The Human Development Index (HDI), since its inception in 1990, has come up with an indicator for each country that aggregates the three dimensions of health (representing how long and fulfilled a life one lives), literacy (representing knowledge) and income (as a proxy for standard of living) into a single dimension. This was an important departure from income-based measures that focused on a single dimension. Before aggregating across dimensions, each indicator was normalized and took values between zero and unity.[1]

Prior to 2010, the approach followed to aggregate was a simple averaging across dimensions. A problem with this method was that a deficit in one dimension will perfectly substitute an equal attainment in another dimension. Income remaining same, this means that a country where both health and education attainments have the same value (say, 0.4 each) will have the same HDI as another country where health is 0.2 and education is 0.6 (a situation not quite uncommon in some of the Sub-Saharan countries reeling under a HIV/AIDS epidemic a few years ago).[2]

In 2010, to address perfect substitutability across dimensions, the calculation of HDI was aggregated by the geometric mean. Usage of the geometric mean also meant that the ordinal ranking across countries would not change if the maximum used for normalizing changed therefore the pegging of a maximum to a goalpost was done away with. Note that this was an advantage of the method, but not a requirement to begin with, definitely not when millennium development goals that can influence the various outcomes that are of relevance in the measure of HDI are themselves pegged to a goalpost.

We propose another alternative method of aggregation by taking the additive inverse of the distance from the ideal. This method also addresses the perfect substitutability across dimensions. In addition, this proposed method satisfies two other conditions. One is that the emphasis across dimensions should be based on their proportionate shortfall from the ideal (note that this ideal is a goalpost and not be understood as a transcendental ideal) or is shortfall sensitive. The other is that the same gap should be considered worse-off at higher levels of attainment. Or, simply put the gaps should decrease as attainment increases.

We impose a set of intuitive properties as axioms. They are as follows.

Monotonicity (M): An increase/decrease in the value of any of the three indicators, value of other two indicators remaining constant, will lead to an increase/decrease in the value of HDI.

Anonymity (A): If there are two situations where the values of the indicators get interchanged then they will give the same HDI value. For instance, if in one situation health is 0.4 and education is 0.6 while in another situation heath is 0.6 and education is 0.4 then both these will give the same HDI value if income remains same. This is a simple symmetry condition and not to be considered as subsitutability.

Normalization (N): If all the three indicators have zero value then the HDI value should be zero while if all the indicators have unity value then the HDI value should be unity. The extremes representing no development and full development, respectively.

Uniformity (U): For a given average attainment, a greater deviation should give a lower HDI value. This is in consonance with uniform development across dimensions, which are considered intrinsic.

Shortfall sensitivity (S): The emphasis for future progress across dimensions should be at least in proportion to the shortfall from the ideal. This means that if education is 0.7 and health is 0.1 (their shortfalls being 0.3 and 0.9) then health should get at least three times the emphasis that is given to education in our future enhancement of capabilities.

Hiatus sensitivity to level (H): The same gap across dimensions at a higher level of attainment should be considered worse-off. Suppose at an average value across dimensions of 0.5 we have education at 0.6 and health at 0.4 and when this average value across dimensions increases to 0.7 then we have education at 0.8 and health at 0.6 then the gap remains same. It has not narrowed down with an increase in attainment and we consider this as worse-off. A corollary in a single dimension will be like this. Consider gender differential in literacy attainment to be 10 percentage points when average literacy rate is 50 per cent. Now, the gender differential remains the same even when literacy rate increases to 80 per cent. Thus, an increase in literacy attainment did not narrow down the differential and we consider this as worse-off.

It so turns out the linear averaging method of aggregation satisfies the first three axioms (or MAN axioms), the geometric mean method of aggregation satisfies the first fours axiom (or MANU axioms, of course, montonicity fails when one of the dimension values is at zero and continues to be so while the others dimension increase) while our proposed method (the displaced ideal method) satisfies all the six axioms (or MANUSH axioms).

We also propose an α-class of measures where special cases turn out to be the linear averaging method when α=1 and our proposed displaced idea method when α=2. Further, when α≥2 then these class of measures satisfy the MANUSH axioms.

Our proposed class of measures can be used in different contexts. It can also consider the dimensions as subgroups. Under such an interpretation, the shortfall sensitivity axiom and the related discussions gives two extremes. One is the leximin ordering (like the Rawlsian scenario) where entire emphasis should be given to the neglected subgroup and then equal emphasis needs to be given after they are equal. The other extreme is to give the entire increment to the better off and leave the worse-off subgroup at their subsistence level till the better off group reaches its maximum and after that efforts can be put to increase the capabilities of the worse-off group. The problem with the latter suggestion is that the better-off group will never reach its maximum so the worse-off will continue to be at its subsistence level. In popular understanding, the two extremes may be referred to as ‘left’ and ‘right’,[3] respectively. To address these extreme positions, many proponents have suggested scale invariance or translation invariance as possible reconciliatory approaches. However, these options would still keep convergence across subgroups at bay. Hence, a proportionate to shortfall approach should be considered as an intermediary position. Of course, we are aware that implementation at the ground level might be different from this measurement exercise, but nevertheless, this will facilitate our understanding.

The word MANUSH means human in many South Asian Languages such as Assamese, Bengali, Marathi and Sanskrit among others. Besides, MANUSH can be rearranged to HUMANS. Thus, we propose the axiom of MANUSH or HUMANS for a human development index.

Prior to 2010, the approach followed to aggregate was a simple averaging across dimensions. A problem with this method was that a deficit in one dimension will perfectly substitute an equal attainment in another dimension. Income remaining same, this means that a country where both health and education attainments have the same value (say, 0.4 each) will have the same HDI as another country where health is 0.2 and education is 0.6 (a situation not quite uncommon in some of the Sub-Saharan countries reeling under a HIV/AIDS epidemic a few years ago).[2]

In 2010, to address perfect substitutability across dimensions, the calculation of HDI was aggregated by the geometric mean. Usage of the geometric mean also meant that the ordinal ranking across countries would not change if the maximum used for normalizing changed therefore the pegging of a maximum to a goalpost was done away with. Note that this was an advantage of the method, but not a requirement to begin with, definitely not when millennium development goals that can influence the various outcomes that are of relevance in the measure of HDI are themselves pegged to a goalpost.

Source: Human Development Index, The Encyclopedia of Earth |

We propose another alternative method of aggregation by taking the additive inverse of the distance from the ideal. This method also addresses the perfect substitutability across dimensions. In addition, this proposed method satisfies two other conditions. One is that the emphasis across dimensions should be based on their proportionate shortfall from the ideal (note that this ideal is a goalpost and not be understood as a transcendental ideal) or is shortfall sensitive. The other is that the same gap should be considered worse-off at higher levels of attainment. Or, simply put the gaps should decrease as attainment increases.

We impose a set of intuitive properties as axioms. They are as follows.

Monotonicity (M): An increase/decrease in the value of any of the three indicators, value of other two indicators remaining constant, will lead to an increase/decrease in the value of HDI.

Anonymity (A): If there are two situations where the values of the indicators get interchanged then they will give the same HDI value. For instance, if in one situation health is 0.4 and education is 0.6 while in another situation heath is 0.6 and education is 0.4 then both these will give the same HDI value if income remains same. This is a simple symmetry condition and not to be considered as subsitutability.

Normalization (N): If all the three indicators have zero value then the HDI value should be zero while if all the indicators have unity value then the HDI value should be unity. The extremes representing no development and full development, respectively.

Uniformity (U): For a given average attainment, a greater deviation should give a lower HDI value. This is in consonance with uniform development across dimensions, which are considered intrinsic.

Shortfall sensitivity (S): The emphasis for future progress across dimensions should be at least in proportion to the shortfall from the ideal. This means that if education is 0.7 and health is 0.1 (their shortfalls being 0.3 and 0.9) then health should get at least three times the emphasis that is given to education in our future enhancement of capabilities.

Hiatus sensitivity to level (H): The same gap across dimensions at a higher level of attainment should be considered worse-off. Suppose at an average value across dimensions of 0.5 we have education at 0.6 and health at 0.4 and when this average value across dimensions increases to 0.7 then we have education at 0.8 and health at 0.6 then the gap remains same. It has not narrowed down with an increase in attainment and we consider this as worse-off. A corollary in a single dimension will be like this. Consider gender differential in literacy attainment to be 10 percentage points when average literacy rate is 50 per cent. Now, the gender differential remains the same even when literacy rate increases to 80 per cent. Thus, an increase in literacy attainment did not narrow down the differential and we consider this as worse-off.

It so turns out the linear averaging method of aggregation satisfies the first three axioms (or MAN axioms), the geometric mean method of aggregation satisfies the first fours axiom (or MANU axioms, of course, montonicity fails when one of the dimension values is at zero and continues to be so while the others dimension increase) while our proposed method (the displaced ideal method) satisfies all the six axioms (or MANUSH axioms).

We also propose an α-class of measures where special cases turn out to be the linear averaging method when α=1 and our proposed displaced idea method when α=2. Further, when α≥2 then these class of measures satisfy the MANUSH axioms.

Our proposed class of measures can be used in different contexts. It can also consider the dimensions as subgroups. Under such an interpretation, the shortfall sensitivity axiom and the related discussions gives two extremes. One is the leximin ordering (like the Rawlsian scenario) where entire emphasis should be given to the neglected subgroup and then equal emphasis needs to be given after they are equal. The other extreme is to give the entire increment to the better off and leave the worse-off subgroup at their subsistence level till the better off group reaches its maximum and after that efforts can be put to increase the capabilities of the worse-off group. The problem with the latter suggestion is that the better-off group will never reach its maximum so the worse-off will continue to be at its subsistence level. In popular understanding, the two extremes may be referred to as ‘left’ and ‘right’,[3] respectively. To address these extreme positions, many proponents have suggested scale invariance or translation invariance as possible reconciliatory approaches. However, these options would still keep convergence across subgroups at bay. Hence, a proportionate to shortfall approach should be considered as an intermediary position. Of course, we are aware that implementation at the ground level might be different from this measurement exercise, but nevertheless, this will facilitate our understanding.

The word MANUSH means human in many South Asian Languages such as Assamese, Bengali, Marathi and Sanskrit among others. Besides, MANUSH can be rearranged to HUMANS. Thus, we propose the axiom of MANUSH or HUMANS for a human development index.

This blog post is based on a working paper working paper co-authored with Hippu Salk Kristle Nathan. In September 2014, this was further revised and put up as another working paper Measuring HDI - the Old, the New and the Elegant: Implications for Multidimensional Development and Social Inclusiveness at the Asia Research Centre, London School of Economics and Political Science (LSE). One can also listen to podcasts of presentation at OPHI or CASE Social Exclusion Seminars at LSE.[4]

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[1] The scales used to measure available information, the normalizing of these information to indicators, the weights given to each of these indicators or their sub-components are important aspects but beyond the scope of the current exercise.

[2] HIV/AIDS denote human immunodeficiency virus/acquired immunodeficiency syndrome.

[3] Note that the Rawlsian scenario identified with the left is not an extreme scenario that takes away from one subgroup in favour of another subgroup. That kind of left extremism, analytically speaking, would be a mirror opposite of the current right. In short, the Rawlsian position is actually a middle position akin to respecting pluralism. This can be achieved through a leximin ordering or by following a proportionate to shortfall approach. These two as also all possibilities within these should be considered as possible Rawlsian approaches.

[4] The antecedents of this works goes back to more than six/seven years and led to two working papers in 2008, viz., An Alternative Approach to Measure HDI, and On a Class of Human Development Index Measures, and a publication in 2010 Progress in Human Development: Are we On the Right Path? (see working paper version). A related work that we propose to do is the Inclusiveness of Human Development in India; also see video below.

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