Interdependent welfare functions and optimal income distribution

The relationship between income distribution and social welfare is empirically analyzed, while explicitly allowing for the interdependence of individual welfare functions. The social welfare function is taken to be an additive function of individual welfare functions of income (WFIs). On the basis of Dutch data it is found that under certain conditions (such as absence of effects of income redistribution on productivity) an equal distribution of incomes is suboptimal. The interdependence of WFIs appears to have a pronounced effect on policy conclusions concerning the desirability of income redistribution vis-à-vis economic growth.     

A POSITIVE AND A NORMATIVE THEORY OF INCOME DISTRIBUTION

A positive theory of income distribution based on assumptions concerning the supply of and demand for each type of productive service is presented. The demand function of the organizers of production may be derived from the maximization of profits with the income scale and the production function as restrictions. A normative theory based on the maximization of a social utility or welfare function is also considered. In the normative theory, production functions and balance equations (some representing compartmentalization of factor markets) are introduced as restrictions and again an income scale results, this time maximizing social welfare. Empirical testing is also considered. The positive theory was developed in part to take into consideration the fact that personal income distributions can reasonably well be described by log normal distributions, and that skill parameters are often normally distributed. Limited testing of the influence of wealth, intelligence, education, and sex suggest that these account for only a small part of the variance in the income distribution. This suggests the need for further research.     

A variable-production generalisation of Lerner's theorem

A government wishes to choose an optimal set of wage rates, but it is uncertain of individual characteristics. All it knows for certain is that each utility function is strictly quasi-concave and that the production function is linear. We assume that it can determine probability distributions, for each individual. of possible utility functions and ability levels. If each of these probability distributions is the same for every individual, expected social welfare is maximised by equalisation of wage rates. But since actual utility functions, and therefore labour supplies, will generally be unequal, incomes will then be unequal.     

Evaluating public goods from individual welfare functions

This paper suggests a method for evaluating public goods based directly on individual welfare functions. A hypothetical project for reducing air pollution is used as an illustration. The paper describes a technique for estimating welfare functions: the approach is applied to a limited experimental sample. The individual marginal rates of substitution between air income are then added up in order to yield estimates of values associated with air quality variations by the group under consideration. Further it is illustrated that the utilization in the aggregation process of heights based on individual differences in marginal utility of income as computed from the welfare functions, can modify the measures of social profitability and yield evaluation more consistent with basis criteria of social welfare.     

Aids,translog, and the Gorman polar form

All budget share models that are linear in the log of income can be derived from Gorman polar form subutility functions. Implications of this result for two-stage budgeting, lifetime utility maximization, and social welfare calculations are discussed.     

Comparisons of heterogeneous distributions and dominance criteria

We are interested in the comparisons of standard-of-living across societies when observations of both income and household structure are available. We generalise the approach of A.B. Atkinson and F. Bourguignon (1987) [3] to the case where the marginal distributions of needs can vary across the household populations under comparison. We assume that a sympathetic observer uses a utilitarian social welfare function in order to rank heterogeneous income distributions. Insofar as any individual can play the role of the observer, we take the unanimity point of view according to which the planner?s judgements have to comply with a certain number of basic normative principles. We impose increasingly restrictive conditions on the household?s utility function and we investigate their effects on the resulting rankings of the distributions. This leads us to propose four dominance criteria that can be used for providing an unambiguous ranking of income distributions for heterogeneous populations.     

Salience and social choice

The axioms of expected utility and discounted utility theory have been tested extensively. In contrast, the axioms of social welfare functions have only been tested in a few questionnaire studies involving choices between hypothetical income distributions. In a controlled experiment with 100 subjects placed in the role of social planners, we test five fundamental properties of social welfare functions to determine the efficacy of traditional social choice models in predicting social planner allocations when presented with choice sets designed to test the axioms of the theory. We find that three properties of the standard social welfare functions tested are systematically violated, producing an Allais paradox, a common ratio effect, and a framing effect in social choice. We find support for scale invariance and a preference for tail-increasing transfers. Our experiment also enables us to test a model of salience-based social choice which predicts the systematic deviations and highlights the close relationship between these anomalies and the classical paradoxes for risk and time.

     

Welfare comparisons with bounded equivalence scales

The paper considers the problem of comparing income distributions for heterogeneous populations. The first contribution of this paper is a precise dominance criterion combined with a simple algorithm for implementing the criterion. This criterion is shown to be equivalent to unanimity among utilitarian social planners whose objectives are compatible with given intervals of equivalence scales. The second contribution of the paper is to show that this criterion is equivalent to dominance for two different families of social welfare functions, one inspired by Atkinson and Bourguignon (in: G.R. Feiwel (Ed.), Arrow and Foundation of the Theory of Economic Policy, Macmillan, London, 1987), in which household utility is a general function of income and needs, and a second family inspired by Ebert (Soc. Choice Welfare 16 (1999) 233), in which household utility is a function of equivalent incomes. Finally, we extend our results to the case where the distributions of needs differ between the two populations being compared.     

A Class of Social Welfare Functions That Depend on Mean Income and Income Polarization

A well‐established strategy for evaluating alternative income distributions is based on the use of an abbreviated social welfare function that depends only on mean income and an inequality index. In keeping with this literature, we study the existence of social welfare functions that can be written as a trade‐off between efficiency and income polarization. This paper proposes a class of social welfare functions consistent with the Esteban and Ray, and Duclos, Esteban and Ray income polarization indices. For this result, we expand the domain for personal preferences to incorporate not only own income but also the well‐being of others. In addition, we link our proposal to the literature on relative satisfaction. The approach is illustrated by an empirical application using the CPS database for the United States in the period 1991–2010.     

Unequal uncertainties and uncertain inequalities: an axiomatic approach

In this paper, we provide an axiomatic characterization of social welfare functions for uncertain incomes. Our most general result is that a small number of reasonable assumptions regarding welfare orderings under uncertainty rule out pure ex ante as well as pure ex post evaluations. Any social welfare function that satisfies these axioms should lie strictly between the ex ante and the ex post evaluations of income distributions. We also provide an axiomatic characterization of the weighted average of the minimum and the maximum of ex post and ex ante evaluations.     

The Optimal Linear Income Tax Model: Utility or Equivalent Income?

This paper compares the use of equivalent income with that of utility, in the social welfare function, in optimal income tax models. Equivalent income is a money metric welfare measure that, unlike utility, is not affected by monotonic transformations of utility. The use of equivalent income is found to produce an optimal tax rate that is more sensitive to the degree of inequality aversion, compared with the use of utility. With Cobb-Douglas and CES utility functions, the optimal tax rate is the same for utility and equivalent income where relative inequality aversion is unity. When using equivalent incomes, the case for high marginal rates does not depend on the assumption of a very low elasticity of substitution between consumption and leisure.     

AN APPROXIMATION FOR THE OPTIMAL LINEAR INCOME TAX RATE

This paper derives a convenient method of calculating an approximation to the optimal tax rate in a linear income tax structure. Individuals are assumed to have Cobb-Douglas preferences and the wage rate distribution is lognormal. First, the optimal tax rate is shown, for a general form of social welfare function, to be the smallest root of a quadratic equation involving a welfare-weighted average wage rate. Second, an approximation to this average is derived for an isoelastic social welfare function. This average depends on the degree of inequality aversion of the welfare function and the coefficient on consumption in individuals' utility functions. Calculations show that the method performs well in comparison with standard simulation methods of computing the optimal tax rate.     

Aggregation and the existence of a social utility function

In a well-known paper Gorman (Econometrica21 (1953)) established that the necessary and sufficient condition for the existence of an aggregate, or social, utility function, independent of the distribution of income, is that all individuals' income consumption paths be parallel straight lines. Recently Chipman (J. Econ. Theory8 (1974)), building on the paper of Hurwicz and Uzawa (in “Preference Utility and Demand”) has shown that if the distribution of income is proportional and individual preferences are homothetic, aggregate consumption behavior obeys the necessary integrability conditions. It is shown here that the consistency of aggregate behavior can be derived from more general conditions than the ones used by Chipman and Gorman. Examples of demand systems from which aggregate behavior implies a social utility function are provided. It is then shown that if individual demand functions are linear in income—a form employed by both Gorman and Chipman—it is not necessary that the distribution of income be fixed.     

The effect on optimal consumption of increased uncertainty in labor income in the multiperiod case

We consider a multiperiod, additive utility, optimal consumption model with a riskless investment and a stochastic labor income. The main result is that for utility functions belonging to the set F, consumption decreases when we go from any sequence of distribution functions representing labor income to a more risky sequence. A concave utility function belongs to F if its first derivative exists everywhere and is convex.     

Status, affluence, and inequality: Rank-based comparisons in games of status

We examine the effects of changes in the income distribution in an economy where agents' utility depends both on consumption and on their rank in the distribution of conspicuous consumption. We introduce a new methodology to compare the behavior of agents that occupy the same rank in the two different income distributions but typically have different levels of incomes. Here, an increase in incomes of the least endowed improves their welfare, yet it also increases social competition, making those in the middle to be worse off—even if they have higher incomes as well. As social competition can be lowered by spreading agents apart in income space, we find that an increase in incomes for all, augmented by (weakly) increased income dispersion, constitutes a sufficient condition for Pareto-improvement. We also show that one can have an increase both in income and relative position but still be worse off.     

Welfare effects of energy subsidy reform in developing countries

We analyze the potential welfare effect of energy subsidy reforms. The income distributions of eleven developing countries from different geographical regions are simulated using the assumption that income is lognormally distributed. We use the concept of the compensating variation to measure how much compensation is required to compensate consumers for a price increase in formerly subsidized goods. The behavior of consumers is modeled by a standard Cobb–Douglas and a quasilinear utility function. In the Cobb–Douglas case, a fixed fraction of income is spent on the subsidized good, which does not change after a price increase. With quasilinear preferences, the optimal amount of the subsidized good does not vary with income, but does change as prices change. We show theoretically and empirically that the required compensating variation can be set below the saved expenditures on subsidies, so a budget neutral reform can have a positive effect on social welfare.     

Size and distribution of incomes as determinants of social welfare

The paper is concerned with the relationship between the inequality of income distributions and the social welfare they imply. Starting-point is an ordering of income distributions in terms of inequality. Based on explicit value judgments about the trade-off between size and distribution of incomes an ordering in terms of social welfare is derived. The welfare functions representing this ordering allow a size-distribution split up. It is shown that the procedure can be reversed. This ordinal approach is considerably more flexible and less arbitrary than most methods proposed in the literature.     

Bivariate income distributions for assessing inequality and poverty under dependent samples

As indicators of social welfare, the incidence of inequality and poverty is of ongoing concern to policy makers and researchers alike. Of particular interest are the changes in inequality and poverty over time, which are typically assessed through the estimation of income distributions. From this, income inequality and poverty measures, along with their differences and standard errors, can be derived and compared. With panel data becoming more frequently used to make such comparisons, traditional methods which treat income distributions from different years independently and estimate them on a univariate basis, fail to capture the dependence inherent in a sample taken from a panel study. Consequently, parameter estimates are likely to be less efficient, and the standard errors for between-year differences in various inequality and poverty measures will be incorrect. This paper addresses the issue of sample dependence by suggesting a number of bivariate distributions, with Singh–Maddala or Dagum marginals, for a partially dependent sample of household income for two years. Specifically, the distributions considered are the bivariate Singh–Maddala distribution, proposed by Takahasi (1965), and bivariate distributions belonging to the copula class of multivariate distributions, which are an increasingly popular approach to modelling joint distributions. Each bivariate income distribution is estimated via full information maximum likelihood using data from the Household, Income and Labour Dynamics in Australia (HILDA) Survey for 2001 and 2005. Parameter estimates for each bivariate income distribution are used to obtain values for mean income and modal income, the Gini inequality coefficient and the headcount ratio poverty measure, along with their differences, enabling the assessment of changes in such measures over time. In addition, the standard errors of each summary measure and their differences, which are of particular interest in this analysis, are calculated using the delta method.     

The Aggregation of Climate Change Damages: a Welfare Theoretic Approach

The economic value of environmental goods is commonly determined using the concepts of willingness to pay (WTP) or willingness to accept (WTA). However, the WTP/WTA observed in different countries (or between individuals) will differ according to socio-economic characteristics, in particular income. This notion of differentiated values for otherwise identical goods (say, a given reduction in mortality risk) has been criticized as unethical, most recently in the context of the 'social cost' chapter of the IPCC Second Assessment Report. These critics argue that, being a function of income, WTP/WTA estimates reflect the unfairness in the current income distribution, and for equity reasons uniform per-unit values should therefore be applied across individuals and countries. This paper analyses the role of equity in the aggregation of climate change damage estimates, using basic tools of welfare economics. It shows one way of how WTP/WTA estimates can be corrected in aggregation if the underlying income distribution is considered unfair. It proposes that in the aggregation process individual estimates be weighted with an equity factor derived from the social welfare and utility functions. Equity weighting can significantly increase aggregate (global) damage figures, although some specifications of weighting functions also imply reduced estimates. The paper also shows that while the postulate of uniform per-unit values is compatible with a wide range of 'reasonable' utility and welfare specifications, there are also cases where the common-value notion is not compatible with defensible welfare concepts.     

Transfer principles in income redistribution

Dalton's principle of transfer for income redistribution says that society will be better off when a unit of income in transferred from a richer to a poorer individual. This paper investigates the conjunction of a natural extension of Dalton's principle — which says that any combination of a socially desirable transfer with its inverse at uniformly higher levels of income will have positive social benefit — and a traditional utilitarian-egalitarian formulation of social welfare. The transfer principles involved in the extension of Dalton's rule correspond to stochastic dominance relations between income distributions with a common mean, and lead to the conclusion that, when ? and g are income distributions, the social welfare of ? is larger than that of g if $\text{Σe}{}^{\mathrm{-ax}}\text{?(x)<Σe}{}^{\mathrm{-ax}}\text{g(x)}$ for all positive a. This conclusion is intimately connected with a measure of income inequality studied by Kolm.