Poverty: Looking for the Real Elasticities

Transcription

1 Poverty: Looking for the Real Elasticities Florent Bresson CERDI - Université d Auvergne First draft March 2006 PLEASE DO NOT QUOTE OR CIRCULATE. Abstract After decades of intensive research on the statistical size distribution of income and despite its empirical weaknesses, the lognormal distribution still enjoys an important popularity in the applied literature on poverty and inequality. In the present study, we try to emphasize the drawbacks of this choice for the calculation of the elasticities of poverty. Using last version of WIID database, we estimate the growth and inequality elasticities of poverty using 1,842 income distributions with a dozen of rival distribution assumptions. Our results confirm that the lognormal distribution is not appropriate for the analysis of poverty. Most of the time, it implies an overestimation of the elasticities and is rather uninformative about the relative impact of growth and redistribution on poverty alleviation. Introduction From an analytical point of view, poverty is directly and only linked to mean income and inequality. As the effects of any variable on poverty are channeled through mean income and its distribution, we have to investigate the sensibility of poverty to variations of these two elements. Since the international community has decided to target his intervention on poverty alleviation, many famous studies like Ravallion (2001) or Dollar & Kraay (2002) have emphasized the importance of growth. Recent papers have also shed light on the role of distribution changes in poverty reduction, like Heltberg (2002), Bourguignon (2002) or Ravallion (2005). Authors agree that growth reduces poverty more efficiently in less inegalitarian countries and that the contribution of inequality reduction to poverty florent.bresson@u-clermont1.fr The document has been realized with L A TEX. All estimations are made with R. 1

2 1 ACCOUNTING VS ANALYTICAL APPROACH alleviation is higher in richer countries. However, there is still no agreement on the real values of growth and inequality elasticities. This is nevertheless an important question for the design of policies aimed at reducing poverty. Among traditional approaches used in the literature, the most attractive seems to be the one which can be called analitycal. Assuming that any observed income distribution can be described by some known distribution, income and inequality elasticities of poverty can be estimated for each observation for given values of per capita income and degree of inequality. Most of theses studies (Quah 2001, Bourguignon 2002, Epaulard 2003, Kalwij & Verschoor 2005, Lopez & Servèn 2006) rely on the lognormal distribution. This is quite an astonishing choice since these authors choose to set aside all the XX th century debates on the statistical size distributions of income. Since the late XIX th century and the pioneering works of Pareto, research has been extremely active to retrieve the functional form which fit perfectly the observed distributions. Practical considerations and considerable influence of the study of Aitchinson & Brown (1957) may still explain the current popularity of the lognormal distribution, but many authors have pointed out his empirical weaknesses and proposed alternative distributions 1. For instance, Bandourian et al. (2002) showed that the lognormal distribution was dramatically outperformed by many alternative functional forms, even in the set of two parameters distributions. If the lognormal distribution is such a poor approximation of income distributions, elasticities obtained through the lognormality hypothesis are questionable. In the present paper, we intend to shed light on the consequences of this traditional choice using alternative distributions which are supposed to exhibit a better fit than the lognormal distribution. Using a data set of income distributions, we conclude that the quality of our predictions can be significantly improved with more flexible functional forms. Moreover, we find that estimated elasticities under the lognormal assumption generally overestimate real elasticities and poorly predicts the trade-off between pure growth and redistribution strategies for poverty alleviation. 1 Accounting vs analytical approach Studies on the relative contributions of growth and inequality reduction to poverty reduction have followed two different pathes 2. The first one can be seen as an accounting approach since it consists in decomposing poverty spells in variations that are respec- 1 A quite comprehensive survey is Kleiber & Kotz (2003). 2 In the following paragraphs, we won t mention the approach which consists in directly regressing the rate of poverty variation on levels and variations of per capita income and income inequality, or some combinations of these elements (Ravallion 1997, Bourguignon 2002, Adams 2002). As Ravallion (1997) himself recognized, this is quite an ad hoc procedure. Not surprisingly, it can produce some erroneous results. Moreover, actual studies only evaluate average elasticities. Some studies showed in that way that lognormality cannot be rejected. However, one may argue that what is true on average can be false in every case. This explains why we concentrate on techniques that enable a direct calculation of elasticities for each observation. 2

3 1 ACCOUNTING VS ANALYTICAL APPROACH tively attributed to changes in mean income and changes in its distribution. The second approach is static. It consists in using the features of the actual distribution of income to calculate the elasticities of poverty to growth and inequality. Contrary to the accounting approach, this analytical approach which has been adopted for the present study, requires some hypotheses about the actual income distribution. As it will be shown in the following sections, the choice of the underlying income distribution function is crucial. Before considering the analytical approach, we would like to emphasize some features of the accounting approach. It was first developed by Jain & Tendulkar (1990), Kakwani & Subbarao (1990) and Huppi & Ravallion (1991), and has been adopted for several case studies by Datt & Ravallion (1992) or recently by Bhanumurthy & Mitra (2004) and Dhongde (2004) 3. Poverty spells are decomposed into a growth component and an inequality component. Roughly speaking, the first one is obtained by measuring the poverty variation due to observed growth, leaving the income distribution unchanged. The second component symmetrically corresponds to the poverty variation resulting from the observed inequality changes, leaving mean income unchanged 4. Problems are encountered if we intend to use the results to infer about the relative performance of growth and inequality reduction on poverty alleviation. Each estimated component of the poverty spell decomposition is the product of a variation rate and a semi-elasticity. If the growth effect is larger than the inequality effect, it may just as well be the result of a higher growth elasticity or of small distribution changes in comparison with observed growth. So, growth and inequality component express both initial conditions and political choices, and cannot be used to draw some political recommendations. For this purpose, we have to estimate elasticities. Elasticities of poverty to growth and inequality reduction can be easily estimated with the accounting approach. Such calculation is done in Lopez & Servèn (2006). If we note GE and IE the growth and inequality effects, µ, P 0 and I the respective initial values of mean income, headcount and inequality index used for the decomposition, we get our desired elasticities for the headcount index through: η µ = GE µ η I = IE I µ, P 0 (1) I. P 0 (2) Contrary to the poverty elasticity of growth, the technique is questionable for the inequality elasticity since the result depends on the index used to calculate the distribution change. So the estimated inequality elasticity of poverty will differ if distributional changes are measured through Gini or Theil coefficients. As any inequality index sum- 3 The theoretical contributions of Tsui (1996), Kakwani (1997) and Shorrocks (1999) are also worth mentioning. 4 We downplay the fact that, depending on the technique used, the decomposition is not necessarily exact since a residual component sometime appears. 3

4 1 ACCOUNTING VS ANALYTICAL APPROACH marizes only some features of the Lorenz curve, it necessarily gives an incomplete view of the evolutions of the income distribution. In particular, Gini coefficients are often criticized because different Lorenz curves can be produced with the same value of the Gini index. In the theoretical case of a change of the distribution with no change of the Gini coefficient, it will simply be impossible to calculate the inequality elasticity. To understand the second drawback of the accounting method, consider two countries with the same initial mean income and degree of inequality. What should we conclude about the inequality elasticity if the income distribution vary in different ways in the two countries, but with the same final inequality index? On figure 1, the two Lorenz curves L ans L are characterized by the same Gini index, but are associated with different values of the headcount index 5 P 0. If we compare the percentage change of the headcount with initial distribution described by the L curve, we may conclude that the inequality elasticity of poverty is higher when the final distribution is L. However these results are quite misleading since they seem to suggest that inequality reduction was a more efficient tool in this country. But, initial conditions were identical in the two countries, so the potential reduction of poverty due to any distributional change should be the same. This simple example shows that the estimated inequality elasticity obtained through equation (2) impressively depends on the way inequality changed. So, the result is informative for the understanding of past events, but it is quite useless if we intend to use the results to realize cross-section comparisons and formulate some general political recommendations. Indeed, such problems are not encountered with the growth elasticity of poverty since it is unique for given initial conditions. Figure 1: Two different evolutions of income distribution and poverty with equal Gini increase. The third problem relating to the accounting approach is data scarcity. The 5 The headcount index is the point where the first derivative of the Lorenz curve is equal to the ratio of the poverty line on mean income. 4

5 2 METHODOLOGY Chen & Ravallion (2004) dataset which is actually the largest database on poverty, contains about 250 observations for only 80 countries during the period Since the time span is often quite large, we should also mention that variations of mean income and its distribution may be quite large. In such cases, it seems difficult to consider the calculated elasticities as marginal effects. The analytical approach is not hampered by these problems. First, the analysis can be done at the macroeconomic level without observations of the poverty levels. The approach is also attractive since elasticities can be deduced from a single observation with relatively few informations. In the case of simple distribution hypothesis like the Pareto or lognormal distributions, per capita income and an inequality measure are sufficient. Second, since a parametric form has to be employed to describe the distribution of incomes, it is often easy to get the derivatives of the distribution so as to get the desired elasticities. Third, as the technique ignores the real evolution of the income distribution of each observation, the calculated inequality elasticities of poverty are standardized, because they all can be obtained through the same marginal transformation of the Lorenz curve. The estimated inequality elasticity is so a potential elasticity and not the result of each country s future redistribution choices. Thus, we can compare the situation of different countries and help choosing between growth or inequality reducing policies to fight poverty. 2 Methodology 2.1 Calculation of the elasticity of poverty In the present paper, we focus on absolute poverty. Any absolute poverty measure is a combination of a poverty line, the mean income and a set of inequality parameters which fully describes the Lorenz curve. With the hypothesis that incomes follow a known distribution, this set of inequality parameters can be reduced to a few ones. To get these parameters, we could use some method of moments as often done with the lognormal distribution. With the help of the per-capita income and the Gini index, we can easily obtain the parameters of two parameters distributions like the Pareto, the lognormal, the gamma, the Weibull or the Fisk distributions. Several reasons led us to forsake this approach. The first one is that one needs more information about inequality for more than two parameters distributions. For large datasets, this supplementary information is usually given by points of the Lorenz curve, but the resolution of the resulting systems of nonlinear equations is generally cumbersome. Second, Gini coefficients are systematically truncated in the available datasets. As we used the points of the Lorenz curve to assess the quality of the fit (cf sec. 4), it appeared that this truncation most of the time increases 5

6 2 METHODOLOGY the size of errors in a significant manner 6. To avoid these shortcomings, we choose to estimate the parameters of our different distributions uniquely from the available points of the Lorenz curve 7. Our preferred measures of poverty are the widely used Foster, Greer & Thorbecke (1984) measures P θ : P θ = z 0 ( ) z y θ f(y)dy, (3) z where y is income, z the poverty line, f is the income density and θ the parameter of inequality aversion. For θ = {0, 1, 2}, P θ is respectively the headcount, poverty gap and severity of poverty index. For the derivation of income and inequality elasticities, we follow Kakwani (1993). With the headcount index, the growth elasticity of poverty, η µ is simply: where µ stands for mean income. For θ 0, the elasticity is: η 0,µ = zf(z) P 0, (4) η θ,µ = θ(p θ 1 P θ ) P θ. (5) For the estimation of the inequality elasticity of poverty, we have to deal with the problem that income distribution can change in various ways. For two-parameter distributions, inequality changes are always unique, and so is the inequality elasticity. For more flexible functional forms, we have to chose how the Lorenz curve should move to get a unique value of the desired elasticity. Kakwani (1993) suggests the following shift of the Lorenz curve: L (p) = L(p) ε ( p L(p) ), (6) where ε indicate a proportional change in the Gini coefficient 8. Such transformation of the Lorenz curve implies Lorenz dominance. So, for negative (positive) value of ε, the situation of the poor never worsen (improve). It is also interesting because it offers the possibility to compare the inequality elasticities obtained from different distributions. The interest is not only practical, since the choice of a particular distribution for poverty analysis is not neutral in terms of the relative importance of growth and inequality 6 Truncations and rounding are also a matter of concern for points of the Lorenz curve, but the loss of precision is less important. 7 With sufficient degrees of freedom, our estimation strategy would also have offered the possibility to get standards errors of the derived elasticities. 8 It can easily be shown that ε can also be interpreted as the same proportional increase of all the standardized moments of the Lorenz curve defined by Aaberge (2000) as: D t = (t + 1) 1 0 p t 1 p L(p) dp. 6

7 2 METHODOLOGY elasticities, and so may reflect some political preferences. With a common strategy for the evolution of inequality, estimated elasticities do not depend on the implicit redistributive policies corresponding to each distribution. On the figure 2 one can observe the differences between several two-parameter distributions which will be used in the present study. In each quadrant, the solid line represents the Lorenz curve corresponding to a Gini coefficient equal to 0.5. The dotted line is the Lorenz curve obtained through a 20% decrease of the Gini index with the same distribution. The dashed line corresponds to an equal increase of the Gini coefficient through Kakwani s transformation. Lognormal Weibull Fisk Pareto Gamma Figure 2: Natural vs Kakwani s transformation of the Lorenz curve for two parameters distributions. In all cases, we can observe that the Lorenz curve resulting from Kakwani s transformation are more skewed toward the upper point of the Lorenz curve than the natural curve corresponding to the same reduction of the Gini coefficient. This means that the poorest benefit more from the fall of income inequality. Since the Foster et al. (FGT) measures are directly linked to the slope and curvature of the Lorenz curve, we should generally obtain greater inequality elasticities of poverty than those that would be naturally derived from a transformation which would preserve the type of the distribution 9. For 9 For example, the natural Gini elasticity of the headcount index under the lognormality assumption 7

8 2 METHODOLOGY the three-parameter distributions that will be used in the study 10, namely the beta of the second kind, the Maddala & Singh (1976) also known as the Burr XII distribution and the Dagum (1977) also called the Burr III distribution, elasticities are not unique. So for a same distribution, "natural" elasticities can be either larger or lower than those corresponding to the Kakwani s transformation. However, since this transformation is particularly "pro-poor", we can reasonably consider that our estimated Gini elasticities will be quite high. From equation (6), Kakwani (1993) proposed the following Gini elasticities: η 0,G = (µ z) f(z), P 0 (7) η θ,g = θ + µ z P θ 1 z P θ θ 0. (8) Another interesting feature of Kakwani s (1993) formula is that the respective importance of growth and inequality elasticities is easily predicted, especially for the headcount index. We note: η 0,µ = z η 0,G µ z (9) η θ,µ z(p θ 1 P θ ) = η θ,g z(p θ 1 P θ ) µp θ 1 θ 0 (10) Equation (9) is particularly interesting since we can see that, for the headcount index, the ratio 11 of the growth elasticity to the Gini elasticity obtained through Kakwani s transformation does not depend on the income distribution. So, it will be the same, whatever distributional assumption is made. As it only depends on per capita income, we can already know that growth policies 12 will be more efficient in terms of poverty reduction that redistributive policies when mean income is less than the poverty line. On the contrary, redistribution is the only effective tool for rich countries. The relationship between mean income and the ratio of the growth and inequality elasticities for a 2$ poverty line is drawn on figure 3. When θ 0, distribution matters. However, we can notice that the ratio is always negative. It can be easily shown that its absolute value decreases with mean income. So is: η 0,G = λ ¼ log z µ σ + σ 2 ½ ¼ σ log 2 z µ σ ½ G σ 2ϕ σ 2 were λ and ϕ represent the hazard rate and density function of the standard normal distribution. 10 We also tried to estimate parameters of the generalized gamma and the four-parameter generalized beta 2 distributions, but results were not satisfying due to non-convergence of the estimators. To understand the link between the distributions used in this study, see McDonald (1984). 11 This is ratio is called Inequality-Growth Trade-off Index (IGTI) in Kakwani (2000). 12 By growth policies, we mean policies that would lead to an increase of mean income with no distributional change. Of course, this is a pure theoretical view since growth always imply some redistribution., 8

9 2 METHODOLOGY elasticiy ratio income per capita Figure 3: Relationship between mean income and the ratio of the elasticities of the headcount index with Kakwani s transformation (2$ poverty line). redistributive policies becomes more and more attractive as per capita income increases. 2.2 Some alternative functional forms for the Lorenz curve In addition to known distribution, we tried to use some ad hoc functional forms for the Lorenz curve. Characterizing a distribution through the direct estimation of the Lorenz curve has been first developed by Kakwani & Podder (1973). Most of the time, these functional forms have been used for descriptive purposes, but Datt & Ravallion (1992) suggested that they could be used to estimate elasticities of poverty. These Lorenz curves can be seen as ad-hoc since they are generally not theoretical grounded the only exception may be Maddala & Singh (1977). However, they generally fit pretty well the data and their estimation is rather easy. Nevertheless, the use of ad hoc Lorenz curves raises some problems. First, the underlying distribution function may not be defined for the value of the poverty line 13. Second, the corresponding distribution function has sometimes no closed form. To calculate the value of the FGT measures and their elasticities, we have to use the following properties of the Lorenz curve 14 : L(p) p = z p=p0 p, (11) 13 We also encounter this problem with the Pareto distribution. 14 More details on the use of ad hoc Lorenz curves for poverty analysis in Datt (1998) 9

10 3 DATA AND RESULTS 2 L(p) p 2 = 1 p=p0 µf(z). (12) Despite the attractiveness of these functional forms, Datt & Ravallion (1992) is the only study in which ad hoc Lorenz curve are used. In the present paper, we included functional forms 15 described by Maddala & Singh (1977), Gaffney, Koo, Obst & Rasche (1980), Arnold & Villaseñor (1989), Kakwani (1980) and Chotikapanich (1993). All the distributions and Lorenz curves used in the present paper are described in tables 1 and 2. 3 Data and results Income distribution data are from the UNU-WIDER World Income Inequality Database (version 2.0a., June 2005). Dropping observations when quality and reference population were not satisfying 16, we get a sample of 1,844 observations for 142 developed and developing countries from 1950 to For each observation we can make use of 6 to 13 points 17 of the Lorenz curve to estimate the parameters of the different Lorenz curves. Most functional forms imply non-linear least squares estimations, but estimators are convergent. To compute our poverty measures and scale parameters, we use PPP per capita income from Penn World Table 6.1. The main characteristics of the database are summarized in tables 13 and 14. For the present exercise, we exclusively work with the traditional 2$PPP poverty line 18, 19. Mean elasticities for the various poverty measures and distributions are reported on table 3. First, it appears that estimated elasticities crucially depend on the functional form which is adopted. Whatever poverty measure we choose, elasticities may greatly vary. In the case of the beta 2 distribution, the average growth elasticity of the headcount index is approximately -5.5 but -1.3 with the Weibull distribution. As noted earlier, distributions generally exhibit different behavior on the tails. As the absolute values of the elasticities of poverty increase rapidly with mean income, the differences stated in table 3 may only result from extreme values of the calculated elasticities. 15 We also tried to estimate the parameters of Castillo et al. (1999) class of Lorenz curve, but estimations were not convergent. 16 In particular, we removed many observations related to urban or rural populations. 17 We add the (0, 0) and (1,1) points since some functional forms for the Lorenz curve do not necessarily respect the conditions L(0) = 0 and L(1) = 1. In our sample, the average number of observations is For international comparisons, the 1$PPP poverty line is also widely used. We prefer using the 2$PPP line because it increases the ratio of the poverty line to mean income. Since our sample includes high income countries, it seems more reasonable to adopt the most meaningful line. We also have to mention that a lower poverty line would increase the number of invalid estimations for the ad hoc Lorenz curves since most of them are not defined z R +. A last reason is that the behavior of most distributions greatly varies at the tails. Our results would presumably be even more heterogeneous with the 1$PPP poverty line. 19 Strictly speaking the exact value is 2.16$ in 1996 PPPs. The poverty line defined for the Millennium Development Goals is fixed for 1993 PPPs, but Penn World Tables 6.1 are based on 1996 values. 10

11 Table 1: The different functional forms used Name Lorenz curve Scale parameter Pareto L(p) = 1 (1 p) 1 1 α y 0 = µ(α 1) α Lognormal L(p) = Φ ( Φ 1 (p) σ ) ȳ = lnµ σ Gamma L(p) = G ( G 1 (p, c, γ), c, γ + 1 ) ρ = µ γ ) Weibull L(p) = G G (W 1 (p, c, β), c, β,1 + 1 β ρ = ( Fisk L(p) = B 1 p, τ, τ 1 ) τ Singh-Maddala L(p) = B 1 (1 (1 p) λ, τ, λ 1 τ ) Dagum L(p) = B 1 (p 1 θ, θ + 1 τ, 1 1 τ ) κ = κ = κ = µ Γ 1+ 1 β µ Γ(1+ 1 τ )Γ(1 1 τ ) µγ(λ) Γ(1+ 1 τ )Γ(λ 1 τ ) µγ(θ) Γ(θ+ 1 τ )Γ(1 1 τ ) Beta 2 L(p) = B G2 ( B2 1 (p, c, λ, θ), c,1, λ + 1, θ 1 ) κ = µγ(θ)γ(λ) Γ(θ+1)Γ(λ 1) Chotikapanich (1993) L(p) = ekp 1 e k 1 µ Gaffney et al. (1980) L(p) = ( 1 (1 p) φ) 1 ζ µ 3 DATA AND RESULTS Maddala & Singh (1977) L(p) = bdp + (1 b + bd)p a + b ( 1 (1 p) d) µ Kakwani (1980) L(p) = p ξp ν (1 p) υ µ ( ) ( ) Arnold & Villaseñor (1989) L(p) = f p 2 L(p) +gl(p)(p 1)+q p L(p) 1 L(p) µ Note: Φ stands for the c.d.f. of the standard normal distribution, c for any constant term, G for the c.d.f. of the Gamma distribution, G G for the c.d.f. of the generalized gamma distribution, W for the c.d.f. of the Weibull distribution, B 1 for the c.d.f. of the Beta distribution of the first kind, B 2 for the c.d.f. of the Beta distribution of the second kind, B G2 for the c.d.f. of the generalized Beta distribution of the second kind. More details on the last distributions in Kleiber & Kotz (2003).

12 3 DATA AND RESULTS Table 2: Calculation of the headcount index for the different functional forms Name Pareto Lognormal Headcount index (c.d.f.) ( α P 0 = 1 z y0) P 0 = Φ ( z µ ) σ Gamma P 0 = G(z, ρ, γ) Weibull P 0 = 1 e z ρ ( Fisk P 0 = 1 + ( ) ) z τ 1 κ Singh-Maddala P 0 = 1 ( 1 + ( ) z τ ) λ κ ( Dagum P 0 = 1 + ( ) ) z τ θ κ Beta 2 P 0 = B 2 (z, κ, λ, θ) Chotikapanich (1993) P 0 = 1 k log ( β z(e k 1) kµ ) Gaffney et al. (1980) φ ζ ( 1 (1 P0 ) φ) 1 ζ 1 (1 P 0 ) φ 1 = z µ Maddala & Singh (1977) bd + a(1 b + bd)p 0 a 1 + bd(1 P 0 ) d 1 = z µ ( ) Kakwani (1980) 1 ξp ν 0 (1 P 0 ) υ ν P 0 υ 1 P 0 = z µ ( ( Arnold & Villaseñor (1989) P 0 = 1 2m n + r g + 2µ)) ( ) ( ) 1 2 z g + 2 z µ m w = f g q 1 m = g 2 4f n = 2fw 4q r = n 2 4mw 2 Note: Φ stands for the c.d.f. of the standard normal distribution, G for the c.d.f. of the Gamma distribution, B 2 for the c.d.f. of the Beta distribution of the second kind. 12

13 Table 3: Mean value of growth and Gini elasticities of P 0, P 1 and P 2 : whole sample. 13 Distribution Growth elasticity Gini elasticity Valid P 0 P 1 P 2 P 0 P 1 P 2 Estimations (%) Pareto Lognormal Gamma Weibull Fisk Beta Singh-Maddala Dagum Arnold & Villaseñor (1989) Chotikapanich (1993) Gaffney et al. (1980) Kakwani (1980) Maddala & Singh (1977) DATA AND RESULTS

14 Table 4: Median value of growth and Gini elasticities of P 0, P 1 and P 2 : whole sample. 14 Distribution Growth elasticity Gini elasticity Valid P 0 P 1 P 2 P 0 P 1 P 2 Estimations (%) Pareto Lognormal Gamma Weibull Fisk Beta Singh-Maddala Dagum Arnold & Villaseñor (1989) Chotikapanich (1993) Gaffney et al. (1980) Kakwani (1980) Maddala & Singh (1977) DATA AND RESULTS

15 3 DATA AND RESULTS To control for these extreme values, we reported the median for each elasticity, functional form and poverty measure in table 4. We still notice significant differences in the estimated elasticities, but values are less heterogeneous. However, one should be careful in interpreting these estimated elasticities. In tables 3 and 4, we can observe some erroneous values. In particular, we observe some positive values for the growth elasticities although growth elasticities are theoretically always negative. These irregular values are due to parameters which do not satisfy the validity conditions of the Lorenz curve (i.e. L(0) = 0, L(1) = 1 and 2 L(p) 0) or intervals of p 2 definitions which do not include the poverty line. The percentage of correct estimations are reported in tables 3 and 4. It appears that the Pareto distribution and the ad hoc functional forms for the Lorenz curve cannot be employed to analyze poverty for the whole sample. As the Pareto distributions and Chotikapanich (1993), Kakwani (1980), Arnold & Villaseñor (1989) Lorenz curves can only be used on a small part of the sample, we temporally exclude them from the set of tools used for the calculation of elasticities. This common sample includes 82.5% of initial observations. For this sub-sample, mean income is lower but difference is not significatively different. The average value of the Gini index is quite equal. Mean and median elasticities are reported in tables 5 and 6. Table 5: Mean value of growth and Gini elasticities of P 0, P 1 and P 2 : valid common sample. Distribution Growth elasticity Gini elasticity P 0 P 1 P 2 P 0 P 1 P 2 Lognormal Gamma Weibull Fisk Beta Singh-Maddala Dagum Gaffney et al. (1980) Maddala & Singh (1977) Common sample: 82.5% of initial observations; mean income: $PPP; mean Gini: In both tables, it appears that the lognormal distribution always provide the largest absolute mean and median values of both growth and Gini elasticities. On the contrary, lowest absolute values are provided by the Weibull distribution. Even if most distributions lead to average growth elasticities close to 2 a common value in the povertyrelated literature whatever poverty measure is considered, we cannot tell which value is the good one. 15

16 4 WHICH DISTRIBUTION SHOULD WE CHOOSE? Table 6: Median value of growth and Gini elasticities of P 0, P 1 and P 2 : valid common sample. Distribution Growth elasticity Gini elasticity P 0 P 1 P 2 P 0 P 1 P 2 Lognormal Gamma Weibull Fisk Beta Singh-Maddala Dagum Gaffney et al. (1980) Maddala & Singh (1977) Common sample: 82.5% of initial observations; mean income: $PPP; mean Gini: Which distribution should we choose? As different distributions imply different results for the elasticities of poverty, the question is now to choose the distribution which corresponds best to the empirical distributions of our data set. We assume that we should converge to the true value of the elasticities as the quality of the fit improves. In the statistical size distribution literature, the classical approach consists in looking at the regression errors. In this way, we calculate the following traditional statistics of goodness-of-fit: scr = sae = N i=1 ( L(p i ) ˆL(p i )) 2, (13) N L(p i ) ˆL(p i ). (14) i=1 A problem with these sums of squared (scr) and absolute (sae) errors is that all errors are given equal weight. As Datt (1998) notes, for the purpose of poverty analysis, we are only interested in errors up to the value of the headcount index. In this way, he proposed using the following partial scr: pscr = n i=1 ( L(p i ) ˆL(p i )) 2, (15) with n corresponding to the first observation of the Lorenz curve where p n ˆP 0. However, in the particular case of the headcount index, we are only interested in the quality of the 16

17 4 WHICH DISTRIBUTION SHOULD WE CHOOSE? fit in the vicinity of the estimated headcount 20. Thus, we propose a measure based on squared error with weight decreasing with the distance from the estimated value of the headcount. To make comparisons feasible between each functional form, we normalize the measure by the sum of weights. It also allows us to do comparisons with the traditional scr statistic. The weighted scr is: ( N i=1 L(p i ) ˆL(p ) 2 (1 i ) pi P 0 ) 2 wscr = N ( i=1 1 pi P 0 ) 2. (16) To compare average quality of the fit for each functional form, we computed the ratios of mean and median statistics for each distribution to the mean and median of the best-fitted distribution. Results are shown in table 7 for the whole sample (including non-valid estimations) and in table 8 for our restricted sample 21. It appears that all twoparameter distributions (Pareto, lognormal, gamma, Weibull, Fisk and Chotikapanich (1993)) performs very poorly in comparison with three-parameter distributions(beta 2, Singh-Maddala, Dagum and Gaffney et al. (1980)) and four-parameter distributions (Maddala & Singh (1977), Arnold & Villaseñor (1989) and Kakwani (1993)). In particular, we can see that the lognormal is about 5 to 10 times less precise than Kakwani (1980) and Maddala & Singh (1977). This classical result is not surprising since a single inequality parameter can hardly account for the observed heterogeneity of income distributions. Despite the obvious superiority of some functional forms, in particular the Kakwani (1980) and Maddala & Singh (1977) forms, we cannot definitively reject the other distributions. If a distribution generally poorly fit the data, it does not mean that fit is systematically poor. To get a more precise picture of the respective performance of each functional form, we ranked the different valid estimations for each observation by their respective value for each statistic. The counts 22 are presented for our different statistics of fit in tables 9 to 11. It appears, that even if two-parameter distributions are most of the time outperformed by more flexible functional forms, they sometime fit better than more flexible functional forms. In particular, the lognormal distribution is the best choice for 43 observations, according to our wscr statistic. So lognormality is not the rule, but it can be the exception. Before turning back to elasticities, we have to notice that the observed ranking inside each family of functional form is rather surprising. An important number of studies 20 Remember that the headcount is defined as the point of the Lorenz curve where the slope is equal to the ratio of the poverty line to mean income, as in equation (11). So its elasticities depend only on the form of the Lorenz curve close to the estimated headcount. 21 Summary statistics for each functional form and statistic are provided in appendix by table 17 and 18 for each sample. 22 The number of observations in table 11 differs from those of tables 9 and 10 since per capita income informations are missing for about 250 observations. Since calculation of the wscr statistic requires this information, the number of observations for the ranks is lower. 17

18 18 Table 7: Ratios of goodness-of-fit. Distribution scr sae wscr Mean Median Mean Median Mean Median Pareto Lognormal Gamma Weibull Fisk Bêta Singh-Maddala Dagum Arnold & Villaseñor (1989) Chotikapanich (1993) Gaffney et al. (1980) Kakwani (1980) Maddala & Singh (1977) WHICH DISTRIBUTION SHOULD WE CHOOSE?

19 5 THE LOGNORMAL CASE AND THE MOST PLAUSIBLE ELASTICITIES Table 8: Ratios of goodness-of-fit (restricted valid sample). Distribution scr sae wscr Mean Median Mean Median Mean Median Log-normale Gamma Weibull Fisk Beta Singh-Maddala Dagum Gaffney et al. (1980) Maddala & Singh (1977) have insisted on the merits of the different functional forms used in the present study but comprehensive comparisons of functional forms for income distributions and Lorenz curve are scarce and rarely mix classical distributions and ad hoc Lorenz curve 23. Using 82 distribution data sets at various years for 23 developed and middle-income countries, Bandourian et al. (2002) observed that the Weibull and Dagum were the best-fitting models for the two- and three-parameter distribution family, when opposed to the gamma, lognormal, beta 1 and Singh-Maddala distributions. Our results suggest that Fisk and lognormal distributions are the best two-parameter models and the Singh-Maddala and Gaffney et al. (1980), the best three-parameter models. For ad hoc Lorenz curves comparisons, Cheong (2002) compared the Kakwani & Podder (1976), Kakwani (1980), Gaffney et al. (1980), Basmann et al. (1990), Fernandez et al. (1991) and Chotikapanich (1993) functional forms on US data from 1977 to 1983 and noticed that Gaffney et al. (1980) and Kakwani (1980) were the most powerful models. In the present study, we find that Kakwani (1980) is the functional form which generally fits best to our sample of distributions. However, for poverty analysis, we should prefer Maddala & Singh (1977) and Gaffney et al. (1980) forms since definition intervals are larger. 5 The lognormal case and the most plausible elasticities In a recent paper, Lopez & Servèn (2006), using the Dollar & Kraay (2002) database, concluded that lognormality cannot be rejected for per capita income distributions. So, even if the lognormal distribution does not fit as well income distributions as other forms do, it may produce reasonable values for the elasticities of poverty. Nevertheless preceding results (cf tables 3 to 6) seem to contradict this assertion. To test this hypothesis, we 23 Most of the time, goodness-of-fit tests of ad hoc Lorenz curves include the lognormal distribution as a benchmark, but never more performing distributions 19

20 20 Table 9: Goodness-of-fit rank: scr statistic. Distribution Pareto Lognormal Gamma Weibull Fisk Beta Singh-Maddala Dagum Arnold & Villaseñor (1989) Chotikapanich (1993) Gaffney et al. (1980) Kakwani (1980) Maddala & Singh (1977) THE LOGNORMAL CASE AND THE MOST PLAUSIBLE ELASTICITIES

21 21 Table 10: Goodness-of-fit rank: sea statistic. Distribution Pareto Lognormal Gamma Weibull Fisk Beta Singh-Maddala Dagum Arnold & Villaseñor (1989) Chotikapanich (1993) Gaffney et al. (1980) Kakwani (1980) Maddala & Singh (1977) THE LOGNORMAL CASE AND THE MOST PLAUSIBLE ELASTICITIES

22 22 Table 11: Goodness-of-fit rank: wscr statistic. Distribution Pareto Lognormal Gamma Weibull Fisk Beta Singh-Maddala Dagum Arnold & Villaseñor (1989) Chotikapanich (1993) Gaffney et al. (1980) Kakwani (1980) Maddala & Singh (1977) THE LOGNORMAL CASE AND THE MOST PLAUSIBLE ELASTICITIES

23 5 THE LOGNORMAL CASE AND THE MOST PLAUSIBLE ELASTICITIES propose a comparison between estimated lognormal elasticities and the values which seems the most plausible for each observation. To get these values, we simply take for each observation the estimated elasticity corresponding to the best fitting model. The composition of these series of elasticities is given by the first column of tables 9 to 11. Thus, we get growth and Gini elasticities series for our different poverty measures from the scr, sae and wscr statistics 24. Summary statistics for these series are given in table 12. Since we are only interested in the quality of the fit in the vicinity of the estimated headcount, the mixed-series based on the wscr criterion will be our preferred series. It appears that lognormal elasticities are most of the time higher than our best-fitting values. On average, the lognormal elasticities is about 1 to 2 percentage points higher in absolute value for the growth elasticities. Differences in Gini elasticities are really striking, in particular for the headcount index. Under the lognormality assumption, the average elasticity is 78.43, twice larger than our estimated elasticities. Such a large difference cannot be attributed to extreme values since the same phenomenon is observed for median values. However, overestimation of Gini elasticities tend to shrink as the θ parameter of the FGT measure increases. Overestimation is a problem, but we can imagine that lognormal elasticities are highly correlated with true elasticities, so that we can find the appropriate values of the desired elasticities again through a simple linear correction. Figure 4 clearly show that the correlation between lognormal and the mixed series based on the wscr criterion is rather low 25 (from 0.3 to 0.5 depending on the elasticity and the goodness-of-fit criterion). The plots also confirm that, on average, the lognormal assumption overestimates, in absolute value, the growth and Gini elasticities of poverty. So, we should be extremely cautious with political recommendations based on the lognormality assumption. Finally, even if the lognormal distribution does not correctly predict the magnitude of each elasticity, we may use it to compare the relative effectiveness of pure growth and pure distributional politics to achieve poverty alleviation. In section 2.1 (equation 9), we showed that under assumption (6) the ratio of the growth to Gini elasticities of the headcount index is independent of the income distribution chosen. This is not the case at higher values of the θ parameter of the FGT class of poverty measures. To investigate whether the lognormal assumption biases policy recommendations towards growth or distributional policies, we can use the ratio of the lognormal elasticities ratio to the one corresponding to our best-fitting estimations. A value greater (lower) than one would indicate that the lognormal assumption bias politics toward growth (inequality reduction). A kernel estimation of the density of this ratio for the P 1 and P 2 measure is 24 As a test of robustness, we also designed series corresponding to estimations that were ranked second according to our different statistics. Result for these second-best series do not differ from those obtained with best-fitting series. 25 The same differences are observed with the mixed series based on the scr and wscr statistics. Plots are reported on figures 8 and 9. 23

24 24 Table 12: Summary statistics for the mixed series based on the scr, sae and wscr statistics. Distribution P 0 P 1 P 2 scr sea wscr scr sea wscr scr sea wscr Growth elasticities Mean Min st quartile Median rd quartile Max e 03 2e 03 2e 03 Std deviation Gini elasticities Mean Min st quartile Median rd quartile Max Std deviation THE LOGNORMAL CASE AND THE MOST PLAUSIBLE ELASTICITIES

25 5 THE LOGNORMAL CASE AND THE MOST PLAUSIBLE ELASTICITIES Growth elasticity of P0 Growth elasticity of P2 Gini elasticity of P1 lognormal wscr elasticities Growth elasticity of P1 lognormal lognormal wscr elasticities Gini elasticity of P0 lognormal lognormal wscr elasticities Gini elasticity of P2 lognormal wscr elasticities wscr elasticities wscr elasticities Note: Vertical and horizontal solid and dashed line respectively correspond to mean and median values. Figure 4: Comparison of the estimated elasticities of poverty between lognormal distribution and wscr series. reported on figure 5. First we note that there is apparently no systematic bias of the lognormal assumption toward a particular kind of poverty reduction policy since mean and median values are close to 1. However, the dispersion of the ratio is quite high, in particular if we consider the P 2 index 26. Under the lognormal hypothesis, we notice that the relative importance of the growth elasticity can be overestimated or underestimated in excess of 50%. Such under- and overestimations may be due to the presence of high income countries in the sample. As poverty defined with a 2 $ PPP poverty line is essentially a concern for low income countries, it seems reasonable to restrict our sample to low income countries. Kernel densities for countries with income per capita lower than $PPP (795 observations) are reported on figure 6. As with the whole sample, we notice that, on average, the lognormal assumption does not lead to any systematic political bias. However, uncertainty is still significant. Finally, to see if the lognormality assumption imply some policy bias depending on the degree of inequality, we restrict our sample to observations with low income income inequalities. Kernel densities for observations with Gini index lower than 0.4 (866 obser- 26 Since our results may be disturbed by the strong presence of developed and eastern Europe countries, we also realized the same exercise on a more balanced sample of 281 observations spanning from 1985 to Results are unchanged. 25

26 5 THE LOGNORMAL CASE AND THE MOST PLAUSIBLE ELASTICITIES P1 P2 Density Note: Vertical solid and dashed line respectively correspond to mean and median values. Figure 5: Kernel density of the ratio of policy bias for P 1 and P 2 (lognormal against best-fitting estimations). P1 P2 Density Note: Vertical solid and dashed line respectively correspond to mean and median values. Figure 6: Kernel density of the ratio of policy bias for P 1 and P 2 (lognormal against best-fitting estimations): low income countries. 26

27 6 CONCLUDING REMARKS vations) are reported in figure 7. In this case, mean and median values suggest an average policy bias toward pure growth policies. For the P 1 index, overestimation is about 15%, and about 20% for the P 2 index. The variance of the ratio is still important. P1 P2 Density Note: Vertical solid and dashed line respectively correspond to mean and median values. Figure 7: Kernel density of the ratio of policy bias for P 1 and P 2 (lognormal against best-fitting estimations): low inequality observations. 6 Concluding remarks In the present paper, we intend to respond to the following questions. Is the lognormal assumption a good assumption for the estimation of elasticities of poverty? What are the real values of the elasticities of poverty? For the first question, we note that the observed heterogeneity of income distributions cannot be handled by functional forms which are characterized by little flexibility. In the present paper we focused on the lognormal distribution since it is the most widely used in the literature. Our results prove that the use of the lognormality assumption often implies an overestimation of estimated elasticities. Moreover, it poorly helps for the design of poverty alleviation policies at the country level since it results in quite uncertain estimations of the poverty-inequality trade-off that policy-makers have to face. We have to admit that we actually cannot answer with certainty to the second question. Accuracy of the results may surely be improved with more flexible functional forms. We also have stressed that inequality elasticities are potential elasticities since it depends on political choices. However, one should question if the Kakwani s transformation is appropriate for observed redistribution policies. As noted earlier, this transformation implies quite spectacular improvement of the situation of the poorest. Less pro-poor redistribution policies would surely be more realistic. So a more complete view of potential inequality elasticities would require finding some class of Lorenz transformations 27