PII: S0167-6296(99)00010-7

Ž.

Journal of Health Economics 18 1999 523–550

www.elsevier.nlrlocatereconbase

The menu-setting problem and subsidized

prices: drug formulary illustration

Todd Olmstead, Richard Zeckhauser

)

HarÕard UniÕersity, Cambridge, MA, USA

Abstract

Ž.

The menu-setting problem MSP determines the goods and services an institution offers

and the prices charged. It appears widely in health care, from choosing the services an

insurance arrangement offers, to selecting the health plans an employer proffers. The

challenge arises because purchases are subsidized, and consumers or their physician

agents may make cost-ineffective choices. The intuitively comprehensible MSP model—

readily solved by computer using actual data—helps structure thinking and support decision

making about such problems. The analysis uses drug formularies—lists of approved drugs

in a plan or institution—to illustrate the framework. q 1999 Elsevier Science B.V. All

rights reserved.

JEL classification: D61 allocative efficiency; Cost-benefit analysis; I10 Health, general; H42 Publicly

provided private goods

Keywords: Menu-setting; Subsidy; Drug formulary; Cost effective; Constrained maximization

1. Introduction

A menu lists the goods or services an institution offers, and the prices it

charges. When a profit-seeking organization, such as a restaurant or mail-order

clothier, sets its menu it cannot simply decide on an item-by-item basis. It must

worry about cannibalization of purchases due to cross-elasticity of demand. The

)

Corresponding author. Kennedy School of Government, Harvard University, 79 JFK St., Cam-

bridge, MA 02138, USA. Tel.: q1-617-495-1174; Fax: q1-617-496-3783; E-mail:

richard_[email protected]

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PII: S0167-6296 99 00010-7

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550524

pasta dish may draw customers from the more expensive and profitable meat

items, and similarly for the clothier with sweaters cheap and cher. Taking

inexpensive items off the menu is not the solution, because they allow profitable

sales to some customers who would not otherwise buy. Here lies the essence of the

Ž.

menu-setting problem MSP : making an item available to one group of persons

will draw off more desirable purchases from other individuals. This paper sets

forth the MSP, identifies its particular salience in the health-care arena, and

illustrates with drug formularies.

The MSP is of greater importance in the health-care arena, and takes on an

additional salient dimension, because so many goods and services are offered at

heavily subsidized prices. The goal in health care, at least implicitly, is to spend

treatment dollars where they will produce significant benefits. However, once

items are subsidized, the menu-setter has a significant concern that inappropriate

purchases will be made. Why is health care subsidized, as opposed say to meals or

clothing? Risk spreading provides the primary argument for subsidy of health-pro-

moting items: they are expensive and random factors make their consumption

desirable. Hence, the providing institution would like to offer the items at little or

no cost to appropriate users, those who would derive significant expected medical

benefits from them, but charge inappropriate users, i.e., those who benefit

marginally, the full costs of provision. Yet once the subsidized item is placed on

the menu, it is available for consumption by those who reap high value and those

who do not. Self-interested patients, or physicians acting on their behalf, will

choose items that are expensive relative to the health benefits secured; in other

words, they will undertake consumption that is not cost effective. There may be

other justifications for subsidy beyond risk spreading, such as paternalism, or that

needed health care is a right. The same conflict between ready affordability for

some and costrineffective consumption for others would apply.

Note, the majority of health care expenditures in most developed nations are

paid for by government. Even in market-oriented United States, employers subsi-

dize their workers’ health care, since it is an employee benefit, not a source of

profitability, and is tax favored. Most care-giving institutions, such as HMOs and

hospitals, are nonprofits. Moreover, even for-profit insurers and providers subsi-

dize health-care purchases at time of sale. This caters to risk aversion, and allows

profit makers to charge higher premiums and government and nonprofits to

produce greater welfare. All these subsidy-offering entities strongly encounter the

MSP: What services should be made available?

We take the objective of the health-care provider, whether government or

private, for-profit or nonprofit, to be to maximize expected health-care outcomes

subject to a constraint on resource expenditure. Setting the constraint appropri-

Presumably, a for-profit entity can charge more if it secures better outcomes; hence it also solves

this constrained maximization problem.

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550 525

ately is a problem left for another day. If all patients desiring a particular

treatment were homogeneous, there would be no problem. All cost-effective

treatments and only such treatments would be placed on the menu. The challenge

is that patients desiring a treatment are heterogeneous; for some the treatment is

well worthwhile, for others not. This paper addresses the problem of setting the

menu to balance effective and wasteful provision. The drug formulary is our

central example. A formulary provides a crisp example of a menu. Its prices are

almost invariably subsidized, implying that inappropriate or excessive use may be

a problem. There is also the possibility of cannibalization, with the use of one drug

replacing another.

Though we focus on the problem of drug formularies, we could have easily

directed attention to health care more generally. As they proceed, readers are

encouraged to apply the MSP to their own areas of interest, such as mental-health

care, or the use of diagnostic technologies, such as MRIs, indeed to any problem

where the expected benefits of particular treatments or procedures vary widely

from person to person. The MSP also applies at a higher level to the menu of

health-plan options that an employer presents to employees. We provide brief

illustrations from these areas before turning to drug formularies.

Most health plans offer their insureds a fixed number of subsidized visits, say

eight, to a mental-health professional per year. For a person with an anxiety

disorder, this may be a roughly appropriate number. But this number may be far

too few for a person with a serious mood or psychotic disorder, and many more

than is worthwhile for milder stress-related conditions. Assuming the health plan

cannot discriminate among such individuals in services offered, it encounters the

MSP.

An MRI machine is a patient magnet. A health plan with such a machine will

incur many uses that are at best of marginal value. But it is difficult for health

plans to direct physicians to use MRIs only when they are cost effective. The

doctor just might get some diagnostic insight from an MRI, and the patient wants

it. In many such cases, the doctor just goes along with a costrineffective use. The

use of diagnostic technologies, such as MRIs, will also depend on how readily

competitive services are available. To some extent, MRIs will substitute for

CATscans, or even regular X-rays. The MSP captures any propensities to substi-

tute among goods or services.

Frequently, the alternative to a procedure is doing nothing at all, but still the

Ž.

MSP applies. If PSA tests screening for prostate cancer are routinely offered to

men over 50, their highest value use, knowledgeable men under 50 may request

Many health plans, recognizing the need to draw distinctions among individuals, now contract out

management of their mental health service to firms that specialize in drawing such distinctions. The

contractors are given considerable latitude in allocating treatments. Essentially, mental health care is

made a non-menu item.

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550526

them. Once a health plan makes flu shots cheaply available, even individuals in

their 20s may request them. Undesired choices may also result when offering a

health plan. An employer might wish to offer a strongly subsidized version of a

generous health plan to allow employees with significant health problems to

secure intense treatment at affordable rates. The strong subsidy may also attract to

the plan people with ordinary health risks, who were not the employer’s intended

recipient group. This ‘you can’t get one without the other’ feature is the essence of

the MSP, which suggests that the MSP is first cousin to the problem of adverse

selection.

We develop the remainder of our analysis in the context of drug formularies,

which are explicit lists of approved pharmaceuticals within a health plan or

institution. Such formularies are used in one form or another by most health care

providers in the US. Their objective is to maximize expected health-care benefits

for a heterogeneous population subject to a constraint on pharmaceutical expendi-

tures. Drug formularies encounter a serious agency problem, given the subsidies

they offer. To spread risks and possibly to pursue distributional goals, patients are

charged much less for drugs than the marginal cost to the provider. Patients, who

Ž.

are allowed to choose their drugs, will not count costs to the provider or plan .

Thus, moral hazard will result in excess expenditures on drugs as judged by the

people covered by the provider, because it is these individuals who ultimately pay

the costs of pharmaceutical use, whether through premiums or reduced subsidies

for other health services.

In this situation, where choosers are both subsidized

and self-interested, how is a provider to set the optimal menu of alternatives?

We present the MSP as a mathematical model that explicitly accounts for

patient heterogeneity, and considers all medical conditions and drugs together

when selecting the formulary. This model is flexible enough to accommodate a

variety of objectives and cost structures, and can be used as a decision support tool

by formulary decision makers. The interest in the model stems from the fact that

drug prices are subsidized. Therefore, marginal cost pricing does not carve the

path to efficiency. The drug formulary example is illustrative of any attempt to

limit resource use by a group of subsidized, heterogeneous individuals choosing

for themselves, such as defining an approved set of procedures for a health plan.

Ž.

As Newhouse 1998 points out, subsidizing drugs may promote allocative efficiency, from the

standpoint of society at large. The social marginal cost of drugs is often quite low relative to the price

charged to a health provider; pharmaceutical companies need to set prices high to recover the fixed

costs of R&D and marketing, and they are able to do so given their market power. Subsidizing

consumers’ purchases brings the prices they face closer to the producer’s marginal cost.

Ž.

Baumgardner 1991 analyzes an HMO setting quantity restrictions to control the use of new,

expensive technologies. He observes that a principal competitive advantage of HMOs is their ability to

control costs in this way.

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550 527

In what follows, we say that individuals choose, recognizing that it is usually the

physician choosing on their behalf.

This paper consists of seven sections. Section 2 provides information about

formularies, including a description and assessment of the methods typically used

to determine them. Section 3 considers the problem of seeking the optimal drugs

to treat a single medical condition, framing it as an MSP, allowing but not

requiring charges to patients. Section 4 extends our single-condition MSP formula-

tions to the multiple-condition problem considering all medical conditions and

drugs simultaneously. Section 5 discusses pricing in MSPs. Section 6 discusses

several limitations on the use of the MSP framework to determine drug formula-

ries, and Section 7 concludes, and relates the analysis back to menus in health care

more generally.

2. Drug formularies

A formulary is a listing of pharmaceuticals permissible to use in a given

institution, such as a hospital or HMO. It is continually revised to reflect the

current judgment of the institution’s medical staff. A formulary system enables the

staff, typically working through a committee, to evaluate and select from among

numerous available drug entities and drug products, those considered most useful

Ž.

in patient care American Society of Hospital Pharmacists, 1986a, 1991 . Formula-

ries are also concerned with cost effectiveness.

The first known formulary in the US was published for the Continental forces

during the American Revolution American Society of Hospital Pharmacists,

1986b . In 1933, the first guidelines for operating a formulary system were

Ž.

formulated by a physician and a pharmacologist Nash et al., 1993 . In 1965, the

development of hospital formularies was mandated by the Joint Commission on

Accreditation of Hospitals Joint Commission on Accreditation of Hospitals,

1965 , and an estimated 95% of managed care plans currently use a formulary

Ž.

Freundlich, 1995 .

There are three basic types of formularies: open, closed or restricted, and

incentive-based. An open formulary serves merely as a guide: a physician may

prescribe any drug, but is encouraged to use the formulary list in prescribing

decisions. In contrast, a closed or restricted formulary lists the drugs that will be

reimbursed by the health care provider; nonformulary drugs will be reimbursed

only if they are authorized prior to prescribing. An incentiÕe-based formulary

represents a hybrid between the open and closed formularies; patients pay a higher

price for nonformulary drugs.

Health-care providers use formularies to generate safe, effective, and cost-con-

scious use of medications for patients. By assuring drug-by-drug review of all

medications, the formulary system can reeducate and remind physicians of alterna-

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550528

tive cost-effective therapies and balance the promotional tactics of pharmaceutical

manufacturers. Formularies can save money through the lowered prices that come

from bulk purchasing and competitive bidding, and by reducing waste and

overuse. By giving physicians in-depth experience with a limited number of drugs

with proven superiority, formularies can reduce the risk to patients of medication

errors and drug reactions which may result from sound-alike drug names and a

proliferation of drugs that caregivers must know and to which patients might be

exposed. The advantages of such a focus are growing given the increased number

of new drugs being marketed, many highly potent with significant side effects, and

the increasing influence of possibly biased and unscientific advertising information

Ž.

U.S. Congress, 1995 .

Like all management tools, formularies possess certain intrinsic limitations.

They cannot, for example, determine whether the therapy of choice should be a

drug product rather than rest or diet, and they have limited power to cope with the

inappropriate use of even the most appropriate medication Rucker and Schiff,

1990 . Moreover, the formulary system has been severely criticized for placing too

much emphasis on cost and too little on quality. A common complaint is that

formularies often include only those drugs that are cost effective for the ‘average

patient’ while overlooking the special needs of individuals Bakst, 1995; U.S.

Congress, 1995 . Even if physicians can seek authorization to prescribe nonformu-

lary drugs, such procedures can delay needed treatment for the patient and

ultimately lead physicians to prescribe a suboptimal formulary drug instead U.S.

Congress, 1995 .

Despite their widespread use in the US, the impact of formularies on the quality

of care for patients has not been well studied. A few controversial studies have

Ž.

indicated that formularies lead to either higher costs Horn et al., 1996 or lower

Ž.

quality of care for some patients U.S. Congress, 1995 , but much work remains to

be done in this area before any sound conclusions can be reached Freundlich,

1995 .

Drug formularies are determined by formulary committees.

The membership

of formulary committees varies, but most include physicians, pharmacists, nursing

representatives, lawyers, and ethicists. Large committees may consist of several

subcommittees, each responsible for reviewing a given therapeutic class of drugs

Ž.

Nash et al., 1993 . Health-care providers typically have formal procedures for

changing their formulary, and most formulary committees meet at least quarterly

Ž.

to vote on the acceptance or elimination of formulary drugs Rascati, 1992 .

Determining an ‘optimal’ drug formulary requires tradeoffs between the cost

and relative effectiveness of numerous pharmaceuticals for the heterogeneous

Ž.

In hospitals, these are often known as pharmacy and therapeutics P&T committees.

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550 529

group of people who are on the health plan. A major difficulty is that once a drug

is placed on the formulary, it will be selected by many individuals—actually, their

doctors often choose on their behalf—even when it is not cost effective.

For example, heart-attack patients can be treated with a variety of thrombolytic

drugs to increase their chance of survival. Two of the most effective drugs are

streptokinase, a drug derived from bacteria that costs US$240 per dose, and tissue

Ž.

plasminogen activator TPA , a genetically engineered substance that costs

US$2400 a dose. Although TPA is ten times more expensive than streptokinase, it

is on average more effective in preventing death. A recent study involving over

41,000 heart-attack patients found that 6.3% of patients treated with TPA died

within 30 days, compared to 7.2% of those treated with streptokinase GUSTO

Investigators, 1993 . The problem is that the relative effectiveness of the two

Ž.

treatments varies across patient groups, and a cost-benefit or cost-effectiveness

analysis of the two treatments will reach different conclusions for different

patients. Table 1 summarizes the results of the study, disaggregated by important

patient subgroups. For all subgroups TPA was statistically significantly superior.

Ž.

TPA is more effective if only slightly than streptokinase in preventing death for

all patient subgroups. Hence, each subgroup would choose TPA and not worry

about cost.

We discuss infarct location and hours to drug therapy first. There are severe

differences in net benefits, suggesting that the plan might prefer to make TPA

available only for some groups. Due to practical restrictions, however, if a plan

makes TPA available to patients with an anterior infarction who arrive at the

emergency room less than 4 hours after the attack—the group whose mortality

reduction is probably the greatest—it must also make it available to patients with

an inferior infarction who arrive at the emergency room five hours after the attack,

though their smaller survival gains may make the choice of TPA cost-ineffective.

Thus, a health-care provider faces the agonizing dilemma of deciding which

thrombolytic drug or drugs to offer its members. If it offers TPA, all rational

Health-care professionals claim that the provider cannot, in practice, offer a given drug to some

patients and deny it to others on the basis of cost effectiveness Basskin, 1998; Caul, 1998; Glomski,

1998; McLaughlin, 1998; Nash, 1998; Sorrenti, 1998 . This restriction may also have some legal bite.

The Patients’ Bill of Rights Act of 1998 states that ‘‘patients should not be discriminated against in

their access to covered health-care services.’’ And a few years ago, a court awarded US$89 million to

the family of a California woman who died after her HMO refused to pay for a controversial

bone-marrow transplant that doctors hoped would cure her breast cancer. The court’s decision hinged

on the fact that the HMO had earlier approved identical treatments for two other women, both of whom

Ž.

testified at the trial as living proof that the therapy might have worked Shoop, 1994 .

The group experiencing greater than 6 hours to drug therapy actually had better results with

Ž.

streptokinase 8.3 vs. 10.4% mortality . Since the group comprised only 4% of the population, the

difference is not statistically significant, and the group is not included in the table.

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550530

Table 1

Ž.

Mortality after thrombolytic drug therapy for heart attacks ns41,021 patients

Ž.

Subgroup Percent of Mortality rate % Absolute improvement using TPA

Ž.

patients lives saved per 1000

Streptokinase TPA

Infarct location

Anterior 39 10.5 8.6 19

Inferior 61 5.3 4.7 6

Hours to drug therapy

0 to 2 27 5.4 4.3 11

2 to 4 51 6.7 5.5 12

4 to 6 19 9.3 8.9 4

()

Age years

- 75 88 5.5 4.4 11

G75 12 20.6 19.3 13

heart-attack patients will select it, at great expense to the provider, making it less

feasible to offer some other desirable drug under a restricted budget, or raising the

cost of the health plan, thereby making it unaffordable to some segments of the

population and less attractive to the rest. If the plan offers streptokinase and not

TPA, all patients will select it, getting less effective treatment than with TPA; but

it may then be feasible for the plan to support some other desirable drug. Or, the

cost of the plan will decrease, becoming affordable to more segments of the

population, or leaving its members with resources for other purposes.

Age presents a more challenging problem. Patients over 75 secure a slightly

greater mortality reduction when they take TPA rather than streptokinase. How-

ever, they have far fewer years to live; hence they benefit much less in terms of

Ž.

measures such as quality-adjusted life years QALYs . The choice of output

measure involves ethical judgments.

The formulary problem becomes more complex when all relevant pharmaceuti-

cals, medical conditions, and patient subgroups are considered simultaneously.

Formulary committees deal with the complexity of this problem by making two

simplifying assumptions that can lead to suboptimal results.

First, instead of considering all drugs simultaneously, committees usually

consider only one therapeutic class of drugs at a time Seaver, 1995; Garrelts,

1997 . Unfortunately, determining the most cost-effective drugs within each

This pattern, and the accompanying ethical challenge, arises whenever the more expensive drug

gives the greatest mortality benefit to those already at the greatest mortality risk.

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550 531

therapeutic class will not necessarily result in the most cost-effective formulary,

because marginal benefits may vary across classes, and drugs provided for one

category may be inappropriately selected by another. In fact, such a localized

procedure is inconsistent with the fundamental objective of welfare economics,

which seeks to maximize the benefits from society’s scarce resources Birch and

Donaldson, 1987 .

Second, committees tend to group together patients with the same disease and

then recommend drugs that are the most cost effective for the ‘average’ patient in

Ž.

the group Bakst, 1995; U.S. Congress, 1995; Garrelts, 1997 . However, the cost

effectiveness of a drug often varies among patient groups. Pharmacogenetic

research, for example, has discovered important differences among racial and

ethnic groups in the metabolism of drugs, in their effectiveness, and in their side

Ž.

effects Levy, 1993 . A major conclusion of this research is that the most

Ž.

cost-effective drug or dosage for treating a given disease often differs by ethnic

and racial group.

Therefore, an assumption that all patients are the same can

result in a formulary that overlooks the special needs of important subpopulations

of patients or spends cost-ineffectively on others.

In the past decade, a number of decision analytic models have been developed

to help committees determine better drug formularies.

Unfortunately, these

models typically evaluate the relative desirability of medications within a given

therapeutic class for an ‘aÕerage’ patient; while they undoubtedly reduce the

amount of subjectivity involved in the formulary decision-making process, they

fail to correct the incorrect simplifying assumptions.

It is clear that formulary committees would benefit from a formal model that

explicitly accounts for patient heterogeneity and that considers all drugs simultane-

ously, both within and across medical conditions. We introduce patient hetero-

geneity in Section 3, and then develop a model with heterogeneity and multiple

medical conditions in Section 4.

Exactly how this welfare economics mandate should be carried out in practice is controversial. The

central question is: How should benefits be aggregated across different groups of individuals? The

usual answer is to use some metric, such as total QALYs gained, as an output measure. Such metrics,

in effect, maximize the expected utility gain of a randomly chosen individual.

Ž.

Levy 1993 provides specific examples of racial and ethnic differences in response to the

following commonly used agents: cardiovascular drugs including beta-blockers, diuretics, and ACE

.Ž .

inhibitors , central nervous system agents including tranquilizers, antidepressants, and neuroleptics ,

Ž.

analgesics including acetaminophen, codeine, and morphine , and alcohol. Other important factors

involved in determining a patient’s response to a medication include age, gender, multiple disease

states, presence of other drugs, and pregnancy.

Clearly, as difficult as it would be to prioritize on, say infarct location, it would be much harder to

do so on a racial basis, even if this resulted in some drugs being precluded for all patient groups.

Ž. Ž. Ž.

See, for example, Kresel et al. 1987 ; Senthilkumaran et al. 1987 ; Cano and Fujita 1988 ; Calvo

Ž. Ž. Ž. Ž. Ž.

et al. 1990 ; Barriere 1991 ; Schumacher 1991 ; Einarson et al. 1995 ; Lee et al. 1995 ; Harvey et

Ž.

al. 1996 .

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550532

3. The optimal formulary for heterogeneous patients with a single medical

condition

We first use our model to determine which drugs to include in the formulary to

treat a single arbitrary medical condition experienced by three groups of patients.

The condition can be treated with any of four different drugs, whose costs and

efficacy for patients with different expected responses to the drugs are shown in

Table 2. For simplicity, we measure efficacy in QALYs. Since patients within a

category may have different medical responses to a drug, QALYs should be

thought of as an expected value. We assume that patients choose the formulary

drug that gives them the most QALYs, neglecting cost, and that the health-care

provider cannot offer a given drug to some patients and deny it to others.

facilitate exposition, we assume that there is one patient in each group equal

numbers of patients in the groups would produce the same result .

If the objective is to maximize total QALYs in the population of patients

suffering from this condition, the obvious solution to the single-condition problem

is to include on the formulary each drug that is best for some patient group. Thus,

the formulary should contain Drugs 1, 2, and 3.

In total, such a formulary will

yield 30 QALYs and cost US$11. In general, if money were no object, such an

approach could be used for as fine a partition of patients as data permit.

In practice, it may be prohibitively expensive to give each patient group its best

drug, but it is not often obvious how to make the necessary tradeoffs between the

cost of the drugs and their relative effectiveness. A simplistic approach used by

formulary committees, and described in the decision-analytic literature cited

earlier, is to rank the drugs according to how they perform on the ‘average’ patient

and then select the one or two highest-ranked drugs for the formulary. Unfortu-

nately, when patients receive differing benefits from drugs, such an approach will

often result in suboptimal formulary decisions. In our example, if the committee

decides to include the best drug to treat an ‘average’ patient, Drug 4 will

undoubtedly be selected because it is ‘good’ for all patient groups, resulting in 18

QALYs at a cost of US$15. But this strategy is both less effective and more

expensive than including Drugs 1, 2, and 3, which provides more QALYs, 30, at

less cost, US$11.

Appropriate optimization considers the number of QALYs attainable for a

given level of expenditure. For example, Fig. 1 shows that 17 QALYs can be

attained for US$3, 23 QALYs for US$5, 26 QALYs for US$9, and 30 QALYs for

Ž.

The assumption that providers cannot discriminate comes from interviews with Basskin 1998 ;

Ž. Ž. Ž. Ž. Ž.

Caul 1998 ; Glomski 1998 ; McLaughlin 1998 ; Nash 1998 ; and Sorrenti 1998 , asking their

Ž.

views of medical practice. Only Bakst 1998 demurred from this opinion, believing it to be too

idealistic.

Patients in group A will select Drug 1, those in B will select Drug 3, and those in C will select

Drug 2.

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550 533

Table 2

Quality-adjusted life years

Patient group Drug 1 Drug 2 Drug 3 Drug 4

A10126

B32106

C41026

Ž.

Unit cost US$ 1 3 7 5

US$11.

Fig. 1 enables committee members to evaluate the cost effectiveness of

a drug within the appropriate context. That is, in the presence of patient hetero-

geneity, the cost effectiveness of a given drug depends on the set of drugs in the

formulary.

With only a few drugs and a few patient groups, it is straightforward to

construct the equivalent of Fig. 1 by hand. However, the problem rapidly becomes

unwieldy as the number of drugs or patient groups increases. Fortunately, the

single-condition problem can be modeled as an MSP for which good solution

techniques already exist.

3.1. Menu-setting problems

MSPs arise when one individual, the setter, puts an array of permissible options

on a menu, from which each of a heterogeneous group of choosers will select their

preferred alternatives. The setter’s task is to define the menu that maximizes her

objective function subject to the choosers’ known preferences and various con-

straints. For clarity, we treat the setter in this paper as female and the choosers

as male. Though we develop our analysis in terms of the MSP generally, our

The respective menus and choices leading to these outcomes are as follows: Drug 1 is offered, all

groups choose it. Drugs 1 and 2 are offered, A’s and B’s choose Drug 1, and C’s choose Drug 2. Drugs

1, 2, and 4 are offered, A’s choose Drug 1, B’s choose Drug 4, and C’s choose Drug 2.

Although MSPs are generally quite common, formal treatment of MSPs did not begin until the

early 1980s, when marketing researchers began publishing articles on the problem of designing an

optimal product line. The ‘MSP’ is known as the ‘product-line design problem’ in the marketing

Ž.

literature. The objective there is to maximize the setter’s profits. Olmstead and Zeckhauser 1996

discovered this problem independently, gave it the menu-setting name, and pioneered the subsidized-

menu problem, the context where purchases are likely to be subsidized and maximizing consumer

welfare or social surplus is likely to be the setter’s objective.

Ž.

MSPs are characterized by a principal–agent relationship in which: 1 there is a group of

Ž.

heterogeneous agents, the choosers, with a known distribution of preferences; 2 there is one principal,

Ž.

the setter, who sets the menu from which each member of the group must choose; 3 each agent gets

Ž.

one choice from the menu; and 4 the agents’ and the principal’s preferences differ. See Pratt and

Ž.

Zeckhauser 1991 for a thorough discussion of agency problems.

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550534

Fig. 1.

primary interest, given our health-care audience, is contexts where price is set

below marginal cost, the subsidized menu problem.

MSPs are solved from the perspective of the setter.

The setter’s objective

function may be purely self-interested; for example, a private-sector firm with

some market power may select a mix of competing products to maximize profits.

In the drug formulary case, the objective function is likely to relate to the welfare

distribution among choosers, with some measure of overall welfare as the maxi-

mand. In the single-condition problem, the formulary committee is the setter; all

patients are the choosers; and the menu consists of the drugs on the formulary.

One possible objective of the formulary committee is to define the menu of drugs

Ž.

that is, the formulary that will maximize the expected ‘health utility’ of a

randomly chosen patient, subject to a budget constraint.

This objective can also

be thought of as maximizing the expected total health benefit from the formulary.

The challenge to efficient allocation within the MSP framework arises when

two conditions apply: first, the choosers are heterogeneous either in their prefer-

ences, or in the setter’s preferences for their choices, and second, the setter’s and

at least some choosers’ preferences differ. For example, the setter might like to

make option a1 available to chooser i, but she will not do so because that would

also make it available to chooser j, who would select it, and she finds the pair of

combinations a1-i and a1-j highly unfavorable; she decides to refuse option a1

to chooser i. If the choosers were not heterogeneous, the setter could put solely

her preferred choice on the menu. If the choosers’ preferences corresponded to her

own, she could put all options on the menu, knowing that the choosers’ selections

would be her own.

Sometimes the choosers are clustered into categories—say

Note that the solution to an MSP tells the setter which specific options already under considera-

tion to include on the menu; it does not tell the setter which options to consider in the first place.

Ž.

This might be the objective selected by a chooser in an ‘original position’ Harsanyi, 1955 ; that is,

before he knew who he would be and before he knew from what conditions he would suffer.

At this stage in the paper, we ignore considerations such as set-up costs, or economies of scale,

which would limit choices to capitalize on decreasing average costs.

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550 535

patients with ear infections—with a menu offered to each category. If the choosers

in each category remain heterogeneous, then each category would present its own

MSP.

The divergence in preferences in the context of the formulary problem arises

because the patients, whose drug purchases are subsidized price below marginal

cost to the provider , will have insufficient incentive to be concerned with the cost

to the provider, hence to their fellow enrollees, of the treatments they receive. In

contrast, the health-care provider must usually meet some budget constraint.

cost effectiveness is a concern, letting each patient or patient group choose for

itself may be a poor system. Table 3, which parallels Table 2, but with different

entries, illustrates this point.

Say that the budget is US$21. The optimal assignment—the one that maxi-

mizes QALYs for the dollars spent—gives Drug 1 to Group A, Drug 3 to Group

B, and Drug 4 to Group C. Thus 35 QALYs are obtained at a cost of US$20.

However, if the menu contained Drugs 1, 3, and 4, B’s would choose Drug 3, as

the optimum requires, but A’s and C’s would deviate, selecting Drugs 4 and 3

respectively, each gaining only 1 QALY at respective additional costs of US$6

and US$5. Expenditures would then be US$31, hence out of reach. If the plan lets

patients choose for themselves, the optimal menu with a budget of US$21 includes

only Drugs 1 and 2. In this case, Group A chooses Drug 1 and Groups B and C

choose Drug 2. Thus 25 QALYs are obtained at a cost of US$21. Interestingly,

B’s and C’s are both worse off under the menu system than they would be if the

plan used an assignment system with the same budget.

3.2. Model for the MSP

We now present a mathematical model of the single-condition formulary

problem. To highlight the menu-setting aspect of the problem, we take drug prices

as fixed, though formularies can in fact often bargain for lower prices, possibly

yielding significant savings. We also allow for a fixed cost in having a drug on the

formulary. Though order costs represent fixed costs, the more consequential fixed

expense that results from including a drug is keeping doctors informed about its

properties.

Let i index the groups of patients with the medical condition, is1,...,m; and

let j index the drugs capable of treating the medical condition that are being

considered for inclusion in the formulary, js1,...,n. Let u sthe expected

Doctors are agents for both the health plan and their patients, which produces a conflict of interest.

Ž.

Presumably, professional ethics the Hippocratic Oath requires that they place their patients’ interest

first, and consider resource expenditure only secondarily. In this spirit, resource-saving concerns, say

by an HMO, may dictate prescribing generic rather than equivalent brand-name drugs in most

circumstances, since there is likely to be little or no therapeutic difference.

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550536

Table 3

Quality-adjusted life years

Patient group Drug 1 Drug 2 Drug 3 Drug 4

A102111

B 2615 3

C 091110

Ž.

Unit cost US$ 1 10 12 7

utility a patient in group i gets from drug j.

Let c s the unit cost of treating a

patient in group i with drug j, let f s the fixed cost of including drug j in the

formulary, and let Bsthe pharmaceutical budget. Let N s the number of patients

in group i.

Let y be a binary variable indicating whether or not drug j is put on the

formulary, and let x be a binary variable indicating whether a patient in group i

chooses drug j. We would normally expect that the formulary’s objective is to

maximize consumer welfare—or equivalently, the expected welfare of a randomly

chosen individual—subject to an expenditure constraint.

The formulation is

then:

ConsumerWelfare:Maximize Nu x 1.0

Ž.

ÝÝ

iijij

, y

is1js1

subject to:

x s 1 ; i 1.1

Ž.

x F y ; i, j 1.2

Ž.

ij j

uxGuy ; i, j 1.3

Ž.

ik ik ij j

mn n

Nc x q fyFB,1.4

Ž.

ÝÝ Ý

iij ij jj

s1js1 js1

and

4

x , y g 0,1 ; i, j.1.5

Ž.

ij j

Ž.

Constraints 1.1 require each patient to choose exactly one formulary drug.

Ž.

Constraints 1.2 ensure that only drugs in the formulary can be chosen by the

The u ’s are assumed to be normalized within individuals so that their addition across patients is

more meaningful, as it is with QALYs. Measures other than QALYs are possible, of course, including

Ž. Ž.

healthy-year equivalents HYEs , willingness-to-pay WTP , and direct utility.

This objective function evaluates welfare within the boundaries of the plan, i.e., for the provider’s

enrollees, and does not consider, for example, society’s resource costs in producing drugs, profits to

drug companies, or pharmaceutical R&D expenditures.

()

T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550 537

Ž.

patients. Constraints 1.3 guarantee that each patient chooses the formulary drug

Ž.

that gives him the most utility. Constraint 1.4 ensures that the budget constraint

Ž.

is not violated. And constraints 1.5 ensure that drugs are either offered or not,

and if offered, either chosen or not.

Ž.

The above formulation 1.0–1.5 parallels a model presented by Green and

Ž. Ž.

Krieger 1985 . Other formulations are also possible. Dobson and Kalish 1993 ,

for example, extend the Green and Krieger model by explicitly considering prices

and costs. The net utility to the patient is defined as the difference between value

and price, and is measured in dollar terms, as opposed to the QALY metric that we

considered above. That is, u s Õ y p , where Õ is the expected dollar value of

ij ij j ij

the drug to the patient, and p is the price of the drug j that is charged to the

patient.

Each drug has an invariant per-unit cost, c , and a fixed cost, f . In this

formulation, prices may vary. The objective is to maximize social surplus, which

equals consumer welfare less the plan’s expenditure on drugs, both quantities

measured in dollar terms. The objective is thus:

mn n

SocialSurplus:Maximize N Õ y cxy fy.2.0

Ž. Ž.

ÝÝ Ý

iij j ij jj

, y, p

is1js1 js1

Ž. Ž.

The equivalents of constraints 1.3 and 1.4 change for this objective. See

Appendix A, which formulates the multiple-condition problem with patient utili-

ties set equal to expected value from the drug less their price.

The MSP framework is very flexible and can be modified to model the

particular requirements of many scenarios. The MSP framework can adopt a

societal, institutional, or patient perspective, depending on how the setter defines

the objective function and the factors she includes in the costs and benefits. The

MSP framework can also allow for a variety of interdependent drug restrictions.

Such restrictions may arise from negotiations with pharmaceutical suppliers and

take the form of ‘eitherror’ constraints i.e., two drugs cannot both appear in the

.Ž

formulary or ‘all or none’ constraints i.e., the setter wishes to offer either all of a

given set of drugs or none of them . If concerns about particular patient groups

One can now obtain a graph like Fig. 1 by iteratively solving the above formulation using

successively larger budgets, B.

This formulation is soundly based in microeconomics: u is a measure of consumer surplus—wil-

lingness-to-pay for the drug minus the price of the drug. Dobson and Kalish note that little work has

been done on measuring monetary utilities.

If the setter wishes to offer Drug 1 or Drug 2, but not both, she adds the constraint: y q y F1. If

the setter wishes to offer either all q of the drugs from a set M or none of them, she adds the

constraint: yjs

q, where

is a binary variable. Many other types of linear integer constraints

g M

may be incorporated within the standard MSP mathematical programming formulation.

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550538

or regulatory constraints make it essential to have a particular drug j in the

formulary, then y can simply be set to 1, and the remaining problem solved.

4. The multiple-condition formulary problem

We now consider all medical conditions and drugs simultaneously.

This is

important, since many drugs can be employed in treating a number of different

conditions. Simply combining the results of single-condition optimizations will not

work. When conditions are examined one at a time, there is no way to allocate the

fixed costs of a drug used to treat multiple conditions. More important, drugs that

may be highly cost effective in treating some conditions may be only marginally

beneficial in treating others, yet still selected for those conditions. Thus Prozac,

which is highly beneficial for individuals suffering from depression or obsessive–

Ž.

compulsive disorder, may offer only marginal positive and possibly negative

benefits for individuals experiencing normal reactions to situations of stress, such

as grief. Similarly Viagra, clearly useful in treating many forms of impotence,

might offer at best mildly enhanced sexual performance to individuals without

dysfunction. Yet, once on the menu, both drugs are likely to be demanded by

marginal beneficiaries.

Mathematically, the extension from the single-condition model to the multiple-

condition model is relatively straightforward. We maintain the assumption that

drugs approved for a given medical condition cannot be withheld from anybody

with that condition. However, we allow drugs to be approved for some medical

conditions and not for others. So, for example, powerful and expensive antibiotics

can be approved for postsurgical recovery but not for low-grade infections.

This

results in a powerful model that combines the MSP framework with the classic

assignment framework. It creates a group of MSPs—one for each medical

condition—connected through either a budget constraint or an objective function.

In practice, some desirable restrictions of drugs to particular conditions or

diagnoses may not be feasible; our formulation below allows for situations where

a drug made available for one condition must be made available for some others.

Ž.

Olmstead and Zeckhauser 1996 consider a variety of nonutilitarian objective functions for MSP

problems, including citizen choice models with plurality, Borda and approval voting, and Rawlsian

Ž.

maximin and leximin objective functions.

The same model that solves the multiple-condition formulary problem can be used to solve the

more general multiple-condition problem—that considers all treatments simultaneously—by replacing

the words ‘drug’ and ‘formulary’ with the words ‘treatment’ and ‘health plan’ in the following

discussion.

Ž.

Frequently drugs are even prescribed for off-book i.e., non-FDA approved uses, though this may

be due more to drug approvals lagging behind experimentation and knowledge, rather than drugs being

deployed where their benefits are marginal.

The decision to restrict a given drug to certain medical conditions must be made by the formulary

committee when setting up the problem.

()

T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550 539

Let K be the set of all r medical conditions to be treated, and let J be the set

of all drugs considered for inclusion in the formulary. Let J : J be the set of all

drugs that may treat condition k, and let K : K be the set of all medical

conditions that may be treated with drug j. Let I be the set of all patient groups

with medical condition k.

Let u s the expected utility a patient in group i

ijk

gets from receiving drug j for condition k, kgK, jgJ , igI . Let c s the

kk ijk

unit cost of treating a patient in group i with drug j for condition k, kgK,

jgJ , igI ; let f sthe fixed cost of including drug j in the formulary, and let

kkj

Bsthe pharmaceutical budget. Let N sthe number of patients in group i with

condition k, kg K, igI .

Let z be a binary variable indicating whether drug j is included on the

formulary, jgJ; let y be a binary variable indicating whether drug j is included

on the formulary and approved to treat condition k, kgK, jg J ; and let x be

kijk

a binary variable indicating whether a patient in group i chooses drug j for

condition k, kgK, jgJ , igI . When the setter’s goal is to maximize consumer

welfare, the formulation is:

ConsumerWelfare:Maximize Nu x 3.0

Ž.

ÝÝÝ

ik ijk ijk

, y, z

kgKjgJigI

subject to:

x F1 ;kgK, ; igI ,3.1

Ž.

ijk k

x Fy ; kgK, ; igI , ; jgJ ,3.2

Ž.

ijk jk k k

uxG uy ;kgK, ; ig I , ; jgJ ,3.3

Ž.

ilk ilk ijk jk k k

Nc x q fzFB,3.4

Ž.

ÝÝÝ Ý

ik ijk ijk j j

gKjgJigIjgJ

4

x ,y ,z g 0,1 ;kgK,; ig I , ; jg J , ;lgJ,3.5

Ž.

ijk jk l k k

and

y Frz ; jg J.3.6

Ž.

jk j

Ž.

There are two significant differences between 3.0–3.6 and its single-condition

Ž.

counterpart, 1.0–1.5 . First, 3.1 is written as an inequality to enable the

provider to avoid treating certain medical conditions altogether. If this is undesir-

The partition of patient groups can vary by medical condition.

If we were dealing with medical treatments in a health plan, this formulation would apply if there

were a unified budget for the plan, covering pharmaceuticals, diagnostic tests, mental health, etc. If

there are separate budgets for different categories, then there would be one budget constraint equation,

equivalent to 3.4, for each category.

()

T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550540

Table 4

Quality-adjusted life years

Condition 1 Condition 2

Patient group Drug 1 Drug 2 Drug 3 Patient group Drug 4 Drug 5 Drug 6

A5710D538

B673E541

C875F563

Ž. Ž.

Unit cost US$ 2 4 6 Unit cost US$ 1 5 10

able, the provider can ensure that a particular medical condition is covered by at

Ž.

least one formulary drug by changing the appropriate constraint in 3.1 to an

Ž.

equality. Second, 3.6 is necessary to model the fixed costs of putting a drug on

the formulary.

We now demonstrate this model in action. We consider two conditions, each

with three possible drugs and three classes of patients. The data are shown in

Table 4.

The setter’s objective is to maximize the total QALYs in the population of

patients suffering from these two conditions, subject to a budget constraint and an

additional equity requirement that at least one drug be offered to treat each

condition.

The results of this problem for the case in which each group contains five

patients are shown graphically in Figs. 2 and 3. For each budget level, Fig. 2

shows the number of QALYs that can be obtained and Fig. 3 shows which drugs

should be offered.

Note that some of the drugs move on and off the formulary

as the budget varies. Such jumps as a response to relaxing a constraint are

common in models like the MSP that contain binary variables and linear con-

straints and objectives.

Figs. 4 and 5 underscore the importance of accurately modeling the size of

each patient group. These figures, which differ significantly from Figs. 2 and 3,

show the results of the problem for the case in which groups A through F contain

10, 2, 3, 1, 7, and 7 patients, respectively. The same number of patients, 15,

suffer from each condition as before; only the mix of patients among the three

groups suffering from each condition has changed . Given the crowding of

outcomes, Fig. 4 does not indicate the drugs associated with each outcome.

When constructing Fig. 3, we broke ties in favor of the least expensive formulary. With a budget

of US$85, for example, the setter can obtain 200 QALYs in the patient population either by offering

Drugs 1, 3, 4, and 5, or by offering Drugs 1, 2, 3, and 4. However, since a formulary comprising Drugs

1, 3, 4, and 5 costs US$85 and a formulary comprising Drugs 1, 2, 3, and 4 costs US$75, the latter

formulary is assumed to be optimal for an US$85 budget.

()

T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550 541

Fig. 2.

Fig. 3.

The optimal formulary differs considerably between these two cases Figs. 3

. wxw x

and 5 over the budget intervals US$55–US$89 and US$95–US$140 . In

Fig. 4.

()

T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550542

Fig. 5.

addition, the QALYs vs. budget ‘curves’ shown in Figs. 1, 2 and 4 are ill-behaved

in that each exhibits both increasing and decreasing returns to scale over various

budget intervals; this effect is especially pronounced in Fig. 4 over the budget

interval US$45–US$94 . Such curves are common in MSPs. While the MSP is, in

general, a difficult problem to solve to optimality, general-purpose math program-

ming software and specialized heuristics can readily find near-optimal solutions to

problems of realistic size.

5. Pricing in MSPs

Ž. Ž .

Whether prices are modeled explicitly, as in 2.0 and Appendix A A.0–A.5 ,

Ž.Ž.

or implicitly, as in 1.0–1.5 and 3.0–3.6 , they are an integral part of the solution

to any MSP. What do we expect these prices to look like? The answer depends on

a number of important factors, including the objective of the setter e.g., social

surplus, consumer welfare maximization, or profit maximization , the degree to

which the setter can price discriminate among the choosers, and the degree of

Ž.

market power enjoyed by the setter e.g., price-taker vs. monopolist . The essence

of the subsidized-menu problem is that prices are set below marginal cost,

presumably to promote risk spreading and, possibly, distributional goals.

Our discussion of pricing applies to MSPs in general, and is not limited to

formulary drug pricing applications. There are a number of other important

The MSP belongs to the class of problems known as NP-complete, which means that the time it

takes to find the optimal solution may grow exponentially with the size of the problem. GAMS and

LINDO are commonly used to solve math programming problems like the multiple-condition MSP.

Ž. Ž.

Dobson and Kalish 1993 and Green and Krieger 1985 provide a technical discussion of specialized

heuristics that can solve large single-condition MSPs to within 1% of optimality in seconds.

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550 543

applications of MSPs in the health-care arena, and we discuss pricing in this

broader context. The most general is what mix of services a provider, such as an

HMO or an insurance plan, should offer. A key problem for such plans is the

challenge that leads to MSPs: once an option is on the menu it will be selected by

individuals who receive little benefit from it, since they pay but a fraction of the

cost. Price could be raised to deter low-value users, but not high-value users, but

that would sacrifice efficiency in risk spreading.

5.1. Profit maximization and price discrimination

The menu-setting formulation is adaptable to a range of objectives, including

profit maximization, the presumed objective of a for-profit medical enterprise. The

setter’s strategy will depend on the extent to which she can price discriminate

among the choosers. If she can perfectly price discriminate, she is a monopolist

who has complete information about her choosers and is permitted to offer them

different menus. Then the setter is no longer solving an MSP, but rather a classic

assignment problem. In practice, such tailoring is rarely feasible in a medical

setting; the same products must be offered at the same price to all individuals, or

at least to broad classes of individuals. For example, all privately insured patients

might pay one price, government-insureds another, and charity patients who

cannot be denied treatment nothing. Moreover, even in a for-profit setting, if

patients are risk averse with respect to monetary expenditures, there will be an

up-front charge with use prices set below marginal cost. Thus, the MSP may

appear even in a for-profit setting.

In many real-world situations, the choosers cluster into categories—say, pa-

tients with knee injuries—with a menu offered to each category. If the choosers

remain heterogeneous within categories, then each category presents its own MSP.

Consider, for example, the problem of a health plan deciding which medical

treatments to offer its enrollees. In practice, patients are probably grouped by

Ž.

medical condition e.g., knee injury, ear infection , and each patient with a certain

medical condition is offered a menu of treatments designed specifically for that

condition. The MSP framework can be adapted to model this type of problem by

defining the decision variables to reflect the feasible chooser-item choice combina-

Ž.Ž .

tions, as in 3.0–3.6 and in A.0–A.5 . The resulting model is a powerful hybrid

of the ‘pure’ MSP and assignment frameworks.

When the setter cannot identify the choosers but knows their distribution of

preferences, then adverse selection prevents her from extracting as much surplus

as above, but she can still price discriminate indirectly. If the setter is a

monopolist, for example, she will provide products of quality lower than the ideal

to the less lucrative segments of the population to minimize the switch to these

products by the more lucrative segments. Alternatively, gold-plated products might

lure and extract rent from the highest value customers, a pattern we see with

luxury wings of hospitals. Of course, such tailoring can be done by altering any of

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T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550544

the products’ attributes, including price. As the market power of the setter

declines, prices will approach marginal costs.

5.2. Marginal cost pricing?

Let us return to the formulation where the setter seeks to maximize social

surplus, as defined by the consumer welfare of participants in the plan. It is

tempting to conjecture that efficiency concerns will lead a benevolent setter to use

marginal cost pricing. However, benevolent setters are often concerned with more

than mere economic efficiency. In many public and nonprofit settings, for

example, prices are heavily subsidized or set at zero. In most medical care

settings, including hospitals, HMOs, and drug formularies, the price to the

consumer at the time of purchase is set well below its marginal cost. Such

subsidies facilitate risk spreading and cater to various conceptions of social

Ž.

welfare such as arguments that health care is a merit good or right .

Distributional concerns may lead a benevolent social planner who can differen-

tiate among choosers to cross-subsidize medical care by charging ‘well-insured’

patients a greater portion of their marginal costs than ‘under-insured’ patients.

Moreover, marginal cost pricing may be ruled out by political considerations. For

example, tuitions at most colleges are subsidized, and few people would prefer to

charge chemistry majors more than English majors, even if the former cost more

to educate.

Thus, while marginal cost pricing may promote efficient resource allocation,

we do not expect to see it in practice in many settings in the health care field.

Virtually all drug formularies subsidize drugs to some extent.

In MSPs with

benevolent setters, marginal cost pricing, or indeed any pricing in pursuit of

efficiency, should be thought of as one desideratum competing with such goals as

political acceptability, distributional concerns, and risk spreading. Even for-profit

medical plans may employ subsidies to promote risk spreading, thereby making

their plans more attractive and permitting them to remain competitive at a higher

premium.

The MSP framework can accommodate real-world constraints on pricing in a

number of ways. For example, if the prices are predetermined e.g., zero or a

nominal copayment , they are in effect set exogenously and treated as parameters

in the MSP.

The resulting problem then seeks the menu that maximizes welfare

Note that in settings where subsidies occur, the setter is usually constrained by available resources

such as the budget, and subsidies often must be covered by the premiums charged to all participants.

See also

, which discusses the drug-pricing problem taking society’s rather than the provider’s

resource costs into account.

In some circumstances, subsidies may be desirable from the standpoint of the plan’s budget

because they will induce drug compliance and thereby reduce future medical costs.

Labor negotiations sometimes set pricing rules for drugs, but the composition of the formulary is

left to the health plan or plans offered by the employer.

()

T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550 545

Ž.

subject to politically or socially determined prices and often a budget constraint .

Instead of exogenously setting the prices, we may wish to set price ceilings on

certain items or to restrict prices for certain classes of drugs or services to be

identical. Each of these situations can be modeled without difficulty by adding

appropriate constraints to the basic MSP model.

6. Limitations of the MSP framework for drug formularies

In this section, we consider a number of limitations to using the menu-setting

framework to determine drug formularies, one of their most immediate areas of

application, and thereby highlight areas for future research.

6.1. Data

A practical limitation of the MSP framework is the need for information about

the benefits and costs of using drugs to treat medical conditions. For example, the

MSP framework requires a global outcome measure to assess health benefits both

within and across medical conditions. Unfortunately, there is no general agreement

on methods for combining the measurement of the different dimensions of health

outcomes such as morbidity and mortality into a unidimensional measurement

scale, and some researchers question whether there can be a single best measure

Ž. Ž.

Detsky, 1994 . For example, Johannesson et al. 1993 and Zeckhauser and

Ž. Ž.

Shepard 1976 recommend using QALYs, Mehrez and Gafni 1989 recommend

Ž. Ž.

using healthy-year equivalents HYEs , Gafni and Birch 1993 recommend using

Ž.

willingness-to-pay WTP , and still others recommend using conjoint analysis to

assess utility directly Chinburapa and Larson, 1989; Maas and Stalpers, 1992;

Hornberger et al., 1995 . None of these measures is universally accepted, which is

not surprising given the distributional consequences and ethical implications of

each.

Even the measurement of costs for drugs can sometimes be difficult and

contentious. Although charges for drugs and services are relatively easy to capture,

the relationship between such charges and underlying resource costs is tenuous in

a system where reimbursement schedules, charity cases, and other factors distort

Ž.

the market Willke, 1995 . While the full firepower of econometrics can be

applied when analyzing cost data, the collection of such data for drugs is poor

Ž.

Willke, 1995 . Fortunately, both the quality and the quantity of data are improv-

ing with the maturation of the field of pharmacoeconomics—which describes and

analyzes both costs and outcomes of drug therapy to health care systems and

In the discussion of the thrombolytic drug data in Table 1, we saw that the choice of output

measure had a salient effect on the relative well-being of ‘under 75’ and ‘over 75.’

()

T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550546

society. In any case, the MSP framework is available for use by formulary

committees where the appropriate data are or might be made available.

6.2. Scope

The MSP framework can be used to determine the formulary that maximizes

the health returns from a given level of pharmaceutical expenditures. By varying

the budget, the user can sketch out the frontier of possible outcomes. But the

framework cannot tell us what the tradeoff rate should be between health outputs

Ž.

for example, QALYs and dollars, which must be known if we are to set drug

budgets appropriately, or maximize social surplus directly.

Moreover, just as we criticized formulary committees for considering one class

of drugs at a time, we too can be criticized for taking an overly narrow view. By

focusing solely on drug expenditures, we assume that expenditures on all other

types of medical treatments are set at optimal levels. A broader perspective would

determine the optimal amounts to spend on medical treatments of all kinds, not

just different drugs. But critics concerned with allocating society’s resources for

all welfare-related activities would consider even this perspective too narrow.

Where to limit the scope of one’s analysis is a political, institutional, and

philosophical question that must be answered for each formulary committee before

it embarks on an MSP analysis.

The models considered here reveal the implications of a wide variety of

assumptions when determining drug formularies. For example, a committee could

examine the relationship between patient benefits and the pharmaceutical budget,

or evaluate the impact of approving a given drug for a particular medical condition

in terms of its benefits for patients and the budget. It could look at the distribu-

tional consequences of assuming a utilitarian objective function, and modify the

model accordingly. The models can be used for extensive sensitivity analysis by

altering data values as well as objective functions.

7. Conclusion

The menu-setting framework, here focused on determining a drug formulary,

helps to address problems in a wide range of health-care settings. Three key

Ž.

characteristics of such settings are that 1 subsidies are common, thereby ruling

out marginal cost pricing as an instrument sufficient to ensure an efficient

Ž. Ž.

outcome, 2 there are multiple medical conditions, and 3 within each condition

patients are heterogeneous with respect to benefits from treatment.

The challenge of the MSP arises because patients choose or have their

physicians choose for them drugs or treatments on a self-interested basis. Hence,

the substantially simpler assignment problem cannot be employed to allocate

patients to treatments. The agency loss of excess expenditure arises. The MSP

()

T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550 547

formulation is designed to curtail agency losses in such situations. In similar

fashion, it can assist the employer who must select the menu of subsidized health

care plans to be offered to employees.

The MSP framework permits optimization in a subsidized situation, but it does

much more. It highlights the payoffs to both expanding and constraining alterna-

tives, and it illuminates the virtues and limitations of using prices to limit demand.

The MSP framework offers insights whenever a principal must design a program

for a heterogeneous group of subsidized, self-interested agents; in effect in the vast

majority of circumstances where health care is offered.

Acknowledgements

We thank Michi Aoi, David Cutler, William Hogan, Joseph Newhouse, John

Pratt, Stephen Schenkel, M.D., Miles Shore, M.D., Charles Slavin, the interview-

ees listed in the references, and the referees. The Taubman Center for State and

Local Government and the Robert Wood Johnson Foundation provided research

support.

Appendix A. The multiple-condition problem with prices

The following formulation of the multiple-condition problem is an extension of

Ž.

objective function 2.0 , with appropriate constraints.

Parameters

K the set of all r medical conditions to be treated

J the set of all drugs considered for addition to the formulary

J the set of all drugs that may treat condition k, J : J

k k

K the set of all medical conditions that may be treated with drug

j, K : K

I the set of all patient groups with medical condition k

Õ the value to patient i, measured in monetary terms, of using

ijk

drug j for condition k, kgK, jg J , igI

p the price to the patient of using drug j for condition k, kgK,

jgJ

u s Õ yp the utility a patient in group i gets from receiving drug j for

ijk ijk jk

condition k, kg K, jgJ , igI

c the unit cost of treating a patient in group i with drug j for

ijk

condition k, kg K, jgJ , igI

f the fixed cost of including drug j in the formulary

B the pharmaceutical budget

()

T. Olmstead, R. ZeckhauserrJournal of Health Economics 18 1999 523–550548

N the expected number of patients in group i with condition k,

kgK, igI

Decision Õariables

z a binary variable indicating if drug j is added to the formu-

lary, jgJ

y a binary variable indicating if drug j is added to the formulary

to treat condition k, kgK, jg J

x a binary variable indicating whether a patient in group i

ijk

chooses drug j for condition k, kgK, jg J , igI .

The objective of the setter is to maximize social surplus. The formulation is:

SocialSurplus:Maximize N Õ y cxy fz A.0

Ž. Ž.

ÝÝÝ Ý

ik ijk ijk ijk j j

, y, z, p

kgKjgJigIjgJ

subject to:

x F1 ;kgK, ; igI A.1

Ž.

ijk k

x Fy ; kgK, ; igI , ; jgJ A.2

Ž.

ijk jk k k

Õ ypxG Õ ypy ; kg K, ; igI , ; jgJ A.3

Ž.Ž. Ž.

ilk jk ilk ijk jk jk k k

4

x ,y ,z g 0,1 ; kgK, ; igI , ; jgJ , ;lgJ A.4

Ž.

ijk jk l k k

y Frz ; jg J.A.5

Ž.

jk j

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