اندازه گیری پویای نابرابری درآمد از مدل های مارکوف : کاربرد برخی از کشورهای اروپایی

In this paper we present a methodology for measuring income inequality dynamically within a Markov model of income evolution. The proposed methodology requires knowledge of the evolution of the population and the averages and medians of the incomes in a country and allows the computation of dynamic inequality indices. The methodology is supported with statistics from Eurostat data applied on France, Germany, Greece and Italy.

The themes of inequality and wealth concentration have been extensively studied by means of static inequality indices, see e.g. Athanasopoulos and Vahid (2003) and Kooleman and Van Doorslaer (2004). One of the most common indices is Theil's Entropy (Theil, 1967). The index is scale independent and invariant to replication of population, it satisfies the strong principle of transfers and it is additively decomposable, see e.g. Athanasopoulos and Vahid (2003) and Cowell (1995). Many papers make use of Markov Chain modelling to describe income dynamics (Bickenbach and Bode, 2003, Quah, 1993 and Quah, 1994). Nevertheless these contributions did not propose measuring inequality through dynamic indices. This drawback has been overcome in the recent paper (D'Amico and Di Biase, 2010), where the authors proposed dynamic inequality indices. This generalisation was made possible by considering a population that evolves over time according to a semi-Markov process and by considering the income of each economic agent as a reward process. Therefore it is possible to justify changes in indices when the population composition varies over time. The model was implemented in D'Amico et al. (2011) to simulate an artificial economic system with immigration; however critical points that occur when handling with real-world applications were never faced. The main problem of applying the model proposed in D'Amico and Di Biase (2010) is that of data availability. Indeed, the model requires microdata concerning income evolution of agents, which are very often unavailable to researchers. To overcome this problem we propose in this paper an application methodology that makes possible the application of the model when only the averages and medians evolution of the incomes in a country are available. The methodology is considered for a Markov Chain model of income evolution. The results showed different types of temporal evolutions of the index in the considered countries, suggesting the strong necessity to implement an economic integration European policy. The paper is organised as follows: Section 2 describes the stochastic model, the Dynamic Theil's Entropy and the computation of its expectation in the transient and in the asymptotic cases. Section 3 presents data and methodology of application. Section 4 provides the results. Finally, some conclusions and further developments can be found in Section 5.

In this paper we explained a stochastic model of income inequality. The application of the model requires microdata series of income that are unfortunately, rarely available to researchers. In order to surmount this difficulty we proposed a procedure that was able to implement the model with incomplete data and with only the accessible means and medians of the incomes. In particular, we analysed Eurostat data of four European countries: France, Germany, Greece and Italy. The results of the application showed different types of temporal evolutions of the index. Indeed, the index can increase and/or decrease and it can be convex and/or concave throughout the considered horizon time. Among the indications provided by the model we would like to highlight the strong necessity to implement an European policy of economic integration, given the very different behaviours of inequality in the studied countries. For these reasons we think that the results are relevant for the adoption of an economic and social policy of inequality maintenance. Possible avenues for future developments of our model include: a) Measuring the effects of the fiscal system on the Dynamic Theil's Entropy. By recovering the tax rates and the tax bases in the different countries, we could reconstruct the gross income distribution and then it should be possible to compute the Dynamic Theil's Entropy based on the gross incomes instead of the net incomes. The difference between the two values could be interpreted as a measure of the reduction of the inequality due to the redistributive effect of the fiscal system; b) Measuring the variability of each income yCj by considering the income as a random variable and then the vector View the MathML sourcey¯(t) as a random process. This major complexity could be addressed by using Markov (or semi-Markov) reward processes.