Unambiguous events and dynamic Choquet preferences

This paper explores the relationship between dynamic consistency and existing notions of unambiguous events for Choquet expected utility preferences. A decision maker is faced with an information structure represented by a filtration. We show that the decision maker’s preferences respect dynamic consistency on a fixed filtration if and only if the last stage of the filtration is composed of unambiguous events in the sense of Nehring (Math Social Sci 38:197–213, 1999). Adopting two axioms, conditional certainty equivalence consistency and constrained dynamic consistency to filtration measurable acts, it is shown that the decision maker respects these two axioms on a fixed filtration if and only if the last stage of the filtration is made up of unambiguous events in the sense of Zhang (Econ Theory 20:159–181, 2002).     

The α-MEU model: A comment

In Ghirardato et al. (2004) [7], Ghirardato, Macheroni and Marinacci propose a method for distinguishing between perceived ambiguity and the decision-maker?s reaction to it. They study a general class of preferences which they refer to as invariant biseparable. This class includes CEU and MEU. They axiomatize a subclass of α-MEU preferences. If attention is restricted to finite state spaces, we show that any α-MEU preference relation, satisfies GMM?s axioms if and only ifα=0 or 1, that is, the preferences must be either maxmin or maxmax. We show by example that these axioms may be satisfied when the state space is [0,1].     

Measures of distance between income distributions

This paper presents a class of measures of distance between two income distributions. The measures are based on distribution functions and evaluate the degree of inequality of income between two populations. This class of measures is derived by an axiomatic approach and it is completely characterized by the properties normally associated with a distance function and a set of axioms.     

Local versus Global Assessment of Mobility*

Mobility indices are popular tools designed to quantify the extent of income changes by aggregating “local” distributional change into a “global” scalar according to some rule. For some mobility measures, this aggregation rule is only implicit in their standard definition. We derive an insightful approximation to the (statistical) aggregation rule for the important class of mobility indices introduced by Shorrocks (Journal of Economic Theory 19 (1978), 376–93) and further generalized by Maasoumi and Zandvakili (Economic Letters 22 (1986), 97–102), which enables us to characterize their normative properties. We also develop methods for estimation and inference. A substantive empirical contribution emerges from the comparison of mobility between the United States and Germany. Our methods reveal why income mobility is higher in Germany than in the United States: Higher German mobility in the bottom of the distribution is combined with an implicitly higher weighting by the mobility index at the bottom.     

STRATEGIC CANDIDACY AND SINGLE‐PEAKEDNESS*

In elections, the voting outcomes are affected by strategic entries of candidates. We study a class of voting rules immune to strategic candidacy. Dutta et al. (2001 ) show that such rules satisfying unanimity are dictatorial if all orderings of candidates are admissible for voters’ preferences. When voters’ preferences are single‐peaked over a political spectrum, there exist non‐dictatorial rules immune to strategic candidacy. An example is the rule selecting the m‐th peak from the left among the peaks of voters’ preferences, where m is any natural number no more than the number of voters. We show that immunity from strategic candidacy with basic axioms fully characterizes the family of the m‐th leftmost peak rules.     

Vulnerability Orderings for Expected Poverty Indices

Vulnerability to poverty refers to the risk of an individual or household falling below the poverty line. A partial vulnerability to poverty ordering is defined to indicate when a situation characterized by uncertainty means that people are more vulnerable to poverty than in another situation. The family of expected poverty indices used in this context contains versions under vulnerability of many well‐known poverty indices. The ordering resembles the Hardy et al. result on the Lorenz partial ordering. A separate ordering for the expected income gap ratio is also investigated. First‐order stochastic dominance is used to indicate when the expected value of the censored returns from the states increases.     

Multidimensional generalized Gini indices

Summary. The axioms that characterize the generalized Gini social evaluation orderings for one-dimensional distributions are extended to the multidimensional attributes case. A social evaluation ordering is shown to have a two-stage aggregation representation if these axioms and a separability assumption are satisfied. In the first stage, the distributions of each attribute are aggregated using generalized Gini social evaluation functions. The functional form of the second-stage aggregator depends on the number of attributes and on which version of a comonotonic additivity axiom is used. The implications of these results for the corresponding multidimensional indices of relative and absolute inequality are also considered.Received: 20 August 2003, Revised: 26 May 2004, JEL Classification Numbers: D63. Correspondence to: John A. WeymarkWe are grateful to our referee for his or her comments.     

Measuring inequality change in an economy with income growth

This paper analyzes how to measure changes in inequality in an economy with income growth. The discussion distinguishes three stylized kinds of economic growth:

• 1.(1) high income sector enrichment,
• 2.(2) low income sector enrichment,
• 3.(3) high income sector enlargement, in which the high income sector expands and absorbs persons from the low income sector.

Th e two enrichment types pose no problem for assessing inequality change in the course of economic growth: for high income sector enrichment growth, inequality might reasonably be said to increase, whereas for low income sector enrichment, inequality might be said to decrease. These adjustments are non-controversial and non-problematical. Where problems arise is in the case of high income sector enlargement growth. In that case, the two alternative approaches have been shown in this paper to yield markedly results:

• 1.(1) The traditional inequality indices generate an inverted-U pattern of inequality. That is, inequality rises in the early stages of high income sector enlargement growth and falls thereafter.
• 2.(2) The new approach suggested here, based on axioms of gap inequality and numerical inequality, generates a U pattern of inequality. That is, inequality falls in the early stages of high income sector enlargement growth and rises thereafter.

The discrepancy between the familiar indices and the alternative approach based on axioms of gap inequality and numerical inequality bears further scrutiny. Two courses of action are possible. One might try to axiomatize inequality in ways that generate an inverted-U pattern in high income sector enlargement growth, thereby rationalizing the continued use of the usual inequality indices with the inverted-U property. Alternatively, one might retain the axioms proposed here, embed them into a more formal structure, and construct a family of inequality indices consistent with them. Others might wish to pursue the first course; I am at work on the second.     

Living Standard,Social Welfare,and the Redistribution of Income in a Heterogeneous Population

The paper considers the redistribution of income in a population of households which may differ by type. The redistribution is based on (differences in) living standard and a principle of between‐type‐progressive transfers (BTPT). We characterize the relationship between a social welfare ordering satisfying the BTPT principle and the concept of living standard the principle is based on. It turns out that there is a close link: The ordering of living standard can be derived from the social welfare ordering. Conversely, the class of welfare orderings fulfilling the BTPT principle for a given concept of living standard can be completely described.     

Group identification

I study a model of group identification in which individuals' opinions as to the membership of a group are aggregated to form a list of group members. Potential aggregation rules are studied through the axiomatic approach. I introduce two axioms, meet separability and join separability, each of which requires the list of members generated by the aggregation rule to be independent of whether the question of membership in a group is separated into questions of membership in two other groups. I use these axioms to characterize a class of one-vote rules, in which one opinion determines whether an individual is considered to be a member of a group. I then show that the only anonymous one-vote rule is self-identification, in which each individual determines for himself whether he is a member of the group.     

Mobility of Top Incomes in Germany

I analyze German top income mobility using micro‐level panel data of personal income tax returns which are highly representative for top income taxpayers for the years 2001–06. Top income mobility is assessed in three dimensions: (i) persistence in top income fractiles and its stability over time, (ii) measures of individual mobility that are not dependent on the fractile size: the degree of mobility between equally sized groups and mobility in ranks, and (iii) mobility's impact on top income shares. Persistence in top income fractiles is comparatively high and fairly stable across the analyzed period. Top income recipients are less prone to downward mobility and see less variation in annual ranks than less rich tax units. Mobility's impact on income concentration is moderate. The top percentile's share is reduced by roughly 5 percent over six years.     

The measurement of rank mobility

In this paper we investigate the problem of measuring social mobility when the social status of individuals is given by their rank. In order to sensibly represent the rank mobility of subgroups within a given society, we address the problem in terms of partial permutation matrices which include standard (“global”) matrices as a special case. We first provide a characterization of a partial ordering on partial matrices which, in the standard case of global matrices, coincides with the well-known “concordance” ordering. We then provide a characterization of an index of rank mobility based on partial matrices and show that, in the standard case of comparing global matrices, it is equivalent to Spearman's ρ index.     

Social orderings for the assignment of indivisible objects

In the assignment problem of indivisible objects with money, we study social ordering functions which satisfy the requirement that social orderings should be independent of changes in preferences over infeasible bundles. We combine this axiom with efficiency, consistency and equity axioms. Our result is that the only social ordering function satisfying those axioms is the leximin function in money utility.     

Nest-monotonic two-stage acts and exponential probability capacities

This paper examines conditions for Choquet expected utility (CEU) to satisfy both the reduction of two-stage acts and the recursion axioms, which are taken for granted in economics. A key idea of this paper is to consider nest-monotonic two-stage acts, which share their rankings of states with those of their reduced one-stage acts. Our main theorem shows that the axioms, one of which is restricted to nest-monotonic two-stage acts, and consequentialism are satisfied if and only if the preference is exponential CEU, which is such that the probability capacity is an exponential transformation of a probability measure. This result indicates that within a specified range of decision problems, exponential CEU is the only form of CEU that derives indifference to the timing of information resolution. Furthermore, the relation between first- and second-stage exponential CEU is characterized both by the f*-Bayesian updating rule and by comonotonic dynamic consistency. Conditions to establish the law of iterated expectation for CEU are also discussed.     

A Note on α‐Gini Measures

This note provides a characterization of α‐Gini inequality measures. These measures generalize the standard Gini index by including one sensitivity parameter α, which captures different value judgments. The α‐Gini measures are shown to be weakly decomposable and unit consistent. Weak decomposition provides within‐group and between‐group inequalities. Unit consistency keeps unchanged the ranking of two income distributions when the income units vary. It is shown that the α‐Gini measures are relevant with either “leftist” or“rightist” views.     

Income Mobility, Temporary and Permanent Poverty

This paper incorporates poverty persistence in a measure of aggreg ate poverty over two-periods by decomposing the Foster et al . (1984) class of poverty measures into those temporarily in poverty and those in poverty in both periods. An additional weight is added to the permanent component in forming an aggregate poverty measure over both periods; this weight reflects the degree of poverty-persistence aversion. The effect on aggregate poverty of mobility between permanent and temporary poverty is found to be unambiguous only in the case of the headcount poverty measure. Simulations are used to investigate the relationship between poverty and mobility. The effects of two different types of mobility (random proportio nal income changes and a systematic regression towards or away from the median) are isolated     

Parental Wealth and the Black–White Mobility Gap in the U.S.

Utilizing longitudinal data from the Panel Study of Income Dynamics (PSID), this paper examines the relationship between parental wealth and intergenerational income mobility for black and white families. I find that total parental wealth is positively associated with upward mobility for low‐income white families, but is not associated with reduced likelihood of downward mobility for white families from the top half of the income distribution. Conversely, I find that total parental wealth does not have the same positive association for low‐income black families, while home ownership may have negative associations with the likelihood of upward mobility for these families. However, for black families from the top half of the income distribution, home equity is associated with a decreased likelihood of downward mobility, suggesting a heterogeneous relationship between home ownership and mobility for black families.     

An extension of Fishburn's theorem on extending orders

P. C. Fishburn [J. Econom. Theory31 (1983), 000-000] has shown that one can extend any well ordering of a set to a linear ordering of the set of all non-empty subsets of that set, while satisfying two axioms denoted (GP¹) and (M¹). By applying the compactness theorem of logic, this note shows that the well ordering assumption can be waived. Definability and well ordering properties of the extension are also discussed and shown to depend on the class of basic orderings considered.     

Optimal consumption and the timing of the resolution of uncertainty

As is well-known, consumers want to accumulate precautionary savings in the face of income risks when their marginal utility is convex (prudence). In this paper, we explore the effect of the timing of the resolution of income uncertainty on savings. An agent faces uncertainty about his income at date t+2. What is the effect of being informed that the uncertainty will be resolved at date t+1 on the consumption at date t? We show that the effect is positive, if and only if, marginal utility is convex (prudence), when either the risk free rate is equal to the rate of pure preference for the present, or when the utility function is HARA. The intuition is that an early resolution of uncertainty allows for time-diversifying the risk. It therefore plays a role similar to a reduction of the income risk, whose effect on savings is negative under prudence.