7.1 Motivations and Objectives

A long-established reasoning for progressive taxation is that all taxpayers should be subjected to equal utility sacrifice in paying taxes. The term “equal sacrifice” may be interpreted in three different ways. According to equal absolute sacrifice, everybody gives up the same amount of utility while paying taxes so that the difference between the utility of before-tax income and the utility of after-tax income is the same across taxpayers (Mill, 1989). Thus, implicit idea under the notion of equal absolute sacrifice is that the social evaluation function is of symmetric utilitarian type. That is why often equal absolute sacrifice rule is referred to as the “utilitarian” form of equal sacrifice (see Lambert and Naughton, 2009).

Equal sacrifice rule of taxation has sometimes been interpreted as sacrifice proportional to one's wellbeing (Cohen Stuart, 1889; Seligman, 1894). In other words, under equal proportional sacrifice everybody should give up the same percentage of utility while paying taxes. Evidently, given the positivity of utilities, equal proportional sacrifice is the same as equal absolute sacrifice in logarithms of utilities. In this case there is a tacit assumption that the social evaluation is of the symmetric Cobb–Douglas category so that under logarithmic transformation of utilities it becomes the symmetric utilitarian sum of logarithms of utilities. Finally, the equal marginal sacrifice principle claims that the total tax burden should be divided out among the taxpayers such that the after-tax marginal utilities of the taxpayers are equalized.

4.1 Motivations and Objectives

Taxation is the principal mechanism through which redistribution of income from the rich to the poor takes place. The analysis of taxation is one of the major pieces of tasks of economists who analyze public sector economics. Taxes can be distinguished with respect to the effects they generate on the distribution of income. While raising revenue for some specific welfare purposes, it is ethically desirable that a tax scheme should exert an equalizing effect on the distribution of income; the rich should bear a greater share of the tax burden than the poor. This makes the after-tax income distribution more equal than its before-tax twin. In the process, the tax burdens come to be more unequally distributed than the pre-tax incomes. These two effects of a tax system are referred to as the “redistributive” effect and the “disproportionality” (or “departure from proportionality”) effect of a tax system. (For a recent discussion on these features of a tax system, see Chakravarty and Sarkar, 2025.)

Now, the basic definition of progression of an income tax system is that the local measure “average rate progression,” the tax liability as a proportion of the income, should rise with income (Pigou, 1928). A local or structural indicator looks at the extent of progression along the income scale; it focuses on income-by-income progression.

2.1 Motivations and Objectives

We have argued in Chapter 1 that a more uniform distribution of income in a population makes individuals better off economically in respect of income. Since high income inequality is likely to generate financial hardship for the lower-income sections of the population and social discontentment and political instability, any society should reduce its high inequality to a reasonably low level.

For determining the level of equality we need a yardstick that summarizes the closeness of incomes of different individuals. Such an indicator gives us a concrete idea about the deviation of the actual distribution from the norm, the distribution of perfectly equal incomes. An adequate indicator of income equality should incorporate interpersonal comparisons so that a redistribution of income from a better-off individual to a worse-off individual, such that the donor does not become poorer than the recipient, generates a better state of incomes, as desired from a social welfare standpoint. Equivalently, we say that equality increases under a progressive transfer of income, a Robin Hood operation. In the literature this notion of value judgment is known as the Pigou–Dalton transfer principle. It represents the egalitarian ethic that higher equality of incomes among individuals is socially preferred. A second value judgment involved in equality evaluation is anonymity; any reordering of incomes does not change the degree of equality – reflecting irrelevance of all characteristics other than income.

The purpose of this coursebook is to establish inter-linkage among three different features of social wellbeing – namely equality, depolarization, and tax progressivity – all based on society's income distributions. Low equality is socially dispreferred since it refers to the accumulation of a highly significant part of the total income of the society in the possession of a few. Depolarization is concerned with the improvement in the level of wellbeing of the middle-income group of the society. The existence of a rich middle class in a society is always desirable, since a wealthy middle-income group contributes highly to the society's economic growth and development in many ways. Tax progressivity investigates the extent to which equality is raised through taxation.

Chapter 1 presents an introductory outline of the materials analyzed in the remaining chapters. Chapter 2 formally defines and analyzes the notion of equality. Chapter 3 provides a rigorous treatment of the concept of depolarization.

In Chapter 4 we discuss different structural or local indicators of tax progressivity that look at the extent of progression at each income point. We look particularly at the redistributive and departure from the proportionality effects of taxation. We also investigate the implication of structural measures with respect to depolarization. One section of the chapter examines the impacts of equal proportionate income growths on revenue and redistributive effects of taxation. Given the before-tax income distribution, the impacts of equal proportionate increase in taxes on structural measures are investigated as well.

6.1 Motivations and Objectives

This chapter begins by considering the problem of acquiring a certain amount of tax from the individuals in a society in conjunction with assigning a specific quantity of subsidy to the same set of individuals in an inequality-minimizing manner, where the tax and the subsidy sizes need not be the same (see Chakravarty and Sarkar, 2022). The tax-collection-subsidy provision scheme we are considering here can be viewed as follows: tax payments are made by the individuals to the government, and subsidy (benefit/transfer) payments are made from the government to the individuals. If the sizes of the total tax and total subsidy are the same, then Fei's (1981) inequality-minimizing solution for a balanced budget plan becomes a particular case of the tax-subsidy allocation program considered here. However, Fei's analysis has a limitation; it does not consider the practical problem where the before- and after-tax total incomes may differ. In general, total tax levied on a set of individuals is higher than the total transfer made to them. The administration may be required to make welfare payments on an absolute basis – for instance, when the individuals in a society get affected by some natural calamity such as cyclone and flood. It may become essential for the administration to spend money on an urgent basis for the maintenance or construction of a public good. All these expenditures are generally financed from the taxes raised from the individuals.

In the theory of distribution in economics, equality assessments have generally been formulated on the basis of the distribution of some individual achievement – health, income, literacy, opportunity, and so on – across the population. Thus, equality is a multifaceted phenomenon. Equality becomes an issue of scrutiny in multiple dimensions, and it is not reducible to equality in one sphere only (Sen, 2009). Although a great many problems have been addressed and resolved, new problems are being posed and analyzed. (For a recent discussion on equality from different perspectives, see Piketty, 2022.)

The choice of a specific dimension of wellbeing is certainly a matter of value judgment. In this book we will consider income as the only dimension of human wellbeing because the main body of the book, Chapters 4–8, will study income taxation and subsidy allocation from different standpoints, covering a wide range of topics. Since people also care about other dimensions, our focused analysis does not express the view that they are less important for human welfare.

Social policy makers often become concerned with low equality because low equality means high income gaps between the rich and the poor. This may give rise to social disorder and lead to civil wars. Following the publication of Russett's (1964) seminal article, many researchers have attempted to determine the relationship between inequality and conflict. Nonetheless, it has sometimes been argued in the literature that the relationship between inequality and conflict is a debatable issue.

3.1 Motivations and Objectives

In the income distribution literature attempts were made to relate the concept of “bipolarization,” “shrinking middle class,” to the population share in some well-defined middle-income group of the society. Thus, in a highly polarized society individuals are disproportionately placed in the upper and lower ends of the income distribution. Such movements of the individuals toward the extremes are likely to have negative consequences on social and political relations. In Chapter 1 we have argued that a well-off and sizable middle-income group of an economy contributes to the wellbeing, peace, and democracy of the society in many ways. Therefore, a desirable objective of a policy planner is to recommend policies that can make the society less bipolarized or, equivalently, more depolarized. The subject of this chapter is to present an analytical discussion on depolarization.

The aforementioned notion of polarization contrasts with the concept of income “multipolar” polarization, suggested by Esteban and Ray (1994) (see also Duclos, Esteban, and Ray, 2004; Esteban and Ray, 2012). The three properties that are taken to be innate to this concept of polarization are (i) polarization is a matter of subgroups, (ii) with two or more subgroups, polarization increases when “within-group” dispersion reduces, and (iii) polarization increases when “between-group” distances rise. Properties (ii) and (iii) represent respectively the “identification”/attachment and “alienation”/detachment components of polarization.

8.1 Motivations and Objectives

As we have contended in Chapter 1, like inequality and polarization, deprivation is also a social bad. Runciman (1966) argued that the magnitude of deprivation perceived by a person for not being promoted is an increasing function of the number of persons who have been promoted. Yitzhaki (1979) considered deprivation with respect to income in the Runciman framework and showed that one credible index of deprivation in a society is the absolute Gini index for the society. Earlier, Sen (1973) argued that in any pair-wise comparison, the income shortfall of the worse-off person from the better-off person may be taken as a size of the degree of depression suffered by the worse off when he compares his position with that of the better off on the income scale. These depression sizes, when averaged across individuals in the society, generate the absolute Gini measure of inequality as a measure of social depression. A similar interpretation holds for the recently revived absolute Bonferroni standard of inequality (Chakravarty, 2007). Thus, when appropriately formulated, a tax policy desiring reduction of the Gini/Bonferroni measure of social deprivation effectively requires a curtailment in the inequality magnitude of before-tax incomes, as measured by the Gini/Bonferroni index.

In this chapter we examine the incidence of taxation on deprivation. More precisely, our objectives in the chapter are to design tax strategies that are intended to reduce social deprivation.

5.1 Motivations and Objectives

In the earlier chapter we were concerned with income-by-income progressivity of a tax schedule. But very often it becomes necessary to get a concrete idea about the extent of overall progressivity of the schedule. For instance, a policy maker may be required to know whether the current system of taxation is sufficiently progressive. Another important issue that arises in this context is the problem of ranking between two or more tax schemes in terms of progressivity. A taxpayer in India may be interested in knowing whether the new tax system is more progressive than the earlier one.

A global or overall indicator of progressivity is a scalar representation of the degree of progressivity implicit under a tax program taken as a whole. In this chapter we make a detailed analytical treatment of tax progressivity on an all-inclusive footing and several related issues.

For checking global progressivity of a tax program, it is natural to expect that it is locally progressive everywhere. The two highly innovative propositions of Jakobsson (1976) suggest that this inquiry can be made along two lines: looking at the equalizing effect of taxation on the income distribution (redistributive effect) and checking to what extent the distribution of tax burdens is more unequally distributed than the incomes from which taxes are collected (departure from proportionality effect). The corresponding families of progressivity indicators are, therefore, consistent respectively with the local metrics “residual income progression” (RIP) and “tax liability progression” (TLP).

Every 5 years, the World Congress of the Econometric Society brings together scholars from around the world. Leading scholars present state-of-the-art overviews of their areas of research, offering newcomers access to key research in economics. Advances in Economics and Econometrics: Twelfth World Congress consists of papers and commentaries presented at the Twelfth World Congress of the Econometric Society. This two-volume set includes surveys and interpretations of key developments in economics and econometrics, and discussions of future directions for a variety of topics, covering both theory and application. The first volume addresses such topics as contract theory, industrial organization, health and human capital, as well as racial justice, while the second volume includes theoretical and applied papers on climate change, time-series econometrics, and causal inference. These papers are invaluable for experienced economists seeking to broaden their knowledge or young economists new to the field.

Every 5 years, the World Congress of the Econometric Society brings together scholars from around the world. Leading scholars present state-of-the-art overviews of their areas of research, offering newcomers access to key research in economics. Advances in Economics and Econometrics: Twelfth World Congress consists of papers and commentaries presented at the Twelfth World Congress of the Econometric Society. This two-volume set includes surveys and interpretations of key developments in economics and econometrics, and discussions of future directions for a variety of topics, covering both theory and application. The first volume addresses such topics as contract theory, industrial organization, health and human capital, as well as racial justice, while the second volume includes theoretical and applied papers on climate change, time-series econometrics, and causal inference. These papers are invaluable for experienced economists seeking to broaden their knowledge or young economists new to the field.