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 principle
--Income inequality (e.g.,
Income
Gini)
--Income poverty (e.g., FGT Income
metric)
Restricted in Scope?
No
Yes (poverty line)
--Inequality metric applied
to health
Health
No
-- QALY CEA with healthbased equity weights
Health
No
--Health concentration
Index
Health
Yes (social class)
--Social welfare functions
Utility
No
-- Multidim. Inequality
Multiple dimensions
No
-- Multidim. Poverty (AF)
Multiple dimensions
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.
The Concentration Index
G3
(100%)
100%
CI/2
50%
G2 (50%)
G1 (5%)
0%
0%
25%
80%
CI = [(2∑Si hi)/Nμ] – 1, with
“relative rank” Si increasing in
status
100%
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
Individual 2
Income
Health
Utility
Program Q
$300K
.55
1.72
$30K
.6
1.55
Program Q*
$200K
.6
1.73
$70K
.55
1.58
Utility
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
(u1(x), u2(x), …, uN(x)) W (u1(y), u2(y), …, uN(y))
• Requires an interpersonally comparable utility
function u(.). ui(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(.): ui(x) > ui(y) iff i
prefers x to y. ui(x) = ui(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 uJim(.), representing Jim’s preferences, and uSue(.)
-- 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. cJim(A): Jim is indifferent between A
and cJim(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 Quo
A
B
EU
Jim 10
90
50
June 90
10
50
Policy
A
B
EU
50-ε 50-ε 50-ε
50-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 wellbeing) and Pareto principle in ranking final outcomes,
but not the ex ante Pareto principle
• We’re not there yet. Scholarship matters!