Socio-economic Health Inequality Indices: A Fuzzy Approach Applied to European Countries

Stepbane Mussard and Maria Noel Pi Alperin

Introduction

Since the seminal work of Yaari (1987), the theory of decision has been particularly interested in the use of rank-dependent indices. This dual theory of choice under risk has led the economists to employ concentration indices (based on households’ ranks in the society) because of their statistical properties and their simple interpretations - see, for instance, Kakwani et al. (1997) and Yitzhaki and Schechtman (2013). They are also commonly used as concentration indices, which are relevant for any given degree of inequality aversion supported by the decision-maker (Yitzhaki, 1983).

Makdissietal. (2013) propose a multi-dimensional socio-economic health inequality index based on the rank-dependent approach (Yaari, 1987) and involving a parameter that accounts for the intensity of health redistribution, that is, an inequality aversion parameter. The authors employ the so- called counting approach, with categorical dimensions, in order to allow each health dimension to be gauged in proportion of the overall amount of the socio-economic health inequality index.

In this chapter, we present a multi-dimensional socio-economic health inequality index, with a rank-dependent structure, based on fuzzy-set theory. This family of indices depends on different health dimensions and on one exogenous risk factor, and reflects two very different behaviours of the social planner. First, the usual attitude towards inequality as embodied by the degree of aversion to inequality (see Yitzhaki, 1983), i.e. the willingness of the social planner to operate redistributive policies toward less healthy people in order to alleviate overall inequality in the society. Second, the degree of risk sensibility of the social planner defined by the interaction between the health dimensions and the risk factor (see Mussard and Pi Alperin, 2020). Risk factors are analysed through the prism of equality of opportunity (see e.g. Dworkin, 1981, 2000; Roemer, 1998; Fleurbaey, 2008). The literature distinguishes between fair and unfair causes of health inequalities. The origin of fair inequalities are risky behaviours which are under an individual’s control (e.g. lifestyle) while unfair inequalities are associated with exogenous risks beyond the individual’s control (e.g. childhood circumstances). In what follows, we will concentrate our analysis on exogenous risks factors.

The theory of fuzzy sets has been widely used to measure health. For example, Pi Alperin and Berzosa (2011) introduce a different way to measure overweight accounting for different intensities of overweight and obesity. Pi Alperin (2016) proposes a synthetic indicator of individual health, including different health dimensions evaluated by diagnosed or reported diseases and limitations on daily activities related to the mental and physical aspects of health. Finally, Mussard and Pi Alperin (2020) show that fuzzy-set theory can be applied to construct socio-economic health inequality indices. The advantage of this theory relies on the fact that each individual can be associated with a given degree of health deprivation, whereas the literature focuses on a clear state of health dimension, i.e. the individual is deprived or not. Then, fuzzy-set theory gives more flexibility to design multi-dimensional socio-economic health inequality indexes. In what follows, we employ this methodology with health dimensions constructed using fuzzy sets instead of categorical health dummy variables.

The chapter is organised as follows. The second section reviews the family of socio-economic health Gini indices based on fuzzy sets. The third section exposes how risk factors are introduced into socio-economic health Gini indexes based on fuzzy sets. A stochastic dominance criterion is also proposed to ensure a robust ranking of health deprivation matrices for a wide range of risk sensibility and inequality aversion parameters. The forth section presents the empirical strategy of the chapter. The fifth section is devoted to an application on European countries. The final section concludes the chapter.

Socio-economic Inequality Indices Based on Fuzzy Sets Health

This section summarises the rank-dependent socio-economic health inequality indices introduced by Mussard and Pi Alperin (2020) and shows how this family of indexes can be extended using the fuzzy-set theory in a multidimensional context.

Multi-dimensional Fuzzy Gini Indices

Let В be a fuzzy sub-set of n individuals of a population P such that any individual in В presents some degree of deprivation in at least one health dimension k, k e (1,...,K). The К variables represent different dimensions of physical and mental aspects of health selected to measure multi-dimensional health inequality. Let yE be an equivalent income distribution such that F(yE) is its cumulative distribution function defined over [0,д], where a is the maximum conceivable equivalent income. The rank p of the individuals in В are issued from F(yE), such that p € [0,1]. Instead of employing the counting approach to measure whether an individual is deprived or not in one dimension of health deprivation (e.g. Alkire and Foster, 2011; or Makdissi and Yazbeck, 2014), we adopt a fuzzy logic approach.

Let HB(p) = (b^(p),...,hk(p),...,hK[p) be, for an individual in В at rank p of the equivalent income distribution, his/her degree of deprivation in each health dimension. The n x К health deprivation matrix is HB(p) = (bv...,hK), where hk denotes the kxh column of HB and HB(p) denotes a row of HB(p) for an individual in В at rank p. If an individual at rank p is totally affected by disease k, then the individual is totally deprived in the &th dimension of health, hk(p) = 1. On the contrary, if an individual at rank p is not affected by disease k, then hk(p) = 0, in this case there is no deprivation in the health dimension k. If the individual at rank p is more or less affected by disease k, then 0 < ht(p) < 1. Thereby,

represents the fC-dimensional profile of deprivation of an individual in В at rank p of the equivalent income distribution.

In order to aggregate the health deprivation over the К dimensions, a generalised mean is employed:1

where вк 6 [0,1] is the weight attached to dimension k (see the next subsection for the weights parametrisation).

Then, 1 - HR (p,a) represents the average health achievement for an individual at rank p (the dual part of the individual deprivation). On this basis, thanks to a rank-dependent function2 v(p), such that v: [0,1] -> [0,1], which embodies the social planner’s behaviour with respect to redistributive principles, the socio-economic health achievement index based on fuzzy sets is given by,

The fuzzy index A(HB,a) is a multi-dimensional extension of the health concentration index (see Erreygers et al., 2012). However, it provides the ability of society to reach a healthy state.

Then, it is possible to derive by duality a socio-economic health inequality index based on fuzzy sets in a multi-dimensional context:

where ^,(H„a)=J (l - HB (p, a)^dp. Following Yitzhaki (1983), rhe

rank-dependent function that distorts the ranks p may be chosen to provide an extended Gini index,

with v the inequality aversion parameter. Therefore,

is a family of socio-economic health Gini index based on fuzzy sets, such that г > 2 outlines health inequality aversion of the social planner, whereas v € (1,2) outlines health inequality loving (a and v will be explained in below). The intensity of the socio-economic health inequality is mainly dependent on the weights associated with each dimension.

Weight Vectors: Standard Approaches in The Fuzzy Sets Literature

In a multi-dimensional context, weights вк are of crucial importance to aggregate the information along dimensions (e.g. Alkire and Foster, 2011). Although one of the most used weight schemes to aggregate health achievement or health deprivation is the equal weights scheme, for many years now the fuzzy-set literature has tackled the problem of finding more relevant weight schemes.

In 1990, Cerioli and Zani proposed weights that take into_account the distribution of the dimensions among the population. Let hk * Obe the arithmetic mean of the kth health dimension. The weight decreases with deprivation, that is, as far as hk increases:

In 1999, Betti and Verma extended the previous weight scheme accounting for the correlation between dimensions. In other words, in the context of health measure, their weight scheme accounts for the prevalence of a given health dimension, from one side, and it limits the influence of highly correlated health dimensions on the synthetic indicator, from the other side. Therefore, Betti and Verma’s weight of dimension k is built as follows:

where 6k depends only on the distribution of hk (coefficient of variation of

hk)>

and where 6kb depends on the correlation between k and the other dimensions:

The index pkk' is Pearson’s correlation coefficient between dimensions k and k’. F(-) denotes the indicator function valued to be 1 if the expression in brackets is true and 0 otherwise, and where pH is a pre-determined cut-off correlation level between the two dimensions. The weight 6b is inversely linked with the intensity of the correlation between hk and the other dimensions. Although one weight scheme must be chosen to compute multidimensional inequality indices, testing for different weighting schemes is important to show the robustness of this indices.

Risk Factors and Robust Ranking of Health Deprivation Matrices

The two-parameter family of the multi-dimensional Gini index based on fuzzy sets, G(HB, a, v), allows one to deal with a variety of parameters a and v, which represent two different behaviours of the social planner (see Mussard and Pi Alperin, 2020): the parameter a is related to the risk sensibility of the social planner, which is derived from the association between the risk dimension and the health dimensions; the parameter v reflects the fact that the social planner is more and more inclined to redistribute resources to individuals at the bottom of the income distribution yE as far as v increases.

Risk Factors and Risk Sensibility

The parameter a should be interpreted as the risk sensibility of the social planner. However, the interpretation of this parameter is different from the usual risk aversion studied in the literature of risk and uncertainty. The social planner may focus on particular health dimensions and judge whether or not these dimensions affect his/her own view about health deprivation. For this purpose, we consider that one dimension is embodied by one specific risk factor. Indeed, the kth dimension of the health deprivation profile of an individual at rank p that is bK(p) is related to, and represents, a given exogenous risk factor.

The parametrisation of a depends on whether an individual is considered or not as totally deprived in all health dimensions. In other words, if

1 - Нв (р,а) = 0, the individual is deprived in all health dimensions (non- healthy). In contrast, if, 1 - HB (p,a) = 1, the individual is not deprived in any health dimension (totally healthy, i.e. the level of achievement is a maximum).

The risk sensibility properties are related to the usual union/intersection approach introduced by Atkinson (2003). The union approach is defined as follows:

It defines the situation where the social planner looks for only one dimension of health in which the health achievement is a maximum. For instance, if there exists one individual at rank p, (of the equivalent income distribution) and one dimension k, except the risk factor, k €{1,...,K-1), such that hk (pt) = 0, then this individual is judged to be totally healthy. The result would be exactly the same for an individual at rank p, being healthy in all dimensions. Furthermore, whether or not those individuals are affected by the risk factor K, that is bK (p;) = 1, for i = 1,2, the social planner would actually judge individuals at rank p] and p, being healthy. We say in this case that the social planner is risk-neutral, in other terms, he/she has no sensibility towards the exogenous risk factor. Note that when an individual at rank p is not affected by the risk factor, then hK (p) = 0. In addition, if this individual is non-healthy in all dimensions, then hk (p) = 1 for all k €{1,...,K). This would provide a healthy state, while the individual is non-healthy. In order to avoid this problem, risk neutrality is defined as follows:

The risk sensibility property based on the intersection approach explains that if an individual is only deprived in risk dimension K, then the social planner judges this person as totally non-healthy, in the same manner as an individual being non-healthy in all dimensions. Let a -» ■», then:

The risk sensibility is therefore the highest possible because the risk factor hK (p) = 1 necessarily implies a non-healthy state. The a parameter allows one to deal with different types of social planners, and more precisely how they behave in accordance with the risk factor K.5

206 S. Mussard and M.N. Pi Alperin Stochastic Dominance

We show that the multi-dimensional socio-economic health Gini indices based on fuzzy-set theory, G(HB, a, v) are consistent with a non-ambiguous ranking between health deprivation matrices.

The willingness of the social planner to setup redistributive actions to alleviate inequality in the society is related to the intensity of the parameter v.4 The parametrisation v = 1 corresponds to Pen’s parade: an additional income for one individual at rank p improves his/her health achievement and therefore the socio-economic health achievement index increases. The parametrisation v = 2 corresponds to a progressive transfer, from one individual at rank p{ to another individual at rank p, (such that p( > p,), which increases the socio-economic health achievement index. Those positional transfers (based on ranks rather than incomes) are generalised in such a way that the more the value of v, the more the intensity of the income transfer towards individuals located at the lower tail of the equivalent income distribution. The class of distortions functions v(p) consistent with those generalised positional transfer principles are such that,

and v[b (1) = 0 for all / = l,...,v (see Aaberge, 2009; Makdissi and Mussard, 2008). The socio-economic health inequality indices based on fuzzy sets satisfying the above property for v(p) are included in the set CY.

Makdissi and Yazbeck (2014) proposed a ranking between health deprivation matrices on the basis of achievement curves. In the same spirit, we propose achievement curves, which depend on the parameter v, and furthermore on the parameter a, the so-called (v,«)-achievement curves. The achievement curve of order (l,a) is defined as, for all a > 0:

The achievement curve of order (l,a) yields the health achievement of one individual at rank p of the population in proportion to the achievement mean. The (v,a)-order achievement curve for any given ve{2, 3,...,) and a > 0 is:

The achievement curve of order (2,a) provides the proportion of the population whose health achievement is not higher than (p). On this basis, a robust ranking between health deprivation matrices may be proposed. Indeed, when the achievement curves do not cross, there is more (or less) inequality for all percentiles p € [0,1]. This result is compatible with particular social planners being more or less inclined to perform income transfer

(with respect to the value of v) and also being more or less sensitive to risk (with respect to the value of a).

Theorem 3.1 in Mussard and Pi Alperin (2020) - For all socio-economic health Gini indices based on fuzzy sets G(HB,a,v) such that G(Hg,a,v)e Qv

with ve{ 1,2,3,___} and a > 0, and for two fuzzy health deprivation matrices

HB and GB, the two following statements are equivalent:

The above result yields a non-ambiguous ranking of multi-dimensional health deprivation matrices with respect to the risk factor and to the intensity of income transfers. To be precise, socio-economic health inequalities depend on the risk insensibility of the social planner a -> 0, on his/her risk sensibility a > 1 and finally on his/her extreme risk sensibility a -» oo. In addition, as far as v increases, income transfers performed to less worthy individuals are taken into account.

Empirical Strategy

This section briefly explains the main data source used in this chapter, presents some descriptive statistics of the 13 countries included in the analysis, and describes the health dimensions selected to analyse multi-dimensional socio-economic health inequalities.

Data

This chapter uses data from waves 5 and 6 Release 7.0.0 of the Survey of Health, Ageing and Retirement in Europe (SHARE; Borsch-Supan, 2019; Maker and Borsch-Supan, 2017; Maker and Borsch-Supan, 2015; Borsch- Supan et al., 2013). SHARE is a multidisciplinary and cross-national panel database collecting micro-data on health, socio-economic status and social and family networks. The objective of the survey is to better understand the ageing process and, in particular, to examine the different ways in which people aged 50 and older live in Europe. The first wave of data was collected in 2004. Wave 5 and wave 6 were collected in 2013 and 2015, respectively. Thirteen European countries (Austria, Belgium, Czech Republic, Denmark, Estonia, France, Germany, Italy, Luxembourg, Slovenia, Spain, Sweden and Switzerland) are analysed in this chapter. In particular, all countries that collected SHARE data in both waves.

The analysis is mainly based on data from wave 6, but we also use some information from wave 5. The main reason is that wave 5 collected information on, for example, the educational level and the country of birth of an individual’s parents. Hence, we matched this information about parents with information on the individual given by wave 6. This matching

Table 13.1 Descriptive statistics

Country

Gender %

Age %

Total

Male

Female

< 65

> 65

Austria

41.86

58.14

32.95

67.05

2962

Germany

46.86

53.14

45.17

54.83

4144

Sweden

45.70

54.30

26.66

73.34

3440

Spain

44.47

55.53

34.45

65.55

4850

Italy

44.48

55.52

34.48

65.52

3579

France

42.07

57.93

36.10

63.90

3069

Denmark

45.41

54.59

45.08

54.92

3312

Switzerland

44.87

55.13

37.33

62.67

2467

Belgium

44.52

55.48

42.18

57.82

4270

Czech Republic

39.56

60.44

34.30

65.70

4300

Luxembourg

45.96

54.04

49.38

50.62

1051

Slovenia

41.09

58.91

41.00

59.00

2166

Estonia

37.90

62.10

30.65

69.35

4509

Total

43.27

56.73

36.85

63.15

44,119

is possible since the information matched is not changing over the various years. Then, only individuals present in both waves are considered in the analysis. In addition, we complete the missing values using imputation techniques. The descriptive statistics of the 13 countries analysed in this chapter are present in Table 13.1.

Health Dimensions

Synthetic indicators of health are constructed following the methodology based on fuzzy sets proposed by Pi Alperin (2016). These indicators aggregate health dimensions reflecting different aspects of mental and physical health of individuals present in the SHARE survey. In particular, three dimensions belong to the mental health aspect (depression, memory and cognition) and six dimensions are part of the physical health (long-term illnesses, limitations in daily activities, limitations in instrumental activities, limitations in general activities, eyesight and hearing). Then, for each health dimension there are healthy individuals, completely non-healthy individuals and individuals characterised by different intensities of health failure.5 More precisely, the synthetic scores are calculated as the weighted mean of the К dimensions of health. In order to determine the weights associated with each health dimension, we follow the procedure defined by Betti and Verma (2008) described previously.6

Socio-economic Health Inequalities by Risk Factors in European Countries

In this section, we provide for each country a ranking of health deprivation matrices by considering one risk factor at a time.

Socio-economic Health Inequality Indices 209 Risk Factors as Circumstances

As was explained previously, in this chapter we follow the literature on equality of opportunity. It is possible then to make a distinction between different origins of risk (Mussard et al., 2018). In our study, we deal with unfair risks related to circumstances beyond an individual’s control. In other words, we consider variables associated with exogenous risk factors that individuals dot not control for such as the level of education of parents or their nationality. These circumstances may increase inequalities in health and, as Roemer (1995) suggests, individuals’ encountering bad consequences should be compensated. Several authors investigate inequality of opportunity in health by analysing the role of circumstances during childhood such as family and social backgrounds (see, for instance, Trannoy et al., 2010; Jusot et al., 2013).

In order to perform the empirical application over European countries, we compute achievement curves over the nine dimensions of health deprivation and one risk factor. Subsequently, an achievement curve is composed of ten dimensions, the last tenth dimension being the exogenous risk factor. These dimensions are averaged using Betti and Verma’s weighting scheme, Equation (13.7)7 Subsequently, for each country we compute one achievement curve for each risk factor. In particular, four exogenous risks factors are analysed associated to the individual’s parents: the nationality of parents, the educational level of parents, the economic situation of the family during childhood and the longevity of parents.8

The parents’ nationality risk factor is valued to be 1 if both parents are immigrants and 0 otherwise. Concerning the parents’ educational level, the risk factor is valued to be 1 if both parents have no education, primary or lower secondary education. In contrast, it is valued to be 0, if one of the parents studied beyond high school. Concerning the economic situation of the family during childhood, individuals were asked whether their family used to have financial difficulties when they were growing up (from birth to aged 15 inclusive). For individuals coming from low-income families or families whose financial situation varied over time, the variable is equal to 1, and 0 otherwise. Finally, the last risk factor concerns the longevity of parents. Individuals in the survey report whether the parents are still alive at the time of the survey and their age at death if applicable. With this information, the longevity risk factor was set up to be equal to 1 if at least one of the parents had short longevity (i.e. those who died younger than the life expectancy of their generation at birth) and equal to 0 for individuals with both parents enjoying longevity.

Comparing the four achievement curves issued from each risk factor, for one given country, provides the inequality of the repartition of health deprivation in such a way that it becomes possible to detect the risk factor that aggravates the level of inequality in the society. When an achievement curve, say AN, for the risk inherent to nationality, lies

  • 1.0 0.9 0.8 I 0.7
  • 0 0.6
  • 1 0.5

E

g 0.4 | 0.3 < 0.2 0.1

Neutrality - equal weight - order 2

  • 0.50
  • 0.40

i

о 0.30

г

о

E

§ 0.20

Neutrality - equal weight - order 3

0.0 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 Ranks

0.9 1.0 0.0 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 Ranks

0.9 1

(a)

Neutral-nat-2 — Neutral-edu-2 — Neutral-econ-2 — Neutral-long-2

(b)

  • — Neutral-nat-3 — Neutral-edu-3
  • — Neutral-econ-3 - Neutral-long-3

Figure 13.1 a. Order 2, a = 0. b. Order 3, a = 0.

nowhere above AE being the achievement curve inherent to the risk of education, this means that the level of inequality in the society due to nationality is higher than that due to education. If the achievement curves cross, then no conclusion can be drawn for the entire society. However, if the crossing point is at rank p‘, the conclusions about inequality is only valid for individuals located before rank p'. If the social planner has no risk sensibility (a = 0), then, for any given type of risk dimension K, the achievement curves provide the same repartition of health deprivation among the population. As an example, for Austria (Figures 13.1a and 13.1b), considering that the social planner is inclined to perform redistributive policies (for instance v = 2,3), the four achievement curves are indistinguishable. Therefore, with regard to the four types of risk, the social planner does not make any distinction between these risk factors. Furthermore, since he/she has no risk sensibility, the repartition of health deprivation is seen to be egalitarian.

We obtain the same result for all countries. In other words, no distinction can be made about risk factors when the social planner has no risk sensibility.

Risk Sensibility

The level of risk sensibility of the social planner increases with parameter a. We suppose an intermediate value of a = 50 [a = 100 may be seen as a maximum). In Table 13.2 below, we expose the results of the ranking of the different risk factors for each country. We begin with the redistribution parameter v = 2. In this case, the social planner makes simple redistribution with progressive transfer from one individual at rank p to another one at rank p1 such that p1 < p (it is a minimal requirement of redistribution). We analyse achievement curves of order (v = 2, a. = 50). If the achievement curves intersect, then no ranking is available (NR in

Table 13.2 Ranking of health inequalities by risk factors (v = 2, a = 50)

Country

Parent’s Exogenous Risk Factors

Nationality

Education

Economic

Longevity

Austria

NR

p = 80%

NR

NR

Germany

NR

NR

NR

NR

Sweden

NR

p = 90%

NR

NR

Spain

NR

NR

NR

NR

Italy

NR

1"

2 s'

NR

France

NR

1*

NR

NR

Denmark

NR

NR

NR

NR

Switzerland

NR

p = 70%

NR

NR

Belgium

NR

p = 75%

NR

NR

Czech Republic

NR

NR

NR

NR

Luxembourg

NR

p = 67%

NR

NR

Slovenia

NR

NR

NR

NR

Estonia

NR

NR

NR

NR

,f Possible crossing between the curves at the bottom of the income distribution

the tables below). If one achievement curve lies nowhere above the other ones, then it corresponds to the highest risk factor that aggravates the socio-economic health inequality in the society (1 in the tables below for the most important risk factor, 2 for the second more important risk factor and so on).

As can be seen in Table 13.2, the health deprivation matrices do not allow the achievement curves related to each risk factor to be fully compared (NR). However, many countries rank parents education level as the main exogenous risk factor that widens socio-economic health inequalities in society, such as in France and Italy. In other countries, the dominance of achievement curves is assessed up to a given percentile p. For instance, in Sweden, the parents education level increases the socio-economic health inequalities for all individuals below the 90th percentile of the equivalent income distribution.

In Table 13.3, the preference of the social planner who contemplates redistribution to less worthy people is increased to v = 3. In this case, the crossing between the achievement curves at the order v = 2 is less likely to be observed. This means that the socio-economic health inequalities inherent to education are, for all individuals located in the percentile interval [0,1], higher than those issued from the other risk factors for Austria, Sweden, Italy, France, Switzerland, Belgium and Luxembourg (Table 13.3).

Note that the economic situation of the family during childhood can be considered as a risk factor in Italy, although the achievement curves cross at the very bottom of the equivalent income distribution.

Table 13.3 Ranking of health inequalities by risk factors (v = 3, a = 50)

Country

Parent’s Exogenous Risk Factors

Nationality

Education

Economic

Longevity

Austria

NR

1

NR

NR

Germany

NR

NR

NR

NR

Sweden

NR

1

NR

NR

Spain

NR

NR

NR

NR

Italy

NR

V

2*

NR

France

NR

1

NR

NR

Denmark

NR

NR

NR

NR

Switzerland

NR

1

NR

NR

Belgium

NR

1

NR

NR

Czech Republic

NR

NR

NR

NR

Luxembourg

NR

1

NR

NR

Slovenia

NR

NR

NR

NR

Estonia

NR

NR

NR

NR

Table 13.4 Ranking of health inequalities by risk factors (v = 2, a = oo)

Country

Parent’s Exogenous Risk Factors

Nationality

Education

Economic

Longevity

Austria

NR

p = 80%

NR

NR

Germany

NR

1

NR

NR

Sweden

NR

p = 80%

NR

NR

Spain

NR

1

NR

NR

Italy

NR

V

2*

NR

France

NR

Iй'

NR

NR

Denmark

NR

NR

NR

NR

Switzerland

NR

p = 80%

NR

NR

Belgium

NR

p = 75%

NR

NR

Czech Republic

NR

NR

1

NR

Luxembourg

2

1

3

4

Slovenia

NR

NR

2*

Iй'

Estonia

NR

NR

NR

NR

Extreme Risk Sensibility

We now suppose that the risk sensibility of the social planner is a maximum (a -» oo) (see Table 13.4). Results are very close to those of Table 13.3. However, new results are itemised. The economic situation of the family during childhood is the first cause of health inequalities in the Czech Republic. It is the second more important risk factor in France after education. In Luxembourg, nationality becomes the second more important risk factor after education, and the economic situation of the family during

Table 13.5 Ranking of health inequalities by risk factors (v = 3, a. = «)

Country

Parent’s Exogenous Risk Factors

Nationality

Education

Economic

Longevity

Austria

NR

1

NR

NR

Germany

NR

NR

1

NR

Sweden

NR

1

NR

NR

Spain

NR

1

NR

NR

Italy

NR

1*

2*

NR

France

NR

1

2

NR

Denmark

NR

NR

NR

NR

Switzerland

NR

1

NR

NR

Belgium

NR

1

NR

NR

Czech Republic

NR

NR

1

NR

Luxembourg

2

1

3

4

Slovenia

NR

NR

2*

Is

Estonia

NR

NR

NR

NR

a. Order 2, a. = °o. b. Order 3, a = «>

Figure 13.2 a. Order 2, a. = °o. b. Order 3, a = «>.

childhood, and their longevity, appears to be ranked 3 and 4, respectively. In Slovenia, longevity is top-ranked; however, we cannot assess this result because of possible crossing between achievement curves at the bottom of the equivalent income distribution.

We obtain exactly the same results when the level of redistribution is increased by one unit v = 3 (Table 13.5).

For each country, the achievement curves are very close to each other, so that the dominance cannot be proven for sure. For instance, in Italy, it is apparent that the achievement curve related to education reveals concentrated inequalities (blue curves in Figures 13.2). Flowever, a crossing between achievement curves appears at p = 15% for v = 2 (Figure 13.2a) and for v = 3 (Figure 13.2b).

214 S. Mussard and M.N. Pi Alperin Conclusion

In this chapter, a family of socio-economic health inequality indexes based on fuzzy sets has been proposed. The advantage of dealing with such a family of indexes is to simulate the behaviour of the social planner in two ways: his/her ability to perform income transfers to improve health achievement, and his/her sensibility to the exogenous risk factor. Furthermore, this family of indexes is rewritten with the aid of achievement curves that yield the ability to perform robust ranking between health deprivation matrices.

The empirical application on 13 European countries shows that the level of education of parents is the main driver of risk that aggravates the socio-economic health inequalities in society. This is particularly true, for example in Austria, Sweden, Spain, Italy, France, Switzerland, Belgium and Luxembourg.

Appendix Al: Degree of Membership of Health Dimensions

Table A.1.1 Depression

Depression scale Euro-d*

Degree of membership

Non-depressed (0 dimension)

0

Between 1 and 11 dimensions

1 - (12 - X )/12

Completely depressed (12 dimensions)

1

* Depression, pessimism, suicidal thought, guilty, sleep, interest, irritability, appetite, tiredness, concentration, enjoyment, tearfulness.

Table A.1.II Memory

Memory

Degree of membership

Four questions have been asked regarding date, day of the week, month and year

Knows all

0

Knows 3 of 4

0.25

Knows 2 of 4

0.50

Knows 1 of 4

0.75

None of them

1

Table A.l.Ill Cognition

Capacity to memorise words

Degree of membership

How many words do you recall?*

I recall more than 15 words

0

I recall more than 1 and less than 16

(16 -X )/14 1

I recall only 1

* This number is the addition between the first trial and the delayed trial.

Table A.1.IV Chronic illness

Long-term illness

Degree of membership

Do you have any long-term health problems, illness, disability or infirmity?

No

0

One

0.75

More than one

1

Table АЛ. V Limitation activities 1

Health and daily activities

Degree of membership

Because of a health problem, do you have difficulty doing any of the following daily activities?*

No

0

Somewhat

1 - (6 - X.)/6

Yes

1

* Dressing, bathing or showering, eating, cutting up the food, walking across a room, getting in or out of bed.

Table A. 1. VI Limitation activities 2

Health and instrumental activities

Degree of membership

Because of a health problem, do you have difficulty doing any of the following instrumental activities?*

No

0

Somewhat

1 - (5 - X.)/5

Yes

0

* Telephone calls, taking medications, managing money, shopping for groceries, preparing a hot meal.

Table A.l.VII Limitation activities 3

Health and general activities

Degree of membership

Because of a health problem, do you have difficulty doing any of the following activities? *

No

0

Somewhat

1 _ (4 - X )/4

Yes

1

* Walking 100 metres, walking across a room, climbing several flights of stairs, climbing one flight of stair.

Table A. 1. VIII Eyesight

Eyesight distance and reading*

Degree of membership

Both are E or VG

0

One is E or VG, the other is G or F

0.15

One is E or VG, the other is P

0.25

Both are G or F

0.30

One is G or F, the other is P

0.60

Both are P

1

E: excellent; VG: very good; G: good; F: fair; P: poor

Table A.1.IX Hearing

Hearing

Degree of membership

Is your hearing8'

Excellent or Very good

0

Good

0.20

Fair

0.50

Poor

1

’•'With or without a hearing aid.

Acknowledgement

This research is part of the HEADYNAP project supported by the National Research Fund, Luxembourg (contract FNRC12/SC/3977324/HEADYNAP) and by core funding for LISER from the Ministry of Higher Education and Research of Luxembourg. This chapter uses data from SHARE Waves 5 and 6 (DOE 10.6103/SHARE.w5.700, 10.6103/SHARE.w6.700). The SHARE data collection has been funded by the European Commission through FP5 (QLK6-CT-2001-00360), FP6 (SHARE- 13: RII-CT-2006-062193, COMPARE: CIT5-CT-2005-028857, SHARELIFE: CIT4-CT- 2006- 028812), FP7 (SHARE-PREP: GA №211909, SHARE-LEAP: GA N"227822, SHARE M4: GA №261982) and Horizon 2020 (SHARE- DEV3: GA №676536, SERISS: GA №654221) and by DG Employment, Social Affairs & Inclusion. Additional funding from the German Ministry of Education and Research, the Max Planck Society for the Advancement of Science, the U.S. National Institute on Aging (U01 AG09740- 13S2, P01 AG005842, P01 AG08291, P30 AG12815, R21 AG025169, Yl-AG-4553- 01,IAG BSR06-11, OGHA 04-064, HHSN271201300071C) and from various national funding sources is gratefully acknowledged (see www.share -project.org).

Notes

  • 1 See the characterisation proposed by Blackorby et al. (1981).
  • 2 See the seminal work of Yaari (1987).
  • 3 See Mussard and Pi Alperin (2016) for an axiomatisation of desirable properties related to the introduction of exogenous risk factors into socio-economic health inequality indices.
  • 4 See Aaberge (2009) for the general case of probability distortion functions .
  • 5 See Appendix Al for a complete description of the construction of each health dimension.
  • 6 All the indicators of this chapter are computed using the MDEPRIV program (see Pi Alperin and Van Kerm, 2009).
  • 7 The results are very similar with the weight scheme proposed by Cerioli and Zani.
  • 8 These variables have been selected as they are frequently used to measure childhood conditions circumstances (see e.g. Jusot et al., 2013; Deutsch, et al., 2016).

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