Rethinking Discounting of Health Benefits in Cost Effectiveness Analysis by Arnab K. Acharya ...

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Rethinking Discounting of Health Benefits in Cost Effectiveness Analysis
by
Arnab K. Acharya
Institute of Development Studies
University of Sussex
Falmer-Brighton
Sussex, BN1 9RE
United Kingdom
[email protected]
and
Christopher JL Murray
Global Programme on Evidence for Health Policy,
WHO Geneva. 1211 Ave. Appia
Geneva, Switzerland
[email protected]
October 2000
Rethinking Discounting of Health Benefits in Cost Effectiveness Analysis
Arnab Acharya and Christopher JL Murray
It is now standard practice in most cost-effectiveness studies to discount future health
benefits at a rate between three and five percent per year (Evans and Hurley 1995). For example,
new cost-effectiveness guidelines from a committee appointed by the United States Public Health
Service (Gold et al. 1996) recommend that QALYs gained in the future should be discounted at
three percent. Likewise, the World Bank (Jamison et al. 1993) recommends that DALYs averted
through interventions should be discounted at three percent per year in cost-effectiveness studies.
Despite the near universal use of discounting in cost-effectiveness analyses, the practice is
widely debated. Public health advocates of prevention programs often feel that devaluing future
health gains through discounting is counter-intuitive. A classic example is the case of hepatitis B
immunization programs where the primary health benefits occur 40 years after immunization.
Using a 3% discount rate the benefits of these programs appear to be nearly one order of
magnitude smaller than if the benefits are not discounted. Challenges to discounting health
benefits or more generally future well-being, however, have also come from philosophers (Parfit
1984, Sidgwick, 1907 etc.) and economists (Anand and Hansen, 1997; Marglin, 1964; Sen,
1964).
By discounting future health states we mean the act of valuing a particular health state at
a nearer time higher than the exact same health state occurring at a later time. . Discounting
future is tantamount to discounting wellbeing. There is no rational reason why someone would
have a preference over today’s wellbeing over that of the future. Although it could be that
individuals discount their own future health status; it does not follow that society should on
social projects that yield value into the future. Discounting of benefits into the future is called
the social discount rate; social discount rate can discount future money returns as well wellbeing
that cannot be measured in terms of money. Our discussion is primarily regarding the discounting
of future wellbeing. Arguments in favor of discounting wellbeing require justification, as prima
facie there seems to be no reason why, say, environmental policies should be made by devaluing
lives saved 10 years from now in comparison to today’s lives. The arguments against
discounting mainly runs by pointing out arguments in favor of discounting are not valid.
Disparate lines of argumentation have been used to justify discounting health benefits:
individuals may through their market behavior express a positive discount rate for their own
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health; rational individuals should discount their own health; individuals may in practice want
society to make decisions affecting future health using a positive discount rate; and rational
social decision-makers may be forced to discount future health benefits to avoid absurd or
inconsistent decisions. The main thrust of each of these arguments is briefly reviewed and then
the paper focuses on two issues through theoretical models: whether or not rational individuals
will always discount future health states and whether or not social evaluation of future health
states be discounted. Most practitioners of cost-effectiveness analysis view these
consequentialist arguments as the strongest basis for discounting future health benefits. All
consequentialist arguments are based on the assumption that the social objective regarding health
is to maximize aggregated health states.
Viscusi and Moore (1989) extrapolate empirically a discount rate for future life years by
examining workers choice between wages and hazardous employment. Marginal rate of
substitution between wage and risk is a function of implicit price of risk, attitude toward risk and
rate of time preference for consumption. They find that educated workers discount future life
years at a lower rate than uneducated workers. Interview studies have also been used to elicit
discount rates for future health, although the findings have demonstrated a wide range of results.
Redelmeir and Heller (1993) studied medical students, house officers and attending physicians to
explore their individual discount rates for different health outcomes. Discount rates calculated
from interview responses ranged from -0.9 to .096 with a mean .033. 15.7% of the rates
calculated from interview responses were greater than 10% and 10% were less than zero.
Individuals may discount their own future health for many reasons including high rates
of pure time preference for utility derived from consumption, distorted perceptions of health
risks, or perception of their health and monetary status. Even if in practice a subset of
individuals discount their own future health, we can ask if rational individuals should be led to
discount their own future health. Does the existence of a positive rate of pure time preference for
consumption automatically imply that individuals should discount their own future health? In
the model developed in section 2 of this paper, we show that this is not necessarily the case.
For cost-effectiveness analyses of health interventions, arguments for or against
discounting health are not usually based on what individuals actually do but, but perhaps what
they express. Cropper et al. investigated individuals social discount rates for health benefits and
found that there is a positive correlation between the two among each individual and further that
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it varies by education and wealth. There are several reasons to suspect that social discount rates
for health benefits may not be an aggregation of individual health discount rates of those active
in the marketplace. First, individuals may have a lower discount rate for social decisions in
which tradeoffs are made between the health of one group and another. Second, there is no a
priori logical reason why the concerns of future generations for their own health should be
ignored in choosing a discount rate for social decisions affecting the health of those generations.
Further, we would need to arrive at a social discount rate through a collective choice rule from
the individual preference orderings regarding health states for people one does not know.
Arrow’s Impossibility Theorem rules out a reasonable collective choice rule. Arguments for
discounting in cost-effectiveness analysis come from sources other than any attempt to impose a
functional form over individual preference ordering regarding discounting health.
The dominant arguments in the literature on cost-effectiveness for discounting are
consequentialist arguments, namely that the consequences of not discounting are unacceptable.
The two main arguments, the time paradox and the consistency argument, are not arguments for
discounting the value of future health per se but arguments that health must be discounted at the
same rate as costs. This combined with the positive returns on investment then requires that
health benefits be discounted, and at the same rate as that of consumption.
The time paradox was first formulated by Keeler and Cretin (1983) who argue that
postponement of expenditure on health would entail higher number of saved lives in the future
years due to interest earned on the saved expenditure. If society values health the same way
every period, then postponement of expenditure would entail improved cost-efficiency as
expenditure on health is pushed forward indefinitely into the future. They show that since cost
borne in present value terms on a project is always less when postponed one year hence and the
health benefit is the same when postponed and not discounted at the same rate, it always pays to
defer a health intervention program. One would thus never implement a health intervention
program. This is called the time paradox that only disappears, Keeler and Cretin claim, if cost of
health and benefit are discounted at same rate. Although Keeler and Cretin model is presented
mathematically, certain assumptions should be made clearer and they do not stand up to closer
scrutiny. Note that discounting at an exponential rate here is seen as a normative stance from a
social point of view.
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Weinstein and Stason (1977) offer an argument based on consistency for discounting
future life years in cost-effectiveness analysis:
“[Since health states] are being valued relative to dollars and, since a dollar in the future
is discounted relative to a present dollar, so must a life year in the future be discounted
relative to a present dollar… It is the discounting of dollar costs, and the assumed steady-
state relation between and the assumed steady-state relation between dollars and health
benefits, that mandates the discounting of health as well as dollars.” (p. 720)
Under this view future health states should be discounted exponentially at the rate of interest in
the economy just as future cost is assessed in its present value. Whether or not individuals
discount future health is immaterial to the policy maker because the consistency argument is
essentially normative.
We will show that the conclusions of the time paradox argument and the consistency
argument depend on stringent unrealistic assumptions and quite contrary to those found in most
optimization models in economics. They can be seen as slightly arbitrary as they would not
apply to all types of projects that affect future health status. For example, environmental projects
that must be undertaken now to yield any benefit at all may save thousands of lives in the future.
Either the time paradox or the consistency argument would not require discounting these lives.
The purpose of this paper is to re-examine the standard arguments for discounting in
cost-effectiveness analyses. More specifically, we wish to answer two questions: First, must
health benefits be discounted at the same rate as costs? Including the specific variant of this
argument that benefits be discounted at the marginal rate of return on investment. Second, if we
conclude that we are not obligated to discount costs and benefits at the same rate, should health
benefits, nevertheless, be discounted at some positive rate? In the next section, Section II, we
clarify the terminology used to describe discounting and time preference. In Section III, we
review the use of discounting in cost-benefit analysis. In Section IV, we develop an expected
utility maximization model of individual behavior to show that rational and reasonable
individuals need not discount her future health. In Section V, a social choice model is developed
and used to show that both the time paradox and the consistency arguments do not hold for
reasonable assumptions about decision constraints. Having concluded that costs and benefits
need not be discounted at the same rate, we readdress the question of whether health should be
discounted in Section VI, the concluding section.
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II. Definitions and Terminology
Individuals may voice that they would accept a lower incremental consumption today
only for a higher incremental consumption a year from today. We call this practice discounting
of future consumption. Such discounting does not entail by itself that consumer’s value the
utility derived from next year’s consumption less than the utility derived from consumption
today. Consumers may prefer incremental consumption now to that in the future for several
reasons.
First, for most consumers the marginal utility derived from a unit of consumption is
declining as consumption levels increase. This relationship can be captured by a concave utility
function over consumption goods. If consumers expect to be better off next year, they will value
a marginal unit of consumption as less valuable next year compared to a marginal increase in
today’s consumption. Of course, if consumers expect to be worse off, ceteris paribus, they may
have a negative discount rate.
Second, real consumers do not live forever and have a probability less than 1 of
surviving to enjoy consumption next year. A rational individual will discount future
consumption by their perceived probability of survival in the future. On average, survival
probabilities would induce discount rates around 1% per year, but the actual and perceived rates
would depend on age, socio-economic status and location.
Third, individuals may prefer incremental consumption today to next year simply
because they are impatient or myopic. This last possibility is known as pure time preference.
All three reasons for discounting we formalize by using a simply intertemporal additive utility
function:
¥
(cid:229)
(1) EU(C)= d tp(t)u(c ),
t
t=0
where U is the utility function for the entire span of life, C denotes a consumption vector, E is
the expectation operator, u is the single period utility function, t denotes time,
(cid:230) (cid:246)
1
d = (cid:231) (cid:247) where a is the rate of pure time preference, and p is the probability of survival
Ł 1+a ł
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from time zero to time t.1 2 Note that in (1), the uncertainty appears through p(t), the probability
of survival.
The standard continuous version of Equation (1) is the following, for now we ignore
survival:
(1¢) (cid:242) Tu(c )e- dtdt.3
t
0
A standard practice is to distinguish between consumption discount factor and the discount rate.
The first is simply the marginal rate of substitution (MRS) between the consumption levels in
two periods and the second is the rate of change in marginal rate of substitution. As MRS is
simply the ratio of the trade one is willing is to undertake for two commodities, distinguished by
the time of availability, it answers the question how much would one be willing to trade today’s
consumption for that which would be available tomorrow. Hence, it can be written as the
following for period t and t+j:
Consumption disocunt factor = - dct = u’(ct+j)e- dj .
dc u’(c )
t+j t
Note that this is simply the ratio of the evaluation or price of the good or the willingness to pay
for good in period t+j and that in period t.
Discount rate is purely a marginal concept; it measures the value of consumption for the
marginal increase in time and consumption. Note that one is willing to pay
R =e- dtu’(c )
t t
for a small amount of good to be delivered at period t; the rate of change in time of this price is
the discount rate:

1The pure rate of time preference is usually called the rate of discount. For heuristic reasons we assume that
there is a constant rate of time preference.
2 Yaari’s (1965) formulation of the problem of optimal savings for an individual with a bequest motive
takes discounting as the rate of time preference and distinctive from the probability of survival. We think
this is misleading; yet, economist have not been careful to distinguish rate of time preference from
discounting (see Kamien and Schwartz, 1991; as this is a text book p. 63).
3 To avoid mathematical complexity we have taken a finite time horizon.
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dR u"(c) dc
t =d + .
R u’(c) dt
t
c
Multiplying through by =1and noting that the elasticity of substitution for consumption can
c
- u"(c)c
be written as s (c)= , we obtain
u’(c)
.
c
(2) r =d +s (c) ,
c
c
where the dot indicates time derivative. Equation (2) is also called the consumption rate of
interest (see Arrow and Kurz, 1970; Little and Mirrlees, 1974; Dasgupta, Mäler and Barrett,
1999). The interest rate for consumption good is composed of pure rate of time preference and
rate of change in valuation of the extra amount of the good multiplied by the rate of change in
consumption. If there is no change in consumption, we obtain the interest rate as the pure rate of
time preference.4 Even if the pure rate of time preference is zero, the term r is positive for any
c
consumer who is risk averse.
A clear interpretation of Equation (2) comes from Broome (1994). The interest rate
simply reflects the decline in price due to increase in production if dc/dt is positive. To the
question “Why should a person value a commodity less just because she will possess it in 1995
rather than 1994?” Broome (1994) answers that it has little to do with the values of the people
who buy present and future commodities. It has to do with the economy’s productive
technology, not with its consumption, if we think of Equation (1¢) as representing the average
consumer and c as the per-capita growth rate. Discount rate simply reflects the price of a dated
commodity; any commodity will have a present price at any particular point in time; the
percentage difference between the present price of a present commodity and the present price of
the same commodity next year is the commodity’s own interest rate. For goods which we value
at a constant marginal utility, the rate of interest is equal to time preference, which can be zero.

4 In discrete time the consumption rate of interest would be MRS between consumption in period t and t+1.
Arrow and Kurz (1970) and Dasgupta, Mäler and Barrett (1999) stress that in Equation (2) we are offering
the consumption discount rate where consumption is the numeraire good; that is consumption’s value is
measured by itself not by some other good.
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The utility function so far is over consumption goods. In cost-benefit analysis all goods
can be valued in terms of consumption goods, the standard cost-benefit analysis can employ
Equation (2), we will examine that issue in the next section. We will develop an equation similar
to Equation (2) for health and examine its implications subsequently. We now turn to use of
interest rate or the discount rate in cost-benefit analysis.
III. Project Evaluation
Whether or not a project that yields future returns should be carried out is determined in
light of how much value the project yields in the future. Given that Equation (2) can represent
the valuation of the future unit of consumption by a representative consumer, how should the
future returns on an investment be valued from a social point of view? Note that in Equation (2)
that part of the second term in the right hand side is the growth rate. If capital is essentially non-
consumption at the present, in a competitive economy r will equal the marginal productivity of
c
capital, for now ignore the issue of indeterminacy. The marginal productivity of capital must be
r as one would be indifferent between sacrificing this amount now; and any return on higher
c
than this would be arbitraged away in a competitive economy.
The rate r is a constant in time if the economy is on a steady-state growth path where
c
capital-labor is constant; the simplest way this can occur is if the economy can be described by
constant returns to scale Cobb-Douglas technology. It is the simplest version of Solow’s neo-
classical growth model. Here per-capita rate of accumulation of capital can converge to an
exogenous growth rate of labor; at this point savings induced is the optimal savings rate if the
path of capital accumulation is that which induces maximum feasible per-capita consumption
(see Burmeister and Dobell, Chapter 1, 1970). In the simplest version of neo-classical growth
theory the interest rate is determined by the rate of growth of labor; technology and rate of
growth determine the capital-labor ratio.
Any project small in comparison to the economy whose productivity is below that of rate
of return of the prevailing capital stock should not be adopted. The criterion of adoption of
project that yields benefits up till time T is the following:
(cid:242) T e- (cid:242)-t f’(k(u))du ‡ 0,
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where f’(k(u)) is the rate of return on capital. One needs to know the short term interest rate
through knowing the return on capital in each period; for that one needs to know the capital
accumulation up to any time t. Project evaluation, then, is only possible at the steady state;
however, some approximation methods are available to characterize returns around the steady
state capital level (Arrow and Kurz, 1970). Only if the economy is in the steady state path,
should the benefits, near and far, be discounted at the same rate.
The rate of interest given above is the consumption rate of interest. At times, however,
this rate may be different than the interest rate producers face. If public investment comes out of
private consumption then the applicable interest rate is the rate that consumers face. No project
with its own interest rate below this interest rate would be adopted. This is in effect how
comparisons are made using internal rate of return. It may be that public investment replaces
private investment; as very often producers and consumers face different rates due to suboptimal
tax system, some adjustment needs to be met. It is beyond the scope of this paper to deal with
this issue. Even with this consideration an argument can be made simply to use consumer rate of
interest.
A strand of argument that is in opposition to use of market discount rate is called the
isolation paradox. It was conceived of separately by Marglin (1963) and Amartya Sen (1967). In
Sen’s version it is nearly the prisoners’ dilemma regarding public investment. The argument
runs as follows: If individuals take on public investment on their own, the outcome yielding to
the future generation is likely to be negligent. However, if all individuals save then there is
positive return to future generation; and each individual will gain positive utility in terms of their
concern for well being of future generations. The trouble is that there is no way to engage
every one to save for investment, what is needed is a mechanism that will induce everyone to
save more. Government savings will then be required. It can be shown that social discount rate
which takes into account the weighted averages of well being of one’s self and one’s cohort in
the current generation and one’s own offspring and that of others’ in the future is smaller than
the private discount rate. The argument by Sen does not depend on altruistic motives over the
welfare of others in the same period or that for the heirs of others in the future.
We have shown so far that if the value of project returns can be determined in terms of
consumption goods then its return can be compared with the rate of return of the present capital
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