On Distributional change, Pro-poor growth and Convergence, with an Application to Non-Income Dimensions in India

Transcription

1 On Distributional change, Pro-poor growth and Convergence, with an Application to Non-Income Dimensions in India Shatakshee Dhongde Georgia Institute of Technology, U.S.A Jacques Silber Bar-Ilan University, Israel A preliminary version of this paper was presented at the 33rd General Conference of the International Association for Research in Income and Wealth (IARIW) that took place in Rotterdam, the Netherlands, on August 24-30, The authors are thankful to Natalie Quinn for her thoughtful discussion of this paper and to Stephan Klasen for his cogent comments. A more recent version of this paper was presented at a seminar given at Beijing Normal University on November 12, The authors are grateful to Li Shi for his very useful remarks. The paper was also presented at the meetings of the Southern Economic Association in Atlanta, November and the authors wish to thank participants for their comments. 1

2 Abstract Using the concept of relative concentration curve this paper proposes new graphical tools and indices to measure distributional change, σ- and β-convergence and the pro-poorness of income growth. A distinction is made between a non-anonymous and an anonymous analysis. The approach is then extended to compute the income pro-poorness of the growth rates of other characteristics, such as education and health. An empirical illustration based on state-wide Indian data on literacy and infant survival levels in 2001 and 2011 highlights the usefulness of the proposed measures. It appears that growth in literacy and infant survival shares was relatively higher in poorer states. Key Words: β-convergence - σ-convergence - Gini index India infant mortality - literacy pro-poor growth - relative concentration curve J.E.L. Classification: D31 I32 O15 2

3 Introduction During the past two decades numerous papers have been written on the convergence hypothesis and there seem to be three competing hypotheses. The first approach, the absolute convergence hypothesis, argues that per capita incomes of countries will converge to one another in the long-run, whatever the initial conditions prevailing in these countries. An alternative view, the conditional convergence hypothesis, contends that there will be convergence, independently of initial conditions, of the per capita incomes of countries only if the structural characteristics (e.g. preferences, technologies) of these countries are identical. Finally there is the concept of club convergence according to which the per capita incomes of countries will converge only if they are identical in their structural characteristics and if their initial conditions are similar. 1 As far as empirical investigations of convergence are concerned, several measures have been proposed to measure such a convergence. There is thus the notion of σ-convergence which generally amounts to computing the standard deviation of the logarithms of incomes at two different times and checking whether the dispersion of incomes decreased over time, in which case the gap between rich and poor countries would have become smaller. But there is also the concept of β -convergence which checks whether countries with lower GDP per capita tend to grow faster than richer ones. Here a distinction is made between a convergence analysis that is unconditional and one that is conditional on a set of relevant variables. While the focus of the literature on convergence is on countries, there exists a somehow similar literature examining to what extent growth has been pro-poor, but here the emphasis is on individuals, not countries. Several definitions have been in fact proposed to 1 See Galor (1996) for details on the distinction between three ways of defining convergence 3

4 characterize pro-poor growth. 2 Some argue that growth is pro-poor when it raises the incomes of the poor (UN, 2000, Ravallion, 2004) whereas others believe that growth can be labeled pro-poor only if it raises the incomes of poor proportionately more than it raises the average income in society (Kakwani, 2004). Furthermore, it is important to make a distinction between studies of pro-poor growth that take an anonymous approach in the sense that they are usually based on cross-sections and do not follow individuals over time and works based on panel data that do not assume such anonymity. These two approaches may in fact lead to conflicting results, as emphasized by Grimm (2007) and Nissanov and Silber (2009). Note also that while studies of pro-poor growth generally look at developing countries and, as a consequence, take an absolute approach to the definition of the poverty line (that is, they assumed a constant poverty line in real terms), when looking at poverty in developed countries (see Deutsch and Silber, 2011) the poverty line is generally defined in relative terms (that is, the poverty line is assumed to be equal to some percentage of the median or mean standardized income). Essama-Nssah and Lambert (2009) suggested a common analytical framework to analyze pro-poor growth which allowed them to decompose their proposed measures of propoorness across income sources or consumption expenditure components. Such a breakdown has important policy implications because, for example, it may help identifying income sources which may be anti-poor. While the previously mentioned studies focused mainly on the income dimensions of poverty, Klasen (2008) and Grosse et al. (2008) extended the analysis to non-income dimensions, deriving growth incidence curves and related pro-poor growth measures to non-income indicators in the domains of education, health and nutrition. More recently 2 See Klasen (2008) for details on different notions of pro-poor growth 4

5 Bérenger and Bresson (2012), using the concept of sequential stochastic dominance, suggested a new way of testing the pro-poor nature of growth for poverty measures, based on both income and other characteristics such as education. The aim of this paper is to suggest a unified approach which links these apparently different notions of convergence and pro-poorness. We use the framework of distributional change and concentration curves to propose indices which measure inequality, pro-poorness of growth and convergence. In addition the present study follows in a certain way the approach taken by Grosse et al. (2007) because it also proposes new tools of analysis to determine whether growth in non-income indicators was pro-poor. This is an important issue because the growing literature on multidimensional poverty has shown that an increase in income does not guarantee an improvement in human development. Reducing income poverty is one of the eight Millennium Development Goals (MDGs). 3 Other MDGs emphasize universal primary education, promoting gender equality and improvement in maternal health, ensuring environmental sustainability and so on. It is hence important to develop a tool box to monitor changes in the distribution of these non-income indicators and their eventual link to income growth. Starting from the concept of distributional change, this paper makes first a distinction between a non-anonymous and an anonymous analysis of pro-poor income growth based on the notion of the elasticity of non-anonymous (anonymous) income at time 1 with respect to the corresponding non-anonymous (anonymous) income at time 0, such an elasticity being measured via the concept of relative concentration curve (see, Kakwani, 1980 for details). These concepts of distributional change and of elasticity of income at one period with respect of that of another period are shown to be related to the notions of σ- and β- convergences. This approach is then extended to compute non-anonymous and anonymous 3 5

6 growth rates of another characteristic, such as the level of education, with respect to income. Each of the concepts previously mentioned is represented by some index related to the famous Gini index and graphical illustrations derived from the concept of Lorenz and relative concentration curves are also given. Empirical illustrations based on Indian state level data confirm the usefulness of the proposed approach. 4 We compile data on adult literacy and infant mortality, two key indicators in the Millennium Development Goals (MDGs). The second MDG emphasizes achieving universal primary education. In India, overall literacy rates have increased from 52 percent in 1991 to about 74 percent in Reducing child mortality is another MDG; the goal is to reduce by two thirds, between 1990 and 2015, the under-five mortality rate in member countries of the United Nations. Compared to the rest of the world, child mortality rates in India have been significantly higher, primarily because of high infant mortality rates. 5 In 1990, the infant mortality rate in India was 80 per 1000 and it declined to about 44 per 1000 in India needs to reduce the rate further to 27 per 1000 by 2015, in order to achieve the MDG. There exists significant regional variation in literacy rates as well as infant mortality rates in India as seen in Figure 1 below. For example, in 2011, infant mortality rate was the lowest (11 per 1000) in Goa and highest (59 per 1000) in Madhya Pradesh. Only a handful of states such as, Goa, Kerala and Tamil Nadu in the south and Manipur and Sikkim in the east had rates lower than the target (27 per 1000). On the other hand, many states (Assam, Madhya Pradesh, Meghalaya, Odisha, Tamil Nadu, Uttar Pradesh) had rates higher (more than 50 per 1000) than those in some of the poorest Sub-Saharan African countries such as Ethiopia, Malawi, Namibia, Rwanda. 4 We use data at the state and not individual level because our approach is particularly useful when the number of observations is relatively small. In such a case traditional econometric approaches to convergence analysis cannot be used. The methodology suggested in the present paper allows one to study convergence and pro-poorness issues even when the number of observations is limited. 5 The infant mortality rate measures the number of children (aged less than one year) who die per 1000 live births 6

7 Figure 1.A. Literacy Rates across Indian states Figure 1.B. Infant Mortality Rates across Indian states The paper is organized as follows. In Section 2, we define indices measuring the inequality in income growth as well as the elasticity of income at time t with respect to income at a previous period. A graphical illustration of these indices is also provided. In Section 3, we extend these indices to measure distributional change in non-income dimensions. An 7

8 empirical application based on Indian data on literacy and infant mortality is then given in Section 4 while Section 5 summarizes the results of our analysis. 2. Measures of Distributional Change for Income Among the numerous algorithms which have been proposed to compute the famous Gini index (see, Yitzhaki, 1998), some have expressed this index in a matrix form (see, Pyatt, 1976, and Silber, 1989). Silber (1989) expresses the Gini index of income inequality I G as I G = e Gs (1) where e is a 1 by n row vector whose elements are the individual population shares ( 1 ), s is n a n by 1 column vector of the income shares s i and G is a n by n square matrix whose typical element g ij is equal to 0 if i = j, to -1 if j > i and to +1 if i > j, n being the number of individuals (or states, countries and so on). Note that the income shares s i have to be ranked by decreasing value of income. The G-matrix approach has been applied to various domains. For instance, Dawkins (2004, 2006) uses it to formulate a Gini index of residential segregation. In this paper, we extend the use of the G-matrix to formulate measures of distributional change, convergence and pro-poor growth Inequality of non-anonymous income growth rates Silber (1995) has shown that the G-matrix approach could be extended to the measurement of distributional change. He proposed two measures of distributional change, one J GP, a population weighted measure, and another one, J GI, which is income-weighted and is defined as follows. Let us call s 0 the vector of the income shares s 0i at time 0 and let us assume that these shares are ranked by decreasing values. Similarly let us call s 1 the vector of the income shares s 1i at time 1, the shares being ranked by their values at time 0. In other words if s 0i is the share of an individual who has rank i at time 0 (remember that rank 1 is given to the richest individual), then s 1i is the income share at time 1 of the individual who 8

9 had rank i at time 0. Below we provide an illustration with 5 individuals and their income shares at time 0 and time 1. Income at time 0 Income share at time 0 (vector s 0 ) Income at time 1 Income share at time 1 (vector s 1 ) A B C D E TOTAL Let us now define a row vector σ 0 of the income shares s oi, these shares being now ranked by decreasing values of the ratios (s 1i s 0i ). Similarly call w 1 the column vector of the income shares s 1i, these shares being also ranked by decreasing values of the ratios (s 1i s 0i ).The ratio (s 1i s 0i ) may also be expressed as (s 1i ) = (y 1i ny 1 s 0i ) = (y 1i y 0i ) = 1+( y i y 0i ) (y 0i ny 0 ) (y 1 y 0 ) 1+( y y 0 ) (2) where y ti is the income of individual i at time t, y t the average income at time t, y i = (y 1i y 0i ) and y = (y 1 y ). 0 The product σ 0 Gw 1 measures in a certain way the inequality of the (non-anonymous) growth rates ( y i y 0i ) and is in fact the measure of distributional change J GI proposed by Silber (1995). Ranked by decreasing values of ) (s 1i s 0i Vector σ 0 Vector w 1 E B D A C σ 0 Gw 1 = 0.45 The index σ 0 Gw 1 may be given a graphical representation. On the horizontal axis we plot the cumulative values of the shares σ 0i, the shares being ranked by increasing values of the ratios (s 1i s 0i ). Similarly on the vertical axis we plot the cumulative values of the shares 9

10 w 1i, the shares being again ranked by increasing values of the ratios (s 1i s 0i ). The index σ 0 Gw 1 is equal to twice the area lying between the curve obtained and the diagonal Elasticity of the non-anonymous incomes: Unconditional β - convergence Let us now continue our analysis of the non-anonymous distributional change and assume that the shares s 0i are ranked by decreasing values, which gives us the row vector s 0. Similarly let us also rank the shares s 1i by decreasing values of the income shares they had at time 0, that is, by decreasing values of the shares s 0i. This gives us the column vector s 1, as it was defined previously. The product s 0 Gs 1 is then a measure of the elasticity of the non-anonymous incomes at time 1 with respect to the values of the non-anonymous incomes at time 0. If this product is positive, it will show that, as a whole, the higher the income share at time 0, the higher the corresponding non-anonymous income share at time 1; if it is negative it will show that the lower the income shares at time 0, the higher the nonanonymous income shares at time 1. Ranked by decreasing values of (s 0i ) Vector s 0 Vector s 1 A B C D E s 0 Gs 1 = Here again we obtain a graphical representation, by plotting on the horizontal axis the cumulative values of the shares s 0i and on the vertical axis the cumulative values of the shares s 1i, both sets of shares being ranked this time by increasing values of the shares s 0i. The curve obtained may cross the diagonal once or several times. It can be shown that if any area below the diagonal is given a positive sign and any area above the diagonal a negative sign, then the sum of these signed areas is equal to half the value of the index s 0 Gs 1. 10

11 It is clear that the index expressed as s 0 Gs 1 measures in a certain way the degree of β - convergence. More precisely when s 0 Gs 1, which varies between -1 and +1, is negative, it indicates that the income growth rates of the poor were higher than those of the rich, so that there was β - convergence. In other words if s 0 Gs 1 is negative, we can say that nonanonymous growth was pro-poor. 6 On the contrary when s 0 Gs 1 is positive, it shows that the income growth rates were higher for the rich than for the poor so that there was income divergence and the non-anonymous growth was somehow pro-rich Inequality of anonymous income growth Until now the whole analysis has been non-anonymous in the sense that we have always compared the income share of an individual at time 1 with his/her income share at time 0. We can however implement an anonymous analysis (which we have evidently to implement if we do not have panel data but only two different cross sections, with the same number of individuals 7 ) in the sense that we would always compare the income of an individual who had rank i at time 1 with the income of the individual who had rank i at time 0, these individuals being generally different. Ranked by decreasing value of s 0 Income share at time 0 (vector s 0 ) Ranked by decreasing value of η 1 A 0.40 E B 0.32 B C 0.20 A D 0.06 C E 0.02 D TOTAL Income share at time 1 (vector η 1 ) Let us now first call, as before, s 0 the vector of the shares s 0i, the latter being ranked by decreasing values at time 0. Similarly let us call η 1 the vector of the shares s 1i, the latter being ranked by decreasing values at time 1. The typical share of the vector η 1 will be 6 Note that we use the term pro-poor growth in a broad sense so that as soon as our index is negative it is pro-poor, even if most of the area above the diagonal may not concern the very poor. 7 If, as is generally the case, the number of observations in both cross-sections is different, it is always possible to draw a random sample of the same size n, from each cross-section. 11

12 denoted as η 1i. Let us now call τ 0 (with typical share τ 0i ) the vector of the shares s 0i where the latter are ranked by decreasing ratios η 1i s 0i and ϕ 1 the vector of the shares η 1i where the latter are also ranked by decreasing ratios η 1i s 0i. The product τ 0 Gϕ 1 measures in a certain way the inequality in the anonymous growth rates, that is, of the growth rates obtained when we compare the i th shares at time 0 and 1, this comparison being done for each i. It should be stressed here that the higher the value of the index τ 0 Gϕ 1, the greater the inequality in the anonymous growth rates (growth rates of the various centiles). A graphical representation of the index τ 0 Gϕ 1 is shown below. Ranked by decreasing values of η 1i s 0i Vector τ 0 Vector ϕ τ 0 Gϕ 1 = Elasticity of the anonymous incomes: Unconditional σ-convergence Finally if we compute the product s 0 Gη 1 we compute the elasticity of the anonymous income shares at time 1 with respect to the anonymous income shares at time 0. In a way the index s 0 Gη 1 corresponds to the concept of σ- convergence. If s 0 Gη 1 is negative, the growth rates of the poor (which may be different individuals at time 1 and 0) were generally higher than those of the rich, so that inequality decreased while if s 0 Gη 1 is positive, growth was pro-rich so that inequality increased. In the present case the poor are not necessarily the same in periods 0 and 1, and similarly for the rich. 12

13 Vector s 0 Decreasing values of (s 0i ) Vector η 1 Decreasing values of (s 1i ) s 0 Gη 1 = Measures of Distributional Change for Non-Income Indicators Assume now that in addition to knowing the incomes at times 0 and 1, we also know, for instance, the educational levels at times 0 and Inequality of Growth and Elasticity for Non-Income Indicators We apply the same definitions of the four indices that have been given in section 2, and compute again four different Gini-related indices. To simplify the notations, we use the same names for the vectors as those used previously in the case of income, but apply them now to educational levels. In the non-anonymous case, we first compute the degree of inequality in the growth rates in educational levels, what was called σ 0 Gw 1 in section 2, when we analyzed the income distributional change. Then we compute an index measuring the degree of β-convergence, that is, non-anonymous pro-poor education growth, what was called s 0 Gs 1 in section 2 in the case of income. In the anonymous case we similarly first compute the degree of inequality in the anonymous growth rates in educational levels of the various centiles, what was called τ 0 Gϕ 1 in section 2 when we analyzed income distributional change. Finally we compute an index measuring the degree of σ-convergence, that is, an index of anonymous pro-poor education growth. 13

14 This corresponds to the index s 0 Gη 1 that we computed in section 2 in the case of income growth Income pro-poorness of Growth in Non-income indicator Let us call µ 0 the vector of educational shares at time 0 ranked by decreasing income at time 0 and µ 1 the vector of educational shares at time 1 ranked by decreasing income at time 0. The product µ 0 Gµ is then an index which measures the relationship between growth rates 1 in educational levels and the corresponding incomes. If the index µ 0 Gµ is positive, it 1 means that the non-anonymous growth rates in educational levels were as a whole higher for higher incomes while if it is negative it implies that the growth rates in educational levels were generally higher for individuals with a low income. Finally let us call, as before, µ 0 the vector of educational shares at time 0 ranked by decreasing income at time 0 and θ 1 the vector of educational shares at time 1 ranked by decreasing income at time 1. The product µ 0 Gθ 1 is then an index which measures the relationship between the growth rates in educational levels of the various centiles and the corresponding incomes of these centiles Convergence conditional on income Starting with the non-anonymous case, we defined previously an index s 0 Gs 1 measuring somehow the degree of (non-anonymous) β-convergence in educational levels, and an index µ 0 Gµ measuring the relationship between the individual growth rates in educational levels 1 and the corresponding individual incomes. The difference between the former and the latter index may then be considered as a measure of the conditional (on income) β-convergence. Given that a negative index is a sign of pro-poorness, we can conclude that if this difference is negative (positive), the growth rates in educational levels were generally higher for 14

15 individuals having low (high) values of the other determinants (income excluded) of these growth rates in individual educational levels. Finally in the anonymous case, we defined an index s 0 Gη 1 measuring somehow the degree of σ-convergence in educational levels and an index µ 0 Gθ 1 measuring the relationship between the growth rates in educational levels of the various centiles and the corresponding incomes of these centiles. The difference between the former and the latter index may then be considered as a measure of the conditional (on income) σ-convergence. If this difference is negative (positive), we conclude that the growth rates in the educational levels of the various centiles were generally higher, the lower (higher) the level of the other non-income determinants of educational levels. Table 1 summarizes the notations used to define the various indices introduced in Sections 2 and Empirical Application We compile data on state-wide literacy rates and infant mortality rates from the two recent rounds of the Indian Census, namely 2001 and Table 2 lists state-wide shares in each dimension. 8 More populous states such as Maharashtra, and Uttar Pradesh had greater shares of the literate population as well as the number of surviving infants, compared to less populous states such as Sikkim, and Andaman and Nicobar Islands Weighted Estimates The Census provides data on state i s population(tp i ) and the literacy rates(lr i ). Using this data, we derive that state s share in total literate population. share of literate adults i = LR ixshare of TP i LR i xshare of TP i (3) 8 Data is available for all states (except Nagaland) and for 4 of the 7 union territories: / 15

16 Similarly, the Census provides data on infant mortality rates. In place of infant mortality rates, we compute child survival rates (as did Grosse et. al., 2008). 9 State i s infant survival rates (SR i ) are derived by taking a linear transformation of the state s infant mortality rates (MR i ) as (SR i = 1000 MR i ). Thus state i s share in number of infants who survived is then calculated using Census data on infant mortality rates (MR i ) and birth rates(br i ). share of survived infants i = SR ixbr i xshare of TP i SR i xbr i xshare of TP i (4) Equation (3) shows that a state s share in the literate population is weighted by its share in total population while Equation 4 indicates that a state s share of survived infants is weighted by its share in total live births. Table 3 contains estimates of the indices proposed in the previous section along with the bootstrap confidence intervals. In the non-anonymous case, we compare the share of a state in 2001 with the share of the same state in 2011, despite the fact that the ranking of the state over the years may have changed. On the other hand, in the anonymous case, we compare the share of a state which had rank i in 2011 with the share of the state which had rank i in 2001, these states being generally, but not necessarily, different. Inequality in the growth rates In the non-anonymous case, the index σ 0 Gw 1 is calculated by ranking state shares by decreasing values of the ratio(s 1i s 0i ). The index varies between 0 (perfect equality) and 1 (maximal inequality). As seen in Table 3, the Gini index of inequality for literacy growth rate is and for infant survival is In the anonymous case, the index τ 0 Gϕ 1, is calculated by ranking state shares by decreasing values of the ratio, η 1i s 0i. The values of the anonymous indices are close to the non-anonymous estimates; the bootstrap confidence intervals show that they are statistically significantly different from each other. Overall, we 9 An improvement in child mortality comes out as a lower value but this lower value is mathematically interpreted as deterioration. Since survival rates are positive entities the interpretation is easier and more intuitive. 16

17 find evidence suggesting low inequality among states in growth rates of literacy and infant survival. Elasticity over time Elasticity of literacy and survival levels in 2011 with respect to their values in 2001 is measured by estimating the index s 0 Gs 1 in the non-anonymous case and by estimating the index s 0 Gη 1 as the anonymous counterpart. The index varies between -1 and +1. The nonanonymous elasticity values are positive and small, for literacy and for infant survival. These values are however statistically significantly different from 0 which suggests mild β-divergence among states. In other words, states with lower (higher) shares in 2001 also had lower (higher) shares in The anonymous elasticity index measures the degree of σ - convergence. For both the indicators, we find that the elasticity values are small and close to the elasticity values in the non-anonymous case. In fact, we find that there is not much difference in the ranking of the states in the non-anonymous and anonymous case. A positive sign of the anonymous index suggests that there was no evidence of σ - convergence in either dimension. Income pro-poorness of the growth rates In order to measure the income pro-poorness of growth rates, we combine data on nonincome dimensions with data on state average incomes. We use the per capita net state domestic product at constant ( ) prices to measure state income levels. In the nonanonymous case, we rank all shares by decreasing values of state average incomes in In the anonymous case, we rank shares in 2001 by decreasing values of income in 2001 and shares in 2011 by decreasing values of income in 2011 (see Figure 2). If any area below the diagonal is given a positive sign and any area above the diagonal a negative sign, then the sum of these areas is equal to half the value of the respective indices µ 0 Gµ and μ 1 0 Gθ 1. The indices measure the relationship between growth rates in literacy or survival levels and 17

18 corresponding income levels. In the non-anonymous case, the curve for literacy and survival rates is largely above the diagonal and the index is negative (Table 3). For the anonymous case, the curve intersects the diagonal several times. The estimates of the indices are negative and though small in magnitude, the values are significantly different from 0. Thus in both cases, the estimated values indicate that growth in literacy and survival levels were slightly pro-poor income. Figure 2. Income Pro-poorness of Growth in Literacy and Infant Survival Graphical representation of the indexµ 0 Gµ 1 : Non-Anonymous Case Graphical representation of the indexµ 0 Gθ 1 : Anonymous Case Conditional Convergence in growth rates In the non-anonymous case, the indices s 0 Gs 1 and µ 0 Gµ both use the same data on states 1 shares. The difference between the two indices is that in the former, the shares are ranked by decreasing values of shares in 2001,(s 0i ) and in the latter, they are ranked by decreasing 18

19 values of income(y 0i ). If we take the difference between the two indices s 0 Gs 1 and µ 0 Gµ, 1 a negative difference suggests conditional (on income) β -convergence. We find that the difference between the two indices, for both indicators, is positive. Thus, growth rates in literacy levels were generally higher for states having high values of the other non-income determinants of these growth rates; same was true for survival levels. Thus the estimates suggest conditional on income β -divergence. In the anonymous case, we take the difference between s 0 Gη 1 and µ 0 Gθ 1 where a negative difference suggests conditional (on income) σ -convergence. As seen in Table 3, the difference is positive and statistically significantly different than 0, suggesting conditional σ -divergence in literacy and survival rates Shapley Decomposition In the discussion so far, we considered states shares in the literate population and in the number of infants who survived. A rise in a state s share in the number of literates may result from an increase in the literacy rates and/or an increase in the state s population share. Similarly a rise in a state s share in the number of surviving infants may be due to an increase in infant survival rates or in the state s share of births. In order to separate the effect of changes in population weights from the effect of changes in the values of the states indicators, we implement the Shapley decomposition procedure (see, Appendix A for details on this procedure). The results of the decomposition for the literacy indices are summarized in Table 4 and for the survival indices in Table 5. For each indicator, the decomposition does not vary much between anonymous and non-anonymous indices. However significant differences are observed between the two indicators. In the case of literacy, the contribution of the change in literacy rates to the overall change in the indices is more than 60 percent. Changes in the population weights are less important. However, in the case of survival indices, the contribution of changes in population weights is much more important (almost 90 percent) 19

20 than that of changes in infant survival rates. The evidence from the Shapley decomposition thus suggests the following. Firstly, the literacy rates among states changed (increased) significantly over the decade. Secondly, there was no such important change when analyzing infant survival levels. Most of the variation in states share of surviving infants was due to variation in the states birth shares. This is not surprising since changes over time in survival rates are by definition small when compared to changes in infant mortality rates Unweighted Estimates We undertook the Shapley value decomposition in order to isolate the impact of changes in literacy and survival rates. Another way of dealing with this issue is to estimate unweighted indices, giving equal weight to each state, regardless of its population size. These estimates are given in Table 6. We find that the weighted and the unweighted inequality indices are comparable for literacy since as seen in the previous subsection, the change in population weights did not play much of a role in explaining the change states share of literates. For survival rates, the unweighted inequality estimates are almost close to zero since the survival rates did not vary significantly over time. For both indicators, the unweighted indices show evidence of (conditional as well as unconditional) β - and σ - convergence among states. Although there is no evidence on convergence when we use weighted indices, we do not believe this is a contradiction because the weighted and unweighted indices measure different types of shares. The weighted indices show that states share in the number of literates (and infants surviving) did not converge over time, whereas the unweighted indices show that states literacy (and infant survival) rates (as a percent of state population) did converge. 10 Note that the same issue arises with literacy since we could have worked with illiteracy rather than literacy rates. We estimate all the indices, using the infant mortality rates and discuss these in Appendix B. 20

21 5. Conclusions In this paper we first introduced new graphical tools to detect β-convergence and σ- convergence even when the number of observations is small. Traditional regression based approaches to test convergence require numerous observations on individual data. A novel feature of our approach is that our indices can be used when data is limited and available in aggregate form such as deciles. We proposed Gini-related indices measuring these two types of convergence (divergence) and showed the link between β-convergence and a nonanonymous analysis of distributional change on one hand, and between σ-convergence and an anonymous analysis of distributional change, on the other hand. Second we gave a graphical interpretation to the concept of income pro-poorness of the growth rates of non-income variables, both in the non-anonymous and anonymous cases. We also derived an index of such an income pro-poorness. Third we proposed a formulation for the computation of conditional (on income) β-convergence and σ- convergence of non-income variables and introduced Gini-related measures of these conditional on income convergences. Empirical illustrations based on the Census data on Indian states were presented to show the relevance of the concepts introduced in this paper. We analyzed distributional changes in two key indicators, namely literacy rates and infant survival rates between 2001 and We estimated unweighted indices as well as population weighted indices and used the Shapley decomposition on the latter. All estimates were provided with confidence intervals. Our results are important especially in the context of the Millennium Development Goals. The goals emphasize the role of non-income indicators in attaining sustainable development. There is significant variation across states in terms of literacy and infant mortality. Our analysis suggests that overall growth in literacy and survival rates was pro- 21

22 poor. Poorer states with low income made more progress in these dimensions. For example, Bihar which had the lowest per capita income in 2001, witnessed one of the fastest growth in the number of literates (24 percent) and in the number of infants surviving their first year (38 percent) between 2001 and There was also evidence on β - and σ -convergence. States with higher share in, say literacy rates in 2001 had a lower share in these rates in For instance, Kerala s share in literacy rate declined from 4.2 percent in 2001 to 3.9 percent in 2011, though it had one of the highest literacy rates (90 and 94 percent respectively) in India. In conclusion, we hope that the proposed indices provide a useful set of measures to analyze distributional changes in similar non-income indicators for policy purposes. 22

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24 Grosse, M., K. Harttgen and S. Klasen (2008) Measuring Pro-Poor Growth in Non-Income Dimensions, World Development 36(6): Kakwani, N. C. (1980) Income inequality and poverty: methods of estimation and policy applications, published for the World Bank by Oxford University Press. Kakwani N., S. Khandker and H. H. Son (2004) "Pro-Poor Growth: Concepts and Measurement with Country Case Studies," Working Paper Number 1, International Poverty Centre, Brasilia. Klasen S. (2008) Economic Growth and Poverty Reduction: Measurement Issues using Income and Non-Income Indicators, World Development 36(3): Lambert, P. and B. Zheng (2010) On the Consistent Measurement of Achievement and Shortfall Inequality, Working Paper 10-01, Department of Economics, University of Colorado Denver. Lasso de la Vega, C. and O. Aristondo (2012) Proposing indicators to measure achievement and shortfall inequality consistently, Journal of Health Economics 31(4): Nissanov, Z. and J Silber (2009) On Pro-Poor Growth and the Measurement of Convergence, Economics Letters 105: Pyatt, G. (1976) On the interpretation and disaggregation of the Gini coefficient, The Economic Journal 86(342): Ravallion, M. (2004) "Pro-Poor Growth: A Primer," World Bank Research Working Paper No Shorrocks, AF (2013), Decomposition Procedures for Distributional Analysis: A Unified Framework Based on the Shapley Value, Journal of Economic Inequality, 11, 1, pp Silber, J. (1989) "Factors Components, Population Subgroups and the Computation of the Gini Index of Inequality," The Review of Economics and Statistics LXXI:

25 Silber, J. (1995) "Horizontal Inequity, the Gini Index and the Measurement of Distributional Change," in C. Dagum and A. Lemmi (eds.), Income Distribution, Social Welfare, Inequality and Poverty, Vol. VI of Research on Economic Inequality, pp Silber, J. (forthcoming) On Inequality in Health and Pro-Poor Development: The Case of Southeast Asia, Journal of Economic Studies. United Nations (2000) A Better World for all, New York, United Nations. Yitzhaki, S. (1998) More than a dozen alternative ways of spelling Gini, Research on Economic Inequality 8:

26 Table 1: Summary of Proposed Indices of Distributional Change Notation Vector Ranked Index Type Measures Range Interpretation s 0 s 0i Decreasing values of the shares s 0i at time 0 s 1 s 1i Decreasing values of the shares s 0i at time 0 s 0 Gs 1 Nonanonymous Elasticity -1 to 1 If the index is negative, there is β-convergence; if it is positive there is β- divergence. σ 0 s 0i Decreasing values of s 1i s 0i w 1 s 1i Decreasing values of s 1i s 0i σ 0 Gw 1 Nonanonymous Inequality 0 to 1 0: Perfect equality 1: Maximal inequality η 1 s 1i Decreasing values of the shares s 1i at time 1 τ 0 s 0i Decreasing values of η 1i s 0i φ 1 η 1i Decreasing values of η 1i s 0i s 0 Gη 1 Anonymous Elasticity -1 to 1 If the index is negative, there is σ-convergence; if it is positive there is σ- divergence. τ 0 Gφ 1 Anonymous Inequality 0 to 1 0: Perfect equality 1: Maximal inequality μ 0 s 0i Decreasing values of incomesy 0i at time 0 μ 0 Gμ 1 Nonanonymous Income Propoorness -1 to 1-1: Pro-poor 1: Not pro-poor μ 1 s 1i Decreasing values of incomes y 0i at time 0 θ 1 s 1i Decreasing values of incomes y 1i at time 1 μ 0 Gθ 1 Anonymous Income Propoorness -1 to 1-1: Pro-poor 1: Not pro-poor 26

27 Table 2: State-wise shares in Literacy and Infant Survival Levels Literacy Infant Survival States AN Islands* Andhra Pradesh Arunachal Pradesh Assam Bihar Chandigarh* Chhattisgarh Goa Gujarat Haryana Himachal Pradesh Jammu Kashmir Jharkand Karnataka Kerala Madhya Pradesh Maharashtra Manipur Meghalaya Mizoram NCT of Delhi* Odisha Puducherry* Punjab Rajasthan Sikkim Tamil Nadu Tripura Uttar Pradesh Uttarakhand West Bengal Total Table 2 shows percent share of each state in total literate population and total number of infants who survived their first year. *Union Territories 27

28 Table 3: Weighted Estimates of Proposed Indices Inequality in growth rates Literacy Infant Survival Non Anonymous Anonymous Non Anonymous Anonymous (0.0422; ) (0.0317; ) ( to ) ( to ) Elasticity over time (0.0027; ) ( to ) ( to ) ( to ) Income Propoorness ( ; ) ( ; ) ( ; ) ( ; ) Conditional Convergence (0.0368; ) (0.0281; ) ( ) show 5% and 95% bootstrap confidence interval (0.0351; ) (0.0314; ) Table 4: Shapley Decomposition of Literacy Rate Indices Inequality in growth rates Change in literacy levels (60.5%) Non Anonymous Change in population (39.5%) Total value Change in literacy levels (61.5%) Anonymous Change in population (38.5%) Total value Elasticity over time (104%) (-4%) (86.4%) (13.6%) Income Propoorness (69.3%) (30.7%) (69%) (31%) Conditional Convergence (73.8%) ( ) shows % contribution (26.2%) (73 %) (27%)

29 Table 5: Shapley Decomposition of Infant Survival Indices Inequality in growth rates Change in survival levels (4.5%) Non Anonymous Change in population (95.5%) Total value Change in survival levels (4.3%) Anonymous Change in population (95.7%) Total value Elasticity over time (11.2%) (88.8%) (11.2%) (88.8%) Income Propoorness (4%) (96%) (2.9%) (97.1%) Conditional Convergence (5.9%) ( ) shows % contribution (94.1%) (5.3%) (94.7%) Table 6: Unweighted Estimates of Proposed Indices Inequality in growth rates Literacy Infant Survival Non Anonymous Anonymous Non Anonymous Anonymous (0.0268; ) (0.0224; ) (0.0054; ) ( ; ) Elasticity over time ( ; ) ( ; ) ( ; ) ( ; ) Income Propoorness ( ; ) ( ; ) ( ; ) ( ; ) Conditional Convergence ( ; ) ( ; ) ( ) show 5% and 95% bootstrap confidence interval ( ; ) ( ; ) 29

30 Appendix A: Shapley decomposition of the literacy and mortality indices Let I( w, l) be the value of one of the indices defined in sections 2 and 3. This index can thus measure the inequality in the growth rates of infant survival or literacy rates between times t = 0 and t = 1, the elasticity of the infant survival rate (literacy rate) at time 1 with respect to the infant survival rate (literacy rate) at time 0 or the degree of pro-poorness of the growth rates of infant survival or literacy rates between times 0 and 1. The index I depends first on the variation that took place over time in the weights used to compute it w, (state shares in the total amount of births in India, for the infant survival rates, and state shares in the total population in India for the literacy rates). But evidently this index I depends also on the change l that was observed over time in the state infant survival rates (literacy rates). Using the well-known principles of the Shapley decomposition (see Chantreuil and Trannoy, 2013, and Shorrocks, 2013 for details), we can define the contribution C w of the change w in the state weights as C w = 1 {[I( w 0; l 0)] [I( w = 0; l 0)]} 2 +( 1 ){[I( w 0; l = 0)] [I( w = 0; l = 0)]} 2 where I( w 0; l 0) means that it is assumed that both the state weights w and the state indicators l varied between times t and t, I( w = 0; l 0) means that it is assumed that the state weights w did not vary while the state indicators l varied between times t and t,and I( w 0; l = 0) means that it is assumed that the state weights w varied while the state indicators l did not vary between times t and t. Obviously I( w = 0; l = 0) will be equal to 0 since neither the weights nor the indicators varied. 30

31 We can similarly define the contribution C l of the change l in the state indicators as C l = 1 {[I( w 0; l 0)] [I( w 0; l = 0)]} 2 +( 1 ) {[I( w = 0; l 0)] [I( w = 0; l = 0)]} 2 It is then easy to check that the sum of both contributions,(c w + C l ) is in fact identical to the value of the index I( w 0; l 0) actually observed (de facto, between times t and t, both the state weights w and the state indicators l varied). Note that although C l isolates the impact of a change in literacy rates, it still depends on the state population weights. 31

32 Appendix B: Results based on Infant Mortality Table B1: Shapley Decomposition of Infant Mortality Indices Inequality in growth rates Change in mortality levels (45.3%) Non Anonymous Change in population (54.7%) Total value Change in mortality levels (55.3%) Anonymous Change in population (44.7%) Total value Elasticity over time (72.0%) (28.0%) (63.3%) (36.7%) Income Propoorness (63.2%) (36.8%) (43.9%) (56.1%) Conditional Convergence (66.3%) ( ) shows % contribution (33.7%) (52.3%) (47.7%) Table B2: Unweighted Estimates of Infant Mortality Non Anonymous Inequality in growth rates (0.0959; 0.135) Elasticity over time ( ; ) Income Pro-poorness (-0.011; ) Conditional Convergence (-0.114; ) ( ) show 5% and 95% bootstrap confidence intervals Anonymous (0.0467; ) ( ; ) ( ; ) ( ; ) Comparing Table 3 and Table B1, we note that the values of the indices using infant mortality are greater than the values based on survival rates, since the variation in mortality is much larger than the variation in survival rates. However, as expected, the qualitative results do not change; we do not find evidence of convergence and growth is largely pro-poor. The Shapley decomposition of the indices based on infant mortality confirms our intuition (Table B2). The decomposition shows that the 32

33 contribution of the change in infant mortality rates is greater than the contribution of the change in population shares. Several papers in recent years have emphasized the fact that when the variable analyzed is a bounded variable, like the infant mortality rate, one will not obtain the same measure of inequality when the computation of the inequality index is based on the infant mortality rate and when it is based on its complement, the infant survival rate. Solutions to this problem have been proposed by Erreygers (2009), Lambert and Zheng (2010), Lasso de la Vega and Aristondo (2012) and Chakravarty et al. (2013); Silber (forthcoming) reviews in detail these papers. In Table B3 below we provide a simple illustration of one of the solutions proposed by Lasso de la Vega and Aristondo (2012). We compute the index I r (x, B max ) = (I(x)I(B max 1 x)) (1 2 ) where x is the infant mortality rate, B max is equal to 1000 and the index I is the Gini index. Estimates in Table B-3 confirm the important impact of changes in states birth weights on the overall change in mortality (survival). Table B3: Shapley decomposition of the Lasso de la Vega and Aristondo index Case analyzed Non anonymous Contribution of changes in infant mortality and survival rates (27%) Contribution of changes in birth weights (73%) Index I r (x, B max ) (0.0498; ) Anonymous (24.2%) (75.8%) (0.0427; ) 33