][− φ π ] + [π φ ][1 + φ π − π ]

m ′( φ )

−πL

− π L −1

L

πH

L

πH −1

H

=

=
=

L

−2

[
[1 − φ

H

H

]
π ]

π L φ − π L 1 − φ −1 − π L + φ −1π L
πHφ

π H −1

π Lφ
πHφ

− πH + φ

(1 − φ )(1 − π )
(1 − φ )(1 − π )
(1 − π )
(1 − π )
−1

L

π H −1

−1

H

π Lφ
πHφ

−πL

−1

−πL

L

π H −1

H

−1

L

−1

H

H

43
So, applying l'Hôpital's Rule:

(12.5) lim
φ→1

π Lφ − π L (1 − π L )
πHφ

π H −1

(1 − π )

=

H

π L (1 − π L )

π H (1 − π H )

Since φ=exp(κLk), lim φ = 1 , so, from (12.5), we can also conclude that:
κ L →1

π Lφ −π L (1 − π L )
π (1 − π L )
(12.6) lim
= L
κ L →0 π φ π H −1 ( − π )
1
π H (1 − π H )
H
H
(12.2) holds if the indemnity and deductible policies are equally efficient. (12.7) says
that as the Ls' cure rate approaches 0, we will be indifferent where π H is approximately
equal to π L. The choice of indemnity or deductible becomes simple. If π H<π L, use a
deductible policy. If π L is more than slightly less than π H, use an indemnity policy.
Theorem 5: The optimal indemnity is always larger than the optimal deductible.
(P.13) i*>d*
Subtracting (P.7) from (P.6):
(13.1) i * −d * =

(φ − 1)( y 0 − π H x ) (φ − 1)( y0 − π H x )
−
1 + (φ − 1)π L
φ − (φ − 1)π H

Since the numerators of the two terms are identical, the sign of this expression equals the
opposite sign of the difference of the denominators:
(13.2) sgn (i * −d *) = sgn ([φ − (φ − 1)π H ] − [1 + (φ − 1)π L ])
= (1 − φ )(1 − π S )
This expression is negative because φ>1 and π S<1. i* is always greater than d*. [Q.E.D.]
Figure 5 depicts equations (1.1), (3.1), and (13.2). There are two horizontal lines
representing expected utility of Ls only under different contracts. The upper of the two,
equivalent to (1.1) shows the Ls’ expected utility if treated equals Ls’ expected utility if

44
they are not treated but receive i*. The lower line shows Ls’ expected utility if they are
treated (at a marginal cost of d*) and if they are not treated. It is easy to see from the
diagram why the i*>d*.

Figure 5: Graphic Representation of Theorem 5
U(-;w)

κL
κL U(-;w)+
(1-κL )U(-;s)

U(-;s)

1-κL

U(y 0 +i*;s)=κL U(y 0 ;w)+(1-κL )U(y 0 ;s)
U(y 0 ;s)=κL U(y 0 -d*;w)+(1-κL )U(y 0 -d*;s)

i*

d*
y0 -d*

y0

y0 +i*

Theorem 6: The relative efficiency of modes F and I is determined solely by the
relative size of the optimal indemnity i* and the cost of treatment x. More formally,
>
>
(P.16) i* = x iff Ûf = Ûi*
<
<
This compares welfare under I (P.1) and under F (P.15), which differ only in their
insurance premiums π Hx+π Li* and π Sx=π Hx+πLx, respectively. Thus, if i*>x, welfare is
higher with full insurance (P.15). If i*<x, welfare is higher with (P.1).