#### Transcript Comment on Bosssert, D’Ambrosio & Grenier paper

```Comment on Bosssert,
D’Ambrosio & Grenier paper
Branko Milanovic,
World Bank
IHDI general approach
• The general approach
𝑊 = μ 1 − 𝑙𝑜𝑠𝑠
• And loss is:
𝑙𝑜𝑠𝑠 = 1 − 𝐸𝐷𝐸(ε)
μ
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•
The question are then:
1) what is the correct ε?
2) should ε be the same for all dimensions?
3) should aggregation of dimensions in a single
index also use the same ε?
• In the past the approach was inequality aversion = 0, W
= μ.
• Now, inequality aversion = 1, and
𝐸𝐷𝐸 = 𝑛 𝑥1 𝑥2 𝑥𝑛
• Which gives the loss function as
𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑚𝑒𝑎𝑛
𝑙𝑜𝑠𝑠 = 1 −
𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑚𝑒𝑎𝑛
• And the new HDI has moved from being W=arithmetic
mean to W=geometric mean of individual outcomes
• But we do not want loss to depend so much on
minimum values (or 0) as to generate loss=1 for
whenever 0 appears
• We modify the scale by dropping zeros
• Certainly not an ideal solution particularly
when 0s are meaningful (as in education).
• But what should be our ε? Is there any way to
choose between 0 and 1, or to take an
intermediate value?
• Do we have any theoretical basis to guide our
choice?
• Perhaps we do…
Rawls
• Take a Rawlsian approach. The loss function is
entirely determined by the position of the lowest
recipient. It becomes
𝑦1
𝑙𝑜𝑠𝑠 = 1 −
μ
• The outcome is the same as with the geometrical
mean when the lowest values is 0.
• Otherwise, it is a “degenerate” case of
geometrical mean with (probably) even stronger
implicit inequality aversion that we currently
have with the geometrical mean.
Roemer
• Or take Roemer’s approach. Let ε depend on
some ratio between effort and circumstance.
The greater the share of effort in total
inequality across a given dimension, the less
inequality aversion.
• Loss function becomes
1
(
𝑥 1−ε𝑗 )1/1−ε𝑗
𝑛
𝑙𝑜𝑠𝑠 = 1 −
μ
• Where εj is specific to each dimension
• One may argue that the role of effort in determining
income is greater than the role of effort in determining
health. That would argue for a lower ε to be attached
to inequality in income
• Moreover, the ratio between effort and circumstance is
not the same for a given dimension in all countries
• That means that ε may be both dimension and country
specific: 150 countries x 3 dimensions = 450 ε.
• The approach would link IHDI to inequality of
opportunity literature
Subjective approach
• Retrieve ε for each dimension from worldwide
surveys of individuals (similarly to the way
income inequality aversion parameters have
been estimated)
• Bottom line: we would have grounded ε in
something more solid (Rawls, inequality of
opportunity, subjective views of people) than
arbitrary selection between 0 and 1.
Several other issues considered by BDG
• Use income in untransformed form. Yes, these
reduces variability.
• Equivalency scales. No. Scales depend on price
structure and price structure varies across
countries (Lanjouw, Milanovic, Paternostro). Even
for EU, it is doubtful that the scales are the same
for all countries (Romania vs. Luxemburg). It also
makes calculations more complex, but it is a
moot point because individual data are unlikely
to be available any time soon.
Another point: methodological nationalism
• Unequal treatment of individuals across the world
• Currently the index is “nationally centered”
• Loss function reflects deviation of an individual
outcome from a national mean
• If that individual is placed in another country, her
contribution will change
• So the same individual outcome is treated differently
depending on the country where it occurs
• Should we look for a global index which could be
decomposable by countries?
• In other words, the yardstick against which the
deviation is measured, should be global not national
Conclusion
• If we cannot move in the short run to Rawlsian
(extreme), Roemerian (difficult to calculate) or
subjective estimate of ε, then the current
selection of ε=1 is preferable because of its
relative simplicity
• The problem of truncation of the variables is an
annoying one but inescapable
• ε=0.5 leads, in my opinion, to a much more
cumbersome expression without any clear gain in
terms of being more grounded or “reasonable”
than ε=1.
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