# Assessing the efficiency of inequality changes in terms of poverty alleviation

Florent Bresson

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Florent Bresson. Assessing the efficiency of inequality changes in terms of poverty alleviation. 2025. hal-05229456

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WORKING PAPER

No. 2025/5

Assessing the efficiency
of inequality changes in terms
of poverty alleviation

Florent Bresson
## Cite this working paper

Bresson, F. "Assessing the efficiency of inequality changes in terms of poverty alleviation", CERDI Working Papers, 2025/5, CERDI.

## The author

### Florent Bresson

Associate Professor of Economics,
Université Clermont Auvergne, CNRS, IRD,
CERDI, F-63000 Clermont-Ferrand, France
Email: florent.bresson@uca.fr

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Ce travail a bénéficié d'une aide de l'Etat français gérée par l'Agence Nationale de la Recherche au titre du programme "France 2030" portant la référence "ANR-16-IDEX-0001".

This work was supported by the Agence Nationale de la Recherche of the French government through the program "France 2030" "ANR-16-IDEX-0001".

CERDI Working Papers are available online at: https://tinyurl.com/2xwfzw8s

Director of Publication: Simone Bertoli

Editor: Catherine Araujo Bonjean

Publisher: Marie Dussol

ISSN: 2114-7957

### Disclaimer

Working papers are not subject to peer review; they constitute research in progress. Responsibility for the contents and opinions expressed in the working papers rests solely with the authors. Comments and suggestions are welcome and should be addressed to the authors.
**Abstract**

This paper introduces a novel analytical approach to assess the efficiency of inequality reduction in poverty alleviation. I develop a general transform of the Lorenz curve that enables the computation of point elasticities of poverty with respect to inequality while perfectly controlling for changes in inequality levels. Unlike previous methods, the suggested approach makes it possible to measure elasticities corresponding to observed inequality changes. It also provides a framework for benchmarking these elasticities. These elasticities are estimated for different subperiods using Brazilian income distribution data from 1981-2022. Results show the increasing powerfulness of inequality reduction as a lever for poverty alleviation. Moreover, while decreased inequality significantly contributed to poverty reduction during Brazil's celebrated poverty reduction period between 2003 and 2014, I observe that the distributional changes were not more pro-poor, compared with later subperiods, yet more efficient than in earlier subperiods in terms of poverty reduction.

**Keywords**

Poverty; inequality; inequality elasticities of poverty; Brazil.

**JEL codes**

D31, D63, O15.
# 1 Introduction

Despite the unprecedented progresses obtained during the Millennium Development Goals (MDG) era, Target 1.1 of the Sustainable Development Goals (SDG), that is eradicating extreme poverty by 2030 at the global level, was an ambitious objective when it was set in 2015 (Ravallion, 2013). Of course, the COVID-19 pandemic has severely undermined the achievement of this goal (World Bank, 2020, 2022), but the pre-COVID-19 trends also stirred up some pessimism regarding the odds of success (Ravallion, 2020). The 2022 *Poverty and Shared Prosperity Report* summarizes this position in an unambiguous manner: *“The goal of reducing global poverty to 3 percent by 2030 was difficult enough to achieve before the current crises. The recent setbacks have put this target nearly out of reach—and there is an urgent need to correct course.”*<fnref n="1" />

What does correcting the course mean? Monetary poverty analysis basically identifies two channels for poverty reduction, namely the increase in the population mean income and favorable changes in the relative distribution of income (Kakwani, 1993, Bourguignon, 2003). Poverty reduction during the MDG era was essentially fueled by large and sustained increases in the average income of most developing countries (Kraay, 2006, Dollar et al., 2015, Bluhm et al., 2018), with China being the most striking example of this successful growth-driven poverty reduction strategy (Chen and Ravallion, 2007, Goh et al., 2009, Ravallion, 2011). Growth in average income is undoubtedly still key for massive poverty alleviation in most low-income countries, but lessening inequality is often regarded as a necessary condition for successful poverty eradication (World Bank, 2016, De Schutter, 2024). Shrinking inequality first directly contributes to poverty alleviation, and its relative contribution is likely to increase as mean income rises (Dağdeviren et al., 2004, Ram, 2007, Bergstrom, 2022). It also strengthens the poverty reduction return of a given growth rate in mean income (Kalwij and Verschoor, 2007, Ravallion, 2007). Finally, it may also boost economic growth (Heltberg, 2004), although the theoretical effect is uncertain (Cerra et al., 2022), and the empirical evidence is mixed regarding this last effect (Banerjee and Duflo, 2003). Recent projections nevertheless suggest that a substantial decline in inequality would make it possible to have extreme poverty below 3% by 2040 (Lakner et al., 2022, World Bank, 2024).

The direct effect of shrinking inequalities on poverty depends both on the magnitude of the inequality change and on the degree of pro-poorness of this change. Indeed, a given decrease in poverty may result from a large but relatively inefficient change in inequality, or be the consequence of a small but highly effective inequality variation. In the present paper, I focus on this efficiency aspect of the poverty effect of inequality changes. This efficiency of poverty reduction is commonly assessed with the help of an inequality elasticity of poverty. Here, I introduce a general transform of the Lorenz curve that enables the generation of any possible Lorenz curve while perfectly controlling the change in the degree of inequality. This makes it possible to propose formulas for the point elasticity of poverty with respect to inequality. The distinctive feature of my approach is that, unlike previous tries (Kakwani, 1993, Bourguignon, 2003, Bresson, 2010), it allows for the computation of

<footnote n="1">¹World Bank (2022, p.8)</footnote>
the value of this elasticity corresponding to observed inequality changes. I also propose a framework to benchmark this elasticity as well as well as decompositions to assess the relative contributions of the incidence, intensity, and inequality dimensions of monetary poverty (Jenkins and Lambert, 1997). I finally use the decomposability property of the Gini index in the case of non-overlapping group to obtain an elasticity of poverty with respect to the between component and an elasticity of poverty with respect to inequality within the *p* poorest percent of the population.

I illustrate my findings using data on the income distribution in Brazil during the period 1981–2022. Brazil is a relevant case for this purpose. First Brazil is known as one of the most unequal economy around the world (Alvaredo et al., 2024). Consequently, theory predicts a potentially large effect of inequality reduction for poverty alleviation. Secondly, starting from the mid 1990s, Brazil effectively succeeded in fighting both inequality and poverty, notably during Lula da Silva’s first presidential period. Indeed, the Gini index fell from 0.6 in 1993 to 0.52 in 2014². At the same time the share of the population living with less than $3.65 (PPP, base 2017) dropped from 39% to 8.7%. So it is worth questioning how inequality changes contributed to reduced poverty and whether changes in the relative distribution was efficient as a channel of poverty alleviation.

Considering first yearly changes in the relative distribution of income, I observe a substantial variability for the inequality elasticity of poverty. Disregarding two values that can be regarded as outliers and sticking to the $3.65 poverty line, the average value of the elasticity is 3.15, 4.89, and 6.00 respectively for the headcount index, the poverty gap index, and the squared poverty gap index, hence stressing the potential gains with respect to poverty alleviation of inequality changes in a country like Brazil. More surprisingly, I do not find evidence of an increasing trend for this elasticity during the considered period, although it is generally assumed that this elasticity should increase as mean income increases. My results also show varying contributions of the incidence, intensity, and inequality dimensions of poverty, hence stressing the heterogeneity of the pattern of relative distribution changes in terms of poverty reduction.

Then, focusing on the subperiod 2003–2014, that is the period of Lula da Silva’s first two presidential mandates and the beginning of Dilma Rousseff’s presidency, I note that the dramatic decrease in poverty during the period was mostly due to the unprecedented decrease in inequality in this country. The pattern of the change in the relative distribution of income was pro-poor during the period, but, compared to other subperiods, it could have been significantly more pro-poor, had the extremely poor benefited from each eliminated point of inequality as they did for instance between 2019 and 2022.

This paper contributes to the macroeconomic literature on the relationship between growth, inequality, and poverty. The poverty-growth-inequality triangle (Bourguignon, 2004) has generated a rich empirical and theoretical literature regarding the relationships between these three concepts (Cerra et al., 2022). Here, I focus on the static direct effects of inequality changes on poverty. It notably helps documenting the connection between the World Bank twin goals, namely ending extreme poverty and promoting shared prosperity (Lakner et al., 2014, World Bank, 2015).

<footnote n="2">²Reported values are our own estimates based on the data used in section 5.</footnote>
Three approaches have been considered in the literature to assess the relevance of the inequality lever for poverty reduction. The first one is an econometric approach that regresses poverty variations on mean income and inequality indicators (Ravallion, 1995, 2001, 2007, Besley and Burgess, 2003, Adams, 2004, Klasen and Misselhorn, 2006, Kalwij and Verschoor, 2007, Fosu, 2017, Bluhm et al., 2018, Bergstrom, 2022). More specifically, when relative variations of poverty are regressed on the relative changes in per-capita income and in inequality, the coefficient associated with this last variable is interpreted as an average value for the inequality elasticity of poverty. Disregarding the difficulty of having a causal interpretation for this coefficient as well as purely technical econometric issues, econometric studies generally assume a constant value or a specific function form for the inequality elasticity of poverty, hence concealing the heterogeneity of inequality changes effects on poverty (Hartmann and Wacker, 2023). The second approach is based on poverty variations decompositions into growth and inequality effects (Kakwani and Subbarao, 1990, Huppi and Ravallion, 1991, Datt and Ravallion, 1992, Kakwani, 2000, Shorrocks, 2013, Aristondo et al., 2023). An inequality elasticity of poverty can then be further estimated by dividing the inequality effect by the relative change in the considered inequality index. The main issue of the approach is that it yields arc elasticities. Arc elasticities can easily be computed but their value depends on the magnitude of the considered change, hence preventing sound comparisons between periods and countries. The last approach is analytical; the inequality elasticity of poverty is derived from the formula of the chosen poverty index under the assumption of a specific transformation of the relative distribution of income (Kakwani, 1993, Bresson, 2010). As with the second method, the approach makes it possible to fully account for initial conditions and to get an estimate of the elasticity for each country-year observation. Moreover, it returns point-elasticities, hence addressing the comparability issue. However, its usefulness for analyzing observed poverty variations relies on the relevance of the assumed transformation of the relative distribution of income.

The present paper extends this last approach while addressing the issue of the appropriateness of the presumed pattern of inequality changes. Indeed, the considered transform may perfectly mimic every observed change in the Lorenz curve, hence yielding values of the inequality elasticity of poverty that reflect true inequality changes. Moreover, the flexibility of the approach as well as the capacity to control the variation of the degree of inequality gives the opportunity to benchmark each estimation so as to assess the more or less pro-poor nature of the redistribution processes at play during the considered period.

The paper also adds to the understanding of poverty changes in Brazil. Being a large and highly unequal country, the Brazilian case has generated numerous empirical studies (see notably Datt and Ravallion, 1992, Kakwani, 1993, Ferreira et al., 2010, de Souza, 2012, Sotomayor, 2021, Wang and Sun, 2021). The inequality channel has often been pointed as a potentially powerful lever for poverty alleviation in this country (Ravallion, 2011) and social policies, notably conditional cash transfer programs, implemented during the first decade of the XXIst century have been praised for having made growth pro-poor during this period (Lustig et al., 2013). My results corroborates the large and growing potential of shrinking income inequality for poverty reduction. However, the last four
decades also indicate that the efficiency of inequality reduction is highly instable in the short run. Indeed, considering the headcount index with a $3.65 poverty line (PPP, base 2017), yearly estimates of the inequality elasticity of poverty range from -34.8 to 9.9. Benchmarking the estimates obtained for larger periods also sheds new lights on the most successful period for poverty reduction in the recent history of Brazil. My results show that, surprisingly, the redistribution pattern between 2003 and 2014, that is since the first election of Lula da Silva as president of Brazil to the start of the political and economic crisis in 2014, essentially favored the non-extreme poor, hence lowering the efficiency of the observed decrease in the Gini index.

The remainder of this paper is structured as follow. Section 2 introduces notations and summarize the main results regarding analytical expressions of poverty elasticities with respect to mean income and inequality. Section 3 offers additional results using decomposability properties of either widely used poverty indices or of the Gini index. The case of relative poverty lines is studied in section 4. The Brazilian application is presented in section 5. Finally, section 6 concludes.

## 2 Notations and the growth and inequality elasticities of poverty

Let $F(z)$ denote the cumulative distribution function (cdf) of income (or consumption, or any other continuous variable that serves as a proxy for individual well-being). The function returns the proportion of the population with income less than $z$. If the chosen value for $z$ is interpreted as the poverty line, $F(z)$ thus corresponds to the most widely used poverty index, namely the headcount ratio $\Theta_0$. Inverting the cdf at the $p$th population percentile yields the corresponding value of the quantile function $y(p) := F^{-1}(p)$ (Gastwirth, 1971). Assuming that income is a non-negative real number, poverty can then be estimated with the help of additively decomposable poverty indices $\Theta$ of the form:

$$ \Theta(z; F) := \int_{0}^{F(z)} \theta(y(p), z) dp, \qquad (1) $$

where $z > 0$. In the present paper, I assume that the poor are those persons whose income is strictly less than the poverty line. For $\Theta$ to be consistent with core poverty measurement axioms (Zheng, 1997), we need $\theta$ in (1) i) to be strictly positive for $y(p) < z \forall p < F(z)$ and zero otherwise, and ii) to be non-increasing everywhere. Most poverty indices, like those proposed by Watts (1968), Foster et al. (1984), and Kraay et al. (2023), are of the form (1) and satisfy all these restrictions. The function $\theta$ then takes the respective functional forms $\theta_{\log}(p, z) := \log z - \log y(p)$, $\theta_{\alpha}(p, z) := (1 - \frac{y(p)}{z})^{\alpha}$ with $\alpha \ge 0$, and $\theta_{P}(p, z) := \frac{z}{y(p)} - 1$.

An income distribution can be fully described by its mean income $\mu$ and its Lorenz curve $L(p)$ since they jointly capture all the moments of the distribution. Figure 1 makes it clear how $\Theta$ is fully determined with the help of $z$, $\mu$, and $L$. Indeed, an interesting property of the Lorenz curve is $\frac{\partial L}{\partial p}|_{p=q} = \frac{y(q)}{\mu}$. A direct implication it that setting $y(p) = z$ makes it possible to identify easily the value of the headcount index. However, the incidence of
<figure><img src="image_4.png" /><figcaption>Figure 1: Lorenz-curve, mean-income, and the three dimensions of poverty.</figcaption></figure>

poverty is not the unique dimension of poverty that is worth consideration when evaluat-
ing poverty. As stressed by numerous authors (Sen, 1976, Jenkins and Lambert, 1997), the
intensity of poverty and the inequality among the poor shall also be taken into account in
order to get a well-behaved poverty index. Both dimensions can also be analyzed thanks
to Figure 1. The intensity of poverty is basically related to the mean income of the poor $\mu_z$.
Using the mean value theorem, it can easily be shown that the value of $\mu_z$, as a proportion
of $\mu$, corresponds to the slope of the ray through the origin and the point $A := (\Theta_0, L(\Theta_0))$.
The inequality dimension is finally captured by the curvature of the Lorenz curve on the
interval $[0, \Theta_0]$. The further the curve deviates from the ray through the origin and A, the
larger is the inequality component of poverty.<fnref n="3" />

Within this analytical framework, changes in poverty can only occur through changes
in the poverty line, the mean income, or the Lorenz curve. Figure 2 illustrates the effect
on the headcount index of either an increase in mean income or in the level of relative
inequality.<fnref n="4" /> A common exercise in poverty analysis is to estimate an additive decomposi-
tion of poverty variations into components that can respectively be attributed to changes
in mean income and changes in the relative distribution of income.<fnref n="5" /> When evaluated in
relative terms, these components can be expressed as the product of an elasticity and the
relative change in either mean income or in some inequality index. This multiplicative de-
composition notably enables the evaluation of the effectiveness of each channel in fighting
poverty with the help of these elasticities.

In the next pages, we discuss how these elasticities can be obtained analytically in the
spirit of Kakwani (1993).

<footnote n="3">³Let $S$ be the area between the Lorenz curve and the ray through the origin and $A$. It can then be shown that $\frac{2S}{\Theta_0 L(\Theta_0)}$ equals the value of the Gini index estimated for the vector of the income of the poor.</footnote>

<footnote n="4">⁴In the first instance, we discard the possibility of having changes in $z$. The case of variable poverty lines is discussed in section 4.</footnote>

<footnote>By the way, it can easily be seen that the effect of an increase in $z$ is symmetric to a decrease in $\mu$, so
that Figure 2a can also be used to illustrate the effect of a change in $z$, provided the new poverty line $z'$ is
such that $\frac{z'}{\mu} = \frac{z}{\mu'}$.</footnote>

<footnote n="5">⁵See notably Kakwani and Subbarao (1990), Huppi and Ravallion (1991), Datt and Ravallion (1992), Kakwani (2000), and Shorrocks (2013).</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 2: The mean income and inequality channels of poverty alleviation.</figcaption></figure>

## 2.1 Kakwani's elasticities

A general expression of the point elasticity $\eta_{\Theta}$ of $\Theta$ with respect to mean income is (Kakwani, 1993):

$$ \eta_{\Theta} = \frac{1}{\Theta} \left[ -\theta(y(\Theta_0), z) z f(z) + \int_{0}^{F(z)} y(p) \frac{\partial \theta}{\partial y} dp \right]. \quad (2) $$

In the case of the headcount index $\Theta_0$, one then gets the formula $\eta_0 = -\frac{zf(z)}{\Theta_0}$, where $f$ stands for the density function associated with $F$. For other members of the family of FGT indices (i.e. $\alpha \ge 1$), the elasticity becomes $\eta_{\alpha} = \alpha \frac{\Theta_{\alpha} - \Theta_{\alpha-1}}{\Theta_{\alpha}}$.6 With the Watts index, formula (2) becomes $\eta_{\log} = -\frac{\Theta_0}{\Theta_{\log}}$. Using $\theta_P$, one finally obtains $\eta_P = -\frac{\Theta_P + \Theta_0}{\Theta_P}$. It is worth pointing that, with the exception of the headcount index that requires an estimation of $f(z)$, data requirement for the computation of these elasticities is very low since they can be computed using aggregate poverty indices.

Let $\Psi$ be a relative inequality index like the Gini index. Assuming that an increase in mean income is associated with a certain change in $\Psi$, a total growth elasticity of poverty $\tau_{\Theta}$ can then be defined as $\tau_{\Theta} := \eta_{\Theta} + \frac{d \log \Psi}{d \log \mu} \varepsilon_{\Theta}^{\Psi}$ where $\varepsilon_{\Theta}^{\Psi}$ is the inequality elasticity of poverty Ravallion (2007).7 It deserves to be stressed that the reference to growth for $\tau_{\Theta}$

<footnote n="6">6 In the case $\alpha = 1$, an alternative expression is $-\frac{\mu z}{z - \mu z}$ (Kakwani, 1993).</footnote>

<footnote n="7">7 During many decades, the relationship between mean income and inequality was dominated by Kuznets's</footnote>
and $\eta_{\Theta}$ is misleading. Indeed, economic growth usually refers to GDP per capita growth while $\tau_{\Theta}$ and $\eta_{\Theta}$ relate to growth in individual income. Moreover, the relationship between growth in GDP per capita and growth in average income is not a one-to-one relationship (Ravallion, 2003, Deaton, 2005). Having this caveat in mind, I nevertheless follows the common practice and name $\tau_{\Theta}$ the (total) growth elasticity of poverty and $\eta_{\Theta}$ the pure-growth elasticity of poverty. Moreover, for the sake of simplicity, I will solely consider the Gini index for the measurement of inequality. Consequently, the superscript in $\varepsilon_{\Theta}^{\Psi}$ can be dropped, so that $\varepsilon_{\Theta}$ shall be read as the Gini elasticity of $\Theta$ for the rest of the paper.

In comparison with $\eta_{\Theta}$, computing a point elasticity with respect to changes in the relative distribution of income is a trickier exercise since there is no unique way for the Lorenz curve to change as a result of a change in $\Psi$ as illustrated by Figures 2b and 2c. The same change in any inequality index can result in different poverty variations, possibly in opposite directions. Putting this issue aside, Kakwani (1993) assumed that the Lorenz curves systematically shifts in a specific manner, so that the resulting Lorenz curve $L^K(p)$ is:

$$L^K(p) = L(p) - \lambda(p - L(p)), \qquad (3)$$

with $\lambda \ge -1$. The corresponding quantile function is:<fnref n="8" />

$$y^K(p) = (1 + \lambda)y(p) - \lambda\mu. \qquad (4)$$

It describes an income distribution whose mean income is the same, but the value of the Gini coefficient $G$ is 100$\lambda$% larger than with the initial income distribution<fnref n="9" />. To represent the change resulting from (3), an alternative representation is Ravallion and Chen's (2003) growth incidence curve (GIC) that plots for each population percentile $p$ the growth rate $g(p)$ of the corresponding income quantile. In the case of Kakwani's transform, its expression is simply:

$$g^K(p) := \frac{y^K(p)}{y(p)} - 1 = \left(1 - \frac{\mu}{y(p)}\right) \lambda. \qquad (5)$$

Lakner et al. (2022) dubbed it the convex GIC since, for $\lambda < 0$, $g^K(p)$ is convex (and decreasing) over the whole unit interval.

Under assumption (3), $\varepsilon_{\Theta}$ is directly related to the first-order derivative of $\Theta$ with respect to $\lambda$. Considering the possibility of $\theta$ being discontinuous at the poverty line, the

<footnote>(1955) theory. However, the empirical evidence (Deininger and Squire, 1996, Chen and Ravallion, 1997) is that changes in inequality are poorly correlated with changes in average income, hence suggesting that $E(\frac{d \log \Psi}{d \log \mu})$ is not significantly different from zero. However, as underlined by (Ravallion, 2016, section 8.2), this lack of correlation essentially means that there is a strong heterogeneity in observed values for $\frac{d \log \Psi}{d \log \mu}$.</footnote>

<footnote n="8">⁸Thanks to the relationship between the slope of the Lorenz curve, the mean income and the quantile function mentioned page 4, we have $y(q) = \mu \frac{\partial L}{\partial p}|_{p=q}$.</footnote>

<footnote n="9">⁹Bresson (2010) showed that with the transform (3), $\lambda$ also corresponds to the relative change in the value of many rank-dependent relative inequality indices including the whole set of S-Gini indices (Donaldson and Weymark, 1980) and those proposed by Aaberge (2000). A simple test of assumption (3) is thus the different members of this family of inequality indices jointly grow or shrink at the same rate.</footnote>
following general formula is then obtained:

$$ \varepsilon_{\Theta}^{K} = \frac{1}{\Theta} \left[ \theta(y(\Theta_0), z)(\mu - z)f(z) + \int_{0}^{\Theta_0} (y(p) - \mu) \frac{\partial \theta}{\partial y} dp \right], \quad (6) $$

where K indicates that assumption (3) holds. For the headcount index and other aforementioned poverty indices, the corresponding formulas are respectively $\varepsilon_0^K = (\mu - z)\frac{f(z)}{\Theta_0}$, $\varepsilon_\alpha^K = \alpha(1 + \frac{\mu-z}{z}\frac{\Theta_{\alpha-1}}{\Theta_\alpha})$, $\varepsilon_{\log}^K = \frac{(\mu-\kappa-1)\Theta_0}{\kappa-1\Theta_{\log}}$, and $\varepsilon_P^K = \frac{z\Theta_0}{\Theta_P}(\frac{\mu}{\kappa-2} - \frac{1}{\kappa-1})$ where $\kappa_a$ is the power mean of order $a$ of the income of the poor.<fnref n="10" /> The sign of $\varepsilon_0^K$ depends only on the sign of difference $\mu - z$. With $\Theta_1$, $\Theta_W$, and $\Theta_P$, the elasticity is necessarily non-negative.<fnref n="11" /> For other members of $\Theta_\alpha$, $\varepsilon_\alpha^K$ is negative whenever $\frac{\Theta_{\alpha-1}-\Theta_\alpha}{\Theta_{\alpha-1}} > \frac{\mu}{z}$. In practice, we generally have $\mu > z$; $\Theta_\alpha$ is then strictly positive.

## 2.2 A more flexible framework

<figure><img src="image_6.png" /></figure>

Note: The value of the headcount index has been computed using the $\$3.65 (2017 PPPs) per person and per day poverty line.

<caption>**Figure 3: Change in the Lorenz curve and usual inequality change assumptions for Brazil, 2016–2017.**</caption>

Kakwani’s hypothesis (3) is very specific since it assumes that the resulting relative income distribution is Lorenz dominated by the initial distribution. According to Ravallion (2022), empirical results do poorly support the hypothesis of Lorenz dominance relationships as the most widespread situation when tracking inequality changes. Figure 3 exemplifies this situation and its implication for the study of the consequences of inequality

$$ \kappa_a := \begin{cases} \left( \frac{1}{\Theta_0} \int_0^{\Theta_0} y(p)^a dp \right)^{\frac{1}{a}} & \text{if } a \neq 0 \\ e^{\frac{1}{\Theta_0} \int_0^{\Theta_0} \log(y(p)) dp} & \text{if } a = 0 \end{cases} \quad (7) $$

<footnote n="10">¹⁰The definition of $\kappa_a$ is:</footnote>

<footnote n="11">¹¹With the poverty gap ratio ($\alpha = 1$), the formula becomes $\frac{\mu-\mu_z}{z-\mu_z}$ (Kakwani, 1993), which is necessarily positive since $\mu > \mu_z$ and $z > \mu_z$. Since $\mu > \mu_z \ge \kappa-1$, the numerator of $\varepsilon_{\log}^K$ is positive, hence the elasticity. Finally, since $\mu \ge \kappa-1 \ge \kappa-2$, the element within parentheses for the expression of $\varepsilon_P^K$ is positive, and so is also the elasticity.</footnote>
changes on poverty. The continuous curve depicts the variation of the Lorenz curve at each population percentile between 2016 and 2017 in Brazil.<fnref n="12" /> The comparison with the curve one should have observed under hypothesis (3) illustrates its limitations in this specific case. Firstly, there is no Lorenz dominance relationship between the two income distributions since the variation is negative for the lower half of the population and then becomes positive for $p \in [0.51, 0.92]$. Secondly, while assumption (3) predicts an increase in the income share of the poor, we observe a sizeable decrease. The same observation holds for another widely used assumption, namely lognormality (Bourguignon, 2003, Bluhm et al., 2018, Bergstrom, 2022). This does not preclude these assumptions to be virtually reasonable on average, but they cannot be regarded as adequate for the study of each single growth spell.

So, presuming that $\varepsilon_{\Theta}^{K}$ provides the best estimate of the true value of $\varepsilon_{\Theta}$ is a heroic assumption. This may notably be true for poverty indices that are sensitive to the shape of the Lorenz curve across all population quantiles up to $F(z)$, such as $\Theta_{2}$, $\Theta_{log}$, and $\Theta_{P}$.

To consider the variety of possible changes in the relative income distribution, hence the uncertainty regarding the value of $\varepsilon_{\Theta}$, I consider the more flexible transform of the Lorenz curve:<fnref n="13" />

$$ \tilde{L}(p) = \left(1 - \frac{\lambda}{\beta}\right) L(p) + \frac{\lambda}{\beta} L'(p) - \lambda(p - L'(p)), \quad (9) $$

where $\beta > 0$ and $\lambda$ still corresponds to the relative change in the value of $G$, so that $\varepsilon_{\Theta}$ can be estimated with the help of the first-order derivative of $\Theta$ to $\lambda$. The function $L'(p)$ depicts an instrumental Lorenz curve that shows the same value for the Gini coefficient as $L(p)$. The first two elements in (9) identify a Gini-preserving change in the relative distribution of income, hence allowing $L(p)$ and $\tilde{L}(p)$ to cross. Consequently, under (9), the initial and final Lorenz curves may not show a Lorenz dominance relationship. The third element captures the proportional change in the value of the Gini index. The parameter $\beta$ is a weighting factor that makes it possible to adjust the relative influence of both components in modifying the relative distribution of income. Equation (3) is a special case of (9) obtained for $L'(p) = L(p) \forall p \in [0, 1]$.

Noting $\tilde{y}(p)$ and $y'(p)$ the quantile functions respectively associated with $\tilde{L}(p)$ and $L'(p)$ when $\mu' = \mu$, and $\tilde{g}$ and $g'(p)$ the corresponding growth rates at quantile $p$, the general expressions of $\tilde{y}(p)$ and $\tilde{g}(p)$ are respectively:

$$ \tilde{y}(p) = y(p) + \lambda \left( \frac{(1 + \beta)y'(p) - y(p)}{\beta} - \mu \right), \quad (10) $$

$$ \tilde{g}(p) = \lambda \left( \frac{(1 + \beta)g'(p) - 1}{\beta} - \frac{\mu}{y(p)} \right). \quad (11) $$

$$ \tilde{L}(p) = (1 - \lambda) L(p) + \lambda L^{*}(p) - \lambda(p - L^{**}(p)), \quad (8) $$

where $L^{*}(p)$ and $L^{**}(p)$ are also an instrumental Lorenz curves whose value of $G$ equals the value observed for $L(p)$. With the adequate choices for $L^{*}(p)$ and $L^{**}(p)$, the transform is virtually equivalent to the transfer suggested in the present paper. However, (9) is more elegant and easier to use, notably for estimation purposes.

<footnote n="12">¹²See section 5 for the data sources and the estimation technique.</footnote>

<footnote n="13">¹³Transform (9) is close to the one proposed in Bresson (2010). In this study, the suggested transform was:</footnote>
It can easily be checked that the new income distribution Lorenz-dominates the initial distribution if and only if $(\frac{(1+\beta)L'(p)-L(p)}{\beta} \le p$. Regarding poverty, it is known that an unambiguous poverty decrease is observed if a restricted first-order dominance relationship of the final income distribution over the initial distribution is observed (Atkinson, 1987), that is if the final cdf is nowhere larger than the initial cdf for any poverty line up to some income threshold $z^+$ regarded as a maximum value for the poverty line. Since a first order dominance relationship is obtained as a result of a sequence of income increments (and no decrement), that is $\tilde{y}(p) \ge y(p)$ and $\exists \hat{p}$ s.t. $\tilde{y}(\hat{p}) \ge y(\hat{p})$, it is clear from (10) that $\varepsilon_{\Theta} \le 0$, whatever $\Theta$ and $z$, if, considering $\lambda > 0$, we have:

$$y'(p) \ge \frac{y(p) + \beta\mu}{1 + \beta} \quad \forall p \le F(z^+). \quad (12)$$

This result can be checked from the general formula for $\varepsilon_{\Theta}$ associated with (9):

$$\varepsilon_{\Theta} = \frac{1}{\Theta} \left[ \theta(y(\Theta_0), z) \left( \mu + \frac{z - (1 + \beta)y'(\Theta_0)}{\beta} \right) f(z) + \int_{0}^{\Theta_0} \left( \frac{(1 + \beta)y'(p) - y(p)}{\beta} - \mu \right) \frac{\partial \theta}{\partial y} dp \right] \quad (13)$$

that is positive whenever the terms between parentheses is negative, that is when condition (12) holds.

It can also be seen that the influence of $\beta$ is not straightforward. Indeed, I observe $\frac{\partial \varepsilon_{\Theta}}{\partial \beta} = \frac{1}{\Theta} \left( \theta(y(\Theta_0), z) \frac{y'(\Theta_0) - z}{\beta^2} f(z) + \int_0^{\Theta_0} \frac{y'(p) - y(p)}{\beta^2} \frac{\partial \theta}{\partial y} dp \right)$. Consequently, the sign of this derivative depends on the sign of $y'(p) - y(p)$ over the interval $[0, \Theta_0]$. In the empirical application provided in section 5, we show that very low values for $\beta$ are often associated with very large values for $|\varepsilon_{\Theta}|$.

Expressions of $\varepsilon_{\Theta}$ for specific poverty indices are presented in the top panel of table 10.

Unlike (3), equation (9) is interesting for estimation purposes since $\beta$, $\lambda$ and $y'(p)$ can be chosen so as to generate observed changes in the relative distribution of income. Let us assume that $\tilde{L}(p)$ refers to the final income distribution of a given observed growth spell and let $\tilde{\lambda}$ indicate the corresponding relative change in $G$. When $\beta = \lambda = \tilde{\lambda}$, (9) boils down to $\tilde{L}(p) = L'(p) - \tilde{\lambda}(p - L'(p))$. The instrumental curve $L'(p)$ then equals $\frac{\tilde{L}(p) + \tilde{\lambda}p}{1 + \tilde{\lambda}}$, and the corresponding quantile function $y'(p)$ is $\frac{\mu}{1 + \tilde{\lambda}} \left( \frac{\tilde{y}(p)}{\mu} + \tilde{\lambda} \right)$. Plugging it into (13) then results in a value of $\eta_{\Theta}$ that perfectly corresponds to the change in inequality associated with the observed growth spell. The empirical application in section 5 essentially relies on this use of transformation (9) with the aforementioned appropriate calibration for $\beta$ and $L'(p)$. A comparison with Kakwani's elasticities and other estimation techniques is provided in appendix C using the Brazilian data considered for section 5. In general, it appears that Kakwani's formulas significantly overestimate the value of $\varepsilon_{\Theta}$ corresponding to observed inequality changes.

Kakwani's elasticity can be used to benchmark observed changes, but its normative value is not straightforward. So, it may be interesting to consider other specific choices for $L'(p)$ that would be more relevant from a social welfare perspective. Valuable candidates
are $L'_{AR}(p)$ and $L'_R(p)$ defined as:<fnref n="14" />

$$L'_{R}(p) = \begin{cases} (1-G)p & \text{if } 0 \le p < 1 \\ 1 & \text{if } p = 1 \end{cases}, \qquad (16)$$

$$L'_{AR}(p) = \begin{cases} 0 & \text{if } 0 \le p < G \\ \frac{1}{1-G}p - \frac{G}{1-G} & \text{if } G \le p \le 1 \end{cases}. \qquad (17)$$

$L'_R$ is the Lorenz curve of an income distribution whose value of the Gini index equals $G$ and that maximizes the income share of poorest part of the population. Using this functional form for $L'$ with transform (9) yields, when $\lambda > 0$, the utmost pro-poor inequality increase, i.e. the increase in inequality is associated with the largest possible income gain for the very poorest. From a Rawlsian perspective, it generates the most desirable way of increasing inequality without changing mean income. On the contrary, $L'_{AR}$ refers to a distribution that, for a given value of the Gini index, maximizes the number of fully deprived individuals. Plugging it into (9) means that, for $\lambda > 0$, the resulting increase in inequality is the most regressive increase one could obtain since it induces transfers from the very poorest.<fnref n="15" /> Consequently, it can be deemed “anti-Rawlsian.” It is worth stressing that $L'_R(p)$ and $L'_{AR}$ can be dubbed respectively as “Rawlsian” and “anti-Rawlsian” only in the case of $\lambda$ being positive. With a negative value for $\lambda$, using $L'_{AR}$ in (9) results in a very pro-poor sequence of transfers while an anti-poor sequence of transfers is obtained with the help of $L'_R(p)$.

$L'_R$ and $L'_{AR}$ do not necessarily generate bounds for $\varepsilon_{\Theta}$. This will be the case only for indices complying with the transfer axiom like $\Theta_2$, $\Theta_{log}$ or $\Theta_P$.<fnref n="16" /> Section 3 illustrates how large can be the interval defined by the elasticities associated with $L'_R$ and $L'_{AR}$, hence stressing the theoretical uncertainty regarding the effect of a given change in the Gini index on poverty.

$$y'_{R}(p) = \begin{cases} (1-G)\mu & \text{if } 0 \le p < 1 \\ \lim_{p \to 1} \left(1 + \frac{pG}{1-p}\right)\mu & \text{if } p = 1 \end{cases}, \qquad (14)$$

$$y'_{AR}(p) = \begin{cases} 0 & \text{if } 0 \le p < G \\ \frac{\mu}{1-G} & \text{if } G \le p \le 1 \end{cases}. \qquad (15)$$

<footnote n="14">¹⁴The corresponding quantile functions are respectively:</footnote>

<footnote n="15">¹⁵I assume income are non-negative. Allowing for negative income makes it possible to consider even more regressive redistribution patterns.</footnote>

<footnote n="16">¹⁶In the case of the headcount index, the index is only sensitive to income changes resulting in entry to or exit from the poverty domain. As a consequence, bounds for $\varepsilon_0$ would require the use of instrumental Lorenz curves inducing transfers around the poverty line. Maximizing the decrease in the headcount ratio for a given inequality change would result in transfers prioritizing income increases for the least poor over income increases for the neediest. This biased incentive scheme is often pointed out as a reason for preferring more compelling indices like the other members of the FGT family of poverty indices. For this reason, it is not worth considering those instrumental curves $L'(p)$ that would identify bounds for $\varepsilon_0$.</footnote>
# 3 Disentangling relative distribution changes

## 3.1 The three ‘I’s of poverty

As stressed earlier, the literature on poverty measurement typically identifies three dimensions of poverty: incidence, intensity, and inequality. This can be clarified by examining alternative expressions of the poverty indices used in this paper. Indeed:<fnref n="17" />

$$ \Theta_1(z; F) = \frac{z - \mu_z}{z} \Theta_0, \qquad (19) $$

$$ \Theta_2(z; F) = \left( \left( \frac{z - \mu_z}{z} \right)^2 + 2 \frac{\mu_z^2}{z^2} GE_{2,z} \right) \Theta_0, \qquad (20) $$

$$ \Theta_{\log}(z; F) = \left( \log \left( \frac{z}{\mu_z} \right) + GE_{0,z} \right) \Theta_0, \qquad (21) $$

$$ \Theta_P(z; F) = \left( (2GE_{-1,z} + 1) \frac{z}{\mu_z} - 1 \right) \Theta_0. \qquad (22) $$

where $\mu_z$ is the average income of the poor and $GE_{\kappa,z}$ is a generalized entropy index computed for the income distribution of the poor. The general formula for this family of inequality indices is (Bourguignon, 1979, Cowell, 1980):

$$ GE_{\kappa,z} := \begin{cases} \frac{1}{(\kappa-1)\kappa\Theta_0} \int_0^{\Theta_0} \left(\frac{y(p)}{\mu_z}\right)^{\kappa} - 1 \, dp & \text{if } \kappa \notin \{0; 1\} \\ \frac{1}{\Theta_0} \int_0^{\Theta_0} \frac{y(p)}{\mu_z} \log\left(\frac{y(p)}{\mu_z}\right) \, dp & \text{if } \kappa = 1 \\ \frac{1}{\Theta_0} \int_0^{\Theta_0} \log\left(\frac{\mu_z}{y(p)}\right) \, dp & \text{if } \kappa = 0 \end{cases}. \qquad (23) $$

Here, these decompositions are used to identify the different channels by which inequality changes, as estimated by $\varepsilon_\Theta$, affect poverty. Let $\Psi_z$ denote an index of inequality among the poor. Assuming no change in either $\mu$ or $z$, I obtain:

$$ \varepsilon_{\Theta} = \pi_{0,\Theta} \varepsilon_0 + \pi_{\Theta,\mu_z} \zeta + \pi_{\Theta,\Psi_z} \nu_{\Psi_z}, \qquad (24) $$

with $\pi_{0,\Theta} := \frac{\partial \Theta}{\partial \Theta_0} \frac{\Theta_0}{\Theta}$, $\pi_{\Theta,\mu_z} := \frac{\partial \Theta}{\partial \mu_z} \frac{\mu_z}{\Theta}$, $\pi_{\Theta,\Psi_z} := \frac{\partial \Theta}{\partial \Psi_z} \frac{\Psi_z}{\Theta}$, $\zeta := \frac{\partial \mu_z}{\partial \lambda} \frac{1}{\mu_z}$, and $\nu_{\Psi_z} := \frac{\partial \Psi_z}{\partial \lambda} \frac{1}{\Psi_z}$.

Equations (19) to (22) show that $\pi_{0,\Theta} = 1$ for $\Theta_\alpha$, $\Theta_\log$, and $\Theta_P$ since these indices are linear functions of $\Theta_0$.<fnref n="18" /> Formulas for $\pi_{\Theta,\mu_z}$ and $\pi_{\Theta,\Psi_z}$ are easily derived from equations (18) to (22) and are presented in Table 1.

An expression of $\zeta$ can be derived from (24) with $\alpha = 1$. Indeed, we then have $\pi_{1,\Psi_z} = 0$,

$$ \Theta_{\alpha}(z; F) = \left(\frac{z - \mu_z}{z}\right)^{\alpha} \left(1 + (\alpha^2 - \alpha)GE_{\alpha,z}^{d}\right) \Theta_0, \qquad (18) $$

where $GE_{\alpha,z}^d$ is a generalized entropy index computed on the series of relative income gaps $d(p) := \frac{z-y(p)}{z}$ for the poor.

<footnote n="17">¹⁷Decompositions (19) and (20) are respectively provided by Sen (1976) and Foster et al. (1984).</footnote>

<footnote>For $\Theta_\alpha$, a more general decomposition has been proposed by Aristondo et al. (2010):</footnote>

<footnote n="18">¹⁸For $\alpha > 2$, this can be seen from equation (18). Actually, it can be shown that every poverty index (1) is a linear function of $\Theta_0$. Let $y_z(p)$ denote the quantile function associated with the distribution of income among the poor. Equation (1) can then be expressed as $\Theta(z; F) = \Theta_0 \int_0^1 \theta(y_z(p), z) dp$.</footnote>
<table><caption>**Table 1: Expressions of $\pi_{\Theta,\mu_z}$ and $\pi_{\Theta,\Psi_z}$ for some poverty indices.**</caption><thead><tr><th>Θ</th><th>π<sub>Θ,μ<sub>z</sub></sub></th><th>π<sub>Θ,Ψ<sub>z</sub></sub></th></tr></thead><tbody><tr><td>Θ<sub>1</sub></td><td>-<sup>μ<sub>z</sub></sup>&frasl;<sub>z-μ<sub>z</sub></sub></td><td>0</td></tr><tr><td>Θ<sub>2</sub></td><td>-<sup>2μ<sub>z</sub>(z-(1+2GE<sub>2,z</sub>)μ<sub>z</sub>)</sup>&frasl;<sub>(z-2μ<sub>z</sub>)z+(1+2GE<sub>2,z</sub>)μ<sub>z</sub><sup>2</sup></sub></td><td><sup>2μ<sub>z</sub><sup>2</sup>GE<sub>2,z</sub></sup>&frasl;<sub>(z-2μ<sub>z</sub>)z+(1+2GE<sub>2,z</sub>)μ<sub>z</sub><sup>2</sup></sub></td></tr><tr><td>Θ<sub>log</sub></td><td>-<sup>1</sup>&frasl;<sub>log(<sup>z</sup>&frasl;<sub>μ<sub>z</sub></sub>) + GE<sub>0,z</sub></sub></td><td><sup>GE<sub>0,z</sub></sup>&frasl;<sub>log(<sup>z</sup>&frasl;<sub>μ<sub>z</sub></sub>) + GE<sub>0,z</sub></sub></td></tr><tr><td>Θ<sub>P</sub></td><td>-<sup>(2GE<sub>-1,z</sub>+1)z</sup>&frasl;<sub>(2GE<sub>-1,z</sub>+1)z-μ<sub>z</sub></sub></td><td><sup>2GE<sub>-1,z</sub>z</sup>&frasl;<sub>(2GE<sub>-1,z</sub>+1)z-μ<sub>z</sub></sub></td></tr></tbody></table>

so that:

$$ \varepsilon_1 = \varepsilon_0 - \frac{\mu_z}{z - \mu_z} \zeta, \qquad (25) $$

$$ \Rightarrow \quad \zeta = (\varepsilon_0 - \varepsilon_1) \frac{z - \mu_z}{\mu_z}, \qquad (26) $$

$$ \Rightarrow \quad = \left( \frac{\beta\mu + z - (1 + \beta)y'(\Theta_0)}{\beta} \right) \frac{(z - \mu_z)f(z)}{\mu_z\Theta_0} + \left( \frac{(1 + \beta)\mu'_z - \beta\mu - \mu_z}{\beta\mu_z} \right). \qquad (27) $$

Under Kakwani's assumption, this simplifies into $\zeta^K = (\mu - z)^{\frac{z-\mu_z}{\mu_z}} \frac{f(z)}{\Theta_0} + \frac{\mu_z-\mu}{\mu_z}$ that is negative if $\mu - z < \frac{\mu-\mu_z}{z-\mu_z} \frac{\Theta_0}{f(z)}$. A sufficient condition for this is $z > \mu$. In this case, the first term of the expression of $\zeta^K$ is negative and captures the effect of poverty line crossings, that is poor individuals leaving poverty as a result of the inequality increase. This results in a decrease in $\mu_z$ since those exiting poverty are relatively better off compared to those remaining in poverty. The second term is negative and catches the average decrease in income for those that were initially poor. When $z < \mu$, the former effect becomes positive as a consequence of the entry in poverty of individuals whose income is larger that those that were initially poor. If the density at the poverty line is large in regard to the headcount index, this may more than offset the latter effect, potentially resulting in $\zeta^K > 0$.

The two effects are also at play when the general transform is considered. However, if $\mu'_z$ is relatively large, the second term in (27) may be positive. This makes it difficult to identify under which conditions $\zeta$ is positive. A sufficient condition for $\zeta$ to be negative is when $\mu \ge z \ge y'(\Theta_0)$ and $\mu_z > \mu'_z$ are simultaneously met.

Using (24) again along with (26) and elasticities from Table 1, I finally obtain:

$$ \nu_{\alpha} = \begin{cases} \left(1 + \frac{1-\frac{\mu_z}{z}}{2GE_{-1,z}}\right)(\varepsilon_{\log} - \varepsilon_0) + \left(1 + \frac{1}{2GE_{-1,z}}\right) \frac{z-\mu_z}{\mu_z}(\varepsilon_0 - \varepsilon_1) & \text{if } \alpha = -1 \\ \left(\frac{\log\left(\frac{z}{\mu_z}\right)}{GE_{0,z}} + 1\right)(\varepsilon_{\log} - \varepsilon_0) + \frac{z-\mu_z}{\mu_z} \frac{\varepsilon_0 - \varepsilon_1}{GE_{0,z}} & \text{if } \alpha = 0 \\ \frac{((z-2\mu_z)z+(1+2GE_{2,z})\mu_z^2)(\varepsilon_2 - \varepsilon_0) + 2(z-(1+2GE_{2,z})\mu_z)(z-\mu_z)(\varepsilon_0 - \varepsilon_1)}{2\mu_z^2 GE_{2,z}} & \text{if } \alpha = 2 \end{cases} \qquad (28) $$

An inspection of these formulas shows that the sign of $\nu_\alpha$ is indeterminate, even under Kakwani's assumption. Three effects are jointly affecting inequality among the poor, namely the movements across the poverty line, the subsequent change in the mean income of the poor, and the change in the income share among those that were initially poor. With transform (3), a positive value for $\lambda$ means an increase in the distance between any pair of poor persons. If $z < \mu$, the average income of those who were initially poor decreases.
Larger distance associated with a lower value for $\mu_z$ both entail larger inequalities among those individuals. However, the decrease in income of the neediest non-poor persons results in poverty entry. This simultaneously increases the average income of the poor and the income dispersion below the poverty line. These opposite effects explain the indeterminate sign for $\nu_\alpha^K$. With transform (9), the situation becomes more complex, as the distances between poor individuals do not necessarily increase, and $\mu_z$ may also decline.

## 3.2 Decompositions of the Gini index

I now consider a different approach for the evaluation of the effects of marginal changes in the relative distribution of income on poverty taking advantage of the decomposability property of both the poverty and the inequality indices considered here when the population is split into non-overlapping groups. More specifically, I assume that there is a population quantile $q \in ]0, 1[$ that is regarded as relevant by the social evaluator for the purpose of splitting the population into two parts, namely the $q$ poorest part and the $1-q$ richest part. In the context of poverty analysis, a natural candidate for $q$ is the value of the headcount index $F(z)$, but other values for $q$ within the unit interval are worth our consideration. This is notably the case of $q = 40\%$ that is used both for the World bank's shared prosperity indicator and SDG 10 (targer 10.1).

It is worth noting that the additive structure of $\Theta$ makes it possible to consider the following decomposition:

$$ \Theta = C_{\Theta}^{0,q} + C_{\Theta}^{q,F(z)}, \qquad (29) $$

where $C_{\Theta}^{a,b} := \int_a^b \theta(y(p), z) dp$ is the contribution to $\Theta$ of those individuals ranging between the population quantiles $[a, b]$. Of course, if $q \le \Theta_0$, we have $C_{\Theta}^{q,F(z)} = 0$.

It is also known that many inequality indices like the Gini index can be additively decomposed into a between and a within components. The between component is the value of the inequality index one would observe if individual incomes within each group were all replaced by the corresponding group average. In the case of the Gini index, the chosen split of the population would then result in a between component $G_q^B$ defined by:

$$ G_q^B := 1 - \frac{(1 - (1 - q)^2)\mu_- + (1 - q)^2\mu_+}{\mu} = (1 - q) \left( \frac{\mu_+}{\mu} - 1 \right), \qquad (30) $$

where $\mu_-$ and $\mu_+$ are the average income respectively for the bottom and top parts of the population. The within component $G_q^W$ is simply a weighted sum of the Gini indices computed for each group with weights being the product of the population shares and of the income shares, that is:

$$ G_q^W := qL(q)G_- + (1 - q)(1 - L(q))G_+, \qquad (31) $$

where $G_-$ ($G_+$) is the value of the Gini index for the bottom (top) part of the income distribution.

Regarding the sensitivity of poverty to changes in the two components of $G$, I note that
the elasticity $\varepsilon_{\Theta}$ is related to the between and within elasticities as follows:<fnref n="19" />

$$ \varepsilon_{\Theta} = \varepsilon_{\Theta,q}^{B} \frac{\partial G_{q}^{B}}{\partial G} \frac{G}{G_{q}^{B}} + \varepsilon_{\Theta,q}^{W} \frac{\partial G_{q}^{W}}{\partial G} \frac{G}{G_{q}^{W}}, \quad (36) $$

where $\varepsilon_{\Theta,q}^B$ ($\varepsilon_{\Theta,q}^W$) denotes the elasticity of $\Theta$ with respect to the between (within) component of $G$ under transform (9) and considering a split at population quantile $q$.

It can easily be shown that transform (3) results in a 100$\lambda\%$ increase in both the between and the within component of the Gini index.<fnref n="20" /> Consequently, $\frac{\partial G^B}{\partial G} \frac{G}{G^B} = \frac{\partial G^W}{\partial G} \frac{G}{G^W} = 1$ so that $\varepsilon_{\Theta}^K$ can be additively decomposed into a between-inequality elasticity $\varepsilon_{\Theta,q}^{KB}$ and a within-inequality elasticity $\varepsilon_{\Theta,q}^{KW}$, that is $\varepsilon_{\Theta}^K = \varepsilon_{\Theta,q}^{KB} + \varepsilon_{\Theta,q}^{KW}$. In the more general case of transform (9), it is necessary to weigh the two elasticities according to the elasticities of the between and within components with respect to $G$.

To get the analytical expressions of the elasticities $\varepsilon_{\Theta,q}^B$ and $\varepsilon_{\Theta,q}^W$, specific transformations of the Lorenz curve have to be considered, namely i) one that changes $G_q^B$ while leaving $G_q^W$ unaffected, and ii) one that results in a variation of $G_q^W$ but keeps $G_q^B$ unchanged. However, expression (31) makes it clear that, except in trivial cases (i.e. $G_- = G_+$ and $q = \frac{1}{2}$, or $G_- = G_+ = 0 \forall q \in ]0, 1[$), it is not possible to find a transform of the Lorenz curves that changes $G_q^B$ while simultaneously preserving $G_q^W$, $G_-$, and $G_+$. Indeed, an increase in $G_q^B$ means a lower value for $L(q)$. If we impose a zero variation in both $G_-$ and $G_+$, the value of $G_q^W$ may either increase or decrease depending on $q$, $G_-$, and $G_+$.

Here, I assume that the demand is larger for transforms that independently consider changes in either $G_q^B$, $G_-$, or $G_+$ while keeping unchanged the two other elements. This means that the corresponding formula for $\varepsilon_{\Theta,q}^B$ introduced in the next paragraphs takes the indirect effect on $G_q^W$ mediated by the change in $L(q)$ into account.

### 3.2.1 Between inequalities

To raise the income share of the top part of the distribution, hence increasing $G_q^B$, while keeping $G_-$ and $G_+$ unchanged, a simple transform implies scaling down all incomes from the bottom $q\%$ of the population and scaling up all income from the remaining top part, so that the resulting income vector can be obtained from the initial vector through a sequence of mean-preserving spreads (Dasgupta et al., 1973) across $y(q)$. The resulting quantile

$$ \varepsilon_{\Theta} = \frac{\partial \Theta G}{\partial G \Theta}, \quad (32) $$

$$ = \left( \frac{\partial \Theta}{\partial G^B} \frac{\partial G^B}{\partial G} + \frac{\partial \Theta}{\partial G^W} \frac{\partial G^W}{\partial G} \right) \frac{G}{\Theta}, \quad (33) $$

$$ = \frac{\partial \Theta}{\partial G^B} \frac{G^B}{\Theta} \frac{\partial G^B}{\partial G} \frac{G}{G^B} + \frac{\partial \Theta}{\partial G^W} \frac{G^W}{\Theta} \frac{\partial G^W}{\partial G} \frac{G}{G^W}, \quad (34) $$

$$ = \varepsilon_{\Theta,q}^{B} \frac{\partial G^{B}}{\partial G} \frac{G}{G^{B}} + \varepsilon_{\Theta,q}^{W} \frac{\partial G^{W}}{\partial G} \frac{G}{G^{W}}. \quad (35) $$

<footnote n="19">¹⁹We have:</footnote>

<footnote n="20">²⁰See appendix A.</footnote>
function is:

$$ \tilde{y}^B(p) = \begin{cases} \left(1 + \lambda^B \left(1 - \frac{\mu}{\mu_{-}}\right)\right) y(p) & \text{if } 0 \le p < q \\ \left(1 + \lambda^B \left(1 - \frac{\mu}{\mu_{+}}\right)\right) y(p) & \text{if } q \le p \le 1 \end{cases}, \qquad (37) $$

where $\lambda^B$ denotes the relative change in $G_q^B$. The corresponding equation for the Lorenz curve is:

$$ \tilde{L}^B(p) = \begin{cases} L(p) - \lambda^B \frac{q-L(q)}{L(q)} L(p) & \text{if } 0 \le p < q \\ L(p) - \lambda^B \frac{q-L(q)}{1-L(q)} (1-L(p)) & \text{if } q \le p \le 1 \end{cases}. \qquad (38) $$

Since the transform implies scaling up or down the income of the poor, there is a straightforward connection with the growth elasticity of poverty $\eta_\Theta$ as can be seen from the general expression for $\varepsilon_{\Theta,q}^B$:

$$ \varepsilon_{\Theta,q}^{B} = \begin{cases} \left(1 - \frac{\mu}{\mu_{-}}\right) \eta_{\Theta} & \text{if } \Theta_{0} < q \\ \left(1 - \frac{\mu}{\mu_{+}}\right) \eta_{\Theta} + \frac{1}{\Theta} \left(\frac{\mu}{\mu_{+}} - \frac{\mu}{\mu_{-}}\right) \int_{0}^{q} y(p) \frac{\partial \theta}{\partial y} dp & \text{if } \Theta_{0} \ge q \end{cases}, \qquad (39) $$

In the case $\Theta_0 < q$, expression (39) is systematically positive since $\eta_\Theta$ is then negative and $\frac{\mu}{\mu_{-}} > 1$. When $\Theta_0 \ge q$, we necessarily have $\varepsilon_{0,q}^B < 0$ because the income of the least poor persons increases. For other indices, I note that the second term in (39) is positive, since $\frac{\mu}{\mu_{+}} < \frac{\mu}{\mu_{-}}$ and $\frac{\partial \theta}{\partial y} \le 0$, but the first term is positively linked to $\eta_\Theta$ whose sign is indeterminate and depend notably on the continuity of $\theta$ at the poverty line. In this situation, the effect of the decrease in income for the very poorest is potentially compensated by the effect of both an increase in income for the least poor and of poverty exits. Consequently, the sign of $\varepsilon_{\Theta,q}^B$ can be either positive or negative, depending notably on the ratio $\frac{q}{\Theta_0}$. The lower it is, the more likely $\varepsilon_{0,q}^B < 0$ is observed.

Expressions for $\Theta_\alpha$, $\Theta_{log}$, and $\Theta_P$ of $\varepsilon_{\Theta,q}^B$ are reported on the top panel of Table 11. The term within square brackets makes it possible to identify under which specific conditions $\varepsilon_{\Theta,q}^B$ becomes negative. For instance, considering $\Theta_1$, we have $\varepsilon_{\Theta,q}^B < 0$ if and only if $\frac{q}{\Theta_0} < \frac{(\mu_+-\mu)\mu_z}{(\mu_+-\mu_-)\mu}$.

In some contexts, researchers and policy makers may be more interested in the poverty effect of a 1% increase in the income share of the bottom $q$% of the population than of a 1% increase in the value of the Gini index or its between component. With transform (38), the relative change in $L(q)$ is $\lambda^B \frac{L(q)-q}{L(q)} = \lambda^B \frac{\mu_+-\mu}{\mu_-}$.21 This makes it possible to propose an expression for a bottom income share elasticity of poverty $\rho_{\Theta,q}^B$, that is the elasticity of $\Theta$ with respect to $L(q)$, whose expression is simply:

$$ \rho_{\Theta,q}^{B} = \begin{cases} \eta_{\Theta} & \text{if } \Theta_{0} < q \\ \eta_{\Theta} - \frac{1}{\Theta} \frac{\mu}{(1-q)\mu_{+}} \int_{q}^{\Theta_{0}} y(p) \frac{\partial \theta}{\partial y} dp & \text{if } \Theta_{0} \ge q \end{cases}, \qquad (40) $$

This elasticity quantifies the poverty effect of a pure change in the income share of

<footnote>21Equation (38) can also be used to generate a distribution showing the same income share for the bottom $q$% of the population as in the final observed distribution while keeping inequalities preserved both within the bottom part and within the top part of the population. If $\tilde{L}$ is the final observed curve, then the appropriate calibration is $\lambda^B = \frac{\tilde{L}(q)-L(q)}{L(q)-q}$.</footnote>
the bottom $q$% of the population (i.e. $\mu$, $G_-$, and $G_+$ are leaved unchanged). It can be compared with the bottom share elasticity of poverty $\rho_{\Theta,q}$ derived from transform (9) and that incorporates the effects of changes in $G_-$ and $G_+$. In this case, the relative variation of $L(q)$ is $\lambda \frac{(1+\beta)L'(q)-L(q)-\beta,q}{\beta L(q)} = \lambda \frac{(1+\beta)\mu'_- - \mu_- - \beta\mu_-}{\beta\mu_-}$. The general expression of $\rho_{\Theta,q}$ is simply:

$$ \rho_{\Theta,q} = \frac{\beta\mu_-}{(1 + \beta)\mu'_- - \mu_- - \beta\mu_-} \varepsilon_{\Theta}. \quad (41) $$

In general, we expect $\rho_{\Theta,q}$ to show the opposite sign when compared with $\varepsilon_{\Theta}$, but this result do not necessarily holds since it is possible for $L(q)$ to raise when $G$ increases under transform (38). Inspection of equation (41) shows that $\operatorname{sgn}(\rho_{\Theta,q}^B) \neq \operatorname{sgn}(\varepsilon_{\Theta})$ if and only if $\mu'_- < \frac{\mu_- + \beta\mu_-}{1+\beta}$.

### 3.2.2 Within inequalities

Let's turn to the effect of changes in either $G_-$ or $G_+$. For this purpose it is necessary to consider the following transforms of the quantile function:

$$ \tilde{y}_{-}(p) := \begin{cases} y(p) + \lambda \left( \frac{(1+\beta)y'_{-}(p)-y(p)}{\beta} - \mu_{-} \right) & \text{if } p < q, \\ y(p) & \text{if } p \ge q, \end{cases} \quad (42) $$

$$ \tilde{y}_{+}(p) := \begin{cases} y(p) & \text{if } p < q \\ y(p) + \lambda \left( \frac{(1+\beta)y'_{+}(p)-y(p)}{\beta} - \mu_{+} \right) & \text{if } p \ge q, \end{cases} \quad (43) $$

where $y'_-(p)$ ($y'_+(p)$) is an instrumental quantile function showing the same values as $y(p)$ regarding $\mu_-$ ($\mu_+$) and $G_-$ ($G_+$). The parameter $\lambda$ now makes it possible to control the relative change of the Gini index among those initially belonging either to the bottom or to the top part of the population.<fnref n="22" />

The corresponding Lorenz curves are:

$$ \tilde{L}^{-}(p) := \begin{cases} \left(1 - \frac{\lambda}{\beta}\right) L(p) + \frac{\lambda}{\beta} L'_{-}(p) - \lambda \left(\frac{L(q)p}{q} - L'_{-}(p)\right) & \text{if } p < q, \\ L(p) & \text{if } p \ge q, \end{cases} \quad (44) $$

$$ \tilde{L}^{+}(p) := \begin{cases} L(p) & \text{if } p < q \\ \left(1 - \frac{\lambda}{\beta}\right) L(p) + \frac{\lambda}{\beta} L'_{+}(p) - \lambda \left(\frac{L(q)-q}{1-q} + \frac{1-L(q)}{1-q}p - L'_{+}(p)\right) & \text{if } p \ge q, \end{cases} \quad (45) $$

where $L'_-(p)$ and $L'_+(p)$ are the Lorenz curve respectively associated with $y_-(p)$ and $y_+(p)$.

<footnote n="22">²²It is worth pointing that in the case $\lambda$ is strictly positive, reranking becomes possible, so that $\lambda$ is an upper bound for the relative increase in $G_-$. The same remark applies when considering increasing inequalities within the top part of the income distribution. This could be a serious limitation if we were considering non-marginal changes in inequality within each population part for the estimation the desired elasticities.</footnote>
The resulting elasticities $\bar{\varepsilon}_{\Theta,q}$ and $\varepsilon_{\Theta,q}^{+}$ are:

$$ \varepsilon_{\bar{\Theta},q} = \begin{cases} \frac{1}{\Theta} \left[ \theta(\Theta_0, z) \left( \mu_- + \frac{z-(1+\beta)y'_-(\Theta_0)}{\beta} \right) f(z) \right. \\ \left. + \int_0^{\Theta_0} \left( \frac{(1+\beta)y'_-(p)-y(p)}{\beta} - \mu_- \right) \frac{\partial \theta}{\partial y} dp \right] & \text{if } \Theta_0 < q, \\ \frac{1}{\Theta} \int_0^q \left( \frac{(1+\beta)y'_-(p)-y(p)}{\beta} - \mu_- \right) \frac{\partial \theta}{\partial y} dp & \text{if } \Theta_0 \ge q \end{cases} \quad (46) $$

$$ \varepsilon_{\Theta,q}^{+} = \begin{cases} 0 & \text{if } \Theta_0 < q \\ \frac{1}{\Theta} \left[ \theta(\Theta_0, z) \left( \mu_+ + \frac{z-(1+\beta)y'_+(\Theta_0)}{\beta} \right) f(z) \right. \\ \left. + \int_q^{\Theta_0} \left( \frac{(1+\beta)y'_+(p)-y(p)}{\beta} - \mu_+ \right) \frac{\partial \theta}{\partial y} dp \right] & \text{if } \Theta_0 \ge q \end{cases} \quad (47) $$

The top part of Tables 12 and 13 show formulas for usual poverty indices. If $\Theta_0 \ge q$, it can be checked that $\bar{\varepsilon}_{0,q} = \varepsilon_{1,q} = 0$. For the headcount index, this is due to the absence of movements across the poverty line. With the poverty gap ratio, the zero value for $\bar{\varepsilon}_{1,q}$ is due to the fact that transform (42) does not change the average income of the poor and that the index is not distribution-sensitive. With distribution-sensitive poverty indices, $\bar{\varepsilon}_{1,q}$ is always strictly positive. On the other hand, when $\Theta_0 < q$, we face the same theoretical uncertainty regarding the sign of $\bar{\varepsilon}_{\Theta,q}$ as for the sign of $\varepsilon_{\Theta}$. For instance $\bar{\varepsilon}_{1,q}$ will be positive unless $\mu'_- > \frac{\beta\mu_+ + \beta\mu_-}{1+\beta}$.

Because of the focus axiom, i.e. poverty does not change as a result of an income increment for the non-poor, $\varepsilon_{\Theta,q}^{+} = 0$ when $\Theta_0 < q$. In the opposite situation, two effects can be identified, relating to the two terms within the square brackets in (47). The first corresponds to moves across the poverty line and its sign is not a priori determined. The second relates to income changes for those who were initially poor; it may either be negative or positive. With the headcount index, only the first effect acts, so that $\varepsilon_{\Theta,q}^{+} > 0$ if $y'_+(\Theta_0) < \frac{\beta\mu_+ + z}{1+\beta}$. No simple rule can be proposed for other poverty indices but a sufficient condition for $\varepsilon_{\Theta,q}^{+} > 0$ is $y'_+(p) < \frac{\beta\mu_+ + y(p)}{1+\beta} \quad \forall p \in [q, \Theta_0]$.

In the case $y(p)$ is used either for $y_-(p)$ or for $y_+(p)$, equations (46) and (47) respectively simplify to:

$$ \varepsilon_{\bar{\Theta},q}^{K-} = \begin{cases} \frac{1}{\Theta} \left[ \theta(F(z),z)(\mu_{-} - z)f(z) + \int_{0}^{F(z)} (y(p) - \mu_{-}) \frac{\partial \theta}{\partial y} dp \right] & \text{if } \Theta_0 < q, \\ \frac{1}{\Theta} \int_{0}^{q} (y(p) - \mu_{-}) \frac{\partial \theta}{\partial y} dp & \text{if } \Theta_0 \ge q \end{cases}, \quad (48) $$

$$ \varepsilon_{\Theta,q}^{K+} = \begin{cases} 0 & \text{if } \Theta_0 < q \\ \frac{1}{\Theta} \left[ \theta(F(z),z)(\mu_{+} - z)f(z) + \int_{q}^{F(z)} (y(p) - \mu_{+}) \frac{\partial \theta}{\partial y} dp \right] & \text{if } \Theta_0 \ge q \end{cases}. \quad (49) $$

Like $\varepsilon_{\bar{\Theta},q}^{B}$, the effect of inequality increments crucially depends on the sign of $\Theta_0 - q$. If $\Theta_0 < q$, the sign of $\varepsilon_{0,q}^{K-}$ is the same as the sign of the difference $\mu_- - z$. While $\Theta_1, \Theta_W$, and $\Theta_P$ are always positive in this case, the sign of $\varepsilon_{\bar{\Theta},q}^{K-}$ with other members of $\Theta_\alpha$ becomes negative if $\frac{\Theta_{\alpha-1}-\Theta_\alpha}{\Theta_{\alpha-1}} > \frac{\mu_-}{z}$. Considering $\varepsilon_{0,q}^{K+}$ along with $\Theta_0 \ge q$, its sign equals sgn($\mu_+ - z$). Inspection of the formulas of $\varepsilon_{1,q}^{K+}$, $\varepsilon_{\log,q}^{K+}$, and $\varepsilon_{P,q}^{K+}$ makes it possible to conclude that they cannot be negative. Once again, a negative value with other members of $\Theta_\alpha$ is possible if and only if $\frac{\mu_+}{z} < \frac{C_{\alpha-1}^{q,\Theta_0} - C_{\alpha}^{q,\Theta_0}}{C_{\alpha-1}^{q,\Theta_0}}$.
# 4 Relative and societal poverty lines

In the previous pages, poverty was assessed using an absolute approach for the definition of the poverty domain. However, numerous countries explicitly consider a relative approach for setting the poverty domain, hence making the poverty line endogenous with respect to the current distribution of income. Moreover, even countries that officially endorse the absolute approach do not necessarily keep their national poverty line fixed. As emphasized notably by Bourguignon and Atkinson (2000) and Chen and Ravallion (2011), above some development threshold, countries generally proceed to upward adjustment of their poverty lines as they become wealthier. Consequently, it is necessary to complement my previous analysis with the indirect effect of relative distribution changes on poverty that is channeled by the change in the value of the poverty line.

Considering practices for the setting of relative poverty lines $z^R$, I alternatively study the following three cases in this paper:

i) the poverty line is a linear expression of mean income, i.e. $z^R = a + b\mu$,<fnref n="23" />

ii) the poverty line is a linear expression of a specific income quantile, i.e. $z^R = a + by(p^R)$,

iii) the poverty line is a linear expression of the Gini index and the mean income, i.e. $z^R = a + b(1 - (1 - 2c)G)\mu$, where $a > 0$ and $\{b, c\} \subset [0, 1]$. The restriction $b \le 1$ is imposed to prevent a uniform positive growth process from resulting in increased poverty (Foster, 1998). Option ii) is clearly the most widespread choice, with $p^R = 50\%$ or 60%, as it is used by most OECD countries as well as many Latin-American countries. Ravallion and Chen (2019) derived option iii) from a simple model of reference income for the definition of a relative poverty line. To the best of our knowledge, it has not been officially adopted by any country around the World, but Ravallion and Chen (2019) showed that it proved to be a very good approximation with $c = 0$, above some GDP per capita threshold, of national poverty lines, considering a mixture of relative and absolute poverty lines for their sample.

Clearly, the case of poverty lines being uniquely defined with respect to mean income is of little interest in the present paper since I consider pure inequality changes, i.e. mean-preserving distributional changes. Consequently, the corresponding inequality elasticities of poverty are simply $\varepsilon_\Theta$, $\varepsilon_{\Theta,q}^B$, $\varepsilon_{\Theta,q}^-$, and $\varepsilon_{\Theta,q}^+$ as defined in (13), (39), (46), and (47). For the remaining two cases, a change in $z^R$ will generally ensue from such a transform like (9). Consequently, the corresponding elasticities generally differs from those obtained for a fixed poverty line. More specifically, the resulting Gini elasticity of relative poverty $\varepsilon_\Theta^R$ is:

$$ \varepsilon_{\Theta}^{R} = \varepsilon_{\Theta} + \frac{1}{\Theta} \frac{\partial \Theta}{\partial z} \frac{\partial z^{R}}{\partial \lambda}. \qquad (50) $$

Since well-behaved poverty indices are necessarily non-decreasing functions of the poverty line (Zheng, 1997), $\frac{\partial \Theta}{\partial z} \ge 0$ and the difference between $\varepsilon_\Theta^R$ and $\varepsilon_\Theta$ only depends on the sign of $\frac{\partial z^R}{\partial \lambda}$. With $z^R$ being a function of the Gini index, the increase in $G$ results in

<footnote n="23">²³The results can easily be extended to any increasing function $h$ of mean income, that is $z^R = h(\mu)$. The generally admitted restriction is that the elasticity of $z^R$ with respect to mean income is within the unit interval (Kilpatrick, 1973).</footnote>
a decrease in $z^R$ so that I necessarily observe $\epsilon_{\Theta}^R \le \epsilon_{\Theta}$ whatever the Lorenz curve transformation used to generate the increase in $G$.

The case of $z^R$ being a function of a given income quantile is more complex because there is no systematic ranking between $\epsilon_{\Theta}^R$ and $\epsilon_{\Theta}$. In the case of transform (3), I obtain $\epsilon_{\Theta}^{KR} \ge \epsilon_{\Theta}^K \Leftrightarrow y(p^R) \ge \mu$. In practice, the chosen value for $p^R$ is often 50% as initially suggested by Fuchs (1967). $\epsilon_{\Theta}^{KR} \le \epsilon_{\Theta}^K$ is then expected since the median income is typically lower than the average income. Considering now the general transform (9), I note $\epsilon_{\Theta}^R \ge \epsilon_{\Theta} \Leftrightarrow y'(p^R) \ge \frac{\beta\mu+y(p^R)}{1+\beta}$.

Expressions of $\epsilon_{\Theta}^R$, $\epsilon_{\Theta}^{RB}$, $\epsilon_{\Theta}^{R-}$, and $\epsilon_{\Theta}^{R+}$ for specific poverty indices are given in the middle and bottom panels of Tables 10 to 13.

# 5 Application to Brazil, 1981–2022

To illustrate the usefulness of the results discussed in the previous section, I consider the case of Brazil during the period 1981–2022. After having described the data, having summarized changes in the income distribution, and presented the most salient economic and political changes during the considered period, I shed light on the instability of the inequality elasticity of poverty and of its components. Then I focus on the 2003-2014 period that showed unprecedented and sustained progress in poverty alleviation to evaluate the role of inequality variation and assess the efficiency of observed relative distribution changes as a lever of poverty reduction.

I also question whether differences in the economic situation and economic policies have resulted in more pro-poor changes in the relative distribution of income.

## 5.1 Data

For this empirical application, I relied on synthetic distributions generated with the help of grouped data—specifically, the mean income and the Lorenz curve. The income distribution for each country and each date is estimated through a two-step procedure. At the first stage, I modeled the income distribution parametrically using the lognormal distributions. At the second stage, I generated a random sample of 100,000 values from the parametric distribution.<fnref n="24" /> To ensure an exact fit with the information provided on the original Lorenz curve, the generated sample is finally adjusted using the procedure introduced by Shorrocks and Wan (2008).

Information on the Lorenz curve and mean income has been obtained from PIP, the recent World Bank's poverty and inequality platform, who, in the Brazilian case, is based on the Pesquisa Nacional por Amostra de Domicilios from up to 2011 and on the Pesquisa Nacional por Amostra de Domicílios Contínua since 2012. Although grouped data from PIP cannot compete with data from household surveys, the utilization of PIP as our dataset source is particularly advantageous due to the high-quality information it provides.<fnref n="25" />

<footnote n="24">²⁴The code used to generate the synthetic income distribution and obtain all the results presented in this paper is available upon request.</footnote>

<footnote n="25">²⁵When referring to high-quality information provided by PIP, I mean the ability to generate income series whose cdfs are nearly indistinguishable from those corresponding to the original data. This does not pre-</footnote>
deed, PIP offers detailed data for all available distributions since 101 points of the Lorenz curve can be computed from the data.²⁶ This feature allows us to employ generated income distributions that closely match the original income series, hence enhancing the accuracy of our analysis. Moreover, in the case of Brazil, PIP offered data on a yearly basis since 1981 with the exception of the years 1991, 2000, and 2010.

There is no official poverty line in Brazil. During the period of interest, IGBE, the Brazilian national statistical institute, adopted various approaches for the poverty line. For instance, poverty was defined with respect of a percentage of the minimum wage during at the beginning of the 2000s. For their 2010 social indicators report, IGBE used a relative poverty line computed as 60% of the median income. Starting from 2017, IGBE computed poverty levels with respect to the international poverty lines suggested by the World Bank. In this paper, I consider an absolute approach for the definition of the poverty line. The chosen values for the poverty line are $3.65 and $6.85 per capita, per day in 2017 PPP, that is the values recommended by the World Bank respectively for lower-middle-income and upper-middle-income countries (Jolliffe et al., 2022). Finally, regarding poverty indices, estimations will be limited to the headcount index ($\Theta_0$), the poverty gap index ($\Theta_1$) and the squared poverty gap index ($\Theta_2$). Since the original Lorenz curves often show zero values for the very poorest population quantiles, hence indicating the presence of zero incomes in the income distribution for some years, it was not possible to use the other indices considered in the theoretical part of the paper.

## 5.2 Growth, inequality, and poverty in Brazil

<figure><img src="image_7.png" /><figcaption>Figure 4: Average income, inequality, and poverty in Brazil, 1981–2022.</figcaption></figure>

I first briefly sketch some trends about aggregate income and well-being between 1981

clude the existence of biases and measurement errors in the original survey data (de Souza, 2012). On the uncertainty regarding for instance the true level of income inequality in Brazil, see Alvaredo et al. (2024).

²⁶https://datacatalog.worldbank.org/search/dataset/0063646/-Poverty-and-Inequality-Platform--PIP---Percentiles. The version of the dataset used for this paper is the one released on the 3rd of May, 2024. Points of the Lorenz curves are not provided in the dataset, but can easily be computed using the average income from each population centiles.
and 2022 in Brazil. Figure 14 shows the average income, inequality, and poverty levels during the considered period.<fnref n="27" /> Growth in average income was somehow chaotic up to 1995. Then a plateau was reached during eight years and a sustained positive growth rate could be observed starting from 2003 up to 2014. From 2014 to 2020, there was no increase in average income and a decline was observed in 2022. Estimated inequalities, as measured by the Gini index, were also varying a lot during the first 15 years and started to decline after 1996, with a steeper slope starting from 2003. In 2016, inequalities started to rise again with a noticeable but seemingly transient drop in 2020. Out of the 37 year-to-year comparisons, a Lorenz dominance relationship was observed only 7 times, hence underlining the potential inadequacy of Kakwani's assumption for predicting changes in Brazil's Lorenz curve for all population quantiles.<fnref n="28" /> Finally, poverty exhibited erratic trends during the 1980s. It declined steadily and significantly from 1992 to 2016 and started to increase slowly after. In 2020, a significant drop in poverty can also be observed. All these observations are in line with results highlighted by Ferreira et al. (2006), Ferreira et al. (2010), Ciaschi et al. (2020), Wang and Sun (2021), and Alvaredo et al. (2024).

The variations highlighted on Figure 4 as well as the economic and political changes stressed in the literature suggest splitting the overall period of observation into five subperiods, namely 1981–1993, 1993–2003, 2003–2014, 2014–2019, and 2019–2022. The role of inequality changes in observed poverty trends can be assessed with the help of the Shapley value decomposition of poverty variations into growth and inequality effects (Shorrocks, 2013).<fnref n="29" /> Results are presented in Table 2. Inequality variations adversely affected poverty during the 1981–1993 and 2014–2019 subperiods, and contributed to poverty reduction in the other subperiods. Their role was extremely volatile across subperiods. Indeed, they accounted for less than a quarter of the observed poverty changes between 1994 and 2002, but almost the totality of the corresponding variation between 2014 and 2019. For most observations, the contribution was small compared with the contribution of the change in average income. It was positive (i.e. it increased poverty) up to the mid-1990s and then became predominantly negative up to 2014. Starting from this year, inequality variations became the most important driver of poverty changes.

The contrasted trajectories sketched in the previous paragraphs are the results of changes in policy measures and in the macroeconomic environment that may be of prime interest for the study of estimated inequalities elasticities of poverty. For this purpose,

Subperiod 1981–1993 corresponds to the last years of the military dictatorship, that ended in 1985, and the beginning of a new democratic era. After two decades of policies marked by an import substitution industrialization strategy and export promotion that resulted in sustained growth and increased foreign indebtedness, the subperiod 1981–1993 was characterized by economic stagnation and high inflation. Due to the large amount of debt and the 1982 Mexican debt crisis, Brazil found it very difficult to access international

<footnote n="27">²⁷Estimated values are also reported on Table 14, in the appendix.</footnote>

<footnote n="28">²⁸More specifically, I observe 1985 $>_{L}$ 1984, 1987 $>_{L}$ 1988, 2006 $>_{L}$ 2005, 2008 $>_{L}$ 2007, 2013 $>_{L}$ 2012, 2015 $>_{L}$ 2016, and 2022 $>_{L}$ 2021 where $j >_{L} k$ indicates that distribution j Lorenz-dominates distribution k.</footnote>

<footnote n="29">²⁹Results for year-to-year variations are displayed on Figure 11 in the appendix. Visual inspection shows the consistency of the chosen split of the whole period into the considered five subperiods.</footnote>
<table><caption>Table 2: Shapley value estimates of the contribution of mean income and inequality changes to observed poverty changes in Brazil, 1981–2022.</caption><thead><tr><th rowspan="2"></th><th colspan="5">z = $3.65</th><th colspan="5">z = $6.85</th></tr><tr><th>1981-1993</th><th>1993-2003</th><th>2003-2014</th><th>2014-2019</th><th>2019-2022</th><th>1981-1993</th><th>1993-2003</th><th>2003-2014</th><th>2014-2019</th><th>2019-2022</th></tr></thead><tbody><tr><td></td><td colspan="10" style="text-align: center;">headcount index</td></tr><tr><td>ΔΘ</td><td>-0.032</td><td>-0.137</td><td>-0.174</td><td>0.021</td><td>-0.024</td><td>-0.028</td><td>-0.154</td><td>-0.243</td><td>0.019</td><td>-0.028</td></tr><tr><td>GE</td><td>-0.052</td><td>-0.115</td><td>-0.108</td><td>-0.001</td><td>-0.002</td><td>-0.047</td><td>-0.127</td><td>-0.172</td><td>-0.002</td><td>-0.003</td></tr><tr><td>IE</td><td>0.020</td><td>-0.022</td><td>-0.066</td><td>0.022</td><td>-0.022</td><td>0.019</td><td>-0.027</td><td>-0.071</td><td>0.021</td><td>-0.025</td></tr><tr><td></td><td colspan="10" style="text-align: center;">poverty gap index</td></tr><tr><td>ΔΘ</td><td>-0.011</td><td>-0.076</td><td>-0.078</td><td>0.015</td><td>-0.014</td><td>-0.022</td><td>-0.110</td><td>-0.144</td><td>0.018</td><td>-0.020</td></tr><tr><td>GE</td><td>-0.029</td><td>-0.060</td><td>-0.045</td><td>-0.000</td><td>-0.001</td><td>-0.039</td><td>-0.090</td><td>-0.092</td><td>-0.001</td><td>-0.002</td></tr><tr><td>IE</td><td>0.017</td><td>-0.016</td><td>-0.033</td><td>0.015</td><td>-0.014</td><td>0.017</td><td>-0.020</td><td>-0.052</td><td>0.019</td><td>-0.019</td></tr><tr><td></td><td colspan="10" style="text-align: center;">squared poverty gap index</td></tr><tr><td>ΔΘ</td><td>-0.004</td><td>-0.052</td><td>-0.047</td><td>0.011</td><td>-0.010</td><td>-0.015</td><td>-0.085</td><td>-0.098</td><td>0.015</td><td>-0.016</td></tr><tr><td>GE</td><td>-0.019</td><td>-0.038</td><td>-0.026</td><td>-0.000</td><td>-0.000</td><td>-0.031</td><td>-0.068</td><td>-0.059</td><td>-0.001</td><td>-0.001</td></tr><tr><td>IE</td><td>0.015</td><td>-0.014</td><td>-0.021</td><td>0.011</td><td>-0.009</td><td>0.016</td><td>-0.017</td><td>-0.038</td><td>0.016</td><td>-0.015</td></tr></tbody></table>

Note: ΔΘ, GE, and IE respectively denotes the absolute variation of poverty, and its growth and inequality components.
financing and had to engage in austerity and adjustment programs. Plans were launched in 1986, 1987, 1989, and 1990 to bring inflation under control but failed. Nevertheless, the 1988 Constitution included several changes regarding social rights, notably with the definition and revision of the minimum wage, as well as guaranteed access to education and social assistance. All these modifications proved to be decisive for poverty and inequality reductions observed in the next decades (de Souza, 2012, Lustig et al., 2013).

Thanks notably to a peg of the real to the US dollar, the 1994 *Plano Real* succeeded in breaking the inflationary spiral. The subperiod 1994–2003, which includes Fernando Henrique Cardoso’s presidency, was also characterized by the privatization of large state-owned companies, tightened tax collection, cuts in government spending, and reduced barriers to international trade. Policy measures were also adopted to fight inequalities and poverty, notably by expanding primary education and fighting malnutrition with the help of the *Bolsa Escola* and *Bolsa Alimentação* programs (Andrews, 2004). The country achieved stabilization, but GDP growth was nevertheless low. In the wake of the 1997 Asian crisis and the 1998 Russian crises, large deficits in the current account resulted in the need for a new adjustment program in 1999.

Lula da Silva’s rise to power in 2003 did not result in a significant change in terms of macroeconomic policies. However, the country benefited from much more favorable international conditions because of the China-led commodity boom, which led to sustained growth up to 2013. Lula da Silva’s first two presidencies are also famous for implementing the *Bolsa Familia* social welfare program that brought together different programs, notably conditional cash transfer programs, that were initiated during Fernando Henrique Cardoso’s presidency, and improved their efficiency. Nevertheless, the literature identifies the decline in wage inequality as the main driver of the notable decline in income inequality during the period. Like many Latin American countries during this decade, the returns to education declined as a result of many factors including the dramatic increase in schooling during the previous decade, a decline in demand for skilled workers as a result of the commodity boom that favored less skilled-labor-intensive sectors, and a decline in the average quality of education (Lustig et al., 2013, 2016). Changes in the revision mode for the minimum wage also contributed significantly to this decline in wage inequalities (Sotomayor, 2021).

This prosperous period ended with the 2014 combined political and economic crises, notably due to decreasing primary commodity prices, unwise fiscal policy measures in favor of the industrial sector, and the discovery of vast corruption schemes involving many parties and members of the political class. The country entered a recession, and unemployment increased sharply. The political turmoil that resulted in Dilma Rousseff’s impeachment in 2016 limited the government’s capacity to respond to the crisis with the appropriate measures. The recovery started in 2017, but in 2019, the Brazilian economy had not returned to its pre-crisis levels in terms of GDP per capita and unemployment.

The last subperiod, namely 2019–2022, is very short and combines the influence of the COVID-19 pandemic and the presidential mandate of Jair Bolsonaro. During this period, a significant reform of the pension system was implemented, as well as other policy measures combining economic liberalization and tax cuts for specific sectors.
## 5.3 The variability of the inequality elasticities of poverty in Brazil

Here, I focus on estimated elasticities associated with observed yearly inequality changes. Estimated values are reported in Table 15. The two most striking features are i) increasingness and ii) variability over the considered period.

In spite of large year-to-year fluctuations, it can be seen that the Gini elasticity of poverty tends to increase during the period, hence confirming that inequality reduction becomes increasingly powerful to fight poverty as the ratio $\bar{\mu}_z$ increases. For instance, considering $\Theta_0$ and the $\$3.65$ poverty line, the average value of $\varepsilon_\Theta$ was 1.96, raised to 3.31 between 2003 and 2014, and finally increased up to 7.29 for the 2019–2022 subperiod. Moving to other poverty indices or using the $\$6.85$ poverty line does not affect this result.

As noted earlier, variability is large. Considering the headcount index, values range from -34.96 to 9.78 (with $z = \$3.65$). Even if I discard these very large elasticities respectively observed for the years 2016 and 2020, my results emphasize the potential difficulty to predict the poverty effect of a given change in inequality. Indeed, the support of the observed distribution of $\varepsilon_0$ is broad and year-to-year variations are substantial. A slightly different picture is obtained when considering the $\$6.85$ poverty line. The range of estimated values is narrower (from -2.68 in 2014 to 3.89 in 2020), and year-to-year variability is less pronounced. Another noteworthy result is that having a decline in poverty as a result of an increase in inequality is not pure theory. Out of the 37 yearly (or quasi-yearly) observed changes, negative values for $\varepsilon_0$ have been estimated five times with the $\$3.65$ poverty line and three times with the $\$6.85$ poverty line.

The estimated elasticities associated with the poverty gap index and the square poverty gap index show a much larger sensitivity of these indices to observed changes in the overall relative distribution in comparison to the estimated values for $\varepsilon_0$, hence a larger variability. Pairwise differences between these elasticities also fluctuate much from year to year. This result is suggestive of varying relative contributions of the different dimensions of poverty to poverty changes linked with inequality variations.

Disentangling observed elasticities $\varepsilon_1$ and $\varepsilon_2$ with the help of equation (24) shows that the incidence, intensity, and inequality channels equally contributed to the noticed variability of these elasticities (Figure 5).<fnref n="30" /> Initial conditions play a minor role in this unpredictability as shown by the relatively constant values of $\pi_{\Theta,\mu_z}$ and $\pi_{\Theta,GE_{2z}}$.<fnref n="31" /> Fluctuations of $\varepsilon_0$, $\zeta$, and $\nu_{GE_{2z}}$ underline the crucial influence of the inequality change pattern. An other noteworthy observation is that the different contributions do not systematically show the same sign. For instance, the value of $\varepsilon_2$ was -0.39 for the short period 1999–2000 with the $\$3.65$ poverty line. The contributions of incidence, intensity, and inequality were respectively 1.98, -2.74, and 0.38. So considering that income inequality decreased during the period, changes in inequality among the poor had a small poverty reduction effect. On the contrary, the magnitude of the effect channeled through the decrease in the headcount index was substantial but more than compensated by the effect of the decrease in the average income of the poor, hence resulting in the negative value of the elasticity.

<footnote n="30">³⁰Elasticities are reported on Table 19, in the appendix.</footnote>

<footnote n="31">³¹See Table 19 in the appendix.</footnote>
<figure><img src="image_8.png" /><figcaption>(b) $\alpha = 1, z = \$6.85.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(c) $\alpha = 2, z = \$3.65.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(d) $\alpha = 2, z = \$6.85.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 5: Contribution of the incidence, intensity, and inequality dimensions to $\varepsilon_\alpha$ for Brazil, 1981–2022.**</figcaption></figure>

Tables 20 and 21 respectively reports estimated values for the between-Gini and the bottom-Gini elasticities of poverty. Values have been computed considering a split of the population at $q=40\%$. The value refers to the threshold used to evaluate progresses with respect to SDG10 and it was also used by the World Bank for its shared prosperity indicator up to 2024 (World Bank, 2015). Unsurprisingly, $\varepsilon_{0q}^B$ crucially depends on the sign of the difference $q - \Theta_0$. The elasticity is close to 0 when the headcount index exceeds 40%, and ranges between 3 and 4.5 otherwise. Turning to $\Theta_1$ and $\Theta_2$ does not alter this finding. Keeping in mind that a unique transform makes it possible to obtain a pure change in $G^B$ (that is keeping $\mu$, $G_-$ and $G_+$ unchanged), this result suggests that initial conditions poorly influence the value of $\varepsilon_{0q}^B$.$^{32}$

A 1% increase in the Gini coefficient for the bottom 40% of the population had in general

<footnote>$^{32}$As shown by equation (39), there is close connection between this elasticity and the average income elasticity of poverty. Table 18, in the appendix, reports estimated values for $\eta_\Theta$ and shows the same relative stability of its value in Brazil during the considered period.</footnote>
no effect up to 1992—this results from $\Theta_0$ generally exceeding 40% during this period—, then modestly decreased poverty between 1993 and 2009, and then increased more substantially poverty starting from 2011 (Table 21) with the $3.65 poverty line. With the $6.85 poverty line, the nullity of $\bar{\varepsilon}_{0,q}$ extends to 2006 and then becomes essentially negative. With $\Theta_1$ and $\Theta_2$, estimated values are usually non-negative and tend to increase during the period.

Can inequality elasticities of poverty be easily predicted? Do their evolutions essentially reflect the mechanical effects of changes in the initial conditions or the evolutions in the patterns of inequality changes? Theory suggests that, the larger the mean income is, the larger shall be the inequality elasticities of poverty (Bourguignon, 2003, Epaulard, 2003). This result is first due to the mechanical influence of lower values for $\Theta$. Under Kakwani's assumption, this effect is reinforced by the larger relative income change at the tails of the income distribution in comparison to income levels close to the mean. Here, even if the Lorenz curve did not necessarily shifted according to Kakwani's assumption, I observe a positive correlation between $\mu$ and $\varepsilon_{\Theta}$.<fnref n="33" /> I also note that the variability of $\varepsilon_{\Theta}$ increases with mean income. So, inequality reduction would become more and more attractive as a way to fight poverty when mean income increases, but it seems that it is at the cost of a larger difficulty to predict the magnitude of the effect.

To estimate the influence of initial conditions and of the magnitude of inequality changes on the value of $\varepsilon_{\Theta}$, I consider the following parsimonious linear model:

$$ \varepsilon_{\alpha,t} = \beta_0 + \beta_1 \mu_t + \beta_2 G_t + \beta_3 \frac{\Delta_{t,t+1} G}{G_t} + u_t, \quad (51) $$

Results are presented in Tables 23 and 24. Columns with odd numbering refer to the whole set of spells; estimations without the years 2014 and 2016 are presented in columns with even numbering to avoid the excessive influence of these two outliers. Whatever the poverty index and the value of the poverty line, the omission of these two observations dramatically improves the fit of the model.

Regarding the influence of initial conditions, a first result relates to the coefficient associated with the Gini index that is statistically significant only in the case of $\varepsilon_0$ with the $6.85$ poverty line. Consequently, if decreasing inequality generally had a direct effect on poverty alleviation in Brazil, there was no indirect effect through an increased efficiency for further inequality reduction (i.e. a larger inequality elasticity of poverty). The evidence of a positive effect of mean income is mixed in the Brazilian case. The corresponding coefficient is not systematically positive, and statistical significance depends on the exclusion of outlier and on the choice of the poverty index. Non-linear specifications with respect to $\mu$ and $G$ (not shown here) do not improve the quality of the fit. So, it is possible that changes in the Brazilian economy and policy decisions have, on average, offset the mechanical positive influence of increasing mean income on $\varepsilon_{\Theta}$.

I also checked whether the value of the Gini elasticity of poverty is correlated with

<footnote n="33">³³If we drop estimates for the year 2014 and 2016, Pearson's correlation coefficient is 0.49, 0.21, and 0.10 respectively for $\Theta_0$, $\Theta_1$, and $\Theta_2$ considering the $3.65$ poverty line. With the $6.85$ poverty line, the values are respectively 0.69, 0.49, and 0.28.</footnote>
variations of the Gini index. As shocks and policy measures that affect the inequality level may not affect the Lorenz curve symmetrically in the case of inequality increases and decreases, it makes sense to add $\frac{\Delta G}{G}$ in model (51). Results are nevertheless disappointing since the coefficient never statistically differs from zero (see Tables 23 and 24). Figure 6 confirm that there is little support for an asymmetric treatment of inequality elasticities of poverty as regards to the sign of $\frac{\Delta G}{G}$. Moreover, it suggests that the magnitude of this relative change may have an effect on $\varepsilon_{\Theta}$. As shown in section 2.2, the value of the elasticity notably depends on $\beta$ which determines the relative contribution of mean-preserving spread sequences (that increase $G$) with respect to Gini-and-mean-preserving changes that may induce Lorenz crossing. These last type of changes may have large effects on poverty, but, by definition, they do not have any effect on the Gini index. Consequently, the magnitude of $\varepsilon_{\Theta}$ increases. So, instead on considering $\frac{\Delta G}{G}$ as a correlate of $\varepsilon_{\Theta}$, it may be preferable to replace it by its absolute value. Figure 6 suggests that the relationship may not be linear. The following specification proved to be the most appropriate:

$$ \varepsilon_{\alpha,t} = \beta_0 + \beta_1 \mu_t + \beta_2 G_t + \beta_3 \left| \frac{\Delta_{t,t+1} G}{G_t} \right|^{-1} + u_t, \quad (52) $$

Model (52) clearly outperforms model (51), notably when the year 2014 and 2016 are not dropped from the sample. The estimated value of $\beta_3$ remains significant in most cases when these two peculiar observations are excluded, but the sign becomes generally positive, i.e. smaller relative changes in the Gini index are associated with a larger expected value of the inequality elasticity of poverty in Brazil between 1981 and 2022. I essentially consider this change in the sign of $\beta_3$ as indicative of a general conclusion regarding the instability of the inequality elasticities of poverty and the difficulty to predict its value.

## 5.4 The efficiency of inequality-based poverty reduction in Brazil between 2003 and 2013

I now focus on the subperiod 2003–2014 to assess whether policy measures and changes in economic conditions at that time resulted in greater efficiency of inequality changes in terms of poverty alleviation.

Table 3 returns the value of the Gini elasticity of poverty for each subperiod. For the 2003–2014 subperiod, the Gini elasticity of poverty was 3.17, 4.84, and 5.92 respectively for the headcount index, the poverty gap index, and the squared poverty gap index. The value was substantially larger than during the previous subperiods, but lower than those observed during the 2014–2019 and 2019–2022 subperiods. Benchmarking the observed elasticities with respect to Rawlsian and anti-Rawlsian scenarios also show that changes in inequality between 2003 and 2014 were very effective regarding the decrease in the headcount index, but performed relatively poorly when considering more comprehensive poverty indices like $\Theta_1$ and $\Theta_2$. In other words, changes in the relative distribution were favorable for poverty reduction but significantly less than a change that would have been fully oriented to extreme poverty relief. In contrast, inequality changes during the subperiod 2014–2019 were highly anti-poor as each additional percent increase in the Gini index
was associated with a large increase in poverty and estimated values of $\varepsilon_{\Theta}$ were relatively close to those obtained with the anti-Rawlsian scenario.

This performance can be assessed with the help of Figure 7. The continuous yellow curve depicts the GIC that would have been observed, had the average income been preserved but the Gini index had increased by 1% using transform (9). The subperiods 1993–2003 and 2003–2004 are compared because they both correspond to sustained decline in inequality and poverty in Brazil. The comparison between the corresponding two normalized GICs shows that the redistribution pattern induced a larger relative increase for most poor quantiles during the second subperiod when compared with the first one. However, redistribution was relatively more favorable for the very poorest part of the population between 1993 and 2003. Because of larger initial inequalities and of a lower observed decrease in inequality, the normalized Rawlsian and anti-Rawlsian scenarios result in larger growth rates (in absolute terms) for the first subperiod in comparison with the second subperiod. I observe that, whatever the considered population quantile below the poverty line, a 1% decrease in the value of the Gini index was associated with an income increment that was closer to the Rawlsian scenario for the first subperiod, hence resulting in a more pro-poor redistribution pattern when compared with the second subperiod.

Table 4 makes it possible to understand this relatively low effectiveness of inequality reduction between 2003 and 2014 in comparison with its potentially large poverty reduction effect. For the poverty gap ratio and the squared poverty gap ratio, it reports the share of observed elasticities that can be associated with changes in the incidence, intensity, and inequality dimensions of poverty using formulas from section 3. Considering the $3.65 poverty line, the subperiod 2003–2014 clearly stands out from the other subperiods due to a greater contribution of the incidence dimension. For instance, it amounts to 54% with $\Theta_2$ for this subperiod, but only 46% for the previous subperiod. In other words, in spite of the impressive results observed with respect to poverty alleviation during the 2000s, the same inequality change could have led to a much greater poverty decrease, had the very poorest benefited more from each eliminated Gini point.

As can be seen from equation (13), the estimated value of $\varepsilon_{\Theta}$ depends on both initial conditions and changes in the relative distribution of income. To control for the effect of initial conditions in our comparison of $\varepsilon_{\Theta}$ between the different subperiods, I estimate counterfactual values for the Gini elasticities of poverty using the counterfactual scenarios underlying the Change-In-Change evaluation framework (Athey and Imbens, 2006). More specifically, I compute the elasticity that would have been observed had the relative income distribution in 2003 changed in the same manner as in other subperiods, that is using the quantile function $y^c(p) := y_{t+s}(F_t(y_{03}(p)))$, where $t$ and $t+s$ refer respectively to the initial and final year of the benchmark subperiod, as the final distribution.

The results are provided in Table 5. The line referring to the subperiod 2003–2014 corresponds to the observed elasticities; other lines refer to counterfactual changes. Compared to the values reported in Table 3, the range of estimated values has dramatically shrunk, hence emphasizing the importance of initial conditions. Nevertheless, significant differences can still be observed depending of the scenario considered for inequality changes. More precisely, I observe that the value of $\varepsilon_{\Theta}$ for the subperiod 2003–2014 is, in
most cases, much larger than those associated with the observed evolutions for previous subperiods, but lower than those corresponding to later subperiods. In other words, the pattern of inequality changes evolved in a way that made poverty more sensitive to variations of the Gini index over the whole period, and the subperiod 2003–2014 represented a key point in this evolution.

These results do not conflict with those obtained with the help of the Rawlsian and anti-Rawlsian scenarios. With this latter approach, differences between subperiod counterfactual elasticities reflect both differences in initial conditions (as given by $y(p)$) and differences in the magnitude of inequality changes (through distinct values of $\beta$). With the CIC-based approach, initial conditions are perfectly controlled for, and discrepancies between the observed elasticity and counterfactual elasticities only reflect diverging patterns regarding the change in relative distribution. So, given its income distribution in 2003, Brazil performed better between 2003 and 2014 than during previous subperiods regarding the efficiency of inequality changes in terms of poverty reduction. Nevertheless, the picture is relatively mixed when the poverty alleviation potential of inequality contraction is considered.

Finally, it is worth pointing that these results are only indicative of increasing association between inequality and poverty variations between 1981 and 2022. They notably left fully unexplained the causes of this modification as it results from both a combination of different policy measures and changes in other factors like international prices, climatic conditions, or technological change. Digging into the sources of more pro-poor inequality changes would hence require further analyses.

# 6 Concluding remarks

What does a 1% increase in GDP per capita imply for poverty reduction? From a theoretical point of view, seeking a single value to answer this question is ultimately unproductive. In the spirit of Ravallion (2022), let's consider the following expression for a GDP elasticity of poverty $\gamma_{\Theta}$:

$$ \gamma_{\Theta} := \frac{\partial \log \Theta}{\partial \log Y} = \left(1 - \frac{\partial \log z}{\partial \log \mu}\right) \eta_{\Theta} \frac{\partial \log \mu}{\partial \log Y} + \left(1 - \frac{\partial \log z}{\partial \log \Psi}\right) \varepsilon_{\Theta} \frac{\partial \log \Psi}{\partial \log Y}, \quad (53) $$

where $Y$ is GDP per capita and $\Psi$ is a (relative) inequality index. Sticking to an absolute definition of poverty results in $\frac{\partial \log z}{\partial \log \mu} = \frac{\partial \log z}{\partial \log \Psi} = 0$. Expression (53) then boils down to $\frac{\partial \log \Theta}{\partial \log Y} = \eta_{\Theta} \frac{\partial \log \mu}{\partial \log Y} + \varepsilon_{\Theta} \frac{\partial \log \Psi}{\partial \log Y}$ which is the value many empirical studies have tried to estimate (Ravallion, 1995, 2007, Kalwij and Verschoor, 2007, Bluhm et al., 2018). The elasticity $\eta_{\Theta}$ has been well established since Kakwani (1993) and is fully predictable as it only depends on initial conditions. Some authors have studied other elements of this relationship, like $\frac{\partial \log \mu}{\partial \log Y}$ (Ravallion, 2003, Deaton, 2005) and $\frac{\partial \log \Psi}{\partial \log Y}$ (Deininger and Squire, 1996, Ravallion, 2001, Dollar and Kraay, 2002). It has notably been pointed out that the instability of these parameters contributes to the variability of $\gamma_{\Theta}$. However, little has been said about the role of $\varepsilon_{\Theta}$ in the unpredictability of the GDP elasticity of poverty. Indeed, apart from Bresson (2010), $\varepsilon_{\Theta}$ has never been treated as a random variable, but rather
as a predetermined function of the current income distribution either by considering rigid hypotheses regarding the income distribution (Bourguignon, 2003, Bergstrom, 2022) or its change (Kakwani, 1993).

In this paper, I follow the approach initiated by Kakwani (1993) and consider analytic expressions of the inequality elasticity of poverty $\varepsilon_{\Theta}$. Ravallion (2022) emphasized the usefulness of Nanak Kakwani's formulas to understand theoretically how growth and inequality elasticities of poverty are likely to change during the development process and how initial income inequalities (average income) determine the value of the growth (inequality) elasticity of poverty. On the other hand, he was more critical regarding their use either for the analysis of observed poverty changes or for projection. Part of his critics related to the way changes in inequality were modeled by Kakwani. Indeed the unique transformation considered assumed systematic Lorenz dominance and drastically reduced the way changes in the relative distribution of income may affect poverty.

I put forward that point elasticities of poverty should not be regarded as relics from the (not so) early ages of poverty analysis. In the present paper, I extend Kakwani's approach and enhance its applicability for both theoretical and empirical analyses. More specifically, I introduce a general transform for the Lorenz curve that makes it possible to generate any change in the Lorenz curve while giving exact formulas for the corresponding value of $\varepsilon_{\Theta}$. These formulas are then applied on Brazilian income distributions between 1981 and 2022. In particular, my results show that $\varepsilon_{\Theta}$ can be highly volatile and quite unpredictable on the sole basis of initial conditions. The considered framework also may also be used to assess the pro-poorness of the pattern of changes in the relative distribution of income. In the case of Brazil, I conclude that changes in inequality during the period of Lula da Silva's first presidential mandates and the beginning of Dilma Rousseff's presidency were clearly pro-poor. However, they were not as effective as they could have been in comparison to both highly pro-poor benchmark scenarios and to observed patterns in inequality changes that prevailed over later subperiods.

This study calls for further research regarding the empirical distribution of the inequality elasticity of poverty. There is a crucial need to know more about the poverty effects of inequality changes around the World before we can appreciate how inequality reduction can complement growth in order to achieve the objective of bringing global poverty below 3% of the World population by the year 2030.
# Appendix

## A Lorenz transform and change in the between and within components of the Gini index

### A.1 Transform (3)

Using the quantile function associated with transform (3) as well as the definition of the between component of the Gini index, we have:

$$G^{KB} = 1 - \frac{(1 - (1 - q)^2)\mu_{-}^{K} + (1 - q)^2\mu_{+}^{K}}{\mu}, \qquad (54)$$

$$= 1 - \frac{(1 - (1 - q)^2)((1 + \lambda)\mu_{-} - \lambda\mu) + (1 - q)^2((1 + \lambda)\mu_{+} - \lambda\mu)}{\mu}, \qquad (55)$$

$$= 1 - \frac{(1 + \lambda)((1 - (1 - q)^2)\mu_{-} + (1 - q)^2\mu_{+}) - \lambda\mu}{\mu}, \qquad (56)$$

$$= (1 + \lambda) \left( 1 - \frac{(1 - (1 - q)^2)\mu_{-} + (1 - q)^2\mu_{+}}{\mu} \right), \qquad (57)$$

$$= (1 + \lambda)G^B. \qquad (58)$$

Transform (3) hence results in the between component of the Gini increasing at the same rate as the total Gini index. Regarding the within component, we can make use of the additive decomposability of the Gini index, that is $G^{KW} = G^K - G^{KB} = (1 + \lambda)G - (1 + \lambda)G^B = (1 + \lambda)G^W$. Consequently, under assumption (3), the within component of the Gini index grows at the same rate the value of the Gini index for the whole population.

### A.2 Transform (9)

Turning to transform (9), we obtain:

$$\tilde{G}^B = 1 - \frac{(1 - (1 - q)^2)\tilde{\mu}_{-} + (1 - q)^2\tilde{\mu}_{+}}{\mu}, \qquad (59)$$

$$= 1 - \frac{(1 - (1 - q)^2) \left( \mu_{-} + \lambda \left( \frac{(1 + \beta)\mu'_{-} - \mu_{-}}{\beta} - \mu \right) \right) + (1 - q)^2 \left( \mu_{+} + \lambda \left( \frac{(1 + \beta)\mu'_{+} - \mu_{+}}{\beta} - \mu \right) \right)}{\mu}, \qquad (60)$$

$$= 1 - \frac{(1 - (1 - q)^2) \left( \mu_{-} + \lambda \left( \frac{(1 + \beta)\mu'_{-} - \mu_{-}}{\beta} - \mu \right) \right) + (1 - q)^2 \left( \mu_{+} + \lambda \left( \frac{(1 + \beta)\mu'_{+} - \mu_{+}}{\beta} - \mu \right) \right)}{\mu}$$

$$+ \frac{(1 - (1 - q)^2)\lambda\mu_{-} + (1 - q)^2\lambda\mu_{+}}{\mu} - \frac{(1 - (1 - q)^2)\lambda\mu_{-} + (1 - q)^2\lambda\mu_{+}}{\mu}, \qquad (61)$$

$$= 1 - \frac{(1 - (1 - q)^2)((1 + \lambda)\mu_{-} - \lambda\mu) + (1 - q)^2((1 + \lambda)\mu_{+} - \lambda\mu)}{\mu}$$

$$- \frac{(1 - (1 - q)^2)\lambda(1 + \beta)\frac{\mu'_{-} - \mu_{-}}{\beta} + (1 - q)^2\lambda(1 + \beta)\frac{\mu'_{+} - \mu_{+}}{\beta}}{\mu}, \qquad (62)$$

$$= (1 + \lambda)G^B - \frac{\lambda(1 + \beta)\frac{(1 - (1 - q)^2)(\mu'_{-} - \mu_{-}) + (1 - q)^2(\mu'_{+} - \mu_{+})}{\beta}}{\mu}, \qquad (63)$$

If $\mu'_{-} = \mu_{-}$ and $\mu'_{+} = \mu_{+}$, the last term of (63) vanishes and $\tilde{G}^B = (1 + \lambda)G^B$, so that (9) results in a 100$\lambda\%$ increase of $G^B$. Because of the restriction on the different parameters, the last term
of (63) equals zero $\forall \lambda$ if and only if:

$$ (1 - (1 - q)^2)(\mu'_{-} - \mu_{-}) + (1 - q)^2(\mu'_{+} - \mu_{+}) = 0. \quad (64) $$

Since, by assumption, $q\mu'_{-} + (1-q)\mu'_{+} = q\mu_{-} + (1-q)\mu_{+} = \mu$, we also must have:

$$ q(\mu'_{-} - \mu_{-}) + (1 - q)(\mu'_{+} - \mu_{+}) = 0. \quad (65) $$

Given the restrictions imposed on $q$, the unique solution of (64) and (65) is $\mu'_{-} = \mu_{-}$ and $\mu'_{+} = \mu_{+}$. These conditions are thus both necessary and sufficient for (9) to result in the Gini and its between component to increase in the same proportion. Because of the additive structure of the Gini index, its within component also increases proportionally, $\tilde{G}^W = \tilde{G} - \tilde{G}^B = (1 + \lambda)G - (1 + \lambda)G^B = (1 + \lambda)G^W$. In the other cases, transform (9) do not yield a proportionate increase in the between and within components of the Gini index.

### A.3 Transform (38)

Thanks to (37), it can be seen that transform (38) results in $\tilde{\mu}_{-}^{B} = \mu_{-} - \lambda(\mu - \mu_{-})$ and $\tilde{\mu}_{+}^{B} = \mu_{+} - \lambda(\mu - \mu_{+})$. The value of the between component of the Gini index thus becomes:

$$ \tilde{G}_{q}^{B} = (1 - q) \left( \frac{\tilde{\mu}_{+}}{\tilde{\mu}} - 1 \right), \quad (66) $$

$$ = (1 - q) \left( \frac{\mu_{+} - \lambda(\mu - \mu_{+})}{\mu} - 1 \right), \quad (67) $$

$$ = (1 - q) \left( \frac{(1 + \lambda)\mu_{+}}{\mu} - (1 + \lambda) \right), \quad (68) $$

$$ = (1 + \lambda)(1 - q) \left( \frac{\mu_{+}}{\tilde{\mu}} - 1 \right), \quad (69) $$

$$ = (1 + \lambda)G_{q}^{B}. \quad (70) $$

With this transform, $\lambda$ can be thus interpreted as the relative variation of the between component of $G$.

Considering now the within component, I obtain:

$$ \tilde{G}_{q}^{W} = q\tilde{L}(q)\tilde{G}_{-} + (1 - q)(1 - \tilde{L}(q))\tilde{G}_{+}, \quad (71) $$

whose value is affected as a result of transform (38) since a positive value of $\lambda$ results in a decrease in $L(q)$. Unless $G_{-} = G_{+}$, the within component will consequently vary.

## B Derivation of the formulas for the inequality elasticities of poverty

It should first be noted that $\varepsilon_{\Theta} := \frac{1}{\Theta} \frac{\partial \Theta}{\partial \lambda} |_{\lambda=0}$. All transforms of the Lorenz curves considered in sections 2 and 3 result in a quantile function that depends on a scale parameter $\lambda$ that can be interpreted as the relative variation of the Gini coefficient or of one of its components ($G^B$, $G_-$, or $G_+$). Let $\bar{y}(p, \lambda)$ be a generic expression of this quantile function. Thanks to Leibniz integral rule, I know that:

$$ \frac{\partial}{\partial \lambda} \int_{0}^{\Theta_{0}} \theta(\bar{y}(p, \lambda), z) dp = \theta(\bar{y}(\Theta_{0}, \lambda), z) \frac{\partial \Theta_{0}}{\partial \lambda} + \int_{0}^{\Theta_{0}} \frac{\partial \theta}{\partial y} \frac{\partial \bar{y}}{\partial \lambda} dp. \quad (72) $$

Regarding the expression of $\frac{\partial \Theta_{0}}{\partial \lambda}$, Kakwani (1993) noted that a change in the poverty line can
be substituted for the change in the distribution of income. Let $\bar{z}(\lambda)$ be this adjusted poverty line, namely the value of the poverty line that generates the same value of the headcount index with the original distribution $y(p)$ as the one obtained with $z$ and $\bar{y}(p, \lambda)$.

Now, I use two known properties of the Lorenz curve, namely:

$$ \frac{\partial L}{\partial p} = \frac{y(p)}{\mu}, \qquad (73) $$

$$ \frac{\partial^2 L}{\partial p^2} = \frac{1}{\mu f(y(p))}. \qquad (74) $$

Evaluating (73) at the headcount index associated with $\bar{z}$ and the initial distribution of income, and differentiating both members of with respect to $\lambda$ results in:

$$ \left. \frac{\partial^2 L}{\partial p^2} \right|_{p=\Theta_0(\bar{z})} \frac{\partial \Theta_0(\bar{z})}{\partial \lambda} = \frac{1}{\mu} \frac{\partial \bar{z}}{\partial \lambda}, \qquad (75) $$

$$ \Rightarrow \qquad \frac{\partial \Theta_0(\bar{z})}{\partial \lambda} = f(\bar{z}) \frac{\partial \bar{z}}{\partial \lambda} \qquad (76) $$

with the help of (74).

Plugging this expression into (72), dividing both terms by $\Theta$, and evaluating the resulting expression for $\lambda = 0$ returns the general expression for the inequality elasticities presented in sections 2 and 3:

$$ \left. \frac{1}{\Theta} \frac{\partial \Theta}{\partial \lambda} \right|_{\lambda=0} = \frac{1}{\Theta} \left( \theta(y(\Theta_0), z) f(z) \left. \frac{\partial \bar{z}}{\partial \lambda} \right|_{\lambda=0} + \int_0^{\Theta_0} \left. \frac{\partial \theta}{\partial y} \frac{\partial \bar{y}}{\partial \lambda} \right|_{\lambda=0} dp \right). \qquad (77) $$

The expression of $\bar{z}(\lambda)$, and consequently of $\left. \frac{\partial \bar{z}}{\partial \lambda} \right|_{\lambda=0}$, is obtained using the definition of $\bar{z}(\lambda)$ and the quantile function associated with the considered transform. Evaluated at $\Theta_0$ with the transformed income distribution, (73) becomes $\left. \frac{\partial \bar{L}}{\partial p} \right|_{p=\Theta_0(\bar{z})} = \frac{z}{\mu}$. By definition of $\bar{z}$, I also have $\left. \frac{\partial \bar{L}}{\partial p} \right|_{p=\Theta_0(\bar{z})} = \frac{\bar{z}}{\mu}$. The quantile function $\tilde{y}(p)$ can be obtained as $\mu \left. \frac{\partial \bar{L}}{\partial p} \right|_{p=\Theta_0(\bar{z})}$ and makes it possible to express $\tilde{y}(p)$ as a function of $y(p)$, hence $z$ as a function of $\bar{z}$.

In the case of transform (9), I have:

$$ z = \bar{z}(\lambda) + \lambda \left( \frac{(1 + \beta)y'(\Theta_0) - \bar{z}(\lambda)}{\beta} - \mu \right), \qquad (78) $$

$$ \Rightarrow \qquad \bar{z}(\lambda) = \frac{\beta z + \lambda(\beta\mu - (1 + \beta)y'(\Theta_0))}{\beta - \lambda}, \qquad (79) $$

$$ \Rightarrow \qquad \left. \frac{\partial \bar{z}}{\partial \lambda} \right|_{\lambda=0} = \frac{\beta^2 \mu - \beta(1 + \beta)y'(\Theta_0) + z\beta + \lambda(\beta\mu - (1 + \beta)y'(\Theta_0))}{(\beta - \lambda)^2}, \qquad (80) $$

$$ = \mu + \frac{z - (1 + \beta)y'(\Theta_0)}{\beta}. \qquad (81) $$

Using this expression in conjunction with (77) and the expression of $\left. \frac{\partial \bar{y}}{\partial \lambda} \right|_{\lambda=0}$ results in the expression of $\varepsilon_{\Theta}$ as given by equation (13).

Considering now transform (38), the relationship between $z$ and $\bar{z}(\lambda)$ becomes:

$$ z = \begin{cases} \left(1 + \lambda \left(1 - \frac{\mu}{\mu_{-}}\right)\right) \bar{z}(\lambda) & \text{if } \Theta_0 < q \\ \left(1 + \lambda \left(1 - \frac{\mu}{\mu_{+}}\right)\right) \bar{z}(\lambda) & \text{if } q \le \Theta_0, \end{cases} \qquad (82) $$
$$ \Rightarrow \quad \bar{z}(\lambda) = \begin{cases} \frac{z}{1+\lambda(1-\frac{\mu}{\mu_{-}})} & \text{if } \Theta_0 < q \\ \frac{z}{1+\lambda(1-\frac{\mu}{\mu_{+}})} & \text{if } q \le \Theta_0, \end{cases} \qquad (83) $$

$$ \Rightarrow \quad \left. \frac{\partial \bar{z}}{\partial \lambda} \right|_{\lambda=0} = \begin{cases} -\left(\frac{1-\frac{\mu}{\mu_{-}}}{1+\lambda(1-\frac{\mu}{\mu_{-}})}z\right)^2 & \text{if } \Theta_0 < q \\ -\left(\frac{1-\frac{\mu}{\mu_{+}}}{1+\lambda(1-\frac{\mu}{\mu_{+}})}z\right)^2 & \text{if } q \le \Theta_0 \end{cases}, \qquad (84) $$

$$ = \begin{cases} -\left(1 - \frac{\mu}{\mu_{-}}\right)z & \text{if } \Theta_0 < q \\ -\left(1 - \frac{\mu}{\mu_{+}}\right)z & \text{if } q \le \Theta_0. \end{cases} \qquad (85) $$

Plugging this expression into (77) and the expression of $\left. \frac{\partial \bar{y}}{\partial \lambda} \right|_{\lambda=0}$ derived from (37) results in the expression of $\bar{\varepsilon}_{\Theta,q}^B$ as given by equation (39).

Considering now transform (44), I obtain:

$$ z = \begin{cases} \bar{z}(\lambda) + \left( \frac{(1+\beta)y'_{-}(\Theta_0) - \bar{z}(\lambda)}{\beta} - \mu_{-} \right) \lambda & \text{if } \Theta_0 < q \\ \bar{z}(\lambda) & \text{if } \Theta_0 \ge q, \end{cases} \qquad (86) $$

$$ \Rightarrow \quad \bar{z}(\lambda) = \begin{cases} \frac{\beta z + (\beta \mu_{-} - (1+\beta)y'_{-}(\Theta_0)) \lambda}{\beta - \lambda} & \text{if } \Theta_0 < q, \\ z & \text{if } \Theta_0 \ge q, \end{cases} \qquad (87) $$

$$ \Rightarrow \quad \left. \frac{\partial \bar{z}}{\partial \lambda} \right|_{\lambda=0} = \begin{cases} \frac{(\beta \mu_{-} - (1+\beta)y'_{-}(\Theta_0))(\beta - \lambda) + \beta z + (\beta \mu_{-} - (1+\beta)y'_{-}(\Theta_0))\lambda}{(\beta - \lambda)^2} & \text{if } \Theta_0 < q, \\ 0 & \text{if } \Theta_0 \ge q \end{cases}, \qquad (88) $$

$$ = \begin{cases} \mu_{-} + \frac{z - (1 + \beta) y'_{-}(\Theta_0)}{\beta} & \text{if } \Theta_0 < q \\ 0 & \text{if } \Theta_0 \ge q. \end{cases} \qquad (89) $$

Using this expression for (77) as well as the expression of $\left. \frac{\partial \bar{y}}{\partial \lambda} \right|_{\lambda=0}$ derived from (42) yields the formula of $\bar{\varepsilon}_{\Theta,q}$ proposed in equation (46).

Finally, I consider transform (45) that generates:

$$ z = \begin{cases} \bar{z}(\lambda) & \text{if } \Theta_0 < q \\ \bar{z}(\lambda) + \lambda \left( \frac{(1+\beta)y'_{+}(p) - \bar{z}(\lambda)}{\beta} - \mu_{+} \right) & \text{if } \Theta_0 \ge q, \end{cases} \qquad (90) $$

$$ \Rightarrow \quad \bar{z}(\lambda) = \begin{cases} z & \text{if } \Theta_0 < q \\ \frac{\beta z + (\beta \mu_{+} - (1+\beta)y'_{+}(\Theta_0))\lambda}{\beta - \lambda} & \text{if } \Theta_0 \ge q, \end{cases} \qquad (91) $$

$$ \Rightarrow \quad \left. \frac{\partial \bar{z}}{\partial \lambda} \right|_{\lambda=0} = \begin{cases} 0 & \text{if } \Theta_0 < q \\ \frac{(\beta \mu_{+} - (1+\beta)y'_{+}(\Theta_0))(\beta - \lambda) + \beta z + (\beta \mu_{+} - (1+\beta)y'_{+}(\Theta_0))\lambda}{(\beta - \lambda)^2} & \text{if } \Theta_0 \ge q \end{cases}, \qquad (92) $$

$$ = \begin{cases} 0 & \text{if } \Theta_0 < q \\ \mu_{+} + \frac{z - (1 + \beta) y'_{+}(\Theta_0)}{\beta} & \text{if } \Theta_0 \ge q. \end{cases} \qquad (93) $$

As for the previous transforms, using this expression for (77) and evaluating $\left. \frac{\partial \bar{y}}{\partial \lambda} \right|_{\lambda=0}$ from (43) makes it possible to obtain the expression of $\varepsilon_{\Theta,q}^+$ proposed in equation (47).
# C Observed and predicted inequality elasticities of poverty

## C.1 Comparison with estimates based on Kakwani's assumption

Table 15 reports the estimated values of the Gini elasticities $\varepsilon_\alpha$ associated with yearly changes in the Lorenz curves for Brazil. These values are obtained considering the appropriate calibrations for $\beta$ and $L'(p)$ discussed in section 2.2.

Are Kakwani's elasticities $\varepsilon_\alpha^K$ good predictions for $\varepsilon_\alpha$?<fnref n="34" /> Figure 8 makes it clear that, even after dropping outliers, Kakwani's assumption performs quite poorly. In the case of Brazil, I observe both significant under-estimations and over-estimations. On average, $\varepsilon_\alpha^K$ tends to overestimate $\varepsilon_\alpha$ over the whole period. However, it can be seen from Figure 8 that the inadequacy of $\varepsilon_\alpha^K$ as an estimate of the observed value of $\varepsilon_\alpha$ has varied a lot through time. Indeed, it could be regarded as a reasonable approximation during the period 1981–1993, but it does not perform in a satisfactory manner for the other subperiods, notably 1994–2002 and 2014–2018. I therefore strongly recommend checking for the validity of assumption (3) before considering $\varepsilon_\alpha^K$ to assess the effect of observed changes in the relative distribution of income on poverty.

Another way of pointing how unduly restrictive is assumption (3) is provided by Figure 9. Here, I simulated the values of $\varepsilon_\alpha$ that would have been obtained for the year 2003, had the relative income distribution (marginally) shifted according to yearly changes observed during the period of interest.<fnref n="35" /> By keeping initial conditions fixed, this makes it possible to illustrate the critical influence of Lorenz change patterns on $\varepsilon_\alpha$, hence its potential large variability. The cdf of the distribution of the simulated elasticities is plot on Figure 9 for each poverty indicator and each poverty line. Unsurprisingly, I effectively note that the range of values for the Gini elasticities of poverty, considering plausible inequality changes, is large, whatever the chosen poverty index. Given historical changes in Brazil, a 90% confidence interval indicates that a 1% increase in the Gini index in 2003 would have resulted in a relative change of the headcount index lying between 0.67% and 11.62% with the $3.65 poverty line. The corresponding intervals are [-6.71,20.65] and [-14.45,25.79] respectively for the poverty gap index and the squared poverty gap index. Figure 9 also shows that $\varepsilon_\alpha^K$ is not necessarily a good estimator of the expected value for $\varepsilon_\alpha$ in the Brazilian case.

## C.2 Comparison with arc elasticities and econometric evaluations

Two other techniques have been also been considered in the literature to estimate elasticities of poverty, namely one based on poverty variation decompositions and one based on econometric regressions. Here, we compare their results with our estimates. It is worth pointing that, unlike the technique described in section 2, the output of both approaches is not point elasticities but arc elasticities.

Considering the approach based on poverty variation decompositions, I consider the technique suggested by Kakwani (2000) and Shorrocks (2013).<fnref n="36" /> The estimated inequality component obtained with the help of the Shapley value technique is then multiplied by $\frac{G}{\Theta\Delta G}$ to obtain an estimate

<footnote n="34">³⁴Estimated values for $\varepsilon_\alpha^K$ are given in table 16 in the appendix.</footnote>

<footnote n="35">³⁵More specifically, I used the counterfactual scenario suggested by Athey and Imbens (2006). In the present case, for a given year $t$ between 1981 and 2021, the counterfactual quantile function is $y^c(p) := y_{t+1}(F_t(y_{03}(p)))$.</footnote>

<footnote>An alternative procedure would have been to apply observed growth incidence curves on the income distribution in 2003, but the presence of zero incomes for many distributions at the beginning of the period prevented using this approach.</footnote>

<footnote n="36">³⁶Estimations have also been performed using the technique proposed by Datt and Ravallion (1992). The corresponding estimated arc elasticities differed only marginally from those obtained with the help of the Shapley decomposition technique.</footnote>
of the arc elasticity $\hat{\varepsilon}_{\Theta}$. In the case of a marginal change in $L$, this approach would theoretically yield the same values as those obtained with (13). Consequently, I expect differences to be larger in the case of large inequality variations. Estimates are reported on Figure 10 and compared with those obtained with formula (13). For most observations, the difference between $\varepsilon_{\alpha}$ and $\hat{\varepsilon}_{\alpha}$ is small, but it deserves to be stressed that it can be large in few cases. This is notably the case for the period 2019–2022 during which large changes in the relative distribution of income occurred.

The econometric approach is not usually considered so as to get a specific value of the inequality elasticity of poverty for each observation. At best, it is supposed to yield a conditional expectation for this parameter, but, in general, a single value is estimated and interpreted as an average value for the inequality elasticity of poverty. Tables 6 and 7 report the values I get for Brazil with the simple model considered in Bourguignon (2003), Kalwij and Verschoor (2007) and Bergstrom (2022), that is:

$$ \frac{\Delta_{t,t+1}\Theta_{\alpha}}{\Theta_{\alpha,t}} = \beta_0 + \beta_1 \frac{\Delta_{t,t+1}\mu}{\mu_t} + \beta_2 \frac{\Delta_{t,t+1}G}{G_t} + u_t, \quad (94) $$

where $u_t$ is the error term and $t$ denotes the initial year. Considering the $3.65$ poverty line, I obtain a coefficient estimate that equals 4.22, 5.74, and 6.46 respectively for $\Theta_0$, $\Theta_1$, and $\Theta_2$.<fnref n="37" /> In each case, a statistically significant overestimation is thus observed when compared with the true value for the average value of $\varepsilon_{\Theta}$ as reported at the bottom of Table 15. Moving to the $6.85$ poverty line does not change this result. On the other hand differences become not statistically significant (but remain economically significant) when the two outliers, namely the years 2014 and 2016 are dropped from the sample. This observed difference between the sample average and the regression coefficient is not surprising as the internal weighing scheme of ordinary least squares estimators, in the presence of essential heterogeneity and asymmetric distributions, can hardly provide coefficients that correspond to the average of the true coefficients associated with each observation. Alternative and more refined specifications (not shown here) do not change this general result. To highlight the poor quality of the fit with model (94) and the necessity to abandon the assumption of constant elasticities of poverty, Tables 8 and 9 show the results I obtain when model (94) is replaced by:

$$ \frac{\Delta_{t,t+1}\Theta_{\alpha}}{\Theta_{\alpha,t}} = \beta_0 + \beta_1\eta_{\Theta}\frac{\Delta_{t,t+1}\mu}{\mu_t} + \beta_2\varepsilon_{\Theta}\frac{\Delta_{t,t+1}G}{G_t} + u_t, \quad (96) $$

Estimated values for $\beta_2$ do not significantly differ from unity at the 5% confidence level, except for the headcount index with the $6.85$ poverty line. As the model essentially returns a first-order Taylor approximation, it is not surprising that the fit is not perfect, but the gain is obvious in comparison with model (94).

# D Additional Figures and tables

$$ \Delta_{t,t+1} \log \Theta_{\alpha} = \beta_0 + \beta_1 \Delta_{t,t+1} \log \mu + \beta_2 \Delta_{t,t+1} G + u_t. \quad (95) $$

Estimates were lower compared to model (94), yet significantly different from the average value of $\varepsilon_{\Theta}$ over the considered period.

<footnote n="37">³⁷As an alternative for model (94), I also considered the model:</footnote>
<figure><img src="image_12.png" /><figcaption>(c) $\alpha = 1, z = \$3.65$.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(d) $\alpha = 1, z = \$6.85$.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(e) $\alpha = 2, z = \$3.65$.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(f) $\alpha = 2, z = \$6.85$.</figcaption></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>Figure 6: Correlation between relative variations of the Gini index and estimated values of $\varepsilon_\alpha$ for Brazil, 1981–2022.</figcaption></figure>
<table><caption>Table 3: Subperiod values of $\epsilon_{\Theta}$ in Brazil, 1981–2022.</caption><thead><tr><th>Period</th><th>Θ<sub>0</sub></th><th>Θ<sub>1</sub></th><th>Θ<sub>2</sub></th></tr></thead><tbody><tr><td></td><td colspan="3" style="text-align: center;">................... z = $3.65 ...................</td></tr><tr><td>1981–1993</td><td>1.09<br/>[1.91,26.51]<br/>(-0.03)</td><td>2.30<br/>[-21.88,36.32]<br/>(0.42)</td><td>3.26<br/>[-45.95,42.25]<br/>(0.56)</td></tr><tr><td>1993–2003</td><td>1.43<br/>[-19.63,2.60]<br/>(0.95)</td><td>2.25<br/>[-21.25,29.38]<br/>(0.46)</td><td>3.10<br/>[-20.39,55.84]<br/>(0.31)</td></tr><tr><td>2003–2014</td><td>3.17<br/>[-7.86,9.23]<br/>(0.65)</td><td>4.84<br/>[-5.68,29.29]<br/>(0.30)</td><td>5.92<br/>[-2.73,48.44]<br/>(0.17)</td></tr><tr><td>2014–2019</td><td>9.65<br/>[-100.33,69.93]<br/>(0.65)</td><td>15.55<br/>[-201.14,80.65]<br/>(0.77)</td><td>18.76<br/>[-297.86,88.51]<br/>(0.82)</td></tr><tr><td>2019–2022</td><td>7.38<br/>[-42.27,89.46]<br/>(0.38)</td><td>12.16<br/>[-35.39,186.27]<br/>(0.21)</td><td>15.57<br/>[-32.15,290.80]<br/>(0.15)</td></tr><tr><td></td><td colspan="3" style="text-align: center;">................... z = $6.85 ...................</td></tr><tr><td>1981–1993</td><td>0.62<br/>[-27.14,6.92]<br/>(0.82)</td><td>1.17<br/>[-1.99,4.26]<br/>(0.51)</td><td>1.72<br/>[-12.58,23.22]<br/>(0.40)</td></tr><tr><td>1993–2003</td><td>1.11<br/>[-5.71,34.38]<br/>(0.17)</td><td>1.48<br/>[-8.23,5.87]<br/>(0.69)</td><td>1.89<br/>[-17.83,18.56]<br/>(0.54)</td></tr><tr><td>2003–2014</td><td>1.31<br/>[-6.66,-0.51]<br/>(1.30)</td><td>2.72<br/>[-7.00,9.07]<br/>(0.60)</td><td>3.70<br/>[-6.34,18.72]<br/>(0.40)</td></tr><tr><td>2014–2014</td><td>2.89<br/>[-20.58,52.96]<br/>(0.32)</td><td>7.01<br/>[-74.22,65.61]<br/>(0.58)</td><td>10.26<br/>[-125.60,72.42]<br/>(0.69)</td></tr><tr><td>2019–2022</td><td>3.27<br/>[-38.98,20.39]<br/>(0.71)</td><td>6.40<br/>[-41.88,76.35]<br/>(0.41)</td><td>9.03<br/>[-39.98,129.80]<br/>(0.29)</td></tr></tbody></table>

Note: sorted values of $\epsilon_{\alpha}^{AR}$ and $\epsilon_{\alpha}^{R}$ within square brackets, value of $\Gamma_{\alpha}$ within parentheses.
<figure><img src="image_18.png" /></figure>

Note: The observed curve corresponds to the annualized growth rate of the quantile during the period; the normalized observed, Rawlsian, and anti-Rawlsian curves refer to the GIC that would have been observed with a constant mean income but a 1% decrease in the value of the Gini index. The vertical dotted line refers to the initial value of the headcount index at the $3.65 poverty line.

<caption>**Figure 7:** Normalized growth incidence curves for Brazil, 1993–2014: observed and benchmark patterns.</caption>

<table><caption>**Table 4:** Relative contribution of the three I's of poverty to observed subperiod inequality elasticities of poverty in Brazil, 1981–2022.</caption><thead><tr><th rowspan="2">Period</th><th colspan="2">Θ<sub>1</sub></th><th colspan="3">Θ<sub>2</sub></th></tr><tr><th>incidence</th><th>intensity</th><th>incidence</th><th>intensity</th><th>inequality</th></tr></thead><tbody><tr><td colspan="6" style="text-align: center;">z = $3.65</td></tr><tr><td>1981–1993</td><td>0.47</td><td>0.53</td><td>0.34</td><td>0.41</td><td>0.25</td></tr><tr><td>1993–2003</td><td>0.64</td><td>0.36</td><td>0.46</td><td>0.28</td><td>0.25</td></tr><tr><td>2003–2014</td><td>0.66</td><td>0.34</td><td>0.54</td><td>0.30</td><td>0.16</td></tr><tr><td>2014–2019</td><td>0.62</td><td>0.38</td><td>0.51</td><td>0.32</td><td>0.16</td></tr><tr><td>2019–2022</td><td>0.61</td><td>0.39</td><td>0.47</td><td>0.33</td><td>0.20</td></tr><tr><td colspan="6" style="text-align: center;">z = $6.85</td></tr><tr><td>1981–1993</td><td>0.53</td><td>0.47</td><td>0.36</td><td>0.39</td><td>0.25</td></tr><tr><td>1993–2003</td><td>0.75</td><td>0.25</td><td>0.59</td><td>0.22</td><td>0.19</td></tr><tr><td>2003–2014</td><td>0.48</td><td>0.52</td><td>0.35</td><td>0.44</td><td>0.21</td></tr><tr><td>2014–2019</td><td>0.41</td><td>0.59</td><td>0.28</td><td>0.43</td><td>0.29</td></tr><tr><td>2019–2022</td><td>0.51</td><td>0.49</td><td>0.36</td><td>0.36</td><td>0.28</td></tr></tbody></table>
<table><caption>Table 5: Estimated values of ε_Θ for the year 2003 with different counterfactual scenarios regarding the change in the relative distribution of income.</caption><thead><tr><th rowspan="2">counterfactual period</th><th colspan="3">z = $3.65</th><th colspan="3">z = $6.85</th></tr><tr><th>Θ<sub>0</sub></th><th>Θ<sub>1</sub></th><th>Θ<sub>2</sub></th><th>Θ<sub>0</sub></th><th>Θ<sub>1</sub></th><th>Θ<sub>2</sub></th></tr></thead><tbody><tr><td>1981–1993</td><td>1.70</td><td>2.71</td><td>3.50</td><td>1.15</td><td>1.65</td><td>2.12</td></tr><tr><td>1993–2003</td><td>1.88</td><td>2.64</td><td>3.42</td><td>1.49</td><td>1.85</td><td>2.24</td></tr><tr><td>2003–2014</td><td>3.18</td><td>4.85</td><td>5.92</td><td>1.31</td><td>2.72</td><td>3.70</td></tr><tr><td>2014–2019</td><td>3.86</td><td>7.68</td><td>9.84</td><td>0.75</td><td>3.19</td><td>5.14</td></tr><tr><td>2019–2022</td><td>3.48</td><td>7.32</td><td>9.42</td><td>1.08</td><td>3.16</td><td>4.97</td></tr></tbody></table>

Note: For instance, line 1 returns the value of ε_Θ if, starting from the 2003 distribution, Brazil had experienced the same change in the relative distribution of income as during the period 1981–1993.
<figure><img src="image_19.png" /><figcaption>(c) $\alpha = 1, z = \$3.65$.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>(d) $\alpha = 1, z = \$6.85$.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>(e) $\alpha = 2, z = \$3.65$.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>(f) $\alpha = 2, z = \$6.85$.</figcaption></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

Note: Triangles correspond to the estimated elasticities for the subperiod. The straight line depicts the diagonal of equality. Dotted lines indicate the average value of the estimated elasticities. To avoid the excessive influence of extreme values, means are computed without the smallest and largest values.

<caption>**Figure 8: Estimated values of $\varepsilon_\alpha$ against $\varepsilon_\alpha^K$ for Brazil, 1981-2022.**</caption>
<figure><img src="image_25.png" /><figcaption>(b) $\alpha = 0, z = \$6.85$.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>(c) $\alpha = 1, z = \$3.65$.</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>(d) $\alpha = 1, z = \$6.85$.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>(e) $\alpha = 2, z = \$3.65$.</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>(f) $\alpha = 2, z = \$6.85$.</figcaption></figure>

<figure><img src="image_30.png" /></figure>

Note: Trimmed means are computed without the smallest and largest values.

<caption>**Figure 9: Distribution of simulated values for $\epsilon_\alpha$ in Brazil, 2003.**</caption>
<figure><img src="image_31.png" /><figcaption>(b) $\alpha = 0, z = \$6.85.$</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>(c) $\alpha = 1, z = \$3.65.$</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>(d) $\alpha = 1, z = \$6.85.$</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>(e) $\alpha = 2, z = \$3.65.$</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>(f) $\alpha = 2, z = \$6.85.$</figcaption></figure>

<figure><img src="image_36.png" /></figure>

Note: Dotted lines indicate the average value of the estimated elasticities. To avoid the excessive influence of extreme values, means are computed without the smallest and largest values.

<caption>**Figure 10: Estimated values of $\epsilon_{\alpha}$ against $\hat{\epsilon}_{\alpha}$ for Brazil, 1981–2022.**</caption>
<table><caption>Table 6: Econometric estimation of the growth and Gini elasticities of poverty, $z = \$3.65$.</caption><thead><tr><th rowspan="3"></th><th colspan="6">Dependent variable</th></tr><tr><th colspan="2">ΔΘ<sub>0</sub>/Θ<sub>0</sub></th><th colspan="2">ΔΘ<sub>1</sub>/Θ<sub>1</sub></th><th colspan="2">ΔΘ<sub>2</sub>/Θ<sub>2</sub></th></tr><tr><th>(1)</th><th>(2)</th><th>(3)</th><th>(4)</th><th>(5)</th><th>(6)</th></tr></thead><tbody><tr><td>Δμ/μ</td><td>-1.145***<br>(0.159)</td><td>-1.144***<br>(0.164)</td><td>-1.482***<br>(0.234)</td><td>-1.478***<br>(0.242)</td><td>-1.652***<br>(0.276)</td><td>-1.646***<br>(0.286)</td></tr><tr><td>ΔG/G</td><td>4.242***<br>(0.659)</td><td>4.239***<br>(0.680)</td><td>5.747***<br>(0.974)</td><td>5.740***<br>(1.004)</td><td>6.467***<br>(1.148)</td><td>6.457***<br>(1.183)</td></tr><tr><td>Const.</td><td>0.033<br>(0.025)</td><td>0.032<br>(0.027)</td><td>0.060<br>(0.037)</td><td>0.059<br>(0.039)</td><td>0.076*<br>(0.044)</td><td>0.073<br>(0.047)</td></tr><tr><td>Obs.</td><td>37</td><td>35</td><td>37</td><td>35</td><td>37</td><td>35</td></tr><tr><td>R<sup>2</sup></td><td>0.667</td><td>0.666</td><td>0.615</td><td>0.613</td><td>0.591</td><td>0.588</td></tr><tr><td>Adj. R<sup>2</sup></td><td>0.648</td><td>0.645</td><td>0.592</td><td>0.589</td><td>0.567</td><td>0.563</td></tr><tr><td>Res. Std. Error</td><td>0.148</td><td>0.152</td><td>0.219</td><td>0.225</td><td>0.258</td><td>0.265</td></tr><tr><td>F Stat.</td><td>34.10***</td><td>31.85***</td><td>27.16***</td><td>25.35***</td><td>24.52***</td><td>22.87***</td></tr></tbody></table>

Note: For column (2), (4), and (6), the observations for the years 2014 and 2016 have been dropped. \*p<0.1; \*\*p<0.05; \*\*\*p<0.01.
<table><caption>Table 7: Econometric estimation of the growth and Gini elasticities of poverty, $z = \$6.85$.</caption><thead><tr><th rowspan="3"></th><th colspan="6">Dependent variable</th></tr><tr><th colspan="2">ΔΘ<sub>0</sub>/Θ<sub>0</sub></th><th colspan="2">ΔΘ<sub>1</sub>/Θ<sub>1</sub></th><th colspan="2">ΔΘ<sub>2</sub>/Θ<sub>2</sub></th></tr><tr><th>(1)</th><th>(2)</th><th>(3)</th><th>(4)</th><th>(5)</th><th>(6)</th></tr></thead><tbody><tr><td>Δμ/μ</td><td>-0.646***<br>(0.076)</td><td>-0.645***<br>(0.079)</td><td>-0.965***<br>(0.121)</td><td>-0.963***<br>(0.125)</td><td>-1.183***<br>(0.162)</td><td>-1.180***<br>(0.168)</td></tr><tr><td>ΔG/G</td><td>2.093***<br>(0.316)</td><td>2.092***<br>(0.326)</td><td>3.385***<br>(0.503)</td><td>3.382***<br>(0.519)</td><td>4.321***<br>(0.674)</td><td>4.316***<br>(0.695)</td></tr><tr><td>Constant</td><td>0.008<br>(0.012)</td><td>0.008<br>(0.013)</td><td>0.022<br>(0.019)</td><td>0.022<br>(0.020)</td><td>0.036<br>(0.026)</td><td>0.035<br>(0.027)</td></tr><tr><td>Obs.</td><td>37</td><td>35</td><td>37</td><td>35</td><td>37</td><td>35</td></tr><tr><td>R<sup>2</sup></td><td>0.715</td><td>0.714</td><td>0.701</td><td>0.699</td><td>0.669</td><td>0.668</td></tr><tr><td>Adj. R<sup>2</sup></td><td>0.699</td><td>0.696</td><td>0.683</td><td>0.680</td><td>0.650</td><td>0.647</td></tr><tr><td>Res. Std. Error</td><td>0.071</td><td>0.073</td><td>0.113</td><td>0.116</td><td>0.151</td><td>0.156</td></tr><tr><td>F Stat.</td><td>42.75***</td><td>39.95***</td><td>39.78***</td><td>37.15***</td><td>34.42***</td><td>32.13***</td></tr></tbody></table>

Note: For column (2), (4), and (6), the observations for the years 2014 and 2016 have been dropped. \*p<0.1; \*\*p<0.05; \*\*\*p<0.01.
<table><caption>Table 8: Econometric estimation of the growth and Gini elasticities of poverty, $z = \$3.65$.</caption><thead><tr><th rowspan="3"></th><th colspan="6">Dependent variable:</th></tr><tr><th colspan="2">ΔΘ<sub>0</sub>/Θ<sub>0</sub></th><th colspan="2">ΔΘ<sub>1</sub>/Θ<sub>1</sub></th><th colspan="2">ΔΘ<sub>2</sub>/Θ<sub>2</sub></th></tr><tr><th>(1)</th><th>(2)</th><th>(3)</th><th>(4)</th><th>(5)</th><th>(6)</th></tr></thead><tbody><tr><td>η<sub>Θ</sub> Δμ/μ</td><td>0.893<sup>***</sup><br>(0.041)</td><td>0.895<sup>***</sup><br>(0.042)</td><td>0.912<sup>***</sup><br>(0.100)</td><td>0.918<sup>***</sup><br>(0.103)</td><td>0.914<sup>***</sup><br>(0.139)</td><td>0.921<sup>***</sup><br>(0.143)</td></tr><tr><td>ε<sub>Θ</sub> ΔG/G</td><td>1.065<sup>***</sup><br>(0.036)</td><td>1.066<sup>***</sup><br>(0.037)</td><td>1.197<sup>***</sup><br>(0.091)</td><td>1.203<sup>***</sup><br>(0.094)</td><td>1.133<sup>***</sup><br>(0.122)</td><td>1.142<sup>***</sup><br>(0.126)</td></tr><tr><td>Const.</td><td>0.017<sup>**</sup><br>(0.007)</td><td>0.018<sup>**</sup><br>(0.007)</td><td>0.050<sup>**</sup><br>(0.020)</td><td>0.054<sup>**</sup><br>(0.022)</td><td>0.071<sup>**</sup><br>(0.031)</td><td>0.076<sup>**</sup><br>(0.033)</td></tr><tr><td>Obs.</td><td>37</td><td>35</td><td>37</td><td>35</td><td>37</td><td>35</td></tr><tr><td>R<sup>2</sup></td><td>0.977</td><td>0.977</td><td>0.881</td><td>0.882</td><td>0.788</td><td>0.788</td></tr><tr><td>Adj. R<sup>2</sup></td><td>0.975</td><td>0.976</td><td>0.874</td><td>0.875</td><td>0.775</td><td>0.775</td></tr><tr><td>Res. Std. Error</td><td>0.039</td><td>0.040</td><td>0.121</td><td>0.124</td><td>0.185</td><td>0.190</td></tr><tr><td>F Stat.</td><td>717.24<sup>***</sup></td><td>681.14<sup>***</sup></td><td>126.32<sup>***</sup></td><td>120.15<sup>***</sup></td><td>63.08<sup>***</sup></td><td>59.63<sup>***</sup></td></tr></tbody></table>

Note: For column (2), (4), and (6), the observations for the years 2014 and 2016 have been dropped. \*p<0.1; \*\*p<0.05; \*\*\*p<0.01
<table><caption>Table 9: Econometric estimation of the growth and Gini elasticities of poverty, $z = \$6.85$.</caption><thead><tr><th colspan="7">Dependent variable:</th></tr><tr><th></th><th colspan="2">ΔΘ<sub>0</sub>/Θ<sub>0</sub></th><th colspan="2">ΔΘ<sub>1</sub>/Θ<sub>1</sub></th><th colspan="2">ΔΘ<sub>2</sub>/Θ<sub>2</sub></th></tr><tr><th></th><th>(1)</th><th>(2)</th><th>(3)</th><th>(4)</th><th>(5)</th><th>(6)</th></tr></thead><tbody><tr><td>η<sub>Θ</sub> Δμ/μ</td><td>0.904***<br>(0.023)</td><td>0.907***<br>(0.024)</td><td>0.898***<br>(0.043)</td><td>0.900***<br>(0.044)</td><td>0.899***<br>(0.066)</td><td>0.903***<br>(0.068)</td></tr><tr><td>ε<sub>Θ</sub> ΔG/G</td><td>1.078***<br>(0.029)</td><td>1.079***<br>(0.029)</td><td>1.065***<br>(0.044)</td><td>1.067***<br>(0.046)</td><td>1.098***<br>(0.065)</td><td>1.100***<br>(0.066)</td></tr><tr><td>Const.</td><td>0.006**<br>(0.002)</td><td>0.007**<br>(0.002)</td><td>0.017***<br>(0.006)</td><td>0.018**<br>(0.007)</td><td>0.030**<br>(0.011)</td><td>0.032**<br>(0.012)</td></tr><tr><td>Obs.</td><td>37</td><td>35</td><td>37</td><td>35</td><td>37</td><td>35</td></tr><tr><td>R<sup>2</sup></td><td>0.989</td><td>0.989</td><td>0.968</td><td>0.969</td><td>0.933</td><td>0.933</td></tr><tr><td>Adj. R<sup>2</sup></td><td>0.988</td><td>0.989</td><td>0.967</td><td>0.967</td><td>0.929</td><td>0.929</td></tr><tr><td>Res. Std. Error</td><td>0.014</td><td>0.014</td><td>0.037</td><td>0.038</td><td>0.068</td><td>0.070</td></tr><tr><td>F Stat.</td><td>1,529.58***</td><td>1,484.71***</td><td>521.24***</td><td>492.63***</td><td>236.45***</td><td>223.78***</td></tr></tbody></table>

Note: For column (2), (4), and (6), the observations for the years 2014 and 2016 have been dropped. \*p<0.1; \*\*p<0.05; \*\*\*p<0.01
<table><caption>Table 10: Expressions of Gini elasticities of poverty for various poverty indices and definitions of the poverty line.</caption><thead><tr><th>Θ</th><th>εΘ</th></tr></thead><tbody><tr><td colspan="2">... absolute poverty line z or relative poverty line z<sup>R</sup> = a + bμ ...</td></tr><tr><td>Θ<sub>0</sub></td><td>(μ + <sup>z-(1+β)y'(Θ<sub>0</sub>)</sup>⁄<sub>β</sub>) <sup>f(z)</sup>⁄<sub>Θ<sub>0</sub></sub></td></tr><tr><td>Θ<sub>α</sub></td><td><sup>α</sup>⁄<sub>zΘ<sub>α</sub></sub> [ (μ + <sup>z</sup>⁄<sub>β</sub>) Θ<sub>α-1</sub> - <sup>z</sup>⁄<sub>β</sub> Θ<sub>α</sub> - <sup>1+β</sup>⁄<sub>β</sub> ∫<sub>0</sub><sup>Θ<sub>0</sub></sup> y'(p) (<sup>z-y(p)</sup>⁄<sub>z</sub>)<sup>α-1</sup> dp ]</td></tr><tr><td>Θ<sub>log</sub></td><td><sup>1</sup>⁄<sub>Θ<sub>log</sub></sub> (<sup>(κ-1+βμ)Θ<sub>0</sub></sup>⁄<sub>βκ-1</sub>) - <sup>1+β</sup>⁄<sub>β</sub> ∫<sub>0</sub><sup>Θ<sub>0</sub></sup> <sup>y'(p)</sup>⁄<sub>y(p)</sub> dp</td></tr><tr><td>Θ<sub>P</sub></td><td><sup>z</sup>⁄<sub>Θ<sub>P</sub></sub> [ (<sup>μ</sup>⁄<sub>κ<sub>-2</sub><sup>2</sup></sub> + <sup>1</sup>⁄<sub>βκ-1</sub>) Θ<sub>0</sub> - <sup>1+β</sup>⁄<sub>β</sub> ∫<sub>0</sub><sup>Θ<sub>0</sub></sup> <sup>y'(p)</sup>⁄<sub>y(p)<sup>2</sup></sub> dp ]</td></tr><tr><td colspan="2">... relative poverty line z<sup>R</sup> = a + by(p<sup>R</sup>) ...</td></tr><tr><td>Θ<sub>0</sub></td><td>(μ + <sup>(1-βb)z<sup>R</sup>-(1+β)y'(Θ<sub>0</sub>)</sup>⁄<sub>β</sub> + <sup>bz<sup>R</sup></sup>⁄<sub>μ</sub> (<sup>(1+β)y'(p<sup>R</sup>)-y(p<sup>R</sup>)</sup>⁄<sub>β</sub> - y(p<sup>R</sup>)) <sup>f(z<sup>R</sup>)</sup>⁄<sub>Θ<sub>0</sub></sub></td></tr><tr><td>Θ<sub>α</sub></td><td><sup>α</sup>⁄<sub>zΘ<sub>α</sub></sub> [ μΘ<sub>α-1</sub> + (<sup>z</sup>⁄<sub>β</sub> + (<sup>(1+β)y'(p<sup>R</sup>)-y(p<sup>R</sup>)</sup>⁄<sub>β</sub> - μ)b ) (Θ<sub>α-1</sub> - Θ<sub>α</sub>) - <sup>(1+β)</sup>⁄<sub>β</sub> ∫<sub>0</sub><sup>Θ<sub>0</sub></sup> y'(p)θ<sub>α-1</sub>(p, z) dp ]</td></tr><tr><td>Θ<sub>log</sub></td><td><sup>1</sup>⁄<sub>Θ<sub>log</sub></sub> [ (<sup>κ-1+βμ</sup>⁄<sub>κ-1</sub> + ((1+β)y'(p<sup>R</sup>) - y(p<sup>R</sup>) - βμ)<sup>b</sup>⁄<sub>z<sup>R</sup></sub> ) <sup>Θ<sub>0</sub></sup>⁄<sub>β</sub> - <sup>1+β</sup>⁄<sub>β</sub> ∫<sub>0</sub><sup>Θ<sub>0</sub></sup> <sup>y'(p)</sup>⁄<sub>y(p)</sub> dp ]</td></tr><tr><td>Θ<sub>P</sub></td><td><sup>Θ<sub>0</sub></sup>⁄<sub>Θ<sub>P</sub></sub> (<sup>z<sup>R</sup>μ</sup>⁄<sub>κ<sub>-2</sub><sup>2</sup></sub> + <sup>z<sup>R</sup>+(<sup>(1+β)y'(p<sup>R</sup>)-y(p<sup>R</sup>)</sup>⁄<sub>β</sub> - βμ)<sup>b</sup></sup>⁄<sub>βκ-1</sub>) - <sup>1+β</sup>⁄<sub>β</sub> <sup>z<sup>R</sup></sup>⁄<sub>Θ<sub>P</sub></sub> ∫<sub>0</sub><sup>Θ<sub>0</sub></sup> <sup>y'(p)</sup>⁄<sub>y(p)<sup>2</sup></sub> dp ]</td></tr><tr><td colspan="2">... relative poverty line z<sup>R</sup> = a + b(1 - (1 - 2c)G)μ ...</td></tr><tr><td>Θ<sub>0</sub></td><td>(1 - (1 - 2c)bG)μ + <sup>z<sup>R</sup>-(1+β)y'(Θ<sub>0</sub>)</sup>⁄<sub>β</sub> <sup>f(z<sup>R</sup>)</sup>⁄<sub>Θ<sub>0</sub></sub></td></tr><tr><td>Θ<sub>α</sub></td><td><sup>α</sup>⁄<sub>z<sup>R</sup>Θ<sub>α</sub></sub> [ μΘ<sub>α-1</sub> + (<sup>z<sup>R</sup></sup>⁄<sub>β</sub> - (1 - 2c)bGμ ) (Θ<sub>α-1</sub> - Θ<sub>α</sub>) - <sup>(1+β)</sup>⁄<sub>β</sub> ∫<sub>0</sub><sup>Θ<sub>0</sub></sup> y'(p)θ<sub>α-1</sub>(p, z<sup>R</sup>) dp ]</td></tr><tr><td>Θ<sub>log</sub></td><td><sup>1</sup>⁄<sub>Θ<sub>log</sub></sub> (<sup>(κ-1+βμ)Θ<sub>0</sub></sup>⁄<sub>βκ-1</sub>) - <sup>(1-2c)bGμΘ<sub>0</sub></sup>⁄<sub>z<sup>R</sup></sub> - <sup>1+β</sup>⁄<sub>β</sub> ∫<sub>0</sub><sup>Θ<sub>0</sub></sup> <sup>y'(p)</sup>⁄<sub>y(p)</sub> dp</td></tr><tr><td>Θ<sub>P</sub></td><td><sup>Θ<sub>0</sub></sup>⁄<sub>Θ<sub>P</sub></sub> (<sup>z<sup>R</sup>μ</sup>⁄<sub>κ<sub>-2</sub><sup>2</sup></sub> + <sup>z<sup>R</sup>-β(1-2c)bGμ</sup>⁄<sub>βκ-1</sub>) - <sup>1+β</sup>⁄<sub>β</sub> <sup>z<sup>R</sup></sup>⁄<sub>Θ<sub>P</sub></sub> ∫<sub>0</sub><sup>Θ<sub>0</sub></sup> <sup>y'(p)</sup>⁄<sub>y(p)<sup>2</sup></sub> dp ]</td></tr></tbody></table>

Note: Θ₀ is the headcount index, Θ<sub>α</sub> are Foster et al.'s (1984) indices (α > 1), Θ<sub>log</sub> is the Watts' (1968) index, and Θ<sub>P</sub> is Kraay et al.'s (2023) index.
<table><caption>Table 11: Expressions of between Gini elasticities of poverty for various poverty indices and definitions of the poverty line.</caption><thead><tr><th>Θ</th><th>Θ<sub>0</sub> &lt; q</th><th>Θ<sub>0</sub> ≥ q</th></tr></thead><tbody><tr><td colspan="3">...absolute poverty line z or relative poverty line z<sup>R</sup> = a + bμ...</td></tr><tr><td>Θ<sub>0</sub></td><td>-<sup>zf(z)</sup>&frasl;<sub>Θ<sub>0</sub></sub>(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>)</td><td>-<sup>zf(z)</sup>&frasl;<sub>Θ<sub>0</sub></sub>(1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>)</td></tr><tr><td>Θ<sub>α</sub></td><td>α<sup>Θ<sub>α</sub> - Θ<sub>α-1</sub></sup>&frasl;<sub>Θ<sub>α</sub></sub>(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>)</td><td><sup>α</sup>&frasl;<sub>Θ<sub>α</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) (Θ<sub>α</sub> - Θ<sub>α-1</sub>) + (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) (C<sub>α</sub><sup>0,q</sup> - C<sub>α-1</sub><sup>0,q</sup>) ]</td></tr><tr><td>Θ<sub>log</sub></td><td>-<sup>Θ<sub>0</sub></sup>&frasl;<sub>Θ<sub>log</sub></sub>(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>)</td><td>-<sup>1</sup>&frasl;<sub>Θ<sub>log</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) Θ<sub>0</sub> + (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) q ]</td></tr><tr><td>Θ<sub>P</sub></td><td>-<sup>Θ<sub>P</sub> + Θ<sub>0</sub></sup>&frasl;<sub>Θ<sub>P</sub></sub>(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>)</td><td>-<sup>1</sup>&frasl;<sub>Θ<sub>P</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) (Θ<sub>P</sub> + Θ<sub>0</sub>) + (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) (C<sub>P</sub><sup>0,q</sup> + q) ]</td></tr><tr><td colspan="3">...relative poverty line z<sup>R</sup> = a + by(p<sup>R</sup>)...</td></tr><tr><td>Θ<sub>0</sub></td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>)</span></span>(by(p<sup>R</sup>) - z<sup>R</sup>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>)</span></span>(by(p<sup>R</sup>) - (1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) z<sup>R</sup>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>)</span></span>(by(p<sup>R</sup>) - z<sup>R</sup>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>)</span></span>(by(p<sup>R</sup>) - z<sup>R</sup>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>)</span></span>(by(p<sup>R</sup>) - z<sup>R</sup>)</td></tr><tr><td>Θ<sub>α</sub></td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>)</span></span>(by(p<sup>R</sup>) - (1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) z<sup>R</sup>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>)</span></span>(by(p<sup>R</sup>) - z<sup>R</sup>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>)</span></span>(by(p<sup>R</sup>) - z<sup>R</sup>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>)</span></span>(by(p<sup>R</sup>) - z<sup>R</sup>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>)</span></span>(by(p<sup>R</sup>) - z<sup>R</sup>)</td></tr><tr><td>Θ<sub>log</sub></td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>)</span></span>(by(p<sup>R</sup>) - <sup>1</sup>&frasl;<sub>z<sup>R</sup></sub>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>)</span></span>(by(p<sup>R</sup>) - <sup>1</sup>&frasl;<sub>z<sup>R</sup></sub>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>)</span></span>(by(p<sup>R</sup>) - <sup>1</sup>&frasl;<sub>z<sup>R</sup></sub>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>)</span></span>(by(p<sup>R</sup>) - <sup>1</sup>&frasl;<sub>z<sup>R</sup></sub>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>)</span></span>(by(p<sup>R</sup>) - <sup>1</sup>&frasl;<sub>z<sup>R</sup></sub>)</td></tr><tr><td>Θ<sub>P</sub></td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>)</span></span>(by(p<sup>R</sup>) - <sup>1</sup>&frasl;<sub>z<sup>R</sup></sub>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>)</span></span>(by(p<sup>R</sup>) - <sup>1</sup>&frasl;<sub>z<sup>R</sup></sub>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>)</span></span>(by(p<sup>R</sup>) - <sup>1</sup>&frasl;<sub>z<sup>R</sup></sub>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>)</span></span>(by(p<sup>R</sup>) - <sup>1</sup>&frasl;<sub>z<sup>R</sup></sub>)</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><span style="display: block; border-bottom: 1px solid black; padding: 0 0.2em;">f(z<sup>R</sup>)</span><span style="display: block; padding: 0 0.2em;">(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>)</span></span>(by(p<sup>R</sup>) - <sup>1</sup>&frasl;<sub>z<sup>R</sup></sub>)</td></tr><tr><td colspan="3">...relative poverty line z<sup>R</sup> = a + b(1 - (1 - 2c)G)μ...</td></tr><tr><td>Θ<sub>0</sub></td><td>-<sup>f(z<sup>R</sup>)</sup>&frasl;<sub>Θ<sub>0</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) z<sup>R</sup> + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>)bμ ]</td><td>-<sup>f(z<sup>R</sup>)</sup>&frasl;<sub>Θ<sub>0</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) z<sup>R</sup> + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>)bμ ]</td><td><sup>α</sup>&frasl;<sub>Θ<sub>α</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] (Θ<sub>α</sub> - Θ<sub>α-1</sub>)</td><td><sup>α</sup>&frasl;<sub>Θ<sub>α</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] (Θ<sub>α</sub> - Θ<sub>α-1</sub>)</td><td><sup>α</sup>&frasl;<sub>Θ<sub>α</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] (Θ<sub>α</sub> - Θ<sub>α-1</sub>)</td><td><sup>α</sup>&frasl;<sub>Θ<sub>α</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] (Θ<sub>α</sub> - Θ<sub>α-1</sub>)</td></tr><tr><td>Θ<sub>α</sub></td><td>α<sup>Θ<sub>α</sub> - Θ<sub>α-1</sub></sup>&frasl;<sub>Θ<sub>α</sub></sub>(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub> + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub>)</td><td><sup>α</sup>&frasl;<sub>Θ<sub>α</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] (Θ<sub>α</sub> - Θ<sub>α-1</sub>)</td><td><sup>α</sup>&frasl;<sub>Θ<sub>α</sub></sub> [ (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) (C<sub>α</sub><sup>0,q</sup> - C<sub>α-1</sub><sup>0,q</sup>) ]</td><td><sup>α</sup>&frasl;<sub>Θ<sub>α</sub></sub> [ (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) (C<sub>α</sub><sup>0,q</sup> - C<sub>α-1</sub><sup>0,q</sup>) ]</td><td><sup>α</sup>&frasl;<sub>Θ<sub>α</sub></sub> [ (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) (C<sub>α</sub><sup>0,q</sup> - C<sub>α-1</sub><sup>0,q</sup>) ]</td><td><sup>α</sup>&frasl;<sub>Θ<sub>α</sub></sub> [ (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) (C<sub>α</sub><sup>0,q</sup> - C<sub>α-1</sub><sup>0,q</sup>) ]</td></tr><tr><td>Θ<sub>log</sub></td><td>-<sup>Θ<sub>0</sub></sup>&frasl;<sub>Θ<sub>log</sub></sub>(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub> + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub>)</td><td>-<sup>1</sup>&frasl;<sub>Θ<sub>log</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] Θ<sub>0</sub> + (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) q ]</td><td><sup>α</sup>&frasl;<sub>Θ<sub>log</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] Θ<sub>0</sub> + (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) q ]</td><td><sup>α</sup>&frasl;<sub>Θ<sub>log</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] Θ<sub>0</sub> + (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) q ]</td><td><sup>α</sup>&frasl;<sub>Θ<sub>log</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] Θ<sub>0</sub> + (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) q ]</td><td><sup>α</sup>&frasl;<sub>Θ<sub>log</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] Θ<sub>0</sub> + (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) q ]</td><td><sup>α</sup>&frasl;<sub>Θ<sub>log</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] Θ<sub>0</sub> + (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) q ]</td></tr><tr><td>Θ<sub>P</sub></td><td>-<sup>Θ<sub>P</sub> + Θ<sub>0</sub></sup>&frasl;<sub>Θ<sub>P</sub></sub>(1 - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub> + (1 - 2c)(G - q<sup>2</sup>G<sub>-</sub> - (1 - q<sup>2</sup>)G<sub>+</sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub>)</td><td>-<sup>1</sup>&frasl;<sub>Θ<sub>P</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] (Θ<sub>P</sub> + Θ<sub>0</sub>) + (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) (C<sub>P</sub><sup>0,q</sup> + q) ]</td><td><sup>α</sup>&frasl;<sub>Θ<sub>P</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] (Θ<sub>P</sub> + Θ<sub>0</sub>) + (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) (C<sub>P</sub><sup>0,q</sup> + q) ]</td><td><sup>α</sup>&frasl;<sub>Θ<sub>P</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] (Θ<sub>P</sub> + Θ<sub>0</sub>) + (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) (C<sub>P</sub><sup>0,q</sup> + q) ]</td><td><sup>α</sup>&frasl;<sub>Θ<sub>P</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] (Θ<sub>P</sub> + Θ<sub>0</sub>) + (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) (C<sub>P</sub><sup>0,q</sup> + q) ]</td><td><sup>α</sup>&frasl;<sub>Θ<sub>P</sub></sub> [ (1 - <sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub>) <sup>bμ</sup>&frasl;<sub>z<sup>R</sup></sub> ] (Θ<sub>P</sub> + Θ<sub>0</sub>) + (<sup>μ</sup>&frasl;<sub>μ<sub>+</sub></sub> - <sup>μ</sup>&frasl;<sub>μ<sub>-</sub></sub>) (C<sub>P</sub><sup>0,q</sup> + q) ]</td></tr></tbody></table>

Note: Θ₀ is the headcount index, Θ_α are Foster et al.'s (1984) indices (α > 1), Θ<sub>log</sub> is the Watts' (1968) index, and Θ<sub>P</sub> is Kraay et al.'s (2023) index.
<table><caption>Table 13: Expressions of $G_+$ elasticities of poverty for various poverty indices and definitions of the poverty line.</caption><thead><tr><th>Θ</th><th>Θ<sub>0</sub> &lt; q</th><th>Θ<sub>0</sub> ≥ q</th></tr></thead><tbody><tr><td colspan="3">...absolute poverty line z or relative poverty line z<sup>R</sup> = a + bμ...</td></tr><tr><td>Θ<sub>0</sub></td><td>0</td><td>(μ<sub>+</sub> + <sup>z-(1+β)y'<sub>+</sub>(Θ<sub>0</sub>)</sup>⁄<sub>β</sub>) <sup>f(z)</sup>⁄<sub>Θ<sub>0</sub></sub></td></tr><tr><td>Θ<sub>α</sub></td><td>0</td><td><sup>α</sup>⁄<sub>zΘ<sub>α</sub></sub> [ (μ<sub>+</sub> + <sup>z</sup>⁄<sub>β</sub>) C<sup>q,Θ<sub>0</sub></sup>⁄<sub>α-1</sub> - <sup>z</sup>⁄<sub>β</sub> C<sup>q,Θ<sub>0</sub></sup>⁄<sub>α</sub> - <sup>(1+β)</sup>⁄<sub>β</sub> ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> y'<sub>+</sub>(p) (<sup>z-y(p)</sup>⁄<sub>z</sub>)<sup>α-1</sup> dp ]</td></tr><tr><td>Θ<sub>log</sub></td><td>0</td><td><sup>1</sup>⁄<sub>Θ<sub>log</sub></sub> (<sup>Θ<sub>0</sub>-q</sup>⁄<sub>β</sub> + ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> <sup>βμ<sub>+</sub>- (1+β)y'<sub>+</sub>(p)</sup>⁄<sub>βy(p)</sub> dp )</td></tr><tr><td>Θ<sub>P</sub></td><td>0</td><td><sup>z</sup>⁄<sub>Θ<sub>P</sub></sub> [ μ<sub>+</sub> ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> <sup>1</sup>⁄<sub>y(p)<sup>2</sup></sub> dp + <sup>C<sup>q,Θ<sub>0</sub>+Θ<sub>0</sub>-q</sup>⁄<sub>zβ</sub> - <sup>1+β</sup>⁄<sub>β</sub> ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> <sup>y'<sub>+</sub>(p)</sup>⁄<sub>y(p)<sup>2</sup></sub> dp ]</td></tr><tr><td colspan="3">...relative poverty line z<sup>R</sup> = a + by(p<sup>R</sup>)...</td></tr><tr><td>Θ<sub>0</sub></td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span> 0 if p<sup>R</sup> &lt; q<br> <span style="font-size: 2.5em; vertical-align: middle;">{</span> (<sup>(1+β)y'<sub>+</sub>(p<sup>R</sup>)-y(p<sup>R</sup>)</sup>⁄<sub>β</sub> - μ<sub>+</sub>) <sup>b f(z<sup>R</sup>)</sup>⁄<sub>Θ<sub>0</sub></sub> if p<sup>R</sup> ≥ q</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span> (μ<sub>+</sub> + <sup>z<sup>R</sup>- (1+β)y'<sub>+</sub>(Θ<sub>0</sub>)</sup>⁄<sub>β</sub>) <sup>f(z<sup>R</sup>)</sup>⁄<sub>Θ<sub>0</sub></sub> if p<sup>R</sup> &lt; q<br> <span style="font-size: 2.5em; vertical-align: middle;">{</span> ((1-b)μ<sub>+</sub> + <sup>z<sup>R</sup>-by(p<sup>R</sup>)- (1+β)y'<sub>+</sub>(Θ<sub>0</sub>)-by'<sub>+</sub>(p<sup>R</sup>)</sup>⁄<sub>β</sub>) <sup>f(z<sup>R</sup>)</sup>⁄<sub>Θ<sub>0</sub></sub> if p<sup>R</sup> ≥ q</td></tr><tr><td>Θ<sub>α</sub></td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span> 0 if p<sup>R</sup> &lt; q<br> <span style="font-size: 2.5em; vertical-align: middle;">{</span> (<sup>(1+β)y'<sub>+</sub>(p<sup>R</sup>)-y(p<sup>R</sup>)</sup>⁄<sub>β</sub> - μ<sub>+</sub>) <sup>(Θ<sub>α</sub>-Θ<sub>α-1</sub>)ab</sup>⁄<sub>z<sup>R</sup>Θ<sub>α</sub></sub> if p<sup>R</sup> ≥ q</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span> <sup>α</sup>⁄<sub>z<sup>R</sup>Θ<sub>α</sub></sub> [ (μ<sub>+</sub> + <sup>z<sup>R</sup></sup>⁄<sub>β</sub>) C<sup>q,Θ<sub>0</sub></sup>⁄<sub>α-1</sub> - <sup>z<sup>R</sup></sup>⁄<sub>β</sub> C<sup>q,Θ<sub>0</sub></sup>⁄<sub>α</sub> - <sup>(1+β)</sup>⁄<sub>β</sub> ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> y'<sub>+</sub>(p) (<sup>z<sup>R</sup>-y(p)</sup>⁄<sub>z<sup>R</sup></sub>)<sup>α-1</sup> dp ] if p<sup>R</sup> &lt; q<br> <span style="font-size: 2.5em; vertical-align: middle;">{</span> <sup>α</sup>⁄<sub>z<sup>R</sup>Θ<sub>α</sub></sub> [ (<sup>(1+β)y'<sub>+</sub>(p<sup>R</sup>)-y(p<sup>R</sup>)</sup>⁄<sub>β</sub> - μ<sub>+</sub>) (Θ<sub>α</sub>-Θ<sub>α-1</sub>)b + (μ<sub>+</sub> + <sup>z<sup>R</sup></sup>⁄<sub>β</sub>) C<sup>q,Θ<sub>0</sub></sup>⁄<sub>α-1</sub> - <sup>z<sup>R</sup></sup>⁄<sub>β</sub> C<sup>q,Θ<sub>0</sub></sup>⁄<sub>α</sub> - <sup>(1+β)</sup>⁄<sub>β</sub> ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> y'<sub>+</sub>(p) (<sup>z<sup>R</sup>-y(p)</sup>⁄<sub>z<sup>R</sup></sub>)<sup>α-1</sup> dp ] if p<sup>R</sup> ≥ q</td></tr><tr><td>Θ<sub>log</sub></td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span> 0 if p<sup>R</sup> &lt; q<br> <span style="font-size: 2.5em; vertical-align: middle;">{</span> (<sup>(1+β)y'<sub>+</sub>(p<sup>R</sup>)-y(p<sup>R</sup>)</sup>⁄<sub>β</sub> - μ<sub>+</sub>) <sup>b Θ<sub>0</sub></sup>⁄<sub>z<sup>R</sup>Θ<sub>log</sub></sub> if p<sup>R</sup> ≥ q</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span> <sup>1</sup>⁄<sub>Θ<sub>log</sub></sub> (<sup>Θ<sub>0</sub>-q</sup>⁄<sub>β</sub> + ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> <sup>βμ<sub>+</sub>- (1+β)y'<sub>+</sub>(p)</sup>⁄<sub>βy(p)</sub> dp ) if p<sup>R</sup> &lt; q<br> <span style="font-size: 2.5em; vertical-align: middle;">{</span> <sup>1</sup>⁄<sub>Θ<sub>log</sub></sub> [ <sup>Θ<sub>0</sub>-q</sup>⁄<sub>β</sub> + (<sup>(1+β)y'<sub>+</sub>(p<sup>R</sup>)-y(p<sup>R</sup>)</sup>⁄<sub>β</sub> - μ<sub>+</sub>) <sup>b Θ<sub>0</sub></sup>⁄<sub>z<sup>R</sup></sub> + ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> <sup>βμ<sub>+</sub>- (1+β)y'<sub>+</sub>(p)</sup>⁄<sub>βy(p)</sub> dp ] if p<sup>R</sup> ≥ q</td></tr><tr><td>Θ<sub>P</sub></td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span> 0 if p<sup>R</sup> &lt; q<br> <span style="font-size: 2.5em; vertical-align: middle;">{</span> (<sup>(1+β)y'<sub>+</sub>(p<sup>R</sup>)-y(p<sup>R</sup>)</sup>⁄<sub>β</sub> - μ<sub>+</sub>) <sup>b Θ<sub>0</sub></sup>⁄<sub>Θ<sub>P</sub>κ-1</sub> if p<sup>R</sup> ≥ q</td><td><span style="font-size: 2.5em; vertical-align: middle;">{</span> <sup>z<sup>R</sup></sup>⁄<sub>Θ<sub>P</sub></sub> [ μ<sub>+</sub> ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> <sup>1</sup>⁄<sub>y(p)<sup>2</sup></sub> dp + <sup>C<sup>q,Θ<sub>0</sub>+Θ<sub>0</sub>-q</sup>⁄<sub>z<sup>R</sup>β</sub> - <sup>1+β</sup>⁄<sub>β</sub> ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> <sup>y'<sub>+</sub>(p)</sup>⁄<sub>y(p)<sup>2</sup></sub> dp ] if p<sup>R</sup> &lt; q<br> <span style="font-size: 2.5em; vertical-align: middle;">{</span> <sup>z<sup>R</sup></sup>⁄<sub>Θ<sub>P</sub></sub> [ μ<sub>+</sub> ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> <sup>1</sup>⁄<sub>y(p)<sup>2</sup></sub> dp + <sup>C<sup>q,Θ<sub>0</sub>+Θ<sub>0</sub>-q</sup>⁄<sub>z<sup>R</sup>β</sub> + (<sup>(1+β)y'<sub>+</sub>(p<sup>R</sup>)-y(p<sup>R</sup>)</sup>⁄<sub>β</sub> - μ<sub>+</sub>) <sup>b Θ<sub>0</sub></sup>⁄<sub>z<sup>R</sup>κ-1</sub> - <sup>1+β</sup>⁄<sub>β</sub> ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> <sup>y'<sub>+</sub>(p)</sup>⁄<sub>y(p)<sup>2</sup></sub> dp ] if p<sup>R</sup> ≥ q</td></tr><tr><td colspan="3">...relative poverty line z<sup>R</sup> = a + b(1 - (1-2c)G)μ...</td></tr><tr><td>Θ<sub>0</sub></td><td>-(1-q)<sup>2</sup>(1-2c)bG<sub>+</sub>μ<sub>+</sub> <sup>f(z<sup>R</sup>)</sup>⁄<sub>Θ<sub>0</sub></sub></td><td>(1 - (1-q)<sup>2</sup>(1-2c)bG<sub>+</sub>)μ<sub>+</sub> + <sup>z<sup>R</sup>- (1+β)y'<sub>+</sub>(Θ<sub>0</sub>)</sup>⁄<sub>β</sub>) <sup>f(z<sup>R</sup>)</sup>⁄<sub>Θ<sub>0</sub></sub></td></tr><tr><td>Θ<sub>α</sub></td><td><sup>(Θ<sub>α</sub>-Θ<sub>α-1</sub>)α(1-q)<sup>2</sup>(1-2c)bG<sub>+</sub>μ<sub>+</sub></sup>⁄<sub>z<sup>R</sup>Θ<sub>α</sub></sub></td><td><sup>α</sup>⁄<sub>z<sup>R</sup>Θ<sub>α</sub></sub> [ (μ<sub>+</sub> + <sup>z<sup>R</sup></sup>⁄<sub>β</sub>) C<sup>q,Θ<sub>0</sub></sup>⁄<sub>α-1</sub> - <sup>z<sup>R</sup></sup>⁄<sub>β</sub> C<sup>q,Θ<sub>0</sub></sup>⁄<sub>α</sub> + (Θ<sub>α</sub>-Θ<sub>α-1</sub>)(1-q)<sup>2</sup>(1-2c)bG<sub>+</sub>μ<sub>+</sub> - <sup>(1+β)</sup>⁄<sub>β</sub> ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> y'<sub>+</sub>(p) (<sup>z<sup>R</sup>-y(p)</sup>⁄<sub>z<sup>R</sup></sub>)<sup>α-1</sup> dp ]</td></tr><tr><td>Θ<sub>log</sub></td><td>- <sup>(1-q)<sup>2</sup>(1-2c)bG<sub>+</sub>μ<sub>+</sub>Θ<sub>0</sub></sup>⁄<sub>z<sup>R</sup>Θ<sub>log</sub></sub></td><td><sup>1</sup>⁄<sub>Θ<sub>log</sub></sub> (<sup>Θ<sub>0</sub>-q</sup>⁄<sub>β</sub> - <sup>(1-q)<sup>2</sup>(1-2c)bG<sub>+</sub>μ<sub>+</sub>Θ<sub>0</sub></sup>⁄<sub>z<sup>R</sup></sub> + ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> <sup>βμ<sub>+</sub>- (1+β)y'<sub>+</sub>(p)</sup>⁄<sub>βy(p)</sub> dp )</td></tr><tr><td>Θ<sub>P</sub></td><td>- <sup>(1-q)<sup>2</sup>(1-2c)bG<sub>+</sub>μ<sub>+</sub>Θ<sub>0</sub></sup>⁄<sub>Θ<sub>P</sub>κ-1</sub></td><td><sup>z<sup>R</sup></sup>⁄<sub>Θ<sub>P</sub></sub> [ μ<sub>+</sub> ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> <sup>1</sup>⁄<sub>y(p)<sup>2</sup></sub> dp + <sup>C<sup>q,Θ<sub>0</sub>+Θ<sub>0</sub>-q</sup>⁄<sub>z<sup>R</sup>β</sub> - <sup>(1-q)<sup>2</sup>(1-2c)bG<sub>+</sub>μ<sub>+</sub>Θ<sub>0</sub></sup>⁄<sub>z<sup>R</sup>κ-1</sub> - <sup>1+β</sup>⁄<sub>β</sub> ∫<sub>q</sub><sup>Θ<sub>0</sub></sup> <sup>y'<sub>+</sub>(p)</sup>⁄<sub>y(p)<sup>2</sup></sub> dp ]</td></tr></tbody></table>

Note: Θ₀ is the headcount index, Θ<sub>α</sub> are Foster et al.'s (1984) indices (α > 1), Θ<sub>log</sub> is the Watts' (1968) index, and Θ<sub>P</sub> is Kraay et al.'s (2023) index.
<table><caption>Table 14: Average income, inequality, and poverty in Brazil, 1981–2022.</caption><thead><tr><th rowspan="2">Year</th><th rowspan="2">μ</th><th rowspan="2">G</th><th colspan="3">z = $3.65</th><th colspan="3">z = $6.85</th></tr><tr><th>Θ<sub>0</sub></th><th>Θ<sub>1</sub></th><th>Θ<sub>2</sub></th><th>Θ<sub>0</sub></th><th>Θ<sub>1</sub></th><th>Θ<sub>2</sub></th></tr></thead><tbody><tr><td>1981</td><td>8.44</td><td>0.580</td><td>0.429</td><td>0.196</td><td>0.119</td><td>0.669</td><td>0.368</td><td>0.246</td></tr><tr><td>1982</td><td>8.60</td><td>0.585</td><td>0.430</td><td>0.198</td><td>0.120</td><td>0.664</td><td>0.368</td><td>0.247</td></tr><tr><td>1983</td><td>7.26</td><td>0.590</td><td>0.509</td><td>0.244</td><td>0.151</td><td>0.721</td><td>0.424</td><td>0.294</td></tr><tr><td>1984</td><td>7.22</td><td>0.584</td><td>0.501</td><td>0.238</td><td>0.145</td><td>0.722</td><td>0.420</td><td>0.289</td></tr><tr><td>1985</td><td>7.61</td><td>0.556</td><td>0.453</td><td>0.208</td><td>0.125</td><td>0.680</td><td>0.382</td><td>0.258</td></tr><tr><td>1986</td><td>13.62</td><td>0.585</td><td>0.260</td><td>0.106</td><td>0.061</td><td>0.495</td><td>0.238</td><td>0.146</td></tr><tr><td>1987</td><td>10.70</td><td>0.597</td><td>0.365</td><td>0.165</td><td>0.100</td><td>0.592</td><td>0.317</td><td>0.209</td></tr><tr><td>1988</td><td>10.54</td><td>0.615</td><td>0.392</td><td>0.184</td><td>0.114</td><td>0.614</td><td>0.338</td><td>0.228</td></tr><tr><td>1989</td><td>12.24</td><td>0.633</td><td>0.364</td><td>0.169</td><td>0.104</td><td>0.584</td><td>0.317</td><td>0.212</td></tr><tr><td>1990</td><td>9.60</td><td>0.605</td><td>0.417</td><td>0.195</td><td>0.119</td><td>0.637</td><td>0.357</td><td>0.241</td></tr><tr><td>1992</td><td>7.64</td><td>0.532</td><td>0.408</td><td>0.193</td><td>0.121</td><td>0.654</td><td>0.356</td><td>0.240</td></tr><tr><td>1993</td><td>9.62</td><td>0.602</td><td>0.397</td><td>0.185</td><td>0.114</td><td>0.641</td><td>0.346</td><td>0.231</td></tr><tr><td>1995</td><td>12.89</td><td>0.596</td><td>0.295</td><td>0.127</td><td>0.075</td><td>0.527</td><td>0.263</td><td>0.167</td></tr><tr><td>1996</td><td>12.96</td><td>0.599</td><td>0.300</td><td>0.135</td><td>0.083</td><td>0.526</td><td>0.268</td><td>0.174</td></tr><tr><td>1997</td><td>12.81</td><td>0.598</td><td>0.304</td><td>0.135</td><td>0.082</td><td>0.531</td><td>0.270</td><td>0.175</td></tr><tr><td>1998</td><td>13.20</td><td>0.596</td><td>0.288</td><td>0.123</td><td>0.073</td><td>0.520</td><td>0.257</td><td>0.163</td></tr><tr><td>1999</td><td>12.38</td><td>0.590</td><td>0.301</td><td>0.129</td><td>0.076</td><td>0.538</td><td>0.267</td><td>0.170</td></tr><tr><td>2001</td><td>13.87</td><td>0.584</td><td>0.262</td><td>0.112</td><td>0.065</td><td>0.480</td><td>0.236</td><td>0.149</td></tr><tr><td>2002</td><td>14.08</td><td>0.581</td><td>0.253</td><td>0.103</td><td>0.058</td><td>0.473</td><td>0.228</td><td>0.140</td></tr><tr><td>2003</td><td>13.40</td><td>0.576</td><td>0.260</td><td>0.109</td><td>0.062</td><td>0.487</td><td>0.236</td><td>0.147</td></tr><tr><td>2004</td><td>13.46</td><td>0.565</td><td>0.244</td><td>0.097</td><td>0.054</td><td>0.473</td><td>0.223</td><td>0.135</td></tr><tr><td>2005</td><td>14.14</td><td>0.563</td><td>0.224</td><td>0.088</td><td>0.048</td><td>0.451</td><td>0.208</td><td>0.124</td></tr><tr><td>2006</td><td>15.19</td><td>0.557</td><td>0.194</td><td>0.074</td><td>0.041</td><td>0.414</td><td>0.184</td><td>0.108</td></tr><tr><td>2007</td><td>15.71</td><td>0.550</td><td>0.180</td><td>0.071</td><td>0.040</td><td>0.390</td><td>0.173</td><td>0.102</td></tr><tr><td>2008</td><td>16.55</td><td>0.541</td><td>0.157</td><td>0.059</td><td>0.033</td><td>0.358</td><td>0.153</td><td>0.088</td></tr><tr><td>2009</td><td>17.06</td><td>0.537</td><td>0.146</td><td>0.056</td><td>0.031</td><td>0.338</td><td>0.144</td><td>0.083</td></tr><tr><td>2011</td><td>18.37</td><td>0.530</td><td>0.120</td><td>0.048</td><td>0.027</td><td>0.299</td><td>0.124</td><td>0.071</td></tr><tr><td>2012</td><td>19.70</td><td>0.535</td><td>0.107</td><td>0.041</td><td>0.022</td><td>0.282</td><td>0.112</td><td>0.063</td></tr><tr><td>2013</td><td>20.54</td><td>0.527</td><td>0.095</td><td>0.034</td><td>0.018</td><td>0.257</td><td>0.100</td><td>0.055</td></tr><tr><td>2014</td><td>20.97</td><td>0.521</td><td>0.087</td><td>0.031</td><td>0.016</td><td>0.244</td><td>0.092</td><td>0.049</td></tr><tr><td>2015</td><td>20.34</td><td>0.520</td><td>0.093</td><td>0.034</td><td>0.018</td><td>0.252</td><td>0.099</td><td>0.054</td></tr><tr><td>2016</td><td>20.33</td><td>0.534</td><td>0.106</td><td>0.042</td><td>0.023</td><td>0.271</td><td>0.110</td><td>0.062</td></tr><tr><td>2017</td><td>20.30</td><td>0.534</td><td>0.110</td><td>0.045</td><td>0.026</td><td>0.271</td><td>0.112</td><td>0.065</td></tr><tr><td>2018</td><td>21.01</td><td>0.539</td><td>0.109</td><td>0.045</td><td>0.025</td><td>0.268</td><td>0.111</td><td>0.064</td></tr><tr><td>2019</td><td>21.11</td><td>0.535</td><td>0.108</td><td>0.045</td><td>0.026</td><td>0.262</td><td>0.110</td><td>0.065</td></tr><tr><td>2020</td><td>21.03</td><td>0.489</td><td>0.053</td><td>0.018</td><td>0.009</td><td>0.187</td><td>0.063</td><td>0.032</td></tr><tr><td>2021</td><td>19.17</td><td>0.530</td><td>0.113</td><td>0.047</td><td>0.027</td><td>0.284</td><td>0.117</td><td>0.068</td></tr><tr><td>2022</td><td>21.33</td><td>0.520</td><td>0.084</td><td>0.031</td><td>0.017</td><td>0.235</td><td>0.090</td><td>0.049</td></tr></tbody></table>

Note: μ is per-capita daily income in $PPP, base 2017.
<figure><img src="image_37.png" /><figcaption>(b) $\alpha = 0, z = \$6.85$.</figcaption></figure>

<figure><img src="image_38.png" /><figcaption>(c) $\alpha = 1, z = \$3.65$.</figcaption></figure>

<figure><img src="image_39.png" /><figcaption>(d) $\alpha = 1, z = \$6.85$.</figcaption></figure>

<figure><img src="image_40.png" /><figcaption>(e) $\alpha = 2, z = \$3.65$.</figcaption></figure>

<figure><img src="image_41.png" /><figcaption>(f) $\alpha = 2, z = \$6.85$.</figcaption></figure>

<figure><img src="image_42.png" /></figure>

Note: The difference between the two curves identifies the growth effect, that is the poverty effect of a pure change in average income.

<caption>**Figure 11: Poverty variations, growth and inequality effects (Shapley decomposition) for Brazil, 1981–2022.**</caption>
<table><caption>Table 15: Estimates of the observed Gini elasticities of FGT indices for Brazil, 1981–2022.</caption><thead><tr><th rowspan="2">Year</th><th colspan="3">z = $3.65</th><th colspan="3">z = $6.85</th></tr><tr><th>Θ<sub>0</sub></th><th>Θ<sub>1</sub></th><th>Θ<sub>2</sub></th><th>Θ<sub>0</sub></th><th>Θ<sub>1</sub></th><th>Θ<sub>2</sub></th></tr></thead><tbody><tr><td>1981</td><td>2.45</td><td>3.63</td><td>3.96</td><td>0.32</td><td>1.65</td><td>2.52</td></tr><tr><td>1982</td><td>2.19</td><td>1.96</td><td>1.30</td><td>0.81</td><td>1.86</td><td>1.94</td></tr><tr><td>1983</td><td>1.82</td><td>3.08</td><td>4.35</td><td>0.11</td><td>1.32</td><td>2.26</td></tr><tr><td>1984</td><td>1.29</td><td>1.55</td><td>1.66</td><td>0.67</td><td>1.16</td><td>1.36</td></tr><tr><td>1985</td><td>0.98</td><td>1.50</td><td>1.77</td><td>0.69</td><td>1.01</td><td>1.24</td></tr><tr><td>1986</td><td>3.73</td><td>6.44</td><td>8.22</td><td>1.10</td><td>3.01</td><td>4.60</td></tr><tr><td>1987</td><td>1.90</td><td>3.18</td><td>4.13</td><td>0.92</td><td>1.85</td><td>2.54</td></tr><tr><td>1988</td><td>2.00</td><td>2.61</td><td>2.81</td><td>1.02</td><td>1.82</td><td>2.21</td></tr><tr><td>1989</td><td>1.98</td><td>3.29</td><td>4.40</td><td>1.14</td><td>2.01</td><td>2.70</td></tr><tr><td>1990</td><td>1.99</td><td>2.54</td><td>2.58</td><td>0.90</td><td>1.69</td><td>2.08</td></tr><tr><td>1992</td><td>1.28</td><td>1.50</td><td>1.55</td><td>0.82</td><td>1.14</td><td>1.30</td></tr><tr><td>1993</td><td>-0.18</td><td>1.29</td><td>3.05</td><td>1.51</td><td>0.41</td><td>1.01</td></tr><tr><td>1995</td><td>4.30</td><td>13.67</td><td>20.57</td><td>0.44</td><td>4.64</td><td>8.88</td></tr><tr><td>1996</td><td>-0.36</td><td>12.38</td><td>20.15</td><td>-2.11</td><td>3.47</td><td>7.46</td></tr><tr><td>1997</td><td>8.48</td><td>15.16</td><td>21.69</td><td>-0.43</td><td>5.69</td><td>10.37</td></tr><tr><td>1998</td><td>2.74</td><td>3.99</td><td>4.71</td><td>2.10</td><td>2.57</td><td>3.11</td></tr><tr><td>1999</td><td>1.98</td><td>-0.67</td><td>-0.39</td><td>3.17</td><td>1.02</td><td>0.41</td></tr><tr><td>2001</td><td>3.37</td><td>12.55</td><td>19.58</td><td>1.10</td><td>4.05</td><td>7.92</td></tr><tr><td>2002</td><td>3.01</td><td>1.18</td><td>-0.27</td><td>1.32</td><td>1.80</td><td>1.60</td></tr><tr><td>2003</td><td>2.84</td><td>5.52</td><td>7.05</td><td>1.30</td><td>2.71</td><td>3.95</td></tr><tr><td>2004</td><td>5.49</td><td>9.71</td><td>12.78</td><td>1.51</td><td>4.95</td><td>6.99</td></tr><tr><td>2005</td><td>3.80</td><td>4.42</td><td>4.56</td><td>1.70</td><td>2.93</td><td>3.67</td></tr><tr><td>2006</td><td>1.90</td><td>-0.18</td><td>-3.29</td><td>2.33</td><td>1.55</td><td>0.65</td></tr><tr><td>2007</td><td>3.44</td><td>5.68</td><td>7.71</td><td>1.77</td><td>3.33</td><td>4.48</td></tr><tr><td>2008</td><td>2.97</td><td>1.68</td><td>0.29</td><td>2.73</td><td>2.62</td><td>2.20</td></tr><tr><td>2009</td><td>4.44</td><td>3.34</td><td>1.34</td><td>2.24</td><td>3.33</td><td>3.36</td></tr><tr><td>2011</td><td>-0.34</td><td>-3.77</td><td>-9.32</td><td>2.95</td><td>0.47</td><td>-1.68</td></tr><tr><td>2012</td><td>3.76</td><td>7.30</td><td>10.23</td><td>2.73</td><td>3.47</td><td>4.92</td></tr><tr><td>2013</td><td>4.84</td><td>6.23</td><td>7.10</td><td>2.09</td><td>4.06</td><td>5.10</td></tr><tr><td>2014</td><td>-15.45</td><td>-37.18</td><td>-50.22</td><td>-2.68</td><td>-9.78</td><td>-19.31</td></tr><tr><td>2015</td><td>5.23</td><td>6.79</td><td>8.28</td><td>2.62</td><td>4.00</td><td>5.27</td></tr><tr><td>2016</td><td>-34.96</td><td>-82.95</td><td>-114.96</td><td>1.43</td><td>-17.78</td><td>-43.34</td></tr><tr><td>2017</td><td>3.75</td><td>4.30</td><td>4.51</td><td>2.61</td><td>3.31</td><td>3.66</td></tr><tr><td>2018</td><td>0.90</td><td>-3.31</td><td>-5.55</td><td>2.20</td><td>-0.35</td><td>-1.81</td></tr><tr><td>2019</td><td>6.93</td><td>10.16</td><td>12.93</td><td>3.30</td><td>5.99</td><td>7.99</td></tr><tr><td>2020</td><td>9.78</td><td>9.71</td><td>9.19</td><td>3.89</td><td>6.51</td><td>7.87</td></tr><tr><td>2021</td><td>5.16</td><td>12.53</td><td>16.32</td><td>2.65</td><td>5.14</td><td>8.21</td></tr><tr><td>mean</td><td>1.61</td><td>1.37</td><td>1.21</td><td>1.43</td><td>1.85</td><td>1.83</td></tr><tr><td>mean*</td><td>3.14</td><td>4.88</td><td>6.00</td><td>1.55</td><td>2.75</td><td>3.72</td></tr></tbody></table>

Note: Elasticities are computed for yearly changes in the Lorenz curves; * is for the average value without the years 2014 and 2016.
<table><caption>Table 16: Estimates of the Gini elasticities of FGT indices ($\epsilon_{\alpha}$) for Brazil, 1981-2022, Kakwani's assumption.</caption><thead><tr><th rowspan="2">Year</th><th colspan="3">z = $3.65</th><th colspan="3">z = $6.85</th></tr><tr><th>Θ<sub>0</sub></th><th>Θ<sub>1</sub></th><th>Θ<sub>2</sub></th><th>Θ<sub>0</sub></th><th>Θ<sub>1</sub></th><th>Θ<sub>2</sub></th></tr></thead><tbody><tr><td>1981</td><td>1.21</td><td>3.87</td><td>6.35</td><td>0.12</td><td>1.42</td><td>2.70</td></tr><tr><td>1982</td><td>1.23</td><td>3.95</td><td>6.49</td><td>0.13</td><td>1.46</td><td>2.76</td></tr><tr><td>1983</td><td>0.76</td><td>3.06</td><td>5.21</td><td>0.02</td><td>1.10</td><td>2.18</td></tr><tr><td>1984</td><td>0.76</td><td>3.06</td><td>5.21</td><td>0.02</td><td>1.09</td><td>2.16</td></tr><tr><td>1985</td><td>0.94</td><td>3.36</td><td>5.62</td><td>0.06</td><td>1.20</td><td>2.33</td></tr><tr><td>1986</td><td>3.38</td><td>7.67</td><td>11.59</td><td>0.78</td><td>3.06</td><td>5.21</td></tr><tr><td>1987</td><td>1.85</td><td>5.26</td><td>8.40</td><td>0.34</td><td>2.05</td><td>3.70</td></tr><tr><td>1988</td><td>1.70</td><td>5.02</td><td>8.08</td><td>0.30</td><td>1.98</td><td>3.60</td></tr><tr><td>1989</td><td>2.23</td><td>6.08</td><td>9.64</td><td>0.46</td><td>2.45</td><td>4.36</td></tr><tr><td>1990</td><td>1.39</td><td>4.48</td><td>7.37</td><td>0.21</td><td>1.72</td><td>3.19</td></tr><tr><td>1992</td><td>1.02</td><td>3.32</td><td>5.50</td><td>0.07</td><td>1.21</td><td>2.35</td></tr><tr><td>1993</td><td>1.54</td><td>4.51</td><td>7.29</td><td>0.24</td><td>1.75</td><td>3.21</td></tr><tr><td>1995</td><td>2.79</td><td>6.88</td><td>10.55</td><td>0.67</td><td>2.77</td><td>4.77</td></tr><tr><td>1996</td><td>2.66</td><td>6.64</td><td>10.31</td><td>0.66</td><td>2.75</td><td>4.74</td></tr><tr><td>1997</td><td>2.65</td><td>6.65</td><td>10.27</td><td>0.63</td><td>2.71</td><td>4.69</td></tr><tr><td>1998</td><td>2.91</td><td>7.11</td><td>10.81</td><td>0.72</td><td>2.88</td><td>4.92</td></tr><tr><td>1999</td><td>2.63</td><td>6.58</td><td>10.09</td><td>0.63</td><td>2.63</td><td>4.54</td></tr><tr><td>2001</td><td>3.14</td><td>7.54</td><td>11.60</td><td>0.81</td><td>3.08</td><td>5.25</td></tr><tr><td>2002</td><td>3.42</td><td>8.05</td><td>12.18</td><td>0.86</td><td>3.20</td><td>5.43</td></tr><tr><td>2003</td><td>3.12</td><td>7.38</td><td>11.34</td><td>0.75</td><td>2.97</td><td>5.07</td></tr><tr><td>2004</td><td>3.40</td><td>7.74</td><td>11.66</td><td>0.80</td><td>3.05</td><td>5.18</td></tr><tr><td>2005</td><td>3.83</td><td>8.36</td><td>12.46</td><td>0.94</td><td>3.31</td><td>5.56</td></tr><tr><td>2006</td><td>4.44</td><td>9.27</td><td>13.57</td><td>1.19</td><td>3.74</td><td>6.15</td></tr><tr><td>2007</td><td>4.67</td><td>9.42</td><td>13.60</td><td>1.29</td><td>3.92</td><td>6.38</td></tr><tr><td>2008</td><td>5.24</td><td>10.38</td><td>14.82</td><td>1.55</td><td>4.33</td><td>6.92</td></tr><tr><td>2009</td><td>5.50</td><td>10.67</td><td>15.19</td><td>1.69</td><td>4.50</td><td>7.18</td></tr><tr><td>2011</td><td>6.27</td><td>11.21</td><td>16.00</td><td>2.05</td><td>5.05</td><td>7.87</td></tr><tr><td>2012</td><td>7.11</td><td>12.44</td><td>18.17</td><td>2.38</td><td>5.71</td><td>8.73</td></tr><tr><td>2013</td><td>7.81</td><td>13.83</td><td>19.78</td><td>2.63</td><td>6.12</td><td>9.37</td></tr><tr><td>2014</td><td>8.13</td><td>14.45</td><td>20.44</td><td>2.86</td><td>6.44</td><td>9.71</td></tr><tr><td>2015</td><td>7.52</td><td>13.34</td><td>19.27</td><td>2.63</td><td>6.04</td><td>9.21</td></tr><tr><td>2016</td><td>7.04</td><td>12.68</td><td>18.55</td><td>2.44</td><td>5.83</td><td>8.96</td></tr><tr><td>2017</td><td>6.59</td><td>12.20</td><td>18.03</td><td>2.42</td><td>5.74</td><td>8.79</td></tr><tr><td>2018</td><td>6.74</td><td>12.62</td><td>18.71</td><td>2.51</td><td>5.99</td><td>9.12</td></tr><tr><td>2019</td><td>6.75</td><td>12.36</td><td>18.55</td><td>2.49</td><td>5.95</td><td>9.11</td></tr><tr><td>2020</td><td>9.70</td><td>15.20</td><td>20.58</td><td>3.62</td><td>7.15</td><td>10.25</td></tr><tr><td>2021</td><td>6.09</td><td>11.18</td><td>17.10</td><td>2.30</td><td>5.37</td><td>8.20</td></tr><tr><td>2022</td><td>8.09</td><td>14.07</td><td>20.05</td><td>2.97</td><td>6.51</td><td>9.78</td></tr><tr><td>mean</td><td>4.11</td><td>8.47</td><td>12.64</td><td>1.24</td><td>3.64</td><td>5.88</td></tr></tbody></table>
<table><caption>Table 17: Arc estimates of the observed Gini elasticities of FGT indices for Brazil, 1981–2022.</caption><thead><tr><th rowspan="2">Year</th><th colspan="3">z = $3.65</th><th colspan="3">z = $6.85</th></tr><tr><th>Θ<sub>0</sub></th><th>Θ<sub>1</sub></th><th>Θ<sub>2</sub></th><th>Θ<sub>0</sub></th><th>Θ<sub>1</sub></th><th>Θ<sub>2</sub></th></tr></thead><tbody><tr><td>1981</td><td>2.52</td><td>3.66</td><td>4.03</td><td>0.30</td><td>1.66</td><td>2.55</td></tr><tr><td>1982</td><td>2.83</td><td>2.33</td><td>1.73</td><td>0.62</td><td>1.81</td><td>2.09</td></tr><tr><td>1983</td><td>1.70</td><td>3.03</td><td>4.28</td><td>0.13</td><td>1.31</td><td>2.24</td></tr><tr><td>1984</td><td>1.14</td><td>1.43</td><td>1.50</td><td>0.75</td><td>1.11</td><td>1.27</td></tr><tr><td>1985</td><td>0.88</td><td>1.30</td><td>1.46</td><td>0.70</td><td>0.97</td><td>1.13</td></tr><tr><td>1986</td><td>3.79</td><td>7.05</td><td>9.59</td><td>0.73</td><td>2.88</td><td>4.74</td></tr><tr><td>1987</td><td>2.02</td><td>3.32</td><td>4.39</td><td>0.99</td><td>1.90</td><td>2.63</td></tr><tr><td>1988</td><td>2.17</td><td>2.62</td><td>2.81</td><td>1.13</td><td>1.86</td><td>2.24</td></tr><tr><td>1989</td><td>1.96</td><td>3.23</td><td>4.27</td><td>1.01</td><td>1.91</td><td>2.61</td></tr><tr><td>1990</td><td>1.80</td><td>2.18</td><td>2.22</td><td>0.83</td><td>1.52</td><td>1.83</td></tr><tr><td>1992</td><td>1.39</td><td>1.62</td><td>1.68</td><td>0.82</td><td>1.22</td><td>1.42</td></tr><tr><td>1993</td><td>-0.34</td><td>1.49</td><td>3.33</td><td>0.56</td><td>0.50</td><td>1.13</td></tr><tr><td>1995</td><td>4.04</td><td>13.78</td><td>21.35</td><td>0.37</td><td>4.65</td><td>9.01</td></tr><tr><td>1996</td><td>-1.20</td><td>12.29</td><td>19.93</td><td>-0.96</td><td>3.43</td><td>7.38</td></tr><tr><td>1997</td><td>6.64</td><td>14.92</td><td>21.02</td><td>-0.09</td><td>5.76</td><td>10.30</td></tr><tr><td>1998</td><td>2.24</td><td>3.95</td><td>4.70</td><td>1.82</td><td>2.51</td><td>3.10</td></tr><tr><td>1999</td><td>0.49</td><td>-0.91</td><td>-0.36</td><td>2.16</td><td>0.74</td><td>0.27</td></tr><tr><td>2001</td><td>3.46</td><td>12.53</td><td>18.92</td><td>0.37</td><td>4.07</td><td>7.86</td></tr><tr><td>2002</td><td>3.01</td><td>1.25</td><td>-0.19</td><td>1.61</td><td>1.80</td><td>1.62</td></tr><tr><td>2003</td><td>3.11</td><td>5.36</td><td>6.60</td><td>1.23</td><td>2.66</td><td>3.84</td></tr><tr><td>2004</td><td>5.30</td><td>9.56</td><td>12.33</td><td>1.90</td><td>4.98</td><td>6.86</td></tr><tr><td>2005</td><td>3.59</td><td>4.07</td><td>4.13</td><td>1.68</td><td>2.87</td><td>3.49</td></tr><tr><td>2006</td><td>2.01</td><td>-0.32</td><td>-3.50</td><td>2.05</td><td>1.42</td><td>0.53</td></tr><tr><td>2007</td><td>3.41</td><td>5.41</td><td>7.24</td><td>1.82</td><td>3.25</td><td>4.27</td></tr><tr><td>2008</td><td>2.86</td><td>1.54</td><td>0.15</td><td>3.09</td><td>2.50</td><td>2.08</td></tr><tr><td>2009</td><td>4.57</td><td>2.75</td><td>0.82</td><td>2.44</td><td>3.17</td><td>3.03</td></tr><tr><td>2011</td><td>-0.13</td><td>-3.84</td><td>-9.24</td><td>2.75</td><td>0.31</td><td>-1.78</td></tr><tr><td>2012</td><td>3.64</td><td>7.10</td><td>9.55</td><td>2.47</td><td>3.31</td><td>4.75</td></tr><tr><td>2013</td><td>4.52</td><td>5.92</td><td>6.65</td><td>2.08</td><td>3.99</td><td>4.88</td></tr><tr><td>2014</td><td>-13.92</td><td>-37.51</td><td>-52.81</td><td>0.16</td><td>-9.65</td><td>-19.49</td></tr><tr><td>2015</td><td>5.13</td><td>7.41</td><td>9.29</td><td>2.61</td><td>4.19</td><td>5.62</td></tr><tr><td>2016</td><td>-36.16</td><td>-83.93</td><td>-120.02</td><td>1.24</td><td>-17.77</td><td>-43.89</td></tr><tr><td>2017</td><td>3.83</td><td>4.34</td><td>4.58</td><td>2.88</td><td>3.33</td><td>3.69</td></tr><tr><td>2018</td><td>0.76</td><td>-3.33</td><td>-5.62</td><td>2.01</td><td>-0.39</td><td>-1.83</td></tr><tr><td>2019</td><td>5.94</td><td>7.09</td><td>7.61</td><td>3.39</td><td>5.04</td><td>5.99</td></tr><tr><td>2020</td><td>10.74</td><td>16.72</td><td>19.52</td><td>4.04</td><td>7.76</td><td>10.99</td></tr><tr><td>2021</td><td>7.11</td><td>11.74</td><td>13.19</td><td>2.14</td><td>5.01</td><td>7.73</td></tr><tr><td>mean</td><td>1.54</td><td>1.38</td><td>1.00</td><td>1.45</td><td>1.83</td><td>1.79</td></tr><tr><td>mean*</td><td>3.06</td><td>4.93</td><td>6.00</td><td>1.50</td><td>2.71</td><td>3.70</td></tr></tbody></table>

Note: Elasticities are computed for yearly changes in the Lorenz curves; * is for the average value without the years 2014 and 2016.