Brice Magdalou

Income distributions and decomposable divergence measures

Inequality indices (i) evaluate the divergence between the income distribution and the hypothetical situation where all individuals have the mean income and (ii) are unambiguously reduced by a Pigou-Dalton progressive transfer. This paper proposes a new approach to evaluate the divergence between any two income distributions, where the second one can be a reference distribution for the first. In the case where the reference distribution is perfectly egalitarian, and uniquely in this case, we assume (i) that any progressive transfer reduces the divergence and (ii) that the divergence can be additively separated between inequality and efficiency loss. We characterize the unique class of decomposable divergence measures consistent with these views, and we derive the associated relative (resp. absolute) subclasses, which express constant relative (resp. absolute) inequality aversion. This approach extends the generalized entropy studied in inequality measurement.

Deprivation, welfare and inequality

We provide a characterization of the generalised satisfaction—in our terminology non-deprivation—quasi-ordering introduced by S.R. Chakravarty (Keio Econ Stud 34:17–32, (1997)) for making welfare comparisons. The non-deprivation quasi-ordering obeys a weaker version of the principle of transfers: welfare improves only for specific combinations of progressive transfers, which impose that the same amount be taken from richer individuals and allocated to one arbitrary poorer individual. We identify the extended Gini social welfare functions that are consistent with this principle and we show that the unanimity of value judgements among this class is identical to the ranking of distributions implied by the non-deprivation quasi-ordering. We extend the approach to the measurement of inequality by considering the corresponding relative and absolute ethical inequality indices.

Mining Matrix Data with Bregman Matrix Divergences for Portfolio Selection

In the early fifties, Markowitz contributed a theory that considerably simplified portfolio choices. He narrowed down the traditional expected utility model and assumed that investors only care for mean and variance. The mean-variance portfolio theory was born. As it name suggests, mean-variance theory is predicated on simple assumptions that are unfortunately seldomly met in real life. Indeed, it is now a well-established fact that for a host of reasons financial returns do not obey Gaussian distributions. This paper first draws on ideas from econometrics, finance and statistics to derive a rigorous generalization of Markowitz’ mean-variance model to a mean-divergence model, lifted to matrix entries, grounded on exponential families of distributions, that we argue is both more realistic and better suited to further developments in learning. The generalized model turns out to heavily rely on Bregman divergences. There has recently been a burst of attention in on-line learning to learn portfolios having limited risk in Markowitz’ setting. In an on-line framework, we then tackle the problem of finding adaptive portfolio strategies based on our generalized model. We devise a learning algorithm based on new matrix generalizations of p-norms to track non stationary target portfolios with limited risk. Theoretical bounds and preliminary experiments over nearly twelve years of S\(\&\)P 500 confirm the validity of the generalized model, the capacity it brings to spot important events that would otherwise be dampened in the mean-variance model, and the potential of the algorithm. Finally, we make an in depth analysis of the matrix divergences and risk premia derived in our model that shed some theoretical light on the ways the risk premium may be blown up as the investor’s portfolio shifts away from a so-called natural market allocation which defines the best (unknown) allocation at the market’s scale.

Opportunities in Higher Education: An Application to France

This paper first sets out criteria to evaluate the inequality of opportunity in higher education. We propose a two-level classification by distinguishing the categorial and the counterfactual approaches. For each approach we consider dominance quasi-orderings and inequality measures. We then apply these criteria to the French higher education system by comparing the situations in 1992 and 2004. Our main findings suggest: (i) that there is inequality of opportunity in each of the two years; and (ii) that the inequality of opportunity tends to increase during the reporting period, especially the opportunity to graduate in the lowest and highest levels of education.

An Oaxaca Decomposition for Nonlinear Models

The widely used Oaxaca decomposition applies to linear models. Extending it to commonly used nonlinear models such as duration models is not straightforward. This paper shows that the original decomposition that uses a linear model can also be obtained by an application of the mean value theorem. By extension, this basis provides a means of obtaining a decomposition formula which applies to nonlinear models which are continuous functions. The detailed decomposition of the explained component is expressed in terms of what are usually referred to as marginal effects. Explicit formulae are provided for the decomposition of some nonlinear models commonly used in applied econometrics including binary choice, duration and Box-Cox models.