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We investigate the relationship between the notion of progressive taxation and inequality reduction under a general version of the concept of inequality equivalence. We consider a two-parameter formalization of the concept of inequality equivalence that both includes, as special cases, the intermediate inequality equivalence and the path-independent/unit-consistent inequality equivalence. Both criteria could range from relative to absolute inequality views as the parameters in the formulation change. For the path-independent/unit-consistent inequality equivalence the condition of nondecreasing average tax rate is necessary and sufficient to guarantee the inequality-reducing effect of taxation for all the inequality views in between the relative and the absolute.

The purpose of this paper is to investigate how to determine optimal investing stopping time in a stochastic environment, such as with stochastic returns, stochastic interest rate and stochastic expected growth rate.

Transformation method was used for solving optimal stopping problem by providing a way to transform path‐dependent problem into a path‐independent one. Based on option pricing theory, optimal investing stopping time was thought of as an optimal executed timing problem of American‐style option.

First, the authors transform a path‐dependent stop timing problem to a path‐independent one with transformation under very general conditions, to directly use the existing conclusion of optimal stopping time literature. Second, when dynamics of capital growth is homogeneous, the authors changed the two dimensional optimal stop timing problem into a single dimension problem based on the assumption of zero exercise costs. Third, the authors investigated the comparative dynamics about asset selling boundary on asset value, state variable and return predictability. With constant discount rate and growth rate, the optimal selling timing depends on the simple comparison between capital cost and growth rate.

The paper's contributions to analysis method may be as follows. The authors demonstrate how to transform a path‐dependent stopping problem into a path‐independent one under general conditions. The transform method in this article can be applied to other path‐dependent optimal stopping problems. In particular, a Riccati ordinary differential equation for the transformation is set up. In most examples commonly met in finance, the equation can be solved explicitly.

The chapter suggests two methodologies to measure inequality of opportunity in health in Israel, an ex-ante and an ex-post approach. In both cases, following the strategy recently suggested by Trannoy, Tubeuf, Jusot, and Devaux (2010), the chapter starts by introducing the production function of health, taking into account circumstances (the father’s years of education, his country of birth, the religion of the individual, his or her country of birth, age and gender) as well as effort variables (the level of education of the individual, his or her occupation and a variable describing his or her smoking habits).

The chapter also suggests then a decomposition of the overall health inequality into a legitimate and an illegitimate component, using the mean logarithmic deviation as inequality index, such a breakdown being applied to both the ex-ante and the ex-post approaches to equality of opportunity.

The aim of this chapter is to shed some light on the behavior of Income Inequality and Inequality of Opportunity over time for 26 European countries. The analysis is carried out using microdata collected by the European Union Statistics on Income and Living Conditions (EU-SILC), which incorporates a wide variety of personal harmonized variables, allowing comparability between countries. The availability of this database for years 2004 and 2010 is particularly relevant to assess changes over time in the main inequality indices and the contribution of circumstances to inequality of opportunity. Furthermore, a bootstrap estimation is performed with the aim of testing whether the differences between both years are statistically significant.

Purpose – Our ultimate goal is to characterize three methodological issues. First, compare the relative performance of alternative estimation methods for long time series, second, estimate the degree of correlation between effort and circumstances, and, third, decompose total inequality into inequality of opportunity and inequality of effort according to an ideal tree.

Methodology – We estimate parametrically and nonparametrically the ex-ante inequality of opportunity in the United States between 1969 and 2007. The degree of correlation between effort and circumstances is computed following the proposal in Björklund et al. (2011). In addition, we decompose total inequality based on an ideal tree with three levels of disaggregation by applying the natural decomposition of the squared coefficient of variation and the Nested Shapley value.

Findings – We find significant differences between the nonparametric and parametric approaches. In particular, our results reveal that considering cross-effects between circumstances may be relevant. Moreover, the degree of correlation between effort and circumstances which has significantly increased over the period 1969 and 2007 in the United States, explains between 5% and 20% of total IO. In addition, race is the main circumstance during the 1970s and 1980s, accounting for more than 50% of the direct IO, while parental education take the lead in the last two decades.

Originality – We modify the parametric specification by considering cross-effects between circumstances. We estimate the degree of correlation between effort and circumstances for long time series. We decompose total inequality according to a three-level hierarchical model.

When health is measured by a bounded variable, differences in health can be presented as levels of attainment or shortfall. Measurement of heath inequality then usually involves the choice of either the attainment or the shortfall distribution, and this choice may affect comparisons of inequality across populations. A number of indices have been introduced to overcome this problem. This chapter proposes a framework in which attainment and shortfall distributions can be jointly analyzed. Joint distributions of attainments and shortfalls are defined from points of view consistent with concerns for relative, absolute or intermediate inequality. Inequality measures invariant according to the corresponding ethical criterion are then applied. A dominance criterion that guarantees unanimous rankings of the joint distributions is also proposed.

A number of multidimensional poverty measures that respect the ordinal nature of dimensions have recently been proposed within the counting approach framework. Besides ensuring a reduction in poverty, however, it is important to monitor distributional changes to ensure that poverty reduction has been inclusive in reaching the poorest. Distributional issues are typically captured by adjusting a poverty measure to be sensitive to inequality among the poor. This approach, however, has certain practical and conceptual limitations. It conflicts, for example, with some policy-relevant measurement features, such as the ability to decompose a measure into dimensions post-identification and does not create an appropriate framework for assessing disparity in poverty across population subgroups. In this chapter, we propose and justify the use of a separate decomposable inequality measure – a positive multiple of “variance” – to capture the distribution of deprivations among the poor and to assess disparity in poverty across population subgroups. We demonstrate the applicability of our approach through two contrasting inter-temporal illustrations using Demographic Health Survey data sets for Haiti and India.

Purpose – The concept of equality of opportunity (EOp) goes back to Roemer (1993, 1998) who argues that a society should guarantee its members equal access to advantage regardless of their circumstances, while holding them responsible for turning that access into actual advantage by the application of effort. First, this chapter investigates how family background influences income acquisition in 17 European countries. Second, it particularly scrutinizes how governments affect EOp through redistributive policies.

Methodology – We apply two different methods in order to measure EOp: the Gini opportunity index defined by Lefranc et al. (2008) and a parametric estimation method. Effective redistribution is measured via income concepts at different stages of the tax-and-transfer schemes.

Findings – We find clear country clustering in terms of EOp for Nordic, Continental European, and Anglo-Saxon countries. For Eastern Europe our results are less definitive. By examining the impact of redistributive policies in the countries under analysis, it can be concluded that both taxes and transfers reduce inequality of opportunity (IOp), with social benefits typically playing a key role. Furthermore, the equalizing impacts of the tax-benefit system on IOp differ substantially from the ones observed in the traditional notion of inequality of outcomes.

Originality – We systematically compare two approaches used to identify the extent of EOp. Our results reveal that both methods yield rather robust country rankings for various circumstance sets. Furthermore, the impact of tax-benefit policies on EOp is rarely addressed in the existing literature. We contribute by focusing on effective redistribution directly related to different income concepts.

Purpose – The aim of this chapter is to analyze the measurement error due to partial observability of circumstances, given the current methodologies used to capture the inequality of opportunity.

Methodology – We limit our analysis to the case in which IOp is measured by using an inequality index and evaluated according to the ex ante and the ex post approaches proposed by Checchi and Peragine (2010). Both approaches rely on the use of two hypothetical distributions obtained by standardizing and smoothing processes so that in one of them all inequalities are due to effort and in the other they are all caused by individuals circumstances. The impact of a change in the observability of circumstances has been analyzed directly in the first case (ex ante approach) and indirectly in the second (ex post approach). Hence, in the latter results are obtained by using a decomposable inequality index, while in the first case the result holds for any Lorenz consistent inequality index.

Findings – We have shown that, in both cases, we can only identify a lower bound of opportunity inequality, and the results obtained when measuring IOp with the two approaches are biased in the same direction.

Originality – The idea that inequality of opportunity, as it is generally measured, is only a lower bound of the real inequality of opportunity has already been hinted in Peragine (2004a), Fleurbaey (2008, Chap. 9), and Ferreira and Gignoux (2011). We attempt to bring this intuition to complete fruition in this chapter by providing clear and distinct demonstrations of this result for both the ex ante and the ex post approach for measuring IOp.

The chapter investigates inequality reducing taxation for various inequality views. Using the general definition of an inequality concept (Ebert, 2004), corresponding definitions of Lorenz dominance, inequality reduction, and measures of tax progression are provided. The framework allows us to simplify and clarify the different approaches found in the literature, to extend this analysis, and to present brief and transparent proofs.