An overall inequality reducing and horizontally equitable tax system with application to Polish data

In the case of homogeneous populations progressive taxation is usually associated with a reduction of inequality in after tax income distribution. In the real world, populations of tax payers are not homogeneous and households differ not only with respect to income but also in other social, demographic and economic characteristics. The aim of the paper is to propose an inequality reducing tax system taking into account, in some way, differences in needs between tax payers. A population of households is considered and each household is distinguished by two attributes: pre-tax income and family size. To compare incomes in this heterogeneous environment an affine transformation is proposed as equivalent income function, from which income dependent equivalence scales are derived. By first, the horizontal equity concern of a tax system has to be faced. Ebert and Lambert (2004) define horizontal equity requirements for income dependent scales. Applying the proposed scales a tax system can be derived and we show that it satisfies the horizontal equity conditions as defined by two authors. Then, we consider the redistributive effects of a tax system by introducing an inequality parameterized measure as in Ebert and Moyes (2000). We show that, when the reference type family is fixed, the application of the suggested equivalence income function is such that the tax system can be overall inequality reducing provided that the reference type tax function is average rate or minimal progressive. As it is known, conditions ensuring that a tax system is overall inequality reducing are not sufficient to guarantee inequality reduction within each set of households with the same size. We derive restrictions which allow the equivalent income function to generate a tax system which is overall inequality reducing and within type inequality reducing. These restrictions concern the tax function for the reference family type and the domain of incomes. To see where these restrictions come from, it is important to note that a tax system which is overall inequality reducing for a fixed reference may well increase inequality (or keep it unaltered) when another reference type is chosen. But, if a tax system is within type inequality reducing, it is also overall inequality reducing for any reference type. Ebert and Moyes (2000) show that only two particular form of equivalence function can be adopted when reference independence is required. The first function yields an income independent absolute scale; the second function yields an income independent relative. The former has to be used when inequality is considered from the absolute point of view, the latter has to be used when an intermediate between absolute and relative concept of inequality is adopted or when relative inequality is considered. Here we suggest only an income transformation function which originates income depending scales. On considering inequality either in absolute or in relative sense, using this function we obtain a tax system which is overall inequality reducing and reference independent if some restrictions hold on income domain, on the set of taxable income and on the tax function for the reference type: we think that these restrictions are not too limitative as it can be understood from the applicative part. The paper is organized as follows. In the first part, following Ebert-Lambert (2004), horizontal equity (HE) conditions are defined in both cases when equivalence scales are income dependent or income independent. Then, following Ebert-Moyes (2000), we summarize what has to be meant for an overall inequality reducing tax system (OIR), for a within type inequality reducing (WIR) tax system and for a reference independent tax system (RI). In the second part the paper we describe the particular affine income transformation function that we adopt and we investigate on the conditions allowing to this function to generate a tax system which is OIR and WIR. In the third part we simulate modifications to the present personal income tax in Poland, with the aim to design a more family oriented system. The paper is completed by an appendix which analyzes the relation among the most important instruments used in the real world to take into account horizontal equity. Quotient, exemption and tax credit are discussed under different hypotheses on tax function