Axioms for health care resource allocation☆
Introduction
In health economic analyses, it is often assumed that an individual's preferences for own health can be represented by an index measuring quality-adjusted life years (QALYs).
As pointed out by Wagstaff (1991) and others, a policy of maximizing the total number of QALYs in society may fail to capture distributive justice. How to adjust for this problem has not only evolved into a major research theme in health economics but it has also attracted attention from other social sciences, e.g. social philosophy and medical ethics. Fundamental to this problem is the question of whether it, at least in theory, is possible in a meaningful way to aggregate individual health indices (QALYs), in particular when individuals not necessarily have identical preferences.
The aim of this paper is to investigate one way of obtaining an explicit link between QALYs at the individual level and distributive justice at the societal level.
In recent literature, different ways of accounting for distributive justice via specific social welfare functions have been suggested. Williams (1997), for example, advocates a concrete interpretation of the fair innings principle which involves the estimation of a social welfare function defined on the distribution of expected QALYs which is a member of the family of CES functions (constant elasticity of substitution). Dolan (1998) suggests a Cobb–Douglas social welfare function on individual QALYs. Because social welfare is then strictly concave in individual QALYs, this reflects a particular concern for people with bad health. However, in a comment to Dolan's article, Johannesson (1999) argues that the approach suggested by Dolan has no theoretical foundation and should not be used as a basis for aggregating QALYs. Johannesson suggests the following alternative framework: each individual has a utility function which depends on the entire distribution of individual QALYs and the social welfare function is then defined as the sum of individual utilities. The idea is that since individual utilities already capture altruism or concerns for distributive justice, there is no need to take these things into account again via the curvature of the social welfare function.1 In Nord, 1993, Nord, 1999 and Nord et al. (1999) variants of the cost-value method are suggested and analyzed. The societal value of a health gain is defined as the product of certain weight functions (the utility gain, a weight determined by the severity of the initial condition, and a weight determined by the potential for health) and the method for health care resource allocation implied is radically different from those suggested by Williams, Dolan or Johannesson.2
It is difficult to evaluate the superiority of one functional form over the other. For instance, different specifications of individual utility and social welfare may represent the same underlying preference relations and concerns for justice in resource allocation; and it may not be clear for some approaches whether key properties such as the Pareto principle are satisfied. Also, the characteristics of these approaches depend crucially on how the individual health utilities (QALYs) have been constructed from observable data (see below). Thus, instead of comparing functions, with properties that are difficult to compare, in this paper we compare underlying axioms which in certain combinations lead to different allocation methods.
In this respect our approach complements that of Bleichrodt (1997) who investigates a situation with choice under uncertainty and aggregation of individual health utilities (interpreted as QALYs). Bleichrodt examines axioms related to decisions under uncertainty, whereas our focus primarily is axioms related to comparisons of deterministic outcomes.
In the above-mentioned papers, health states and life years do not enter the models explicitly, but abstract one-dimensional individual health utility indices are the primitives of the model. However, when we speak of justice in health and health care, this is usually in terms of individual entitlement to life years and health states, not abstract health utilities. There are many possible ways of constructing an individual one-dimensional health index from relevant underlying data. For instance, a health index can be ‘timeless’ or depend on life years, and there are various ways of estimating parameters (rating-scales, standard gambles, time trade-offs, etc.). In other words, it is difficult to evaluate the reasonableness of methods based on the maximization of some function of health indices when it is not explicitly described how these health indices are derived from underlying data.
We characterize allocation methods by axioms which have fairly simple and intuitive interpretations in this particular context of health care resource allocation. These axioms are formulated as principles of distributive justice in particular ‘simple’ dilemmas, for example, involving comparisons of certain gains in life years for two different people at the same health state. Once having established a principle of justice in these cases, social welfare evaluations of arbitrary distributions of (changes in) life years and health states in society turn out as a consequence of the underlying axioms.
The plan of the paper is as follows. In Section 2, the necessary definitions are provided and the health care resource allocation problem that underlies these definitions is outlined. In Section 3, we derive an axiomatic characterization of QALY measures as individual utility functions in a deterministic framework. In Section 4, we give sufficient conditions for the existence of a continuous social welfare function being a function of individual QALYs only. Based on the representation of individual preferences, we derive axiomatic characterizations of two specific solutions to the health care resource allocation problem. The symmetric Cobb–Douglas solution is characterized by an anonymity axiom in terms of the distribution of individual life years for any fixed distribution of health states; an alternative characterization by means of equal social preferences for relative gains in life years for individuals at some health state is also obtained (Section 4.1). By replacing this axiom with an axiom of non-age dependence in terms of social preferences for gains in life years for individuals at a particular health state, we obtain a characterization of a utilitarian solution with certain weights (Section 4.2). Further, we identify two characterizations of the class of social welfare functions which can be written as a weighted sum of power transformations of individual QALYs (Section 4.3). The latter results are obtained as applications of a theorem by Bergson (Burk, 1936). Section 5 considers two extensions of the model, involving uncertainty and non-chronic health states, respectively, and explains how the same axioms for distributive justice (restricted to comparisons of cases with deterministic chronic health states) can be used to obtain similar characterizations of welfare functions (Sections 5.1 Uncertainty, 5.2 Non-chronic health). Finally, we briefly discuss status quo biases (Section 5.3), and provide some concluding remarks (Section 5.4).
Section snippets
Health care resource allocation
Let A be a set of conceivable health states of an individual. We think of a health state as a list of factors describing observable aspects of individual well-being. We do not make any assumptions on the mathematical structure of A, but in two cases we impose a certain richness condition (Theorems 3 and 8). We assume that an individual is always in the same health state throughout life. The health profile of individual i is a pair containing health state ai and prospective lifetime
Individual QALYs
QALY-based individual utility functions have been characterized in the literature by regularity conditions on preferences over health lotteries, see Pliskin et al. (1980), Bleichrodt et al. (1997) and Miyamoto et al. (1998). In this section, we derive a simple characterization of QALY-based individual preferences in a situation of choice under certainty.
We make the assumption that if an individual obtains zero lifetime, health state does not matter. This property has been referred to as the
Aggregating QALYs
Having introduced assumptions on individual preferences for own health, we now go on to aggregation of QALYs.
We have imposed a rather restrictive structure on preferences for own health. On the other hand, we do not assume that individuals have symmetric preferences (with one exception—Theorem 8). The possibility that individuals have different views on the trade-off between lifetime and health states is accounted for, and health states may even be ranked differently by the individuals.
The
Discussion
We have so far considered the simplest possible relevant model where the health profile of an individual is summarized in two pieces of information, health state and life years. We discuss in turn extensions of the model to incorporate uncertainty and to incorporate non-chronic health states, respectively.
Acknowledgements
The paper has benefitted much from comments by two anonymous referees. I also thank Kristian Schultz Hansen, Jens Leth Hougaard, Hans Keiding and Alan Williams for helpful advice and comments. The usual disclaimer applies.
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An earlier version was presented at a seminar at the Health Economics Research Centre, University of Oxford, January 2002, and at the 4th European Conference on Health Economics, Paris, July 2002.