Inequality-adjusted HDI: the basics · 2017-11-06 · Inequality-adjusted HDI: the basics Introduction Introduction: What is the inequality-adjusted HDI I The HDI is insensitive to

Inequality-adjusted HDI: the basics


Gaston Yalonetzky

March 2011


Table of contents

Introduction

The computation and decomposition of the IHDI

Some properties of the IHDI

The interpretation and policy aspects of the IHDI

Limitations of the IHDI


Introduction

Introduction: What is the inequality-adjusted HDI

I The HDI is insensitive to the degree of inequality within eachdimension.

I The IHDI is a generalization of the HDI that takes intoaccount, for every dimension, both the average achievementand the way it is distributed across the population.

I The IHDI is based on the measures of Foster, Lopez-Calvaand Szekely (2005), which in turn are based on the inequalityindices of Atkinson (1970).

I Since the IHDI is never higher than the HDI, the former isinterpreted as ”actual human development”, penalized byinequality, while the latter means ”potential humandevelopment” (should inequality be completely suppressed).


Introduction







Introduction







Introduction








The basic formulas of the IHDI: The Atkinson statistic forthe geometric mean

When the inequality aversion parameter, ε, is equal to 1, thestatistic Ax is:

Ax = 1−∏N

n=1 x1Nn

x= 1−

N√

x1...xN

x

Where: x = 1N

∑Nn=1 xn



The basic formulas of the IHDI: The Atkinson statistic forthe geometric mean

When the inequality aversion parameter, ε, is equal to 1, thestatistic Ax is:

Ax = 1−∏N

n=1 x1Nn

x= 1−

N√

x1...xN

x

Where: x = 1N

∑Nn=1 xn



The basic formulas of the IHDI: Relationship betweenmean attainment and inequality

The geometric mean penalizes the arithmetic mean by the degreeof inequality: ︷︸︸︷

x = x(1− Ax) = N√

x1...xN

Similarly, the inequality-adjusted dimension index is obtained fromthe HDI dimension index, by penalizing it with the inequality loss:

IIx = Ix(1− Ax)



The basic formulas of the IHDI: Relationship betweenmean attainment and inequality

The geometric mean penalizes the arithmetic mean by the degreeof inequality: ︷︸︸︷

x = x(1− Ax) = N√

x1...xN

Similarly, the inequality-adjusted dimension index is obtained fromthe HDI dimension index, by penalizing it with the inequality loss:

IIx = Ix(1− Ax)



The basic formulas of the IHDI: Computing the IHDIThen the dimension-specific, inequality-adjusted measures areaggregated using, again, the geometric mean.

IHDI = D√

II1 ...IID

In the specific case of the HDR:

IHDI = 3√

IILifeIIEducation

IIIncome

Notice that, by contrast, the HDI is computed using the averages(and the normalizations):

HDI = 3√

ILife IEducationIIncome

In both cases, the IHDI and the HDI, are sensitive to inequalityacross the normalized dimensions. They penalize ”unbalancedrelative development”.




IHDI = D√

II1 ...IID


IHDI = 3√

IILifeIIEducation

IIIncome


HDI = 3√






IHDI = D√

II1 ...IID


IHDI = 3√

IILifeIIEducation

IIIncome


HDI = 3√






IHDI = D√

II1 ...IID


IHDI = 3√

IILifeIIEducation

IIIncome


HDI = 3√


In both cases, the IHDI and the HDI, are sensitive to inequalityacross the normalized dimensions.

They penalize ”unbalancedrelative development”.




IHDI = D√

II1 ...IID


IHDI = 3√

IILifeIIEducation

IIIncome


HDI = 3√





The basic formulas of the IHDI: the loss

Notice the following interesting relationship between the IHDI andthe HDI:

IHDI = 3√

(1− ALife)(1− AEducation)(1− AIncome)HDI

The HDI represents the highest possible level, the ”potential”, thatthe IHDI could get if one could freely transfer achievements acrosspeople, in order to eliminate within-dimension inequality. From thisrelationship we can compute the percentage loss in ”potentialHDI” due to inequality:

Loss = 1− IHDI

HDI= 1− 3

√(1− ALife)(1− AEducation)(1− AIncome)





IHDI = 3√


The HDI represents the highest possible level, the ”potential”, thatthe IHDI could get if one could freely transfer achievements acrosspeople, in order to eliminate within-dimension inequality.

From thisrelationship we can compute the percentage loss in ”potentialHDI” due to inequality:

Loss = 1− IHDI

HDI= 1− 3






IHDI = 3√


The HDI represents the highest possible level, the ”potential”, thatthe IHDI could get if one could freely transfer achievements acrosspeople, in order to eliminate within-dimension inequality. From thisrelationship we can compute the percentage loss in ”potentialHDI” due to inequality:

Loss = 1− IHDI

HDI= 1− 3




The basic formulas of the IHDI: Example



The IHDI versus the HDI: an Example



The basic formulas of the IHDI: Decomposition of the loss

The Loss function can be decomposed into an approximation of thesum of the dimension’s inequality contributions by noticing that:

Loss ≈1

D

D∑d=1

Ad

Then the contribution of inequality in, say, education, can becomputed as:

CEducation =AEducation∑D

d=1 Ad





Loss ≈1

D

D∑d=1

Ad



d=1 Ad





Loss ≈1

D

D∑d=1

Ad



d=1 Ad



Decomposition of the loss: Example



The basic formulas of the IHDI: Data aspects andrequirements

I Q: What data do we need?

A: Ideally, disaggregated at the individuallevel.

I Q: All the variables from the same dataset? A: Not necessary for theIHDI. Because it is not sensitive to the association of dimensions, there isno need to rely on the same dataset for all the variables.

I Q: What if I have, say, municipal data for some variable? A: Inequalitywill be underestimated for that variable and then for the whole IHDI.

I Q: I want to use an income measure, shall I take its log? A: Taking thelog of income compresses inequality. It’s better to work with the levels.However, for flexible options see the appendix of the HDR 2010.

I What if I have zeroes or negative values? A: The IHDI cannot becomputed sensibly with negative values and one zero suffices for Ax = 1!Disposing of those observations or imputing values may be necessary.The HDR 2010 replaces them with the minimum value of the bottom 0.5percentile distribution of positive incomes. It may be necessary to assessthe impact of imputations on inequality.




I Q: What data do we need? A: Ideally, disaggregated at the individuallevel.









I Q: All the variables from the same dataset?

A: Not necessary for theIHDI. Because it is not sensitive to the association of dimensions, there isno need to rely on the same dataset for all the variables.

















I Q: What if I have, say, municipal data for some variable?

A: Inequalitywill be underestimated for that variable and then for the whole IHDI.

















I Q: I want to use an income measure, shall I take its log?

A: Taking thelog of income compresses inequality. It’s better to work with the levels.However, for flexible options see the appendix of the HDR 2010.

















I What if I have zeroes or negative values?

A: The IHDI cannot becomputed sensibly with negative values and one zero suffices for Ax = 1!Disposing of those observations or imputing values may be necessary.The HDR 2010 replaces them with the minimum value of the bottom 0.5percentile distribution of positive incomes. It may be necessary to assessthe impact of imputations on inequality.











Properties of the IHDI

The following are properties relevant to the IHDI:

I Sensitivity to progressive rank-preserving transfers within dimensions. TheIHDI increases when a richer person transfers part of its achievement to apoorer person, and their pairwise ranks do not change.

I Path-independence. If x = Ix then the IHDI would also be path

independent, i.e. IHDI =∏D

d=1

∏Nn=1 x

1NDid

I Sub-group consistency. Notice that: x(1− Ax) =∏G

j=1[∏Nj

i=1 x1Nj

i ]NjN





I Sensitivity to progressive rank-preserving transfers within dimensions.

TheIHDI increases when a richer person transfers part of its achievement to apoorer person, and their pairwise ranks do not change.



d=1

∏Nn=1 x

1NDid


j=1[∏Nj

i=1 x1Nj

i ]NjN








d=1

∏Nn=1 x

1NDid


j=1[∏Nj

i=1 x1Nj

i ]NjN






I Path-independence.

If x = Ix then the IHDI would also be path


d=1

∏Nn=1 x

1NDid


j=1[∏Nj

i=1 x1Nj

i ]NjN








d=1

∏Nn=1 x

1NDid


j=1[∏Nj

i=1 x1Nj

i ]NjN








d=1

∏Nn=1 x

1NDid

I Sub-group consistency.

Notice that: x(1− Ax) =∏G

j=1[∏Nj

i=1 x1Nj

i ]NjN








d=1

∏Nn=1 x

1NDid


j=1[∏Nj

i=1 x1Nj

i ]NjN




Some more properties:

I Individual scale invariance.

Multiplying one variable by a constant, doesnot change 1− Ax although the IHDIs get re-scaled, yet rankings arepreserved, percentage differences are preserved, and even the Lossfunction remains unchanged.

I Independence of standarized values. Standarization (i.e. division of thedimensional achievement by a common value across countries, e.g. thatof one specific country) does not affect country rankings.

I Consistency over time. Since scale revisions do not affect relative andpercentage rankings, then these remain consistent over time (i.e. they donot themselves constitute a source of change).





I Individual scale invariance. Multiplying one variable by a constant, doesnot change 1− Ax although the IHDIs get re-scaled, yet rankings arepreserved, percentage differences are preserved, and even the Lossfunction remains unchanged.








I Independence of standarized values.

Standarization (i.e. division of thedimensional achievement by a common value across countries, e.g. thatof one specific country) does not affect country rankings.















I Consistency over time.

Since scale revisions do not affect relative andpercentage rankings, then these remain consistent over time (i.e. they donot themselves constitute a source of change).










Some interpretation and policy aspects of IHDII As mentioned, the IHDI is a generalization of the HDI that takes into

account the degree of inequality in the distributions of the dimenisons.

The difference between the HDI (”potential HD”) and the IHDI (”actualHD”) reflects the loss in wellbeing due to the unequal distribution ofattainments. These comparisons are related to old wellbeingmeasurement theory (e.g. Atkinson, Sen’s metric).

I The inequality captured by the IHDI includes all sources, i.e. it is aninequality of ”outcomes” that is not helpful to analyze unequalopportunities.

I These measures are descriptive. They draw the attention to situations inwhich inequality is prominent vis-a-vis mean attainment in explainingtrends in wellbeing. But they do not say anything about what should bedone. Inequality and mean attainments may be in substitute orcomplementary relationships (e.g. poverty traps).

I The IHDI should not be compared to a Gini coefficient. Discounting themean attainment with a Gini coefficient, µ(x)[1− G(x)], has been done(Anand and Sen, 1993; Hicks, 1997), but, unlike the FLS measures, theGini is generally not sub-group consistent and it is not path-independent.




account the degree of inequality in the distributions of the dimenisons.The difference between the HDI (”potential HD”) and the IHDI (”actualHD”) reflects the loss in wellbeing due to the unequal distribution ofattainments. These comparisons are related to old wellbeingmeasurement theory (e.g. Atkinson, Sen’s metric).
















I These measures are descriptive.

They draw the attention to situations inwhich inequality is prominent vis-a-vis mean attainment in explainingtrends in wellbeing. But they do not say anything about what should bedone. Inequality and mean attainments may be in substitute orcomplementary relationships (e.g. poverty traps).







I These measures are descriptive. They draw the attention to situations inwhich inequality is prominent vis-a-vis mean attainment in explainingtrends in wellbeing.

But they do not say anything about what should bedone. Inequality and mean attainments may be in substitute orcomplementary relationships (e.g. poverty traps).















I The IHDI should not be compared to a Gini coefficient.

Discounting themean attainment with a Gini coefficient, µ(x)[1− G(x)], has been done(Anand and Sen, 1993; Hicks, 1997), but, unlike the FLS measures, theGini is generally not sub-group consistent and it is not path-independent.











I Lack of association sensitivity.

I Zero and negative values.




I Lack of association sensitivity.

I Zero and negative values.

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Inequality-adjusted HDI: the basics · 2017-11-06 · Inequality-adjusted HDI: the basics Introduction Introduction: What is the inequality-adjusted HDI I The HDI is insensitive to