EuroEconomica, Vol 34, No 2 (2015)
EuroEconomica
Issue 2(34)/2015 ISSN: 15828859
Fuzzy multidimensional inequality measurement. Policies to reduce inequality in Tunisia
Lamia HASNAOUI^{1}, Besma BELHADJ^{2}
^{1}University of Tunis El Manar, Tunis, Tunisia, hasnaoui.lamia@yahoo.fr
^{2}University of Carthage, Tunis, Tunisia, Besma.Kaabi@isg.rnu.tn
Abstract. This article debates a systematic treatment of the conceptual frame work of the multidimensional fuzzy measurement of inequality. Fuzzy logic is a type of multivalued logic consequential from fuzzy set theory. The introduction of the dimension relative to the human existence like health, energy and housing provides considerable enrichment to our understanding of inequality and its causes. In fact, we propose a multidimensional fuzzy measurement with the membership functions of inequality then we suggested some policies to reduce inequality. An application based on individual wellbeing data from Tunisian households in 2010 is presented to illustrate the use of the proposed index.
Keywords: income distribution, inequality, Fuzzy approach
1 Introduction
The income inequality is the disparate distribution of household income across the different participants in an economy. It varies between societies, historical periods, economic structures and systems. The phenomenon can refer to cross sectional distribution of income or wealth at any particular period or to the lifetime income and wealth over longer periods of time.^{ }There are a variety of numerical indices for measuring economic inequality. Economic inequality refers to how economic metrics are distributed among individuals in a group, among groups in a population, or among countries or regions. Economists usually think of three metrics of economic disparity: wealth inequality, income inequality, and consumption. The issue of economic inequality can implicate notions of equity, equality of outcome, and equality of opportunity.
Indeed, the causes of income inequality can vary considerably by region, gender, education and social status. Economists are divided as to whether income equality is ultimately positive or negative and what are the implications of such disparity. In fact, income distribution has always been a central concern of economic theory and economic policy. Therefore, many researches on the measurement of this concept are based on Gini index. Differences in national income equality are measured by the national Gini coefficient. The Gini coefficient is a number between 0 and 1, where 0 corresponds with perfect equality (where everyone has the similar income) and 1 corresponds with absolute inequality (where one person has all the income, and everyone else has zero income). Indeed, economists have developed new Gini decomposition.
Bhattacharga and Mahalabonis (1967) introduce a Gini decomposition in order to study regional disparities. Eugene et al (2002) defined and studied the betanornal distribution highlighting great flexibility in modeling not only symmetric heavytailed distribution, but also skewed and bimodal distribution. Zenga (2007) proposed an alternative curve based on the ratios of lower and higher means. Akinsete et al (2008) introduced the betapareto distribution. BarretoSouza et al (2009) proposed and studied the proprieties of the betageneralized exponential distribution. Pescimet et al (2010) introduce the betageneralized halfnormal distribution. Moreover, inequality in access to resources, and in particular in education, may inhibit reductions in child mortality and fertility rates and prevents the expansion in education of the next generation. These important development policy goals are closely linked to the educational attainment levels of the female population (Summers, 1994; Murthi et al., 1995). To the extent that these linkages exist, gender bias in education may thus prevent progress in the improvement in the wellbeing of the people in a considerable number of developing countries. Noglo (2014) aims to measure and analyze the inequalities in the distribution of nonmonetary wealth in Togo. The decomposition of the Gini index in Shapleyvalue showed evidence that the withingroups inequality is preponderant and policies must give priority to it. This paper debated concerning a systematic treatment of the conceptual frame work of the measurement of inequality. An alternative approaches are evaluated based on the fuzzy approach. Fuzzy logic is a type of multivalued logic consequential from fuzzy set theory to deal with probable reasoning rather than the precise one. In binary logic, in contrast to fuzzy logic, the variables can have a membership value equal to 0 or 1. In fuzzy set theory with fuzzy logic the set membership values can range inclusively between 0 and 1. Indeed, the degree of truth of a statement can range between 0 and 1 and is not constrained to the two true values (true (1), false (0)) the same as in classic predicate logic (Novák et al., 1990). And when linguistic variables are used, those degrees may well be managed by definite functions as discussed below. Zadeh (1965) has proposed fuzzy sets theory, as a consequence of this theory the fuzzy logic has emerged. Wilkinson (1963) has introduced the concept without using the term and therefore preceded fuzzy set theory. However, our main problem is what is the level of inequality in Tunisia and what is the appropriate politic to reduce it? Therefore, in order to take full advantage of this approach, it is of interest to calculate the rate of inequality by attribute and across regions. An application based on individual wellbeing data from Tunisian households in 2010 is presented to illustrate use of the proposed concepts. The rest of the paper is organized as follows: Section 2 presents the distribution of income per quantile. Section 3 presents the multidimensional inequality measure augmented by the fuzzy approach. Section 4 explores the empirical illustration, the most important results and policies to reduce inequality. Section 6 deals with the conclusion.
2 Distribution of income per quantile
In this section, we measure the monetary inequalities by using the distribution curve that shows how the consumption (or income) is divided in the population and across regions. The division of the distribution in quantile shows how the share of income are divided through of the same size of population groups, ranging from the poorest to the richest and present how this distribution evolves between regions. Due to lack of data, the illustration is limited to a measure using data from Tunisia in 2010. The information used is supplied from the household survey data come from the 11281 Tunisian household survey conducted by the Tunisian Institute of Statistics (2010). A brief summary the total annual expenditure variable is given in Tables (1).
Table 1. Summary statistics of the total annual expenditure variable (2010)
Minimum 
First quantile 
Median 
Mean 
Third quantile 
Maximum 
259 
5328 
8486 
10580 
13230 
197000 
The survey provides demographic characteristics of households. In order to take into account diverse geographical and socioeconomic characteristics of regions in Tunisia, we split the country into 7 different homogenous regions. Tunisia is usually subdivided into four naturals regions: The Greater Tunis, North East, North West, Centre East, Centre West, South East and South West. These investigations are carried out about the household including food consumption and nutrition, level of household economy, employment, population, housing conditions and literacy.
In Tunisia, we have two classes (figure1) therefore we noticed the absence of the middle class. The definition of the middle class indicate that the first criterion used to define its contours is the level of income or expenditure in deed we have other criteria such as socioprofessional category, the level of actual consumption of durable goods, the size of the family, residential space, employment status, heritage and perception of belonging to the middle class.
Figure1.Distribution of income in 2010
Figure 1 shows that the percentage of middle class in Tunisia including poor class is reached about 90 percent of the population. This class split into three classes the middle class with law spender capacity can easily switch in poverty it is about 12 percent, the intermediate middle class and the superior middle class it is about 80 percent. We did not notice a high level of inequity across regions. The government must work hard to bring development back to center stage. Our main concern in this quest is the urgent need to decrease the levels of inequalities. Development policies may help our country to develop the middle class. Our country need to design wholeofgovernment strategies to reduce inequalities. Any of these strategies should rest on three most important pillars: The first challenge is to create new productive and rewarding jobs. Second, we have to investment in education and skills development to promote employment and employability. Finally, reforming tax and benefit policies can also progress the distribution of income. Indeed, the choice of this welfare indicator depends on other factors. Inequality is a multidimensional phenomenon. They should in principle account different aspects of economic and social life, such as food, housing, clothing or the possibility of pursuing a profession, the state of health and education, for example. When admits inequalities integrate social and demographic components as life expectancy or literacy, this has undoubtedly affect the measurement of inequality, particularly in the context of regional comparisons, where coverage health or education, for example, can differ from one region to another. Indeed, it is interest to consider the level of inequality in a multidimensional form. We present in the section below an alternative approaches are evaluated based on the fuzzy approach. In order to take full advantage of it, we propose an index augmented by this approach.
3 Multidimensional inequity fuzzy measure
To measure inequity, we suggest the multidimensional inequality index by attribute k, defined by specific functions as discussed below. To define this index, we introduce the membership function from the fuzzy approach. This function can be inclusively range between 0 and 1.The membership function may be managed by specific functions as discussed below. This membership function is defined by the gap between the median , which is applied to the number of considered units, and the medial , which is applied to the importance of possessed character .
We suppose that and
The medial and median are expressed respectively as follows:
_{ }
and indicate respectively the percentage of payroll and employee.
The membership function measures the degree of the inequality by the socioeconomic attribute k.
_{ } (1)
If _{ }then concentration null
If then presence of concentration
If then strong concentration
The income inequality index, across individuals, is:
_{ } (2)
This measure of inequality is ranged from 0 refers to the perfect equality to 1, which refers to perfect inequality. While this coefficient has many desirable properties mean independence, population size independence, symmetry, and PigouDalton Transfer sensitivity it cannot easily be decomposed to show the sources of inequality. This inequity index is not entirely satisfactory. To see this, consider the axioms that make a good measure of income inequality, namely:
Axiom 1: Mean independence
This means that if all incomes were doubled, the measure would not change. The index satisfies this.
Axiom 2: Population size independence
If the population were to change, the measure of inequality should not change, ceteris paribus. The index satisfies this too.
Axiom 3: Symmetry
If you and I swap incomes, there should be no change in the measure of inequality. The index satisfies this. For example, if x’ is obtained from y’ by permutation of income, therefore
Axiom 4: PigouDalton Transfer sensitivity
Under this criterion, the transfer of income from rich to poor reduces measured inequality. The index satisfies this too. We suppose a distribution of income x’ obtained from a distribution y’ where we make a handoff progressive income. X is obtained from y by a single progressive transfer if there is i and r such as for all
The index satisfies this too.
Axiom 5: Decomposability
This means that inequality may be broken down by population groups or income sources or in other dimensions. The index is decomposable or additive across groups. That is, the total index of society is equal to the sum of the coefficients of its subgroups.
Axiom 6: Statistical testability
One should be able to test for the significance of changes in the index over time. This is less of a problem than it used to be because confidence intervals can typically be generated using bootstrap techniques.
Axiom 7: the relative invariance property
The relative invariance property is the homogeneity of degree zero of a function. . This property allows as overcoming the different nature units.
4 Empirical illustration
In this section, the data used is the same as in the section 2. The survey also provides demographic characteristics of households. In order to take into account diverse geographical and socioeconomic characteristics of regions in Tunisia, in this section, we divided also the country into 7 different homogenous regions to create use of the characteristics of different regions in Tunis; we split the households according to their location with respect to 7 different regions.
Table 2. The measurement of inequity by regions

Great Tunis 
North East 
North West 
Center East 
Centre West 
South East 
South West 
Consumption 
0.5853 
0.52 
0.511 
0.6726 
0.5863 
0.56 
0.609 
Alimentation 
0.336 
0.038 
0.049 
0.0318 
0.065 
0.0359 
0.1557 
Alcohol and tobacco 
0.0019 
0.0066 
0.0033 
0.003 
0.0062 
0.0017 
0.0027 
Dress 
0.0049 
0.0241 
0.0291 
0.017 
0.0171 
0.00141 
0.028 
Housing and energy 
0.1925 
0.1231 
0.2204 
0.2423 
0.212 
0.2476 
0.1534 
Furniture 
0.0132 
0.0182 
0.0127 
0.0152 
0.0246 
0.0093 
0.0174 
Health and care 
0.0232 
0.063 
0.0235 
0.038 
0.037 
0.0316 
0.013 
Transport 
0.06311 
0.06 
0.0282 
0.113 
0.306 
0.0494 
0.131 
Telecom 
0.0074 
0.0057 
0.0063 
0.0197 
0.0049 
0.0047 
0.0025 
Entertainment 
0.00181 
0.0036 
0.0019 
0.0052 
0.0013 
0.0011 
0.0016 
Education 
0.0018 
0.0024 
0.0013 
0.0014 
0.0035 
0.0006 
0.0007 
Hotel 
0.00303 
0.0016 
0.0006 
0.0014 
0.002 
0.0024 
0.00167 
Other expenditure 
0 
0.00066 
0.0049 
0.0079 
0.0033 
0.0069 
0.00072 
The region of North West (table 2, column 3) has relatively low levels of consumption reached about 51.1 percent compared to the other regions. And the region of Centre East (table 2, column 4) presented the highest inequity index reached about 67.26 percent. The inequality of the condition of life refers to the inequality of access to goods and services that satisfy the needs of household like foods, health, education. Access to education has always been one of the historical claims of the regions. A number of economists have argued that inequality leads to economic instability. One mechanism by which this happens is that the rich consume a smaller proportion of their income than the poor. Meanwhile, as inequality grows, individuals facing low or declining relative incomes maintain their consumption through borrowing.
The region of South West (table 2, column 7) has relatively low levels of inequality index of education reached about 0.07percent compared to the other regions. And the region of Centre West (table 2, column 5) presented the highest index reached about 0.35percent. We remark that we don’t have high difference in the rate of inequality but the governments have to investment in education and skills development to promote employment and employability. The Tunisian politic of education was regarded as the prototype of success, we must take attention to the problem of mass illiteracy. The principal reform in Tunisia will be devoted to developing and restructuring higher education and driven by technological skills and needs. In fact, the government effort must increase in rural primary education not only have large economic returns, but also contribute to equity in rural and interior areas to facilitate the access to best education. The number of pilot school increase over this decade. Indeed, we can analyses the education like investment as describe in human capital theory. Schultz (1961), Becker (1975) and Mincer (1958) showed that the most efficient technology is adopted more rapidly by the richly endowed economies for highly skilled workers. According to this authors, the level of education increase the economic growth rate, by accelerating the assimilation of technical progress. In fact, if we invest in education, we realize the wage gain over the long term. While the national perspective, this investment increase the economic growth rate and reduce inequality.
The region of South East (table 2, column 6) presents the highest level of inequality index of energy reached about 24.76 percent. Indeed, the region of North East (table 2, column 2) presents the low level it is about 12.31 percent of inequality of housing and energy. Moreover, it presents the highest level of inequality of health and care, reached about 6.3 percent. In fact, the region of South West (table 2, column 7) presents the low level, reached about 1.3 percent. The region of the Center East (table 2, column 5) has an important index of inequality of transport about 30.6percent compared to the other regions as the North West (table 2, column 1) registered only modest level, reached only 2.82percent. We notice that the level of the index of inequality is important but the main interesting is the quality of this services energy, health and care and transport so we can talk about the degree of poverty in these regions. The information conducted by the Tunisian Statistical Institute in 2010 prove that regional disparities are exacerbated by the concentration of public services, investment and economic activities in the coastal region, therefore the interior regions are less well served in terms of public services. It is the same with regard to the number of hospital beds, and the illiteracy rate is 30percent in rural areas against 15percent in urban areas and 23percent national. Excessive concentration of Directors attended the neglect of public services, including basic services in areas of the interior. Similarly, the concentration of economic activity in the regions of North East and Central East has led to regional disparities, the coastal region with 75percent of nonfarm employment. The governorate of Tunis has absorbed 23percent of all job vacancies in 2010 against only 4percent in the South east, although this region accounts for 9percent of the population. Without a doubt, Provision and equal access to highquality public services and strengthening social protection programmes will help provide equal opportunities for all citizens. For many developing countries and emerging economies, reducing inequalities will require tackling informality, expanding the social security system, and reducing underemployment.
Development of rural and interior regions infrastructure is a key of rural and economic life. But nowadays, Tunisian’s government gave less priority to rural and interiors regions infrastructure in promoting agriculture production rural Nonagricultural employment and improved living standards for the rural population. To develop the interior regions, we must introduce electricity, it can profoundly affect the interior rural regions. It may expend the production hours in the day. The television and radios extend accessible free entertainment and education indeed government must invest too in telecommunication. In fact, government spending on roads electrification, education and other public investment in rural areas contributed to rapid growth in agricultural production. Consequently, policies to promote agriculture production and to promote growth led to increase inequality in the North West regions.
Many strategies should rest on most important pillar which is the reforming tax and benefit policies can also progress the distribution of income. We must go slowly for fear of practicing a tax optimization policy so if we tax the rich a lot it encourages them to go abroad or refrain from investing. Therefore, we can reduce inequality through the creation of employment, encouraging investors to invest and to ameliorate employment in Tunisia and following an appropriate fiscal policy. Certainty, taxes coming from rich are not always efficient and adequate but they are necessary from the political and social point of view. Consequently, the main solution is the use of a redistributive policy to promote the redistribution of wealth in favor of the poorest, to curb inequality of opportunity, to a high quality education for the most disadvantaged and to facilitate access to employment. We can also reduce inequality by giving more chance to jobless to create their own business through tax havens and encouraging them with micro credit with low interest rates .the government is obliged to use these main solutions to reduce unemployment therefore reducing inequality.
5 Conclusion
The analysis of inequality is frequently seen as a unidimensional form. In fact, it is interest to consider the level of inequality in a multidimensional form they should in principle account different aspects of economic and social life. Nowadays, the choice of the welfare indicator depends on the standard of living and other factors, such as health, education level and transport. In this paper, a multidimensional form of inequality is considered. We have presented a systematic treatment of the conceptual frame work of the measurement of inequality augmented by the fuzzy approach. Fuzzy logic is a type of multivalued logic consequential from fuzzy set theory to deal with probable reasoning rather than the precise one. Our main anxiety in this search is the urgent need to reduce the levels of inequalities and to develop the quality of services. Development policies may help our country to develop the middle class. Proposed solutions range from the practical but weak, like raising the minimum wage, to the fanciful, like global wealth tax. Our country need to design wholeofgovernment strategies to reduce inequalities. Any of these strategies should rest on three most important pillars. The first challenge is to create new productive and rewarding jobs. Second, we have to investment in education and skills development to promote employment and employability. In fact, if we invest in education, we realize the wage gain over the long term and my reduce poverty too. While the national perspective, this investment increase the economic growth rate and reduce inequality. Finally, reforming tax and benefit policies can also progress the distribution of income.
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