Human Development Index: The Old, The New and The Elegant

Srijit Mishra and Hippu Salk Kristle Nathan

Seminar at National Institute of Public Finance and Policy, New Delhi

10 July 2013

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Focus of the study

• NOT rationale behind choosing these 3 indicators

• NOT how the indicators are measured and scaled

• NOT how the indicators are normalized and weighed

INVESTIGATES the appropriateness of the known two measures of HDI

proposes an alternative technique

Inverse of the Euclidean Distance from Ideal

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HDI – the old (till 2009)

Life Expectancy at birth1. A long and healthy life

2. Knowledge

3. Ability to achieve decent

standard living

Adult literary rate (2/3)

Gross enrolment ratio (1/3)

3 dimensions –

h

GDP per capita (PPP)

e

y

3

HDILA = 1/3 (h) + 1/3 (e) + 1/3 (y) 0 ≤ h, e, y ≤ 1

Iso- HDILA lines

e

h

k

j

(1,1)

(0,0)

Iso-HDI lines – old HDI (perfect-substitutability)

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HDI – the new (from 2010)

Life Expectancy at birth1. A long and healthy life

2. Knowledge

3. Ability to achieve decent

standard living

0 ≤ h, e, y ≤ 1

Mean years of schooling: adults (2/3)

Expected years of schooling: children (1/3)

3 dimensions –

h

GNI per capita (PPP)

e

y

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HDIGM = (h * e * y)1/3

Iso- HDIGM lines

k

j

(1,1)

(0,0)

e

h

Iso-HDI lines – new method

6

Displaced Ideal

Zeleny (1974)

better system should have less distance from “ideal”.

dj

dk

j

k

e

h

jd22 )1()1( jj he

ej ek

hk

hj

HDIDIj > HDIDI

k if and only if dj < dk

HDIDIj = HDIDI

k if and only if dj = dk

HDIDIj < HDIDI

k if and only if dj > dk

I (1.0, 1.0)

Additive Inverse of distance

=

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HDI – the elegant (proposed)

1. A long and healthy life

2. Knowledge

3. Ability to achieve decent

standard living

0 ≤ h, e, y ≤ 1

3 dimensions –

h

e

y

HDIDI = 1-(√((h2 + e2 + y2)/3))

8

Iso- HDIDI lines

k

j

(1,1)

(0,0)

e

h

Iso-HDI lines – elegant method

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A measure of HDI should be greater (lower) if the index value in one dimension

is greater (lower) with indices value remaining constant in all other dimension.

LA, GM and DI satisfy

I

k

Iso-HDILA

Iso-HDIGM

e

h

1

O

For any random country k in

Zone A hk ≥ hj , ek ≥ ej (hk=hj or ek=ej)

LA: HDIk > HDIj

GM: HDIk > HDIj

DI: HDIk > HDIj

Zone B hk ≤ hj , ek ≤ ej (hk=hj or ek=ej)

LA: HDIk < HDIj

LA: HDIk < HDIj

DI: HDIk < HDIj

Iso-HDIDI

Axiom M: Monotonicity

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Axiom A: Anonymity

A measure of HDI should be indifferent to swapping of values across dimensions.

LA, GM and DI satisfy Anonymity

LA: hj + ej = hj’ + ej’

GM: hj * ej = hj’ + ej’

Note that this is a statistical property and does not invoke substitution between dimensions

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DI: dj = dj’

A measure of HDI should have a minimum and a maximum i.e. HDI (0,1)

LA, GM and DI satisfy this;

for GM the value will be zero if any dimension has no development

e

y

hI

e=1

y =1

h=1

O

Axiom N: Normalization

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HDI = 0: NO development (h = 0, e = 0, y = 0) - “Origin”

HDI = 1: COMPLETE development (h = 1, e = 1, y = 1) – “Ideal”

Axiom U: Uniformity

LA fails, GM and DI satisfy

Illustration (1)

Uniform to Non-Uniform

j(0.5, 0.5) dj =√(0.50) GM =√(0.25)

j’(0.6,0.4) dj’= √(0.52) GM =√(0.24)

Change in HDI:

HDILAj = HDILA

j’

HDIGMj > HDIGM

j’

HDIDIj > HDIDI

j’

Illustration (2)

Non-Uniform to Uniform

k(0.8, 0.4) dk =√0.80 GM =√(0.32)

k’(0.6,0.6) dk’=√0.72 GM =√(0.36)

Change in HDI:

HDILAk = HDILA

k’

HDIGMk < HDIGM

k’

HDIDIk < HDIDI

k’

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

e

O (0, 0)

j

j’

h

k

k’Line of equality

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I (1,1)

j k

h

e

Movement is

along the direction

proportion to shortfall

Ideal paths

Iso-HDI lines

Axiom S: Signalling

DI satisfies

LA and GM fail

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Axiom H: Hiatus sensitivity

e

h Equal gap at higher

attainment should be

considered worse off

DI satisfies

LA and GM fail

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M: Monotonicity A: Annonimity N: Normalization U: Uniformity S: Signalling H: Hiatus Sensitivity

LAGM

DI

MANUSH Axioms

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HDI rank: revisit with GM and DI

LA ~ GM

Top 54 countries did not change ranks

Does not affect the Top

Shift to GM was virtually no shift for high HD countries

LA ~ GM ~ DI Does affect the Top

COUNTRY Health Index

Education Index

Income Index

Rank LA

Rank GM

Rank DI

Rank Diff

Range

Ireland 0.890 0.993 0.994 5 5 15 -10 0.1035Japan 0.954 0.956 0.959 8 8 3 +5 0.0127United States 0.881 0.971 1.000 12 12 21 -9 0.1192Italy 0.922 0.958 0.944 20 20 12 +8 0.0367

Along expected lines

Source: UNDP (2008)

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HDI rank: revisit with GM and DI

LA ~ GM Does it affect the Bottom?

COUNTRY Health Index

Education Index

Income Index

Rank LA

Rank GM

Rank DI

Rank Diff

Range

Congo 0.346 0.56 0.328 168 169 169 -1 0.2319Ethiopia 0.446 0.38 0.393 169 166 168 3 0.0668Chad 0.423 0.296 0.444 170 170 170 0 0.1478Central African Republic 0.311 0.423 0.418 171 171 171 0 0.1119Mozambique 0.296 0.435 0.421 172 172 172 0 0.1383Mali 0.469 0.282 0.39 173 173 173 0 0.1868Niger 0.513 0.267 0.343 174 175 175 -1 0.2460Guinea-Bissau 0.347 0.421 0.353 175 174 174 1 0.0734Burkina Faso 0.44 0.255 0.417 176 176 176 0 0.1847Sierra Leone 0.28 0.381 0.348 177 177 177 0 0.1014

DI too captures change in rank at the bottom

The values of the indices influence the ranking, rather than aggregation technique

Source: UNDP (2008)

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HDI rank: revisit with GM and DI

LA ~ GM

Source: UNDP (2008)

The difference was also captured by DI

Countries where GM differed from LA most

COUNTRY Health Index

Education Index

Income Index

Rank LA

Rank GM

Rank DI

Rank Diff

Range

Lesotho 0.293 0.768 0.585 138 148 148 -10 0.4747Swaziland 0.265 0.730 0.647 141 151 151 -10 0.4645Zimbabwe 0.265 0.770 0.503 151 157 157 -6 0.5055Papua New Guinea 0.532 0.518 0.541 145 140 139 5 0.0234Togo 0.547 0.538 0.453 152 147 147 5 0.0940

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Class of HDI

• Mα =1-DαI ; • D αI=(1/nΣ(1-xj) α)1/α where (j=1,…,n)• For α≥1, Mα

satisfies the axioms of– monotonicity, ∂Mα/∂xj>0 for all j. – Anonymity, (xj,wj) swap values with (xi,wi), i≠j – normalization, Mα[0,1], and

• For α>12, M α satisfies the axioms of – uniformity (position penalty) and – signaling (path penalty)– hiatus sensitivity

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References• Anand, S and Sen, A (1994) Human Development Index:

Methodology and Measurement, Human Development Report 1994.• Mishra, Srijit and Nathan, Hippu Salk Kristle (2008) On A Class of

Human Development Index Measures, WP-2008-020, Indira Gandhi Institute of Development Research, Mumbai.

• Nathan, Hippu Salk Kristle and Mishra, Srijit (2010), Progress in Human Development: Are we On the Right Path? International Journal of Economic Policy in Emerging Economies, Vol.3. No. 3, 199-221.

• Nathan, Hippu Salk Kristle, Mishra, Srijit and Reddy, B. Sudhakara (2008), An Alternative Measure of HDI, WP-2008-001, Indira Gandhi Institute of Development Research, Mumbai

• Nathan, Hippu Salk Kristle and Mishra, Srijit (forthcoming), Group Differential for attainment and failure indicators, Journal of International Development.

• UNDP (2008), Human Development Report 2007/08: Fighting Climate Change: Human Solidarity in a Divided World, Oxford University Press, New Delhi

• UNDP (2010), Human Development Report 2010: The Real Wealth of Nations; Pathways to Human Development, Oxford University Press, New Delhi

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