Income tax was introduced in France in 1914, taking effect in 1916 to finance the war effort. It was retained because the state needed funding for its growing responsibilities, including education, justice and transport infrastructure. Moreover, the development of social policies aimed at reducing inequality only increased the need to call on citizens to supplement state revenues.
Opinions differ sharply on how citizens should be taxed. Should taxation be efficient or fair? And what do those terms even mean? Should households be taxed, as they are in France, or should taxation be individual, as advocated by the French economist Thomas Piketty?
When the function jumps
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Some principles are universally accepted. For example, taxation should be progressive: not only should those who earn more pay more, but, given identical family circumstances, the percentage of income taken in tax should also increase with income. Likewise, an increase in earnings must leave an individual with a higher income even after tax.
What do these principles mean in terms of functions? Let x be a household's income and f (x) the annual amount levied. The first condition requires not only that the function f be increasing, but also that the function g, defined by f (x) / x, be increasing. The second principle requires the slope of the graph of f to be less than 1 everywhere and, above all, the graph to be continuous.
