On various ways of measuring pro-poor growth

Deutsch, Joseph; Silber, Jacques Article On various ways of measuring pro-poor growth Economics: The Open-Access, Open-Assessment E-Journal Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Deutsch, Joseph; Silber, Jacques (2011) : On various ways of measuring pro-poor growth, Economics: The Open-Access, Open-Assessment E-Journal, ISSN 1864-6042, Kiel Institute for the World Economy (IfW), Kiel, Vol. 5, Iss. 2011-13, pp. 1-57, https://doi.org/10.5018/economics-ejournal.ja.2011-13 This Version is available at: https://hdl.handle.net/10419/50735 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc/2.0/de/deed.en Vol. 5, 2011-9 | August 1, 2011 | http://dx.doi.org/10.5018/economics-ejournal.ja.2011-9 Vol. 5, 2011-11 | August ???, 2011 | http://dx.doi.org/10.5018/economics-ejournal.ja.2011-11 Vol. 5, 2011-13 | September 9, 2011 | http://dx.doi.org/10.5018/economics-ejournal.ja.2011-13 On Various Ways of Measuring Pro-Poor Growth Joseph Deutsch and Jacques Silber Bar-Ilan University, Ramat-Gan Abstract This paper examines three possible approaches to pro-poor growth. The first one assumes that the poverty line remains constant in real terms over time. The second perspective examines the case where the poverty line is equal to half the median of the income distribution but assumes that such a poverty line is determined exogenously. Finally we also propose a third type of decomposition of the change in poverty, one which is obtained when the poverty line is assumed to be endogenous. In addition, whatever the assumption made concerning the poverty line, we take both a relative and an absolute approach to inequality measurement when defining pro-poor growth. With a relative approach to pro-poor growth it is assumed that inequality does not to vary when all incomes are multiplied by a constant whereas, with an absolute approach to pro-poor growth, inequality is supposed not to vary when an equal sum is added to all incomes. The empirical illustration covers the period 1990–2006 in Israel and the analysis is based on the use of the FGT poverty index. It turns out that the assumptions made concerning the way the poverty line is defined and the choice between a relative and an absolute approach to pro-poor growth greatly affect the results. As a whole however growth was pro-rich in Israel during the 1990–2006 period. Special Issue The Measurement of Inequality and Well-Being: New Perspectives JEL I32, O15 Keywords Inequality; Israel; pro-poor growth; Watts index Correspondence Jacques Silber, Department of Economics, Bar-Ilan University, 52900 Ramat-Gan, Israel; e-mail: [email protected] Citation Joseph Deutsch and Jacques Silber (2011). On Various Ways of Measuring Pro-Poor Growth. Economics: The Open-Access, Open-Assessment E-Journal, Vol. 5, 2011-13. http://dx.doi.org/10.5018/economicsejournal.ja.2011-13 © Author(s) 2011. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany www.economics-ejournal.org 1 1 Introduction The concept of pro-poor growth has become very popular during the last decade. It reflects the idea that economic growth should affect all the segments of society and this is why the term “inclusive growth” is also often used. There are however various ways of understanding the term “pro-poor”. Some would argue that growth is pro-poor when it raises the incomes of the poor. Others consider 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 (see, Kakwani et al., 2004, and Ravallion, 2004, for more details on these two approaches). Dollar and Kraay (2002) thus found, on the basis of a large cross-country data set, that the incomes of the individuals who belong to the two poorest deciles of the income distribution rise on average at the same rate as the mean income. Another issue that should be stressed is that most studies of pro-poor growth take an anonymous approach in the sense that they are usually based on crosssections and do not follow individuals over time, as would have been possible, had panel data been available (see, however Grimm, 2007, and Nissanov and Silber, 2009, for an approach to the topic that does not assume anonymity). Finally because studies on pro-poor growth have generally looked at developing countries they took an absolute approach to the definition of the poverty line in the sense that they assumed a constant (in real terms) poverty line. When looking at poverty in developed countries one tends however to define the poverty line in relative terms, that is, to assume that it is equal to some percentage of the median or mean standardized income. In such a case the issue of pro-poor growth becomes clearly quite different. The present paper aims precisely at checking whether growth was pro-poor in a country which is considered today as a developed country, Israel. We start (Section II) with a short review of the literature on pro-poor growth measurement. Then in a more methodological section (Section III) we make a distinction between three possible approaches to pro-poor growth analysis. The first one assumes that the poverty line remains constant in real terms over time (what was earlier labeled the absolute approach to poverty). The second perspective examines the case where the poverty line is equal to half the median of the income distribution (what was called previously the relative approach to poverty) but assumes that such a poverty line is determined exogenously. Since such a www.economics-ejournal.org 2 definition of the poverty line implies in fact that it will vary whenever there is growth or inequality change, we also propose a third type of decomposition of the change in poverty, the one which one obtains when the poverty line is assumed to be endogenous. In addition, whatever the assumption made concerning the poverty line, we took both a relative and an absolute approach to inequality measurement when defining pro-poor growth. With a relative approach to pro-poor growth it is assumed that inequality does not to vary when all incomes are multiplied by a constant whereas, with an absolute approach to pro-poor growth, inequality is supposed not to vary when an equal sum is added to all incomes. Section IV is then devoted to an empirical illustration based on the annual Israeli income surveys, during the period 1990–2006. We first draw the Growth Incidence Curve (see, Ravallion and Chen, 2003) for the standardized net income, the latter being equal to the ratio of the household's net income over the square root of the number of individuals in the household (see, Buhman et al., 1988, for more details on such an approach to the issue of equivalence scales). Then we give the results of the various decompositions of the changes over time in the poverty index. As poverty index we selected the popular FGT index. Concluding comments are given in Section V. 2 On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature During the past ten years or so there have been various suggestions concerning the way one could check whether economic growth was in favor of the poor. The present section gives a quick survey of the different proposals that have appeared in the literature to measure “pro-poor growth”. Before reviewing these contributions a distinction should be made between an absolute and a relative approach to this topic. Thus some studies (see, Baulch and McCulloch, 2002, or Kakwani and Pernia, 2000) consider that growth will be propoor if poverty falls more than it would have fallen, had all incomes grown at the same rate. This is therefore a “relative approach” in the sense that a pro-poor growth requires that the incomes of the poor grow at a higher rate than those of the non-poor. www.economics-ejournal.org 3 It is however also possible to take an "absolute approach" to poverty. In such a case growth will be assumed to be “pro-poor” if the standard living of the poor people has improved. Whatever approach one selects it should be clear that the answer to the question “was growth “pro-poor”?” will depend on the measure of poverty that is selected and the poverty line that is adopted. 2.1 The Ravallion and Chen (2003) Definition of “Pro-poor” Growth Ravallion and Chen (2003) have proposed an interesting tool to measure the impact of growth on poverty. They called it the “Growth Incidence Curve” (GIC) and it is defined as follows. On the horizontal axis plot the various percentiles of the income (or consumption) distribution1. As a consequence at the 50th percentile the Growth Incidence Curve will indicate the growth rate of the median income. Clearly if the curve is above the horizontal axis at all points up to some percentile p ~, we can conclude that poverty has fallen when it is measured via the headcount ratio and the poverty line is not greater than p ~ (see, Atkinson, 1987). Note that the area under the growth incidence curve up to the headcount ratio will give the total growth in incomes of the poor during the period under analysis. Ravallion and Chen (2003) have thus defined the “pro-poor growth rate” as the mean growth rate of the poor. They have also shown that it is equal to the change in the Watts poverty index per unit of time, divided by the headcount ratio. There is clearly a difference between this mean growth rate of the poor and the growth rate of the mean income (consumption) of the poor. 2.2 The Baulch and McCulloch (2002) Approach “Pro-poor” growth may be also analyzed from a different angle. Since an index of poverty can usually be expressed as a function of the mean of the distribution of the variable on the basis of which this index is computed and of the Lorenz curve _________________________ 1 Naturally, if one works with data collected at the household level, the variable should be some standardized income or consumption level, the normalization depending on the equivalence scale that is chosen. www.economics-ejournal.org 4 corresponding to this distribution, it is generally possible to decompose a change in poverty (in the poverty index) into elements measuring respectively the impact of the growth rate of the mean income (consumption), that of the changes in the distribution (variations in the degree of inequality of the distribution) and generally some interaction effect (see, for example, Datt and Ravallion, 1992). Then growth will be defined as “distribution neutral” (corresponding to a flat Growth Incidence Curve) if the redistribution component that was just mentioned is nil whereas it will be “pro-poor” if this redistribution component is negative. In other words Baulch and McCulloch (2002) derive their measure of pro-poor growth by comparing the actual distribution of income with the one that would have been observed, had there been no change in the distribution of incomes (that is, had growth been “distribution-neutral”). Note that Kakwani (2000) proposed a decomposition which does not include any interaction effect (see, Appendix A for more details) 2.3 The Kakwani and Pernia (2000) Approach These authors defined first what they called the total poverty elasticity of growth, that is, the percentage change in poverty when the growth in the mean income (consumption) is equal to 1%. They then defined a second elasticity which measures the percentage change in poverty that is observed when the growth in mean income (consumption) is equal to 1% and there is no change over time in relative inequality. For Kakwani and Pernia (2000) the Pro-Poor Growth index (PPGI) is equal to the ratio of these two elasticities and they concluded that growth is pro-poor if this ratio PPGI is greater than one. Note that if there is negative growth, growth will be defined as pro-poor in relative terms if the relative loss in income from negative growth is smaller for the poor than for the non-poor, that is if the ratio PPGI is smaller than one (see, Appendix A for more details on this approach). www.economics-ejournal.org 5 2.4 The Approach of Kakwani and Son (2002) It may be observed that the concept of PPGI that was just defined does not take into account the actual level of growth that is observed. This is why Kakwani and Son (2002) have defined what they call the “poverty equivalent growth rate” )(PEGR . The PEGR refers to the growth rate that would result in the same level of poverty reduction as the one actually observed, assuming there had been no change in inequality during the growth process. Growth will therefore be assumed to be pro-poor if the PEGRis higher than the actual growth rate. If the PEGR is positive but smaller than the actual growth rate, it implies that growth is accompanied by an increase in inequality but a reduction in poverty is still observed. In such a case Kakwani et al. (2004) talk about a “trickle down” process where the poor receive proportionally less benefits from growth than the non-poor. Finally, if the PEGR is negative, we have the case where positive economic growth leads to an increase in poverty. 2.5 The Approach of Son (2004) Son (2004) defined what she called a poverty growth curve (PGC). It is defined as follows. Let )(pg refer to the growth rate of the mean income (consumption) of the bottom p percent of the population. By plotting )(pg on the vertical axis against pon the horizontal axis one obtains what Son (2003) called a Poverty Growth Curve. It should be clear that if 0)( fpg (0)( ppg ) for all p , poverty decreased (increased) during the period under examination. If )(pg is greater than the average growth rate for all %100pp, one can conclude that growth was pro-poor. If )(pg is positive for all %100pp but smaller than the average growth rate, one can then conclude that growth reduced poverty but during the period inequality increased. Such a situation could refer to what has been called a “trickle down growth”, a situation where growth reduces poverty but the benefits of growth are smaller for the poor than for the non-poor. Finally if )(pg is negative for all %100pp, we have a situation where the increase in inequality more than “compensates” growth so that the net effect of growth is to increase poverty, a situation which corresponds to what has been called “immiserizing growth”. www.economics-ejournal.org 6 One may wonder what difference there is between a Growth Incidence Curve (GIC) and a Poverty Growth Curve (PGC). As stressed by Son (2003) it can be shown that the GIC is derived from first-order stochastic dominance while the PGC is based on second-order stochastic dominance. Since second-order stochastic dominance is more likely to hold than first-order, the PGC should provide more conclusive results, although it is based on stronger assumptions. Son (2004) emphasizes another potential advantage of the PGC. Since the GIC is based on individual data while the PGC implies estimating the growth rate of the mean income (consumption) up to the th p percentile, the latter procedure is somehow less prone to measurement errors. 3 Methodological Considerations for the Empirical Implementation While most of the studies mentioned previously used formulations based on a continuous approach to the topic, we prefer to work in discrete terms, among other reasons because we also want to check the existence of pro-poor growth over periods that are longer than a year. In what follows a distinction will be made between a relative and an absolute approach to pro-poor growth. Moreover we will also examine several cases, as far as the definition of the poverty line is concerned. As is well known, a distinction has to be made between an absolute and a relative poverty line. An absolute poverty line is a threshold expressed in terms of a level of expenditures or income (in real terms) that covers basic needs. Such a poverty line will therefore not depend on the rate of economic growth and will not vary when there is an increase in living standards. A relative poverty line, on the contrary, depends on the rate of economic growth since usually it is defined as being equal to some percentage of the average or median income. Both approaches can naturally be criticized. On one hand an absolute poverty line does not take into account the fact that what is viewed as basic needs varies over time and that people do make comparisons between their standard of living and that of others. On the other hand when a relative poverty line is adopted, poverty can never disappear and if there is economic growth together with an increase in income inequality, one may observe at the same time an increase in relative poverty and a decrease in absolute poverty. www.economics-ejournal.org 7 But even if one adopts a relative poverty line so that the latter will vary over time, it can be considered as exogenous or assumed to be endogenously determined (e.g. when it is defined as being equal to half the median of the relevant income distribution). In the following subsection we examine the case of relative pro-poor growth, assuming the poverty line remains constant over time. The cases of absolute pro-poor growth when it is assumed that there is a constant poverty line and that where the poverty line varies over time are examined in Appendix C. 3.1 The Case of Relative Pro-poor Growth, Assuming the Poverty Line Remains Constant over Time Let },...,{}{ 1n xxx = and },...,{}{ 1n yyy = represent the vector of incomes at times 0 and 1 and let )(x θ and )(y θ refer to the poverty index at times 0 and 1. Finally let ))(/( x θ θ Δrefer to the relative change in the poverty index between times 0 and 1, with )()()( xy θ θ θ − =Δ . Assuming no change in the poverty line z, the relative change θ θ /)(Δ in the poverty index will now be expressed as )),/((/)( R Ixxg ΔΔ=Δ θθ (1) where x is the mean income of the distribution given by }{x, )( xyx −= Δ is the difference between the average income at times 0 and 1 and R I refers to some relative measure of income inequality of the distribution given by )(x. Using the concept of Shapley decomposition (see, Shorrocks, 1999, Sastre and Trannoy, 2002, and Appendix B, for more details), θ θ /)( Δ may be written as )()/()/( R ICxxC Δ+Δ=Δ θθ (2) where )/( xxC Δrefers to the contribution of the relative change over time in the average income and )( R IC Δto the contribution of the change in relative inequality between time 0 and time 1. The contribution )/( xxC Δ may itself be expressed as www.economics-ejournal.org 14 Table 1: The Relative Approach to Pro-poor Growth, with a Constant Poverty Line Decomposition of the Actual Percentage Change in Poverty Indices into “Pure Growth” and “Pure Inequality Change” Components—the Case of Standardized Net Income Period Actual percentage change in the FGT poverty index (with the parameter α equal to 2) Hypothetical percentage change in the FGT poverty index (with the parameter α equal to 2), assuming growth without inequality change Hypothetical percentage change in FGT poverty index (with the parameter α equal to 2), assuming there was only a change in inequality and no growth Total Gr In 1990-1991 0.021 0.015 0.005 1991-1992 0.055 -0.081 0.136 1992-1993 -0.068 0.041 -0.109 1993-1994 -0.046 -0.201 0.156 1994-1995 -0.286 -0.076 -0.210 1995-1996 0.125 0.094 0.031 1996-1997 0.073 -0.213 0.286 1997-1998 -0.153 -0.060 -0.092 1998-1999 -0.182 -0.166 -0.016 1999-2000 -0.097 -0.072 -0.025 2000-2001 -0.085 -0.071 -0.014 2001-2002 0.516 0.218 0.298 2002-2003 -0.005 -0.018 0.013 2003-2004 0.101 -0.074 0.175 2004-2005 -0.097 -0.121 0.024 2005-2006 -0.086 -0.071 -0.015 1990-2000 -0.483 -0.539 0.056 2000-2003 0.380 0.108 0.272 2003-2006 -0.092 -0.263 0.171 1990-2006 -0.352 -0.719 0.367 www.economics-ejournal.org 15 Table 2: The Relative Approach to Pro-poor Growth, with a Constant Poverty Line Annual Measures of Pro-poor Growth—the Case of Standardized Net Income Period Actual Poverty Elasticity of Growth (FGT index with the parameter α equal to 2) Hypothetical Poverty Elasticity of Growth, assuming no change in inequality (FGT index with the parameter α equal to 2) Pro-Poor Growth Index (FGT index with the parameter α equal to 2) Poverty Equivalent Growth Rate (FGT index with the parameter α equal to 2) Actual Growth Rate of Standardized Net Income δ η PPGI PEGR 1990-1991 -3.641 -2.687 1.355 -0.008 -0.006 1991-1992 1.843 -2.718 -0.678 -0.020 0.030 1992-1993 4.318 -2.631 -1.641 0.026 -0.016 1993-1994 -0.580 -2.561 0.227 0.018 0.079 1994-1995 -9.263 -2.460 3.765 0.116 0.031 1995-1996 -4.413 -3.325 1.327 -0.038 -0.028 1996-1997 0.954 -2.775 -0.344 -0.026 0.077 1997-1998 -6.170 -2.440 2.529 0.063 0.025 1998-1999 -2.683 -2.447 1.097 0.075 0.068 1999-2000 -3.596 -2.683 1.340 0.036 0.027 2000-2001 -3.189 -2.668 1.195 0.032 0.027 2001-2002 -7.905 -3.346 2.363 -0.154 -0.065 2002-2003 -0.731 -2.491 0.294 0.002 0.007 2003-2004 3.507 -2.559 -1.371 -0.039 0.029 2004-2005 -1.802 -2.247 0.802 0.043 0.054 2005-2006 -2.759 -2.272 1.214 0.038 0.031 1990-2000 -1.522 -1.699 0.896 0.284 2000-2003 -11.364 -3.231 3.517 -0.118 2003-2006 -0.777 -2.228 0.349 0.041 1990-2006 -0.831 -1.697 0.490 0.207 www.economics-ejournal.org 16 2-The Absolute Approach to Pro-poor Growth: In Table 3 we decompose again the growth rate in poverty into two components, one reflecting “pure growth” and the other the impact of inequality change on poverty, but this time an absolute approach to inequality is taken. Let us first concentrate our attention on the years where poverty significantly increased, that Table 3: The Absolute Approach to Pro-poor Growth, with a Constant Poverty Line Decomposition of the Actual Percentage Change in Poverty Indices into “Pure Growth” and “Pure Inequality Change” Components—the Case of Standardized Net Income Period Actual percentage change in the FGT poverty index (with the parameter α equal to 2) Hypothetical percentage change in the FGT poverty index (with the parameter α equal to 2), assuming growth without inequality change Hypothetical percentage change in FGT poverty index (with the parameter α equal to 2), assuming there was only a change in inequality and no growth Total Gr In 1990-1991 0.021 0.054 -0.033 1991-1992 0.055 -0.287 0.341 1992-1993 -0.068 0.147 -0.214 1993-1994 -0.046 -0.745 0.699 1994-1995 -0.286 -0.283 -0.003 1995-1996 0.125 0.346 -0.221 1996-1997 0.073 -0.850 0.923 1997-1998 -0.153 -0.251 0.099 1998-1999 -0.182 -0.724 0.541 1999-2000 -0.097 -0.322 0.225 2000-2001 -0.085 -0.326 0.241 2001-2002 0.516 1.032 -0.516 2002-2003 -0.005 -0.083 0.078 2003-2004 0.101 -0.348 0.449 2004-2005 -0.097 -0.607 0.510 2005-2006 -0.086 -0.368 0.282 www.economics-ejournal.org 17 is, 2001–2002 and eventually 2003–2004. In 2001–2002 it appears that "pure growth" had the main impact on poverty, its effect being actually much higher than the overall change in poverty. This was a year when the per capita standardized net income decreased by 6.5% (see, Table 2) and although the change in inequality would per se have led to a decrease in poverty this effect did not compensate the strong effect of the negative growth on poverty. Note that this is due to the fact that since growth was negative, richer people, in “dollar terms” lost more than poorer people so that an absolute approach to pro-poor growth should indicate that inequality change per se should have decreased poverty. The picture is different in 2003–2004. There “pure growth” per se would have led to a decrease in poverty and inequality change to an increase. This was a year where the average standardized net income increased by 2.9% (see, Table 2) and clearly the absolute (in "dollar terms") increase in this income was higher for the rich than the poor so that inequality change per se would have led to an increase in poverty, and this effect of inequality change was in fact stronger than the pro-poor effect of “pure growth”. If we now take a look at the year where poverty most decreased (in 1994– 1995) we observe (see, Table 3) that in that year the decline in poverty was only the consequence of “pure growth”. We have also computed the values of the measures PPGI and PEGRof pro-poor growth when an absolute approach to the topic is taken and when working with standardized net income4. It turns out that the Pro-Poor Growth Index ()PPGI was never greater than one and that the only years (1992–1993, 1995– 1996 and 2001–2002) in which the Poverty Equivalent Growth Rate )(PEGR was higher than the actual growth rate (or less negative) were years where actual growth was negative. 4.2.2 The Case of a Poverty Line That Varies over Time but Is Exogenous: 1-The Relative Approach to Pro-poor Growth: The results of this investigation are given in Table 4. Note first that once the poverty line is allowed to vary over time, it is less likely that poverty will decrease, as it did in the case of a constant poverty line. As mentioned previously _________________________ 4 The results may be obtained upon request from the authors. www.economics-ejournal.org 18 Table 4: Decomposition of the Actual Percentage Change in Poverty Indices into Components Reflecting Respectively: “Pure Growth”, “Pure Inequality Change” and Variations in the (Exogenous) Poverty Line—the Case of Standardized Net Income and of a Relative Approach to Pro-poor Growth Period Actual percentage change in the FGT index Hypothetical percentage change in the FGT index, assuming there was only growth Hypothetical percentage change in the FGT index, assuming there was only a change in inequality Hypothetical percentage change in the FGT index, assuming there was only a change in the poverty line 1990-2000 -0.020 -0.757 0.075 0.662 2000-2003 0.232 0.102 0.228 -0.099 2003-2006 0.138 -0.302 0.185 0.255 1990-2006 0.373 -1.108 0.542 0.939 we assumed that the poverty line was equal at times 0 and 1 to half the median of the standardized net income, although at this stage we still consider this poverty line as being exogenously determined. Table 4 shows thus that over the whole period 1990–2006 the FGT index increased by 37.3%. When we decompose these overall variations in poverty we also observe the important contribution of changes in the poverty line. If the only change that had taken place during the period 1990– 2006 had been “pure growth” the FGT index would have decreased by 111%. On the other hand if there had been only a change in the poverty line (with no growth and inequality change), poverty, measured via the FGT index, would have increased by 94%. Finally the change in inequality would, ceteris paribus, have induced an increase of 54%. Table 4 indicates in fact that whatever the period examined, the change inequality would have led to an increase in poverty (assuming no growth and no change in the poverty line). 2-The Absolute Approach to Pro-poor Growth: This case is examined in Table 5 which shows for the last three years of the period examined the decomposition of variations in the poverty indices in the case of both an absolute and a relative approach. The absolute approach is in fact not fit to analyze longer periods because, in periods of negative growth, the average dollar www.economics-ejournal.org 19 decrease in income may be big enough to bring the poorest individuals to a potential negative income, assuming there was only “pure growth”. When comparing the absolute and relative approaches we observe for example during the period 2005–2006 where there was a decrease of 2.6% in the FGT poverty index, that, according to the relative approach, “pure growth” would per se have led to a 7.5% decrease in the FGT index while the “pure inequality change” would have led to an additional decrease of 1.4% in the FGT index. There was however a countervailing effect of the change in the poverty line. The absolute approach on the contrary indicates that the “pure growth” effect should, per se, have led to a decrease of 33.1% in the value of the FGT poverty index while the pure effect of (absolute) inequality change would have led to a 24.2% increase in poverty, most likely because the dollar increase in the incomes of the rich was much higher than the dollar change in the income of the poor. The effect of the variation in the poverty line is evidently the same as in that of the relative case. This illustration shows therefore very clearly how relevant a distinction between a relative and an absolute approach to pro-poor growth is. Table 5: Decomposition of the Actual Percentage Change in Poverty Indices into Components Reflecting Respectively: “Pure Growth”, “Pure Inequality Change” and Variations in the (Exogenous) Poverty Line—the Case of Standardized Net Income: Comparing the Relative and Absolute Approach to Pro-poor Growth (Annual Changes) for Selected Years Approach selected Period Actual percentage change in the FGT index Hypothetical percentage change in the FGT index, assuming there was only growth Hypothetical percentage change in the FGT index, assuming there was only a change in inequality Hypothetical percentage change in the FGT index, assuming there was only a change in the poverty line Absolute 2003-2004 0.155 -0.331 0.423 0.064 Absolute 2004-2005 0.011 -0.571 0.474 0.108 Absolute 2005-2006 -0.026 -0.332 0.242 0.063 Relative 2003-2004 0.155 -0.076 0.168 0.064 Relative 2004-2005 0.011 -0.129 0.031 0.109 Relative 2005-2006 -0.026 -0.075 -0.014 0.064 www.economics-ejournal.org 20 4.2.3 The Case of a Poverty Line That Varies over Time but Is Endogenous: 1-The Relative Approach to Pro-poor Growth: As mentioned previously, once the poverty line is defined as being equal to half the median of the distribution of net standardized income, we cannot consider a change over time in the poverty line as exogenous. It has to be the consequence of either “pure growth” or/and “inequality change”. Once a change in the poverty line is considered as endogenous, we are, once again, left with a simple decomposition of the variation over time in the poverty index into a component reflecting the impact of “pure growth” and one that is the consequence of a “pure inequality change”. The results of this type of analysis are given in Table 6. Note that if one takes a relative approach to pro-poor growth, it turns out that, when a change in the poverty line is considered as being endogenous, there is practically no “pure growth” effect. Such a result was expected because when there is no change in inequality and the poverty line is equal to half the median of the distribution, no important change in poverty should occur. Table 6: Decomposition of the Actual Percentage Change in Poverty Indices into Components Reflecting Respectively “Pure Growth” and “Pure Inequality Change”—the Case of an Endogenous Poverty Line, Standardized Net Income and a Relative Approach to Pro-poor Growth (Annual Changes) Period Actual percentage change in the FGT index Hypothetical percentage change in the FGT index, assuming there was only growth Hypothetical percentage change in the FGT index, assuming there was only a change in inequality 1990-2000 -0.020 0.000 -0.020 2000-2003 0.232 -0.001 0.232 2003-2006 0.138 0.001 0.137 1990-2006 0.373 0.000 0.372 www.economics-ejournal.org 21 Table 7: Decomposition of the Actual Percentage Change in Poverty Indices into Components Reflecting Respectively “Pure Growth” and “Pure Inequality Change”—the Case of an Endogenous Poverty Line and Standardized Net Income: Comparing the Relative and Absolute Approaches for Annual Changes for Selected Years Approach selected Period Actual percentage change in the FGT index Hypothetical percentage change in the FGT index, assuming there was only growth Hypothetical percentage change in the FGT index, assuming there was only a change in inequality Absolute 2003-2004 0.155 -0.239 0.395 Absolute 2004-2005 0.011 -0.416 0.427 Absolute 2005-2006 -0.026 -0.241 0.215 Relative 2003-2004 0.155 0.001 0.154 Relative 2004-2005 0.011 0.000 0.010 Relative 2005-2006 -0.026 -0.001 -0.025 2-The Absolute Approach to Pro-poor Growth: The results of this type of analysis are given in Table 7. Note first that if one takes an absolute approach to pro-poor growth, when a change in the poverty line is considered as being endogenous, there is now both a “pure growth” as well as a “pure inequality change” effect. Such a result was expected because equal additions to all incomes induce indeed a translation of the income distribution and hence of the median and since the poverty line is defined as being equal to half the median, the data will now reveal different amounts of poverty5. Let us take a look, for example, at the period 2003–2004 during which the FGT index rose by 15.5%. The relative approach to pro-poor growth indicates (see, Table 7) that this was only the consequence of inequality change while the absolute approach shows that _________________________ 5 This is simple to show in the case where poverty is measured via the headcount ratio. Assume, for simplicity, three incomes equal respectively to 100, 300 and 600. The poverty line will be equal to 150 and hence a third of the individuals will be poor. Assume now equal additions of 200 to all incomes. The new distribution is {300, 500, 800} and the new poverty line is 250, so that no individual is poor. www.economics-ejournal.org 22 the “pure growth” effect should have led to a decrease in poverty but the impact of (absolute) inequality change was very strong and of opposite direction (this was a period of growth and hence the incomes of rich people increase, in dollar terms, more than those of poor people). So here again we see an important difference between what one may conclude on the basis of a relative and an absolute approach to inequality. 5 Concluding Comments This paper attempted to check whether growth in Israel was pro-poor during the period 1990–2006. It used concepts that have appeared recently in the literature on pro-poor growth, such as that of Pro-Poor Growth Index and Poverty Equivalent Growth Rate. Three basic scenarios were examined. In the first one it was assumed that the poverty line was constant in real terms and hence did not vary over time. The second scenario supposed that the poverty line was equal to half the median of the relevant income distribution. It therefore varied over time but we still assumed that it could be considered as exogenous. The last case we examined was that in which the poverty line was still assumed to be equal to half the median but we considered this poverty line as endogenous. We each time checked whether the change in growth was mainly the consequence of “pure growth” or whether it was also influenced by changes in inequality. Under the second scenario we obviously had also a third possible impact, that of an exogenous change in poverty. Whatever the scenario under scrutiny, we always took first a relative approach to pro-poor growth, that is, one where inequality is assumed not to vary when all incomes are multiplied by a constant, second an absolute approach to pro-poor growth, that is, one where inequality is supposed not to change when an equal sum is added to all incomes. We analyzed the distribution of the standardized net income and selected as poverty measure the FGT index. The following conclusions may be drawn. First it turns out that the assumptions made concerning the way the poverty line was defined and the choice between a relative and an absolute approach to pro-poor growth greatly affected the results. Second it turns out that during the period 1990–2006 as whole growth in Israel was pro-rich rather than pro-poor. This result is obtained when drawing Growth Incidence as well as Poverty Growth Curves. Such a conclusion is www.economics-ejournal.org 23 however not true for the annual growth since years of pro-poor growth alternated with period of pro-rich growth. Third, assuming a constant poverty line in real terms and taking a relative approach to poverty, it appears that the annual Poverty Equivalent Growth Rate ( PEGR ) was generally smaller than the average growth rate (but not always). For broader periods the PEGR was however never greater than the growth rate of the average net standardized income. Fourth, assuming still a constant poverty line but taking an absolute approach to pro-poor growth, we observed that the main impact on poverty was generally that of “pure growth”. Fifth, when it is assumed that the poverty line varies over time but is exogenous, and if one takes a relative approach to pro-poor growth, we observed a very important contribution of changes in the poverty line to the overall change in the FGT index. Finally, when it is assumed that the poverty line varies over time but is endogenous and if one takes a relative approach to pro-poor growth, it turns out, and this was expected, that there is practically no “pure growth” effect. Acknowledgement: The authors acknowledge the financial support of the van Leer Jerusalem Institute and of the Adar Foundation of the Department of Economics at BarIlan University. www.economics-ejournal.org 30 where )(xf is the density function of income x and zis the poverty line. In addition let us assume that 0)/( pxP ∂ ∂, 0)/( 22 fxP ∂∂ , 0),( =zzP and ),( xzP is a homogenous function of degree zero in zand x . It can then be shown, on the basis of Atkinson's theorems (1987), that if 0))(( ≥Δ pL μ for all p , then 0 ≤ Δ θ for all poverty lines and the class of poverty measures that has just been defined (poverty measures that are: nondecreasing, anonymous and obey the principle of transfer). As stressed by Son (2003), it should be clear that if the generalized Lorenz curve shifts upward (downward), one can conclude that poverty decreased (increased). This result is the basis for the derivation by Son (2003) of the concept of poverty growth curves. Let us before remember that the height of the Lorenz curve )(pL may be expressed as μ μ p pL p =)( (A-10) where evidently )(pL refers to the share in total income (consumption) of the p percent poorest individuals in the population while p μ is the mean income (consumption) of these p percent poorest individuals ( μ , as before, represents the average income or consumption in the whole population). Taking logarithms on both sides of (A-10) we then derive that )())(()( pLnpLLnLn p−= μ μ (A-11) If we now take the first difference in (A-11) we obtain (see, Son, 2003) ))(()( pLLnpg μ Δ= (A-12) where )()( p Lnpg μ Δ= is the growth rate of the mean income (consumption) of the bottom p percent of the population. By plotting )(pg on the vertical axis against p on the horizontal axis one obtains what Son (2003) called a Poverty Growth Curve. It should now be clear that if 0)( fpg (0)( ppg ) for all p , poverty decreased (increased) during the period under examination. Note that (A-12) may also be expressed as ))(()( pLLnpg Δ+= γ (A-13) www.economics-ejournal.org 31 where )( μ γ LnΔ= is the growth rate of the mean income (consumption) in the whole population. Expression (A-13) clearly implies that if γ f)(pg for all %100pp, growth was pro-poor since this implies that 0))(( fpLLn Δ , that is the entire Lorenz curve shifted upward (inequality decreased). If γ pp )(0 pg for all %100pp, we can conclude that growth reduced poverty but during the period inequality increased. Such a situation could refer to what has been called a “trickle down growth”, a situation where growth reduces poverty but the benefits of growth are smaller for the poor than for the non-poor. Finally if 0)( ppg for all %100pp (assuming 0fg) we have a situation where the increase in inequality more than "compensates" growth so that the net effect of growth is to increase poverty, a situation which corresponds to what has been called “immiserizing growth”. b) Comparing Poverty Growth Curves and Growth Incidence Curves: Call p xthe income (consumption) level of an individual that is located at the th p percentile. Since it is well known (see, Kakwani, 1980) that the derivative )(' pL of the Lorenz curve may be expressed as )/()(' μ p xpL = (A-14) we derive that )(pLxp μ = . If we now take the logarithms of the latter expression and then its first difference we end up with ))('()( pLLnpr Δ+= γ (A-15) where )(pr refers to the growth rate of the income (consumption) of the individual located at the th p percentile. The plot of )(pr on the vertical axis against that of the cumulative percentages p on the horizontal axis gives us precisely what Ravallion and Chen (2003) called the Growth Incidence Curve. The higher this curve is, the greater the reduction in poverty. One may wonder what difference there is between a Growth Incidence Curve (GIC) and a Poverty Growth Curve (PGC). As stressed by Son (2003) and mentioned previously, the GIC is derived from first-order stochastic dominance www.economics-ejournal.org 32 while the PGC is based on second-order stochastic dominance. Since second-order stochastic dominance is more likely to hold than first-order, the PGC should provide more conclusive results (although, as stressed previously, it is based on stronger assumptions). Son (2004) emphasizes another potential advantage of the PGC. The estimation of )(pr is based on individual data while that of )(pg implies estimating the growth rate of the mean income (consumption) up to the th p percentile, a procedure which is somehow less prone to measurement errors. c) Using the Poverty Growth Curve to Derive an Index of Pro-Poor Growth: As explained previously, the higher the PGC, the greater the reduction in poverty. This is why Kakwani and Son (2006) proposed to use the area under the PGC as a measure of pro-poor growth. More precisely, integrating (A-3) on both sides, they defined a new pro-poor growth rate *g as ∫∫ Δ+== 1 0 1 0)(ln)(* dppLdppgg γ (A-16) It is well known that the Gini index G may be expressed as ∫−= 1 0})]([{2 dppLpG (A-17) Let us now similarly define an inequality index *G as ∫−= 1 0}))](ln()[ln({2*)ln( dppLpG (A-18) One may then prove (see, Kakwani and Son, 2006) that *)ln()2/1(* Gg Δ−= γ (A-19) Expression (A-19) implies that growth is pro-poor if 0*)ln()2/1( pG Δ . www.economics-ejournal.org 33 Appendix B: A Short Summary of the Concept of Shapley Decomposition Let ),( baF be a function depending on two variables a and b. Such a function need not be linear. Although Chantreuil and Trannoy (1999) and Sastre and Trannoy (2002) limited their application of the Shapley value to the decomposition of income inequality, Shorrocks (1999) has shown that such a decomposition could be applied to any function. The idea of the Shapley value is to consider all the possible sequences allowing us to eliminate the variables a and b. Let us start with the elimination of the variable a. This variable may be the first one or the second one to be eliminated. If it is eliminated first, the function ),( baF will become equal to )(bF since the variable a has been eliminated so that in this case the contribution of a to the function ),( baF is equal to ),( baF – )(bF . If the variable a is the second one to be eliminated the function F will then be equal to )(aF . Since both elimination sequences are possible and assuming the probability of these two sequences is the same, we may conclude that the contribution )(aC of the variable a to the function ),( baF is equal to )()2/1()](),()[2/1()( aFbFbaFaC + −= (B-1) Similarly one can prove that the contribution )(bC of the variable bto the function ),( baF is )()2/1()](),()[2/1()( bFaFbaFbC + −= (B-2) Combining (B-1) and (B-2) we observe that ),()()( baFbCaC =+ (B-3) www.economics-ejournal.org 34 Appendix C: The Case of Absolute Pro-Poor Growth When the Poverty Line Is Constant and That Where the Poverty Line Varies over Time I) The Cases of Absolute Pro-Poor Growth When the Poverty Line Is Constant over Time: Using the notations of section III-A let us now express the absolute change )( θ Δ in the poverty index as ),()( A Ixf ΔΔ=Δ θ (C-1) where A I refers to some absolute measure of income inequality and A I Δto the change in this measure of inequality. Using again the concept of Shapley decomposition, )( θ Δ may be written as )()()( A ICxC Δ+Δ=Δ θ (C-2) where )( xC Δrefers to the contribution of the change over time in the average income and )( A IC Δto the contribution of the change in absolute inequality between time 0 and time 1. The contribution )( xC Δ may itself be expressed as )]}0;0()0;0([ )]0;0()0;0(){[2/1()( =Δ=ΔΔ−=Δ≠ΔΔ+ ≠Δ=ΔΔ−≠Δ≠ΔΔ=Δ AA AA IxIx IxIxxC θθ θθ (C-3) Similarly the contribution )( A IC Δ may be written as )]}0;0()0;0([ )]0;0()0;0(){[2/1()( =Δ=ΔΔ−≠Δ=ΔΔ+ =Δ≠ΔΔ−≠Δ≠ΔΔ=Δ AA AAA IxIx IxIxIC θθ θθ (C-4) Combining (C-3) and (C-4) we observe that www.economics-ejournal.org 35 })]({})({[ })]({})({[})]({})({[ )]0;0([ )]0;0([)()( xy xxxy Ix IxICxC A AA θθ θθθθ θ θ −= −−−= =Δ=ΔΔ− ≠Δ≠ΔΔ=Δ+Δ (C-5) Let us now define the expression )0;0( =Δ≠ΔΔ A Ix θ . It is easy to derive that this expression may be also expressed as })({})({ xxx θ θ −Δ+ where })({ xx Δ + θ refers to the poverty rate which is observed in a distribution where the incomes are equal to the original incomes at time 0 plus an amount xyx − =Δ assumed to have been added to every individual. Similarly the expression )0;0( ≠Δ=ΔΔ A Ix θ may be also written as })({})({ xxy θ θ −Δ− where })({ xy Δ − θ refers to the poverty rate which is observed in a distribution where the incomes are equal to the difference between the incomes at time 1 and a sum xyx − = Δ assumed to have been deducted from every individual. We therefore end up with })]}({})({[})]({})({){[2/1( })]]}({})({[[})]({})({[[ })]]({})({[})]({})((){[[2/1()( xxxxyy xxxxx xxyxyxC θθθθ θθθθ θ θ θ θ −Δ++Δ−−= −−−Δ++ − Δ − − −=Δ (C-6) Similarly we can write that })]}({})({[})]({})({){[2/1( })]]}({})({[})]({})({[[ })]]({})({[})]({})({){[[2/1()( xxyxxy xxxxy xxxxyIC A θθθθ θθθθ θθθθ −Δ−+Δ+−= −−−Δ−+ −Δ+−−=Δ (C-7) Combining (C-6) and (C-7) we observe again that, as expected, })({})({)()( xyICxC A θθ −=Δ+Δ (C-8) www.economics-ejournal.org 36 The actual relative change in the poverty index observed between times 0 and 1 will therefore be expressed as AA InGrd +=)/( θθ (C-9) where θ /)( xCGrAΔ= (C-10) and θ /)( AA ICIn Δ= (C-11) where θ , as before is the value of the poverty index at the original period 0. Using (C-9), (C-10), and (C-11) we will now call ∂ the total poverty elasticity of growth, that is, the percentage change in poverty )/( θ θ dwhen the growth in the mean income (consumption) is equal to 1%. Similarly, following Kakwani and Son (2008) call A η the percentage change in poverty ( A Gr ) that is observed when the growth in mean income (consumption) is equal to 1% and there is no change over time in absolute inequality. Kakwani and Son (2008) have then defined the Absolute Pro-Poor Growth Index )( A PPGI as A A PPGI η δ = (C-12) The measure A η is also called the neutral absolute growth elasticity of poverty, that is, the elasticity of poverty with respect to growth when the benefits of growth are shared equally, in the absolute sense of equality, by all the members of society. We may therefore conclude (see, Kakwani and Son, 2008) that growth is pro-poor in the absolute sense if A PPGI is greater than one. Note that if growth is negative, growth will be defined as absolute pro-poor if A PPGI is less than one (the absolute loss of income resulting from negative growth would be smaller for the poor than for the non-poor. Finally Kakwani and Son (2008) have also defined a “poverty equivalent growth rate” ( A PEGR ) in the case where an absolute approach to inequality is www.economics-ejournal.org 37 adopted. A PEGR refers to the growth rate that would result in the same level of poverty reduction as the one actually observed, assuming there had been no change in absolute inequality during the growth process. Let us, as before, call γ the actual growth rate (of the mean income) and let us call A γ the growth rate that would have been observed had there been no change in absolute inequality. If growth is neutral in the absolute sense (that is, when there is growth without change in absolute inequality) the relative change in poverty will be expressed as AA γη . Here again we would like this hypothetical relative change in poverty to be equal to the one which is actually observed and is equal to δγ . It is then easy to conclude that if AA γη = δγ , we must have γγ η δ γ ×=== )()( A A AA PPGIPEGR (C-13) Expression (C-13) implies that growth is pro-poor if A γ is greater than γ . II) The Cases of Relative and Absolute Pro-Poor Growth When the Poverty Line Varies over Time but Is Exogenously Determined: A) The Case of Relative Pro-Poor Growth: Let, as before, },...,{}{ 1n xxx = and },...,{}{ 1n yyy = represent the vector of incomes at times 0 and 1 and let ),( x zx θ and ),( y zy θ refer to the poverty index at times 0 and 1, x zand y z being the corresponding poverty lines. Finally let )),(/( x zx θ θ Δrefer to the relative change in the poverty index between times 0 and 1, with ),(),()( xy zxzy θ θ θ − = Δ. The relative change )),(/( x zx θ θ Δin the poverty index will therefore be expressed as )),/(,()),(/( R xIxxzgzx ΔΔΔ=Δ θθ (C-14) where x is the mean income of the distribution given by }{x, )( xyx −= Δ is the difference between the average income at times 0 and 1, R I refers to some relative www.economics-ejournal.org 38 measure of income inequality of the distribution given by }{x, R I Δ to the change in relative inequality and zΔto the change in the poverty line. Using the concept of Shapley decomposition, )),(/( x zx θ θ Δ may be written as )()()/()),(/( zCICxxCzx R xΔ+Δ+Δ=Δ θθ (C-15) where )/( xxC Δrefers to the contribution of the relative change over time in the average income, )( R IC Δto the contribution of the change in relative inequality and )( zC Δto the contribution of the change in the poverty line between time 0 and time 1. Let us now express the contribution )/( xxC Δ . It may be written as )]}0;0;0)/(()),(/[( )]0;0;0)/(()),(/){[(6/2( )]}0;0;0)/(()),(/[( )]0;0;0)/(()),(/){[(6/1( )]}0;0;0)/(()),(/[( )]0;0;0)/(()),(/){[(6/1( )]}0;0; 0)/(()),(/[( )]0;0;0)/(()),(/){[(6/2()/( =Δ=Δ=ΔΔ− =Δ=Δ≠ΔΔ+ ≠Δ=Δ=ΔΔ− ≠Δ=Δ≠ΔΔ+ =Δ≠Δ=ΔΔ− =Δ≠Δ≠ΔΔ+ ≠Δ≠Δ=ΔΔ− ≠Δ≠Δ≠ΔΔ=Δ zIxxwithz x zIxxwithzx zIxxwithzx zIxxwithzx zIxxwithzx zIxxwithzx zIxxwithzx zIxxwithzxxxC R x R x R x R x R x R x R x R x θθ θθ θθ θθ θθ θθ θθ θθ (C-16) Similarly the contribution )( R IC Δof the change in relative inequality may be written as www.economics-ejournal.org 39 )]}0;0;0)/(()),(/[( )]0;0;0)/(()),(/){[(6/2( )]}0;0;0)/(()),(/[( )]0;0;0)/(()),(/){[(6/1( )]}0;0;0)/(()),(/[( )]0;0;0)/(()),(/){[(6/1( )]}0;0;0)/(()),(/[( )]0;0;0)/(()),(/){[(6/2()( =Δ=Δ=ΔΔ− =Δ≠Δ=ΔΔ+ ≠Δ=Δ=ΔΔ− ≠Δ≠Δ=ΔΔ+ =Δ=Δ≠ΔΔ− =Δ≠Δ≠ΔΔ+ ≠Δ=Δ≠ΔΔ− ≠Δ≠Δ≠ΔΔ=Δ zIxxwithzx zIxxwithzx zIxxwithzx zIxxwithzx zIxxwithzx zIxxwithzx zIxxwithzx zIxxwithzxIC R x R x R x R x R x R x R x R x R θθ θθ θθ θθ θθ θθ θθ θθ (C-17) Finally the contribution of the change in the poverty line will be expressed as )]}0;0;0)/(()),(/[( )]0;0;0)/(()),(/){[(6/2( )]}0;0;0)/(()),(/[( )]0;0;0)/(()),(/){[(6/1( )]}0;0;0)/(()),(/[( )]0;0;0)/(()),(/){[(6/1( )]}0;0; 0)/(()),(/[( )]0;0;0)/(()),(/){[(6/2()( =Δ=Δ=ΔΔ− ≠Δ=Δ=ΔΔ+ =Δ=Δ≠ΔΔ− ≠Δ=Δ≠ΔΔ+ =Δ≠Δ=ΔΔ− ≠Δ≠Δ=ΔΔ+ =Δ≠Δ≠ΔΔ− ≠Δ≠Δ≠ΔΔ=Δ zIxxwithzx zIxxwithzx zIxxwithzx zIxxwithzx zIxxwithzx zIxxwithzx zIxxwithzx zIxxwithzxzC R x R x R x R x R x R x R x R x θθ θθ θθ θθ θθ θθ θθ θθ (C-18) Combining (C-16), (C-17) and (C-18) we observe that })],({/})),({}),({[( })],({/})),({}),({[(})],({/})),({}),({[( )]0;0;0)/(()),(/[( )]0;0;0)/(()),(/[( )()()/( xxy xxxxxy R x R x R zxzxzy zxzxzxzxzxzy zIxxwithzx zIxxwithzx zCICxxC θθθ θθθθθθ θθ θθ −= −−−= =Δ=Δ=ΔΔ− ≠Δ≠Δ≠ΔΔ= Δ+Δ+Δ (C-19) Let us now first define the expression www.economics-ejournal.org 46 })],({/})),({}),({[( })],({/})),({}),({[( })],({/})),({}),({[( )]0;0;0)(()),(/[( )]0;0;0)(()),(/[( )()()/( xxy xxx xxy A x A x A zxzxzy zxzxzx zxzxzy zIxwithzx zIxwithzx zCICxxC θθθ θθθ θθθ θθ θθ −= −− −= =Δ=Δ=ΔΔ− ≠Δ≠Δ≠ΔΔ= Δ+Δ+Δ (C-33) Let us now first define the expression )]0;0;0)(()),(/( ≠Δ≠Δ≠ΔΔ zIxwithzx A x θθ . It may clearly be written as ),(/)),(),(( xxy zxzxzy θ θ θ − Similarly the expression )]0;0;0)(()),(/( =Δ=Δ=ΔΔ zIxwithzx A x θθ may be written as 0),(/)),(),(( = −xxx zxzxzx θ θ θ Now let us define the expression ()0;0;0)((),(/ =Δ=Δ≠ΔΔ x A xzIxwithzx θθ . It is easy to derive that this expression may be also expressed as }),({/}),({}),({( xxx zxzxzxx θ θ θ − Δ+ where })),({( x zxx Δ+ θ refers to the poverty rate which is observed in a distribution where the incomes are equal to the original incomes to which a sum xΔ, equal to the difference between the average incomes at times 1 and 0, has been added to each individual's income at time 0 and where the poverty line is the one observed at time 0. It is should be clear that if the same amount of money x Δ is added to all the incomes, then by definition absolute inequality will have remained constant. Similarly the expression www.economics-ejournal.org 47 )0;0;0)(()),(/( ≠Δ=Δ≠ΔΔ zIxwithzx A x θθ may be written as }),({/)}),({)})),({(( xxy zxzxzxx θ θ θ −Δ+ where )})),({( y zxx Δ+ θ refers to the poverty rate which is observed in a distribution where the incomes are equal to the incomes observed at time 0 to which an equal sum xΔ has been added to all incomes, assuming the poverty line is that observed at time 1. Let us now define the expression )0;0;0)(()),(/( =Δ≠Δ≠ΔΔ zIxwithzx A x θθ . It may be written as ),(/}),{},{( xxx zxzxzy θ θ θ − Similarly we will define the expression )0;0;0)(()),(/( ≠Δ≠Δ=ΔΔ zIxwithzx A x θθ as },{/}),{}),{(( xxy zxzxzxy θ θ θ − Δ− where )}{( xy Δ− θ refers to a hypothetical distribution at time 0 where the individual incomes would be those actually observed at time 1 from which an equal amount xΔ would have been deduced from each individual income. It is easy to see that the expression )0;0;0)(()),(/( =Δ≠Δ=ΔΔ zIxwithzx A x θθ will be written as },{/}),{}),{(( xxx zxzxzxy θ θ θ − Δ− Finally one should observe that the expression )0;0;0)(()),(/( =Δ=Δ=ΔΔ zIxwithzx A x θθ will be written as 0),(/)),(),(( = −xxx zxzxzx θ θ θ We therefore end up with www.economics-ejournal.org 48 )]},(/)),(),([( })],({/}),({})),({(){[(6/2( )]},(/)),(),([( })],({/)}),({)})),({(){[(6/1( )]},{/}),{}),{([( )],(/}),{},{){[(6/1( }]},{/}),{}),{([( )],(/)),(),(){[(6 /2( )( xxx xxx xxy xxy xxx xxx xxy xxy zxzxzx zxzxzxx zxzxzx zxzxzxx zxzxzxy zxzxzy zxzxzxy zxzxzy xC θθθ θθθ θθθ θθθ θθθ θθθ θθθ θθθ −− −Δ++ −− −Δ++ −Δ−− −+ −Δ−− −= Δ (C-34) )],()),()[6/1( )]),((),()[6/1( )],()),()[6/2( )]),((),()[6/2()( yy xx xx yy zxzxx zxyzy zxzxx zxyzyxC θθ θθ θθ θ θ −Δ++ Δ−−+ −Δ++ Δ − − =Δ↔ (C-35) Similarly we can write that )]},(/)),(),([( ],{/}),{}),{(){[(6/2( })]},({/}),({})),({([( )],(/}),{},{){[(6/1( )]},(/)),(),([( }],{/}),{}),{(){[(6/1( })]},({/)}),({)})),({([( )],(/)),(),(){[(6 /2( )( xxx xxx xxx xxx xxy xxy xxy xxy A zxzxzx zxzxzxy zxzxzxx zxzxzy zxzxzx zxzxzxy zxzxzxx zxzxzy IC θθθ θθθ θθθ θθθ θθθ θθθ θθθ θθθ −− −Δ−+ −Δ+− −+ −− −Δ−+ −Δ+− −= Δ (C-36) so that www.economics-ejournal.org 49 ],{/}),{}),{(){[(6/2( )],(/})),{(},{){[(6/1( }],{/}),{}),{(){[(6/1( )],(/)),((),(){[(6/2()( xxx xxx xyy xyy A zxzxzxy zxzxxzy zxzxzxy zxzxxzyIC θθθ θθθ θθθ θθθ −Δ−+ Δ+−+ −Δ−+ Δ+−=Δ (C-37) Finally we can also write that )]},(/)),(),([( )],(/)),(),(){[(6/2( })]},({/}),({})),({([( })],({/)}),({)})),({(){[(6/1( )]},{/}),{}),{([( }],{/}),{}),{(){[(6/1( )]},(/}),{},{[( )],(/)),(),(){[(6 /2( )( xxx xxy xxx xxy xxx xxy xxx xxy zxzxzx zxzxzx zxzxzxx zxzxzxx zxzxzxy zxzxzxy zxzxzy zxzxzy zC θθθ θθθ θθθ θθθ θθθ θθθ θθθ θθθ −− −+ −Δ+− −Δ++ −Δ−− −Δ−+ −− −= Δ (C-38) The latter expression may be also written as )],(/)),(),(){[(6/2( })]}),({(()})),({(){[(6/1( }})]),{((})),{(){[(6/1( )]},(/)),(),(){[(6/2()( xxy xy xy xxy zxzxzx zxxzxx zxyzxy zxzyzyzC θθθ θθ θθ θ θ θ −+ Δ+−Δ++ Δ−−Δ−+ −=Δ (C-39) It is then easy to observe that if we sum the three expressions )(),(),( zCICxC AΔΔΔ given in (C-35), (C-37) and (C-39) we obtain www.economics-ejournal.org 50 )],(/)),(),(){[(6/2( )})]}),({(()})),({(){[(6/1( }})]),{((})),{(){[(6/1( )]},(/)),(),(){[(6/2( ],{/}),{}),{(){[(6/2( )],(/})),{(},{){[(6/1( }],{/}),{}),{(){[(6/1( )] ,(/))),((),(){[(6/2( )],()),(()[6/1( )]),((),()[6/1( )],()),(()[6/2( )]),((),()[6/2( )()()( xxy xy xy xxy xxx xxx xyy xyy yy xx xx yy A zxzxzx zxxzxx zxyzxy zxzyzy zxzxzxy zxzxxzy zxzxzxy zxzxxzy zxzxx zxyzy zxzxx zxyzy zCICxC θθθ θθ θθ θθθ θθθ θθθ θθθ θθθ θθ θθ θθ θθ −+ Δ+−Δ++ Δ−−Δ−+ −+ −Δ−+ Δ+−+ −Δ−+ Δ+−+ −Δ++ Δ−−+ −Δ++ Δ−−= Δ+Δ+Δ (C-40) which amounts, as expected, to writing )],(/)),(),([()()()( xxy AzxzxzyzCICxC θθθ −=Δ+Δ+Δ (C-41) III) The Cases of Relative and Absolute Pro-Poor Growth When the Poverty Line Varies over Time but Is Endogenously Determined A) Measuring Relative Pro-Poor Growth (in discrete terms) when the poverty line is variable and endogenous: Let, as before, },...,{}{ 1n xxx = and },...,{}{ 1n yyy = represent the vector of incomes at times 0 and 1 and let )(x θ and )(y θ refer to the poverty index at times 0 and 1. Finally let ))(/( x θ θ Δrefer to the relative change in the poverty index between times 0 and 1, with )()()( xy θ θ θ − = Δ. The relative change θ θ /)(Δ in the poverty index will now be expressed as www.economics-ejournal.org 51 )),/(()( R Ixxg ΔΔ=Δ θ (C-42) where x is the mean income of the distribution given by }{x, )( xyx −= Δ is the difference between the average income at times 0 and 1 and R I refers to some relative measure of income inequality of the distribution given by )(x. The poverty line will always be assumed to be equal to half the median of the corresponding distribution. Using the concept of Shapley decomposition, )( θ Δ may be written as )()/()/( R ICxxC Δ+Δ=Δ θθ (C-43) where )/( xxC Δrefers to the contribution of the relative change over time in the average income and )( R IC Δto the contribution of the change in relative inequality between time 0 and time 1. The contribution )/( xxC Δ may itself be expressed as )]}0;0)/(()/( )0;0)/(()/[( )]0;0)/(( )/()0;0)/(()/){[(2/1()/( =Δ=ΔΔ− =Δ≠ΔΔ+ ≠Δ=Δ Δ−≠Δ≠ΔΔ=Δ R R R R Ixxwith Ixxwith Ixxwith IxxwithxxC θθ θθ θθθθ (C-44) Similarly the contribution )( R IC Δ may be written as )]}0;0)/(( )/()0;0)/(()/[( )]0;0)/(( )/()0;0)/(()/){[(2/1()( =Δ=Δ Δ−≠Δ=ΔΔ+ =Δ≠Δ Δ−≠Δ≠ΔΔ=Δ R R R RR Ixxwith Ixxwith Ixxwith IxxwithIC θθθθ θθθθ (C-45) Combining (C-44) and (C-45) we observe that www.economics-ejournal.org 52 })]({/}))({})({[( })]({/}))({})({[( })]({/}))({})({[( )]0;0)/(()/[( )]0;0)/(()/[( )()/( xxy xxx xxy Ixxwith Ixxwith ICxxC R R R θθθ θθθ θθθ θθ θθ −= −− −= =Δ=ΔΔ− ≠Δ≠ΔΔ= Δ+Δ (C-46) Let us first define the expression ( ).0;0)/(()/ ≠Δ≠ΔΔ R Ixxwith θθ It may be written as )},({/))},({)},({( xxy zxzxzy θ θ θ − Let us now define the expression ( )0;0)/(()/ =Δ≠ΔΔ R Ixxwith θθ . It is easy to see that it may be written as )},({/)},({))},1(({( )1( xxkx zxzxzkx θ θ θ −+ + where )/( xxk Δ= and ))},1(({ )1( kx zkx + + θ refers to the poverty rate which is observed in a distribution where the incomes are equal to the original incomes multiplied by a factor k equal to the growth rate of the average income between times 0 and 1 and the poverty line is equal to half the median of the distribution )}1({ kx +. It should be clear that if all the incomes are multiplied by the same constantk, by definition relative inequality will have remained constant. Similarly the expression )0;0)/(()/( ≠Δ=ΔΔ R Ixxwith θθ may be written as )},({/)},({)))},1/(({(( )1/( xxky zxzxzky θ θ θ −+ + where )))},1/(({( )1/( ky zky + + θ refers to the poverty rate which is observed in a distribution where the incomes are equal to the incomes observed at time 1 divided by one plus the growth rate of the average income between time 0 and time 1, )1/( ky z+being the poverty line (half the median) corresponding to this distribution )}1/({ ky +. Finally )0;0)/(()/( =Δ=ΔΔ R Ixxwith θθ will evidently be expressed as 0)},({/)},({)},({( = −xxx zxzxzx θ θ θ We therefore end up with www.economics-ejournal.org 53 )]}},({/))},({))},1(({[( )]},({/)))},1/(({()},({){[(2/1( )]}},({/))},({)},({[( )]},({/))},({))},1(({){[(2/1( )]]}},({/))},({)))},1/(({([( )]},({/))},({)},({){[(2/1( )/( )1( )1/( )1( )1/( xxkx xkyy xxx xxkx xxky xxy zxzxzkx zxzkyzy zxzxzx zxzxzkx zxzxzky zxzxzy x x C θθθ θθθ θθθ θθθ θθθ θθθ −++ +−= −− −++ −+− −= Δ + + + + (C-47) Similarly we can write that )]}},({/))},({))},1/(({[( )]},({/))},1(({)},({){[(2/1( )]}},({/))},({)},({[( )]},({/))},({))},1/(({){[(2/1( )]]}},({/))},({))},1((({[( )]},({/))},({)},({){[(2/1()( )1/( )1( )1/( )1( xxky xkxy xxx xxky xxkx xxy R zxzxzky zxzkxzy zxzxzx zxzxzky zxzxzkx zxzxzyIC θθθ θθθ θθθ θθθ θθθ θθθ −++ +−= −− −++ −+− −=Δ + + + + (C-48) Combining (C-47) and (C-48) we observe again that, as expected, })({})({)()( xyICxC A θθ −=Δ+Δ (C-49) The actual relative change in the poverty index observed between times 0 and 1 will therefore be expressed as AA InGrd +=)/( θθ (C-50) where θ /)( xCGrAΔ= (C-51) www.economics-ejournal.org 54 and θ /)( AA ICIn Δ= (C-52) where θ , as before is the value of the poverty index at the original period 0. B) Measuring Absolute Pro-Poor Growth (in discrete terms) when the poverty line is variable and endogenous: Let, as before, },...,{}{ 1n xxx = and },...,{}{ 1n yyy = represent the vector of incomes at times 0 and 1 and let )},({ x zx θ and )},({ y zy θ refer to the poverty index at times 0 and 1. Throughout this section it is assumed that the poverty line varies over time and is endogenous because it is always defined as being equal to half the median of the corresponding distribution. Finally let ))},({/( x zx θ θ Δrefer to the relative change in the poverty index between times 0 and 1, with )},({)},({)( xy zxzy θ θ θ − = Δ. The absolute change )( θ Δ in the poverty index will now be expressed as ),()( A Ixf ΔΔ=Δ θ (C-53) where x is the mean income of the distribution given by }{x,xΔis the change over time in the average income, so that xyx − = Δ and A I refers to some absolute measure of income inequality of the distribution given by )(x. Using the concept of Shapley decomposition, )( θ Δ may be written as )()()( A ICxC Δ+Δ=Δ θ (C-54) where )( xC Δrefers to the contribution of the change over time in the average income and )( A IC Δto the contribution of the change in absolute inequality between time 0 and time 1. The contribution )( xC Δ may itself be expressed as )]}0;0()0;0([ )]0;0()0;0(){[2/1()( =Δ=ΔΔ−=Δ≠ΔΔ+ ≠Δ=ΔΔ−≠Δ≠ΔΔ=Δ AA AA IxIx IxIxxC θθ θθ (C-55) www.economics-ejournal.org 55 Similarly the contribution )( A IC Δ may be written as )]}0;0()0;0([ )]0;0()0;0(){[2/1()( =Δ=ΔΔ−≠Δ=ΔΔ+ =Δ≠ΔΔ−≠Δ≠ΔΔ=Δ AA AAA IxIx IxIxIC θθ θθ (C-56) Combining (C-55) and (C-56) we observe that })]({})({[})]({})({[})]({})({[ )]0;0([)]0;0([ )()( xyxxxy IxIx ICxC AA A θθθθθθ θθ −=−−−= =Δ=ΔΔ−≠Δ≠ΔΔ= Δ+Δ (C-57) Let us first define the expression )0;0( ≠Δ≠ΔΔ A Ix θ . It may be written as )},({)},({ xy zxzy θ θ −. Let us now define the expression )0;0( =Δ≠ΔΔ A Ix θ . It is easy to derive that this expression may be also expressed as )},({)},({ xxx zxzxx θ θ −Δ+ Δ+ where )},({ xx zxx Δ+ Δ + θ refers to the poverty rate which is observed in a distribution where the incomes are equal to the original incomes at time 0 to which an amount equal to xyx − = Δ has been assumed to have been added to every individual and where the poverty line xx zΔ+ is equal to half the median of the distribution }.{ xx Δ + Similarly the expression )0;0( ≠Δ=ΔΔ A Ix θ may be also written as )},({)},({ xxy zxzxy θ θ −Δ− Δ− where )},({ xy zxy Δ− Δ − θ refers to the poverty rate which is observed in a distribution where the incomes are equal to the incomes at time 1 minus the difference x Δ between the average income at time 1 and at time 0 ( xyx −=Δ ) and where the poverty line is equal to half the median of the distribution }{ xy Δ −. We therefore end up with