On the existence of a probability measure compatible with a total preorder on a Boolean algebra

If an economic agent's beliefs about the relative likelihood of events are characterized by a total preorder ? on the algebra $\text{A}$ of events, the problem arises to know under which conditions, ? is representable by a probability measure. Here we show that there exists a probability measure compatible with a total preorder on a Boolean algebra, if and only if, the Boolean algebra is well bounded, weakly Archimedean, and perfectly separable, this last condition substituting for Villegas' monotone condition used in Chateauneuf and Jaffray (1984); if σ-additivity is required. Villegas' monotone condition, must merely be added.     

Inequality and Crime: Separating the Effects of Permanent and Transitory Income*

Earlier studies on income inequality and crime have typically used total income or total earnings. However, it is quite likely that it is the changes in permanent rather than in transitory income that affects crime rates. The purpose of this paper is therefore to disentangle the two effects by, first, estimating region‐specific inequality in permanent and transitory income and, second, estimating crime equations with the two separate income components as explanatory variables. The results indicate that it is important to separate the two effects; while an increase in the inequality in permanent income yields a positive and significant effect on total crimes and three different property crimes, an increase in the inequality in transitory income has no significant effect. Using a traditional, aggregate, measure of income yields insignificant effects on crime.     

Expected utility theory on mixture spaces without the completeness axiom

A mixture preorder is a preorder on a mixture space (such as a convex set) that is compatible with the mixing operation. In decision theoretic terms, it satisfies the central expected utility axiom of strong independence. We consider when a mixture preorder has a multi-representation that consists of real-valued, mixture-preserving functions. If it does, it must satisfy the mixture continuity axiom of Herstein and Milnor (1953). Mixture continuity is sufficient for a mixture-preserving multi-representation when the dimension of the mixture space is countable, but not when it is uncountable. Our strongest positive result is that mixture continuity is sufficient in conjunction with a novel axiom we call countable domination, which constrains the order complexity of the mixture preorder in terms of its Archimedean structure. We also consider what happens when the mixture space is given its natural weak topology. Continuity (having closed upper and lower sets) and closedness (having a closed graph) are stronger than mixture continuity. We show that continuity is necessary but not sufficient for a mixture preorder to have a mixture-preserving multi-representation. Closedness is also necessary; we leave it as an open question whether it is sufficient. We end with results concerning the existence of mixture-preserving multi-representations that consist entirely of strictly increasing functions, and a uniqueness result.     

The Computational Complexity of Rationalizing Behavior

We study the computational complexity of rationalizing choice behavior. We do so by analyzing two polar cases, and a number of intermediate ones. In our most structured case, that is where choice behavior is defined in universal choice domains and satisfies the “weak axiom of revealed preference,” finding the complete preorder rationalizing choice behavior is a simple matter. In the polar case, where no restriction whatsoever is imposed, either on choice behavior or on choice domain, finding a collection of complete preorders that rationalizes behavior turns out to be intractable. We also show that the task of finding the rationalizing complete preorders is equivalent to a graph problem. This allows the search for existing algorithms in the graph theory literature, for the rationalization of choice.     

A simple voting scheme generates all binary relations on finite sets

A simple head-to-head voting scheme in which voters hold complete and transitive preferences over alternatives generates all binary relations on finite sets. The minimal number of voters required to generate a binary relation provides a measure of complexity for binary relations. Complexity so defined tells us, by how much a given binary relation fails to qualify as a total preorder.     

Estimating a Parsimonious Model of Inequality Aversion in Stackelberg Duopoly Experiments*

In the Stackelberg duopoly experiments in Huck et al. (2001) , nearly half of the followers’ behaviours are inconsistent with conventional prediction. We use a test in which the conventional self‐interested model is nested as a special case of an inequality aversion model. Maximum likelihood methods applied to the Huck et al. (2001) data set reject the self‐interested model. We find that almost 40% of the players have disadvantageous inequality aversion that is statistically different from zero and economically significant, but advantageous inequality aversion is relatively unimportant. These estimates provide support for a more parsimonious model with no advantageous inequality aversion.     

Measuring regional inequality: to weight or not to weight?

Measuring regional inequality: to weight or not to weight? Spatial Economic Analysis. When estimating regional inequality, many economists use inequality indices weighted by regions’ proportion of the national population. Although this approach is widespread, its adequacy has not received attention in the regional science literature. This paper proves that such an approach is conceptually inconsistent, yielding an estimate of interpersonal inequality among the whole population of the country rather than an estimate of regional inequality. Nevertheless, as a measure of interpersonal inequality, such an estimate is very rough (even misleading) and does not always have an intuitive interpretation. Moreover, population-weighted inequality indices do not meet the requirements for an adequate measure of inequality.     

Inequality for Wage Earners and Self‐Employed: Evidence from Panel Data*

In this paper we highlight the importance of analysing the evolution of income inequality separately for employees and self‐employed workers. Using Spanish panel data on income and consumption for the period 1987–96, we find noticeable differences across these groups in the evolution of income inequality, and in the relative importance of the transitory and permanent components of income variance. The evolution of inequality is mainly explained by movements in the transitory component for the self‐employed and by the permanent component for the employees. Our results suggest that different policies should be implemented for each group.     

Edgeworth expansions and normalizing transforms for inequality measures

Finite sample distributions of studentized inequality measures differ substantially from their asymptotic normal distribution in terms of location and skewness. We study these aspects formally by deriving the second-order expansion of the first and third cumulant of the studentized inequality measure. We state distribution-free expressions for the bias and skewness coefficients. In the second part we improve over first-order theory by deriving Edgeworth expansions and normalizing transforms. These normalizing transforms are designed to eliminate the second-order term in the distributional expansion of the studentized transform and converge to the Gaussian limit at rate O(n⁻¹)$O\left(n^{-1}\right)$. This leads to improved confidence intervals and applying a subsequent bootstrap leads to a further improvement to order O(n^−3/2)$O\left(n^{-3/2}\right)$. We illustrate our procedure with an application to regional inequality measurement in Côte d’Ivoire.     

Mis-measurement of inequality: a critical reflection and new insights

This article documents that the Gini index is an insufficient measure of inequality and, according to the traditional logic of interpretation, that it may lead to incorrect deductions. Since, apart from concentration, it cannot grasp other relevant features of inequality like heterogeneity and asymmetry—which, beyond its intensity, allow for considering the direction of inequality too—we suggest using the less known Zanardi index of asymmetry of the Lorenz curve as an appropriate measure of inequality. Our findings are supported with estimates from the Luxembourg Income Study Database.

     

Bounding quantile demand functions using revealed preference inequalities

This paper develops a new approach to the estimation of consumer demand models with unobserved heterogeneity subject to revealed preference inequality restrictions. Particular attention is given to nonseparable heterogeneity. The inequality restrictions are used to identify bounds on counterfactual demand. A nonparametric estimator for these bounds is developed and asymptotic properties are derived. An empirical application using data from the UK Family Expenditure Survey illustrates the usefulness of the methods.     

A fuzzy AHP approach based on the concept of possibility extent

The main purpose of this paper is to develop a fuzzy AHP method for tackling the uncertainty and imprecision existing in multi-criteria decision process. The proposed method uses fuzzy pair-wise comparison judgments in place of exact numerical values of the comparison ratios. The geometric mean technique is used to integrate all decision-makers’ opinions and construct the fuzzy positive reciprocal matrices. The algebraic operations of triangular fuzzy numbers are utilized to calculate the fuzzy suitability indices of all alternatives. The extent analysis method is used to compute the degree of possibility of priority among fuzzy suitability indices. Besides, two principles are presented to solve the multi-criteria decision problem in a fuzzy decision environment. Principle I provides a partial preorder, and Principle II gives a total preorder on the set of the possible alternatives. Finally, a numerical example of selecting the company with optimal performance in performing customer relationship management is used to demonstrate the decision process of proposed method.     

Income distribution and inequality measurement: The problem of extreme values

We examine the statistical performance of inequality indices in the presence of extreme values in the data and show that these indices are very sensitive to the properties of the income distribution. Estimation and inference can be dramatically affected, especially when the tail of the income distribution is heavy, even when standard bootstrap methods are employed. However, use of appropriate semiparametric methods for modelling the upper tail can greatly improve the performance of even those inequality indices that are normally considered particularly sensitive to extreme values.     

(Hemi)continuity of additive preference preorders

It is shown that the two common notions of topological continuity for preference preorders, which require closed contour sets and a closed graph respectively, are equivalent even when completeness is not assumed, provided that the domain is a normed linear space or a topological group and the preorder is additive.     

Understanding Italian Inequality Trends*

This article develops a unifying framework for analysing the effects of: (i) the changing distribution of individual incomes by main factor sources, (ii) the increasing participation of wives in the labour force, and (iii) the changing distribution of family types on the distinctive trends towards inequality in equivalent household income in Italy between 1977 and 2004. Changes in the distribution of work and pension incomes explain most of the trend. The higher average likelihood of wage‐earning wives had an unequalizing effect on households on the left tail of the income distribution. Little is explained by the changing distribution of family types.     

The ‘postcode lottery’ of the Italian public health bill analysed with the hierarchy Stochastic Multiobjective Acceptability Analysis

Over the last two decades, due to strong decentralization and widespread budget constraints, the Italian co-payment for health care has become a way to finance public health. This phenomenon has provoked a continuous increase of private costs of public health and an evident regional heterogeneity. As a result, a pervasive spatial inequality of access to public health care is becoming increasingly clear. The aim of this paper is to measure this inequality, mainly determined by the differences among regional co-payment prices. Access, equity, and needs are all part of the phenomenon ‘inequality of access’, and they are difficult to define and measure in health care. For this reason, most of the previously proposed measurement methods have inherent limitations and have prompted us to use an innovative approach focused exclusively on the supply side. In particular, we focus only on the cost of health benefits (co-payment). From a methodological perspective, we use a recent new version of the Stochastic Multiobjective Acceptability Analysis (SMAA), which is a methodology mainly used to build composite indicators of multidimensional phenomena out of the market. In order to deal with the hierarchical structure of the Italian health care system, we use the Hierarchy Stochastic Multiobjective Acceptability Analysis (HSMAA), which takes into account the uncertainty with respect to the weights assigned to the considered criteria, as in the standard SMAA, but also the uncertainty with respect to the weights assigned to the considered sub-criteria. Applying for the first time HSMAA to measure inequality allows us to create a unique index for each region and then to make a classification among them. The results show that, since there are different prices for the same health benefits among different regions, there are strong spatial inequalities in the cost of the Essential Levels of health care in Italy.     

Hierarchical analysis: Classification with ordinal object dissimilarities

Summary Lerman [1970] has demonstrated, that the dissimilarity indices normally used in data analysis are identical up to strictly monotone transformationsf:R ₊R ₊ if the data are nominal and each set of attribute scores is finite.In that case he proposes to use a preorder between pairs of objects to express similarity or dissimilarity, in order to avoid inconsistent classification results that might occur, if clustering schemes which are not monotone invariant are applied to a quantitative index. Here it is shown, how a hierarchy on the objects can be calculated if such a preorder relation is given.     

DOES GOVERNMENT SPENDING AFFECT INCOME INEQUALITY? A META-REGRESSION ANALYSIS

In this paper findings of a meta-regression analysis are presented exploring the effects of government spending on income inequality, with a particular focus on low- and middle-income countries. We identify a total of 84 separate studies containing over 900 estimates of the effect of one or more measures of spending on one or more measures of income inequality. The results show some evidence of a moderate negative relationship between government spending and income inequality, which is strongest for social welfare and other social spending, and when using the Gini coefficient or the top income share as the measure of inequality. However, both the size and direction of the estimated relationship between government spending and income inequality is affected by a range of other factors, including the control variables and estimation method used. We also find evidence of publication bias, in that negative estimates of the relationship appear to be under-reported in the literature.     

Does income inequality lead to banking crises in developing countries? Empirical evidence from cross-country panel data

This study empirically examines whether increasing income inequality results in banking crises using panel data for 68 countries covering the years 1973 to 2010. The results show that developing countries with high inequality tend to have higher levels of domestic credit and that domestic credit booms increase the probability of banking crises. We also find that developing economies display direct channels from inequality to banking crises without an association with credit booms. We find no consistent evidence that income inequality contributes to banking crises in advanced economies. In developing countries, the probability of banking crises increases dramatically as income inequality levels increase: The probability of a systemic banking crisis within three years is 9.5% when the Gini is as low as 0.2 in developing countries and increases to 57.4% when the Gini is 0.4. These results are robust to several specifications.     

DOES EDUCATION REDUCE INCOME INEQUALITY? A META‐REGRESSION ANALYSIS

This paper reexamines the effects of education on inequality through a comprehensive meta‐regression analysis of the extant empirical literature. We find that education affects the two tails of the distribution of income: Education reduces the income share of top earners and increases the share of the bottom earners. Education has been particularly effective in reducing inequality in Africa. Some of the results suggest that secondary schooling appears to have a stronger effect than primary schooling, though this finding is not always robust. The heterogeneity in reported estimates can be largely explained by differences in the specification of the econometric model and measure of inequality and education.