Download Rethinking Discounting of Health Benefits in Cost Effectiveness

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
2
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 steadystate 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.
5
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:
(1)
EU (C )=
∞
∑δ
t
p (t )u (ct ) ,
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,
 1 
 where α is the rate of pure time preference, and p is the probability of survival
δ =
1 + α 
6
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:
T
(1′)
∫
0
u (ct )e −δt dt .3
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:
− δj
− dct u ’(ct + j ) e
=
.
Consumption disocunt factor =
dct + j
u ’(ct )
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
Rt = e −δt u ’(ct )
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:
1
The 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.
7
dRt
u" (c) dc
=δ +
.
Rt
u ’(c) dt
c
= 1 and noting that the elasticity of substitution for consumption can
c
− u" (c) c
, we obtain
be written as σ (c) =
u ’(c)
Multiplying through by
.
(2)
c
rc = δ + σ (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 rc is positive for any
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 nonconsumption at the present, in a competitive economy rc will equal the marginal productivity of
capital, for now ignore the issue of indeterminacy. The marginal productivity of capital must be
rc as one would be indifferent between sacrificing this amount now; and any return on higher
than this would be arbitraged away in a competitive economy.
The rate rc is a constant in time if the economy is on a steady-state growth path where
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 neoclassical 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:
T
∫
t
− f ’( k (u )) du
e ∫−
≥0,
9
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
10
stock.5 Note that Equation (2) can be defined for nearly any commodity for an individual as
mentioned earlier. Some good if valued at a constant rate can turn out to have, in the absence of
positive rate of time preference, a zero change in price or zero rate of interest. Health could be
such a good. Further, note that many health projects are public goods.
IV. Should Rational Individuals Discount Their Own Health?
Should we expect rational individuals to discount prospects of future survival and
health? In the following section, we develop a simple three period model to show that if
individuals value their lives essentially because they are able to enjoy consumption there is no
unambiguous reason why they should discount life years.
The results of the three period model presented below depend critically on how health
and survival are captured in an expression for utility. If utility is a function of survival and
consumption and standard assumptions of additive separability are made then given health and
consumption are perfect substitutes:
U = f (c, l ) such that U = u (c)+q (l ).
The implications of this additively separable form of a utility function are quite profound. If q is
concave then there must be declining marginal utility of increased health and the willingness-topay for health improvement must be declining as health improves. This form of a utility function
appears to be counter-intuitive. If survival or health is reduced to zero, no amount of
consumption can compensate this particular reduction in probability. A more plausible form of
the utility function makes survival and consumption inseparable such that:
U = u(c) p.
Generalizing to include health state as well as survival, we can arrive at a multiplicative form of
the utility function such that:
U = kpu(c),
5
The model is essentially a one-good model. Increasing degree of complexity can be introduced.
11
where k is quality of health. In the latter form k and p should be distinguished, perhaps by never
allowing k to become zeros or negative. There is a straightforward interpretation of p being
zero—death; it is not so clear what we can mean by stating that the quality of health is zero. It is
not postulated that k is decreasing in time. The equation above is consistent with the notion of
quality-adjusted life-years used in cost-effectiveness analysis.
III.1
A simple model
We examine in a three period model the decision to undertake a medical intervention
whose impact comes about in the future. A medical intervention impacts as to improve the
probability of survival in some period. An individual derives satisfaction from consumption of
goods in each period; there is a probability attached to be able to survive any given period. A
derived function from a von-Neumann-Morganstern utility function is the following:
(3)
V (C , P) = U = u(c0 ) + δ p1 u(c1 ) + δ 2 p2 u(c2 ), 6
6
Following Levhari and Mirman (1977) equation (3) can be justified more formally. Let qt be the
probability of dying at the end of the period, then a stream of consumption produces the expected utility
expressed by:

T
t
∑ q ∑ δ
t
t =0
T
Note that
∑q
t
t
i =0

u ( ct )  .

= 1. Through a change in the order of a summation, we can write:
t =0
T
pt = ∑ q i .
i =t
pt is the probability of dying from period t on; i.e. the probability of surviving until that time. Finally, we
obtain:
T
∑pδ
t =o
t
t
u ( ct ) .
To clarify understanding of this function, take the hypergeometric distribution qt = (1-q)tq, t=0,1,.., then
∞
pt =
∑q
i
= (1 − q) t .
i =t
Here the individual would maximize the function:
12
where P indicates a probability distribution and C denotes the stream of consumption. Here, pt
is essentially the probability of surviving period t and the sum of these values is life expectancy.
Health is modeled as probability of survival in (3); and any intervention affects these
probabilities. The marginal rate of substitution between the probability of survival in the two
different periods is the following:
dp1 u(c2 )δ
=
.
dp2
u(c1 )
Although dp1/dp2 is a function of time preference for consumption, it is also a function of the
level of consumption. The probability of survival at any point in time, pt, is influenced by health
expenditure that can be undertaken at any time prior to t or at t. For sake of convenience we
impose the condition that all medical expenditure will take place at time 0.
We ask the question what would be the largest amount of expenditure--a reduction in
consumption--the decision maker would be willing to incur in order to achieve an increment in
survival probability in any given period. Denote πt to indicate the incremental probability of
survival in any given period, t = 1 and 2. For an increment in period 1, there is no increase in
period 2. And for a period 2 increase symmetry holds. Hence, for either πt , π is the incremental
increase. The maximum willingness to pay for a health intervention that achieves the
incremental probability is simply the difference in the aggregate life-time utility converted to
dollars terms due to the intervention and that without the intervention.7
∞
∑
(1 − q) t α t u (ct ) =
t =0
∞
∑ [(1 − q)α ] u(c ),
t
t
t =0
thus, increasing the discount factor. Note that summing pt’’s is the life expectancy, for this example:
∞
∑p
t
= 1 / q = life exp ec tan cy = µ .
t =0
7
Two medical interventions that can be undertaken suffice to explain the scenario we have in mind. BCG
vaccination takes effect in one's 30s to prevent tuberculosis and hepatitis-B which afflicts at the age of 40.
13
Let P1 and P2 denote two distributions of survivorship:
P1 = {p1 + π, p2} and P2 = {p1 , p2 + π}.
It is obvious that P1 and P2 yield the same life expectancy. We ask would the consumer pay more
for P1 or P2? If there is no necessary reason to favor P1 over P2, then we cannot claim that
individuals necessarily discounts future life years. We examine two scenarios. First where the
consumer has no access to credit markets but receives wages commensurate with growth in the
economy. We allow consumption smoothing in the second model through the possibility of
savings.
III.1a. No access to credit market
Suppose that expenditure occurs in period 0, denoted M, to induce a health effect by
affecting probability of survival in period 1, we then re-write equation (1′) as a function M1 also
(1″)
V ( P, Y − M 1 , π 1 ) = EU = u( y 0 − M ) + δ ( p1 + π ) u( y1 ) + δ 2 p2 u( y 2 ).
The willingness to pay to achieve P1 is defined as that value of M1, which equates to (2’) and (2),
formally:
V ( P 1 , Y − M1 ) = V ( P, Y ) .
Similarly, we define M2, to achieve P2.
An example shows that there is no necessary reason to believe that M1 > M2.
To obtain the
maximum expenditure the decision-maker is willing to incur, we need to derive an explicit value
for M. Supposing that the incremental increase in survival probability occurs in period 1, then M
would be that value which equates equation (1′) and equation (1″). Through a second order
Taylor approximation of U(y0-M) and rearrangement, we can show that equating (1′) and (2″),
we obtain:
And imagine that the mutually exclusive two possibilities: the person has some chance of being afflicted
with tuberculosis and zero chance of being afflicted with hepatitis-B. Now imagine the reverse.
14
(4)
u’( y 0 )( − M ) +
M2
u"( y 0 ) + πδ u( y1 ) = 0 .8
2
Similarly for the incremental probability in period 2 where π = π1 =π2.
To solve for this value we impose some structure to the model. First let
u ( c) =
(5)
c 1−γ
,
1− γ
where 0 < γ < 1. Assume that the decision maker lives in a dynamic economy where income
grows at the rate of r. Using (5) and the growth rate of the economy and recall that δ =
1
,
1+ α
the feasible root is the following:
(6)
y
Mt = t
γ
t


1−γ
 1 + 2γ π  (1 + r )  − 1 , for t = 1 and 2. 9


1− γ  1+ α 


Equation (6), M, serves as a general equation for the willingness to pay for the increment of π in
 (1 + r ) 1−γ 
∂ M
 as R; and it can be show that
> 0. If R>1,
∂ R
 1+α 
the corresponding period. Denote 
it is increasing in time; hence M2 > M1. If R < 1, then income grows at a slower rate; hence, M1 >
M2. Thing to note here is that if r is significantly higher than α, then M2>M1. This is because
individuals value quality of life in years lived and the actual chance to enjoy that period. The
willingness to pay for an incremental increase in probability of survival is increasing, that is
∂ M
> 0 ; yet this increases at a decreasing rate—the marginal willingness to pay for an
∂π
incremental increase in probability of survival is falling. Hence, the change in price of health
8
With the Taylor approximation, the equality is the following:
M2
u( y 0 ) + u’( y 0 ) (− M ) +
u"( y 0 ) + ( p1 + π 1 )u( y1 ) + p2 u( y 2 )
2
= u( y 0 ) + p1u( y1 ) + p2 u( y 2 ) .
9
We assumed that M< y0, then only the negative root is possible when solving the quadratic Equation (6).
15
can be higher than it is in the future. Thus in case of health the change in price can be easily be
the opposite of that of (2) which is the change in price of consumer good.
Note that willingness to pay is higher for a smaller discount rate, a finding consistent
with empirical findings in Viscusi and Moore (1989) who report that those with small discount
rate require greater compensation for incremental increase in probability of death. A factor that
has not been included in equation (1′) is the deterioration of health through time; this is the
variable k in Equation (3). If k is monotonically decreasing in time, it may result in person
favoring earlier years. Yet, there is plenty of reason to believe that k is not monotonic in time
and perhaps only that it is always small near death. But this again points to the fact that the only
reason a person would favor the present over any future years is that the quality of life may be
better in the present year.
III.1b. Consumption smoothing
In this formulation we allow for consumption smoothing by making a credit market
available which affords a return at the rate of to the decision maker. Denote the value of a stream
of consumption generated by distribution P and wealth level W0 --the wealth at period 0--as V(W0
, P). Now suppose that V(W0 ,P1) > V(W0 , P2) then to obtain P1 a person would pay higher
premium than he would for P2. In this sense the willingness to pay would be higher for P1.
Wealth evolves according to:
(7)
Wt +1 = (Wt − ct )r
where r is 1 plus the available rate of return, c and W is the consumption and wealth levels at
time t as indicated by the subscript. There is no distinct period income. Our task is to obtain a
value function for the stream of consumption a utility maximizer will choose. The problem is
solved through dynamic programming, in an iterative manner for equation (5). The value
function we intend solve, using (2), is the following:
V0 (W0 ) = max{ u(c0 ) + δ p1V1 (W1 )} .
c0
16
By a well known theorem in the dynamic programming literature, the functional form in V0 and
V1 is the same with respect to W. By noting that (7) can be rewritten as
ct =
rWt +1 −Wt
r
we write the following equation known as the Bellman equation:
V (W0 ) = max { u(W0 ) + δ p1 V (W1 )} .
(8)
W1
The analytical solution for (8) are obtained for the two distinct utility functions; for (5) noting
that the value function is a function of P, we obtain the following:
(W r )
V (W , P) =
2 1−γ
0
(9)
1−γ
0
((δp ) )
where A =
(1 + (δp )
−1 / γ 1−γ
2
−1 / γ
2
+ δp 2
r
)
1−γ
(
)
 (δp A) −1 / γ 1−γ + δp A 
1
1
,

−1 / γ 2 1−γ 
r )

 (1 + (δp1 A)
.
To understand P better note that as stated before
exp ected life =
∞
∑p
i
= µ.
i =1
For any period n, if the chance of surviving under P1 up until that period is at least as great as it
is under P2, then P1 favors earlier health than later health more than does P2 and can be written as
n
∑
p 1i ≥
i =1
n
∑p
2
i .
i =1
Levahri and Mirman (1977) for this model compare savings—defined as the residual of
consumption in the first period--for both distributions; they show that the comparison is
ambiguous. That it is possible that savings can be either lower or higher under P1 than it is under
P2. The import of this observation is that if the returns on savings are high enough then
17
individuals are willing to gamble and forego earlier consumption in favor of later consumption.
Levhari and Mirman (1977) note that ‘uncertainty discounting’ will not always be strong enough
to outweigh the advantages of future possibilities due to the high rate of return on investment.
There is no reason to think that a consumer would always have a positive willingness to pay to
assure health in earlier period over that in a later period. We can show this result using (9).
Through numerical examples, we can confirm the results shown by Levhari and Mirman
(1977). For our purpose we need to show that there would be no reason for any individual to
prefer a distribution that favors health status in the near future than to some time later. To show
this consider two distribution with survival at the initial period at 1, with positive probability for
two more periods: t = {1,2}; we are interested in the decision made at period 0. We will consider
the utility function above, have two distributions and vary γ, r, and δ:
P1 = {.75, .25} and P2 = {.625, .375}
The first distribution favors earlier health over later health more than does the second
distribution. We will show that willingness to accept P1 over P2 is not necessarily positive.
Hence, there is no reason to accept that individuals must value increases in current health status
more than future health status. We need to show that it is not necessarily the case that V(W0;
P1)>V(W0; P2). Inequality flows in both directions. First consider r=1.1, γ = .5 and δ = .95;
compared to the market the consumer is patient. We can show that the consumer will spend in
the initial period nearly .887 portion of his/her wealth under P1 and under P2 it is .883; and
2.157W0 = V(W0; P1)<V( W0; P2) = 2.164W0. The consumer decreases his/her consumption, as
the future in the later period is more attractive. Now consider the situation where the consumer
is rather impatient compared to the market: r=1.06, δ=.95, γ=.7; we have greater concavity as
well. One can show that in this situation with P1, we have consumption at the initial period of
.74 and with P2 we have .76 of his/her wealth. Further, 4.16W0 = V(W0; P1)>V(W0; P2) = 4.09W0.
There is no reason to think that a distribution favoring earlier health is preferred.
In the section we have argued is that it is not necessarily true that individual will express
preference for current healthy years over future healthy years. Although agents devalue
consumption at a positive rate, they would still be willing to pay a higher price to receive an
incremental increase in survival at some future period than they would for a nearer period.
18
III.2 Implications of the consumer model for a social decision-maker
If it is not clear that a rational individual would discount future health status or future life
years, then what does it imply for cost-effectiveness analysis? We can start answering this
question by first asking what would ensue if individual did discount future years of life.
If individuals do discount their future health status, should the policy maker also? There
may be some justification for discounting so long as only the welfare of a particular patient is
concerned. But a social planner even dealing with only a single cohort will find that the discount
rates across the society is unlikely to be the same. If all medical care is delivered in perfect
markets then prices would reflect some sort of average of the discount rates. However, as is the
case frequently, a third party payer is likely to be the one which develops some sort of costeffectiveness league tables. It is difficult to see how a market with the prevailing insurance
systems can reflect an average discount rate for individuals in society. A positive rate of time
preference among all may warrant some discounting in developing the cost-effectiveness priority
list for those intervention whose effects are borne out some years later as long as the planner is
only dealing with that cohort.
Most authors do not argue that social discount rate should be based on the individual
discount rate or time preference; there are no representative agent models which show
discounting is rational. Instead, they point to the absurd consequences of not discounting future
benefits in an essentially utilitarian framework where aggregate health is maximized. 10 Yet,
there is some confusion as to how health appears in the utility function; for example, in
Lipscomb et al. (1996) health appears as consumption good for individuals:
“…economically rational individual will attempt to adjust investment in the commodity
called ‘health’ so that, over time, the (marginal) rate of time preference equals the (real)
consumption rate of interest i.….The model [depicting individual rational consumer]
predicts that [the] individual will seek opportunities to trade present consumption (whether
health itself or other goods and services) for future health; she could accomplish this by
saving money at the rate i and then using the proceeds to buy health” (1996, p. 229).
Thus one is tempted to accept the market rate of interest as the rate at which one ought to
discount future health status.
10
Although an allocation mechanism that yields maximization of aggregated health states is Pareto optimal,
there may be other allocations that do not yield maximization of health states but are Pareto optimal.
19
Health is not like any other good; for one thing the class of admissible utility functions
for health is more limited than for most other goods. There may be a lexicographic element to
health states. We have shown that it is highly rational and reasonable for individuals not to
discount future health. The act of discounting future health in when carrying out CEAs cannot
be justified through individual behavior. Recognizing that empirical observations do not lend
support to the existence of positive rate of interest for individuals, Lipscomb et al. (1996) draw
attention to the prescriptive aspects cost-effectiveness analysis. They write, “it is reasonable for
the social decision maker to act as if individuals tend, on average, to discount future health
effects at the same (real) rate they use to discount money and other easily transferable
commodities” (p.228).
Some economists (see Feldstein, 1964; Marglin, 1963; Sen, 1967), as we noted earlier
have argued that the society's desired level of national savings as expressed by individuals is
likely to be determined by a different parameter than that which determines personal savings
decisions. This rate could reflect people's concern for the good of the society as a whole and the
good of the generations to come. Health is essentially a private good; and the literature has
essentially been concerned with detailing the role of discounting when we seek to maximize
aggregate health, a social welfare function for which no economic justification is given.
Assuming that this function can be justified, the argument can be made that some type of public
funds are always involved in health care, what then is the justification for discounting future
health states?
IV. The planning problem and discounting
To the best of our knowledge, we know of no author who proposes a collective choice
rule which derives a social rate of time preference for health nor the maximand, the aggregate of
health states across individuals. It is not argued in the literature that social time preference
ought to be an average of individual rate of time preference for future health states. Instead, the
arguments that are most readily used can be classified as consequentialist arguments in that they
argue that without a positive rate of time preference absurd policy conclusions would follow.11
We take issue with some of these arguments.
20
Implicit in many arguments is the view that society may view aggregate health stock as
any other good in the economy and simply treats it like consumption. The social planner may
eschew a concave function over the health of the society. However, health expenditure may
affect health in a concave manner in that marginal expenditure on health has a decreasing effect.
It could very easily have a marginal effect of zero beyond a certain level of threshold
expenditure.12
Supposing that the planner has a finite planning time horizon due to political necessity,
does it follow that she is induced to discount future health states? To answer this question we
develop a simple model to show that if health expenditure affects health in a concave manner
then the planner will be induced to undertake some health expenditure every time period even
without discounting. The exercise seeks to explore the effect on health expenditure if society
were to discount at different rates the utility from consumption and health. If it turns out that the
consequence of this asymmetry is unappealing, we may indeed prefer to discount health while
performing cost-effectiveness analysis.
IV.1.
The Simple Planning Problem
We present a plausible simple planning problem to develop our argument. The planner
is essentially the government who must decide between health and all other goods. We do not
present a full account of government behavior as that would be outside the scope of this paper;
instead we develop a simple partial equilibrium two-period model. The government expenditure
can undertake health expenditure in terms of both preventive and curative care, all other types of
care, and invest its fund. We also allow government to borrow but it is required to balance its
budget.
Let the welfare relation be additive; c denotes consumption, h denotes health
expenditure in curative care and p denotes preventive care. The welfare function W can be
written as
(10)
W ( ct , l t ) = u ( ct ) + l t
11
This is similar to the usual requirement that a positive rate of time preference be used in infinite horizon
growth models because not doing so would either lead to the non-existence of a solution or an absurd high
rate of savings. As Anand and Sen (1994) point out argument for use of a positive rate of time preference
in designing social policy often hinges on convenience rather than on reflection.
12
This could very easily be construed as equivalent to Ramsey’s consumption “bliss” point, where any
further consumption for the society yielded no incremental benefit.
21
where the subscript t indicates time period, t = {0,1} and l indicates health. This form of the
welfare function in which utility from health is linear and utility is a concave function of
consumption implies that a person values an increase in a unit of health more as consumption
increase:
(11)
dct
1
=
.
dlt
u’(ct )
Equation (11) is the marginal rate substitution between a unit of health and a unit of consumption
and is independent of the level of health but not independent of the level of consumption. This
is so because society does not experience decreasing marginal utility from health status of the
entire population.
There are two component to l: a function g that transforms curative expenditure h to
health and similarly, function m that transforms expenditure on preventive care p undertaken in
the previous period. All functional forms are concave and have positive first derivatives. Hence,
(10) transforms to the following:
(10′)
W (ct , ht , pt −1 )=u (ct )+ g (ht )+m( pt −1 ).
Assume that u, g and m are all concave in their arguments. The government decides upon
consumption and health; investment decisions can be thought of yielding future social goods but
essentially can be converted to money. The budget constraint for this model is similar to the
budget constraint of two-period consumption decision model. In the model the government can
borrow in the first period against its future investment earning; for sake of simplicity we will
simply assume that there is only first period lump sum allocation for the government
expenditures. Given the above discussion the budget constraint is the following:
(12)
f ( I ) + Y0 = c0 + p0 + h0 + h1 + c1 ,
with f is the functional form over investment I and Y is the initial budget. Note also that I = Y0 –
e, where e simply stands for the complete set of expenditures in the first period.
22
Our task is to solve the following problem using (10′), (11) and (12):
(13) max imize L =u (c1 )+ g (h1 )+δ u (c 2 )+δ
h
(g (h2 )+m( p1 ) )+
c1 ,c2 , p1 ,h1 , h2
λ (Y0 + f ( I )−c1 −c 2 −h1 −h2 − p1 ),
where λ is the Lagrangian multiplier and δ and δh are the social rate of time preference for
consumption and health respectively. Hence, health and consumption are discounted differently.
For simplicity we will assume that functional form for preventive and curative care are the same;
hence g = m.
The central argument made by Keeler and Cretin is that without discounting the planner
will end up spending nothing on health in the first period due to the fact that postponement of
consumption will yield higher return and thus increased ability to purchase health. This
argument ignores that expenditure smoothing is natural as a solution to the problem stated in
(13). The first order conditions for (13) are the following:
(14)
i.
u’(c1 ) = λ (1 + f ’( I ))
ii. g ’(h1 ) = λ (1 + f ’( I ))
iii.
δ h g ’( p1 )=λ (1 + f ’( I ))
iv. δ u’(c2 ) = λ
v. δ h g ’(h2 )=λ
vi. Y0 + f ( I ) = c1 + c2 + h1 + h2 + p1 .
The second order conditions are guaranteed by concavity; the system is determined. The
problem is further simplified by noting that the value of the return on marginal investment must
equal the prevailing market rate of interest, r. We can also say that if the rate of time preference
is higher (less) than or (the same as) the rate of return on investment then consumption in period
1 is greater (less) than or (the same as) consumption in period 2. The exact same relation holds
for curative care in the two periods. In case the health rate of time preference is zero, that is δh is
zero, health expenditure is higher in the second period than it is for the first period. Further,
expenditure on prevention and the first period expenditure on health are equal; thus the health
status in second period is better in the second period. The cost effectiveness of a extra dollar
spent on curative care is
23
1
.
g ’(ht )
It is immediate that for δh > 1/(1+r), which is the case if consumption is discounted at a higher
rate than health, the cost-effectiveness ratio cut-off point is higher in period 1 than in period 0:
(15)
1
1
<
g ’(h0 ) g ’(h1 )
Hence, the acceptable programs can be worse in period 1 than in period 0; lesser cost-effective
programs are accepted in the second period. However, we do not have the extreme result of
Keeler and Cretin.
The marginal rate of substitution between the consumption between the two periods is
the following:
−
u ’(c0 )
dc1
=
= 1 + f ’( I ).
dc0 u ’(c1 ) δ
Similarly, although -dl2/dl1 =1 from Equation (10) in the form embedded in (13), it is
straightforward to show from (ii), (iii) and (v) that at the equilibrium the tradeoff in expenditure
on health between the adjacent periods is the following:
(16)
1 + f ’( I )
,
1 + 1 + f ’( I )
which is less than 1+f′(I),the tradeoff between two periods for consumption. The opportunity
cost of foregoing expenditure on health today for the next time period is smaller than it is for
consumption. This is not due to discounting nor the additive nature of welfare function but due
to the fact there are two instruments for affecting health in period 1 and only one in period 0. As
one has to start somewhere, expenditure on current health is likely to fall below next period’s
health expenditure due to the availability of preventive care.
An interesting special case is when the marginal effect on health of a dollar expenditure
is a constant, say, at γ; and for now assume that health is not at all discounted. In this case the
equalities in (14-ii, iii, v) do not hold; if γ is significantly greater than any of the right side
24
equation then all expenditure will be spent on health. Since it is unlikely that health expenditure
is the only sort of expenditure the planner will undertake, the only interesting conditions without
any discounting of health are the following:
(14a)
ii.
g ’(h1 ) =γ < λ (1 + f ’( I ))
v.
iii. g ’( p) = γ < λ (1 + f ’( I ))
g ’(h2 ) = γ ≤ λ .
But if this is the case nothing is spent on health unless equation (v) is an equality; and in the case
of equality we observe all health expenditure on the second period. With discounting at the
consumption level, we observe that only sensible equality comes again from equation (v); i.e.,
δg ’(h1 ) ≥ γ before any other health expenditure takes place. This unique inequality is more
likely to be the case at the consumption rate of discount than if discounting was smaller for
health; i.e., δ < δh. Hence, even with discounting all health expenditure takes place in the second
period. Thus by (14a), we would have a Keeler and Cretin type of result even with discounting.
What drives their result is not the discount rate of zero but the equal opportunity of interventions
between different time periods without any other types of expenditure.
IV.2 Implications for Time Paradox
Given that most planning horizon are finite, time paradox is not sufficient reason to
believe in discounting. However, the use of finite time horizon itself is a form of discounting.
That is, beyond a point in time, the welfare generated in subsequent period is discounted to zero;
hence, the pure rate of time preference is infinite beyond this point. Furthermore we can impute
explicit transversality conditions to show concern beyond the planning horizon. We think this is
an accurate description of a planning problem. Planning model with infinite time horizons can
ensure health expenditure and consumption smoothing with additional assumptions. A theorem
in the mathematics of dynamic optimization can establish that if health is simply a concave
function of expenditure in an infinite time horizon model, the solution exist only if there is rate
of time preference to obtain a closed bounded subspace of the payoff space. However, it is not
required that health be discounted at the same rate as money is. Concavity alone in a finite
horizon model can assure that a solution exist. A discount rate of zero would require additional
assumption.
25
One of the main assumptions in Keeler and Cretin is the following: “If P is a feasible
program then P′, achieved by delaying P one period, is also feasible.” The meaning of this as we
take is that if intervention A in period t costing C is yields benefit level E, so it must also be in
period t+1 because the health condition of the population remains the same. It may be that if
opportunity to intervene in period t is missed all health condition may require C′>C to produce E.
Hence, the marginal expenditure in period t+1 does not yield the same effectiveness. This is
essentially the concavity of assumption on the effect of health expenditure.
Health expenditure do not exhibit constant returns to scale, in fact the marginal return
from health expenditure usually falls quite drastically and very quickly. In the Keeler and Cretin
world the opportunity for health interventions last indefinitely through time. However, the
additive time separability form embedded in Equation (13) is the right form because health
intervention need to be timely; that is, the opportunity for carrying out health intervention do not
exist indefinitely through time. Apart from the concavity, the functional form of g and m could
well vary in time. It may be that because of high level of consumption in the future due to
growth in the economy with healthier diets and habits, cleaner environment—developed without
concern for health--marginal return to health expenditure may drastically be different in the
future periods. Hence, postponing health expenditure simply means missed opportunity to carry
out interventions.
Empirically there are good grounds to believe that there is a secular trend toward a
reduction in ailments that can be treated at low cost. Hence, the health production function is
likely to be such that the marginal benefit out of expenditure on health is likely to decline.
Ramsey had postulated that society will reach a bliss point in terms of consumption in that
marginal benefit out of further consumption is zero. Our idea that health expenditure rapidly
approaches a low effectiveness is similar to this idea. Conditions such as bliss point or declining
opportunity to use low cost intervention entails that increasing health expenditure produces no
gains in increase in utility, allowing us to adopt an infinite horizon model.
V. Related arguments for discounting
Lipscomb et al. (1996) argue that discounting benefits and cost at the same rate leads to
resource allocation in a time neutral fashion (p. 221). One of the aims of allocation of health
dollars, they argue, is to allocate expenditure in different time periods where some identical
26
interventions will be carried out to treat patients under identical conditions. CEA should be such
that it treats patients suffering from a similar ailment in different period the same way. There
are two observations one can make regarding the time neutral argument. First, time neutrality is
essentially a statement about rights of a patient. It is reasonable to be convinced of this, one can
simply not apply health maximization approach in this context and use one of many rights based
argument for health care (see Dworkin, 1994).
Second, more importantly the time neutral argument can withstand the concavity
assumption. Recall from Equation (15) that cost-effectiveness ratio is larger in the period 1 than
it is in the period 0 if time preference on health is lower than the rate of return on monetary
investment, usually the growth rate. So what is denied in period 0 can be made available in
period 1. This violates time neutrality. However, strictly it does not; money simply in the next
period is more abundant due to growth rate in the economy; hence, the equivalency of cost
between the two period does not hold. Time neutrality demands that patient condition is the same
through out time, it must also demand that real cost is the same. Suppose the real cost of a
treatment once cheap rises, then proponents of time neutrality argument would not recommend
that that same intervention be continued. If the argument is that the patient should still be treated
even with the cost rise because the only relevant the sameness is the patient’s condition, then the
time neutrality no longer stems from maximization of health status given a fixed a budget.
There might be some reason to think that current health expenditure must be spent
current health and those condition that current expenditure can affect later. The budget should
not be extended over to the future period. The time neutrality argument seeks to incorporate into
CEA, the larger decision process of allocating health and consumption goods across time.
Since the aim of CEA process is to compare the desirability of one health intervention against
that of another, the trade-off between consumption and health remains fixed. Yet, in the future
this cannot be fixed. Any decision involving future expenditure on health depends on the trade
off in the future between consumption and health in the future. Further any calculation that
evaluates future health in terms of the present health depends on the willingness to trade between
four variables: current consumption and health and future consumption and health. CEA must
assume that this task has already been done. To discount future health is to double count the
trade off between future health and current consumption and health.
27
Discounting cost alone on the other hand reflects growth of consumption goods in the
economy. With growth the marginal utility out of consumption is decreasing where as increase in
health status cannot be said to produce decreasing marginal utility. Further as indicated in
equations (10) and (11), the marginal rate of substitution between health and consumption will
rise as consumption increases.13 As shown in Equation (12), this is due to the fact that utility
arising out of consumption is a concave relation where as utility rising out of health is linear. To
allocate interventions across cohorts in a time neutral way, as Lipscomb et al (1996) write it, we
should discount neither consumption nor health states since the cost effectiveness ratio across
time remain the same.
This ignores the fact when we discount consumption we are not totally
discounting the utility out of consumption, but completely the expenditure on consumption.
A further point made by Lipscomb et al. (1996) is that under a veil of ignorance an
individual would prefer that a social planner is time neutral so that cost per a unit of some
measure of health states for the same intervention remain the same across time. This ignores the
fact a person behind this veil will recognize that he would value health and consumption tradeoff differently in different generations due to the secular trend of rising income and decreasing
marginal utility of consumption. It is unlikely that such a claim would be made from the veil as
it might not be enforceable once outside the veil.
VI. Conclusions
The thrust of the argument has been that in a finite time period, there is no reason to
believe that an individual will value future health states any less her present health state. In fact
they may value it more if they expect a large rise in income. Further, we have shown that there is
no reason to postulate a positive rate of time preference by which the society discounts future
health states of individuals. We have used plausible utility functions in (3) and (10) to show our
results.
The finite planning horizon models seems to be quite appropriate for individuals since
they do not live forever. Olson and Bailey (1981) had shown that empirical behavior is
consistent with consumer behavior in infinitely lived agent if the rate of time preference is
positive. The argument also holds for any finitely long horizon. We have allowed a positive rate
of time preference over consumption and shown that individuals need not discount future health
13
This is also true in section III, as individuals accumulate more consumption good, they value health more,
28
states. Health is modeled multiplicatively as the probability of survival of the individual. We do
not treat health as another commodity; that possibility, however, is not excluded from (3).
The finite period model in section IV does not require a positive rate of time preference.
An infinite period model is perhaps more appropriate to show equity over generations. In that
case we would argue health states would not require discounting even for technical reason
because it is quite appropriate to think that there is something similar to a bliss point regarding
health expenditure in that health benefits after a threshold of expenditure is likely to approach
zero. This view is much more acceptable than Ramsey’s bliss point for consumption.
One further main point of this paper is that when carrying out CEA it should not matter
whether or not there is a social rate of time preference over consumption as long as the utility out
of consumption is concave and health is linear. Rising consumption in the future implies that in
the future the consumer is willing to forego consumption in favor of health at a greater level than
he would in the present. Hence, without any explicit positive rate of time preference the
consumer is willing to value foregone consumption less in the future than he does in the present.
She would do no such thing over health states. This result holds for individual assessment over
future cost-effectiveness of medical interventions. This may imply that even with an infinite
period model where one is likely to use an exponential rate of discount on the future utility, we
may wish to discount cost at a higher rate than health status.
since good health is means to enjoy the higher consumption.
29
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