The impossibility of a progressive tax structure

When households are identical in all respects other than income, a non-decreasing average tax rate is both necessary and sufficient for post-tax incomes to be always at least as equal as pre-tax incomes. This paper investigates whether this well known result can be extended to households with different needs. A tax structure which never increases inequality (in the Lorenz sense) is shown to be unable to discriminate between household types. If, in addition, the tax structure never reduces the relative income of the poorest member of the neediest group of households, then the same proportional tax must be applied to all households, and redistribution is impossible either within or between groups of households with differing needs. A similar conclusion is reached using conventional equivalence scale arguments, and also applies when the parameters of the tax schedules are determined endogenously. We discuss the significance of this impossibility result and its implications for tax design.