progressivity, detailing the pros and cons of each approach, and illustrating their use drawing on results from the literature, where possible for IDA countries. Section 6 then considers the use of aggregate revenue data to help assess tax progressivity. Drawing on the preceding sections, Section 7 then proposes a set of indicators to use to assess the progressivity of tax systems in IDA countries.

# 1 Defining and measuring progressivity

The commonly accepted definition of a progressive tax is one where the average tax rate increases as income (or other relevant welfare metric<sup>5</sup> ) increases. Meanwhile, for a regressive tax the average tax rate decreases as income increases, and for a proportional tax the average tax rate is constant as income increases. Progressivity can equivalently be specified as requiring the marginal tax rate to be higher than the average tax rate (see, e.g., Musgrave and Thin, 1948).

This concept of progressivity derives from concepts of vertical equity, in particular the well-known “abilityto-pay” principle, as well as “equal sacrifice” principles. The ability-to-pay principle, which has its origins in Smith’s (1776) _Wealth of Nations_ , states that someone with greater ability to pay tax should pay more tax (although how much more is not articulated).<sup>6</sup> Ability-to-pay is typically measured in relation to income. Equal sacrifice, which originates from Mill (1848), instead is linked more closely with utility. The principles of equal _absolute_ sacrifice and equal _proportional_ sacrifice, respectively, require that taxpayers should face the same absolute and proportional reduction, respectively, in utility from the imposition of a tax. Assuming that the marginal utility of income decreases with income, both concepts imply that richer households should pay more in tax than poorer households, although they do not necessarily imply that they should pay more as a proportion of their income, as this will depend on the underlying utility functions (see, e.g., Young, 1987).<sup>7</sup> It is important to note that the “increasing average tax rate” definition of progressivity requires not just that the total tax liability increases with income, but that it increases more than proportionally with income.

When it comes to _measuring_ progressivity, this basic definition becomes problematic. In particular, progressivity is defined relative to a point in the income (or other relevant welfare metric) distribution. This makes it difficult to assess (and compare) the overall degree of progressivity of a tax (and is especially problematic where the average tax rate varies non-monotonically over the relevant distribution, as may occur, e.g., with the VAT or excises). A second problem with the basic definition is that it is vague regarding exactly what is being measured: the progressivity of the tax rate structure,<sup>8</sup> or the progressivity of the tax

> 5 It is necessary to choose a welfare metric to rank households from poor to rich. While typically for an income tax this will be income or equivalized income, for the VAT it makes more sense to measure the tax burden relative to (equivalized) expenditure (see Thomas, 2022; IFS, 2011).

> 6 Adam Smith proposes both the benefits principle and ability-to-pay principle. As noted by Musgrave and Musgrave (1989), the benefits principle derives also from earlier work by, amongst others, Locke and Hobbes.

> 7 Theoretically, progressivity metrics can be based on the change in utility induced by a tax (e.g., compensating or equivalent variations), but, in practice, metrics are generally based on tax paid as per the basic increasing average tax rate definition.

> 8 In this paper, we use the term “rate structure” in a broad sense, referring to the combined impact of all tax rules in determining the tax liability including not just the progressive rate schedule, but also any applicable allowances, deductions and tax credits.

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