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.

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Fig. 1
Fig. 2

Notes

  1. 1.

    Health concentration indices do not necessarily pass this upside down test. However, socioeconomic health inequality indices that respect this test allow for higher aversion to socioeconomic health inequality. Khaled et al. [21] coin this ethical view pro-extreme rank transfer sensitivity.

  2. 2.

    Strictly speaking, the authors postulate that the value of the health index is the opposite of the ill health index. The reader is referred to Bosmans [7] and to Kjellsson et al. [22] for some criticisms about the use of this axiom. Precisely, Bosmans [7] proposes a weaker axiom to enlarge the class of health inequality indices respecting consistency (another version of the mirror axiom).

  3. 3.

    See also [3] for the counting approach in the case of poverty measurement.

  4. 4.

    As pointed out by Makdissi and Yazbeck [27], \(\Upsilon (\mathbf {H}(\mathbf {p}))\) is robust to any given monotonic transformation of \(\mathbf {H}\) avoiding a particular view being imposed on the transformation of health status.

  5. 5.

    See [14] and [10]. The transvariation means that the inequality between individuals, computed on the basis of income gaps, has a reverse sign compared with that of the mean income gap of the group which they belong to.

  6. 6.

    See [35] for the link between achievement and concentration indices in the bivariate setting.

  7. 7.

    See [38] for an application to socioeconomic health inequalities in Canada for the standard case \(s=2\).

  8. 8.

    If the number of type i individuals is denoted \(N_i\), such that \(N=\sum _iN_i\), then \(\kappa _i=N_i/N\).

  9. 9.

    For simplicity, we can set \(\phi _i \equiv \phi\). Alternatively, it is be possible to set:

    $$\begin{aligned} \phi _i(\mathbf {H}(\mathbf {p})) = \frac{K-\Upsilon _i(\mathbf {H}(\mathbf {p})) \Theta '_i}{K}, \end{aligned}$$

    where \(\Upsilon _i\) captures the health failure of type i people, so, in this case, the identification function would be specific to each group in the same manner than the weight function \(\Theta '_i\).

  10. 10.

    Compared with the usual approach, we do not define \(v_i:[0,1] \longrightarrow [0,1]\) and particularly we do not impose \(\int _0^1 v_i(p)dp =1\), to be consistent with Definition 3.1, see Sect. 3.

  11. 11.

    Strictly speaking, Aaberge [1] defined the downward positional principle and the upward one. In what follows, we use the upward principle only, which states that the decision maker prefers the poor to become richer (\(v_i^{(3)}(\cdot ) \leqslant 0\)). Note that the analysis could have been done with the downward principle that states that the social planner prefers the rich to become poorer (\(v_i^{(3)} (\cdot ) \geqslant 0\)).

  12. 12.

    Note that anonymity of \(A(\mathbf {H})\) is ensured whenever \(\phi\) is a symmetric function in the different health categories k. Note that this requirement of symmetry is different from that studied in Sect. 5.

  13. 13.

    Regressive means that the transfer occurs from a lower rank individual to a higher rank one.

  14. 14.

    Note that this curve has been used independently by Khaled et al. [21], the so-called Generalized Health Concentration Curves. In addition, they use the terminology Health Concentration Curves for achievement curves.

  15. 15.

    The reader is referred to [21] for the use of non-parametric estimator of achievement curves to perform dominance tests.

  16. 16.

    We use \(f(\mu _\phi )\) instead of \(f(\mu _\phi ,s)\).

  17. 17.

    It is noteworthy that (MIR) would not be respected even if \(\mu _{\phi _i}=c\) is a constant. Indeed, if each \(\alpha _i>0\), it is possible to show that \(\int _0^1w_i(p,s)\mathrm{d}p = \int _{0}^{1} \left( 1- \alpha _is(1-p)^{s-1}\right) \mathrm{d}p\ne 0\), and in this case (MIR) cannot be fulfilled.

  18. 18.

    If \(\alpha _i=8\) for all \(i=1,\ldots ,n\), then the maximal value is \(\mathrm{GS}(\mathbf {H})=1\) and the minimal one is \(\mathrm{GS}(\mathbf {H})=-1\).

  19. 19.

    SHARE; [6]; [5], and [29].

  20. 20.

    The blue collar group includes: service workers and shop and market sales workers; skilled agricultural or fishery workers; craft and related trades workers; plant and machine operators or assemblers; elementary occupations.

  21. 21.

    The white collar group includes individuals working as: legislators, senior officials or managers; professionals; technicians or associate professionals; clerks; armed forces.

  22. 22.

    Two different items were included in the mental health dimension: memory (the ability of people to think about things) and depression (twelve different aspects of a depression’s symptoms). In contrast, seven items have been considered to capture the physical health dimension of individuals: long-term illnesses (having any long-term health problem, illness, or infirmity), other illnesses (a list of 14 health conditions), limitation on daily activities 1 and 2 (difficulties with various activities because of health problems and difficulties with various basic daily activities, respectively), weight problems (overweight, obesity, and underweight), eyesight (eyesight distance and reading), and hearing (quality of hearing).

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Acknowledgements

The Support of Chrome, Lameta, and Liser is gratefully acknowledged. This paper uses data from SHARE Wave 5 (DOI: 10.6103/SHARE.w5.100). The SHARE data collection has been primarily funded by the European Commission through FP5 (QLK6-CT-2001-00360), FP6 (SHARE-I3: RII-CT-2006- 062193, COMPARE: CIT5-CT-2005-028857, SHARELIFE: CIT4-CT-2006-028812), and FP7 (SHAREPREP: N\(^\circ\)211909, SHARE-LEAP: N\(^\circ\)227822, SHARE M4: N\(^\circ\)261982). Additional funding from the U.S. National Institute on Aging (U01_AG09740-13S2, P01_AG005842, P01_AG08291, P30_AG12815, R21_AG025169, Y1-AG-4553-01, IAG_BSR06-11, and OGHA_04-064) and from various national funding sources is gratefully acknowledged (see www.share-project.org).\(\ddag\) This research is part of the HEDYNAP project supported by the Luxembourg Fonds National de la Recherche (contract C12/SC/3977324/HEADYNAP/Pi Alperin) and by core funding from Liser from the Ministry of Higher Education and Research of Luxembourg.

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Correspondence to Stéphane Mussard.

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Stéphane Mussard is the Research fellow MRE University of Montpellier, Grédi University of Sherbrooke, and Liser Luxembourg.

Maria Noel Pi Alperin: This research is part of the HEADYNAP project supported by the Luxembourg Fonds National de la Recherche (contract C12/SC/3977324/HEADYNAP/Pi Alperin) and by core funding from Liser from the Ministry of Higher Education and Research of Luxembourg.

Appendix: Proof

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:

  1. (i)

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

  2. (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:

  1. (i)

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

  2. (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. \(\square\)

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Mussard, S., Pi Alperin, M.N. & Thireau, V. Health inequality indices and exogenous risk factors: an illustration on Luxembourgish workers. Eur J Health Econ 19, 1285–1301 (2018). https://doi.org/10.1007/s10198-018-0973-3

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Keywords

  • Concentration
  • Dominance
  • Health inequality
  • Mirror
  • Symmetry
  • SHARE
  • Luxembourg

JEL Classification

  • D6
  • I1