# Measuring pro-poor growth

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2 94 M. Ravallion, S. Chen / Economics Letters 78 (2003) change in the Lorenz curve. However, it is possible that while the distributional changes were pro-poor, there was no absolute gain to the poor. Equally well, pro-rich distributional shifts may have come with large absolute gains to the poor. A more direct approach is to look at growth rates for the poor. It is common to compare mean incomes across the distribution ranked by income; this is sometimes called Pen s parade (following Pen, 1971). To assess whether growth is pro-poor, a natural step from Pen s parade is to calculate the 2 growth rate in the mean of the poorest quintile (say). Taking this idea a step further, we define a growth incidence curve, showing how the growth rate for a given quantile varies across quantiles ranked by income. The following section defines this curve and discusses its properties. Starting from the Watts (1968) index of the level of poverty, we derive in Section 3 a measure of the rate of pro-poor growth by integration on the growth incidence curve. Our measure is the mean growth rate for the poor (as distinct from the growth rate in the mean for the poor). This can be interpreted as the ordinary growth rate in the mean scaled up or down according to whether the distributional changes were pro-poor. Section 4 illustrates these ideas using data for China in the 1990s. 2. The growth incidence curve Let F ( y) denote the cumulative distribution function (CDF) of income (or expenditure), giving the t proportion of the population with income less than y at date t. Inverting the CDF at the pth quantile gives the income of that quantile: y ( p) 5 F ( p) 5 L 9( p)m ( y 9( p). 0) (1) 21 t t t t t (following Gastwirth, 1971), where L t( p) is the Lorenz curve (with slope L t9 ( p)) and mt is the mean; for example, y t(0.5) is the median. Letting p vary from zero to one yields a version of Pen s parade that is sometimes called the quantile function (see, for example, Moyes, 1999). Comparing two dates, t21 and t, the growth rate in income of the pth quantile is g t( p) 5 [y t( p)/ y t21( p)] 2 1. Letting p vary from zero to one, g t( p) traces out what we will call the growth incidence curve (GIC). It follows from Eq. (1) that: L 9t ( p) g t( p) 5 ]]](gt1 1) 2 1 (2) L 9 ( p) t21 where gt5 (m t/m t21) 2 1 is the growth rate in m t. It is evident from Eq. (2) that if the Lorenz curve does not change then g t( p) 5 gt for all p. Also g t( p). gt if and only if y t( p)/mt is increasing over time. If g t( p) is a decreasing (increasing) function for all p then inequality falls (rises) over time for all inequality measures satisfying the Pigou Dalton transfer principle. (This follows from well-known results on tax progressivity and inequality; see for example Eichhorn et al., 1984). If the GIC lies 1 For example, Chen and Ravallion (2001) find that the rate of poverty reduction in the developing world as a whole over would have been slightly lower if not for the changes in the aggregate Lorenz curve. The slight improvement in overall distribution from the point of view of the poor was almost solely due to economic growth in China. 2 For example, Dollar and Kraay (2000) test whether aggregate growth is good for the poor by calculating the growth rate in the mean of the poorest quintile.

3 M. Ravallion, S. Chen / Economics Letters 78 (2003) above zero everywhere ( g t( p). 0 for all p) then there is first-order dominance (FOD) of the distribution at date t over t21. If the GIC switches sign then one cannot in general infer whether 3 higher-order dominance holds by looking at the GIC alone. 3. Measuring pro-poor growth We assume that a measure of the rate of pro-poor growth should satisfy the following conditions: Axiom 1. The measure should be consistent with the direction of change in poverty, in that a positive 4 (negative) rate of pro-poor growth implies a reduction (increase) in poverty. Axiom 2. The measure of poverty implicit in the measure of pro-poor growth should satisfy the standard axioms for poverty measurement. In the literature following Sen (1976), three axioms have been widely agreed to be essential for any poverty measure, namely the focus axiom (the measure is invariant to income changes for the non-poor), the monotonicity axiom (any income loss to the poor increases poverty), and the transfer axiom (inequality-reducing transfers amongst the poor are poverty reducing). A further axiom identified in the literature is additive decomposability (aggregate poverty can be written as a population weighted mean of the poverty measures across disjoint subgroups). This implies sub-group consistency (if poverty increases in any sub-group then it must increase in the aggregate ceteris paribus) (Foster and Shorrocks, 1991). The popular headcount index of poverty is Ht5 F t(z) where z is the poverty line. This clearly fails the monotonicity and transfer axioms. There are numerous measures satisfying the focus, monotonicity, transfer and decomposability axioms (Atkinson, 1987, provides a compilation and references). The first index to do so was proposed by Watts (1968); Zheng (1993) identifies a larger set of axioms for which the Watts index emerges as the unique poverty measure. By a change of variables, the original Watts index can be written in terms of the quantile function as follows: W H t t5e log[z/y t( p)] dp 0 (3) Thus, on differentiating with respect to time, we see that: Ht dwt d log y t( p) ]] ]]]] dt Ht 2 5E dp 5E g ( p)dp (4) dt 0 0 t 3 An exception is when the overall mean rises and the GIC is decreasing in p; then there is clearly second-order dominance. More generally, second-order dominance is tested by integrating over either the quantile function (Shorrocks, 1983), or its inverse, the CDF. 4 In the context of the inter-temporal aggregation of growth rates, Kakwani (1997) argues that the growth rate should be consistent with an aggregate welfare function defined on mean incomes over time.

4 96 M. Ravallion, S. Chen / Economics Letters 78 (2003) (noting that y t(h t) 5 z), i.e. the area under the GIC up to the headcount index gives (minus one times) the change in the Watts index. On dividing throughout by H t, Eq. (4) motivates measuring the rate of pro-poor growth by the Ht mean growth rate for the poor: e0 g t( p) dp/h t. Notice that this collapses to the growth rate in the overall mean (g t) if all incomes grow at the same rate, in which case the rate of change in the Watts index is Htg t(from Eq. (4)). Thus our measure of the rate of pro-poor growth is the actual growth rate multiplied by the ratio of the actual change in the Watts index to the change that would have been observed with the same growth rate but no change in inequality. Also note that our measure is not the same as the growth rate in the mean income of the poor, as often used in applied work. That measure does not correspond to a poverty measure that satisfies either the monotonicity or transfer axioms. If an initially poor person above the mean escapes poverty then the growth rate in the mean for the poor will be negative; yet poverty has fallen. This is avoided if one fixes H (by calculating the growth rate of the mean for the poorest quintile, say), but then the measure fails the transfer axiom. 4. An illustration for China in the 1990s Fig. 1 gives our estimate of China s GIC for We have calculated this from detailed grouped distributions for rural and urban areas separately; the distributions were constructed to our 5 specification by China s National Bureau of Statistics. Urban and Rural Consumer Price Indices have Fig. 1. Growth incidence curve for China, The distributions published in the China Statistical Yearbook (for example, NBS, 2000) are less than ideal for our purpose since they do not give mean income by class intervals and are quite aggregated (more so in some years than others).

5 M. Ravallion, S. Chen / Economics Letters 78 (2003) been applied to the urban and rural distributions prior to aggregation, assuming a 10% differential in the cost-of-living between urban and rural areas at the base date. (Sensitivity was tested to a 20% differential and zero differential, but these changes shifted the GIC only slightly). We then used parameterized Lorenz curves to calculate mean income at each quantile; we tested both the general elliptical and the incomplete beta specifications (Datt and Ravallion, 1992), and found that the former 6 gave a better fit. However, with household-level data one can calculate the GIC directly without using 7 parameterized Lorenz curves or other interpolation methods. There is first order dominance. Thus poverty has fallen no matter where one draws the poverty line or what poverty measure one uses within a broad class (Atkinson, 1987; Foster and Shorrocks, 1988). The curve is also strictly increasing over all quantiles, implying that inequality rose. The annualized percentage increase in income per capita is estimated to have been about 3% for the poorest percentile, rising to 10% for the richest. In calculating the mean growth rate for the poor with discrete data it makes sense to define the poor as those living below the poverty line at the initial date t21, in keeping with the common practice of measuring performance relative to the base date. (This does not matter in Eq. (4), given that the calculus is based on infinitely small changes). We also normalize by H t21, so that our measure can be interpreted as the mean growth rate for the poorest H t21%. The rate of pro-poor growth is then the annualized change in the Watts index divided by the initial headcount index. Table 1 gives our estimates for a range of poverty lines; for example, the rate of pro-poor growth is 3.9% for H The mean growth rate over the entire distribution is 5.5%. The growth rate in t21 the mean is 6.2% per annum. We repeated these calculations for sub-periods, , , All GIC s showed the same pattern except , which is given in Fig. 2. The GIC changed dramatically Table 1 Growth rates p Growth rate in the mean (% per annum) Mean growth rate for the poorest p% (% per annum): A self-contained computer program is available from the authors to perform these calculations. 7 The empirical quantile function can be readily constructed from household-level data using, for example, the pctile command in STATA. STATA code is available from the authors for doing the calculations in this paper from household-level data.

6 98 M. Ravallion, S. Chen / Economics Letters 78 (2003) Fig. 2. Growth incidence curve for China, in this period, taking on an inverted U shape, with highest growth rates observed at around the 25th 8 percentile. The rate of pro-poor growth for this sub-period is 10.0% per annum (H50.15) above the ordinary growth rate of 8.2%. The rate of pro-poor growth was far higher in this sub-period, and the distributional shifts were more pro-poor. 5. Conclusions For the purpose of monitoring the gains to the poor from economic growth, the growth rate in mean consumption or income of the poor has the drawback that it is inconsistent with one or more standard axioms for measuring the level of poverty. This paper has argued that a better measure of pro-poor growth is the mean growth rate of the poor, which indicates the direction of change in a theoretically defensible measure of the level of poverty, namely the Watts index. The proposed measure of pro-poor growth can be derived from a growth incidence curve giving rates of growth by quantiles of the distribution of income. This curve is also of interest in its own right, as a means of describing how the gains from growth were distributed. China s growth process in the 1990s has been used to illustrate our proposed measure of pro-poor growth. Over , the ordinary growth rate of household income per capita was 6.2% per annum. The growth rate by quantile varied from 3% for the poorest percentile to 10% for the richest, 8 A likely reason is the substantial increase in the government s purchase price for foodgrains in 1994 (World Bank, 1997). Arguably, this was not a sustainable change in relative prices. But it does appear to have entailed a substantial temporary shift in distribution, given that farmers are known to be concentrated around the lower end of the distribution of income in China (Ravallion and Chen, 1999).

7 M. Ravallion, S. Chen / Economics Letters 78 (2003) while the rate of pro-poor growth was around 4%. The pattern was reversed for a few years in the mid-1990s. References Atkinson, A.B., On the measurement of poverty. Econometrica 55, Chen, S., Ravallion, M., How did the world s poorest fare in the 1990s, Policy Research Working Paper. World Bank, Washington DC, Datt, G., Ravallion, M., Growth and redistribution components of changes in poverty: A decomposition with application to Brazil and India. Journal of Development Economics 38, Dollar, D., Kraay, A., Growth is good for the poor, Policy Research Working Paper. World Bank. Gastwirth, J.L., A general definition of the Lorenz curve. Econometrica 39, Eichhorn, W., Funke, H., Richter, W.F., Tax progression and inequality of income distribution. Journal of Mathematical Economics 13, Foster, J., Shorrocks, A.F., Poverty orderings. Econometrica 56, Foster, J., Shorrocks, A.F., Subgroup consistent poverty indices. Econometrica 59, Kakwani, N., Growth rates of per-capita income and aggregate welfare: An international comparison. Review of Economics and Statistics 79, Moyes, P., Stochastic dominance and the Lorenz curve. In: Silber, J. (Ed.), Handbook on Income Inequality Measurement. Kluwer Academic Publishers, Boston. National Bureau of Statistics, China Statistical Yearbook. China Statistics Press, Beijing. Pen, J., Income Distribution. Praeger Publishers, New York. Ravallion, M., Chen, S., When economic reform is faster than statistical reform: Measuring and explaining inequality in rural China. Oxford Bulletin of Economics and Statistics 61, Sen, A.K., Poverty: An ordinal approach to measurement. Econometrica 44, Shorrocks, A.F., Ranking income distributions. Economica 50, Watts, H.W., An economic definition of poverty. In: Moynihan, D.P. (Ed.), On Understanding Poverty. Basic Books, New York. World Bank, China 2020: Sharing Rising Income. World Bank, Washington DC. Zheng, B., An axiomatic characterization of the Watts index. Economics Letters 42,

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