The Atkinson theorem is formulated in a rigorous way. Societal utility of an income distribution is to be maximized for an additive utility function such that the mean income is preserved. Then, for finite distributions with rational probabilities, (1) majorization, (2) finite sequences of Pigou-Dalton transfers, (3) the Lorenz order and (4) the convex stochastic order are equivalent. When distributions are no longer finite, majorization and finite sequences of Pigou-Dalton transfers refer to approximate distributions. With these concepts, the Atkinson theorem can be shown to also hold for general distributions.
Inverse formulations of the Atkinson theorem are given additionally. Switching between convex and concave utility functions can hence be thought of as balancing the income distribution of a society, depending on the perceived social state of a society, sometimes too much inequality, sometimes too little.
Utility Function Income Distribution General Distribution Lorenz Curve Inequality Measure
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