Mathematical control of income tax revenue distribution for the US joint tax returns: 1996-2008

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2013-02-04

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Andrianjafinandrasana, Misaina Navaloniana

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Abstract

This thesis introduces time dimension into income taxation. Economic theory derives optimal income taxation by explaining individual behaviors through utility and social welfaremaximization. We argue that observing collective behaviors through the distribution of number of returns and the revenue themselves offers a viable alternative. Our study is confined toUS joint returns individual income tax from 1996 to 2008. We introduce a combination of cubic spline fit to data and Pareto distribution to estimate the revenue and number of returns distributions. Normalization admits comparative study of year to year changes in those distributions. We discover that the tax burden of low income taxpayers has decreased consistently, while the conventional rule of inflation indexation of tax bracket only is applied. This is an instability in the sense that the distribution of revenue transforms into a qualitatively different distribution over time. We derive the control of the instability. Control using brackets only implies optimal brackets arewidened over the peak of the revenue and shifted to the right for higher brackets. Control using rates only implies an overall reduction of marginal tax rates. Control using both brackets andmarginal rates combines these two features. Some brackets are removed and consequently marginal rates for some taxpayers are reduced. Dependency of the distribution of number of returns to tax liability and tax parameters proves to have insignificant effect on the optimal tax parameters. Fairness is defined in the literature as tax liability per unit income. Control of the instability togetherwith a uniformchange in fairness is addressed. There is a trade-off between controlling instability and ensuring constant change in the fairness of the tax system. We derive the optimal brackets and marginal rates that minimize this trade-off. As expected, the general rule for optimality is that marginal tax rates are barely changed to provide constant change in fairness. Yet, small adjustment of the brackets provides control of the instability.

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