Effects on Taxation on the Forecasting of Income Inequality : Evidence from Germany , Greece , and Italy

Work supported by MIUR research grant PRIN 2008, “Risparmio pensionistico e benefici previdenziali privati: scelte individuali, rischi per il gestore” (“Retirement savings and private pension benefits: individual choices, risks for the operator”), Scientific director: prof Ermanno Pitacco, University of Trieste, Italy. Effects on Taxation on the Forecasting of Income Inequality: Evidence from Germany, Greece, and Italy

Income inequality can be measured by means of econometric indices.The most common are the Gini index and Theil's entropy.
Henri Theil (1967) introduced Theil's Entropy as an alternative measure of income inequality.This index is popular because of its several desirable properties.Of these properties, we use additively decomposability inequality to obtain the values of the indices related to the three European countries of interest.In the whole population, the Theil's index can be additively decomposed to the inequality between the subgroups and to a weighted average of the inequality within each group.
Guglielmo D'Amico and Giuseppe Di Biase (2010) recently generalized Theil's Entropy by taking into account the random time evolution of the agents' incomes.This led to the formulas for the general dynamic inequality and the concentration indices.
In this paper, we investigate the impact of the fiscal system on wealth redistribution in the population for each of the considered countries.We could recover the gross income distributions by using the individual income tax rates in these countries over the net income distributions.Then, by following the methodological approach by D'Amico, Di Biase, and Raimondo Manca (2012), we compute the Dynamic Theil's Entropy.
Fiscal policy could be a fundamental tool of macroeconomic policy that can decrease income inequality by favoring the redistribution of the wealth.PANOECONOMICUS, 2013, 6, pp. 707-723 The paper is organized as follows.Section 2 briefly describes the formula for Dynamic Theil's Entropy.Section 3 presents the data and the methodology of the application of the model.Section 4 shows the results.Conclusions are provided in Section 5.

Literature Survey
In literature, we find two lines of research on the study of the changes of income distribution.One line of research regards the study of static econometric inequality indices, whereas the other proposes income dynamic models.
The themes of inequality and wealth concentration in a subclass of an economic system have acquired high relevance and they have been studied extensively.
Anthony B. Atkinson (1970) specified the form of the social welfare function to emphasize that any measure of inequality involves judgments about social welfare.
François J. Bourguignon (1979) deeply studied the Theil's coefficient and the logarithm of the arithmetic mean over the geometric mean.Frank Cowell (1980) investigated many general decomposable measures.
Anthony F. Shorrocks (1984) proved that the decomposable inequality measures must be monotonic transformations of the additively decomposable indices under weak assumptions.
More recently, Oscar Bajo and Rafael Salas (2002) provided a connection between concentration and inequality measures by putting a particular emphasis on the general entropy inequality indices.These theoretical aspects were developed by Casilda L. De La Vega and Ana M. Urrutia (2005) and Hamid Shahrestani and Bijan Bidabad (2010).The former generalized the decomposability notion and studied those inequality measures that admit a path independent multiplicative decomposition.The latter designed a model to estimate the functional parameters of the Lorenz curve.The Lorenz curve is a graphical representation of the cumulative income distribution function.
Among the significant applications we mention the study by George Athanasopoulos and Farshid Vahid (2003) on the changes in income inequality of individuals in Australia by assessing their statistical significance through the bootstrap method.Moreover, Xander Kooleman and Eddy Van Doorslaer (2004) added an intuitive understanding of the concept of the index for measuring relative inequality with an application of health-related measures by income and, finally, Ranjpour Reza and Karini T. Zahra (2008) analyzed income convergence in ten of the new members of the European Union toward the average European Union per capita income.
In parallel, the other research theme concerns modelling the dynamics of income time.The most common approach relies on the use of the Markov chain.
In particular, in a series of papers, Danny T. Quah (1993Quah ( , 1994Quah ( , and 1996) proposed the use of the Markov chain to verify the wealth convergence hypothesis and to estimate the transition matrix for the world's cross-country distribution of per capita incomes.
Frank Bickenbach and Eckhardt Bode (2003) tested the Markov chain hypothesis of income dynamics by using data for the states of the US from 1929 to 2000.PANOECONOMICUS, 2013, 6, pp. 707-723 With the aim of producing an approach that unifies the previous two lines of research, D'Amico and Di Biase (2010) proposed the use of a semi-Markov process to compute inequality indices dynamically.Indeed, with the dynamic inequality indices, it is possible to justify the changes in the indices even when the population composition varies over time, which is a limitation of the static inequality measures.Actually, the static measures represent a simple snapshot of the economy and further, they are unable to capture the source of income variability over time.Additionally the static approach disregards the dependence within the sequences of incomes and is not able to describe the population dynamic given that it does not make use of a transition probability matrix that describes the agents' income evolution.
The economic inequality measures provide information that can be useful to promote the process of economic integration among different countries.A concrete example is the process of European integration, which was well exposed by Božo Stojanović (2011).
Moreover, it is well recognized that economic integration can be promoted by an adequate fiscal policy, see for example, Miroslav Prokopijević (2010) and Philip Arestis (2011).
Based on the above subjects and the related literature, in this paper we try to fill the gap by deriving an income dynamic model that takes into account the effects of fiscal policy on wealth inequality.

Dynamic Theil's Entropy
Nowadays, it is important to forecast income inequality.A simple way to achieve this is by considering the pattern of a generic static inequality index, such as, Theil's Entropy.
However, the use of the dynamic version of the Theil's index is highly instructive with respect to the consideration of the pattern of the static inequality index.The static approach indeed presents at least two disadvantages with respect to our approach.
First, it disregards the dependence within the sequences of incomes owing to the presence of an underlying Markov chain that drives income evolution.
Second, the estimation of all transition probabilities that are required in our approach provides valuable information about the evolution of the population shares in each class and thus, the mobility in the distribution.Policy makers could use this additional information for planning welfare policies.
Let us assume a system of N economic agents that at each time , t  each produces a quantity ( ) We classify each agent by allocating it, at each time through an allocation map, see Step 2 in Section 3 for a specific choice of the map.This permits the recovering of N time series of states of E : ( ) and suppose that such time series are realizations of a discrete time Markov chain with a transition probability matrix P whose element ij p denotes the probability that an agent now allocated in i C will enter the next allocation j C . For It represents the average wealth of agents in class j C .We assume that each time an agent is in state j C , it produces a wealth equal to We define the process a n t a n t a n t a n t  which describes the time evolution of the shares of income among the classes of population: .
In its static form, Theil's entropy was defined in Theil (1967) by .
Given the population configuration n at time 0 t  and the vector of average incomes ( ) y t , the Dynamic Theil's Entropy is the stochastic process: .
The first moment can be evaluated by: where p.c. is the set of all possible population configurations, ( ) 1 (0) ( ) p are the t-step transition probabilities of the Markov chain.
Notice that the static form of the Theil's entropy makes provision for share of incomes that are numbers, whereas that of the dynamic Theil's entropy uses stochastic processes.Additional details on the Dynamic Theil's Entropy can be found in D'Amico and Di Biase (2010).

Data and Methodology of Application
To apply the model, we should retrieve the series of microdata concerning agent income evolution.If the individual sample paths (1) are observed, then, the transition probability matrix P can be estimated by its maximum likelihood estimator; see for example, Theodore W. Anderson and Leo A. Goodman (1957).From the theoretical point of view, the use of microdata poses no problem, but from a practical point of view, the analyst should consider that it is computationally and economically not convenient to work with all sample paths of agents in a country because they are of the order of tens of millions and therefore, sampling techniques should be applied.
PANOECONOMICUS, 2013, 6, pp.707-723 For these reasons, we propose here a methodology based only on some macrodata, from which we recover the estimation of the probability matrix P and the expected value of the Dynamic Theil's Entropy   ( ) e E T t .The procedure is simple and can be replicated because input data are widely available.It is important to point out that our procedure, based on macrodata, could be combined with microdata to improve the accuracy of estimation when using the microdata.This is confirmed by previous studies in combining macro-and micro-data in microeconometric models as stated by Guido Imbens and Tony Lancaster (1994).They affirmed "...the reason (of the improvement) is that the aggregate variables give information with little or no sampling error about the averages of the micro variables." We used macrodata downloaded from the Eurostat website (http://epp.eurostat.ec.europe.eu/) on the averages and medians of the net incomes, (see Table 1).
and the corresponding probability distribution function is and the median is

 
Med X e   .For example, by using data in Table 1 for Germany in 2005 and solving the system of equations: and, consequently, the distribution of the income within the country.
Step 2: Building the states We consider a five states-Markov chain model and allocate each agent in one of the states according to the following rules: If an economic agent has less than a quarter of the country's average per capita income ( 0.25 r ), then it is considered poor (class 1 C ); If the agent has an income between a quarter and one-half of the country's average per capita income, then it is considered medium-poor (class 2 C ); If the agent has an income between one-half and the country's average per capita income, then it is considered medium (class 3 C ); If the agent has an income between the country's average per capita in- come and its double, then it is considered medium-rich (class 4 C ); If the agent has an income more than the double of the country's average per capita income ( 2 r ), then it is considered rich (class 5 C ).The bounds are those proposed by Quah (1996)  and represent a reasonable subdivision.

Step 6: Recovering the gross income distribution
For each country, we retrieved the tax rates of the year 2011 and the tax bases.We report them in Table 2.This table summarizes the fiscal systems in force in Germany, Greece, and Italy in 2011.Indeed, the table shows the percentage of taxes computed over the gross income brackets reported on the same line that people in each country have to pay.We considered the following computations to reconstruct the gross income distribution.
Denote by i b the maximum tax base of the i-th range of income and i t the tax rate of the i-th range of income.Compute the maximum net income for the i-th range of income as and the maximum cumulated income for the i-th range of income as . Finally, given a net income d of the year 2005 such that , the corresponding gross income is given by Step As explained in the previous steps, the methodology of application assumes that a time homogeneous Markov chain with the transition probability matrix P describes the income dynamic.Under this assumption the probability

 
agent is in at time agent was in at time 1 is independent of t .This implies that, P is the probabilistic representation of the economy for any time t once the matrix P was estimated with data from 2005 to 2008.Then, to execute forecasts starting from 2012, we need to fix the fiscal system in force in 2011 and assume that it will not change.

Results
We implemented the procedure described in Section 3 for each country.

P
Matrix 1 P expresses the transition probabilities related to the net income dy- namic, whereas matrix 2 P refers to the gross income dynamic.For example, the value 0.896 in 1 P expresses the probability that an agent in the class of net income 1 C in this year will get to the class of net income 1 C in the next year and the value 0.078 in 2 P expresses the probability that an agent being in this year in the class of gross income 5 C will get to the class of gross income 4 C in the next year.
As can be seen from the matrices 1 P and 2 P , the income dynamic processes show high persistence, indeed, the matrices are both diagonally dominant and monotonic.
Figure 1 shows the forecasting for the Dynamic Theil's Entropy for the net and gross incomes for the next 10 years in Germany.Both forecasts express a decreasing inequality.This means that the economy of Germany is moving towards a fairer wealth allocation among the classes.The introduction of the fiscal system decelerates this convergence towards a fairer income distribution.In particular the comparison between the values computed on the net income (grey line) and those computed on the gross income (black line) allows us to understand how the fiscal system affects the wealth distribution in Germany.

Greece
As for Greece, the transition probability matrices are: 0.99450 0.00521 0.00029 0.00000 0.00000 0.00001 0.99864 0.00135 0.00000 0.00000 0.00000 0.00003 0.99806 0.00190 0.00000 0.00000 0.00000 0.00251 0.99746 0.00003 0.00000 0.00000 0.00019 0.00016 0.99965 0.99800 0.00120 0.00080 0.00000 0.00000 0.00100 0.99800 0.00100 0.00000 0.00000 0.00000 0.00440 0.98700 0.00490 0.00000 0.00000 0.00000 0.00300 0.99400 0.00300 0.00000 0.00000 0.00020 0.00040 0.99940 The values show a very strong immobility in Greek society.Matrices 3 P and 4 P show that each class represents a cluster.Indeed the probabilities of migration among the classes are very low.This implies that poor agents tend to remain poor and rich agents tend to remain rich.Both matrices 3 P and 4 P are both diagonally dominant and monotonic.It is noteworthy that the introduction of the fiscal system tends to increase the probability to move towards a poorer class in the next period in spite of a very small quantity of probability.
Legend: x-axis is t (time in years), y-axis is the expectation of the Dynamic Theil's Index.

Figure 2 Dynamic Theil's Index for Greece
Figure 2 shows the forecasting of the expectation of the Dynamic Theil's Entropy for the net and gross incomes in Greece.The inequality in the net income (grey line) distribution is approximately constant in time with a tendency to rise slowly.The inequality in the gross income (black line) distribution is markedly lower and exhibits a decreasing path.

P
The matrices are diagonally dominant and monotonic as well.The graphical results shown in Figure 3 lead to the following remarks.The inequality is growing in time for both the net income (grey line) and gross income (black line).Among the considered countries, Italy registers the fastest growth in terms of inequality.The introduction of the fiscal system does not mitigate this increase and it is unable to invert this trend.
A decision maker involved in the policy of economic planning aiming to reduce the initial inequality value of 0.52 has to consider that, ceteris paribus, the inequality will grow further than 0.52 and, therefore, an intervention that is more effective should be planned.
Legend: x-axis is t (time in years), y-axis is the expectation of the Dynamic Theil's Index.
Source: Authors' estimations.The inequality in the net income was higher than the inequality in the gross income for all the considered countries and this is a surprising result.The reason is that, in our model, the Theil's index related to the net and gross distributions are computed by changing the size of the population classes.This happens because, for example in the poorest class 1 C (net), we allocated all the agents having less than one-quarter of the net country's average per capita income.Instead when working with the gross distribution, in the class 1 C (gross), we allocated all the agents having less than one-quarter of the gross country's average per capita income.The number of economic agents that belong to class 1 C (net) and class 1 C (gross) is not the same depending on the specific tax rates and tax bases of the countries.This implies a different population configuration for all the classes and for all the years.Obviously, this fact influences the transition probability matrices and the dynamic Theil's entropy.On the contrary, if we had built the model by considering one of the following two alternative schemes: a. fix the monetary bounds of the classes ( 0.25 , 0.5 , , and 2 r r r r) to be the same in both the case of net and gross incomes; and b. fix the monetary bounds such that the numbers of agents in i C (net) and i C (gross) are equal; then, certainly, the inequality in the net distribution would be less than the inequality in the gross distribution.

Conclusions
In this paper, we have explained a stochastic model of income inequality.The application of the model requires microdata series of income.We have proposed a general procedure based only on the averages and the medians of the income of each country.These data are widely available and can be considered very accurate.In particular, we analyzed the Eurostat data concerning the three European countries: Germany, Greece and Italy.
We retrieved the tax rates and tax bases for these countries and then, we recovered the gross income distributions.We applied the model to compare the inequalities in the net and gross income distributions.We could measure the wealth redistributive effects caused by the fiscal system from the distances between the curves in the figures.
The results of the application showed different types of temporal evolution of the index.In Germany, the index, forecasting on the gross and net distributions, reveals a decreasing and convex shape.In Greece, the inequality forecast of the net distribution is quite flat, whereas that which is computed on the gross distribution is decreasing.The two curves tend to move away from each other.On the contrary, in Italy, we expect the inequality to grow in time for the net and gross distributions.
Moreover, since the model provides a forecast of the inequality in the population on a given horizon of time, a policy maker can use it to implement a fiscal system that is able to achieve the desired inequality.A study toward this objective will be our next work.
the income class of the r-th agent at time t .Individual sam- ple paths are observed in the data (1), which are often called microdata.Set   1,..., ; 0,...
share of income due to class j C : 1 we compute the average net income of its agents as the conditional expectation of the income distribution recovered in Step 1 given that the income is in class k C : 4: Iteration of steps 2 and 3 for the years 2006, 2007, and 2008 The iteration of Steps 2 and 3 for the years 2006, 2007, and 2008 produces the aggregate data of the number of individuals in each state at each year for each country.

Figure 1
Figure 1 Dynamic Theil's Index for Germany

Figure 3
Figure 3 Dynamic Theil's Index for Italy

Table 2
Individual Income Tax Rates Source: Authors' representation with data from http://www.worldwide-tax.com.
It is noteworthy that, in the net income case, approximately 25% of the agents in class 5