# «Edited by ANNE MASON Research Fellow, Centre for Health Economics University of York and ADRIAN TOWSE Director, Ofﬁce of Health Economics Radcliffe ...»

The third stage was to look at the effect of across-class and across-age fair innings weights. Since those in social classes I and II at birth are already expected to achieve the (provisional) fair innings of 61 QALYs, whereas those in social classes IV and V are not expected to achieve this until they reach age 64 years, the impact of social class is found to be larger than the impact of age.* In other words, if the comparison is between young people from social classes I and II and old people from social classes IV and V, the effect of class will be much stronger, so that the older group will get a larger fair innings weight unless they are over 80 years and have achieved an ELQ of above 67 QALYs (see Table 4 of Williams, 1997). Also note that all Alan’s examples are for marginal health beneﬁts where the current population lifetime health prospects apply to the patients in question. Instead, in the case of young people from social classes I and II and old people from social classes IV and V who are all about to die without treatment, then since the former group has much smaller ELQ without treatment compared to the latter group, their weights will be larger to reﬂect this. This time, it is the age factor that determines the weights rather than the class factor.

* In this respect, it is interesting why Alan should have promoted the fair innings weights as a tool for intergenerational equity rather than interclass equity.

## 92 THE IDEAS AND INFLUENCE OF ALAN WILLIAMS

Fair innings weights and the health-related social welfare function* The general setting While the above ideas were explored in the Health Economics 1997 paper, the latter did not include an explicit exposition on how exactly the static and dynamic weights are to be derived, or how they are related to each other. This section is an attempt to ﬁll that gap.Suppose there is an extra QALY that could be given to one of two patient groups of the same size at equal cost, and that the only difference between these two groups is their lifetime health prospect without treatment, such that those in one group have 70 ELQ(0) while those in the other group have 50 ELQ. The two axes of Figure 8.1 represent ELQ(0) of the two groups. It shows the current situation at point P, where ELQ(0) of the two population subgroups a and b are Ha0 and Hb0 respectively. Point A represents the average of the two ELQs, or ‘overall health’. Point P´ is the situation where Ha0 and Hb0 are interchanged. Assuming symmetry, the level of social welfare at points P and P´ are identical. Now, suppose that members of the general public are presented with a description of the present situation P and asked how much overall health, in terms of QALYs, they would be willing to forego (WTF) for an equal distribution between the two population subgroups. This will lead to the determination of the level of the fair innings: i.e. average health minus the WTF.† Point E on the 45° line is where both populations have an ELQ(0) equal to the level of fair innings. The implication is that people are indifferent between the three points P, P´, and E, and thus these points must lie on the same social welfare contour. Figure 8.2 illustrates such a curve.

While there can be more than one speciﬁcation that yields an iso-welfare curve through these three points, let us assume a health-related social welfare function (HRSWF) that is increasing in subgroup health and has constant

**elasticity of substitution so that:**

where W represents the level of health-related social welfare, and r represents the curvature of the HRSWF so that when there is aversion to inequality, then r –1 and the iso-welfare curve becomes convex to the origin. This value can be obtained from observed values for FI, Hax, and Hbx. When WTF = 0, then r = –1. This is the case of the classical utilitarian HRSWF, which implies neutrality over distribution, and individual QALY gains are summed together with no weights attached.

The α and 1 – α parameters indicate the rates at which the health of the two * This section draws on a Health Economists’ Study Group (HESG) paper presented at Newcastle some time ago (Tsuchiya, 2000b).

† In other words, the fair innings is the equally distributed equivalent health.

## BEING REASONABLE ABOUT EQUITY AND FAIRNESS 93

** FIGURE 8.1 The present situation and WTF.**

avg, average expected lifetime QALYs = overall health; FI, the fair innings; WTF, willingness to forego overall health for more equal distribution; A, the point at which both parties achieve average health; E, the point at which both parties achieve the fair innings.

** FIGURE 8.2 The social welfare contour.**

A, the point at which both parties achieve average health; E, the point at which both parties achieve the fair innings; SS, the social welfare contour.

## 94 THE IDEAS AND INFLUENCE OF ALAN WILLIAMS

population subgroups enter the social welfare calculus. So, if the health of the two population subgroups were perceived to be of different social worth (e.g.because one group is responsible for their own poor health), then α ≠ 1 – α to reﬂect this. Where the assumption is that neither party is responsible for the difference in lifetime health, α = 1 – α = ½. While on the one hand r inﬂuences the rate at which the health of a population subgroup affects social welfare based on how healthy that group is compared to the other, regardless of all other attributes of these groups, α on the other hand inﬂuences the rate at which the health of the subgroups affects social welfare depending on who these people are, regardless of their levels of health relative to each other.*

* There may be several factors that affect the value of α and whether or not α = 1 – α. One candidate is the effect of individual choice and liability, so that for example one group will be smokers and the other group will be non-smokers. If, by appropriate means, it can be established that different people are responsible by varying degrees for some aspects of their own lifetime health, and if this magnitude is quantiﬁed, then the α parameter can be estimated and incorporated in the model.

## BEING REASONABLE ABOUT EQUITY AND FAIRNESS 95

** FIGURE 8.3 Static weights for subgroups a and b.**

A, the point at which both parties achieve average health; E, the point at which both parties achieve the fair innings; SS, the social welfare contour; line TT is the tangent to the contour at point P.

Note that the subscript of SW includes the reference subgroup, reﬂecting the fact that this weight is relative to the health of the comparator subgroup, in this case b, and therefore, not independent of how other subgroups fare in terms of ELQ. Further, it does not specify age. These subgroup weights thus obtained are static in the sense that they are calculated based on ELQ(0) regardless of present age, past QALYs, present QALE, or present ELQ of those involved, and thus everybody in the same subgroup is given the same weight. Static weights have been deﬁned in the 1997 paper and calculated with reference to ELQ at birth, but these may also be calculated for ELQ at any given age.

The dynamic model: subgroup-and-age-speciﬁc weights The dynamic weight speciﬁc to population subgroup n at age x can reﬂect the increase in ELQ with age by basing the calculation on Hnx instead of Hn0.

Unless one acquires a disability that is signiﬁcant both in terms of severity and duration, most people’s ELQ improves with age (or with survival). Dynamic weights take into account the effect of increasing ELQ through survival, so that the larger one’s ELQ, the smaller will be the weight given to one’s marginal health improvement. Incorporation of such weights into cost–QALY analyses will imply that, other things being equal, the older one is, the healthcare treatment one receives has to clear an increasingly lower cost per QALY threshold than that for a younger candidate patient. Although the fair innings weights can be interpreted as age weights under an ‘other-things-being-equal’

## 96 THE IDEAS AND INFLUENCE OF ALAN WILLIAMS

clause, strictly speaking, they are not age weights but weights by ELQ.* The actual weight is calculated with reference to the standard prospect of achieving a given fair innings, which makes this weight relatively independent of ELQ of other subgroups so long as the level of the fair innings itself remains the same. The dynamic weight (DW ) is calculated by substituting FI r (the level**of fair innings given r) for Hb and Hnx for Ha of [eq. 2b]:**

** DWnx = [FIr /Hnx ](1 + r) [eq. 3].**

This is the most general formula that allows to deal with cases where n 2

**without any adjustments. Further, the relative subgroup static weight SWn:m**

at birth can be obtained by dividing DWn0 by DWm0.

** Figure 8.4 depicts three cases where the ﬁxed prospect of achieving the fair innings is represented on the horizontal axes (subgroup b), and ELQ at age x of subgroup a is (1) smaller than, (2) equal to, and (3) larger than the fair innings.**

The ﬁgure shows two things: that DWnx is larger (smaller) than 1 when ELQ is smaller (larger) than the fair innings; and that the calculation of DWnx corresponding to different ELQs involves more than one contour of the same SWF.

The 1997 paper justiﬁes this involvement of different levels of W to obtain dynamic weights by an argument which in effect states that: since it is unlikely that we will actually ﬁnd ourselves on the production possibility frontier, the frontier can be ignored. It can also be argued that since it is the MRS between two potential health improvements, and since MRS belongs to the realm of preferences and not resources or production technologies, the production possibility frontier is irrelevant even when it is known for certain that the point lies outside the frontier. The strength of these arguments at the practical level may depend on how feasible it is to picture being at point (Hax,FIr). Where the difference in health between the subgroups is very large and the subgroup in question is the healthier one, this point may become increasingly implausible, not on theoretical grounds, but on practical grounds (because Hbx is so much lower than FIr).

One may further want to question the legitimacy of comparing, for example, DWa20 with DWb60: i.e. the dynamic weight given to 20 year olds in subgroup a, and the dynamic weight given to 60 year olds in subgroup b.

These two weights will be derived from the same SWF, but from two different contours of this: i.e. DWa20 is based on the gradient of the contour at (Ha20,FIr) and DWb60 comes from the gradient of another contour at (FIr, Hb60). Does it make sense to compare gradients from two different contours, and therefore * For instance, an individual that acquires a signiﬁcant permanent disability at middle age will record a sharp drop in ELQ at that point, and therefore his/her weight will sharply increase accordingly. Further, if the disability stays stable, his/her weights will gradually decrease from then onwards, with age, but at a slower rate than the weights of those without permanent disabilities.

## BEING REASONABLE ABOUT EQUITY AND FAIRNESS 97

** FIGURE 8.4 Dynamic weights for subgroup a.**

relate to different levels of social welfare? However, on a theoretical level, it is always possible to identify a point (Ha,Hb), where Ha = Ha20 and Hb = Hb60, and, since static weights do not refer to age x per se, SWa:b at this point will coincide with the ratio between DWa20 and DWb60. In other words, the gradient of the contour through this point will correspond to comparing DWa20 and DWb60, and therefore, the direct comparison of these two can legitimately be made. At an empirical level, the concern translates to the issue of whether or not the value of r is constant for all possible combinations of (Ha,Hb). In other words, whether or not the preferences elicited from the general public will be as well behaved as expected by theory. This is an important issue that goes beyond the scope of this paper.

Fair innings and the sex problem: from health to wider well-being When faced with the question why equity of health and not equity of health

**care, Alan seems to have had two thoughts. The ﬁrst is mentioned above:**

because improving health is the ﬁnal objective of healthcare services. The equity objective is equality of health, where equity of health care might be a means towards achieving this end. The second is because health measured in ELQ is the variable that seems to best reﬂect the people’s well-being. Those who are richer, better educated, better nourished, with more socio-economic opportunities, more privileged, and happier all live longer and healthier lives than those who are not... but there is a caveat to this: provided the comparison is within the same sex group. Women in most societies live longer (and have larger ELQ) than men, although they are poorer, less educated,