Intertemporal poverty comparisons

Abstract

The paper deals with poverty orderings when multidimensional attributes exhibit some degree of comparability. The paper focuses on an important special case of this, that is, comparisons of poverty that make use of incomes at different time periods. The ordering criteria extend the power of earlier multidimensional dominance tests by making (reasonable) assumptions on the relative marginal contributions of each time dimension to poverty. Inter alia, this involves drawing on natural symmetry and asymmetry assumptions as well as on the mean/variability framework commonly used in the risk literature. The resulting procedures make it possible to check for the robustness of poverty comparisons to choices of intertemporal aggregation procedures and to areas of intertemporal poverty frontiers. The results are illustrated using a rich sample of 23 European countries over 2006–2009.

Appendices

Appendix 1: proof of propositions from Sect. 2

Let $z_1(x_2)$ and $z_2(x_1)$ be respectively the value of the first and second-period income, such that $\lambda \bigr (x_1,z_2(x_1)\bigr )=0$ and $\lambda \bigr (z_1(x_2),x_2\bigr )=0$. Thus, $z_1\bigr (z_2(x_1)\bigr )=x_1$, and $z_1$ is then the inverse of $z_2$. Let $z^*$ be the value of income such that $\lambda (z^*,z^*)=0$. We then can define a two-period poverty index as a sum of low $x_1$ (with respect to $x_2$) and of low $x_2$ (with respect to $x_1$) time poverty:

$$\begin{aligned} P(\lambda )&= \int _0^{z^*} \int _{x_2}^{z_1(x_2)}\pi (x_1,x_2,\lambda )f(x_1,x_2) d x_{1} d x_{2}\nonumber \\&+ \int _0^{z^*} \int _{x_1}^{z_2(x_1)}\pi (x_1,x_2;\lambda ) f(x_1,x_2) d x_{2} d x_{1}. \end{aligned}$$

(46)

We first proceed with the first part of the right-hand term of (46). Integrating that expression by parts with respect to $x_1$, we find:

$$\begin{aligned}&\int _0^{z^*} \int _{x_2}^{z_1(x_2)}\pi (x_1,x_2,\lambda )f(x_1,x_2) d x_{1} d x_{2}\nonumber \\&\quad = \int _0^{z^*} \bigl [\pi (x_1,x_2)F(x_1|x_2)\bigr ]_{x_1=x_2}^{x_1=z_1(x_2)} f(x_2)\; dx_2 \nonumber \\&\quad \quad -\int _0^{z^*} \int _{x_2}^{z_1(x_2)} \pi ^{(1)}(x_1,x_2) F(x_1|x_2) f(x_2)\; dx_1dx_2. \end{aligned}$$

(47)

Rearranging the first element of (47), we find

$$\begin{aligned}&\int _0^{z^*} \left[ \pi (x_1,x_2)F(x_1|x_2)\right] _{x_1=x_2}^{x_1=z_1(x_2)} f(x_2)\; dx_2 \nonumber \\&\quad = \int _0^{z^*} \Bigr (\pi \bigr (z_1(x_2),x_2\bigr ) F\bigr (z_1(x_2)|x_2\bigr ) - \pi (x_2,x_2)F(x_1=x_2|x_2)\Bigr ) f(x_2)\; dx_2 \end{aligned}$$

(48)

$$\begin{aligned}&\quad = - \int _0^{z^*} \pi (x_2,x_2)F(x_1=x_2|x_2)f(x_2)\; dx_2, \end{aligned}$$

(49)

since $\pi \bigr (z_1(x_2),x_2\bigr )=0$.

To integrate the second part of the right-hand term of (47) by parts with respect to $x_2$, let $K(x_2)=\int _{x_2}^{z_1(x_2)} \pi ^{(1)}(x_1,x_2)$ $F(x_1,x_2)\; dx_1$. We then get:

$$\begin{aligned} \frac{\partial K(x_2)}{\partial x_2}&= z_1'(x_2)\pi ^{(1)}\big (z_1(x_2),x_2\big ) F\big (z_1(x_2),x_2\big )\nonumber \\&\quad - \pi ^{(1)}(x_2,x_2) F(x_2,x_2) \nonumber \\&\quad + \int _{x_2}^{z_1(x_2)} \pi ^{(1,2)}(x_1,x_2) F(x_1,x_2)\; dx_1\nonumber \\&\quad + \int _{x_2}^{z_1(x_2)} \pi ^{(1)}(x_1,x_2) F(x_1|x_2)f(x_2)\; dx_1. \end{aligned}$$

(50)

Integrating that expression along $x_2$ and over $[0,z^*]$ and rearranging, we have:

$$\begin{aligned}&\int _0^{z^*}\int _{x_2}^{z_1(x_2)} \pi ^{(1)}(x_1,x_2) F(x_1|x_2)f(x_2)\; dx_1dx_2 \end{aligned}$$

(51)

$$\begin{aligned}&= \,\bigr [K(x_2)\bigr ]_0^{z^*} - \int _0^{z^*} z_1'(x_2)\pi ^{(1)}\bigr (z_1(x_2),x_2\bigr ) F\bigr (z_1(x_2),x_2\bigr )\; dx_2 \nonumber \\&\quad + \int _0^{z^*}\pi ^{(1)}(x_2,x_2) F(x_2,x_2)\; dx_2 - \int _0^{z^*}\int _{x_2}^{z_1(x_2)} \pi ^{(1,2)}(x_1,x_2) F(x_1,x_2)\; dx_1dx_2, \end{aligned}$$

(52)

$$\begin{aligned}&= - \int _0^{z^*} z_1'(x_2)\pi ^{(1)}\bigr (z_1(x_2),x_2\bigr ) F\bigr (z_1(x_2),x_2\bigr )\; dx_2 \nonumber \\&\quad + \int _0^{z^*}\pi ^{(1)}(x_2,x_2) F(x_2,x_2)\; dx_2 - \int _0^{z^*}\int _{x_2}^{z_1(x_2)} \pi ^{(1,2)}(x_1,x_2) F(x_1,x_2)\; dx_1dx_2, \end{aligned}$$

(53)

since $z_1(z^*) = z^*$ (hence $K(z^*)=0$) and $F(x_1,0)=0$ $\forall x_1$ (hence $K(0)=0$). Using (49) and (53), we obtain:

$$\begin{aligned}&\int _0^{z^*} \int _{x_2}^{z_1(x_2)}\pi (x_1,x_2,\lambda )f(x_1,x_2) d x_{1} d x_{2} \nonumber \\&= - \int _0^{z^*} \pi (x_2,x_2)F(x_1=x_2|x_2)f(x_2)\; dx_2 \nonumber \\&\quad +\int _0^{z^*} z_1'(x_2)\pi ^{(1)}\big (z_1(x_2),x_2\big ) F\big (z_1(x_2),x_2\big )\; dx_2 \nonumber \\&\quad - \int _0^{z^*}\pi ^{(1)}(x_2,x_2) F(x_2,x_2)\; dx_2 + \int _0^{z^*}\int _{x_2}^{z_1(x_2)} \pi ^{(1,2)}(x_1,x_2) F(x_1,x_2)\; dx_1dx_2. \end{aligned}$$

(54)

Proceeding similarly with the second part of the right-hand term of (46) and adding the above, we obtain:

$$\begin{aligned} P(\lambda )&= - \int _0^{z^*} \pi (x_2,x_2)F(x_1=x_2|x_2)f(x_2)\; dx_2\nonumber \\&\quad + \int _0^{z^*} z_1'(x_2)\pi ^{(1)}\big (z_1(x_2),x_2\big ) F\big (z_1(x_2),x_2\big )\; dx_2 \nonumber \\&\quad - \int _0^{z^*}\pi ^{(1)}(x_2,x_2) F(x_2,x_2)\; dx_2\nonumber \\&\quad + \int _0^{z^*}\int _{x_2}^{z_1(x_2)} \pi ^{(1,2)}(x_1,x_2) F(x_1,x_2)\; dx_1dx_2 \nonumber \\&\quad - \int _0^{z^*} \pi (x_1,x_1)F(x_2=x_1|x_1)f(x_1)\; dx_1 \nonumber \\&\quad +\int _0^{z^*} z_2'(x_1)\pi ^{(2)}\big (x_1,z_2(x_1)\big ) F\big (x_1,z_2(x_1)\big )\; dx_1 \nonumber \\&\quad - \int _0^{z^*}\pi ^{(2)}(x_1,x_1) F(x_1,x_1)\; dx_1\nonumber \\&\quad + \int _0^{z^*}\int _{x_1}^{z_2(x_1)} \pi ^{(1,2)}(x_1,x_2) F(x_1,x_2)\; dx_2dx_1. \end{aligned}$$

(55)

It can be observed that $F(x_2=x_1|x_1)f(x_1) = \frac{\partial F(x_1,x_1)}{\partial x_1} - F(x_1|x_2=x_1)f(x_2=x_1)$, so that:

$$\begin{aligned}&\int _0^{z^*} \pi (x_1,x_1)F(x_2=x_1|x_1)f(x_1)\; dx_1 \nonumber \\&= \int _0^{z^*} \pi (x_1,x_1) \frac{\partial F(x_1,x_1)}{\partial x_1}\; dx_1 - \int _0^{z^*} \pi (x_1,x_1) F(x_1|x_2=x_1)f(x_2=x_1)\; dx_1 \end{aligned}$$

(56)

$$\begin{aligned}&= \left[ \pi (x_1,x_1)F(x_1,x_1)\right] _0^{z^*} -\int _0^{z^*} \left( \pi ^{(1)}(x_1,x_1)+\pi ^{(2)}(x_1,x_1)\right) F(x_1,x_1)\; dx_1\nonumber \\&\quad - \int _0^{z^*} \pi (x_1,x_1) F(x_1|x_2=x_1)f(x_2=x_1)\; dx_1 \end{aligned}$$

(57)

$$\begin{aligned}&= -\int _0^{z^*} \left( \pi ^{(1)}(x_1,x_1)+\pi ^{(2)}(x_1,x_1)\right) F(x_1,x_1)\; dx_1\nonumber \\&\quad - \int _0^{z^*} \pi (x_2,x_2) F(x_1=x_2|x_2)f(x_2)\; dx_2. \end{aligned}$$

(58)

Using that result and changing the integration variable in $\int _0^{z^*}\pi ^{(2)}(x_2,x_2)$ $F(x_2,x_2)\; dx_2$, we then have:

$$\begin{aligned} P(\lambda )&= \int _0^{z^*} z_2'(x_1)\pi ^{(2)}\big (x_1,z_2(x_1)\big ) F\big (x_1,z_2(x_1)\big )\; dx_1 \nonumber \\&\quad + \int _0^{z^*} z_1'(x_2)\pi ^{(1)}\big (z_1(x_2),x_2\big ) F\big (z_1(x_2),x_2\big )\; dx_2 \nonumber \\&\quad + \int _0^{z^*}\int _{x_1}^{z_2(x_1)} \pi ^{(1,2)}(x_1,x_2) F(x_1,x_2)\; dx_2dx_1 \nonumber \\&\quad + \int _0^{z^*}\int _{x_2}^{z_1(x_2)} \pi ^{(1,2)}(x_1,x_2) F(x_1,x_2)\; dx_1dx_2. \end{aligned}$$

(59)

1.1 Proof of Proposition 2

Symmetry implies the following properties:

$$\begin{aligned} \pi ^{(1)}(x_1,x_2)&= \pi ^{(2)}(x_2,x_1) \quad \forall x_1,x_2,\end{aligned}$$

(60)

$$\begin{aligned} \pi ^{(1,2)}(x_1,x_2)&=\pi ^{(1,2)}(x_2,x_1) \quad \forall x_1, x_2. \end{aligned}$$

(61)

At the poverty frontier, we also have $\lambda ({x_1},{x_2})=0$ and $\pi ^{(x_2)}\bigr (z_1(x_2),x_2\bigr ) = 0$. Since:

$$\begin{aligned} \pi ^{(x_2)}\bigr (z_1(x_2),x_2\bigr ) = z_1'(x_2)\pi ^{(1)}\bigr (z_1(x_2),x_2\bigr ) + \pi ^{(2)}\bigr (z_1(x_2),x_2\bigr ), \end{aligned}$$

(62)

we obtain:

$$\begin{aligned} z_1'(x_2)\pi ^{(1)}\bigr (z_1(x_2),x_2\bigr ) = - \pi ^{(2)}\bigr (z_1(x_2),x_2\bigr ). \end{aligned}$$

(63)

Symmetry also leads to $z_1(x_2)=z_2(x_2)$ and $z_1'(x_2) = z_2'(x_2)$. Using (60), we find that:

$$\begin{aligned} z_1'(x_2)\pi ^{(1)}\bigr (z_1(x_2),x_2\bigr ) = z_2'(x_2)\pi ^{(2)}\bigr (x_2, z_2(x_2)\bigr ). \end{aligned}$$

(64)

From the expression of $P(\lambda )$ in (59), the symmetry assumptions therefore lead to:

$$\begin{aligned} P(\lambda )&= \int _0^{z^*} z_1'(x_2)\pi ^{(1)}\big (z_1(x_2),x_2\big ) \Bigr (F\big (z_1(x_2),x_2\big ) + F\bigr (x_2,z_1(x_2)\bigr )\Bigr )\; dx_2 \end{aligned}$$

(65)

$$\begin{aligned}&\quad + \int _0^{z^*}\int _{x_2}^{z_1(x_2)} \pi ^{(1,2)}(x_1,x_2) \bigr (F(x_1,x_2)+ F(x_2,x_1)\bigr )\; dx_1dx_2 \end{aligned}$$

(66)

The necessary and sufficient conditions for Proposition 2 follow upon inspection.

1.2 Proof of Proposition 3

With asymmetry, we assume that $z_2(x_1) \ge z_1(x_1)$ for all $x_1\in [0,z^*]$. Equation (46) can then be rewritten as:

$$\begin{aligned} P(\lambda )&= \int _0^{z^*} \int _{x_1}^{z_2(x_1)}\pi (x_1,x_2;\lambda ) f(x_1,x_2)\; d x_2 d x_1 \nonumber \\&\quad + \int _0^{z^*} \int _{x_2}^{z_2(x_2)}\pi (x_1,x_2,\lambda )f(x_1,x_2)\; d x_1 d x_2. \end{aligned}$$

(67)

Equation (59) then becomes:

$$\begin{aligned} P(\lambda )&= \int _0^{z^*} z_2'(x_1)\pi ^{(2)}\big (x_1,z_2(x_1)\big ) F\big (x_1,z_2(x_1)\big )\; dx_1 \nonumber \\&\quad + \int _0^{z^*} z_2'(x_2)\pi ^{(1)}\big (z_2(x_2),x_2\big ) F\big (z_2(x_2),x_2\big )\; dx_2 \nonumber \\&\quad + \int _0^{z^*}\int _{x_1}^{z_2(x_1)} \pi ^{(1,2)}(x_1,x_2) F(x_1,x_2)\; dx_2dx_1 \nonumber \\&\quad + \int _0^{z^*}\int _{x_2}^{z_2(x_2)} \pi ^{(1,2)}(x_1,x_2) F(x_1,x_2)\; dx_1dx_2. \end{aligned}$$

(68)

We obtain:

$$\begin{aligned} P(\lambda )&= \int _0^{z^*} z_2'(x_1)\pi ^{(2)}\big (z_2(x_1),x_1\big ) \left[ F\big (z_2(x_1),x_1\big ) +F\big (x_1,z_2(x_1)\big )\right] \; dx_1\nonumber \\&\quad + \int _0^{z^*} z_2'(x_1)\left[ \pi ^{(1)}\big (x_1,z_2(x_1)\big )-\pi ^{(2)}\big (z_2(x_1),x_1\big )\right] F\big (x_1,z_2(x_1)\big )\; dx_1 \nonumber \\&\quad +\int _0^{z^*}\int _{x_2}^{z_2({x_2})} \pi ^{(1,2)}({x_1},{x_2}) \left[ F(x_1,x_2)+F(x_2,x_1)\right] dx_1 dx_2 \nonumber \\&\quad +\int _0^{z^*}\int _{x_1}^{z_2({x_1})} \left[ \pi ^{(1,2)}({x_1},{x_2})-\pi ^{(1,2)}({x_2},{x_1}) \right] F(x_1,x_2) dx_2 dx_1, \end{aligned}$$

(69)

with, by assumption, $\pi ^{(1)}\big (x_1,z_2(x_1)\big )-\pi ^{(2)}\big (z_2(x_1),x_1\big )\le 0$ and $\pi ^{(1,2)}({x_1},{x_2})-\pi ^{(1,2)}({x_2},{x_1})\ge 0$. The second and fourth terms of the right-hand side of (69) account for the first condition of Proposition 3, while the first and third terms account for its second condition.

Appendix 2: proof of propositions from Sect. 3

Let the lowest value of mean income on the mean/variability poverty frontier be obtained for $\tau _\eta =0$ at $\mu =z^*$, so that $\tilde{\lambda }(z^*,0) = \lambda (z^*,z^*)=0$. At this point, it is also necessary to differentiate between the cases of $x_1<x_2$ and $x_1>x_2$. Let $\tau _\eta ^{z1}(\mu )\,(\tau _\eta ^{z2}(\mu ))$ be the value of $\tau _\eta$ such that $\tilde{\lambda }\big (\mu ,\tau _\eta ^z(\mu )\big )=0$ when $x_1<x_2$ ($x_1>x_2$). Since individuals are supposed to be poor $\forall \tau ^\eta$ if $\mu \le z^*,\, \tau _\eta ^{z1}(\mu )$ and $\tau _\eta ^{z2}(\mu )$ are defined on the intervals $[z^*,+\infty )$. Due to the monotonicity assumptions, $\frac{\partial \tau _\eta ^{z1}}{\partial \mu }\le 0$ and $\frac{\partial \tau _\eta ^{z2}}{\partial \mu }\le 0$.

Let $q:=\text {prob}(x_1<x_2)$ and $\rho _1\,(\rho _2)$ be the individual poverty measure to be applied when $x_1<x_2\,(x_1>x_2)$. Let $f_1\,(f_2)$ denote the joint density function of $\mu$ and $\tau _\eta$, conditional on $x_1<x_2\,(x_1>x_2)$. The same notation applies for the cdf, conditional cdf and marginal cdf and marginal density functions. With the above, lifetime poverty defined in (19) can alternatively be defined as:

$$\begin{aligned} \tilde{P}(\tilde{\lambda } )&= q\int _0^{z^*}\int _{-\mu ^{1-\eta }}^0 \rho _1(\mu ,\tau _\eta ,\tilde{\lambda }) f_1(\mu ,\tau _\eta )\; d\tau _\eta d\mu \nonumber \\&\quad + q\int _{z^*}^{+\infty }\int _{-\mu ^{1-\eta }}^{\tau _\eta ^{z1}(\mu )} \rho _1(\mu ,\tau _\eta ,\tilde{\lambda }) f_1(\mu ,\tau _\eta )\; d\tau _\eta d\mu \\&\quad + (1-q)\int _0^{z^*}\int _{-\mu ^{1-\eta }}^0 \rho _2(\mu ,\tau _\eta ,\tilde{\lambda }) f_2(\mu ,\tau _\eta )\; d\tau _\eta d\mu \nonumber \\&\quad + (1-q)\int _{z^*}^{+\infty }\int _{-\mu ^{1-\eta }}^{\tau _\eta ^{z2}(\mu )} \rho _2(\mu ,\tau _\eta ,\tilde{\lambda }) f_2(\mu ,\tau _\eta )\; d\tau _\eta d\mu . \nonumber \end{aligned}$$

(70)

For convenience, $\tilde{\lambda }$ is dropped from the expression of $\rho$. We first consider the first and third elements of the right-hand term of (70) and, integrating by parts, find $\forall j=1,2$:

$$\begin{aligned}&\int _0^{z^*}\int _{-\mu ^{1-\eta }}^0 \rho _j(\mu ,\tau _\eta ) f_j(\mu ,\tau _\eta ) d\tau _\eta d\mu \nonumber \\&= \int _0^{z^*} \bigl [\rho _j(\mu ,\tau _\eta )F_j(\tau _\eta |\mu )\bigr ]_{\tau _\eta =-\mu ^{1-\eta }}^{\tau _\eta =0} f_j(\mu )\; d\mu \nonumber \\&-\int _0^{z^*} \int _{-\mu ^{1-\eta }}^0 \rho _j^{(2)}(\mu ,\tau _\eta )F_j(\tau _\eta |\mu ) f_j(\mu )\; d\tau _\eta d\mu . \end{aligned}$$

(71)

As $F_j(\tau _\eta =-\mu ^{1-\eta }|\mu )=0$ and $F_j(\tau _\eta =0|\mu )=1$, the first element on the right-hand side of (71) can be expressed as:

$$\begin{aligned}&\int _0^{z^*}\bigl [\rho _j(\mu ,\tau _\eta )F_j(\tau _\eta |\mu ) \bigr ]_{\tau _\eta =-\mu ^{1-\eta }}^{\tau _\eta =0} f_j(\mu )\; d\mu&\nonumber \\&= \int _0^{z^*} \rho _j(\mu ,0)f_j(\mu ) \; d\mu , \nonumber \\&= \left[ \rho _j(\mu ,0)F_j(\mu )\right] _{\mu =0}^{\mu =z^*} - \int _0^{z^*} \rho _j^{(1)}(\mu ,0)F_j(\mu ) \; d\mu \nonumber \\&= - \int _0^{z^*} \rho _j^{(1)}(\mu ,0)F_j(\mu ) \; d\mu , \end{aligned}$$

(72)

since $F_j(\mu =0) = 0$ and the function $\rho _j$ is zero at the poverty frontier ($\rho _j(z^*,0)=0$).

We now can turn to the second element of the right-hand term of (71). Define $Q_j(\mu ) = \int _{-\mu ^{1-\eta }}^0 \rho _j^{(2)}(\mu ,\tau _\eta )F_j(\mu ,\tau _\eta )\; d\tau _\eta$. We have:

$$\begin{aligned} \frac{\partial Q_j}{\partial \mu }&= (1-\eta )\mu ^{-\eta }\rho _j^{(2)} \left( \mu ,-\mu ^{1-\eta }\right) F_j\left( \mu ,-\mu ^{1-\eta }\right) \nonumber \\&\quad + \int _{-\mu ^{1-\eta }}^0\rho _j^{(1,2)}(\mu ,\tau _\eta )F_j(\mu ,\tau _\eta )\; d\tau _\eta \nonumber \\&\quad + \int _{-\mu ^{1-\eta }}^0 \rho _j^{(2)}(\mu ,\tau _\eta )F_j(\tau _\eta |\mu ) f_j(\mu )\; d\tau _\eta \nonumber \\&= \int _{-\mu ^{1-\eta }}^0\rho _j^{(1,2)}(\mu ,\tau _\eta )F_j(\mu ,\tau _\eta )\; d\tau _\eta \nonumber \\&\quad + \int _{-\mu ^{1-\eta }}^0 \rho _j^{(2)}(\mu ,\tau _\eta )F_j(\tau _\eta |\mu ) f_j(\mu )\;d\tau _\eta , \end{aligned}$$

(73)

since $F_j(\mu ,-\mu ^{1-\eta })=0$. Integrating that expression along $\mu$ and over $[0,z^*]$ and rearranging, we have:

$$\begin{aligned}&\int _0^{z^*} \int _{-\mu ^{1-\eta }}^0 \rho _j^{(2)}(\mu ,\tau _\eta )F_j(\tau _\eta |\mu ) f_j(\mu )\; d\tau _\eta d\mu \nonumber \\&\quad = \left[ Q_j(\mu )\right] _0^{z^*} - \int _0^{z^*} \int _{-\mu ^{1-\eta }}^0 \rho _j^{(1,2)}(\mu ,\tau _\eta )F_j(\mu ,\tau _\eta )\; d\tau _\eta d\mu \nonumber \\&\quad = \int _{{-z^*}^{1-\eta }}^0 \rho _j^{(2)}(z^*,\tau _\eta )F_j(z^*,\tau _\eta )\; d\tau _\eta \nonumber \\&\quad - \int _0^{z^*} \int _{-\mu ^{1-\eta }}^0 \rho _j^{(1,2)}(\mu ,\tau _\eta )F_j(\mu ,\tau _\eta )\; d\tau _\eta d\mu . \end{aligned}$$

(74)

We then consider the second and fourth elements on the right-hand side of (70) and, using once again integration by parts, find:

$$\begin{aligned}&\int _{z^*}^{+\infty }\int _{-\mu ^{1-\eta }}^{\tau _\eta ^{zj}(\mu )} \rho _j(\mu ,\tau _\eta ) f_j(\mu ,\tau _\eta )\; d\tau _\eta d\mu \nonumber \\&\quad = \int _{z^*}^{+\infty } \left[ \rho _j(\mu ,\tau _\eta )F_j(\tau _\eta |\mu ) \right] _{\tau _\eta =-\mu ^{1-\eta }}^{\tau _\eta =\tau _\eta ^{zj}(\mu )} f_j(\mu )\; d\mu \nonumber \\&\quad \quad - \int _{z^*}^{+\infty }\int _{-\mu ^{1-\eta }}^{\tau _\eta ^{zj}(\mu )} \rho _j^{(2)}(\mu ,\tau _\eta ) F_j(\tau _\eta |\mu )f_j(\mu )\; d\tau _\eta d\mu \nonumber \\&\quad =- \int _{z^*}^{+\infty }\int _{-\mu ^{1-\eta }}^{\tau _\eta ^{zj}(\mu )} \rho _j^{(2)}(\mu ,\tau _\eta ) F_j(\tau _\eta |\mu )f_j(\mu )\; d\tau _\eta d\mu , \end{aligned}$$

(75)

as $\rho _j\big (\mu ,\tau _\eta ^{zj}(\mu )\big )=0$ and $F_j(\tau _\eta =-\mu ^{1-\eta }|\mu )=0$.

Let $R_j(\mu ) = \int _{-\mu ^{1-\eta }}^{\tau _\eta ^{zj}(\mu )} \rho _j^{(2)}(\mu ,\tau _\eta )F_j(\mu ,\tau _\eta )\; d\tau _\eta$. We have:

$$\begin{aligned} \frac{\partial R_j}{\partial \mu }&= {\tau _\eta ^{zj}}'(\mu )\rho _j^{(2)} \big (\mu ,\tau _\eta ^{zj}(\mu )\big )F_j(\mu ,\tau _\eta ^{zj}(\mu )\big )\nonumber \\&\quad +(1-\eta )\mu ^{-\eta }\rho _j^{(2)}(\mu ,-\mu ^{1-\eta })F_j(\mu ,-\mu ^{1-\eta })\nonumber \\&\quad + \int _{-\mu ^{1-\eta }}^{\tau _\eta ^{zj}(\mu )} \rho _j^{(1,2)}(\mu ,\tau _\eta )F_j(\mu ,\tau _\eta )\; d\tau _\eta + \int _{-\mu ^{1-\eta }}^{\tau _\eta ^{zj}(\mu )} \rho _j^{(2)}(\mu ,\tau _\eta ) F_j(\tau _\eta |\mu ) f_j(\mu )\; d\tau _\eta , \nonumber \\&= {\tau _\eta ^{zj}}'(\mu )\rho _j^{(2)}\big (\mu ,\tau _\eta ^{zj}(\mu ) \big )F_j(\mu ,\tau _\eta ^{zj}(\mu )\big ) \nonumber \\&\quad \!+\! \int _{-\mu ^{1-\eta }}^{\tau _\eta ^{zj}(\mu )} \rho _j^{(1,2)}(\mu ,\tau _\eta )F_j(\mu ,\tau _\eta )\; d\tau _\eta \!+\! \int _{-\mu ^{1-\eta }}^{\tau _\eta ^{zj}(\mu )} \rho _j^{(2)}(\mu ,\tau _\eta )F_j(\tau _\eta |\mu ) f_j(\mu )\; d\tau _\eta . \end{aligned}$$

(76)

Integrating that expression along $\mu$ and over $[z^*,+\infty ]$ and rearranging, we have:

$$\begin{aligned}&\int _{z^*}^{+\infty } \int _{-\mu ^{1-\eta }}^{\tau _\eta ^{zj}(\mu )} \rho _j^{(2)}(\mu ,\tau _\eta )F_j(\tau _\eta |\mu ) f_j(\mu )\; d\tau _\eta d\mu \nonumber \\&= \left[ R_j(\mu )\right] _{z^*}^{+\infty } - \int _{z^*}^{+\infty } {\tau _\eta ^{zj}}'(\mu )\rho _j^{(2)}\big (\mu ,\tau _\eta ^{zj}(\mu )\big ) F_j(\mu ,\tau _\eta ^{zj}(\mu )\big )\; d\mu \nonumber \\&\quad - \int _{z^*}^{+\infty } \int _{-\mu ^{1-\eta }}^{\tau _\eta ^{zj}(\mu )} \rho _j^{(1,2)}(\mu ,\tau _\eta )F_j(\mu ,\tau _\eta )\; d\tau _\eta d\mu \nonumber \\&= \int _{-\infty ^{1-\eta }}^{\tau _\eta ^{zj}(+\infty )} \rho _j^{(2)}(+\infty ,\tau _\eta )F_j(+\infty ,\tau _\eta )\; d\tau _\eta - \int _{-{z^*}^{1-\eta }}^{0} \rho _j^{(2)}(z^*,\tau _\eta )F_j(z^*,\tau _\eta )\; d\tau _\eta \nonumber \\&\quad - \int _{z^*}^{+\infty }{\tau _\eta ^{zj}}'(\mu )\rho _j^{(2)}\big (\mu , \tau _\eta ^{zj}(\mu )\big )F_j(\mu ,\tau _\eta ^{zj}(\mu )\big )\; d\mu \nonumber \\&\quad - \int _{z^*}^{+\infty } \int _{-\mu ^{1-\eta }}^{\tau _\eta ^{zj}(\mu )} \rho _j^{(1,2)}(\mu ,\tau _\eta )F_j(\mu ,\tau _\eta )\; d\tau _\eta d\mu . \end{aligned}$$

(77)

Using (72), (74), (77), and $\rho _1(\mu ,0)=\rho _2(\mu ,0)\, \forall \mu$ , we finally obtain the following expression for $P(\lambda )$:

$$\begin{aligned} \tilde{P}(\tilde{\lambda })&= - \int _0^{z^*} \rho _2^{(1)}(\mu ,0)F(\mu ) \; d\mu - q \int _{-\infty ^{1-\eta }}^{\tau _\eta ^{z1}(+\infty )} \rho _1^{(2)}(+\infty ,\tau _\eta )F_1(+\infty ,\tau _\eta )\; d\tau _\eta \\&\quad + q\int _{z^*}^{+\infty }{\tau _\eta ^{z1}}'(\mu )\rho _1^{(2)}\big (\mu , \tau _\eta ^{z1}(\mu )\big )F_1(\mu ,\tau _\eta ^{z1}(\mu )\big )\; d\mu \nonumber \\&\quad + q\int _0^{z^*} \int _{-\mu ^{1-\eta }}^0 \rho _1^{(1,2)}(\mu ,\tau _\eta )F_1(\mu ,\tau _\eta )\; d\tau _\eta d\mu \nonumber \\&\quad + q\int _{z^*}^{+\infty } \int _{-\mu ^{1-\eta }}^{\tau _\eta ^{z1}(\mu )} \rho _1^{(1,2)}(\mu ,\tau _\eta )F_1(\mu ,\tau _\eta )\; d\tau _\eta d\mu \nonumber \\&\quad - (1- q)\int _{-\infty ^{1-\eta }}^{\tau _\eta ^{z2}(+\infty )} \rho _2^{(2)}(+\infty ,\tau _\eta )F_2(+\infty ,\tau _\eta )\; d\tau _\eta \nonumber \\&\quad + (1- q)\int _{z^*}^{+\infty }{\tau _\eta ^{z2}}'(\mu )\rho _2^{(2)} \big (\mu ,\tau _\eta ^{z2}(\mu )\big )F_2(\mu ,\tau _\eta ^{z2}(\mu )\big )\; d\mu \nonumber \\&\quad + (1- q)\int _0^{z^*} \int _{-\mu ^{1-\eta }}^0 \rho _2^{(1,2)}(\mu ,\tau _\eta )F_2(\mu ,\tau _\eta )\; d\tau _\eta d\mu \nonumber \\&\quad + (1-q)\int _{z^*}^{+\infty } \int _{-\mu ^{1-\eta }}^{\tau _\eta ^{z2}(\mu )} \rho _2^{(1,2)}(\mu ,\tau _\eta )F_2(\mu ,\tau _\eta )\; d\tau _\eta d\mu . \nonumber \end{aligned}$$

(78)

Proposition 4 then follows directly from  (78) by inspection.

1.1 Proof of Proposition 6

Letting $\rho _1(\mu ,\tau _\eta ) = \rho _2(\mu ,\tau _\eta )$ $\forall (\mu ,\tau _\eta )\in \varGamma _S(\tilde{\lambda })$, it follows that $\forall (\mu ,\tau _\eta )\in \varGamma _S(\tilde{\lambda })$:

$$\begin{aligned} \rho _1^{(2)}(\mu ,\tau _\eta )&= \rho _2^{(2)}(\mu ,\tau _\eta ), \end{aligned}$$

(79)

$$\begin{aligned} \rho _1^{(1,2)}(\mu ,\tau _\eta )&= \rho _2^{(1,2)}(\mu ,\tau _\eta ). \end{aligned}$$

(80)

Moreover, $\tau _\eta ^{z1}(\mu )=\tau _\eta ^{z2}(\mu )$, so that ${\tau _\eta ^{z1}}'(\mu )={\tau _\eta ^{z2}}'(\mu )$. Equation (78) can then be rewritten as:

$$\begin{aligned} \tilde{P}(\tilde{\lambda })&= - \int _0^{z^*} \rho _1^{(1)}(\mu ,0)F(\mu ) \; d\mu - \int _{-\infty ^{1-\eta }}^{\tau _\eta ^{z1}(+\infty )} \rho _1^{(2)}(+\infty ,\tau _\eta )F(+\infty ,\tau _\eta )\; d\tau _\eta \nonumber \\&\quad + \int _{z^*}^{+\infty }{\tau _\eta ^{z1}}'(\mu )\rho _1^{(2)} \big (\mu ,\tau _\eta ^{z1}(\mu )\big )F(\mu ,\tau _\eta ^{z1}(\mu )\big )\; d\mu \nonumber \\&\quad + \int _0^{z^*} \int _{-\mu ^{1-\eta }}^0 \rho _1^{(1,2)}(\mu ,\tau _\eta )F(\mu ,\tau _\eta )\; d\tau _\eta d\mu \nonumber \\&\quad + \int _{z^*}^{+\infty } \int _{-\mu ^{1-\eta }}^{\tau _\eta ^z(\mu )} \rho _1^{(1,2)}(\mu ,\tau _\eta )F(\mu ,\tau _\eta )\; d\tau _\eta d\mu . \end{aligned}$$

(81)

The rest of the proof follows by inspection.

1.2 Proof of Proposition 7

With asymmetry, it is assumed that $\tau _\eta ^{z1}(\mu ) \ge \tau _\eta ^{z2}(\mu )$ for all $\mu \in [z^*,+\infty )$. As a consequence, Eq. (46) can be rewritten as:

$$\begin{aligned} \tilde{P}(\tilde{\lambda } )&= q\int _0^{z^*}\int _{-\mu ^{1-\eta }}^0 \rho _1(\mu ,\tau _\eta ,\tilde{\lambda }) f_1(\mu ,\tau _\eta )\; d\tau _\eta d\mu \nonumber \\&\quad + q\int _{z^*}^{+\infty }\int _{-\mu ^{1-\eta }}^{\tau _\eta ^{z1}(\mu )} \rho _1(\mu ,\tau _\eta ,\tilde{\lambda }) f_1(\mu ,\tau _\eta )\; d\tau _\eta d\mu \\&\quad + (1-q)\int _0^{z^*}\int _{-\mu ^{1-\eta }}^0 \rho _2(\mu ,\tau _\eta ,\tilde{\lambda }) f_2(\mu ,\tau _\eta )\; d\tau _\eta d\mu \nonumber \\&\quad + (1-q)\int _{z^*}^{+\infty }\int _{-\mu ^{1-\eta }}^{\tau _\eta ^{z1}(\mu )} \rho _2(\mu ,\tau _\eta ,\tilde{\lambda }) f_2(\mu ,\tau _\eta )\; d\tau _\eta d\mu . \nonumber \end{aligned}$$

(82)

Noting that $\rho _1^{(2)}(\mu ,\tau _\eta )-\rho _2^{(2)}(\mu ,\tau _\eta ) \le 0$ and $\rho _1^{(1,2)}(\mu ,\tau _\eta )-\rho _2^{(1,2)}(\mu ,\tau _\eta )\ge 0$, we obtain:

$$\begin{aligned} \tilde{P}(\tilde{\lambda })&= - \int _0^{z^*} \rho _2^{(1)}(\mu ,0)F(\mu ) \; d\mu - \int _{-\infty ^{1-\eta }}^{\tau _\eta ^{z1}(+\infty )} \rho _2^{(2)}(+\infty ,\tau _\eta )F(+\infty ,\tau _\eta )\; d\tau _\eta \\&\quad - q\int _{-\infty ^{1-\eta }}^{\tau _\eta ^{z1}(+\infty )} \left( \rho _1^{(2)}(+\infty ,\tau _\eta )-\rho _2^{(2)}(+\infty ,\tau _\eta ) \right) F_1(+\infty ,\tau _\eta )\; d\tau _\eta \nonumber \\&\quad + \int _{z^*}^{+\infty }{\tau _\eta ^{z1}}'(\mu )\rho _2^{(2)}\big (\mu , \tau _\eta ^{z1}(\mu )\big )F(\mu ,\tau _\eta ^{z1}(\mu )\big )\; d\mu \nonumber \\&\quad + q\int _{z^*}^{+\infty }{\tau _\eta ^{z1}}'(\mu )\left( \rho _1^{(2)} \big (\mu ,\tau _\eta ^{z1}(\mu )\big )-\rho _2^{(2)}\big (\mu ,\tau _\eta ^{z1}(\mu ) \big )\right) F_1(\mu ,\tau _\eta ^{z1}(\mu )\big )\; d\mu \nonumber \\&\quad + \int _0^{z^*} \int _{-\mu ^{1-\eta }}^0 \rho _2^{(1,2)}(\mu ,\tau _\eta )F(\mu ,\tau _\eta )\; d\tau _\eta d\mu \nonumber \\&\quad + \int _0^{z^*} \int _{-\mu ^{1-\eta }}^0 \left( \rho _1^{(1,2)}(\mu ,\tau _\eta ) -\rho _2^{(1,2)}(\mu ,\tau _\eta )\right) F_1(\mu ,\tau _\eta )\; d\tau _\eta d\mu \nonumber \\&\quad + \int _{z^*}^{+\infty } \int _{-\mu ^{1-\eta }}^{\tau _\eta ^{z1}(\mu )} \rho _2^{(1,2)}(\mu ,\tau _\eta )F(\mu ,\tau _\eta )\; d\tau _\eta d\mu \nonumber \\&\quad + q\int _{z^*}^{+\infty } \int _{-\mu ^{1-\eta }}^{\tau _\eta ^{z1}(\mu )} \left( \rho _1^{(1,2)}(\mu ,\tau _\eta )-\rho _2^{(1,2)}(\mu ,\tau _\eta )\right) F_1(\mu , \tau _\eta )\; d\tau _\eta d\mu . \nonumber \end{aligned}$$

(83)

The rest of the proof follows by inspection.

Appendix 3: intersection of the different classes of poverty measures

1.1 The derivations of additional restrictions on the individual poverty measure $\pi$ and $\rho$

We first consider the conditions that $\rho$ must obey so that members of $\ddot{\varPi }(\lambda ^+)$ are also members of $\tilde{\varPi }(\tilde{\lambda }^+)$. First, we have $\pi ^{(x_i)}(x_1,x_2)\le 0$,$\forall i=1,2$, so that we should also observe $\rho _t^{(x_i)}(\mu ,\tau _\eta )\le 0$ $\forall t$. When $x_i$ is not the lowest income, an income increment increases variability ($|\tau _\eta |$ rises) while increasing mean income, so that the net effect of this is a priori not known. Assuming $x_1$ to be the lowest income, if that net effect is supposed to correspond to a decrease in the level of poverty, we should have:

$$\begin{aligned}&\rho _1^{(x_2)}(\mu ,\tau _\eta ) = \rho _1^{(1)}(\mu ,\tau _\eta )\mu ^{(x_2)} + \rho _1^{(2)}(\mu ,\tau _\eta )\tau ^{(x_2)}_\eta \le 0 \end{aligned}$$

(84)

$$\begin{aligned}&\Rightarrow \frac{1}{2}\rho _1^{(1)}(\mu ,\tau _\eta ) +\frac{1}{2}\mu ^{-\eta -1}\left( \frac{\eta }{2}(x_2-x_1) -\mu \right) \rho _1^{(2)}(\mu ,\tau _\eta ) \le 0 \end{aligned}$$

(85)

$$\begin{aligned}&\Rightarrow \frac{1}{2}\rho _1^{(1)}(\mu ,\tau _\eta ) - \frac{1}{2}\left( \mu ^{-\eta }+\frac{\eta \tau _\eta }{\mu }\right) \rho _1^{(2)}(\mu ,\tau _\eta ) \le 0 \end{aligned}$$

(86)

$$\begin{aligned}&\Rightarrow \rho _1^{(1)}(\mu ,\tau _\eta ) \le \left( \mu ^{-\eta }+\frac{\eta \tau _\eta }{\mu }\right) \rho _1^{(2)}(\mu ,\tau _\eta ). \end{aligned}$$

(87)

In the same manner, it would also be necessary to observe $\rho ^{(x_1,x_2)}(\mu ,\tau _\eta )\ge 0$. Still supposing $x_1$ to be the lower income, we have:

$$\begin{aligned} \rho _1^{(x_1,x_2)}(\mu ,\tau _\eta )&= \frac{1}{4}\rho _1^{(1,1)}(\mu ,\tau _\eta ) +\frac{1}{4}\mu ^{-\eta -1}\left( \frac{\eta }{2}(x_2-x_1)+\mu \right) \rho _1^{(1,2)}(\mu ,\tau _\eta ) \nonumber \\&\quad - \frac{\eta (\eta +1)}{2}2^{\eta -1}(x_1+x_2)^{-\eta -2}(x_2-x_1)\rho _1^{(2)}(\mu ,\tau _\eta ) \nonumber \\&\quad +\frac{1}{2}\mu ^{-\eta -1}\left( \frac{\eta }{2}(x_2-x_1)-\mu \right) \left( \frac{1}{2}\rho _1^{(1,2)}(\mu ,\tau _\eta )\right. \nonumber \\&\qquad \left. +\frac{1}{2}\mu ^{-\eta -1}\left( \frac{\eta }{2}(x_2-x_1)+\mu \right) \rho _1^{(2,2)}(\mu ,\tau _\eta )\right) \end{aligned}$$

(88)

$$\begin{aligned}&=\frac{1}{4}\rho _1^{(1,1)}(\mu ,\tau _\eta ) +\frac{\eta }{4}\mu ^{-\eta -1}(x_2-x_1)\rho _1^{(1,2)}(\mu ,\tau _\eta ) \nonumber \\&\quad - \frac{\eta (\eta +1)}{2}2^{\eta -1}(x_1+x_2)^{-\eta -2}(x_2-x_1)\rho _1^{(2)}(\mu ,\tau _\eta ) \nonumber \\&\quad +\frac{1}{4}\mu ^{-2(\eta +1)}\left( \frac{\eta ^2}{4}(x_2-x_1)^2-\mu ^2\right) \rho _1^{(2,2)}(\mu ,\tau _\eta ) \end{aligned}$$

(89)

$$\begin{aligned}&=\frac{1}{4}\rho _1^{(1,1)}(\mu ,\tau _\eta ) - \frac{1}{2}\frac{\eta \tau _\eta }{\mu }\rho _1^{(1,2)}(\mu ,\tau _\eta ) - \frac{\eta (\eta +1)\tau _\eta }{8\mu ^2}\rho _1^{(2)}(\mu ,\tau _\eta ) \nonumber \\&\quad + \frac{1}{4}\left( \left( \frac{\eta \tau _\eta }{\mu }\right) ^2-\mu ^{-2\eta }\right) \rho _1^{(2,2)}(\mu ,\tau _\eta ). \end{aligned}$$

(90)

Considering the case of absolute variability aversion ($\eta =0$), $\rho _t$ must exhibit the following two properties:

$$\begin{aligned} {\left\{ \begin{array}{ll} \rho _t^{(1)}(\mu ,\tau _0) \le \rho _t^{(2)}(\mu ,\tau _0), \\ \rho _t^{(1,1)}(\mu ,\tau _0) \ge \rho _t^{(2,2)}(\mu ,\tau _0). \end{array}\right. } \end{aligned}$$

(91)

With relative variability aversion ($\eta =1$), the conditions become:

$$\begin{aligned} {\left\{ \begin{array}{ll} \rho _t^{(1)}(\mu ,\tau _1) \le \frac{1+\tau _1}{\mu }\rho _t^{(2)}(\mu ,\tau _1), \\ \rho _t^{(1,1)}(\mu ,\tau _1) \ge \frac{2\tau _1}{\mu }\rho _t^{(1,2)}(\mu ,\tau _1) + \frac{\tau _1}{\mu ^2}\rho _t^{(2)}(\mu ,\tau _1) + \frac{1-\tau _1^2}{\mu ^2}\rho _t^{(2,2)}(\mu ,\tau _1). \end{array}\right. } \end{aligned}$$

(92)

It can also be shown that $x_1=\mu +\tau _\eta \mu ^\eta$ and $x_2=\mu -\tau _\eta \mu ^\eta$ if $x_1<x_2$. It is then possible to compute $\pi ^{(\mu )},\, \pi ^{(\tau _\eta )}$ and $\pi ^{(\mu ,\tau _\eta )}$ to see what conditions have to be met so that $\pi$ respects the conditions imposed on $\rho$. First, considering the derivatives of $\pi$ with respect to mean income, we should observe:

$$\begin{aligned} \pi ^{(\mu )}(x_1,x_2)= \pi ^{(1)}(x_1,x_2)x_1^{(\mu )} + \pi ^{(2)}(x_1,x_2)x_2^{(\mu )}&\le 0 \end{aligned}$$

(93)

$$\begin{aligned} \Rightarrow (1+\eta \tau _\eta \mu ^{\eta -1})\pi ^{(1)}(x_1,x_2) + (1-\eta \tau _\eta \mu ^{\eta -1})\pi ^{(2)}(x_1,x_2)&\le 0. \end{aligned}$$

(94)

That condition is always fulfilled since $1+\eta \tau _\eta \mu ^{\eta -1}$ and $1-\eta \tau _\eta \mu ^{\eta -1}$ are positive for $\eta \in [0,1]$, and $\pi ^{(1)}(x_1,x_2)$ and $\pi ^{(2)}(x_1,x_2)$ are also non-negative. The result is intuitive. Increasing the mean without altering variability implies increasing income at both periods, so that poverty should logically fall.

Considering now a decrease in variability without a change in mean income, things are less clear since such a change raises the lower income but decreases the higher one. It is then necessary to consider the net sum of those opposite effects. Since $\rho _t^{(2)}(\mu ,\tau _\eta )\le 0$, we should obtain:

$$\begin{aligned}&\pi ^{(\tau _\eta )}(x_1,x_2) = \mu ^\eta \pi ^{(1)}(x_1,x_2) - \mu ^\eta \pi ^{(2)}(x_1,x_2) \le 0 \end{aligned}$$

(95)

$$\begin{aligned}&\Rightarrow \pi ^{(1)}(x_1,x_2) \le \pi ^{(2)}(x_1,x_2). \end{aligned}$$

(96)

Finally, $\pi$ has to be defined so as to respect $\pi ^{(\mu ,\tau _\eta )}(x_1,x_2)\ge 0$. We have:

$$\begin{aligned}&\pi ^{(\mu ,\tau _\eta )}(x_1,x_2)\nonumber \\&\quad = \eta \mu ^{\eta -1}\pi ^{(1)}(x_1,x_2) + \big (1+\eta \tau _\eta \mu ^{\eta -1}\big )\left( \mu ^\eta \pi ^{(1,1)}(x_1,x_2) -\mu ^\eta \pi ^{(1,2)}(x_1,x_2)\right) \nonumber \\&\quad \quad -\eta \mu ^{\eta -1}\pi ^{(2)}(x_1,x_2) + \big (1-\eta \tau _\eta \mu ^{\eta -1}\big ) \left( \mu ^\eta \pi ^{(1,2)}(x_1,x_2)-\mu ^\eta \pi ^{(2,2)}(x_1,x_2)\right) \end{aligned}$$

(97)

$$\begin{aligned}&\quad = \eta \mu ^{\eta -1}\left( \pi ^{(1)}(x_1,x_2)-\pi ^{(2)}(x_1,x_2)\right) \nonumber \\&\qquad + \mu ^\eta \big (1+\eta \tau _\eta \mu ^{\eta -1}\big )\left( \pi ^{(1,1)}(x_1,x_2) -\pi ^{(1,2)}(x_1,x_2)\right) \nonumber \\&\quad \quad +\mu ^\eta \big (1-\eta \tau _\eta \mu ^{\eta -1}\big )\left( \pi ^{(1,2)}(x_1,x_2) -\pi ^{(2,2)}(x_1,x_2)\right) . \end{aligned}$$

(98)

With absolute variability aversion ($\eta =0$), $\pi$ should be such that:

$$\begin{aligned} {\left\{ \begin{array}{ll} \pi ^{(1)}(x_1,x_2) \le \pi ^{(2)}(x_1,x_2), \\ \pi ^{(1,1)}(x_1,x_2) \ge \pi ^{(2,2)}(x_1,x_2). \end{array}\right. } \end{aligned}$$

(99)

With relative variability aversion ($\eta =1$), we obtain, for $x_1<x_2$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \pi ^{(1)}(x_1,x_2) \le \pi ^{(2)}(x_1,x_2), \\ \pi ^{(1)}(x_1,x_2)-\pi ^{(2)}(x_1,x_2) + \mu (1+\tau _1)\left( \pi ^{(1,1)}(x_1,x_2)-\pi ^{(1,2)}(x_1,x_2)\right) \\ +\mu (1-\tau _1)\left( \pi ^{(1,2)}(x_1,x_2)-\pi ^{(2,2)}(x_1,x_2)\right) \ge 0. \end{array}\right. } \end{aligned}$$

(100)

1.2 Proof of Proposition 8 and Corollary 1

We have shown that it is possible to impose restrictions on the derivatives of both $\pi (x_1,x_2)$ and $\rho (\mu ,\tau _\eta )$ to obtain measures that are included in both $\tilde{\varPi }(\tilde{\lambda }^+)$ and $\ddot{\varPi }(\lambda ^+)$. Since the class of poverty measures $\breve{\varPi }_\eta (\tilde{\lambda }^+)$ is not empty, any measure $P(\tilde{\lambda })\in \breve{\varPi }_\eta (\tilde{\lambda }^+)$ can equally be expressed using equation (8) or equation (19). Consequently, both  (59) and (78) are valid expressions for $P(\tilde{\lambda })$. For Proposition 8 not to hold, it would be necessary to show that one can find two distributions $A$ and $B$ such that $A\; \ddot{\succcurlyeq }_{\tilde{\lambda }^+} \;B$ and $B\tilde{\succcurlyeq }_{\eta ,\tilde{\lambda }^+}\; A$. However, with the restrictions imposed on the classes $\ddot{\varPi }(\lambda ^+)$ and $\tilde{\varPi }(\tilde{\lambda }^+)$, such a situation would imply that for any poverty measure in $\breve{\varPi }_\eta (\tilde{\lambda }^+)$, the difference $P_A(\tilde{\lambda }) - P_B(\tilde{\lambda })$ should simultaneously be non-negative and non-positive. This will happen if and only if $P^{1,1}_A(x_1,x_2) = P^{1,1}_B(x_1,x_2),\, \forall (x_1,x_2)\in \varGamma (\tilde{\lambda })$. This proves Proposition 8.

The demonstration of Corollary 1 is straightforward. As long as the class of poverty measures $\breve{\varPi }_\eta (\tilde{\lambda }^+)$ is not empty, observing dominance with respect to either $\ddot{\varPi }(\lambda ^+)$ or $\tilde{\varPi }(\tilde{\lambda }^+)$ precludes observing an opposite strong dominance relationship with the other class of poverty measures, as both classes include $\breve{\varPi }_\eta (\tilde{\lambda }^+)$.

Appendix 4: additional tables

See Tables 1, 2, 3.

Table 1 First-order dominance tests for intertemporal poverty indices (using income-defined indices)

Table 2 Second-order dominance tests for intertemporal poverty indices (using income-defined indices)

Table 3 First-order dominance tests with perfect intertemporal income pooling