Analysis of Household Income in Poland Based on the Zenga Distribution and Selected Income Inequality Measure

Abstract Research background: A lot of research has been directed at describing empirical distributions by using a theoretical model. In the literature there are proposals for various types of mathematical functions. In 2010 Zenga proposed a new three-parameter model for economic size distribution which possesses interesting statistical properties which can be used to model income, wealth and financial variables. Purpose: The aim of this paper is to apply the Zenga model to income distributions in Poland by voivodeship. Research methodology: The basis for the calculations presented in the paper has been based on the individual data coming from a random sample obtained within a Household Budget Survey conducted by the central Statistical Office in the year 2014. The parameters estimates of the Zenga distribution were obtained by means of the D’Addario’s invariants methods, mainly with the Pietra index. Results: The results of the conducted approximations, presented in the paper confirmed the good consistency of the Zenga distribution with the empirical income distribution in Poland, both in total and for households. Novelty: The study contributed to the application of a new three-parameter income distribution model to describe income distributions in Poland.


Introduction
The analysis of the income distribution has a long history. Its evolution has been considered essential to explain not only the causes but also the potential consequences of inequality and poverty. For a majority of countries we can observe similar, characteristic shapes of the income distribution. For many countries income distributions are single-modal with right-sided asymmetry. A lot of research has also been directed at describing empirical distributions by using a theoretical model. This approach can be useful for many reasons. Knowledge of the model of income distribution, which is a simple approximation of empirical distribution and knowledge of tendency of its parameters development may be used to predict the behavior of a particular variable in a following period of time. Nevertheless, the approximation of the empirical wage and income distributions by means of theoretical curves can smooth the irregularities coming from the method of data collecting. While choosing a statistical model it is important both, to find out a theoretical distribution function that would characterize empirical frequency distribution and to choose suitable methods to calculate the parameters of the model. In the literature there are proposals for various types of mathematical functions. Especially the threeparameter distributions have represented a good approximation of income distribution for many countries. Very high accuracy with empirical distribution is characteristic for the Dagum and Singh-Maddala models. These models are the three-parameter density functions with relatively simple analytical forms. Various theoretical distributions for example Pareto, logarithmic-normal, and gamma were used to approximate the income distributions of a Polish population. In the literature, many papers perform this problem for various families of distributions (see Wiśniewski, 1934;Vielrose, 1960;Lange, 1967;Kordos, 1968;Kordos, 1973;Kot, 1999;Kot 2000). The analysis of the income and wage distribution in Poland confirmed that the Dagum and Singh-Maddala distribution are particularly well fitted to empirical data. This observation is widely documented in the literature (see Brzeziński, 2013;Jędrzejczak, 1993;Jędrzejczak, 2006;Łukasiewicz, Orłowski, 2004;Ostasiewicz, 2013;Salamaga, 2016).
In 2010, Zenga proposed a new three-parameter model for distribution by size. The model is a Beta-mixture defined for non-negative distributions that has positive skewness and a Paretian right-tail. Its parameters separately control the location and inequality. In this model μ is the scale parameter and is equal to the expect value, α and θ are shape parameters that the inequality depends on. Studies performed in various countries show that the Zenga distribution exhibits high conformance to the empirical distributions of incomes. This distribution has not been used so far to analyze income distributions in Poland. The aim of this paper is to apply Zenga distribution for Polish Voivodeships.
The parameter α is an inverse inequality indicator and θ is a direct inequality indicator.
In particular the bigger value of the parameter α the less unequal the distribution and the bigger the value, the more unequal the distribution (Porro, 2015). The expected value E(X) is always equal to the parameter µ.
The moment of order r for the Zenga distribution is defined as follows: where ℕ, < + 1, > 0 In the case θ > 0, the distribution function F(x: μ; α; θ) is described by the equation where: is the incomplete beta function.    It is easy to see in graphs 1-4 how many behaviours it can have, changing the values of the parameters. The new density allows for a wider variety of shapes than the traditional three-parameter models of income distributions as for example the Dagum and Singh-Maddala distribution. The distribution depends on three parameters: in particular µ is a scale parameter, α in an inverse inequality indicator and it controls the tails of the distribution, while θ is a direct inequality indicator and it controls the distribution around the expected value µ (Arcagni, Porro, 2013).

Measures of income inequality
The most popular measures of income inequality are the Gini coefficient and Zenga coefficient. The Gini coefficient is half the relative mean difference and is usually defined while being based on the Lorentz curve. The Zenga coefficient is defined and based on the Zenga curve.
The graphs of the Lorenz curve for Zenga distribution can be obtained by putting in the abscissa F(x) and inordinate axis H(x), for a sufficient number of values of x. It is worth noting here that these measures are invariant to scale transformation, so it is enough to consider the case f(x: 1; α; θ). We know that: The first incomplete moment of the Zenga density f(x: μ; α; θ) is given by The graphs of Zenga's point inequality can be obtained putting in abscissa F(x: 1; α; θ) and A(x) on the ordinate axis for a sufficient number of values of x.
The Gini inequality index G can be obtained by numerical integration. Similarly, the Zenga inequality measure I can be obtained by numerical integration of the Zenga point inequality curve for the entire income range. See Zenga (2007aZenga ( , 2007b for further details about the Zenga inequality measure.

Results
The basis for the calculations presented in the paper has been based on the individual data To find a degree of adjustment of a theoretical distribution to the empirical one we have calculated the goodness of fit measures: the Mortara index A 1 , the quadratic K. Pearson indexA 2 and the modified quadratic index A 2 ' which are described by the following formulas: Where j n and ˆj n are respectively the observed and estimated frequencies of the j-th interval.
The coefficient of distributions similarity W p for the empirical data arranged into a grouped frequency distribution with s-class intervals can be calculated by the formula: where w i and w i ' represent empirical and theoretical frequencies, respectively.
The results of the approximation of the empirical income distributions in Poland by means of the Zenga model, together with the goodness of the fit measures A 1 , A 2 , A' 2 and W p are presented in Table 1 and in Figure 5. The approximating methods for the Zenga distribution have been analysed in the paper (Zenga, 2010).    Table 2 corresponding to the areas below the curves are included.      In terms of mean income levels, the largest discrepancy was observed between the mazowieckie and warmińsko-mazurskie voivodeships, while the highest difference in inequality levels we observed between the mazowieckie and lubuskie voivodeships (G maz = 0.3426, Z maz = 0.6783; G lubus = 0.2940, Z lubus = 0.6240). The comparison between the L(p) and I(p) curves highlights the important results. In Figure 9 we can discover that the inequality curves I(p) for wielkopolskie and pomorskie lies below the curve for mazowieckie for the entire income range. It is worth mentioning that, income inequality patterns for warmińsko-mazurskie, wielkopolskie and the national Total are very similar. Furthermore, we can observe that the inequality for mazowieckie is always beneath the national Total. In figures 1-3 we can observe that the inequality curves L(p) for wielkopolskie, warmińsko-mazurskie and Total lies above mazowieckie. It is easy to see that the considerable coherence of the results summarized in (Table 2) and (Figures 6-11) confirm the adequacy of both these tools to measure the inequality degree for polish HBS.

Conclusions
In this work we use a new three-parameter model of income distribution which seems to provide a better fit to the experimental data. The Zenga density function can take on various shapes and this attribute allows a good fitting also for small incomes. This feature of the Zenga density function allows for a better fitting to experimental data than other probability density functions, which have been proposed as income distribution models.
The application of the Zenga model is better than the existing approaches to income analysis because the Zenga distribution parameters can be interpreted in economic terms. The μ parameter is interpreted as an arithmetic mean to individual income or an arithmetic household income mean, and other parameters α and θ can be interpreted in terms of income inequality.
The results of the conducted approximations, presented in the paper confirmed the good consistency of the Zenga distribution with the empirical income distribution in Poland, both in total and for voivodeships and households. The results of the calculations concerning the level of distribution inequality, revealed differences between the Polish voivodeships. Results concerning the distributions of incomes of Polish households for the year 2014 are an extension of our previous paper (Jędrzejczak, Trzcińska, 2018).
The Zenga distribution meets the criteria set for the theoretical income distribution, and the transparent economic interpretation of the Zenga density parameters is an additional argument for using this model to describe a Polish Household Budget Survey.