by André Casajus[*] and Andreas Hoffmann
Estonia was the first European country to introduce a flat tax on income in 1994. Many others followed. For example, Hungary successfully introduced a flat tax in 2012. In the U.S., some of the States (e.g. Pennsylvania) have introduced a flat tax on income. As in Germany, however, the federal income tax in the U.S. is still progressive. We believe the case for the flat tax is strong. Presenting an axiomatic justification for the flat tax as a redistribution rule, this post suggests that you need to accept only a few basic properties to favor a flat tax for income redistribution.
In 1845, McCulloch, author of the most extensive and systematic treatment of public finance in the classical literature, made the following case for the flat tax:
“The moment you abandon the cardinal principle of extracting from all individuals the same proportion of their income or of their property, you are at sea without rudder or compass, and there is no amount of injustice and folly you may not commit.” (McCulloch, p. 143).
He was followed by notable others: J. Mill (1848, Vol. V, II), F. Hayek (1960, XX), M. Friedman (1962, X), and more recently R. Hall and A. Rabushka (1985) and (1996) provided arguments in favor of some version of the flat tax.
For example, Hayek argued that the flat tax is the only tax arrangement that is rule-compatible and not arbitrary. The flat tax is non-discriminatory. A flat tax prevents the majority from “imposing upon a minority whatever burden it regards as right” (Hayek 2011, p. 441).
Rather than revisiting the discussion on the flat tax at length, we shall present Casajus (2015)’s axiomatic justification for the flat tax in a very simple model of redistribution in a society:
Consider a society consisting of a fixed number of members. Any member is fully described by her income in a certain tax period. A redistribution rule assigns to any list of incomes a corresponding list of post-redistribution rewards. There is nothing more.
You only have to accept the following properties of a redistribution rule to favor a flat tax.
A redistribution rule is efficient if the sum of rewards after redistribution equals the total income before redistribution. The very idea of re-distribution suggests that the sum of individual rewards after redistribution should not be greater than before. In addition, efficiency requires that redistribution comes at the lowest, at best (in the model) at no cost.
A redistribution rule is symmetric if any two members of the society with the same income end up with the same reward after redistribution. If taxation depends only on income, symmetry is a rather natural requirement.
A redistribution rule is monotonic if whenever both the society’s total income and the income of a particular member of the society do not decrease (between two tax periods), then this member’s reward after redistribution does not decrease. The intuition behind this property is as follows. Non-decreasing total income guarantees that no member of the society necessarily has to be rewarded less than before the change. For a member whose income does not decrease at the same time, monotonicity ensures that she actually is not awarded less, which seems to be desirable.
Now, there are redistribution rules that satisfy all three properties. An example is the trivial redistribution rule that leaves any member just her individual income, i.e., there is no redistribution. Another example is the egalitarian redistribution rule that gives an equal share of the total income to all members of the society.
The trivial rule and the egalitarian rule are, of course, the extreme cases of the following class of redistribution rules: First, the individual income is taxed at a fixed rate. Second, the total tax revenue is split equally among the members of the society. While the trivial rule corresponds to a zero tax rate, the egalitarian rule corresponds to a tax rate of 100 percent. All these redistribution rules satisfy efficiency, symmetry, and monotonicity.
For societies with more than two members, Casajus (2015a)[†] shows that no other redistribution rules satisfy all three properties. That is, if you regard the presented properties of redistribution as desirable, redistribution should be done by a flat tax (0%–100%) combined with a uniform basic income paid from the tax revenue. Redistribution of income in this fashion has been suggested by Milner (1920), for example.
The tax rate itself is not determined by the three properties. If you dislike the idea of a basic income, choose the appropriate tax rate.
[†] For more, see the following papers: Casajus (2015b) restricts attention to non-negative income. Casajus (2016) considers other properties: efficiency, positivity and differential monotonicity. Yokote and Casajus (2017) use a weak version of differential monotonicity to justify a form of the flat tax.