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The Mathematical Relation Between the Income Density Function and the Measurement of Income Inequality
Daniel B. Levine
Neil M. Singer
Abstract
This paper presents a general formalism for calculating the effect of taxes on income distribution, and the resultant effect on income inequality. We first derive a closed form expression for income inequality (defined from a Lorenz curve) in terms of the income density function. By way of illustration, we use this expression to calculate the effect of a proportional and a lump sum tax on income inequality in a simple exponential income distribution. The results show that the effect of a lump sum tax imposed after a proportional tax is a function of the proportional tax rate, even though the proportional tax itself does not change inequality.
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