The concept of justice in taxation has been associated with the concepts underlying income inequality and its measurement. I take a fresh look at this connection in the light of recent UK experience and of results in the field of inequality measurement. This is developed in the context of several alternative empirical indicators of economic well-being and alternative approaches to characterising inequality comparisons.

The principles guiding tax policy should usually include some conception of distributional justice, and the design of policy should apply these principles in a consistent fashion. Part of the literature on the economics of distributional justice is closely connected to the literature on economic inequality through different threads that focus on different dimensions of inequality, in other words, through different aspects of the justice-in-distribution issue. The main connection involves conventional inequality measurement, but there are also connections to the analysis of income mobility. However, while there might be wide acceptance of the weak proposition that tax progressivity is associated with justice, can tax justice be directly linked with inequality? If so, in what way are changes in inequality linked to changes in the apparent justice of the tax system?

The issues involved are illustrated using a broad-brush picture of taxes and benefits in the UK during the first twenty years of this century. What appears to have happened to the progressivity of government intervention in personal incomes in the light of changes in the structure of taxation and benefits? What might have been expected to have happened to inequality? The paper is organised as follows. In sections 2 and 3 we examine the standard pragmatic and analytical approaches to quantifying progressivity. Section 4 applies this analysis to the recent history of income distribution and redistribution in the UK. Section 5 discusses general issues raised by the analysis and concludes.

There are parallels between approaches to inequality and approaches to tax design. In both fields there is concern for efficiency and equity at the heart of the analysis. In both fields there is a sub-literature that is based on intuition, one that is based on utilitarian welfare analysis and one that makes appeal to an axiomatic method. Some of the early literature on tax design and tax reform show this: it starts from the premise that the justice of the tax system is appropriately assessed in terms of distributional outcomes and the progressivity of the tax schedules that lead to those outcomes.

The connection of tax progressivity with inequality involves two principal elements.

The first is the ‘income’ concept of inequality analysis, which corresponds to the tax base in the taxation literature. According to the Haig-Simons principle, [

The second element is how the taxation component is incorporated into the distributional analysis. The principal idea is that the tax system, extended to cover also the system of benefits (income support), is codified in laws and codes which determine the amount of taxes to be paid, or benefits received, by each citizen. The net result of this intervention by the state is a quantity given by

This

An example of the effect of individual interventions by government agencies on the income distribution appears in the summary data published by UK’s Office for National Statistics (ONS). The advantage of these data is that they provide easily accessible information that enables one to get a clear picture of the way key metrics of distribution have changed in the UK for an interesting range of income definitions. The ONS provides series on the following five income concepts:

Original income: essentially market income plus private pensions

Gross income: the line above plus public cash benefits (including state pensions)

Disposable income: the line above minus direct taxes (including income tax, national insurance and council tax payments)

Post-tax income: the line above minus indirect taxes (including value- added tax, alcohol and tobacco duties)

Final income: the line above plus public non-cash benefits (including health and education)

A summary of the inequality outcomes for the income concepts 1–5 along with consumption over the early 21st century is shown in

Gini history for five income concepts and expenditure.

Should tax justice be identified with progressive taxation? Although it makes sense to connect the concepts of progressive taxation and redistributive taxation, a simple identification of the two concepts is not possible because of the multiple definitions of progressivity; ‘a progressive tax system should be defined as one where the average rate of taxation increases with income before tax. The degree of progression, however, is often referred to by politicians and economists with no precise meaning attached to it’ [

The role of these concepts is examined in section 3.

The way tax-benefit progressivity is conventionally measured can be shown with two applications of Lorenz-curve analysis.

For a population of _{1}, _{2}, …, _{n}_{i}_{n}_{i}/Y_{n}_{i}/T_{n}_{i}_{i}/Y_{n}_{i}/T_{n}_{i}/Y_{n}

Lorenz curves of income and tax payments.

Tax-liability progression is not ideal for understanding the connection from the policy intervention to distributional outcomes. For this purpose, residual-income progression is straightforward and transparent. ‘Residual income’ is income after intervention by the tax-benefit system, _{1}, _{2}, …, _{n}

Lorenz curves for five concepts of income.

In this case the distributional summary depicted in

Three qualifications should be mentioned. The first is that several of the income adjustments described above come, in practice, from imputations (based on household characteristics) rather than direct observations (this is done in the case of non-cash-benefits). Second, the application of some taxes or benefits may cause reordering among income recipients. So, when comparing post-intervention income distributions with pre-intervention distributions it is argued that this reordering should be taken into account: instead of representing the distribution of post-intervention income as _{1}, _{2}, …, _{n}

What is the basis for choosing the standard pragmatic measures or the Lorenz-curve empirics as the appropriate way to consider tax progression? Consider how the progressivity of τ might be characterised in terms of social welfare.

This idea was already present in the seminal Feldstein article [

income-regarding so that, other things being equal, an increase in any person’s income increases social welfare;

equity-regarding so that, other things being equal, a transfer to a richer person from a poorer person would always reduce welfare (the “transfer principle” [

A basic result for a social-welfare function that is both income-regarding and equity-regarding is this: if distribution

A neat result then connects the residual-progression of tax-benefit schedules to the concept of Lorenz dominance [

This result can be deepened and extended by considering the relationship amongst the following three propositions which have a clear practical interpretation:

“The average tax rate never decreases with

“Disposable income

“Inequality of

These three propositions are linked together in the following simple result:

This theorem is fundamental to the relationship between the tax-benefit schedule and inequality; it provides insight on the logical consequence of progressivity for distributional justice. Proposition 1 is a “greatest burden on those with the broadest shoulders” property: the tax-benefit schedule τ is designed in such a way that the ratio τ(

So, the Lorenz-dominance relation, introduced as an intuitive idea in section 2.2, is at the heart of the analysis connecting tax progressivity and tax justice, interpreted in terms of inequality reduction. The analysis needs to be developed in two directions that involve distributional justice questions associated with income taxation. The two directions concern issues that are assumed away when the simple Lorenz-dominance criterion is used.

The first of these focuses on the conventional assumption that inequality remains unchanged under proportionate changes in all incomes. It sidesteps the issue of how inequality comparisons should be made at different levels of real income. Suppose Austria has a higher income per head than Belgium but that the two countries have the same level of inequality. What changes in real income in the two countries would leave this inequality judgment unaltered? The same income growth for everyone in the two countries (relative inequality comparisons)? Or the same dollar increase for everyone in the two countries (absolute inequality comparisons)?

It is usual to distinguish between “vertical inequity,” corresponding to the disparities in incomes that were considered earlier using Lorenz analysis, and “horizontal inequity” corresponding to the idea that government intervention should treat people with the same circumstances in the same way: “equal treatment of equals.” In principle, income recipients with the same level of income and the same circumstances should be liable for the same taxes or transfers. In practice, a tax-benefit system can alter the rankings of the people in the income distribution [

Because taxes are complex and taxpayers are diverse it is unrealistic to expect that the outcome in terms of disposable incomes can be represented as a known determinate function τ. Those with the same income may pay different taxes arising from different behaviour rather than different circumstances. Incomes derived from assets may present valuation problems that can appear to introduce an element of randomness into the relationship between pre-intervention and post-intervention income. The effect of deductions may undermine progressivity [

For a complete analysis a systematic method of aggregating the information about progressivity at each income level is needed. This could be achieved by one of two routes.

It is useful to be able to say that, under conditions A, B and C, one tax-benefit schedule displays more progressivity overall than another. One method for doing this might be to extend the pragmatic method in section 2.2 to the ranking of distributions [

In inequality analysis it is convenient to aggregate the information in a distribution into a single index of inequality, in addition to using Lorenz curves. Similarly, it would be convenient to have a single index of tax-benefit inequity that captures the impact of taxes and benefits on vertical and horizontal inequity. This approach might seem arbitrary, as with the pragmatic approach to progressivity discussed in section 2.2. If one uses an explicit social-welfare function to evaluate income distributions, then it would seem natural to compute the aggregate effects of a progressive schedule on individual income-recipients using in terms of social welfare [

A potential alternative to the social-welfare approach to the aggregate measurement problem could be based on other ways of comparing two distributions. The progressivity problem uses the idea of a reference distribution from which one may quantify the distance from the actual to the reference distribution [

Let us examine the tax-benefit progressivity concepts in the light of recent UK history.

The data used are the same as for

Consider how the tax and benefit system has converted original income to final income, both as one combined operation and in the four notional separate stages identified within the ONS data. Using the same ONS data, the elasticity formulas for tax-benefit progressivity in section 2.1 can be estimated using discrete approximations; again, this can be done for the overall impact of the interventions, or for the four notional interventions computed by ONS.

The picture of redistribution is already evident from the pair of snapshots in

The picture of trends from

Changes in original income distribution: detail.

Changes in final income distribution: detail.

The concept used is that of residual progression, introduced in section 2.1.

Residual progression overall.

Components of residual progression at beginning, middle and end of period.

As shown in section 3 one of the key issues in understanding the effective progressivity of government interventions is the behaviour of the implied average tax rate (“ATR”, where benefits are included as negative taxes).

ATR profiles: 2000–2020.

ATR profiles at each stage of redistribution.

Pragmatic approaches and formal analysis can help to inform judgments about tax progressivity and its relation to income inequality.

Pragmatic approaches focus on two questions: (1) What is the distribution of tax payments? (2) How much do taxes and benefits shift the Lorenz curve? Both these questions tell us something useful, but more is needed to derive conclusions that are based on the established principles of welfare economics. The second question is key to understanding progressivity and inequality.

There is a simple connection between distributional justice and tax progressivity through the basic welfare economics of distributional issues: Lorenz orderings of post-intervention distributions are connected to measures of tax progression. A sharp conclusion on the social-welfare impact of a tax schedule can be drawn by determining whether it is true that the average tax rate is always increasing and post-intervention income is always increasing with pre-intervention income. The first problem is that the simple results sweep aside difficulties raised by the heterogeneity of income recipients and the complexity of tax schedules. This has been addressed in the literature on horizontal inequity covering not only the issue of tax-induced re-rankings in the income distribution but other issues that are less straightforwardly resolved. They may be suitably addressed by further theoretical and econometric developments on the comparisons of pairs of distributions. The second problem is whether “more progressive” necessarily implies “more just.” We know the conditions under which “more progressive” implies “more equal” in terms of outcome, but that is not quite the same thing. To clarify this issue we need to take a position on how, if at all, the criteria for inequality comparisons change when real incomes change (“vertical equity”) and whether distributive justice is to be applied ex ante or ex post (“horizontal equity”).

The experience of the UK in the early 21st century reveals that, as in the 20th century, the major redistributive effect comes from the benefit side of government intervention rather than taxation. Even though it appears that there has been an increase in the share of the richest towards the end of the period and even though the outcome in terms of tax changes has been a reduction in the progressivity of the tax system, the cash-benefit component of government intervention has driven the pattern of progression in the opposite direction.

This suggests three take-aways for the design of policy:

Distributional justice in taxation should be viewed not only in terms of the way taxes are raised but also in terms of the way the proceeds are spent on cash benefits and non-cash benefits. The experience of the UK and many other countries suggests that much of the overall progressivity of government intervention comes from the spending side rather than the revenue-raising side of the intervention activity. It is possible to have a substantial impact on inequality by focusing on the way benefits are structured rather than only worrying about, say, top rates of personal tax.

However, on the tax side, a shift from direct taxation (usually a progressive component) to indirect taxation (a regressive or neutral component) will reduce overall progressivity and worsen the distributional outcomes.

Both the top and the bottom of the distribution are important for appraising the overall progressivity of the tax-benefit system and its consequences for inequality and distributional justice.

See Feldstein’s seminal paper [

Musgrave and Thin [

An alternative pragmatic method uses a modified Lorenz diagram with share of income on the horizontal axis and share of net tax paid on the vertical axis [

Distribution

The Institute for Fiscal Studies uses a different methodology that computes the impact of indirect taxes as a proportion of expenditure rather than as a proportion of income as done in the ONS calculations [

The difficulty of applying a method of equivalisation to achieve this in practice is considered at the end of section 3.2.

The underlying intuition for this result is this: if the elasticity of _{i}/x_{i–1}

There are few inequality indices that have implied contour maps that allow them to be adapted both as relative and as absolute inequality indices; the Gini coefficient (a relative measure) is one of them: multiply the Gini by mean income and you obtain the Absolute Gini; the variance (an absolute measure) is another: take the square root of the variance and divide by the mean and you get the coefficient of variation.

See

See

I am grateful to Tianxang Zheng and Xueyan Zhang for research assistance. I also wish to thank the editor and reviewers for helpful suggestions.

The author has no competing interests to declare.