Multi-dimensional Human Development Measures : Trade-offs and Inequality presented by Jaya Krishnakumar University of Geneva UNDP Workshop on Measuring Human Development June 14, 2013 GIZ, Eschborn, Frankfurt
Outline Introduction Multi-dimensional indices and their properties An extreme but practically imposing index Trade-offs Inequality and its contributing factors Concluding remarks
Introduction Objectives : propose an index which is simple to compute, which satisfies most theoretically desirable properties and which has tough policy implications discuss trade-offs in this and other less strict alternatives present a regression-based methodology for accounting for inequality in human development apply the methodology to a human development index Questions of interest How to choose a measure of Human Development? How to account for its inequality?
Main references Bourguignon, F. and S. Chakravarty (2003), Journal of Economic Inequality. Chakravarty (2003), Review of Development Economics. Krishnakumar, J. (2007), Journal of Human Development and Capabilities. Krishnakumar, J. and A.L. Nagar (2008), Social Indicators Research. Krishnakumar, J. and P. Ballon (2008), World Development. Krishnakumar, J. (2013), UNESCO Encyclopedia of Life Support Systems. Krishnakumar, J. (2013), Working paper. Maasoumi, E. and G. Nickelsburg (1988), Journal of Business and Economic Statistics. Seth, S. (2009), Journal of Human Development and Capabilities.
Multi-dimensional indices Any multi-dimensional index should address the following points. Many dimensions Multiple indicators within a dimension Observations at the individual (micro) level Aggregation over individuals Aggregation over dimensions Decomposability between groups Comparability over time Comparability across countries Welfare interpretation Inequality within a group/population Inequality between groups Inequality due to various factors
Multi-dimensional indices (contd.) Many dimensions: Say J dimensions (j = 1,..., J). Many indicators per dimension: Let us say we have n j indicators in dimension j, j = 1,..., J. Micro-level observations: We assume that we have observations on N individuals (i = 1,..., N). Let us first take a generic individual and see how to arrive at an aggregate index over dimensions. We can also imagine the individual to be a country.
Desirable properties Desirable properties for a uni-dimensional index. Normalisation: I (x j, m j, M j ) = { 0 if xj = m j 1 if x j = M j Monotonicity: Any increase in x j increases I ( ). Translation invariance : I (x j, m j, M j ) = I (x j + c, m j + c, M j + c), where c is any scalar such that m j + c 0. Homogeneity: I (x j, m j, M j ) = I (cx j, cm j, cm j ), for any c > 0. Concavity: I (x j + δ, m j, M j ) I (x j, m j, M j ) is decreasing in δ, where m j < x j + δ < M j. Let z j = x j m j M j m j z j satisfies all except Concavity a j = z r j j = 1,..., J 0 < r < 1 satisfies all of them.
Desirable properties (contd.) Properties for a multi-dimensional well-being index W. Continuity: W ( ) is continuous at all points. Normalisation: W (A) = z when A ij = z i, j. Linear homogeneity: W (λa) = λw (A), λ being a scalar. Symmetry in people: If individual achievements are permuted then W does not change Symmetry in dimension: Permuting dimensions does not change W. Population replication invariance: If A is replicated several times in a large population, then W does not change. Monotonicity: An increment in a single element of A (for one person in one dimension) without changing all the other elements increases W.
Desirable properties (contd.) Properties for a multi-dimensional index (contd.). Subgroup consistency: If the overall well-being of a group increases and everything else remains the same, W increases. Path independence: W takes the same value whether one aggregates over individuals first and then over dimensions or whether one does the other way around. Consistency in Aggregation (CIA) I (a 1 + b 1,..., a J + b J ) = I (a 1,..., a J ) + I (b 1,..., b J ) where a j is defined earlier as zj r and b j is similarly defined for another measure y j ( ) r y j m j b j = j = 1,..., J M j m j One can imagine a j and b j to be component measures.
Desirable properties (contd.) Let us define a general aggregation function I ( ): I = I (z 1,..., z J ) Some special cases: HDI (satisfies all properties except CIA) HDI = 1 J z j J j=1 Generalised HDI (satisfies all properties) Generalised HDI = 1 J a j = 1 J J j=1 J j=1 Generalised Mean (satisfies all properties except CIA) 1 Another Generalised HDI = 1 r J a j J j=1 z r j
A new proposal for consistency in aggregation A closer look at CIA ( ) xj m r ( ) j yj m r j a j + b j = + M j m j M j m j So sum (mean) of transformed normalised values. A new aggregation property : generalised CIA (G-CIA) I [I (a 1, b 1 ),..., I (a J, b J )] = I [I (a 1,..., a J ), I (b 1,..., b J )] We call this new property the generalised CIA as the function that is used to aggregate dimensions is also used to aggregate components of a dimension. The generalised mean (GM) satisfies G-CIA and all the other properties except for path independence. The GM has another important statistical property. The distribution of the generalised mean over dimensions in a population is the closest to each of the different dimensional distributions that compose this aggregate index.
Multiple indicators per dimension First of all, we emphasise the need for a multi-level approach: First aggregate indicators within a (sub-) dimension and then aggregate across (sub-)dimensions We recommend a dashboard approach like the one followed by the Joint Research Center of the European Commission (JRC Dashboard of Sustainability). But due to the high collinearity among indicators within a given dimension, it is more appropriate to use statistical procedures that take account of this and extract the underlying dimensional well-being. E.g. Principal components, Factor Analysis, MIMIC model, Structural Equation Models It is easy to combine statistical procedures for aggregating within dimensions with normative procedures such as generalised mean for aggregating across dimensions.
An extreme but practically imposing index: The Min Index An index which is highly demanding in terms of actual progress for its improvement. I (z 1,...z J ) = min{z 1,...z J } Interpretation: We take the worst performing dimension value as the index value. The min index satisfies all the properties 1 10 except for CIA (see below). Monotonicity and subgroup consistency are only exceptionally satisfied (when it is the value of the worst performing dimension that increases). It is one of the few aggregate indices that satisfy path independence.
The Min Index (contd.) Further, it satisfies our generalised CIA. min (min(z 1,..., z J ), min(y 1,..., y J ))) = min (min(z 1, y 1 ),..., min(z J, y J )) It is subgroup decomposable. Subgroup decomposability: I (A 1 (G 1 ), A 2 (G 2 )) = I (I (A 1 ), I (A 2 )) where G 1 is a group of individuals and G 2 is another group, and A 1 and A 2 are their respecting achievement matrices. For our min index we have: min(z 1, Z 2 ) = min(min(z 1 ), min(z 2 )) One can also have a weighted min version: I (z 1,..., z J ) = min(w 1 z 1,..., w J z J ) which will have the same properties as the unweighted version.
Welfare interpretation Welfare of a country = performance in its worst dimension If we take it to the micro-level, it is the (normalised) value of the most deprived person in her worst dimension. Dimensions are not substitutable. A good performance in one dimension cannot compensate a bad performance in another dimension. That is why we call it imposing. There is no way of showing improvement in the index unless there is an improvement of the situation of the most deprived person in her most deprived dimension. Policy implication: Short run : attend to the worst performing dimension Long-run: attend to all dimensions because any of them could become the worst otherwise!
A practical variant For practical purposes, if it is too strong a condition to take the value of the most deprived person in her most deprived dimension, then one can relax it a bit and consider the following variant. For policy targeting, one can take the most deprived group of individuals rather than the most deprived single individual and/or the most deprived set of dimensions rather than the most deprived single dimension. In this case it is better to take the weighted version as we are introducing substitutability between individuals and between dimensions that are among the worst ones.
Trade-offs Min Index : No Trade-offs Why? Because we are concerned with such fundamental dimensions as health, education etc. One cannot trade one for the other; one should have all these basic well-being aspects. Literature has an extensive discussion on whether the dimensions are substitutes or complements in the context of correlation increasing/decreasing transfers and their effect on HD. We believe this has no meaning for two reasons: We cannot treat the dimensions as substitutes as they are basic; similarly neither are they complements. This discussion is generally done for CES-type aggregates (generalised means). In this case, one can only have either ALL substitutes or ALL complements depending on the value of r. This cannot hold in reality. So better to consider them as independent. Note that this does not mean that the dimensions are not correlated but only that one cannot trade one for another.
Limited Trade-offs In general the trade-off between two dimensions j and k for a well-being index I (z 1,.., z J ) is given by the marginal rate of substitution MRS j,k = I (z 1,...,z J ) z j I (z 1,...,z J ) z k For the (unweighted) generalised mean this is equal to: MRS j,k = z k z j ( ) r 1 zj = = z k ( ) 1 r zk z j Thus we see that MRS depends on the levels of z j and z k. Various cases are possible.
Limited Trade-offs (contd.) Case 1. 0 < r < 1 or 1 r > 0 a. Better performance in dimension j relative to dimension k i.e. ( zj z k ) > 1. In this case, a unit loss in z j needs a compensation of less than unity in z k. Because society has more of z j and because substitutability is not high (0 < r < 1) (i.e. people are not willing to substitute), one unit of z k has a higher social value than that of z j. b. Better performance in dimension k relative to dimension j i.e. ( zj z k ) < 1. In this case, a unit loss in z j needs a compensation of more than unity in z k. Because society has more of z k and because substitutability is not high (0 < r < 1), one unit of z k has less social value than that of z j. Case 2. r > 1 i.e. r 1 > 0 Not socially acceptable
Market prices as trade-offs (contd.) If prices are available for dimensions, then market trade off will be given by MRS j,k = p j p k. Using prices as weights the social marginal rate of substitution will be: Say 0 < r < 1 a. If z j > z k then ( ) r 1 zj p j MRS j,k = = z k p k ( zj z k ) r 1 pj p k < p j p k ( zk z j ) 1 r p j p k i.e. social MRS is less than the market trade-off. Society values z j less than z k because it has more of z j compared to z k (and because they are not so substitutable) and hence the relative social value is less than the relative market value. b. If z j < z k then the opposite happens and the social MRS is more than the market trade-off as z j being the more deprived dimension in the society, a unit loss in z j has a bigger social value (loss) and more than the market trade-off is traded to compensate for it.
Inequality Let us now assume we have a satisfactory index of HD. Now we ask Q1. How unequally is well-being (measured by this index) distributed within and between countries? Q2. What are the contributing factors to this inequality? A1. Report directly the inequality in HD at the country level and at the global level (rather than adjusting it for taking account of inequality). There are many strongly Lorenz-consistent inequality indices such as generalised entropy measures and Atkinosn s index that satisfy many useful properties like decomposability into within and between groups. A2. Here we advocate the following regression-based approach based on Shorrocks, Fields, and Bigotta, Krishnakumar and Rani.
Inequality decomposition by contributing factors Let us say we explain HD by a certain number of economic, social, political, institutional, demographic and natural factors in the form of a regression. HD i = K k=1 β k X i,k + ε i = X i β + ε i i = 1,..., N Then the share s k of each explanatory variable X k as s k = cov(β kx k, HD) V (HD) k = 1,..., K (1) These shares are the same for any inequality measure provided the chosen measure satisfies Shorrocks six assumptions. A notable exception is Atkinson s index (see below).
Inequality decomposition by contributing factors (contd.) The asymptotic variances of the above shares can be derived as ( [ ]) d Σ Σι n (ŝ s) N 0; ι Σ ι Σι By an appropriate transformation, the shares for Atkinson s index are given by: s k = S k A H = f 1 ( S k ) A H = f 1 ( s k AH ) A H = f 1 ( s k f (A H )) A H The asymptotic variances of these shares are given by: Asy.V (s) = J[Asy.V ( s)]j say Q
What are the contributing factors to inequality to HD? HD i = β 1 + β 1Econ i + β 2Soc i + β 3Pol i + β 4Demo i + β 6Inst i + β 7Nat i + ε i, i = 1,..., N Economic: Investment, openness, credit access, urbanisation, infrastructure - energy, transport, ICT (internet, mobile etc.) Social: Public expenditure on health, education, hospital beds per capita, teacher/student ration, insurance coverage for old age, sickness, maternity, work injury, unemployment etc. Political: political stability, democracy autocracy measures, voice and accountability etc. Institutional: Government effectiveness, bureaucracy quality, control of corruption, rule of law etc. Demographic: percentage of older population, youth bulge, gender ratio etc. Natural: quality of air, carbon emissions, micro-pollutants, exposure to natural disasters etc.
Table: Illustration for income inequality in India Urban Rural Shares (%) Shares (%) 1983 1993-94 2004-05 1983 1993-94 2004-05 Age -5.3 *** -2.7 *** 0.8 *** 0.8 *** 1.5 *** 3.3 *** (0.022) (0.017) (0.024) (0.023) (0.020) (0.015) Gender 0.5 *** 0.1 *** 0.02 *** -0.02 *** - 0.1 *** (0.037) (0.008) (0.008) (0.002) (0.002) Household size 21.3 *** 18.8 *** 14.3 *** 7.0 *** 7.4 *** 11.7 *** (0.182) (0.060) (0.062) (0.031) (0.017) (0.019) Land Ownership 4.5 *** 4.3 *** 4.8 *** (0.041) (0.031) (0.020) Social Group 1.1 *** 0.7 *** 1.2 *** 3.1 *** 2.5 *** 2.3 *** (0.002) (0.025) (0.027) (0.028) (0.019) (0.011) Religion 0.8 *** 0.1 *** 0.3 *** 0.5 *** 0.3 *** 1.4 *** (0.051) (0.028) (0.030) (0.026) (0.018) (0.017) State dummies 3.2 *** 3.4 *** 2.8 *** 6.9 *** 9.3 *** 10.8 *** (0.093) (0.034) (0.036) (0.051) (0.031) (0.026) Household head s Industry 1.3 *** 1.0 *** 2.3 *** 2.4 *** 2.6 *** 1.6 *** (0.210) (0.082) (0.104) (0.043) (0.038) (0.029) Education 21.1 *** 26.3 *** 28.0 *** 8.1 *** 10.3 *** 11.2 *** (0.302) (0.088) (0.092) (0.194) (0.108) (0.053) Employment status 3.9 *** 3.2 *** 2.7 *** 4.2 *** 5.1 *** 4.2 *** (0.332) (0.060) (0.122) (0.100) (0.085) (0.051) Residuals 52.1 *** 49.1 *** 47.6 *** 62.5 *** 56.8 *** 48.5 *** (0.199) (0.075) (0.079) (0.059) (0.044) (0.036) Sum 100.0 100.0 100.0 100.0 100.0 100.0 Atkinson (e=2) 0.623 0.302 0.326 0.330 0.219 0.189
Concluding remarks A multi-dimensional HD index which is simple, powerful and theoretically desirable. Possibility of combining statistical and normative procedures in presence of multiple indicators per dimension. No trade-offs = Min index Limited trade-offs = Generalised mean index Comparison between market trade-offs and social trade-offs. Assess contributions to inequality in Human Development of possible causal factors using a regression-based method.