Equity, Pigou-Dalton, and Pareto Matthew Adler Duke University Harvard UC Conference, April 2013

Overview This talk is about methodology. Many equity metrics exist. Which one should we use to assess UC, or the inclusion of particular programs under UC? The Pigou-Dalton (PD) principle, in some form, is the heart of any equity metric Standard health equity metrics satisfy the PD principle with respect to health. Arguably, this is problematic. Well-being also depends on non-health attributes. Moreover, these metrics can violate the Pareto principle. An equity-regarding SWF applied to interpersonally comparable utility would be better. But constructing utility is not easy.

The Pigou-Dalton Principle PD: A non-leaky, gap-diminishing transfer of an appropriate currency, from someone with more to someone with less, is an improvement. (2, 11, 13, 20, 30) ≻ (2, 8, 16, 20, 30) PD is Pareto’s “kinder, gentler” cousin: the heart of equity, just as Pareto is the heart of efficiency. Pareto: If everyone weakly prefers an outcome, and some strictly prefer it, the outcome is better One difference: PD comes in many versions. See Adler, “Equity By the Numbers,” shortly on SSRN

The PD Principle and Equity Metrics Type of Metric Currency for PD principleRestricted in Scope? --Income inequality (e.g., Gini) --Income poverty (e.g., FGT metric) Income No Yes (poverty line) --Inequality metric applied to health -- QALY CEA with health- based equity weights --Health concentration Index HealthHealth No Yes (social class) --Social welfare functions UtilityNo -- Multidim. Inequality -- Multidim. Poverty (AF) Multiple dimensions No Yes (poverty line)

Standard Health Equity Metrics Inequality metrics applied to health. Every standard inequality metric (Gini, coefficient of variation, Theil, Atkinson) applied to a given “currency” satisfies the PD principle in that currency QALY CEA with greater weights for QALY changes to those in poorer health If program P and P* have the very same costs, and the same QALY gain, but P* delivers this to individuals at a lower QALY level, P* is chosen The Health Concentration Index. Looks at the extent to which health is concentrated in higher-status social groups (e.g., SES or racial groups). Satisfies PD principle in “restricted” form, with respect to health transfers from those at higher to those at lower status.

G1 (5%) G2 (50%) G3 (100%) 0%25%80% 100% CI/2 100% 50% 0% The Concentration Index CI = [(2∑S i h i )/Nμ] – 1, with “relative rank” S i increasing in status

Why apply the PD principle to health? The best justification for the PD principle is that someone at lower well-being—whose life is worse—has a stronger claim to a well-being improvement, modulo responsibility. (Adler, Well-Being and Fair Distribution, ch. 5) But someone at a lower health level might be at a greater well-being level, given non-health attributes (not just income, but leisure, social life, status, environmental quality, happiness) A non-leaky transfer in health might be leaky in well-being All the health equity metrics can violate the Pareto principle: a structural failure

QALY CEA vs. Pareto Principle Individual 1 Income Health Utility Individual 2 Income Health Utility Program Q$300K $30K Program Q*$200K $70K Each individual has an additive utility function u = log(income)/4.7 + health. Program Q* has an infinite incremental C/E ratio, with or without equity weights for those in poorer health, so CEA always chooses Q. But both individuals are better off with Q*. (Conflict with Pareto can arise whenever individuals have heterogeneous tradeoff rates b/w health and income; this u provides a simple example.)

The SWF Framework The SWF approach adopts some rule W for ranking utility vectors. It then says: outcome x is morally at least as good as y (x ≽ M y) iff (u 1 (x), u 2 (x), …, u N (x)) W (u 1 (y), u 2 (y), …, u N (y)) Requires an interpersonally comparable utility function u(.). u i (x) is in principle a function of all of i’s attributes in x (health, income, etc.)

The SWF Framework, Pareto, and PD A “preference-respecting” u(.): u i (x) > u i (y) iff i prefers x to y. u i (x) = u i (y) iff i is indifferent If u(.) is “preference respecting” and W is increasing in each individual’s utility, W satisfies the Pareto principle The rank-weighted SWF, continuous prioritarian SWF, and leximin SWF are increasing in each individual’s utility and satisfy PD

The continuous prioritarian SWF

The tricky part How to construct a preference-respecting u(.), if individuals differ in their preferences over attribute bundles (health & non-health attributes ). Jim might prefer A to A*, Sue A* to A ---Pooling utility functions. We do the SWF analysis using both u Jim (.), representing Jim’s preferences, and u Sue (.) -- Extended/higher-order preferences If R is a first-order preference, an extended preference is a ranking of (A, R) bundles. u(.) might represent these extended preferences (and thus assign utilities to individuals as a function of their attributes and first-order preferences) -- Equivalent income. c Jim (A): Jim is indifferent between A and c Jim (A) bundled with “reference” non-income attributes

Ex Ante Pareto: A Caution The SWF approach combined with a preference-respecting u(.) can satisfy both the Pareto and PD principle (in well-being) at the level of outcomes, but the ex ante Pareto principle is more suspect Significance for UC: health insurance choices that are universally preferred ex ante are not necessarily endorsed by the SWF approach

The ex ante Pareto principle and stochastic dominance Status QuoPolicy AB EU ABEU Jim ε 50-ε 50-ε June ε 50-ε 50-ε The continuous prioritarian, rank-weighted, and leximin SWFs all say that the policy is sure to produce a better outcome (for sufficiently small ε). And yet the status quo is ex ante Pareto superior

The Theoretical Ideal Rank UC policies (in practice, with risky outcomes) using the expected value of a PD-respecting, increasing-in-utilities SWF—applied to interpersonally comparable, preference-respecting utilities that are a function of individuals’ health and non-health attributes. This approach satisfies the PD principle (in well- being) and Pareto principle in ranking final outcomes, but not the ex ante Pareto principle We’re not there yet. Scholarship matters!