Journal of Health Economics 26 (2007) 426–429

Health inequality and non-monotonicity of the healthrelated social welfare function: A rejoinder

Indranil Dutta a,b,∗a UNU-WIDER, Helsinki, Finland

b Department of Economics, University of Sheffield, Sheffield, UK

Received 1 November 2006; accepted 14 November 2006Available online 8 December 2006

Abstract

This rejoinder demonstrates that the arguments made in AT (2006) are not robust and the case againstnon-monotonic HRSWF and their empirics remain overwhelming.© 2006 Elsevier B.V. All rights reserved.

JEL classification: D39; D63; I10

Keywords: Health inequality; Non-monotonicity; Social welfare function

This rejoinder will comment on the reply by Abasolo and Tsuchiya (in press) (henceforth AT(2006)) to my note (Dutta, in press) where I had raised several issues about the non-monotonichealth related social welfare function (HRSWF) suggested in Abasolo and Tsuchiya (2004) (here-after AT (2004)). I agree that HRSWFs should take in to account inequality with respect to healthachievements; this, apart from being reasonable, is also well accepted for the standard social wel-fare functions (Sen, 1997). Where I disagree with AT (2004) is whether this inequality aversionthat the HRSWF should possess, must be to such an extent so as to make it non-monotonic. Inthe following, I show that the arguments made in AT (2006) are not robust and the case againstnon-monotonic HRSWF and their empirics remain overwhelming.

Violating monotonicity implies that even if we are able to improve the health achievementof one individual without reducing (or changing) the health achievements of any other individ-uals in society, it may not be desirable to do so because the resulting increase in inequality

∗ Correspondence address: UNU-WIDER, Katajanokanlaituri 6 B, 00160 Helsinki, Finland. Tel.: +358 9 61599212;fax: +358 9 61599333.

E-mail address: dutta@wider.unu.edu.

0167-6296/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.jhealeco.2006.11.002

I. Dutta / Journal of Health Economics 26 (2007) 426–429 427

may be too high, so that the status-quo yields a higher level of the HRSWF. Let us illustratethe issue with an example. Consider a society of two individuals A and B, who are identi-cal in all aspects, except that each are affected by an unique disease, which reduces their lifeexpectancies to 25 years. Now suppose a cure has been found for the disease that inflicts per-son A, at no cost to person B, which would increase person A’s life expectancy to 75 years.Non-monotonicity would say that although person B’s life expectancy is no way reduced, whileperson A’s is highly improved, we should not let person A avail themself of this cure since theresulting inequality of 50 years may outweigh the gains of the increased life expectancy of A.In other words, non-montonicity is leading to a case where reduction in life expectancy of anindividual, without improving the life expectancies of others, can result in an increase in HRSWF.Elsewhere this feature has been referred to as ‘levelling down’ (Broome, 2002). However, asBroome (2002, p. 136) argues, “levelling down cannot possibly be a good thing” based on the“principle of personal good”, since “no change can be good unless it is good for someone”. Hence,he proposes an HRSWF that accounts for both the efficiency (or goodness) and the inequality(or the fairness) but does not violate monotonicity. Writing on a similar issue, Sen (2004, p. 25)indicates that ‘levelling down’ may not be desirable even from a health equity perspective.1 There-fore, as a principle, non-montonicity on which AT (2004) base their measure, is not particularlyendearing.

In trying to defend their measure from the several criticisms made by Dutta (in press), AT(2006) point out that the “appropriateness of scale invariance” which Dutta (in press) has usedas one of his criticism “has been questioned”. However, the reference they cite of Amiel andCowell (1999) has questioned scale invariance in the context of inequality measurement (Amieland Cowell, 1999, p. 35–38), not in respect to a social welfare function.2 Although AT (2006)agree that scale invariance is widely accepted, yet they seem to be unwilling to accept its negativeimplications for their measure, which Dutta (in press) points out as a change in the orderingresulting from scale changes (from months to weeks). Obviously their measure is non-homotheticand will lead to different results as the scales are changed, but to say that their measure “like anyother HRSWF is defined relative to a specific unit of measurement (say years) . . . and is notapplicable to health measured in months” is incorrect. One can easily construct HRSWFs that arescale invariant (hence it does not matter whether health is measured in months or years). However,Dutta (in press) also raised several criticisms of the AT (2004) measure even when their measureis homothetic. In particular, it is shown that there can be a situation where as inequality increases,their non-monotonic measure will show an increase in the HRSWF, which goes against the wholemotivation of their measure.

Dutta (in press) had used simple examples to illustrate the problems with the AT (2004)measure. Curiously enough AT (2006) mentions that the examples do not have much empiricalsupport and that is primarily because the specification considered by Dutta (in press) (i.e. c = 2)implies that “HRSWF is maximised when hi = hj” and that “monotonicity will be violated unlesshealth is distributed equally”.3 This is part misleading and part incorrect. It is misleading because

1 In fact Sen’s (2004, p. 25) example is stronger than the one used here. He examines the case where, in an effort toreduce health inequality, a rich person is prevented from buying a cure that a poor person cannot afford. Even so he statesthat “it would be hard to argue” that “this is a better situation overall”.

2 Interestingly, when it comes to monotonicity of the social welfare function (which is the main issue here) Amiel andCowell (1999, p. 66) in their survey find that the majority respondents agree with it atleast in the income space.

3 Note that according to AT (2004, p. 322) under the specification used by Dutta (in press) (i.e. α = 1 and β = 2) thereshould be no restrictions on c.

428 I. Dutta / Journal of Health Economics 26 (2007) 426–429

“HRSWF is maximised when hi = hj” will be true for any value of c (except c = 0), not just c = 2since (∂W/∂c = 0) implies hi = hj for any c > 0. It is incorrect because Dutta (in press) shows forthe non-homothetic case the AT measure will rank twice the improvement in life expectancy tobe better than no improvement in life expectancy. It is also incorrect to claim that the parametersconsidered by Dutta (in press) do not have empirical support since neither AT (2004) nor AT(2006) provide any empirically valid parameters. As mentioned earlier, the criticisms they seemto raise have nothing to do with the parameter values; rather it has everything to do with thefunctional form of the AT measure.

For the empirical part, AT (2006) claim that the violation of the median voter is sufficientto rule out Rawlsian preferences. It is, however, important to recall that in four out of the fivechoices presented by AT (2004) in their survey, the median voter does not violate the Rawlsiantype preference as has been pointed out in Dutta (in press). In this context, it is not apparentwhy AT (2006) considers the choice (between (+2, +2) and (0, +2)) where the Rawlsian typepreference is violated by the median voter to be preeminent compared to the other choices whereit is satisfied. Instead if we proceed with the natural assumption that each of the five choices areequally important, then for 80 per cent of the choices the median voter does not violate non-monotonicity. From this perspective the claim of non-monotonicity in the data certainly loosesits credence. In fact Olsen (2004) refers to other empirical studies in health that find no viola-tion of monotonicity for the HRSWF. In addition there is also the issue of income effects thatAT (2004) bring in to the empirical analysis of the HRSWF, apart from the health dimension.While income (or socioeconomic status) has a strong association with health, the HRSWF spec-ification in AT (2004, p. 319) does not reflect that, and hence is misspecified relative to the datathey use.

In conclusion, the acceptability of monotonicity crucially depends on whether one is measuringoverall welfare or inequality. As mentioned in Dutta (in press), we do not expect monotonicity tobe satisfied by a reasonable inequality measure, but it has some salience when it comes to a socialwelfare function. Perhaps the problem with both AT (2004) and AT (2006) is that they are unclearon whether they want to use their measure to capture only inequality or whether to measure socialwelfare in general. What I have tried to establish is that from a social welfare perspective there areserious problems with the AT (2004) measure and there are also enough reasons to cast seriousdoubts about the validity of their empirical result of non-monotonic HRSWF.

Acknowledgement

I am grateful to James Foster and S. Subramanian for valuable discussions and to AdamSwallow for his comments. The usual disclaimer applies.

References

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Abasolo, I., Tsuchiya, A., 2004. Exploring social welfare functions and violation of monotonicity: an example frominequalities in health. Journal of Health Economics 23, 313–329.

Amiel, Y., Cowell, A., 1999. Thinking about Inequality. Cambridge University Press, Cambridge.Broome, J., 2002. Fairness, goodness and levelling down. In: Murray, Christopher.J.L., Salomon, Joshua.A., Mathers,

Colin.D., Lopez, Alan.D. (Eds.), Summary Measures of Population Health: Concepts, Ethics, Measurement andApplications. World Health Organization, pp. 135–137.

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