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On the Watts Multidimensional Poverty Index and its Decomposition
World Development Vol. 36, No. 6, pp. 1067–1077, 2008 Ó 2008 Elsevier Ltd. All rights reserved 0305-750X/$ - see front matter www.elsevier.com/locate/worlddev
doi:10.1016/j.worlddev.2007.10.003
On the Watts Multidimensional Poverty Index and its Decomposition SATYA R. CHAKRAVARTY Indian Statistical Institute, Kolkata, India JOSEPH DEUTSCH and JACQUES SILBER Bar-Ilan University, Ramat-Gan, Israel
*
Summary. — The multidimensional extension of the Watts poverty index may be expressed as a function of five determinants measuring, respectively, the impacts of what are defined in the paper as the Watts poverty gap ratio, the Theil–Bourguignon index of inequality among the poor, the overall headcount ratio, the weights of the various dimensions, and some measure of correlation between the various dimensions. Using the Shapley decomposition, we apply this index to world data on the per capita GDP, life expectancy, and literacy rates and derive the contributions of the five determinants defined above to the variation of this index during 1993–2002. Ó 2008 Elsevier Ltd. All rights reserved. Key words — decomposition, multidimensional poverty, Watts index
1. INTRODUCTION Removal of poverty has been and continues to be one of the primary aims of economic policy in a large number of countries. Therefore, targeting poverty alleviation is still a very important issue in many countries. It is thus necessary to know the dimensions of poverty and the process through which it seems to be aggravated. A natural question that arises in this context is how to quantify the extent of poverty. In a pioneering contribution, Sen (1976) regarded the poverty measurement problem as involving two exercises: (i) the identification of the poor and (ii) the aggregation of the characteristics of the poor into an overall indicator that quantifies the extent of poverty. In the literature, the first problem is mostly solved by the income method, which requires the specification of a poverty line representing the income required for a subsistence standard of living. A person is said to be poor if his income falls below the poverty line. On the aggregation issue, Sen (1976) criticized two crude indicators of poverty, the headcount ratio (proportion of
persons with incomes below the poverty line) and the income gap ratio (the difference between the poverty line and the average income of the poor, expressed as a proportion of the poverty line), because they remain unaltered under a transfer of income between two poor persons and the former also does not change if a poor person becomes poorer due to a reduction in his income. Sen (1976) also characterized axiomatically a more sophisticated index of poverty. 1 However, the well-being of a population and hence its poverty, which is a manifestation of insufficient well-being, is a multidimensional phenomenon, income being only one of the
* We are grateful to an anonymous referee for his/her comments and suggestions. We also thank him/her for bringing our attention to the equivalence between Eqns. (8) and (9). Chakravarty wishes to thank the Bocconi University, Milan, Italy, and Silber the Fundacio´n de ´ a Aplicada (FEDEA), Madrid, Estudios de Economı Spain, for support. Final revision accepted: October 5, 2007.
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many attributes on which the well-being depends. Examples of such attributes are food, housing, clothing, education, health, provision of public goods, and so on. While it is true that with a higher income a person is able to improve the position of some of his non-monetary attributes, it may as well be the case that markets for certain attributes do not exist, for instance, in the case of some public goods. Examples are flood control program and malaria prevention program in an underdeveloped country. (See Bourguignon & Chakravarty, 2003; Ravallion, 1996; Tsui, 2002). There are additional arguments for viewing poverty from a multidimensional perspective. In the basic needs approach development is regarded as an improvement in the array of human needs, not just as growth of income alone (Streeten, 1981). There is a debate about the importance of low income as a determinant of under nutrition (Lipton & Ravallion, 1995). In the capability-functioning approach, where functionings deal with what a person can ultimately do and capabilities indicate a person’s freedom with respect to functionings (Sen, 1985, 1992), poverty is regarded as a problem of capability failure. Functionings here are closely approximated by attributes like literacy, life expectancy, etc. An example of a multidimensional index of poverty in the capability failure framework is the human poverty index suggested by the UNDP (1997). It aggregates the deprivations in the living standard of a population in terms of three basic dimensions of life, namely, decent living standard, educational attainment rate, and life expectancy at birth. In view of the above discussion, we assume that each person is characterized by a vector of basic need attributes and a direct method of identification of poor checks if the person has ‘‘minimally acceptable levels’’ (Sen, 1992, p. 139) of this set of basic needs. Therefore, the direct method considers poverty from a multidimensional perspective, more precisely, in terms of shortfalls of attribute quantities from respective threshold levels. These threshold levels are determined independently of the attribute distributions. Since the direct method ‘‘is not based on particular assumptions of consumer behavior which may or may not be accurate,’’ ‘‘it is superior to the income method’’ (Sen, 1981, p. 26). If direct information on different attributes are not available, one can adopt the income method, ‘‘so that the income method is at most a second best’’ (Sen, 1981, p. 26).
Following earlier work on the characterization of the multidimensional extension of the Watts (1968) poverty index (see, Chakravarty & Silber, 2008; Maasoumi & Lugo, 2008; Tsui, 2002), we first (in Section 2) show that this multidimensional extension of the Watts poverty index may be expressed as a function of five determinants which are, respectively, what we define as the Watts income gap ratio, the Bourguignon (1979)–Theil (1967) index of inequality among the poor, the overall headcount ratio, the weights of the various dimensions, and some measure of correlation between the various dimensions. This decomposition should allow us to investigate the relationship between poverty and its determinants, more precisely, to identify the different causal factors of poverty. Then (in Section 3), using the well-known Shapley (1953) decomposition, we derive the contribution of each of these five determinants to the overall change in the Watts index. The paper ends (Section 4) by providing a numerical illustration based on data on the per capita GDP, life expectancy, and literacy rates in various countries of the world in 1993 and 2002. Concluding comments are given in Section 5. 2. THE FIVE DETERMINANTS OF THE WATTS MULTIDIMENSIONAL POVERTY INDEX The objective of this section is to show that the multidimensional extension of the Watts poverty index may be expressed as a function of five determinants. The unidimensional Watts poverty index is defined as P WU ¼ ð1=nÞ
np X
logðc=si Þ;
ð1Þ
i¼1
where n is the total number of individuals, np is the number of poor, c is the poverty line and si is the (positive) income of individual i. We assume without loss of generality that the first np individuals with incomes s1 ; s2 ; . . . ; snp are poor. This index may also be written as "n p X ð1=np Þ logðc=sp Þ P WU ¼ ðnp =nÞ i¼1
# np X ð1=np Þ logðsp =si Þ ; þ i¼1
ð2Þ
ON THE WATTS MULTIDIMENSIONAL POVERTY INDEXAND ITS DECOMPOSITION
where sp is the mean income of the np poor individuals. The Bourguignon (1979)–Theil (1967) mean logarithmic deviation index of inequality is defined as L ¼ log sa ð1=nÞ
n X
P WM ðX ; cÞ
"n # pj m X X ðnpj =np Þ ð1=npj Þ logðcj =spj Þ ¼ ðnp =nÞ
(
j¼1
þ log si ;
m X
"
ðnpj =np Þ
j¼1
ð3Þ
i¼1 npj X
#)
ð1=npj Þ logðspj =sij Þ
ð8Þ
where sa is the arithmetic mean of the incomes. Therefore, the Bourguignon–Theil index of income inequality among the poor (Lp) can be written as np X
log si :
ð4Þ
i¼1
We may now recall that another poverty index, the income gap ratio IGR, can be expressed as np X ðc si Þ=ðnp cÞ IGR ¼ i¼1
where spj is the mean value of indicator j among those individuals who are considered as poor with respect to indicator j, npj (more precisely, npj(X)) is the number of individuals who are poor with respect to indicator j in the attribute matrix X and (from now on) c stands for the vector (c1, c2, . . . , cm) of poverty limits for different dimensions. We show in Appendix A that PWM(X;c) given by (8) is equivalent to P WM ðX ; cÞ ¼ ð1=nÞ
"
¼ ð1=nÞ ð5Þ
A close look at the first expression in brackets on the R.H.S. of (2) shows that it is conceptually similar to the income gap ratio since it will also be equal to zero when the incomes of all the poor individuals are identical to the poverty line. We could call this index the ‘‘Watts poverty gap ratio’’ and denote it by PW,PGR, where
logðcj =sij Þ
i¼1
npj X m X i¼1
i¼1
P W;PGR ¼
npj m X X j¼1
# np X ðsi =cÞ ¼ 1 ðsp =cÞ: ¼ ð1=np Þ np
np X
;
i¼1
i¼1
Lp ¼ log sp ð1=np Þ
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logðcj =sij Þ:
ð9Þ
j¼1
Following Maasoumi and Lugo (2008) we can rewrite this expression as P WM ðX ; cÞ ¼ ð1=nÞ
n X
pi ;
ð10Þ
i¼1
where pi ¼
m X
logðcj = minðsij ; cj ÞÞ
ð11Þ
j¼1
ð1=np Þ logðc=sp Þ ¼ logðc=sp Þ:
ð6Þ
i¼1
One may observe that PW,PGR corresponds more or less to the percentage gap between the poverty line c and the mean income of the poor sp. Combining expressions (2), (4), and (6) we finally end up with P WU ¼ H ðP W;PGR þ Lp Þ;
ð7Þ
where H denotes the headcount ratio (np/n). Following earlier work by Chakravarty and Silber (2008) expression (7) may be extended to the multidimensional case so that the multidimensional Watts poverty index PWM(X;c) can be written as
is the implicit individual poverty function. This shows that PWM(X; c) bears a close resemblance to the following index characterized by Tsui (2002) when its parameter dj, which takes on non-negative values, is the same for all j attributes, where j = 1, 2, . . . , m: ! n X m cj 1X dj log P T ðX ; cÞ ¼ : n i¼1 j¼1 minðxij ; cj Þ ð12Þ As such it is a normalized version of the Maasoumi and Lugo, 2008 first class of IT poverty indices based on an ‘‘aggregate poverty line approach’’ when strong focus is used.
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It is clear that expression (8) may also be expressed as a multidimensional generalization of (7) and written as " # m X ðnpj =np ÞðP W;PGRj þ Lpj Þ ; P WM ðX ; cÞ ¼ H j¼1
DP WM ðX ; cÞ ( " ¼
H 1 g1 (
P W;PGRj ¼
npj X ð1=npj Þ logðcj =spj Þ;
ð14Þ
i¼1
and Lpj ¼
npj X ð1=npj Þ logðspj =sij Þ:
ð15Þ
i¼1
Note that PW,PGRj refers, in fact, to the Watts poverty gap ratio for indicator j, while Lpj represents the Bourguignon–Theil index of inequality for indicator j. It should be stressed that np is the union of all the npj’s so that the sum over all indicators j of all the npj’s may be greater than np(X), the number of poor in X. This is why we need to introduce P a normalizing factor g expressed as g ¼ ð j npj =np Þ and then rewrite (13) as " ! ! m m X X X P WM ðX ; cÞ ¼ H npj =np npj = npj j¼1
#
j
j¼1
ðP W;PGRj þ Lpj Þ ;
or, in short, as
"
P WM ðX ; cÞ ¼ H g
m X
ð16Þ
# rj ðP W;PGRj þ Lpj Þ ;
ð17Þ
j¼1
P
P with g ¼ ð j npj =np Þ and rj ¼ ðnpj = j npj Þ. The indicator g may be considered as measuring the degree of intersection of the various sets of poor, that is, as measuring the correlation between the different poverty dimensions. 3. DERIVING THE EXPRESSION FOR THE CHANGE IN THE WATTS MULTIDIMENSIONAL POVERTY INDEX Let us now add subscripts 1 and 0 to refer to the period in which multidimensional poverty is measured. The change DPWM between the values of the Watts multidimensional index at times 0 and 1 can then be expressed as
#) rj1 ðP W;PGRj1 þ Lpj1 Þ
j¼1
H 0 g0
m X
#) rj0 ðP W;PGRj0 þ Lpj0 Þ
;
j¼1
ð13Þ with
"
m X
ð18Þ () DP WM ðX ; cÞ ¼ f ðDH ; Dg; Drj ; DP W;PGRj ; DLpj Þ;
ð19Þ
where the operator D refers to the variation between times 0 and 1 of the five variables that appear in (19). In Appendix B we apply the Shapley decomposition to (18) to derive the contributions of the change in the headcount ratio H, P in the parameter g, in the weights rj ¼ ðqj = j qj Þ, in each of the Watts poverty gap ratios PW,PGRj, and in each of the Bourguignon–Theil measures of the inequality among the poor Lpj to the overall variation in the Watts multidimensional poverty index. Table 1. Basic data on poverty by dimensions Indicator
1993
Total number of countries Total ‘‘World Population’’ Number of poor countries by life expectancy Number of poor countries by literacy Number of poor countries by per capita GDP Poor population by life expectancy Poor population by literacy Poor population by per capita GDP Share of the poor population in the total ‘‘World Population’’ according to the life expectancy dimension Share of the poor population in the total ‘‘World Population’’ according to the literacy dimension Share of the poor population in the total ‘‘World Population’’ according to the per capita GDP dimension Share of the poor population according to the three criteria together (‘‘union’’ of the separate poverty rates for each of the three dimensions)
164 164 5346.9 5998.0 45 45
2002
40
28
44
38
684.0
735.9
1682.2 692.8 1616.1 778.2 12.8% 12.3%
31.5% 11.6%
30.2% 13.0%
36.1% 19.6%
ON THE WATTS MULTIDIMENSIONAL POVERTY INDEXAND ITS DECOMPOSITION
4. THE EMPIRICAL RESULTS We have applied this decomposition technique to data on the per capita GDP, life expectancy, and literacy rates of the countries for which the figures were available (164 countries representing a population of 5.3469 billions of individuals in 1992 and 5.9980 in 2002). The data were obtained from the World Development Reports for the years 1994 and 2003. As is well known these three variables are the main elements determining the Human Development Index (HDI) which is computed every year by the World Development Programme. The index HDI depends also on school enrollment rates but we have not taken this variable into account in order to maximize the number of countries for which data were available. For each of the three dimensions that we selected we had to determine a ‘‘poverty line.’’ For life expectancy we decided that any country in which life expectancy was smaller than 60 years should be considered as a ‘‘poor country’’ from the point of view of this dimension. Similarly, whenever the literacy rate in a country was smaller than 60% that country was ‘‘labeled’’ poor as far as the literacy dimension is concerned. Finally, for the per capita GDP we did not adopt the 1$ or 2$ a day criterion which is often adopted by international agencies. We rather decided that any country in which the per capita GDP was smaller than 5$ day should be classified as poor from the point of view of income (per capita GDP). This corresponds to an annual per capita GDP of $1825.
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Table 1 gives some basic information on the poverty rates by dimension as well as on the overall poverty rate which corresponds to the ‘‘union’’ of the individuals considered as poor on the various dimensions. Note that all the computations are ‘‘population-weighted’’ which means that the weight of each country corresponds to its weight in the overall population and that if a country’s indicator is below the poverty threshold for each dimension, each individual in this country will be assumed to be poor. It thus appears that 45 countries were poor in both 1993 and 2002 according to the life expectancy dimension. This corresponded to 12.8% of the ‘‘World Population’’ in 1993 and 12.3% in 2002. As far as the literacy rate is concerned, there were 40 poor countries in 1993 and 28 in 2002, the corresponding shares of the ‘‘World Population’’ being, respectively, 30.2% and 13.0%. This sharp decrease is in great part due to the fact that in 2002 India was no more considered as poor in this dimension. Finally, for the dimension given by the per capita GDP we observe that 44 countries were poor in 1993 and 38 in 2002. The corresponding population shares were 30.2% and 13.0%, respectively. Note that for this dimension too India ceased to be poor in 2002. In the first stage of the analysis we estimated the components of the breakdown of the change in the value of the Watts index during 1993–2002 on the basis of all the countries for which data were available. Table 2 gives some information on the value of the indicators used to compute the Watts multidimensional
Table 2. Values in 1993 and 2003 of the determinants of the multidimensional Watts poverty index, broken down by poverty dimensions (all countries included) Headcount ratio, H0
Year 1993 0.36062
Headcount ratio, H1
Year 2002 0.19602
Coefficient, g0
Poverty dimension
Weight of the poverty dimension, rj0
‘‘Watts poverty gap ratio’’ for the dimension, PW,PGRj0
Theil–Bourguignon index of inequality among the poor for the dimension, Lpj0
2.06529
Per capita GDP Life expectancy Literacy rate
0.17176 0.42242 0.40582
0.14197 0.25263 0.43930
0.00313 0.01838 0.03461
Coefficient, g1
Poverty dimension
Weight of the poverty dimension, rj1
‘‘Watts poverty gap ratio’’ for the dimension, PW,PGRj1
Theil–Bourguignon index of inequality among the poor for the dimension, Lpj1
1.87709
Per capita GDP Life expectancy Literacy rate
0.33345 0.31392 0.35262
0.20608 0.31571 0.44018
0.00959 0.02845 0.06130
0.00857 0.02180 0.01668 0.01795 0.13128 0.24707
0.11579
0.11153
Contribution of the Theil–Bourguignon index of inequality among the poor, Lpj Contribution of the ‘‘Watts poverty gap ratio,’’ PW,PGRj Contribution of the weights, rj of the poverty dimensions Contribution of the coefficient, g Contribution of the headcount ratio, H In 2003 In 1993
Variation during Contributions of the various components to the overall change in the Watts multidimensional poverty index 1993–2003 in the calue of the Watts multi- dimensional poverty index, DW
poverty index. Thus we observe that the weights given to the various dimensions are not equal and varied quite a lot during 1993– 2002. We recall that these weights rj (see the definition of rj that appears just after Eqn. (17)) are equal to the ratio of the number of poor computed on the basis of dimension j and the sum of the number of poor computed on the basis of the different dimensions. It appears that in 1993 the percentage of poor on the basis of the per capita GDP was much smaller than that computed on the basis of the two other dimensions since the weight rj corresponding to the per capita GDP was equal to 17.2%, while the weights corresponding to life expectancy and the literacy rates were, respectively, equal to 42.2% and 40.6%. In 2002, on the contrary, the three weights were almost identical, being, respectively, equal to 33.3% (per capita GDP), 31.4% (life expectancy), and 35.3% (literacy rate). As far as the Watts poverty gap ratios are concerned, we may observe that in each case they are in fact approximately equal to the percentage difference between the poverty line for the indicator under review and the mean value of this indicator among those considered as poor. We thus observe that in 1993 this percentage gap was equal to 14.2% for the per capita GDP, 25.3% for the life expectancy, and 43.9% for the literacy rate. All these gaps increased during 1993–2002 though not proportionately. Thus, in 2002 the percentage gap between the poverty line and the average value of the indicator among the poor was equal to 20.6% for the per capita GDP, 31.6% for the life expectancy, and 44.0% for the literacy rate. For the Theil–Bourguignon inequality index among the poor, which can be considered as giving the gap in percent between the arithmetic and geometric means of the distribution among the poor of the indicator corresponding to each dimension, we observe a greater degree of inequality, in both years, for the literacy rates than for the other two dimensions. Thus, in 1993 (2002) this percentage gap was equal to 0.3% (1.0%) for the per capita GDP, 1.8% (2.8%) for the life expectancy, and 3.5% (6.1%) for the literacy rate. Let us finally take a look at the value of the normalizing coefficient g whose definition was given just after Eqn. (17). This coefficient is equal to the ratio of the sum over the ‘‘union’’ of the number of poor computed on the basis of the three dimensions to the total number of poor. It should be clear that the greater the
Table 3. Results of the Shapley decomposition of the poverty change between 1993 and 2002 (all countries included)
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Value of the Watts multidimensional poverty index
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ON THE WATTS MULTIDIMENSIONAL POVERTY INDEXAND ITS DECOMPOSITION
correlation between the individuals classified as poor according to the different dimensions, the smaller is g. Since we observe in Table 2 that the coefficient g decreased during 1993–2002 from 2.065 to 1.877, we may conclude that the correlation increased over the period 1993–2002, meaning that those countries found to be poor according to one dimension are more likely to be classified as poor according to another dimension in 2002 than in 1993. Table 3 gives the contributions of the various determinants of the multidimensional Watts poverty index to the overall variation of the index observed during 1993–2002, where the contributions have been computed on the basis of the Shapley decomposition (see Appendix B). We observe that world poverty, computed on the basis of the three dimensions given by the per capita GDP, life expectancy, and literacy rate, decreased by close to 50% during 1993– 2002 (from 0.247 to 0.131). The whole change was in fact a consequence of the decrease in the overall headcount ratio (see Table 1). The contributions of the other determinants (normalization coefficient g, the weights rj of the poverty dimensions, the Watts income gap ratio, and the Theil–Bourguignon index of inequality among the poor) were quite small and canceled out (see Table 3). In the second stage of the analysis we excluded China and India whose weight in the world population is very high. The results of these estimations are given in Tables 4 and 5.
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It then appears that both in 1993 and in 2002 the weights of the three dimensions were almost equal. (We recall that the weight of a given dimension is equal to the ratio of the number of the poor computed on the basis of that dimension to the sum of the number of poor computed on the basis of the different dimensions). We also observe that whereas when all countries are included, the share of the poor (all dimensions included) in the world population decreased significantly during 1993–2002 (from 36.1% to 19.6%), it slightly increased (from 31.6% to 32.2%) when China and India are excluded. As far as the Watts poverty gap ratios for different dimensions are concerned, we observe a significant increase for the per capita GDP (from 0.14 to 0.33), a slight decrease for the life expectancy (from 0.36 to 0.31), and a significant decrease for the literacy rate (from 0.51 to 0.35). Similar trends were observed for the Theil–Bourguignon index of inequality among the poor since the index increased for the per capita GDP and decreased for the other two dimensions. Table 5 finally gives the contribution of each of the five determinants of the multidimensional Watts poverty index to its change during 1993–2002, when China and India are excluded. The results are quite different from what was observed when these two countries were included in the analysis. The decrease in the Watts index was much smaller (from 0.252 to 0.216) and more than two-thirds of this
Table 4. Values in 1993 and 2002 of the determinants of the multidimensional Watts poverty index, broken down by poverty dimensions (all countries but China and India) Headcount ratio, H0
Year 1993 0.31600
Headcount ratio, H1
Year 2002 0.32179
Coefficient, g0
Poverty dimension
Weight of the poverty dimension, rj0
‘‘Watts poverty gap ratio’’ for the dimension, PW,PGRj0
Theil–Bourguignon index of inequality among the poor for the dimension, Lpj0
2.12263
Per capita GDP Life expectancy Literacy rate
0.31386 0.35823 0.32790
0.14197 0.35685 0.51020
0.00313 0.03034 0.07403
Coefficient, g1
Poverty dimension
Weight of the poverty dimension, rj1
‘‘Watts poverty gap ratio’’ for the dimension, PW,PGRj1
Theil–Bourguignon index of inequality among the poor for the dimension Lpj1
1.87709
Per capita GDP Life expectancy Literacy rate
0.33345 0.31392 0.35262
0.20608 0.31571 0.44018
0.00959 0.02845 0.06130
0.00183 0.01079 0.00047 0.00424 0.03657 0.21551 0.25208
0.02866
Contribution of the Theil– Bourguignon index of inequality among the poor, Lpj Contribution of the ‘‘Watts income gap ratio,’’ PW,PGRj Contribution of the weights, rj of the poverty dimensions Contribution of the headcount ratio, H DW In 2002
Contribution of the coefficient, g
decrease were due to a decrease in the value of the parameter g, that is, to an increase in the degree of correlation between the three dimensions of poverty on which this analysis is based. The other component which played a role in the decrease in the Watts index is the change in the Watts poverty gap ratio, which, as we saw, decreased for the life expectancy and the literacy rate and increased for the per capita GDP.
5. CONCLUDING REMARKS
In 1993
Value of the Watts multidimensional poverty index
Variation during 1993 and 2002 in the value of the Watts multidimensional poverty index
Contributions of the various components to the overall change in the Watts multidimensional poverty index
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Table 5. Results of the Shapley decomposition of the poverty change during 1993–2002 (all countries but China and India)
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In this paper we have shown that a multidimensional extension of the Watts poverty index may be expressed as a function of five determinants which are, respectively, what we define as the Watts poverty gap ratio, the Bourguignon (1979)–Theil (1967) index of inequality among the poor, the overall headcount ratio, the weights of the various dimensions, and some measure of correlation between the various dimensions of poverty. This enables us to identify the causal factors of poverty and hence to implement poverty reduction policies. We then explained how the Shapley decomposition allows one to derive the contribution of each of these five determinants to the change over time in the value of the multidimensional extension of the Watts poverty index. An empirical illustration of the index was then provided using the per capita GDP, life expectancy, and literacy rate data for several countries for the periods 1993–2002. We observed that when all the countries are included most of the change in world poverty during 1993–2002 was related to the decrease in the overall headcount ratio. However when China and India are excluded, two-thirds of the decrease in the Watts index are due to an increase in the correlation between the various dimensions and one-third to changes in the Watts poverty gap ratio. NOTES 1. Several contributions suggested alternatives to and variations of the Sen index. See, for example, Takayama (1979), Blackorby and Donaldson (1980), Kakwani (1980), Clark, Hamming, and Ulph (1981), Chakravarty (1983a, 1983b, 1997), Foster, Greer, and Thorbecke (1984), Thon (1983), and Shorrocks (1995).
ON THE WATTS MULTIDIMENSIONAL POVERTY INDEXAND ITS DECOMPOSITION
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REFERENCES Blackorby, C., & Donaldson, D. (1980). Ethical indices for the measurement of poverty. Econometrica, 58, 1053–1060. Bourguignon, F. (1979). Decomposable income inequality measures. Econometrica, 47, 901–920. Bourguignon, F., & Chakravarty, S. R. (2003). The measurement of multidimensional poverty. Journal of Economic Inequality, 1, 25–49. Chakravarty, S. R. (1983a). A new index of poverty. Mathematical Social Sciences, 6, 307–313. Chakravarty, S. R. (1983b). Ethically flexible measures of poverty. Canadian Journal of Economics, 16, 74–85. Chakravarty, S. R. (1997). On Shorrocks’ reinvestigation of the Sen poverty index. Econometrica, 65, 1241–1242. Chakravarty, S. R., & Silber, J. (2008). Measuring multidimensional poverty: The axiomatic approach. In N. Kakwani, & J. Silber (Eds.), Quantitative approaches to multidimensional poverty measurement. London: Palgrave MacMillan. Clark, S., Hamming, R., & Ulph, D. (1981). On indices for the measurement of poverty. Economic Journal, 91, 515–526. Foster, J. E., Greer, J., & Thorbecke, E. (1984). A class of decomposable poverty measures. Econometrica, 42, 761–766. Kakwani, N. C. (1980). On a class of poverty measures. Econometrica, 48, 437–446. Lipton, M., & Ravallion, M. (1995). Poverty and policy. In J. Behrman, & T. N. Srinivasan (Eds.), Handbook of development economics (Vol. 3). Amsterdam: North-Holland. Maasoumi, E., & Lugo, M. A. (2008). The information basis of multivariate poverty assessments. In N. Kakwani, & J. Silber (Eds.), Quantitative approaches to multidimensional poverty measurement. London: Palgrave MacMillan. Ravallion, M. (1996). Issues in measuring and modeling poverty. Economic Journal, 106, 1328–1343. Sastre, M., & Trannoy, A. (2002). Shapley inequality decomposition by factor components: Some methodological issues. Journal of Economics, (Suppl. 9), 51–89. Sen, A. K. (1976). Poverty an ordinal approach to measurement. Econometrica, 44, 219–231. Sen, A. K. (1981). Poverty and famines. Oxford: Clarendon Press. Sen, A. K. (1985). Commodities and capabilities. Amsterdam: North-Holland. Sen, A. K. (1992). Inequality re-examined. Cambridge, MA: Harvard University Press. Shapley, L. S. (1953). A value for n-person games. In H. W. Kuhn, & A. W. Tucker (Eds.), Contributions to the theory of games II, Annals of mathematics studies, Princeton studies. Princeton: Princeton University Press. Shorrocks, A. F. (1980). The class of additively decomposable inequality measures. Econometrica, 48, 613–625. Shorrocks, A. F. (1995). Revisiting the Sen poverty index. Econometrica, 63, 1225–1230.
Shorrocks, A. F. (1999). Decomposition procedures for distributional analysis. a unified framework based on the Shapley value. mimeo, University of Essex. Streeten, P. (1981). First things first: Meeting basic human needs in developing countries. Yew York: Oxford University Press. Takayama, N. (1979). Poverty, income inequality and their measures: Professor Sen’s axiomatic approach reconsidered. Econometrica, 47, 749–759. Theil, H. (1967). Economics and information theory. North Holland: Amsterdam. Thon, D. (1983). A poverty measure. Indian Economic Journal, 30, 55–70. Tsui, K. Y. (2002). Multidimensional poverty indices. Social Choice and Welfare, 19, 69–93. UNDP (1997). Human development report. Oxford: Oxford University Press. Watts, H. (1968). An economic definition of poverty. In D. P. Moynihan (Ed.), On understanding poverty. New York: Basic Books.
APPENDIX A. EQUIVALENCE BETWEEN (8) AND (9) P WM ðX ; cÞ
(
"n # pj m X X ðnpj =np Þ ð1=npj Þ logðcj =spj Þ ¼ ðnp =nÞ j¼1
þ
m X
ðnpj =np Þ
j¼1
"n pj X
i¼1
#)
ð1=npj Þ logðspj =sij Þ
i¼1
or,
P WM ðX ; cÞ
(
"n # pj m X X ðnpj =np Þð1=npj Þ logðcj =spj Þ ¼ ðnp =nÞ j¼1 i¼1 " npj m X X ðnpj =np Þ ð1=npj Þ logðspj Þ þ j¼1
ð1=npj Þ
npj X
#)i¼1 logðsij Þ
i¼1
or,
P WM ðX ; cÞ
(
¼ ðnp =nÞ
m X ðnpj =np Þð1=npj Þ½npj logðcj =spj Þ j¼1
m X þ ðnpj =np Þ ð1=npj Þnpj logðspj Þ j¼1
ð1=npj Þ
npj X i¼1
#) logðsij Þ
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WORLD DEVELOPMENT
There are obviously 3! = 6 ways of ordering these three determinants a, b, and c: ða; b; cÞ; ða; c; bÞ; ðb; a; cÞ; ðb; c; aÞ; ðc; a; bÞ; ðc; b; aÞ:
or,
P WM ðX ; cÞ
( m X ¼ ðnp =nÞ ðnpj =np Þðlogðcj =spj ÞÞ j¼1
þ
"
npj m X X ðnpj =np Þ logðspj Þ ð1=npj Þ logðsij Þ
#)
i¼1
j¼1
or,
P WM ðX ; cÞ
(
¼ ð1=nÞ
m X
npj ðlogðcj =spj ÞÞ
j¼1
þ
m X
"
npj logðspj Þ ð1=npj Þ
j¼1
npj X
#) logðsij Þ
i¼1
or,
P WM ðX ; cÞ
( m m X X ¼ ð1=nÞ npj logðcj Þ npj logðspj Þ j¼1
þ
m X
j¼1
) npj m X X npj logðspj Þ ðnpj =npj Þ logðsij Þ
j¼1
i¼1
j¼1
Hence
P WM ðX ; cÞ
(
¼ ð1=nÞ
m X
npj logðcj Þ
j¼1
¼ ð1=nÞ
j¼1
npj m X X j¼1
npj m X X
) logðsij Þ
i¼1
logðcj =sij Þ:
i¼1
APPENDIX B. ON THE CONCEPT OF SHAPLEY DECOMPOSITION AND ITS APPLICATION TO ANALYZING CHANGES IN THE WATTS MULTIDIMENSIONAL POVERTY INDEX The Shapley (1953) decomposition is a technique borrowed from game theory, but it has been extended to applied economics by Shorrocks (1999) and Sastre and Trannoy (2002). Let us explain it briefly. Assume that an indicator I is a function of three determinants a, b, c and is written as I = I(a, b, c). I could be an index of inequality but, more generally, any linear or nonlinear function of the concerned variables is allowable.
ðB:1Þ Each of these three determinants may be eliminated first, second, or third. The respective (marginal) contributions of the determinants a, b, c will hence be a function of all the possible ways in which each of these determinants may be eliminated. Let, for example, C(a) be the marginal contribution of a to the indicator I(a, b, c). If a is eliminated first its contribution to the overall value of the indicator I will be expressed as I(a, b, c) I(b, c), where I(b, c) corresponds to the case where a is equal to zero. Since expression B.1 indicates that there are two cases in which a appears first and may thus be eliminated first, we will give a weight of (2/6) to this possibility. If a is eliminated second, it implies that another determinant has been eliminated first (and been assumed to be equal to 0). Expression (B.1) indicates that the two cases in which this possibility occurs are (b, a, c) and (c, a, b). In the first case the contribution of a will be I(a, c) I(c), while in the second it is I(a, b) I(b). To each of these two cases we evidently give a weight of (1/6). Finally, if a is eliminated third, it implies that both b and c are assumed to be equal to 0. Expression (B.1) indicates that there are two such cases, namely, (b, c, a) and (c, b, a). Since we may assume that when each of the three determinants is equal to 0, the indicator I is equal to 0, we may say that the contribution of a in this case will be equal to I(a) 0 = I(a) and evidently we have to give a weight of (2/6) to such a possibility since there are two such cases. We may therefore summarize what we have just explained by stating that the marginal contribution C(a) of the determinant a to the overall value of the indicator I is CðaÞ ¼ ð2=6Þ½Iða; b; cÞ Iðb; cÞ þ ð1=6Þ ½Iða; cÞ IðcÞ þ ð1=6Þ½Iða; bÞ IðbÞ þ ð2=6ÞIðaÞ:
ðB:2Þ
One can similarly determine the marginal contribution C(b) of b and C(c) of c and then find out that Iða; b; cÞ ¼ CðaÞ þ CðbÞ þ CðcÞ: ðB:3Þ This Shapley decomposition may be also applied in a similar way to the case where one wants to understand the respective contribu-
ON THE WATTS MULTIDIMENSIONAL POVERTY INDEXAND ITS DECOMPOSITION
tions to the change over time in the value of the indicator I. We denote this change in I by DI, and Da, Db, and Dc represent respectively the variations over time in the values of the three determinants a, b, and c. Since in (19) the change in the multidimensional Watts poverty index is expressed as a function of five variables, we have to extend the Shapley decomposition described above to the five determinants. For simplicity, we denote the five determinants DH, Dg, Drj, DPW,PGR,j, DLpj of the change in the Watts index that appear in (19) by a, b, c, d, and e, respectively. More precisely, a = DH, b = Dg, c = Drj, d = DPW,PGR,j, and e = DLpj where c, d, and e refer to the simultaneous changes in all the dimensions j of Drj, DPW,PGR,j, and DLpj. It is then easy to show that the contribution C(a) of a in such a case, assuming that I refers to the multidimensional Watts index, becomes CðaÞ ¼ ð24=120Þ½Iða; b; c; d; eÞ Iðb; c; d; eÞ þ ð6=120Þ½Iða; c; d; eÞ Iðc; d; eÞ þ ð6=120Þ½Iða; b; d; eÞ Iðb; d; eÞ þ ð6=120Þ½Iða; b; c; eÞ Iðb; c; eÞ þ ð6=120Þ½Iða; b; c; dÞ Iðb; c; dÞ þ ð4=120Þ½Iða; b; cÞ Iðb; cÞ þ ð4=120Þ½Iða; b; dÞ Iðb; dÞ þ ð4=120Þ½Iða; b; eÞ Iðb; eÞ þ ð4=120Þ½Iða; c; dÞ Iðc; dÞ þ ð4=120Þ½Iða; c; eÞ Iðc; eÞ þ ð4=120Þ½Iða; d; eÞ Iðd; eÞ þ ð6=120Þ½Iða; bÞ IðbÞ þ ð6=120Þ½Iða; cÞ IðcÞ þ ð6=120Þ½Iða; dÞ IðdÞ þ ð6=120Þ½Iða; eÞ IðeÞ þ ð24=120ÞIðaÞ:
ðB:4Þ
One can similarly compute the marginal contributions of b, c, d, and e and here also the sum of the contributions of a, b, c, d, and e is equal
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to the value of the indicator I. To understand well (B.4) it should be remembered that when a determinant appears in one of the lines of (B.4) it means that it is different from 0 whereas if it does not appear it means it is equal to 0. Thus, for example, the line before the last one, which is given by (6/120) [I(a, e) I(e)], should be translated as ð6=120Þ½DP WM ðDH 6¼ 0; Dg ¼ 0; Drj ¼ 0; DP W;PGRj ¼ 0; DLpj 6¼ 0Þ DP WM ðDH ¼ 0; Dg ¼ 0; Drj ¼ 0; DP W;PGRj ¼ 0; DLpj 6¼ 0Þ:
ðB:5Þ
Note that in (B.5), DH = 0 means that when computing the change in the value of the Watts index one has to assume that the headcount ratio did not change between times 0 and 1, whereas DH 5 0 will mean that headcount ratio changed. Similar interpretations hold concerning the changes Dg, Drj, DPW,PGR,j, and DLpj. We note that the decomposition coefficients, that is, the weights assigned to the marginal contributions of the different determinants, depend on the number of possible orderings of the determinants. Hence they are independent of basic attributes as well as across terms. We note also the independence of the population weights rj’s in the attribute-wise decomposition of the Watts index given by (17). This is, in fact, an extension of Shorrocks’ (1980) arguments for subgroup decomposable income inequality indices to the multidimensional case. He showed that only for two inequality indices, namely, the Theil (1967) entropy index and the Bourguignon–Theil mean logarithmic deviation index, the decomposition coefficients are independent of the between group contribution. The poverty index we use here is based on the multidimensional extension of the latter index. Finally, it may be worthwhile to mention that the Shapley decomposition technique becomes applicable to a wide class of poverty indices that share the same structure as the one given by (8). This becomes evident from the equivalence between (8) and (9).
Available online at www.sciencedirect.com
doi:10.1016/j.worlddev.2007.10.003
On the Watts Multidimensional Poverty Index and its Decomposition SATYA R. CHAKRAVARTY Indian Statistical Institute, Kolkata, India JOSEPH DEUTSCH and JACQUES SILBER Bar-Ilan University, Ramat-Gan, Israel
*
Summary. — The multidimensional extension of the Watts poverty index may be expressed as a function of five determinants measuring, respectively, the impacts of what are defined in the paper as the Watts poverty gap ratio, the Theil–Bourguignon index of inequality among the poor, the overall headcount ratio, the weights of the various dimensions, and some measure of correlation between the various dimensions. Using the Shapley decomposition, we apply this index to world data on the per capita GDP, life expectancy, and literacy rates and derive the contributions of the five determinants defined above to the variation of this index during 1993–2002. Ó 2008 Elsevier Ltd. All rights reserved. Key words — decomposition, multidimensional poverty, Watts index
1. INTRODUCTION Removal of poverty has been and continues to be one of the primary aims of economic policy in a large number of countries. Therefore, targeting poverty alleviation is still a very important issue in many countries. It is thus necessary to know the dimensions of poverty and the process through which it seems to be aggravated. A natural question that arises in this context is how to quantify the extent of poverty. In a pioneering contribution, Sen (1976) regarded the poverty measurement problem as involving two exercises: (i) the identification of the poor and (ii) the aggregation of the characteristics of the poor into an overall indicator that quantifies the extent of poverty. In the literature, the first problem is mostly solved by the income method, which requires the specification of a poverty line representing the income required for a subsistence standard of living. A person is said to be poor if his income falls below the poverty line. On the aggregation issue, Sen (1976) criticized two crude indicators of poverty, the headcount ratio (proportion of
persons with incomes below the poverty line) and the income gap ratio (the difference between the poverty line and the average income of the poor, expressed as a proportion of the poverty line), because they remain unaltered under a transfer of income between two poor persons and the former also does not change if a poor person becomes poorer due to a reduction in his income. Sen (1976) also characterized axiomatically a more sophisticated index of poverty. 1 However, the well-being of a population and hence its poverty, which is a manifestation of insufficient well-being, is a multidimensional phenomenon, income being only one of the
* We are grateful to an anonymous referee for his/her comments and suggestions. We also thank him/her for bringing our attention to the equivalence between Eqns. (8) and (9). Chakravarty wishes to thank the Bocconi University, Milan, Italy, and Silber the Fundacio´n de ´ a Aplicada (FEDEA), Madrid, Estudios de Economı Spain, for support. Final revision accepted: October 5, 2007.
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WORLD DEVELOPMENT
many attributes on which the well-being depends. Examples of such attributes are food, housing, clothing, education, health, provision of public goods, and so on. While it is true that with a higher income a person is able to improve the position of some of his non-monetary attributes, it may as well be the case that markets for certain attributes do not exist, for instance, in the case of some public goods. Examples are flood control program and malaria prevention program in an underdeveloped country. (See Bourguignon & Chakravarty, 2003; Ravallion, 1996; Tsui, 2002). There are additional arguments for viewing poverty from a multidimensional perspective. In the basic needs approach development is regarded as an improvement in the array of human needs, not just as growth of income alone (Streeten, 1981). There is a debate about the importance of low income as a determinant of under nutrition (Lipton & Ravallion, 1995). In the capability-functioning approach, where functionings deal with what a person can ultimately do and capabilities indicate a person’s freedom with respect to functionings (Sen, 1985, 1992), poverty is regarded as a problem of capability failure. Functionings here are closely approximated by attributes like literacy, life expectancy, etc. An example of a multidimensional index of poverty in the capability failure framework is the human poverty index suggested by the UNDP (1997). It aggregates the deprivations in the living standard of a population in terms of three basic dimensions of life, namely, decent living standard, educational attainment rate, and life expectancy at birth. In view of the above discussion, we assume that each person is characterized by a vector of basic need attributes and a direct method of identification of poor checks if the person has ‘‘minimally acceptable levels’’ (Sen, 1992, p. 139) of this set of basic needs. Therefore, the direct method considers poverty from a multidimensional perspective, more precisely, in terms of shortfalls of attribute quantities from respective threshold levels. These threshold levels are determined independently of the attribute distributions. Since the direct method ‘‘is not based on particular assumptions of consumer behavior which may or may not be accurate,’’ ‘‘it is superior to the income method’’ (Sen, 1981, p. 26). If direct information on different attributes are not available, one can adopt the income method, ‘‘so that the income method is at most a second best’’ (Sen, 1981, p. 26).
Following earlier work on the characterization of the multidimensional extension of the Watts (1968) poverty index (see, Chakravarty & Silber, 2008; Maasoumi & Lugo, 2008; Tsui, 2002), we first (in Section 2) show that this multidimensional extension of the Watts poverty index may be expressed as a function of five determinants which are, respectively, what we define as the Watts income gap ratio, the Bourguignon (1979)–Theil (1967) index of inequality among the poor, the overall headcount ratio, the weights of the various dimensions, and some measure of correlation between the various dimensions. This decomposition should allow us to investigate the relationship between poverty and its determinants, more precisely, to identify the different causal factors of poverty. Then (in Section 3), using the well-known Shapley (1953) decomposition, we derive the contribution of each of these five determinants to the overall change in the Watts index. The paper ends (Section 4) by providing a numerical illustration based on data on the per capita GDP, life expectancy, and literacy rates in various countries of the world in 1993 and 2002. Concluding comments are given in Section 5. 2. THE FIVE DETERMINANTS OF THE WATTS MULTIDIMENSIONAL POVERTY INDEX The objective of this section is to show that the multidimensional extension of the Watts poverty index may be expressed as a function of five determinants. The unidimensional Watts poverty index is defined as P WU ¼ ð1=nÞ
np X
logðc=si Þ;
ð1Þ
i¼1
where n is the total number of individuals, np is the number of poor, c is the poverty line and si is the (positive) income of individual i. We assume without loss of generality that the first np individuals with incomes s1 ; s2 ; . . . ; snp are poor. This index may also be written as "n p X ð1=np Þ logðc=sp Þ P WU ¼ ðnp =nÞ i¼1
# np X ð1=np Þ logðsp =si Þ ; þ i¼1
ð2Þ
ON THE WATTS MULTIDIMENSIONAL POVERTY INDEXAND ITS DECOMPOSITION
where sp is the mean income of the np poor individuals. The Bourguignon (1979)–Theil (1967) mean logarithmic deviation index of inequality is defined as L ¼ log sa ð1=nÞ
n X
P WM ðX ; cÞ
"n # pj m X X ðnpj =np Þ ð1=npj Þ logðcj =spj Þ ¼ ðnp =nÞ
(
j¼1
þ log si ;
m X
"
ðnpj =np Þ
j¼1
ð3Þ
i¼1 npj X
#)
ð1=npj Þ logðspj =sij Þ
ð8Þ
where sa is the arithmetic mean of the incomes. Therefore, the Bourguignon–Theil index of income inequality among the poor (Lp) can be written as np X
log si :
ð4Þ
i¼1
We may now recall that another poverty index, the income gap ratio IGR, can be expressed as np X ðc si Þ=ðnp cÞ IGR ¼ i¼1
where spj is the mean value of indicator j among those individuals who are considered as poor with respect to indicator j, npj (more precisely, npj(X)) is the number of individuals who are poor with respect to indicator j in the attribute matrix X and (from now on) c stands for the vector (c1, c2, . . . , cm) of poverty limits for different dimensions. We show in Appendix A that PWM(X;c) given by (8) is equivalent to P WM ðX ; cÞ ¼ ð1=nÞ
"
¼ ð1=nÞ ð5Þ
A close look at the first expression in brackets on the R.H.S. of (2) shows that it is conceptually similar to the income gap ratio since it will also be equal to zero when the incomes of all the poor individuals are identical to the poverty line. We could call this index the ‘‘Watts poverty gap ratio’’ and denote it by PW,PGR, where
logðcj =sij Þ
i¼1
npj X m X i¼1
i¼1
P W;PGR ¼
npj m X X j¼1
# np X ðsi =cÞ ¼ 1 ðsp =cÞ: ¼ ð1=np Þ np
np X
;
i¼1
i¼1
Lp ¼ log sp ð1=np Þ
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logðcj =sij Þ:
ð9Þ
j¼1
Following Maasoumi and Lugo (2008) we can rewrite this expression as P WM ðX ; cÞ ¼ ð1=nÞ
n X
pi ;
ð10Þ
i¼1
where pi ¼
m X
logðcj = minðsij ; cj ÞÞ
ð11Þ
j¼1
ð1=np Þ logðc=sp Þ ¼ logðc=sp Þ:
ð6Þ
i¼1
One may observe that PW,PGR corresponds more or less to the percentage gap between the poverty line c and the mean income of the poor sp. Combining expressions (2), (4), and (6) we finally end up with P WU ¼ H ðP W;PGR þ Lp Þ;
ð7Þ
where H denotes the headcount ratio (np/n). Following earlier work by Chakravarty and Silber (2008) expression (7) may be extended to the multidimensional case so that the multidimensional Watts poverty index PWM(X;c) can be written as
is the implicit individual poverty function. This shows that PWM(X; c) bears a close resemblance to the following index characterized by Tsui (2002) when its parameter dj, which takes on non-negative values, is the same for all j attributes, where j = 1, 2, . . . , m: ! n X m cj 1X dj log P T ðX ; cÞ ¼ : n i¼1 j¼1 minðxij ; cj Þ ð12Þ As such it is a normalized version of the Maasoumi and Lugo, 2008 first class of IT poverty indices based on an ‘‘aggregate poverty line approach’’ when strong focus is used.
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WORLD DEVELOPMENT
It is clear that expression (8) may also be expressed as a multidimensional generalization of (7) and written as " # m X ðnpj =np ÞðP W;PGRj þ Lpj Þ ; P WM ðX ; cÞ ¼ H j¼1
DP WM ðX ; cÞ ( " ¼
H 1 g1 (
P W;PGRj ¼
npj X ð1=npj Þ logðcj =spj Þ;
ð14Þ
i¼1
and Lpj ¼
npj X ð1=npj Þ logðspj =sij Þ:
ð15Þ
i¼1
Note that PW,PGRj refers, in fact, to the Watts poverty gap ratio for indicator j, while Lpj represents the Bourguignon–Theil index of inequality for indicator j. It should be stressed that np is the union of all the npj’s so that the sum over all indicators j of all the npj’s may be greater than np(X), the number of poor in X. This is why we need to introduce P a normalizing factor g expressed as g ¼ ð j npj =np Þ and then rewrite (13) as " ! ! m m X X X P WM ðX ; cÞ ¼ H npj =np npj = npj j¼1
#
j
j¼1
ðP W;PGRj þ Lpj Þ ;
or, in short, as
"
P WM ðX ; cÞ ¼ H g
m X
ð16Þ
# rj ðP W;PGRj þ Lpj Þ ;
ð17Þ
j¼1
P
P with g ¼ ð j npj =np Þ and rj ¼ ðnpj = j npj Þ. The indicator g may be considered as measuring the degree of intersection of the various sets of poor, that is, as measuring the correlation between the different poverty dimensions. 3. DERIVING THE EXPRESSION FOR THE CHANGE IN THE WATTS MULTIDIMENSIONAL POVERTY INDEX Let us now add subscripts 1 and 0 to refer to the period in which multidimensional poverty is measured. The change DPWM between the values of the Watts multidimensional index at times 0 and 1 can then be expressed as
#) rj1 ðP W;PGRj1 þ Lpj1 Þ
j¼1
H 0 g0
m X
#) rj0 ðP W;PGRj0 þ Lpj0 Þ
;
j¼1
ð13Þ with
"
m X
ð18Þ () DP WM ðX ; cÞ ¼ f ðDH ; Dg; Drj ; DP W;PGRj ; DLpj Þ;
ð19Þ
where the operator D refers to the variation between times 0 and 1 of the five variables that appear in (19). In Appendix B we apply the Shapley decomposition to (18) to derive the contributions of the change in the headcount ratio H, P in the parameter g, in the weights rj ¼ ðqj = j qj Þ, in each of the Watts poverty gap ratios PW,PGRj, and in each of the Bourguignon–Theil measures of the inequality among the poor Lpj to the overall variation in the Watts multidimensional poverty index. Table 1. Basic data on poverty by dimensions Indicator
1993
Total number of countries Total ‘‘World Population’’ Number of poor countries by life expectancy Number of poor countries by literacy Number of poor countries by per capita GDP Poor population by life expectancy Poor population by literacy Poor population by per capita GDP Share of the poor population in the total ‘‘World Population’’ according to the life expectancy dimension Share of the poor population in the total ‘‘World Population’’ according to the literacy dimension Share of the poor population in the total ‘‘World Population’’ according to the per capita GDP dimension Share of the poor population according to the three criteria together (‘‘union’’ of the separate poverty rates for each of the three dimensions)
164 164 5346.9 5998.0 45 45
2002
40
28
44
38
684.0
735.9
1682.2 692.8 1616.1 778.2 12.8% 12.3%
31.5% 11.6%
30.2% 13.0%
36.1% 19.6%
ON THE WATTS MULTIDIMENSIONAL POVERTY INDEXAND ITS DECOMPOSITION
4. THE EMPIRICAL RESULTS We have applied this decomposition technique to data on the per capita GDP, life expectancy, and literacy rates of the countries for which the figures were available (164 countries representing a population of 5.3469 billions of individuals in 1992 and 5.9980 in 2002). The data were obtained from the World Development Reports for the years 1994 and 2003. As is well known these three variables are the main elements determining the Human Development Index (HDI) which is computed every year by the World Development Programme. The index HDI depends also on school enrollment rates but we have not taken this variable into account in order to maximize the number of countries for which data were available. For each of the three dimensions that we selected we had to determine a ‘‘poverty line.’’ For life expectancy we decided that any country in which life expectancy was smaller than 60 years should be considered as a ‘‘poor country’’ from the point of view of this dimension. Similarly, whenever the literacy rate in a country was smaller than 60% that country was ‘‘labeled’’ poor as far as the literacy dimension is concerned. Finally, for the per capita GDP we did not adopt the 1$ or 2$ a day criterion which is often adopted by international agencies. We rather decided that any country in which the per capita GDP was smaller than 5$ day should be classified as poor from the point of view of income (per capita GDP). This corresponds to an annual per capita GDP of $1825.
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Table 1 gives some basic information on the poverty rates by dimension as well as on the overall poverty rate which corresponds to the ‘‘union’’ of the individuals considered as poor on the various dimensions. Note that all the computations are ‘‘population-weighted’’ which means that the weight of each country corresponds to its weight in the overall population and that if a country’s indicator is below the poverty threshold for each dimension, each individual in this country will be assumed to be poor. It thus appears that 45 countries were poor in both 1993 and 2002 according to the life expectancy dimension. This corresponded to 12.8% of the ‘‘World Population’’ in 1993 and 12.3% in 2002. As far as the literacy rate is concerned, there were 40 poor countries in 1993 and 28 in 2002, the corresponding shares of the ‘‘World Population’’ being, respectively, 30.2% and 13.0%. This sharp decrease is in great part due to the fact that in 2002 India was no more considered as poor in this dimension. Finally, for the dimension given by the per capita GDP we observe that 44 countries were poor in 1993 and 38 in 2002. The corresponding population shares were 30.2% and 13.0%, respectively. Note that for this dimension too India ceased to be poor in 2002. In the first stage of the analysis we estimated the components of the breakdown of the change in the value of the Watts index during 1993–2002 on the basis of all the countries for which data were available. Table 2 gives some information on the value of the indicators used to compute the Watts multidimensional
Table 2. Values in 1993 and 2003 of the determinants of the multidimensional Watts poverty index, broken down by poverty dimensions (all countries included) Headcount ratio, H0
Year 1993 0.36062
Headcount ratio, H1
Year 2002 0.19602
Coefficient, g0
Poverty dimension
Weight of the poverty dimension, rj0
‘‘Watts poverty gap ratio’’ for the dimension, PW,PGRj0
Theil–Bourguignon index of inequality among the poor for the dimension, Lpj0
2.06529
Per capita GDP Life expectancy Literacy rate
0.17176 0.42242 0.40582
0.14197 0.25263 0.43930
0.00313 0.01838 0.03461
Coefficient, g1
Poverty dimension
Weight of the poverty dimension, rj1
‘‘Watts poverty gap ratio’’ for the dimension, PW,PGRj1
Theil–Bourguignon index of inequality among the poor for the dimension, Lpj1
1.87709
Per capita GDP Life expectancy Literacy rate
0.33345 0.31392 0.35262
0.20608 0.31571 0.44018
0.00959 0.02845 0.06130
0.00857 0.02180 0.01668 0.01795 0.13128 0.24707
0.11579
0.11153
Contribution of the Theil–Bourguignon index of inequality among the poor, Lpj Contribution of the ‘‘Watts poverty gap ratio,’’ PW,PGRj Contribution of the weights, rj of the poverty dimensions Contribution of the coefficient, g Contribution of the headcount ratio, H In 2003 In 1993
Variation during Contributions of the various components to the overall change in the Watts multidimensional poverty index 1993–2003 in the calue of the Watts multi- dimensional poverty index, DW
poverty index. Thus we observe that the weights given to the various dimensions are not equal and varied quite a lot during 1993– 2002. We recall that these weights rj (see the definition of rj that appears just after Eqn. (17)) are equal to the ratio of the number of poor computed on the basis of dimension j and the sum of the number of poor computed on the basis of the different dimensions. It appears that in 1993 the percentage of poor on the basis of the per capita GDP was much smaller than that computed on the basis of the two other dimensions since the weight rj corresponding to the per capita GDP was equal to 17.2%, while the weights corresponding to life expectancy and the literacy rates were, respectively, equal to 42.2% and 40.6%. In 2002, on the contrary, the three weights were almost identical, being, respectively, equal to 33.3% (per capita GDP), 31.4% (life expectancy), and 35.3% (literacy rate). As far as the Watts poverty gap ratios are concerned, we may observe that in each case they are in fact approximately equal to the percentage difference between the poverty line for the indicator under review and the mean value of this indicator among those considered as poor. We thus observe that in 1993 this percentage gap was equal to 14.2% for the per capita GDP, 25.3% for the life expectancy, and 43.9% for the literacy rate. All these gaps increased during 1993–2002 though not proportionately. Thus, in 2002 the percentage gap between the poverty line and the average value of the indicator among the poor was equal to 20.6% for the per capita GDP, 31.6% for the life expectancy, and 44.0% for the literacy rate. For the Theil–Bourguignon inequality index among the poor, which can be considered as giving the gap in percent between the arithmetic and geometric means of the distribution among the poor of the indicator corresponding to each dimension, we observe a greater degree of inequality, in both years, for the literacy rates than for the other two dimensions. Thus, in 1993 (2002) this percentage gap was equal to 0.3% (1.0%) for the per capita GDP, 1.8% (2.8%) for the life expectancy, and 3.5% (6.1%) for the literacy rate. Let us finally take a look at the value of the normalizing coefficient g whose definition was given just after Eqn. (17). This coefficient is equal to the ratio of the sum over the ‘‘union’’ of the number of poor computed on the basis of the three dimensions to the total number of poor. It should be clear that the greater the
Table 3. Results of the Shapley decomposition of the poverty change between 1993 and 2002 (all countries included)
WORLD DEVELOPMENT
Value of the Watts multidimensional poverty index
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ON THE WATTS MULTIDIMENSIONAL POVERTY INDEXAND ITS DECOMPOSITION
correlation between the individuals classified as poor according to the different dimensions, the smaller is g. Since we observe in Table 2 that the coefficient g decreased during 1993–2002 from 2.065 to 1.877, we may conclude that the correlation increased over the period 1993–2002, meaning that those countries found to be poor according to one dimension are more likely to be classified as poor according to another dimension in 2002 than in 1993. Table 3 gives the contributions of the various determinants of the multidimensional Watts poverty index to the overall variation of the index observed during 1993–2002, where the contributions have been computed on the basis of the Shapley decomposition (see Appendix B). We observe that world poverty, computed on the basis of the three dimensions given by the per capita GDP, life expectancy, and literacy rate, decreased by close to 50% during 1993– 2002 (from 0.247 to 0.131). The whole change was in fact a consequence of the decrease in the overall headcount ratio (see Table 1). The contributions of the other determinants (normalization coefficient g, the weights rj of the poverty dimensions, the Watts income gap ratio, and the Theil–Bourguignon index of inequality among the poor) were quite small and canceled out (see Table 3). In the second stage of the analysis we excluded China and India whose weight in the world population is very high. The results of these estimations are given in Tables 4 and 5.
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It then appears that both in 1993 and in 2002 the weights of the three dimensions were almost equal. (We recall that the weight of a given dimension is equal to the ratio of the number of the poor computed on the basis of that dimension to the sum of the number of poor computed on the basis of the different dimensions). We also observe that whereas when all countries are included, the share of the poor (all dimensions included) in the world population decreased significantly during 1993–2002 (from 36.1% to 19.6%), it slightly increased (from 31.6% to 32.2%) when China and India are excluded. As far as the Watts poverty gap ratios for different dimensions are concerned, we observe a significant increase for the per capita GDP (from 0.14 to 0.33), a slight decrease for the life expectancy (from 0.36 to 0.31), and a significant decrease for the literacy rate (from 0.51 to 0.35). Similar trends were observed for the Theil–Bourguignon index of inequality among the poor since the index increased for the per capita GDP and decreased for the other two dimensions. Table 5 finally gives the contribution of each of the five determinants of the multidimensional Watts poverty index to its change during 1993–2002, when China and India are excluded. The results are quite different from what was observed when these two countries were included in the analysis. The decrease in the Watts index was much smaller (from 0.252 to 0.216) and more than two-thirds of this
Table 4. Values in 1993 and 2002 of the determinants of the multidimensional Watts poverty index, broken down by poverty dimensions (all countries but China and India) Headcount ratio, H0
Year 1993 0.31600
Headcount ratio, H1
Year 2002 0.32179
Coefficient, g0
Poverty dimension
Weight of the poverty dimension, rj0
‘‘Watts poverty gap ratio’’ for the dimension, PW,PGRj0
Theil–Bourguignon index of inequality among the poor for the dimension, Lpj0
2.12263
Per capita GDP Life expectancy Literacy rate
0.31386 0.35823 0.32790
0.14197 0.35685 0.51020
0.00313 0.03034 0.07403
Coefficient, g1
Poverty dimension
Weight of the poverty dimension, rj1
‘‘Watts poverty gap ratio’’ for the dimension, PW,PGRj1
Theil–Bourguignon index of inequality among the poor for the dimension Lpj1
1.87709
Per capita GDP Life expectancy Literacy rate
0.33345 0.31392 0.35262
0.20608 0.31571 0.44018
0.00959 0.02845 0.06130
0.00183 0.01079 0.00047 0.00424 0.03657 0.21551 0.25208
0.02866
Contribution of the Theil– Bourguignon index of inequality among the poor, Lpj Contribution of the ‘‘Watts income gap ratio,’’ PW,PGRj Contribution of the weights, rj of the poverty dimensions Contribution of the headcount ratio, H DW In 2002
Contribution of the coefficient, g
decrease were due to a decrease in the value of the parameter g, that is, to an increase in the degree of correlation between the three dimensions of poverty on which this analysis is based. The other component which played a role in the decrease in the Watts index is the change in the Watts poverty gap ratio, which, as we saw, decreased for the life expectancy and the literacy rate and increased for the per capita GDP.
5. CONCLUDING REMARKS
In 1993
Value of the Watts multidimensional poverty index
Variation during 1993 and 2002 in the value of the Watts multidimensional poverty index
Contributions of the various components to the overall change in the Watts multidimensional poverty index
WORLD DEVELOPMENT
Table 5. Results of the Shapley decomposition of the poverty change during 1993–2002 (all countries but China and India)
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In this paper we have shown that a multidimensional extension of the Watts poverty index may be expressed as a function of five determinants which are, respectively, what we define as the Watts poverty gap ratio, the Bourguignon (1979)–Theil (1967) index of inequality among the poor, the overall headcount ratio, the weights of the various dimensions, and some measure of correlation between the various dimensions of poverty. This enables us to identify the causal factors of poverty and hence to implement poverty reduction policies. We then explained how the Shapley decomposition allows one to derive the contribution of each of these five determinants to the change over time in the value of the multidimensional extension of the Watts poverty index. An empirical illustration of the index was then provided using the per capita GDP, life expectancy, and literacy rate data for several countries for the periods 1993–2002. We observed that when all the countries are included most of the change in world poverty during 1993–2002 was related to the decrease in the overall headcount ratio. However when China and India are excluded, two-thirds of the decrease in the Watts index are due to an increase in the correlation between the various dimensions and one-third to changes in the Watts poverty gap ratio. NOTES 1. Several contributions suggested alternatives to and variations of the Sen index. See, for example, Takayama (1979), Blackorby and Donaldson (1980), Kakwani (1980), Clark, Hamming, and Ulph (1981), Chakravarty (1983a, 1983b, 1997), Foster, Greer, and Thorbecke (1984), Thon (1983), and Shorrocks (1995).
ON THE WATTS MULTIDIMENSIONAL POVERTY INDEXAND ITS DECOMPOSITION
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REFERENCES Blackorby, C., & Donaldson, D. (1980). Ethical indices for the measurement of poverty. Econometrica, 58, 1053–1060. Bourguignon, F. (1979). Decomposable income inequality measures. Econometrica, 47, 901–920. Bourguignon, F., & Chakravarty, S. R. (2003). The measurement of multidimensional poverty. Journal of Economic Inequality, 1, 25–49. Chakravarty, S. R. (1983a). A new index of poverty. Mathematical Social Sciences, 6, 307–313. Chakravarty, S. R. (1983b). Ethically flexible measures of poverty. Canadian Journal of Economics, 16, 74–85. Chakravarty, S. R. (1997). On Shorrocks’ reinvestigation of the Sen poverty index. Econometrica, 65, 1241–1242. Chakravarty, S. R., & Silber, J. (2008). Measuring multidimensional poverty: The axiomatic approach. In N. Kakwani, & J. Silber (Eds.), Quantitative approaches to multidimensional poverty measurement. London: Palgrave MacMillan. Clark, S., Hamming, R., & Ulph, D. (1981). On indices for the measurement of poverty. Economic Journal, 91, 515–526. Foster, J. E., Greer, J., & Thorbecke, E. (1984). A class of decomposable poverty measures. Econometrica, 42, 761–766. Kakwani, N. C. (1980). On a class of poverty measures. Econometrica, 48, 437–446. Lipton, M., & Ravallion, M. (1995). Poverty and policy. In J. Behrman, & T. N. Srinivasan (Eds.), Handbook of development economics (Vol. 3). Amsterdam: North-Holland. Maasoumi, E., & Lugo, M. A. (2008). The information basis of multivariate poverty assessments. In N. Kakwani, & J. Silber (Eds.), Quantitative approaches to multidimensional poverty measurement. London: Palgrave MacMillan. Ravallion, M. (1996). Issues in measuring and modeling poverty. Economic Journal, 106, 1328–1343. Sastre, M., & Trannoy, A. (2002). Shapley inequality decomposition by factor components: Some methodological issues. Journal of Economics, (Suppl. 9), 51–89. Sen, A. K. (1976). Poverty an ordinal approach to measurement. Econometrica, 44, 219–231. Sen, A. K. (1981). Poverty and famines. Oxford: Clarendon Press. Sen, A. K. (1985). Commodities and capabilities. Amsterdam: North-Holland. Sen, A. K. (1992). Inequality re-examined. Cambridge, MA: Harvard University Press. Shapley, L. S. (1953). A value for n-person games. In H. W. Kuhn, & A. W. Tucker (Eds.), Contributions to the theory of games II, Annals of mathematics studies, Princeton studies. Princeton: Princeton University Press. Shorrocks, A. F. (1980). The class of additively decomposable inequality measures. Econometrica, 48, 613–625. Shorrocks, A. F. (1995). Revisiting the Sen poverty index. Econometrica, 63, 1225–1230.
Shorrocks, A. F. (1999). Decomposition procedures for distributional analysis. a unified framework based on the Shapley value. mimeo, University of Essex. Streeten, P. (1981). First things first: Meeting basic human needs in developing countries. Yew York: Oxford University Press. Takayama, N. (1979). Poverty, income inequality and their measures: Professor Sen’s axiomatic approach reconsidered. Econometrica, 47, 749–759. Theil, H. (1967). Economics and information theory. North Holland: Amsterdam. Thon, D. (1983). A poverty measure. Indian Economic Journal, 30, 55–70. Tsui, K. Y. (2002). Multidimensional poverty indices. Social Choice and Welfare, 19, 69–93. UNDP (1997). Human development report. Oxford: Oxford University Press. Watts, H. (1968). An economic definition of poverty. In D. P. Moynihan (Ed.), On understanding poverty. New York: Basic Books.
APPENDIX A. EQUIVALENCE BETWEEN (8) AND (9) P WM ðX ; cÞ
(
"n # pj m X X ðnpj =np Þ ð1=npj Þ logðcj =spj Þ ¼ ðnp =nÞ j¼1
þ
m X
ðnpj =np Þ
j¼1
"n pj X
i¼1
#)
ð1=npj Þ logðspj =sij Þ
i¼1
or,
P WM ðX ; cÞ
(
"n # pj m X X ðnpj =np Þð1=npj Þ logðcj =spj Þ ¼ ðnp =nÞ j¼1 i¼1 " npj m X X ðnpj =np Þ ð1=npj Þ logðspj Þ þ j¼1
ð1=npj Þ
npj X
#)i¼1 logðsij Þ
i¼1
or,
P WM ðX ; cÞ
(
¼ ðnp =nÞ
m X ðnpj =np Þð1=npj Þ½npj logðcj =spj Þ j¼1
m X þ ðnpj =np Þ ð1=npj Þnpj logðspj Þ j¼1
ð1=npj Þ
npj X i¼1
#) logðsij Þ
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WORLD DEVELOPMENT
There are obviously 3! = 6 ways of ordering these three determinants a, b, and c: ða; b; cÞ; ða; c; bÞ; ðb; a; cÞ; ðb; c; aÞ; ðc; a; bÞ; ðc; b; aÞ:
or,
P WM ðX ; cÞ
( m X ¼ ðnp =nÞ ðnpj =np Þðlogðcj =spj ÞÞ j¼1
þ
"
npj m X X ðnpj =np Þ logðspj Þ ð1=npj Þ logðsij Þ
#)
i¼1
j¼1
or,
P WM ðX ; cÞ
(
¼ ð1=nÞ
m X
npj ðlogðcj =spj ÞÞ
j¼1
þ
m X
"
npj logðspj Þ ð1=npj Þ
j¼1
npj X
#) logðsij Þ
i¼1
or,
P WM ðX ; cÞ
( m m X X ¼ ð1=nÞ npj logðcj Þ npj logðspj Þ j¼1
þ
m X
j¼1
) npj m X X npj logðspj Þ ðnpj =npj Þ logðsij Þ
j¼1
i¼1
j¼1
Hence
P WM ðX ; cÞ
(
¼ ð1=nÞ
m X
npj logðcj Þ
j¼1
¼ ð1=nÞ
j¼1
npj m X X j¼1
npj m X X
) logðsij Þ
i¼1
logðcj =sij Þ:
i¼1
APPENDIX B. ON THE CONCEPT OF SHAPLEY DECOMPOSITION AND ITS APPLICATION TO ANALYZING CHANGES IN THE WATTS MULTIDIMENSIONAL POVERTY INDEX The Shapley (1953) decomposition is a technique borrowed from game theory, but it has been extended to applied economics by Shorrocks (1999) and Sastre and Trannoy (2002). Let us explain it briefly. Assume that an indicator I is a function of three determinants a, b, c and is written as I = I(a, b, c). I could be an index of inequality but, more generally, any linear or nonlinear function of the concerned variables is allowable.
ðB:1Þ Each of these three determinants may be eliminated first, second, or third. The respective (marginal) contributions of the determinants a, b, c will hence be a function of all the possible ways in which each of these determinants may be eliminated. Let, for example, C(a) be the marginal contribution of a to the indicator I(a, b, c). If a is eliminated first its contribution to the overall value of the indicator I will be expressed as I(a, b, c) I(b, c), where I(b, c) corresponds to the case where a is equal to zero. Since expression B.1 indicates that there are two cases in which a appears first and may thus be eliminated first, we will give a weight of (2/6) to this possibility. If a is eliminated second, it implies that another determinant has been eliminated first (and been assumed to be equal to 0). Expression (B.1) indicates that the two cases in which this possibility occurs are (b, a, c) and (c, a, b). In the first case the contribution of a will be I(a, c) I(c), while in the second it is I(a, b) I(b). To each of these two cases we evidently give a weight of (1/6). Finally, if a is eliminated third, it implies that both b and c are assumed to be equal to 0. Expression (B.1) indicates that there are two such cases, namely, (b, c, a) and (c, b, a). Since we may assume that when each of the three determinants is equal to 0, the indicator I is equal to 0, we may say that the contribution of a in this case will be equal to I(a) 0 = I(a) and evidently we have to give a weight of (2/6) to such a possibility since there are two such cases. We may therefore summarize what we have just explained by stating that the marginal contribution C(a) of the determinant a to the overall value of the indicator I is CðaÞ ¼ ð2=6Þ½Iða; b; cÞ Iðb; cÞ þ ð1=6Þ ½Iða; cÞ IðcÞ þ ð1=6Þ½Iða; bÞ IðbÞ þ ð2=6ÞIðaÞ:
ðB:2Þ
One can similarly determine the marginal contribution C(b) of b and C(c) of c and then find out that Iða; b; cÞ ¼ CðaÞ þ CðbÞ þ CðcÞ: ðB:3Þ This Shapley decomposition may be also applied in a similar way to the case where one wants to understand the respective contribu-
ON THE WATTS MULTIDIMENSIONAL POVERTY INDEXAND ITS DECOMPOSITION
tions to the change over time in the value of the indicator I. We denote this change in I by DI, and Da, Db, and Dc represent respectively the variations over time in the values of the three determinants a, b, and c. Since in (19) the change in the multidimensional Watts poverty index is expressed as a function of five variables, we have to extend the Shapley decomposition described above to the five determinants. For simplicity, we denote the five determinants DH, Dg, Drj, DPW,PGR,j, DLpj of the change in the Watts index that appear in (19) by a, b, c, d, and e, respectively. More precisely, a = DH, b = Dg, c = Drj, d = DPW,PGR,j, and e = DLpj where c, d, and e refer to the simultaneous changes in all the dimensions j of Drj, DPW,PGR,j, and DLpj. It is then easy to show that the contribution C(a) of a in such a case, assuming that I refers to the multidimensional Watts index, becomes CðaÞ ¼ ð24=120Þ½Iða; b; c; d; eÞ Iðb; c; d; eÞ þ ð6=120Þ½Iða; c; d; eÞ Iðc; d; eÞ þ ð6=120Þ½Iða; b; d; eÞ Iðb; d; eÞ þ ð6=120Þ½Iða; b; c; eÞ Iðb; c; eÞ þ ð6=120Þ½Iða; b; c; dÞ Iðb; c; dÞ þ ð4=120Þ½Iða; b; cÞ Iðb; cÞ þ ð4=120Þ½Iða; b; dÞ Iðb; dÞ þ ð4=120Þ½Iða; b; eÞ Iðb; eÞ þ ð4=120Þ½Iða; c; dÞ Iðc; dÞ þ ð4=120Þ½Iða; c; eÞ Iðc; eÞ þ ð4=120Þ½Iða; d; eÞ Iðd; eÞ þ ð6=120Þ½Iða; bÞ IðbÞ þ ð6=120Þ½Iða; cÞ IðcÞ þ ð6=120Þ½Iða; dÞ IðdÞ þ ð6=120Þ½Iða; eÞ IðeÞ þ ð24=120ÞIðaÞ:
ðB:4Þ
One can similarly compute the marginal contributions of b, c, d, and e and here also the sum of the contributions of a, b, c, d, and e is equal
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to the value of the indicator I. To understand well (B.4) it should be remembered that when a determinant appears in one of the lines of (B.4) it means that it is different from 0 whereas if it does not appear it means it is equal to 0. Thus, for example, the line before the last one, which is given by (6/120) [I(a, e) I(e)], should be translated as ð6=120Þ½DP WM ðDH 6¼ 0; Dg ¼ 0; Drj ¼ 0; DP W;PGRj ¼ 0; DLpj 6¼ 0Þ DP WM ðDH ¼ 0; Dg ¼ 0; Drj ¼ 0; DP W;PGRj ¼ 0; DLpj 6¼ 0Þ:
ðB:5Þ
Note that in (B.5), DH = 0 means that when computing the change in the value of the Watts index one has to assume that the headcount ratio did not change between times 0 and 1, whereas DH 5 0 will mean that headcount ratio changed. Similar interpretations hold concerning the changes Dg, Drj, DPW,PGR,j, and DLpj. We note that the decomposition coefficients, that is, the weights assigned to the marginal contributions of the different determinants, depend on the number of possible orderings of the determinants. Hence they are independent of basic attributes as well as across terms. We note also the independence of the population weights rj’s in the attribute-wise decomposition of the Watts index given by (17). This is, in fact, an extension of Shorrocks’ (1980) arguments for subgroup decomposable income inequality indices to the multidimensional case. He showed that only for two inequality indices, namely, the Theil (1967) entropy index and the Bourguignon–Theil mean logarithmic deviation index, the decomposition coefficients are independent of the between group contribution. The poverty index we use here is based on the multidimensional extension of the latter index. Finally, it may be worthwhile to mention that the Shapley decomposition technique becomes applicable to a wide class of poverty indices that share the same structure as the one given by (8). This becomes evident from the equivalence between (8) and (9).
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