Health inequality indices and exogenous risk factors: an illustration on Luxembourgish workers

Abstract

This paper proposes an aggregable family of multidimensional concentration indices which is characterized in order to be consistent with a property of exogenous risk factors, i.e. health risks for which agents are not responsible for. It is shown that those indices are of interest when individuals face different risk factors, whereas traditional indices fail to deal with heterogeneous agents. In this respect, necessary and suficient conditions are stated in order to rank two health distributions thanks to the generalized concentration curves. An illustration is performed using a sample of individuals living in Luxembourg aged 50 and older.

Appendix: Proof

Theorem 4.1 Under Definition 3.1, for all health achievement indices $A\left( \mathbf {H}\right) \in \Omega ^{s}$, where $s\in \left\{ 1,2,3,\ldots \right\}$, respecting (2) and (4), the two following statements are equivalent:

• (i) $A(\tilde{\mathbf {H}}) \geqslant A\left( \mathbf {H}\right)$

• (ii) $\sum _{i=1}^{l} \kappa _i \left[ \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{s}\left( p\right) - \mathrm{GAC}_{\mathbf {H},i}^{s}\left( p\right) \right] \geqslant 0, \ \forall l \in \{1,\ldots ,n\}, \ \forall p\in \left[ 0,1\right]$.

Proof

Sufficiency. $\square$

Note that:

$$\begin{aligned} A(\tilde{\mathbf {H}}) - A\left( \mathbf {H}\right)&= \sum _{i=1}^n\kappa _i \int _{0}^{1} v_i(p) \big [ \phi _i(\tilde{\mathbf {H}}(\mathbf {p})) - \phi _i(\mathbf {H}(\mathbf {p}))\big ] dp \\&= \sum _{i=1}^n\kappa _i \int _{0}^{1} v_i(p) \left[ \frac{K-\Upsilon _i(\tilde{\mathbf {H}}(\mathbf {p})) \Theta '_i}{K} - \frac{K-\Upsilon _i(\mathbf {H}(\mathbf {p})) \Theta '_i}{K}\right] \mathrm{d}p. \end{aligned}$$

Since $GAC_{\mathbf {H},i}^{1}\left( p\right) :=(K-\Upsilon _i(\mathbf {H}(\mathbf {p})) \Theta '_i)/K$,

$$\begin{aligned} A(\tilde{\mathbf {H}}) - A\left( \mathbf {H}\right) = \sum _{i=1}^n\kappa _i \int _{0}^{1} v_i(p) \big [ \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{1}\left( p\right) - \mathrm{GAC}_{\mathbf {H},i}^{1}\left( p\right) \big ] \mathrm{d}p. \end{aligned}$$

Integrating s times by parts $\int _{0}^{1} v_i(p) \mathrm{GAC}_{\mathbf {H},i}^{1}\left( p\right) dp$ such that $v_i^{(\ell )}(1)=0$ for all $\ell \in \{0,1,2,\ldots \}$ and $\mathrm{GAC}_{\mathbf {H},i}^{2}\left( 1\right) =0$ by construction, we get that:

$$\begin{aligned} \int _{0}^{1} v_i(p) \mathrm{GAC}_{\mathbf {H},i}^{1}\left( p\right) dp = \left( -1\right) ^{s-1}\int _{0}^{1} v_i^{\left( s-1\right) }\left( p\right) \mathrm{GAC}_{\mathbf {H},i}^{s}\left( p\right) \mathrm{d}p, \end{aligned}$$

and in the same manner,

$$\begin{aligned} \int _{0}^{1} v_i(p) \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{1}\left( p\right) \mathrm{d}p = \left( -1\right) ^{s-1}\int _{0}^{1} v_i^{\left( s-1\right) }\left( p\right) \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{s}\left( p\right) \mathrm{d}p. \end{aligned}$$

Hence,

$$\begin{aligned} A(\tilde{\mathbf {H}}) - A\left( \mathbf {H}\right) = \sum _{i=1}^n \kappa _i \left( -1\right) ^{s-1}\int _{0}^{1} v_i^{\left( s-1\right) }\left( p\right) \big [ \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{s}\left( p\right) - \mathrm{GAC}_{\mathbf {H},i}^{s}\left( p\right) \big ] \mathrm{d}p. \end{aligned}$$

(1)

Now, we use Abel’s Lemma usually employed in the literature of sequential dominance.

Abel’s lemma (see, e.g., [18]: Let $\{x_j\}_{j=1}^{n}$ and $\{y_i\}_{i=1}^{n}$ be sequences of real numbers. If $x_n \geqslant x_{n-1} \geqslant \cdots \geqslant x_1 \geqslant 0$, then $\sum _{i=j}^{n}y_i \geqslant 0$ $\forall j$ is a sufficient condition for $\sum _{i=1}^{n}x_iy_i \geqslant 0$. Contrary to this, if $x_n \leqslant x_{n-1} \leqslant \ldots \leqslant x_1 \leqslant 0$, then $\sum _{i=j}^{n}y_i \geqslant 0$ $\forall j$ is also a sufficient condition for $\sum _{i=1}^{n}x_iy_i \leqslant 0$.

Following Abel’s Lemma, $\sum _{i=1}^{l} \kappa _i \left[ \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{s}\left( p\right) - \mathrm{GAC}_{\mathbf {H},i}^{s}\left( p\right) \right] \geqslant 0$, for all $l \in \{1,\ldots ,n\}$ is a sufficient condition to get $A(\tilde{\mathbf {H}}) \geqslant A\left( \mathbf {H}\right)$.

Necessity. From Lemma 3.1, even if we relax (5), we can use polynomial functions such as:

$$\begin{aligned} v_i^{\left( s-2\right) }\left( p\right) =\left\{ \begin{array}{cc} \alpha _i\left( -1\right) ^{s-2}\epsilon &{} p\leqslant \overline{p} \\ \alpha _i\left( -1\right) ^{s-2}\left( \overline{p}+\epsilon -p\right) &{} \overline{p} <p\leqslant \overline{p}+\epsilon \\ 0 &{} p>\overline{p}+\epsilon \\ 0 &{} \text{ if } s = 1 \end{array} \right. , \end{aligned}$$

with $\alpha _1 \geqslant \cdots \geqslant \alpha _n \geqslant 0$ by Definition 3.1 of exogenous risk factors (to match condition (i) in the case where $s=2)$. By the differentiability assumption included in the set $\Omega ^s$, we get for all agent types $i=1,\ldots ,n$:

$$\begin{aligned} v_i^{\left( s-1\right) }\left( p\right) =\left\{ \begin{array}{cc} 0 &{} p \leqslant \overline{p} \\ \alpha _i\left( -1\right) ^{s-1} &{} \overline{p} < p \leqslant \overline{p}+\epsilon \\ 0 &{} p > \overline{p}+\epsilon \end{array} \right. . \end{aligned}$$

(2)

Suppose by contradiction that $\sum _{i=1}^{l}\kappa _i [\mathrm{GAC}_{\tilde{\mathbf {H}},i}^{s}\left( p\right) - \mathrm{GAC}_{\mathbf {H},i}^{s}\left( p\right) ] < 0$ for all $l=1,\ldots ,n$ on an interval $\left[ \overline{p},\overline{p}+\epsilon \right]$ for some $\epsilon$ close to 0. Thereby, substituting equation (2) into (1), according to $\alpha _1 \geqslant \cdots \geqslant \alpha _n \geqslant 0$, it is easy to show that $A(\tilde{\mathbf {H}}) - A\left( \mathbf {H}\right) < 0$, i.e., a contradiction.

Theorem 4.2

Under Definition 3.1, for all aggregable socioeconomic health inequality indices $I\left( \mathbf {H}\right) \in \Xi ^{s}$, with $s\in \left\{ 1,2,3,\ldots \right\}$, respecting (3) and (6), the two following statements are equivalent:

• (i) $I(\tilde{\mathbf {H}}) \leqslant I\left( \mathbf {H}\right)$

• (ii) $\sum _{i=1}^{l} \theta _i\left[ \mathrm{AC}_{\mathbf {H},i}^{s}\left( p\right) - \mathrm{AC}_{\tilde{\mathbf {H}},i}^{s}\left( p\right) \right] \leqslant 0, \ \forall l \in \{1,\ldots ,n\}, \ \forall p\in \left[ 0,1\right]$.

Proof

The proof goes along the line of Theorem 4.1:

$$\begin{aligned} I(\tilde{\mathbf {H}}) - I\left( \mathbf {H}\right)&= \sum _{i=1}^n \theta _i \int _{0}^{1} v_i\left( p\right) \left[ \left( 1-\frac{\phi _i(\tilde{\mathbf {H}}(\mathbf {p}))}{\mu _{\phi _i(\tilde{\mathbf {H}})}}\right) - \left( 1-\frac{\phi _i(\mathbf {H}(\mathbf {p}))}{\mu _{\phi _i(\mathbf {H})}}\right) \right] \mathrm{d}p\\&= \sum _{i=1}^n \theta _i \int _{0}^{1} v_i\left( p\right) \left[ -\left( \frac{K-\Upsilon _i(\tilde{\mathbf {H}}(\mathbf {p})) \Theta '_i}{K\mu _{\phi _i(\tilde{\mathbf {H}})}}\right) + \left( \frac{K-\Upsilon _i(\mathbf {H}(\mathbf {p})) \Theta '_i}{K\mu _{\phi _i(\mathbf {H})}}\right) \right] \mathrm{d}p \\&= \sum _{i=1}^n \theta _i \int _{0}^{1} v_i\left( p\right) \left[ \mathrm{AC}_{\mathbf {H},i}^{1}(p) - \mathrm{AC}_{\tilde{\mathbf {H}},i}^{1}(p) \right] \mathrm{d}p. \end{aligned}$$

Integrating successively by parts the previous expression yields:

$$\begin{aligned} I(\tilde{\mathbf {H}}) - I\left( \mathbf {H}\right) = \sum _{i=1}^n \theta _i (-1)^{s-1}\int _{0}^{1} v_i^{(s-1)}\left( p\right) \frac{1}{K}\left[ \mathrm{AC}_{\mathbf {H},i}^{s}(p) - \mathrm{AC}_{\tilde{\mathbf {H}},i}^{s}(p)\right] \mathrm{d}p . \end{aligned}$$

(3)

The remainder of the proof relies on that of Theorem 4.1. $\square$

Theorem 5.1 Under Definition 3.1, for all aggregable socioeconomic health inequality indices $\mathrm{GC}(\mathbf {H}) \in \mathcal {M}$ respecting (MIR) but not (SYM), such that $w_i(p,s)=\alpha _i-\alpha _i s(1-p)^{s-1}=\alpha _i-v_i(p)$ and $s\in \left\{ 1,2,3,\ldots \right\}$, sufficient conditions for $\mathrm{GC}(\tilde{\mathbf {H}}) \leqslant \mathrm{GC}\left( \mathbf {H}\right)$ are:

$$\begin{aligned}&\sum _{i=1}^{l} \theta _i\left[ \mathrm{GAC}_{\mathbf {H},i}^{s}\left( p\right) - \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{s}\left( p\right) \right] \leqslant 0, \ \forall l \in \{1,\ldots ,n\}, \ \forall p\in \left[ 0,1\right] , \end{aligned}$$

and in addition, if $s\geqslant 4$,

$$\begin{aligned} \sum _{i=1}^l \theta _i \big [ \mathrm{GAC}_{\mathbf {H},i}^{u}(1) - \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{u}(1)\big ] \leqslant 0, \ \forall u \in \{3,4,\ldots ,s-1\}, \ \forall l\in \{1,\ldots ,n\}. \end{aligned}$$

Proof

Setting for short $w_i(p):=\alpha _i-\alpha _i s(1-p)^{s-1}$ yields:

$$\begin{aligned} \mathrm{GC}(\tilde{\mathbf {H}}) - \mathrm{GC}\left( \mathbf {H}\right)&= \sum _{i=1}^{n} \theta _i \frac{s^{s/(s-1)}}{s-1}\int _{0}^{1} \alpha _i\left( 1- s(1-p)^{s-1}\right) \left[ \phi _i(\tilde{\mathbf {H}}(\mathbf {p})) - \phi _i(\mathbf {H}(\mathbf {p})) \right] \mathrm{d}p\\&= \sum _{i=1}^n \theta _i \frac{s^{s/(s-1)}}{s-1} \int _{0}^{1} w_i\left( p\right) \left[ \frac{K-\Upsilon _i(\tilde{\mathbf {H}}(\mathbf {p})) \Theta '_i}{K} - \frac{K-\Upsilon _i(\mathbf {H}(\mathbf {p})) \Theta '_i}{K}\right] \mathrm{d}p \\&= \sum _{i=1}^n \theta _i \frac{s^{s/(s-1)}}{s-1} \int _{0}^{1} w_i\left( p\right) \big [ \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{1}\left( p\right) - \mathrm{GAC}_{\mathbf {H},i}^{1}\left( p\right) \big ]\mathrm{d}p. \end{aligned}$$

Integrating by parts $w_i(p) \mathrm{GAC}_{\mathbf {H},i}^{1}\left( p\right)$ provides:

$$\begin{aligned} \int _{0}^{1} w_i(p) \mathrm{GAC}_{\mathbf {H},i}^{1}(p) \mathrm{d}p&= \left| w_i(p)\mathrm{GAC}_{\mathbf {H},i}^{2}(p)\right| _{0}^{1} - \int _{0}^{1} w^{(1)}(p) \mathrm{GAC}_{\mathbf {H},i}^{2}(p) \mathrm{d}p \\&= w_i^{(0)}(1)\mathrm{GAC}_{\mathbf {H},i}^{2}(1) - \int _{0}^{1} w^{(1)}(p) \mathrm{GAC}_{\mathbf {H},i}^{2}(p) \mathrm{d}p. \end{aligned}$$

After integrating successively by parts, since $\mathrm{GAC}_{\mathbf {H},i}^{s}(0) =0$ for all $s \in \{1,2,3,\ldots \}$, we obtain:

$$\begin{aligned} \int _{0}^{1} w_i(p) \mathrm{GAC}_{\mathbf {H},i}^{1}(p) \mathrm{d}p = \sum _{u=2}^s (-1)^{u-2} w_i^{(u-2)}(1)\mathrm{GAC}_{\mathbf {H},i}^{u}(1) + \int _{0}^{1} (-1)^{s-1} w_i\left( p\right) ^{s-1} \mathrm{GAC}_{\mathbf {H},i}^{s}\left( p\right) . \end{aligned}$$

By definition $\mathrm{GAC}_{\mathbf {H},i}^{2}(1)=1$ for all $i=1,\ldots ,n$, hence:

$$\begin{aligned} \mathrm{GC}(\tilde{\mathbf {H}}) - \mathrm{GC}\left( \mathbf {H}\right) =&\sum _{i=1}^n \theta _i \frac{s^{s/(s-1)}}{s-1} \int _{0}^{1} (-1)^{s-1} w_i\left( p\right) ^{s-1}\big [ \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{s}\left( p\right) - \mathrm{GAC}_{\mathbf {H},i}^{s}\left( p\right) \big ]dp \\&+ \sum _{u=3}^{s} \sum _{i=1}^{n} \theta _i \frac{s^{s/(s-1)}}{s-1} (-1)^{u-2} w_i^{(u-2)}(1)\big [ \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{u}(1) - \mathrm{GAC}_{\mathbf {H},i}^{u}(1) \big ]. \end{aligned}$$

By Definition 3.1, $\alpha _1 \geqslant \cdots \geqslant \alpha _n$ ensures vertical equity. By virtue of horizontal equity, we have $(-1)^{s-1}w_1(p)^{s-1} \leqslant \cdots \leqslant (-1)^{s-1}w_n(p)^{s-1} \leqslant 0$ for all $p\in [0,1]$. By Abel’s lemma, a sufficient condition for the negativity of the first sum in the equation above is $\sum _{i=1}^l \theta _i \big [ \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{s}\left( p\right) - \mathrm{GAC}_{\mathbf {H},i}^{s}\left( p\right) \big ] \geqslant 0$ for all $l\in \{1,\ldots ,n\}$. In addition, for the negativity of the double sum in the equation above, Abel’s lemma yields $\sum _{i=1}^l \theta _i \big [ \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{u}(1) - \mathrm{GAC}_{\mathbf {H},i}^{u}(1) \big ] \geqslant 0$ for all $u \in \{3,4,\ldots ,s\}$ and for all $l\in \{1,\ldots ,n\}$. Note that, for $s=3$, the condition $\sum _{i=1}^l \theta _i \big [ \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{u}(1) - GAC_{\mathbf {H},i}^{u}(1) \big ] \geqslant 0$ does not have to be checked, since it is included in $\sum _{i=1}^l \theta _i \big [ \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{s}\left( p\right) - \mathrm{GAC}_{\mathbf {H},i}^{s}\left( p\right) \big ] \geqslant 0$ where $p\in [0,1]$. $\square$

Theorem 5.2 Under vertical equity of Definition 3.1, for all aggregable socioeconomic health inequality indices $GS\left( \mathbf {H}\right) \in \mathcal {M}$ respecting (MIR) and (SYM), such that $w_i(p,s)=v_i(p)=\alpha _i\left( p-\frac{1}{2}\right) ^{s-1}$, sufficient conditions for $GS(\tilde{\mathbf {H}}) \leqslant GS\left( \mathbf {H}\right)$ are:

$$\begin{aligned} \sum _{i=1}^{l} \theta _i\left[ GAC_{\mathbf {H},i}^{s}\left( p\right) - GAC_{\tilde{\mathbf {H}},i}^{s}\left( p\right) \right] \leqslant 0, \quad \forall l \in \{1,\ldots ,n\}, \ \forall p\in \left[ 0,1\right] , \ s =2. \end{aligned}$$

Proof

Setting $v_i(p)=\alpha _i\left( p-\frac{1}{2}\right) ^{s-1}$, we get that:

$$\begin{aligned} \mathrm{GS}(\tilde{\mathbf {H}}) - \mathrm{GS}\left( \mathbf {H}\right)&= \sum _{i=1}^n \theta _i \int _{0}^{1} v_i\left( p\right) \left[ \phi _i(\tilde{\mathbf {H}}(\mathbf {p})) - \phi _i(\mathbf {H}(\mathbf {p})) \right] dp \\&= \sum _{i=1}^n\theta _i \int _{0}^{1} v_i(p) \left[ \frac{K-\Upsilon _i(\tilde{\mathbf {H}}(\mathbf {p})) \Theta '_i}{K} - \frac{K-\Upsilon _i(\mathbf {H}(\mathbf {p})) \Theta '_i}{K}\right] dp. \\&=\sum _{i=1}^n\theta _i \int _{0}^{1} v_i(p) \big [ \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{1}\left( p\right) - \mathrm{GAC}_{\mathbf {H},i}^{1}\left( p\right) \big ] dp. \end{aligned}$$

As in Theorem 5.1, integrating successively by parts, we obtain:

$$\begin{aligned} \int _{0}^{1} v_i(p) \mathrm{GAC}_{\mathbf {H},i}^{1}(p) \mathrm{d}p = \sum _{u=2}^s (-1)^{u-2} v_i^{(u-2)}(1)GAC_{\mathbf {H},i}^{u}(1) + \int _{0}^{1} (-1)^{s-1} v_i\left( p\right) ^{s-1} GAC_{\mathbf {H},i}^{s}\left( p\right) . \end{aligned}$$

This entails:

$$\begin{aligned} \mathrm{GS}(\tilde{\mathbf {H}}) - \mathrm{GS}\left( \mathbf {H}\right) =&\sum _{i=1}^n \theta _i \left( -1\right) ^{s-1}\int _{0}^{1} v_i^{\left( s-1\right) }\left( p\right) \big [ \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{s}\left( p\right) - \mathrm{GAC}_{\mathbf {H},i}^{s}\left( p\right) \big ] \mathrm{d}p \end{aligned}$$

(4)

$$\begin{aligned}&+ \sum _{u=3}^{s} \sum _{i=1}^{n} \theta _i (-1)^{u-2}v_i^{(u-2)}(1) \big [ \mathrm{GAC}_{\tilde{\mathbf {H}},i}^{u}(1) - \mathrm{GAC}_{\mathbf {H},i}^{u}(1) \big ]. \end{aligned}$$

(5)

Note that for $1 \leqslant k \leqslant s-1$ and $s\in \left\{ 2,4,6,\ldots \right\}$ for the respect of (MIR) and (SYM), we have

$$\begin{aligned} v_i^{(k)}(p) = \left[ \prod _{r=1}^k(s-r)\right] \left( p-\frac{1}{2}\right) ^{s-k-1}. \end{aligned}$$

Case 1: $p \geqslant \frac{1}{2}$. Clearly, $v_i^{\left( k\right) }(p) \geqslant 0$.

Case 2: $p \leqslant \frac{1}{2}$. Since s is even, then the $(s-1)$ order of the derivative of $v_i(\cdot )$ is odd. Then, k is odd, implying that $s-k-1$ is even. Then, $v_i^{(k)}(p) \geqslant 0$ for all $s\in \{2,4,\ldots \}$.

By Definition 3.1, $\alpha _1 \geqslant \cdots \geqslant \alpha _n$ to match vertical equity. Hence $v_1(p)^{(s-1)} \geqslant \cdots \geqslant v_n(p)^{(s-1)} \geqslant 0$ for all $p\in [0,1]$, and so $(-1)^{s-1}v_1(p)^{(s-1)} \leqslant \cdots \leqslant (-1)^{s-1}v_n(p)^{(s-1)} \leqslant 0$. Following Abel’s Lemma, $\sum _{i=1}^{l} \theta _i \left[ GAC_{\tilde{\mathbf {H}},i}^{s}\left( p\right) - GAC_{\mathbf {H},i}^{s}\left( p\right) \right] \geqslant 0$, for all $l \in \{1,\ldots ,n\}$ is a sufficient condition to get the negativity of the sum in (4). For the negativity of the sum in (5), the sufficient condition is $\sum _{i=1}^{l} \theta _i \left[ GAC_{\tilde{\mathbf {H}},j}^{u}\left( 1\right) - GAC_{\mathbf {H},i}^{u}\left( 1\right) \right] \geqslant 0$, for all $l \in \{1,\ldots ,n\}$ and for $u = 3$ only. Indeed, for u being even, such that $u=4=s$, we get $(-1)^{u-2}v_i^{(u-2)}(1) \geqslant 0$, which is a contradiction of Abel’s lemma used for (4). However, if $u=3=s$, in this case, s is odd, so that (MIR) and (SYM) does not hold. Hence, the value of s is reduced to $s=2$, so that the condition (5) becomes irrelevant. $\square$