1 Ž. Journal of Health Economics Life-cycle preferences over consumption and health: when is cost-effectiveness analysis equivalent to cost benefit analysis? Han Bleichrodt a,), John Quiggin b a ima, Erasmus UniÕersity, PO Box 738, 3000 DR Rotterdam, Netherlands b Department of Economics, James Cook UniÕersity, ownsõille, Australia Received 8 October 998; received in revised form 25 March 999 Abstract his paper studies life-cycle preferences over consumption and health status. We show that cost-effectiveness analysis is consistent with cost benefit analysis if the lifetime utility function is additive over time, multiplicative in the utility of consumption and the utility of health status, and if the utility of consumption is constant over time. We derive the conditions under which the lifetime utility function takes this form, both under expected utility theory and under rank-dependent utility theory, which is currently the most important nonexpected utility theory. If cost-effectiveness analysis is consistent with cost benefit analysis, it is possible to derive tractable expressions for the willingness to pay for quality-adjusted life-years Ž QALYs.. he willingness to pay for QALYs depends on wealth, remaining life expectancy, health status, and the possibilities for intertemporal substitution of consumption. q 999 Elsevier Science B.V. All rights reserved. JEL classification: I0; D6 Keywords: Economic evaluation; Value of life; Health; Utility theory; Willingness to pay ) Corresponding author. el.: q ; fax: q ; r99r$ - see front matter q 999 Elsevier Science B.V. All rights reserved. Ž. PII: S
2 682 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics Introduction he Ž non-. equivalence of cost-effectiveness analysis and cost benefit analysis has received much attention in the literature on the economic evaluation of health care Ž Phelps and Mushlin, 99; Johannesson, 995a; Donaldson, his attention emanates from a concern about the theoretical properties of cost-effectiveness analysis. Notwithstanding the fact that it is currently the most common tool in the economic evaluation of health care, cost-effectiveness analysis, unlike cost benefit analysis, has no foundation in economic welfare theory. he most widely used outcome measure in cost-effectiveness analysis are quality-adjusted life-years Ž QALYs.. here is a well-established literature describing the conditions under which QALY-based decision making is consistent with preferences over lifetime health profiles ŽPliskin et al., 980; Bleichrodt, 995; Bleichrodt and Quiggin, 997; Bleichrodt et al., 997; Miyamoto et al., 998; Ried, Much less is known about the consistency between QALY-based decision making and individual preferences when lifetime utility depends not only on health status, but also on consumption. In this paper, we derive a set of conditions that is both necessary and sufficient for the consistency of QALY-based decision making with life-cycle preferences over consumption and health status. Cost benefit analysis is always consistent with life-cycle preferences over consumption and health status. his follows because cost benefit analysis imposes no assumptions on the lifetime utility function over consumption and health status. Consequently, to derive the conditions under which cost-effectiveness analysis is consistent with life-cycle preferences over consumption and health simultaneously answers the question under which conditions cost-effectiveness analysis is equivalent to cost benefit analysis. wo recent papers by Garber and Phelps Ž 997. and Meltzer Ž 997. also analyzed the allocation of health resources within a life-cycle framework, where both health status and consumption are arguments of the utility function. hese papers focus on the question when the cost side of cost-effectiveness analysis is consistent with life-cycle preferences over consumption and health status. heir main concern is whether future consumption should be regarded as a cost of life-saving medical interventions. Although Garber and Phelps assume that the health argument of the utility function may be represented in terms of QALYs, neither they nor Meltzer consider the conditions under which an individual concerned about both health status and general consumption would seek to maximise QALYs. his paper complements the papers by Garber and Phelps and by Meltzer by focusing on the question when the outcome side of cost-effective- hroughout the text, we interchangeably use the terms cost-effectiveness analysis and QALY-based decision making. Our central result, derived in Section 2, can be applied to other outcome measures as well. We briefly comment on other outcome measures in Section 5.
3 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics ness analysis, i.e., QALY maximisation, is consistent with life-cycle preferences. hroughout the paper, we assume that costs are measured in a consistent way and we restrict attention to the valuation of outcomes. he paper is structured as follows. In Section 2, we derive the central result of this paper, that QALY maximization is consistent with life-cycle preferences only if utility is additive over time, if the one-period utility functions can be multiplicatively decomposed in a utility function over consumption and a utility function over health status, and if the utility of consumption is constant over time. Section 3 provides an axiomatic analysis of the conditions under which the utility function is additive over time and the one-period utility functions are multiplicative in the utility of consumption and the utility of health status. An axiomatic analysis of this model is provided both under expected utility theory and under rank-dependent utility theory, currently the most important nonexpected utility model. Section 3 is technical, but we have tried to increase its accessibility by displaying each step in the derivation separately, by making ample use of examples to illustrate the conditions that are successively introduced, and by deferring all technical assumptions and proofs to Appendix A. In Section 4, we derive constancy of consumption. We show that under the utility function of Section 3, consumption is constant if an individual s rate of time preference is equal to the interest rate. Section 4 further examines the implications for the valuation of longevity and the willingness to pay for a QALY gained in case cost-effectiveness analysis is consistent with cost benefit analysis, i.e., in case the model of Section 3 holds with constant consumption. Johannesson Ž 995a. has argued that cost-effectiveness can only be a useful tool if information is available on the willingness to pay for a QALY gained. o date, no such information exists. We derive expressions for the willingness to pay for a QALY gained both when life duration is certain and when life duration is risky. Section 5 concludes. 2. he central result Our aim is to show under which conditions QALY-based decision making is consistent with life-cycle preferences over consumption and health status. he general idea of our argument follows from the uniqueness properties of the additive utility function and a special property of health. Let Ž h,...,h. denote a sequence of health states, where ht stands for health status in time period t. Similarly, ŽŽ c, h.,..., Ž c, h.. denotes a sequence of pairs of consumption level and health status, where Ž c, h. t t denotes consumption level and health status in period t. he first step in our derivation is to realize that according to the QALY model Ž. the utility of the sequence h,...,h is equal to Ý qh Ž., where qh Ž. t t is the quality weight or utility of health status in period t. wo things should be noted about the QALY utility function. First, it is additive and, second, the one-period
4 684 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics utility functions are identical. It is well known from the literature on additive representations Ž e.g., Wakker, 989. that if the one-period utility functions are additive then they are unique up to positive linear transformations. hat is, if qh Ž. and q X Ž h. t t are both additive one-period utility functions then qž h saqbq X t. Ž ht. Ž. where a can be any real number and b is a strictly positive real number. Note that because the qh Ž. t are unique up to positive linear transformations, we are free to choose their scale and location. hroughout, we set the utility of death equal to zero. Suppose now, that an individual is not only concerned about the sequence of health states, but also cares about consumption. hat is, we now study preferences over the sequence ŽŽ c, h.,..., Ž c, h... We assume that an individual always prefers more consumption to less as long as health status is better than death. Consequently, utility is strictly increasing in consumption. If the utility function over ŽŽ c,h.,..., Ž c,h.. is consistent with the QALY utility function then we know by the argument of the preceding paragraph that it must be a positive linear transformation thereof. Hence, it must have the following form: Ž. Ý Ž t t. U Ž c,h.,..., Ž c,h. s ÕŽ c. qž h. qw Ž c. Ž 2. where Õ is a utility function over consumption that must be strictly positive ŽÕŽ c. corresponds to b in Eq. Ž.. and the wt are period-specific utility functions over consumptions that are real-valued Žthe w correspond to a in Eq. Ž.. t. We next employ a special characteristic of the life-cycle model for consumption and health, namely, the fact that an individual will derive no more utility from consumption once he has died. Bequest motives may lead individuals to consume amounts less than their full wealth. However, it is reasonable to suppose, as is commonly done in economic analyses of bequest motives, that any utility of bequests is additively separable from UŽŽ c, h.,..., Ž c, h... herefore, the existence and form of the utility of bequests has no implications for the form of UŽŽ c,h.,..., Ž c,h... By our scaling convention, qž death. is equal to zero and it follows from Eq. Ž. 2 that the wž. t c must be equal to zero to capture the property that the individual derives no more utility from consumption after he has died. Consequently, life-cycle preferences over consumption and health status are consistent with QALY maximization only if: U Ž c,h.,..., Ž c,h. s ÕŽ c. qž h. Ž 3. Ž. Ý t Eq. Ž. 3 has three interesting properties. First, the utility of consumption is constant over the individual s life-cycle. Because we assume that utility is strictly increasing in consumption, it follows that consumption itself is constant over the individual s life-cycle. Second, the utility of consumption is positive. If there exists a subsistence level of consumption beyond which additional life-years are
5 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics negatively valued, then consumption has to be above this subsistence level. his is not a major restriction. Rosen Ž 988. has shown that if consumption falls below the subsistence level, an individual will convexify his preferences by randomizing between death and survival at a consumption level which is higher than the subsistence level. Hence, Eq. Ž. 3 captures all cases of economic interest. Most importantly, in Eq. Ž. 3, the utility of health status is multiplied by the utility of consumption. Consequently, a given gain in quality of life will be more appreciated at higher levels of consumption. his implies that in the allocation of health care resources, larger welfare gains can be obtained by devoting resources to those individuals who have a high level of general consumption. A comparable result was derived by Pratt and Zeckhauser Ž 996. for the valuation of risk reductions. his result is ethically troubling, since an important motive for the use of cost-effectiveness analysis is the desire to avoid the adverse distributional implications commonly seen to arise from applications of cost benefit analysis. However, the need for a multiplicative utility structure shows that, if cost-effectiveness analysis is to be placed within the realm of econonomic welfare theory, such implications cannot be escaped. In Section 3, we present an axiomatic analysis of the model Ž. Ý U Ž c,h.,..., Ž c,h. s ÕŽ c. qž h. Ž 4. t t his model is slightly more general than Eq. Ž. 3 because consumption is allowed to vary over time. If consumption varies over the life-cycle, an optimal allocation of health resources will assign more weight to health status in high consumption years. Conversely, other things equal, an individual with preferences that can be described by Eq. Ž. 4 will choose higher levels of consumption in those years in which his health status is high. he selection of a constant consumption level has to follow from an optimization problem which requires specification of the budget constraint. We present such an optimization problem in Section 4. () 3. Characterizations of Eq. 4 We focus on preferences under risk. Preferences under certainty follow by restricting attention to riskless lotteries: lotteries that yield an outcome with probability one, i.e., with certainty. Life-cycle preferences are first analyzed under expected utility theory, which is still the dominant theory in health utility measurement. However, it is by now widely accepted that people do not behave according to expected utility theory Ž Camerer, Several nonexpected utility theories have been developed among which rank-dependent utility theory ŽQuig- gin, 982; Yaari, 987; Quiggin and Wakker, 994. is currently the most influential theory. We also derive Eq. Ž. 4 under rank-dependent utility theory. We start with the case where utility is not discounted. he modeling of discounting is similar under expected utility and under rank-dependent utility and
6 686 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics we leave it to the end of the axiomatic analysis. We characterize two discounting models: constant rate discounting and a general discounting model which is consistent with the most important alternatives for constant rate discounted utility that have been proposed in the literature on intertemporal preferences ŽLoewen- stein and Prelec, 992; Harvey, Expected utility theory 3... Some notation In this subsection, we introduce the main concepts used in our axiomatic analysis. o improve accessibility, we have moved all technical assumptions regarding these concepts to Appendix A. We assume that there are points in time. We express this by saying that there is a set Ss,...,4 of time points. he set of all outcomes, i.e., sequences wž c,h.,..., Ž c,h.x is denoted by X. For ease of notation, we sometimes refer to pairs Ž c,h. as y. he consumption levels c are in each period elements of a set t t t t C q, which consists of all attainable consumption levels. he plus sign serves as a reminder that we only consider consumption levels that are above the subsistence level. he health states ht are elements of a set H, which consists of all attainable health states. We assume that H consists only of those health states that are at least as good as death. Under expected utility, the axiomatic analysis also holds if health states worse than death are included. Under rank-dependent utility, however, the generalization to health states worse than death is more arduous. We assume that the sets C q and H are equal in each time period. his assumption is made for convenience and does not restrict our analysis. he analysis can straightforwardly be extended to cover cases where C q and H vary over time Preference conditions and representation theorem w Ž. Let P be the set of all lotteries over X. A typical element of P is p, x ; Ž m. x Ž....; p, x which gives outcome x swžc, h.,..., Žc, h.x m with probability Ž2. p, outcome x swžc 2,h 2.,..., Žc 2,h 2.x with probability p 2, etc. and m is any natural number. In medical decision making, lotteries can be interpreted as treatments, the outcomes of which are risky. We assume that the set P contains all riskless lotteries, which are of the type w, x x. An individual is assumed to have preferences over P. We denote the individual s preference relation over P by #, which stands for at least as preferred as. We write x%y if it is true that x#y but not y#x. hat is, x%y means that x is strictly preferred to y. We write x;y if both x#y and y#x are true. hat is, x;y means that x is indifferent to y. A preference relation over outcomes can be derived from the preference relation over lotteries by restricting attention to riskless lotteries. We assume that the preference relation over outcomes satisfies monotonicity with respect to consumption: higher consumption levels are strictly preferred to lower levels as
7 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics long as health status is unequal to death. We also assume a sort of monotonicity with respect to health status: if health state h is preferred to health state h 2 for a given level of consumption cgc q, then h is preferred to h 2 for all consumption levels. A function V is said to represent the preference relation #, if for any two lotteries P and P that belong to the set P, the individual considers P at least 2 as preferred as P2 if and only if the value of V at P is at least as great as the value of V at P. his definition is expressed mathematically as: P #P if and 2 2 only if VŽ P. GVŽ P. 2. We assume that the expected utility axioms Žvon Neumann and Morgenstern, 953; Jensen, 967. hold. hen, the preference relation # is represented by the following function: m Ž. Žm. Ži. m i is Ž. Ý EU p, x ;...;p, x s puž x. Ž 5. where U is a utility function over outcomes. he utility function in Eq. Ž. 5 is still entirely general. Our aim is to give a set of conditions that ensure that the utility function can be written as Ý ÕŽ c. qž h. t t. We derive this representation by imposing four conditions. he first condition, marginality, ensures that U is a weighted additive function ŽŽ. Ž.. U c,h,..., c,h sý UŽ c,h. of one-period utility functions U. t t t t Definition : he preference relation over P satisfies marginality if for all P, P gp with equal marginal probability distributions over y,...,y :P ;P. 2 2 An example may clarify the effect of marginality. Consider a simple model in X which there are only two time periods. Let y be the outcome Ž20,000, good Y health., i.e., a consumption level of US$20,000 and good health, and let y be the outcome Ž 5000, bad health.. Consider the two lotteries P and P2 displayed in able. P yields a probability r2 of obtaining y Y in both periods and a probability r2 of obtaining y X in both periods. Marginality implies that the individual is indifferent between P and P 2. his follows, because both P and P yield in each time period a probability r2 of y Y and a probability r2 of y X. 2 Hence, P and P2 have equal marginal probability distributions over yand y 2. he above example shows that marginality excludes all complementarity between time periods. It might well be that the individual prefers P to P2 because he dislikes variation in his consumption and health status levels. Such an able An example of marginality Lottery Outcome with probability r2 Outcome with probability r2 Y Y X X P Ž y, y. Ž y, y. Y X X Y P Ž y, y. Ž y, y. 2
8 688 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics aversion to variation is not permitted by marginality. Alternatively, the individual might prefer P2 to P because he wants to avoid the possibility that he obtains the bad outcome y X in both periods. Such a preference for catastrophe avoidance is also excluded by marginality. he next step in our derivation is to make the one-period utility functions identical. hat is, we have to impose a condition which allows the representation ŽŽ. Ž.. of preferences by the utility function U c,h,..., c,h sý Uc,h Ž. t t.before we introduce the condition that achieves this end, symmetry, we define a new concept. A permutation function p Ž. t is a function that specifies a rearrangement of the time periods. For example, if p Ž. t ss then the point t is moved to point s in the rearrangement of the time periods. Definition 2: he preference relation over P satisfies symmetry if for all t g S, for all ygy with h unequal to death, and for all permutation functions p it is true that Ž y,...,y.; Ž y,...,y.. p Ž. p Ž. Let y X and y Y be as in the previous example. hen, by symmetry, the individual is indifferent between the outcome x X which yields y X in the first period and y Y in the second period and the outcome x Y which yields y Y in the first period and y X in the second. his follows because x Y can be obtained from x X by applying the permutation p Ž. s2 and p Ž. 2 s. Symmetry renders irrelevant the point in time at which a particular outcome occurs. If symmetry holds, the individual s preferences are unaffected when a permutation of the time periods turns a decreasing sequence, i.e., a sequence in which outcomes become worse over time, like x Y in the above example, into an increasing sequence, i.e., a sequence in which outcomes improve over time, like x X in the above example. Symmetry is at odds with the common assumption that people have positive time preference, i.e., prefer good outcomes to occur sooner rather than later. If people have positive time preference then they always prefer decreasing sequences to increasing sequences. Later in this section, we show how symmetry can be replaced by conditions that are compatible with positive time preference. Marginality and symmetry ensure that the utility function over sequences of consumption and health status is additively decomposable over time and that the one-period utility functions are equal for each time period. he final two conditions, standard gamble invariance and the zero condition, establish that the one-period utility function can be multiplicatively decomposed, i.e., it can be written as Uc, Ž h. s ÕŽ c. qž h.. Standard gamble invariance implies that Uc, Ž h. t t t t t t can be written as ÕŽ c. qž h. q wž c. t t t t. he zero condition then allows to set wž c. s0 for all t and for all consumption levels. t t Definition 3: he preference relation over P satisfies standard gamble inõariance X if for all c, c gc q : Ž c,h.# w p, Žc,h X.; yp, Žc,h Y.x if and only if Žc X,h.# w p, Žc X,h X.;yp, Žc X,h Y.x with all lotteries elements of P.
9 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics o illustrate, let h, h X, and h Y be three health states such that h X is better than h which is better than h Y. For example, h can be a mild form of asthma, h X no asthma, and h Y a severe form of asthma. Suppose that annual consumption is held fixed at a given level, say US$20,000, and that at this consumption level the individual considers having a mild form of asthma for certain at least as good as undergoing a risky treatment which yields a probability p of no asthma and a probability yp of a severe form of asthma. hen standard gamble invariance says that the individual should still consider a mild form of asthma for certain at least as good as undergoing the risky treatment if consumption is held fixed at another level, say US$5,000. his example illustrates the effect of standard gamble invariance. Standard gamble invariance enables consideration of preferences over health states irrespective of the level at which consumption is held fixed. Standard gamble invariance is typically invoked in health utility measurement. In assessing the utility of a health state by the standard gamble Ž orrance, 986., it is commonly assumed that life-years can be held fixed and that the value at which life-years are held fixed does not affect preferences. his idea is similar to standard gamble invariance as we use it here. he only difference is that we propose to hold consumption fixed instead of life-years in the assessment of preferences over health states. Because standard gamble invariance allows consideration of preferences over health status with consumption held constant, it also allows the definition of a separate utility function over health status which does not depend on consumption. In Appendix A, we prove that imposing standard gamble invariance on top of marginality and symmetry implies that the one-period utility functions Uc, Ž h. t t can be written as wž c. qõž c. qž h. with wž c. real and ÕŽ c. t t t t t t t positive. his model differs from Eq. Ž. 4 by the terms wž c. t t. o complete our characterization, we impose a condition that implies that wž c. t t s0 for all t and for all c t. he condition that has this effect, the zero condition, is a condition that is naturally satisfied in the medical context. In words, the zero condition says that a person derives no more utility from consumption once he has died. Formally, the zero condition is defined as: Definition 4: he preference relation satisfies the zero condition if for all X consumption levels c, c gc q : Ž c, death.; Žc X, death.. Ž. heorem summarizes the derivation of Eq. 4. A formal proof of heorem is given in Appendix A. heorem : Under expected utility theory, the following two statements are equivalent: Ž. i Life-cycle preferences are consistent with the maximization of wž. Ž.x U c h,..., c, h s Ý ÕŽ c. qž h.. Ž ii. t t he preference relation # over P satisfies marginality, symmetry, standard gamble invariance, and the zero condition.
10 690 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics An assessment of the conditions of heorem Before moving on to the characterization of Eq. Ž. 4 under rank-dependent utility theory, we briefly comment on the descriptive validity of the conditions described in Section he conditions are new in the context of life-cycle preferences over consumption and health. herefore, they have not been subjected to empirical tests and we can only speculate about their empirical contents. We believe that marginality and symmetry are the most restrictive conditions of the characterization. he zero condition is unobjectionable in the medical context. Standard gamble invariance is implied by the stronger condition of utility independence. Utility independence is commonly assumed both in medical decision analysis Ž orrance et al., 982; orrance et al., 995; orrance et al., 996. and in general decision analysis Ž Keeney and Raiffa, 976. and it is believed to be a reasonable condition in most decision contexts. Moreover, there is some evidence that preferences over health status and duration satisfy utility independence Ž Miyamoto and Eraker, 988; Bleichrodt and Johannesson, Marginality excludes all complementarity between time periods. his is a strong restriction. In the example presented in Section 3..2, marginality implies that people are indifferent with respect to variations in consumption and in health status. However, empirical evidence has shown that people have a tendency to overweigh their status quo or endowment and are averse to changes therein Ž Samuelson and Zeckhauser, 988: Kahneman et al., Such a tendency is incompatible with marginality. In Section 3.2, we show that under rank-dependent utility theory marginality is no longer imposed and is replaced by an alternative condition. his may enhance the descriptive validity of Eq. Ž. 4. Finally, symmetry excludes positive time preference. Positive time preference is commonly assumed in economic analyses and is confirmed in empirical studies. he existence of positive time preference calls for the replacement of symmetry. his is the topic of Section Rank-dependent utility theory Some notation he main difference between rank-dependent utility theory and expected utility theory is that rank-dependent utility theory does not assume linearity of the utility of a lottery in probability. Under rank-dependent utility theory, preferences over lotteries are represented by the functional m Ž. Žm. Ži. m i is Ž. Ý RDU p, x ;...;p, x s p UŽ x. Ž 6. where the p i are decision weights that depend on, but are in general not equal to the probabilities. If the decision weights are equal to the probabilities, i.e., p is pi for all i, then rank-dependent utility theory is equal to expected utility theory. hat is, rank-dependent utility theory includes expected utility theory as a special case.
11 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics In comparison with expected utility theory, there is one change in the mathematical framework. Remember that under expected utility we studied preferences over the set of lotteries P. Under rank-dependent utility theory, we consider Ž Ž. preferences over a subset of P, the set of rank-ordered lotteries. A lottery p, x ; Ž m....; p, x. m is said to be a rank-ordered lottery if its outcomes are ranked in decreasing order of preference, i.e., x Ž. #x Ž2. #...#x Ž m.. For example, the lottery w p, full health; y p, deathx is rank-ordered, because full health is a better health state than death. However, the lottery w p, death; y p, full healthx is not rank-ordered. We denote the set of rank-ordered lotteries by P x. he set of rank ordered lotteries contains all riskless lotteries. his follows because each outcome is at least as preferred as itself: x# x for all x g X. Consequently, we can define preferences over outcomes by restricting attention to the subset of riskless lotteries. We introduce one new notation. Let A be a subset s,...,t4 of the set of time points Ss,..., 4. Obviously, FsFtF. ByaA x, we denote the outcome Ž x,...,x,a,...,a, x,...,x. sy s t tq, i.e., the outcome that is obtained when the elements x,...,x of the sequence x sž x,...,x. s t are replaced by the corre- sponding elements a,...,a of the sequence Ž a,...,a. s t. For example, let be 40, let S be 25,...,304, let x be the constant sequence yielding a consumption level of US$20,000 and good health in each period and let a be the constant sequence yielding a consumption level of US$5000 and bad health in each period. hen, a x is the outcome yielding a consumption level of US$20,000 and good A health from periods to 24, a consumption level of US$5000 and bad health from periods 25 to 30, and a consumption level of US$20,000 and good health from periods 3 to 40. In case the set A consistst of just one point in time, say point t, then we write axinstead t of aax Preference conditions and representation theorem We use the same steps as in Section 3..2 to characterize the representation Ý ÕŽ c. qž h. t t. Hence, the first step is to find a condition that allows utility to be written as the additive sum of the one-period utility functions, i.e., ŽŽ. Ž.. U c, h,..., c, h s Ý UŽ c, h. t t t. Under expected utility theory, marginality served this purpose. Marginality is no longer available under rank-dependent utility theory, because it requires that the evaluation function for lotteries is linear in probabily. As we have explained, this is not necessarily true under rank-dependent utility theory. herefore, marginality cannot be invoked and we have to find an alternative condition which achieves the additive decomposition under rank-dependent utility theory. he following condition, generalized utility independence, has this effect. Definition 5: he preference relation # on P x satisfies generalized utility independence if for all subsets A of S: w p,a z;yp,b y x# w p,c z;yp,d yx A A A A
12 692 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics if and only if w p,a w;yp,b Õ x# w p,c w;yp,d Õx A A A A with all lotteries elements of Px. An example may clarify the effect of generalized utility independence. For ease of notation, we assume that health status is held constant, say at full health, and we will suppress it from our notation. Let there be two time periods and let As 4. Consider the four lotteries displayed in able 2. Here, P is the lottery yielding US$20,000 in both periods with probability p and US$5,000 in both periods with probability yp. It is easily verified that P corresponds to w p,aa z; yp,b y x, P to w p,c z;yp,d y x, P to w p,a w;yp,b Õ x A 2 A A 3 A A, and P4 to w p,c w; y p,d Õx A A if a s z s US$20,000, c s US$30,000, b s y s w s US$5,000, and dsõsus$0,000. Hence, if P is at least as good as P, for 2 example, because the individual dislikes variation in consumption over time, then generalized utility independence implies that P3 should also be at least as preferred as P 4. Observe that the probability distribution over what happens in the second period is identical for P and P Ž 2 a probability of r2 of 20,000 and a probability of r2 of 5,000. and also for P and P Ž 3 4 a probability of r2 of 5,000 and a probability of r2 of 0,000.. Hence, generalized utility independence implies that if two lotteries have identical probability distributions over the outcomes in a given period, then the individual will ignore the outcomes occurring in this period. Individual preferences are only affected by periods in which lotteries have different probability distributions over the outcomes. Generalized utility independence is a strengthening of utility independence, which says that the individual will ignore periods in which outcomes are common and certain. hat is, utility independence says that the individual will ignore those periods for which riskless lotteries coincide. Utility independence can be derived from generalized utility independence by setting z s y and Õ s w in Definitions. Generalized utility independence extends utility independence by saying that preferences are unaffected by those periods for which lotteries in general, i.e., not only riskless lotteries, coincide. As remarked before, it is widely believed that preferences approximately satisfy utility independence both in medical decisions and in other decision contexts. Because generalized utility independence extends utility independence in a natural able 2 An example of generalized utility independence Lottery Outcome with probability p Outcome with probability y p P Ž 20,000, 20,000. Ž 5,000, 5,000. P Ž 30,000, 20,000. Ž 0,000, 5, P Ž 20,000, 5,000. Ž 5,000, 0, P Ž 30,000, 5,000. Ž 0,000, 0,000. 4
13 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics way, it may be expected that generalized utility independence describes life-cycle preferences over consumption and health reasonably well. he other three conditions used under expected utility theory in characterizing Eq. Ž. 4 can still be used under rank-dependent utility theory. heorem 2 summarizes the derivation. A formal proof of heorem 2 is given in Appendix A. heorem 2: Under rank-dependent utility theory, the following two statements are equivalent: Ž. i Life-cycle preferences are consistent with the maximization of wž. Ž.x U c,h,..., c,h sý ÕŽ c. qž h.. Ž ii. t t he preference relation # on Px satisfies generalized utility independence. Further # satisfies symmetry, standard gamble invariance, and the zero-condition Discounting he general discounting model he characterizations presented in heorems and 2 imply that people give equal weights to different time periods, i.e., they are timing neutral. In economic evaluations of health care, it is more common to assume that people have positive time preference and discount future time periods. Empirical evidence is also supportive of the existence of positive time preference for health ŽOlsen, 993a; Cairns, 994; Chapman, In this section, we characterize two models that incorporate time preference. In one model, we impose no restrictions on the discount weights. herefore, this model is consistent with most discounting models that have been proposed in the literature on intertemporal preferences ŽLoewenstein and Prelec, 992; Harvey, he other model is the constant rate discounting model, which is commonly applied in cost-effectiveness analysis and also underlies the analyses by Garber and Phelps Ž 997. and Meltzer Ž iming neutrality is a consequence of imposing symmetry. herefore, symmetry has to be dropped to allow for time preference. However, symmetry also served to select identical one-period utility functions in heorems and 2. If symmetry is dropped, the one-period utility functions are no longer necessarily identical. Hence, another condition has to be imposed which ensures that the one-period utility functions can be chosen identical. he following condition, trade-off consistency, serves this end. Definition 6: he preference relation # on X satisfies trade-off consistency if for all s, tgs: ifž a xub y. and Ž c x#d y. and Ž a Õ#b w. then Ž c Õ#d w,. s s s s t t t t with all outcomes elements of X. rade-off consistency can be explained in terms of strength of preference. Suppose that a is strictly preferred to b, and that c is strictly preferred to d. hen, the preference Ž a xu by. implies that to obtain the strictly preferred outcome a s s
14 694 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics instead of b in period s is not sufficient to outweigh getting x instead of y in all periods other than s. he preference Ž c x# d y. s s, however, implies that to obtain the strictly preferred outcome c instead of d in period s is sufficient to outweigh getting x instead of y in the other periods. hese two preferences, therefore, indicate that in period s the strength of preference of c over d must be at least as great as the strength of preference of a over b. rade-off consistency claims that if the strength of preference of c over d is at least as great as the strength of preference of a over b in period s, then there does not exist another period t in which the strength of preference of c over d is smaller than the strength of preference of a over b. hat is, if we observe that obtaining a instead of b in period t is sufficient to outweigh getting Õ instead of w in all other periods then it must be that obtaining c instead of d is also sufficient. rade-off consistency thus ensures that utility differences are ordered similarly in different periods. his implies that the one-period utility functions are cardinally equivalent and can be chosen identical. rade-off consistency does not imply that each period gets the same weight. Under expected utility theory, trade-off consistency implies in combination with marginality that the utility function can be represented by Ý l Uc, Ž h. t t t. Under rank-dependent utility theory, this represen- tation follows from trade-off consistency and generalized utility independence. Additionally imposing standard gamble variance and the zero condition gives the wž. Ž.x general discounting model U c,h,..., c,h sý l ÕŽ c. qž h.. t t t he constant rate discounted utility model he constant rate discounted utility model, UwŽ c, h.,..., Ž c, h.x s ty Ý b ÕŽ c. qž h. t t can be obtained by imposing one additional condition, stationarity, on the general multiplicative discounting model of Section It is easily verified that in the constant rate discounted utility model, the ratio between the weights assigned to utility in period t and to utility in period s is equal to b tys. he ratio between the weights assigned to utility in period tqe and to utility in period sqe is also equal to b tys. his implies that in the constant rate discounted utility model only the difference in timing between the outcomes, i.e., tys, affects preferences, but not the positions in time at which the outcomes occur. Stationarity formalizes the idea that preferences depend only on the difference in timing and not on the exact timing of the outcomes. he definition of stationarity is as follows. Definition 7: he individual preference relation # on X satisfies stationarity if there exists a common outcome qgy with health status unequal to death, such that for all x, y gy, with health status unequal to death: Ž x,...,x,q. t t y # Ž y,...,y,q. if and only if Ž q, x,...,x.# Ž q, y,...,y.. y y y In words, stationarity says that preferences over outcomes are unaffected if we move the common outcome from the last to the first period and delay all other
15 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics outcomes with one period. Note that the differences in timing between the x are t unaffected by this permutation of outcomes in time. heorem 3 summarizes Sections 3.3. and A formal proof is given in Appendix A. heorem 3: If we replace symmetry by trade-off consistency in heorems and 2 then the general multiplicative discounting model UwŽ c, h.,..., Ž c, h.x s Ý l ÕŽ c. qž h. t t t represents life-cycle preferences over consumption and health status. If stationarity is imposed in addition then the constant rate discounted wž. Ž.x ty utility model U c, h,..., c, h s Ý b ÕŽ c. qž h. is representing. t t An assessment of the conditions o conclude the axiomatic analysis, we comment on the empirical contents of the conditions used to characterize the two discounted utility models. here exist no tests of trade-off consistency in the medical context. he main effect of trade-off consistency is to impose an additive representation. In an additive representation, preferences over outcomes in one period are independent of Ž common. outcomes in all other periods. herefore, trade-off consistency is most likely to hold in decision contexts where complementarity between periods does not affect preferences over outcomes. Studies that have tested stationarity yield negative results ŽCairns and van der Pol, 997; Bleichrodt and Johannesson, hese studies find that people not only pay attention to differences in timing between outcomes, i.e., to their relative position in time, but also to the moment at which the outcomes occur, i.e., their absolute position in time. he general pattern that emerges from the literature on intertemporal preferences for health is that people are more timing averse, in the sense that their implied rate of time preference is higher, for delays that occur in the near future than for delays that occur in the more distant future. hat is, the difference between year s and year t is discounted more than the difference between year sqe and year tqe. his finding suggests that other discounting models than the constant rate discounted utility model may be more appropriate. If trade-off consistency holds then these models can be derived as special cases of the general discounted utility model. 4. Constant consumption, the willingness to pay for QALYs, and the valuation of longevity In this section, we derive when consumption is constant over time. As shown in Section 2, constant consumption combined with Eq. Ž. 4 implies that cost-effectiveness analysis is consistent with life-cycle preferences over consumption and health
16 696 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics status, or, which is equivalent, that cost-effectiveness analysis is consistent with cost benefit analysis. If consumption is constant and Eq. Ž. 4 holds, tractable expressions can be derived for the willingness to pay for QALYs and for the valuation of longevity. We extend the models proposed by Rosen Ž 988. by including health status in the analysis. We start with the deterministic case where the individual knows his life duration with certainty. he deterministic case illustrates the essential ideas while keeping the analysis relatively straightforward. We then turn to the more realistic stochastic case. 4.. Certainty 4... he optimization problem Consider an individual whose preferences can be described by Eq. Ž. 4 with constant rate discounting: Ý t t ty Ž qa. Us ÕŽ c. qž h. Ž 7. where a is the individual s constant rate of time preference. We assume that Õ, the utility function over consumption, is strictly increasing and concave. he first derivative of Õ with respect to c is denoted by Õ c. he individual s wealth consists of initial wealth W and a fixed annual labor income w. he individual allocates his wealth between consumption and medical expenditures, which we denote by m. In each period, medical expenditures are a function of the sequence of quality of life levels Žqh Ž.,...,qh Ž.. and duration : m sgžžqž h.,...,qh Ž..,. t. We assume that, for all t, the first derivative of medical expenditures with respect to qh Ž. t, denoted by gh t is positive. he first derivative of medical expenditures with respect to duration, g, is also positive. he individual faces a pure capital market at which he can borrow and invest at interest rate r. he individual cannot die in debt and has no heirs. Under these assumptions, the individual s budget constraint becomes: Ý Ý t t Wqw ty s wc qm x ty Ž 8. Ž qr. Ž qr. he Lagrangian expression for this problem is: Ý Ls ÕŽ c. qž h. ql Wqw Ž qa. Ž qr. t t ty ty Ý 5 t t Ž qr. ty ½ y Ž c qm. Ž 9. Ý
17 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics and the first order conditions are: ÕcŽ ct. qž ht. sl ty ty Ž qa. Ž qr. for all tg w, x Ž 0a. ÕŽ ct. slg for all tg w, x Ž 0b ty h ty. t Ž qa. Ž qr. ÕŽ c. qž h. slwcqmyw4 y Ž qa. Ž qr. Ý y q g Ž 0c ty. Ž qr. Under the assumption that the rate of time preference equals the interest rate Ž asr., it follows from Eqs. Ž 0a. and Ž 0b. that that the individual will choose c t and h such that they are constant throughout w, x t. Constancy of health status in turn implies constancy of medical expenditures. his analysis answers the question when an individual whose preferences can be described by Eq. Ž. 4 will select a constant consumption profile: he will do so when his rate of time preference equals the interest rate. Hence, given Eq. Ž. 4, cost-effectiveness analysis is consistent with cost benefit analysis if the individual s rate of time preference is equal to the interest rate. his observation has important implications for the discussion whether the discount rate for health benefits can be different from the interest rate Ž the discount rate for costs. Ž Keeler and Cretin, 983; Olsen, 993b.. Our analysis shows that if the purpose is to achieve equivalence between cost-effectiveness analysis and cost benefit analysis then health benefits should be discounted at the interest rate, that is, health benefits should be discounted at the same rate as costs he willingness to pay for a QALY gained We now derive the willingness to pay for a QALY gained if cost-effectiveness analysis is consistent with cost benefit analysis. hat is, we assume Eq. Ž. 4 and equality of the rate of time preference and the interest rate. We consider the case where a QALY is gained through an increase in quality of life with duration constant. he case where quality of life is held constant and duration changes is discussed in Section Because equality of the rate of time preference and the interest rate implies that qh Ž. t is constant for all t, gh t is constant for all t. Denote the constant value of qh Ž. by q and the constant value of g by g. Rewriting Eq. Ž 0b. gives: t ht h Õ c Ž. sl Ž. g h
18 698 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics which we substitute in Eq. Ž 0a. to give: ÕŽ c. c gh s s Ž 2. ÕcŽ c. q q where s ÕccrÕ is the elasticity of the utility function for consumption. he parameter reflects the possibilities for intertemporal substitution of consumption. he higher the, the better are the possibilities for intertemporal substitution. As tends to unity, ÕŽ c. becomes more linear in c and the individual is less concerned about the distribution of consumption over time than in the value of aggregate consumption. In the limiting case, where is equal to one, the individual is not interested in the distribution of consumption over time, but only cares about total consumption. he term gh indicates how medical expenditures change as a result of a change in quality of life. Hence, gh can be interpreted as the marginal cost of an additional unit of quality of life. A Ž discounted. QALY is gained if quality of life Ž increases with zsr Ý ŽrŽ qr. ty.. units. his expression displays that the higher the interest rate the greater the future gain in quality of life has to be in order to gain a QALY. Substitution of z in Eq. Ž 2. defines the willingness to pay for a QALY gained. he willingness to pay for a QALY gained depends on four factors. It is increasing in consumption and the interest rate and decreasing in quality of life Žceteris paribus, individuals in worse health are willing to pay more for improvements in quality of life., and the possibilities for intertemporal substitution. In providing a numerical illustration, we will avoid the complications associated with discounting by focusing on immediate improvements in health. Empirical estimates of are in the range Žhaler and Rosen, 975; Rosen, We assume that is equal to Consider a person whose annual consumption is US$20,000 and whose quality of life initially is equal to 0.8. It follows that this individual will be willing to pay about US$00,000 per QALY for immediate improvements in health he willingness to pay for longeõity We now derive the willingness to pay for longevity. Substituting Eq. Ž. in Eq. Ž 0c. and rearranging gives: y c q Ž wym. g s Ž 3. A where y y As Ž qr. Ý s Ž qr ty Ý. Ž qr. ty
19 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics he term g reflects the responsiveness of medical expenditures to changes in longevity. hat is, g indicates the marginal cost of additional units of life or the willingness to pay for longevity. Eq. Ž 3. displays that the willingness to pay for longevity increases in consumption and decreases in the interest rate Žthe higher the interest rate the less valued are future life years. and in. he negative sign of the elasticity follows from the fact that, ceteris paribus, additional longevity does not change the individual s total consumption throughout his life, but it changes the distribution of consumption. Because the individual s concern about the distribution of consumption decreases with, his willingness to pay for longevity decreases with. Finally, the willingness to pay for longevity also increases in the surplus of the annual wage rate over the annual medical expenditures. During the additional time that the individual lives he is able to create more wealth. he higher his wage rate the more wealth he creates and the greater his possibilities for increasing consumption. he term Ž wym. does not appear in Section 4..2 where we considered QALY gains through increases in quality of life, because there duration was held constant and, hence, the individual s wealth was unaffected Risk he optimization problem We now turn to the more realistic case where the individual is uncertain about his life duration. Let ft be the probability of living for t time periods and let Ft be the cumulative probability of surviving until time period t at most. hat is, FtsÝ ty ss f s. hen, the probability of being alive at the beginning of time period t, denoted by S t, is equal to yf t. he period-specific death rate rt is defined as the probability of dying during time period t given that one has survived up to time period t. he period-specific death rate is a conditional probability defined as ftrs t. Hence, St and rt are related as follows: ty StsŁ Ž yrs. Ž 4. ss We assume that the individual maximizes expected utility. For each t, ifhe lives exactly t periods, then his utility is defined by Eq. Ž. 4 with constant rate discounting and s t. herefore, the individual s expected utility is equal to: ` Ý t t t ty Ž qa. EUs S ÕŽ c. qž h. Ž 5. We assume that the individual participates with a cohort of identical individuals in an actuarially fair annuity system Ž Yaari, 965; Rosen, Each individual hands over his wealth to an insurance company in exchange for a contract that ensures him his optimal consumption and medical expenditure bundles until death.
20 700 ( ) H. Bleichrodt, J. QuigginrJournal of Health Economics Obviously, individual choices are restricted by the total wealth available. he budget constraint facing each individual is based on overall life expectancy. If initial wealth is positive then those individuals who die early effectively subsidize the insurance pool and the claims of the individuals who live longer than expected are financed out of these subsidies. Under these assumptions, the individual s budget constraint becomes: ` Ý ` Ý Wqw St s S Ž c qm. Ž 6 ty t t t ty. Ž qr. Ž qr. where the medical expenditures are in each period a function of the infinite sequences of the quality of life levels Žqh Ž.,qŽ h., and the age-specific death rates Ž r,r,...: m sgžžqž h.,qž h.,..., Ž r,r, t 2 2. he first derivatives of medical expenditures with respect to qh Ž. t and rt are denoted by gh and g r, t t respectively. Maximization of expected utility, Eq. Ž 5., subject to the budget constraint, Eq. Ž 5., yields the following first order conditions, in which common terms St have been cancelled. ÕcŽ ct. qž ht. sl for all tg w,`x Ž 7a ty ty. Ž qa. Ž qr. ÕŽ ct. slg for all tg w,`x Ž 7b ty h ty. t Ž qa. Ž qr. ` St Õ c ty yrs Ž qa. ` St sl½ yrs Ž qr. Ý t t Ž. qž h. Ý ty t t ` r Ý s t ty Ž qr. 5 Ž c qm yw. yg S for all sg w,`x Ž 7c. he first order conditions in the decision problem under risk, Eqs. Ž 7a. and Ž 7b. are identical with the conditions under certainty, Eqs. Ž 0a. and Ž 0b., because the term S occurs both in Eqs. Ž 5. and Ž 6. t and cancels out. Hence, the problem under risk becomes similar to that under certainty and it follows immediately that the optimal paths for consumption and quality of life are constant only if the rate of time preference equals the interest rate. Hence, the conclusion derived under certainty, that cost-effectiveness analysis will only be equivalent to cost benefit analysis if the individual s rate of time preference is equal to the interest rate, is still valid if life duration is uncertain.
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