What is optimum income tax?

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Many mathematicians and economists have investigated the idea that optimum taxation rates can be mathematically derived. However, depending on the underlying assumptions different conclusions can be reached.

We can illustrate the situation with a simple example. Suppose tax rates are nil. Zero tax causes no disincentive to work (and therefore to produce), however, the government gets nothing. So in any scenario other than anarchist-libertarian, a zero tax rate is not optimum.

Now consider the opposite extreme, with tax rates at 100%. There is no incentive to work at all, so again the government gets nothing. Obviously, there must be an optimal income tax rate somewhere between 0% and 100% which maximizes the government’s income. Heuristically tax rates of around 20% maximize government income by providing for a reasonable tax take without disincentivizing productive work too much.

One of the first to analyze optimum tax in detail was James Mirrlees from Cambridge University, who got a Nobel Prize for his efforts. In 1971 he reached the somewhat surprising conclusion that it was best for society as a whole if the most productive people (assumed to be those with the highest income) were hardly taxed at all, and that more tax was taken from those who earned less.

This unintuitive result arises because if the maximum incentive is provided for the most productive people to maximize their production, it greatly increases the productivity of everyone else, and leads to an increased overall tax take, leading to reduced financial pain for everyone.

The notion that the people with the highest incomes should not be taxed does not sit comfortably with many people (even though that’s what often occurs in practice, because the richest people are readily able to structure their affairs to minimise tax).

Other economists and mathematicians have sought to build on the work of Mirrlees by including other factors in their calculations. Emmanuel Saez has attempted to refine the work of Mirrlees by separately considering the elasticities of earning at each level of income. Waly Wane has argued that even rich people don’t want to be surrounded by poverty (PDF), and if that factor is taken into account then the optimum income tax schedule would incorporate negative marginal tax for less skilled individuals. In other words, for every dollar they earn, the government gives them a bit extra.

Others disregard the conclusions of Mirrlees entirely, on the basis that the highest earnings may result more from luck than from the ability to maximize productivity better than anyone else.

The one thing most studies agree on is that the politician’s “natural urge” to increase tax rates does not necessarily lead to an increased tax take or to a better society.

A corresponding analysis can be applied to consumption taxes such as sales tax. In 1927 Frank Ramsey proposed that the most efficient form of sales tax would heavily tax items with inelastic demand (such as milk), and only lightly tax items with highly elastic demand (such as papaya juice). Otherwise, demand would drop for highly-taxed luxury goods with elastic demand, and the overall tax take would be reduced.

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