The role of diminishing marginal utility in the ordinal and cardinal utility theories

We illustrate the role of the law of diminishing marginal utility in the two main modern utility theories, the ordinal and cardinal utility theories, using a generalised total utility function. In short, the ordinal utility theory, in which utility is immeasurable, must abandon the law of diminishing marginal utility; the cardinal utility theory, although able to retain this law, suffers from keeping the unrealistic view of utility measurability, which Samuelson criticises as “infinitely improbable.” A new utility theory with the advantages of the two mentioned theories (i.e. the notions of both diminishing marginal utility and utility immeasurability) but without the disadvantages (i.e. the law of diminishing marginal utility is excluded and utility is measurable) therefore still seems to be a Holy Grail deserving search and development by economists.     

A family of ordinal solutions to bargaining problems with many players

A solution to bargaining problems is ordinal when it is covariant with respect to order-preserving transformations of utility. Shapley has constructed an ordinal, symmetric, efficient solution to three-player problems. Here, we extend Shapley's solution in two directions. First, we extend it to more than three players. Second, we show that this extension lends itself to the construction of a continuum of ordinal, symmetric, efficient solutions. The construction makes use of ordinal path-valued solutions that were suggested and studied by O'Neil et al. [Games Econ. Behav. 48 (2004) 139–153].     

Prioritarian Poverty Comparisons with Cardinal and Ordinal Attributes

The ethical view of prioritarianism holds that if an extra bundle of attributes is to be allocated to either of two individuals, then priority should be given to the worse off among the two. We consider multidimensional poverty comparisons with cardinal and ordinal attributes and propose three axioms that operationalize the prioritarian view. Each priority axiom, in combination with a handful of standard properties, characterizes a class of poverty measures.     

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.     

Allowing disagreement in evaluations of social welfare

There is no consensus on how to measure interpersonally comparable, cardinal utility. Despite of this, people repeatedly make welfare evaluations in their everyday lives. However, people do not always agree on such evaluations, and this is one important reason for political disagreements. Thus, to keep in control of the normative premises, decision makers may prefer information which can be used as input to an arbitrary social welfare function to information which is the output from a social welfare function specified by the analyst. In this paper we try to identify and simplify sufficient welfare indicators; information which enables decision makers to arrive at welfare evaluations of social states or projects, according to their own ethical beliefs. Our conclusion is that providing factual information about different population groups, their social state, size, and characteristics, may be better for this purpose than the more traditional approach of focusing on ordinal utility information.     

Additive utility

Summary We put the theory of cardinal and additive utilities on the same kind of simple foundation as the theory of ordinal utility. We give necessary and sufficient conditions for preferences to have continuous cardinal or additive utility functions, on connected topological spaces. Basing our proofs on fundamental algebraic theorems yields new techniques that allow us to give simple proofs of earlier results (cf. [4, 14] and to provide a basis for new results [10].The assistance of the National Science Foundation, Grant SES-8510620, is gratefully acknowledged.     

Uniqueness of stationary equilibria in bargaining one-dimensional policies under (super) majority rules

We consider negotiations selecting one-dimensional policies. Individuals have instantaneous preferences represented by continuous, concave and single-peaked utility functions, and they are impatient. Decisions arise from a bargaining game with random proposers and (super) majority approval, ranging from the simple majority up to unanimity. We provide sufficient conditions that guarantee the existence of a unique stationary subgame perfect equilibrium, and we provide its explicit characterization. The uniqueness of the equilibrium permits an analysis of the set of Pareto optimal voting rules. For symmetric distributions of peaks and uniform recognition probabilities unanimity is the unanimously preferred majority rule.     

Instrumental cardinal concerns for social status in two-sided matching with non-transferable utility

In this paper we apply the instrumental approach to social preferences in order to distinguish among various shapes of preferences for social status. In particular, we consider the shape of reduced preferences that emerge in the equilibrium of a two-sided matching model with non-transferable utility. Cole et al., 1992, Cole et al., 1995 show that, under full observability of potential mates׳ attributes, instrumental concerns for social status are ordinal, i.e., only one׳s own rank in the distribution of attributes matters. We show that when we depart from full observability, instrumental concerns for social status become cardinal, i.e., also other features of the distribution of attributes matter. We also show that the actual shape of cardinal concerns depends on how individuals can deal with the informational asymmetry, alternatively leading to upward concerns, i.e., making comparisons with higher rank people; downward concerns, i.e., making comparisons with lower rank people; or bidirectional concerns, i.e, being both upward and downward.     

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.     

贫困按要素分解:方法与例证

本文构建了两个贫困分解框架,一个用于分解贫困的水平,另一个用于分解贫困的变化,这两个方法的独特之处是建立了贫困与生产要素(即资源禀赋)总量及其分配之间的数量关系。具体地说,一个给定的贫困发生率可以被分解为要素或资源短缺的贡献,外加要素分配不均等的贡献。贫困发生率的变化则可分解为由要素的水平变化引起的,或由要素的不均等分配变化引起的。将这两种分解方法用于中国农村的数据,我们发现要素的不均等分配而非资源短缺是导致贫困和贫困变化的主要原因。     

What is a commodity? Two axiomatic answers

Summary. We identify conditions under which preferences over subsets of a consumption world can be reduced to preferences over bundles of "commodities". We distinguish ordinal bundles, whose coordinates are defined up to monotone transformations, from cardinal bundles, whose coordinates are defined up to positive linear transformations.Received: 27 March 2002, Revised: 17 March 2003, JEL Classification Numbers: D11.I am grateful to S. Barberá, C. Chambers, K. Nehring, and O. Sprumont for stimulating conversations and useful remarks. I also thank a referee for helpful comments, and the Social Sciences and Humanities Research Council of Canada for support.     

Value-Added Trade and Empirical Distributions of RCA Indices

Revealed Comparative Advantage (RCA) indices aid in identification of the sectors in which countries reveal comparative advantage or disadvantage. Apart from serving such a dichotomous measure, the RCA indices are frequently employed as cardinal or ordinal measures over time. Application of the indices for comparative analyses calls attention towards the distributions of RCA indices, which must reasonably be stable over time, sectors and countries. Stability of index distributions facilitates the usage of indices as cardinal or ordinal measures over time. The present paper therefore analyses the empirical distributions of RCA indices to determine their suitability. However, such an analysis would be incomplete if the implications for RCA indices due to growing significance of global supply chains are not recognized. Hence apart from analyzing the distributions of gross trade based RCA indices, the distributions of domestic value-added in export based indices are also examined, and the differences are noted. Similar extensive analyses on the distributions of RCA indices are lacking in the literature. In this sense, the present paper makes an important contribution to the existing literature on RCA indices.

     

Self‐Assessed Social Position and Poverty

In this article, we use microlevel data extracted from the 2006 Japanese General Social Surveys to analyze the relationships between self‐assessed social position and socioeconomic factors such as income and poverty. We provide the posterior results of the estimation of the Bayesian multivariate ordered probit model and propose an inequality measure for self‐assessed social position on the basis of the posterior results. We call the inequality measure “regret” and show that the distributions of regret differ for people above and below the poverty line.     

The Impact of Transfers and Taxes on Alternative Poverty Indexes

Changes in the headcount rate are the standard metric for gauging how public transfers and taxes affect US poverty. An alternative strategy, one theoretically more appealing and complete, is to rely on distribution-sensitive indexes (Sen 1976, 1981). How would policy's measured impacts change if such an approach were to be used? This study provides empirical evidence using three selected poverty indexes from the class developed by Foster et al . (1984). Pre- and post-policy values of each index are estimated for the total population and for twenty-three demographic sub-groups using data from March Current Population Surveys covering the period 1992 to 1998. The results indicate that the alternative indexes produce consistent ordinal rankings of policy's impact. (In contrast, the measured cardinal effects of policy differ substantially across indexes.) The empirical evidence has a clear implication for anti-poverty policy: government transfers and taxes are effective in lowering poverty headcount rates, in reducing the depth of poverty and in lessening the relative deprivation among the poor.     

A cardinal approach to straightforward probabilistic mechanisms

In every probabilistic mechanism, society selects an alternative, through a random device, out of a subset of indifferent alternatives. Consequently, in this context individuals face uncertainty and value the different lotteries on alternatives by their expected utility, so that they make use of a Von Neumann-Morgenstern cardinal utility function. Surprisingly, the social choice approach to probabilistic mechanisms assumes the use of ballots which preclude the complete expression of behaviour towards risk: individuals can only announce their ordinal preferences, or an approximation of their cardinal preferences, since in any case only a finite number of representations of preferences is available. This paper attempts to study voting systems in which individuals can express the cardinality of their preferences by assigning weights to the alternatives. It is shown that by voting with ballots which reflect weighting a new class of straightforward probabilistic mechanisms is defined, and that this class strictly contains the class of probabilistic straightforward mechanism designed by Gibbard.     

RANKING INCOME DISTRIBUTIONS WHEN NEEDS DIFFER

We derive criteria for ranking income distributions where households differ in equity-relevant non-income characteristics ('needs'). using methods which do not require cardinal specifications of equivalence scales. We consider comparisons for situations where the distributions of needs differ (eg. Cross nationally or intertemporally), building on the results of Atkinson and Bourguignon (1987) and Atkinson (1992). The modifications required when the individual rather than the household is the income-receiving unit are also discussed. We illustrate the methods with an analysis of changes in social welfare and poverty in the U.K. between 1981 and 1986.     

REDUCING INEQUALITIES AMONG UNEQUALS

This article establishes an equivalence between four incomplete rankings of distributions of income among agents who are vertically differentiated with respect to some nonincome characteristic (health, household size, etc.). The first ranking is the possibility of going from one distribution to the other by a finite sequence of income transfers from richer and more highly ranked agents to poorer and less highly ranked ones. The second ranking is the unanimity among utilitarian planners who assume that agents' marginal utility of income is decreasing with respect to both income and the source of vertical differentiation. The third ranking is the Bourguignon (Journal of Econometrics, 42 (1989), 67–80) Ordered Poverty Gap dominance criterion. The fourth ranking is a new dominance criterion based on cumulative lowest incomes.     

Ordinal vs cardinal status: Two examples

We demonstrate that in models where agents have concerns for status the model predictions can drastically change depending on whether status is modelled as an ordinal or cardinal magnitude. As a proof, we show that two well known theoretical findings are not robust to the substitution of ordinal status with cardinal status [Frank, R.H., The Demand for Unobservable and Other Positional Goods. American Economic Review, (75):101–116, 1985.] and viceversa [Clark, A. and Oswald, R.J., Comparison-Concave Utility and Following Behavior in Social and Economic Settings. Journal of Public Economics, (70):133–155, 1998.].     

A note on small income effects

This note provides an alternative sufficient condition for the small income effect result that is first shown by Vives [Small income effects: a Marshallian theory of consumer surplus and downward sloping demand, Rev. Econ. Stud. 54(1) (1987) 87-103]. The condition is stated by ordinal terms only, whereas Vives assumes cardinal properties of utility representation. Second, as its application, we provide a sufficient condition for the preference being asymptotically quasi-linear, in a two good economy where the second good is a composition of a large number of goods.     

Aggregation of semiorders: intransitive indifference makes a difference

Summary A semiorder can be thought of as a binary relationP for which there is a utilityu representing it in the following sense:xPy iffu(x) –u(y) > 1. We argue that weak orders (for which indifference is transitive) can not be considered a successful approximation of semiorders; for instance, a utility function representing a semiorder in the manner mentioned above is almost unique, i.e. cardinal and not only ordinal. In this paper we deal with semiorders on a product space and their relation to given semiorders on the original spaces. Following the intuition of Rubinstein we find surprising results: with the appropriate framework, it turns out that a Savage-type expected utility requires significantly weaker axioms than it does in the context of weak orders.We wish to thank Tatsuro Ichiishi, Jorge Nieto, Ariel Rubinstein, Efraim Sadka and especially David Schmeidler and anonymous referees for stimulating discussions and comments. I. Gilboa received partial financial support from NSF grants nos. IRI-8814672 and SES-9113108, as well as from the Alfred P. Sloan Foundation.