metric.<sup>10</sup> For VAT, although some studies have used income as the welfare metric, this can produce misleading results, and the use of consumption should be preferred (see, e.g., Thomas, 2022; IFS, 2011).

Even once income (or consumption) is chosen as the base for the welfare metric, the question then arises whether it should be household income or individual income. Studies have taken both approaches, however there is a broad consensus that individual income should be preferred. Choosing the individual rather than the household as the unit of analysis ensures that equal weighting is given in the analysis to the welfare of each individual. In contrast, use of the household as the unit of analysis would mean that the welfare of a one-person household would be given as much weight as that of a large household with many individuals. That approach, while often adopted for ease of computation, is difficult to justify for welfare analysis.

The key difficulty in applying the individual as the unit of analysis is the fact that income (and consumption) is often only provided in surveys on a household basis. To adjust the unit of analysis to the individual it is therefore necessary to multiply household survey weights by household size. Implicit in this approach is the assumption of equal sharing of resources within a family so that the measured welfare of each household member (whether income or expenditure) is identical.

Prior to adjusting the welfare metric to an individual basis, there is also a strong case to make adjustments to ensure that households with different compositions are comparable. This is referred to as equivalization and involves dividing household income by an “equivalence scale”. The simplest option is to just divide household income by the number of individuals in the household. However, this adjustment does not account for differing degrees of need within a household (e.g., a child typically needs less food than an adult to maintain the same welfare level). Additionally, households can be expected to benefit from economies of scale in relation to heating and certain other costs. Two common equivalence scales used are the square root scale and the OECD-modified scale. The square root scale simply takes the square root of household size. The “OECD modified” scale gives a fixed weighting of 1 to the first adult household member, 0.5 to the second and additional household members aged 14 and over, and 0.3 to each child under 14.<sup>11</sup>

# 2.2 Global indicators of effective progressivity<sup>12</sup>

Musgrave and Thin (1948) first proposed a global measure of effective progressivity as the ratio of (one minus) the Gini coefficient of post-tax income to (one minus) the Gini coefficient of pre-tax income. However, more recent literature has distinguished the concepts of progressivity and redistribution more distinctly. Indeed, Kakwani (1977) pointed out that the Musgrave and Thin (1948) metric was actually a 10 Although there are strong arguments that consumption provides a better reflection of the wellbeing of poor households, the fact that income is the base of the tax to be modelled creates a natural preference for income data as this will maximize the accuracy of the tax modelling. 11 A more elaborate option is the parametric scale introduced by Cutler and Katz (1992). The scale is calculated as: � , where is the equivalent size of household , measures the degree of need of children relative to adults; specifies economies of scale in consumption; αα is the number of adults in household and ii aa,ii cc,ii ii is the number of children. This scale explicitly allows for adjustment of need between adults and children, and for mm = �nn + θθnn mm ii θθ aa,ii cc,ii economies of scale with increases in need-adjusted household size. A further benefit of the approach is that its αα nn ii nn explicit nature easily enables sensitivity analysis. See Creedy and Sleeman (2005) for a detailed comparison of different equivalence scales.

> 12 This subsection draws heavily on Thomas (2020).

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