Unequal inequalities. II

This paper analyzes properties of measures of inequality, applied to income inequalities but meaningful for practically any measure of dispersion in economics. We call n the number of persons, i the person's index, x_i person i's income, $\text{x}\text{= Σ(}\text{x}{}_{\mathrm{i}}\text{n}\text{)}$ the average income, x the vector of the x_i's or income distribution, I(x) a real-valued function of x which is the measure (or index) of inequality.Part I (Sects. I–V), which appeared in the last issue of this journal, analyzed several structures or properties, and specific forms, of I. We distinguished several I's: the measures of inequality per person (or “absolute”) I^a, per pound (or “relative”) $\text{I}{}^{\mathrm{r}}\text{=}\text{I}{}^{\mathrm{a}}\text{x}$, and total nI^a. We presented several possible properties of inequality measures, such as: I = 0 if all x_i's are equal (“zero at equality”), I > 0 otherwise (“positivity out of equality”), symmetry of I for x (“impartiality”), $\text{((}\text{?I}\text{?x}{}_{\mathrm{i}}\text{) ? (}\text{?I}\text{?x}{}_{\mathrm{i}}\text{))(x}{}_{\mathrm{i}}\text{? x}{}_{\mathrm{j}}\text{) > 0}$ for x_i ≠ x_j (“rectifiance” of the function I, or “transfers principle,” this being the strict form whereas the weak one is with sign ?), the fact that $\text{(}\text{?(}\text{x}\text{? I}{}^{\mathrm{a}}\text{)}\text{?}{}_{\mathrm{i}}\text{)}\text{(}\text{?(}\text{x}\text{? I}{}^{\mathrm{a}}\text{)}\text{?}{}_{\mathrm{j}}\text{)}$ does not depend upon x_k for k ≠ i,j (“welfare independence,” or, for short, “independence”). Rectifiance plus symmetry is Schur-convexity. Independence plus symmetry plus zero at equality implies that $\text{x}\text{≡}\text{x}\text{? I}{}^{\mathrm{a}}\text{= ?}{}^{\mathrm{?1}}\text{[(}\text{1}\text{n}\text{) Σ ?(x}{}_{\mathrm{i}}\text{)]}$ where $\text{x}$ is the “equal equivalent income”; and we will show that, these three properties being satisfied, the following ones are equivalent to each other: positivity out of equality, rectifiance, quasi-convexity, ?'s concavity.Part I largely focused on the study of six related specific measures of inequality, which in particular possess all the above properties: ?, α, and Ξ being positive parameters, they are $\text{I}{}_{\mathrm{c}}{}^{\mathrm{a}}\text{=}\text{x}\text{+ξ ? [(}\text{1}\text{n}\text{∑ (x}{}_{\mathrm{i}}\text{+ ξ)}{}^{\mathrm{1?epsi;}}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>1</mtext><mtext>??</mtext>}}$, $\text{I}{}_{\mathrm{c}}{}^{\mathrm{a}}\text{=}\text{x}\text{+ξ ? ∏ (x}{}_{\mathrm{i}}\text{+ ξ)}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}$, $\text{I}{}_{\mathrm{c}}{}^{\mathrm{r}}\text{=}\text{I}{}_{\mathrm{c}}{}^{\mathrm{a}}\text{x}$, I_r=I_c^r for ξ=O, $\text{I}{}_{\mathrm{r}}{}^{\mathrm{a}}\text{=}\text{x}\text{I}{}_{\mathrm{r}}\text{=I}{}_{\mathrm{c}}{}^{\mathrm{a}}$ for ξ=, $\text{I}{}_{\mathrm{l}}\text{=(}\text{1}\text{α}\text{)}\text{log}\text{[(}\text{1}\text{n}\text{) ∑ eα·(x?x}{}_{\mathrm{i}}\text{)]}$ and $\text{I}{}_{\mathrm{l}}{}^{\mathrm{r}}\text{=}\text{I}{}_{\mathrm{l}}\text{x}$. Lower indices c, r, l respectively stand for “centrist,” “rightist,” and “leftist” measures of inequality. I_r and I_l are invariant under respectively equiproportional variation in, or equal addition to, all incomes; measures which have the first of these two properties are said to be “intensive.”We now consider different and more general measures, and other properties. We first reconcile the last two properties by dropping the “indepencence” one (Section VI.). Then, we analyze another mildly equalitarian property, the “principle of diminishing transfers” (Section VII). Section VIII turns to the relations between inequality measures and Lorenz and concentration curves. We then consider the effect on inequality of additions of incomes, and we analyze the properties of “diminishing equality” (Section IX). The effect of unions of populations is the topic of Section X. Finally, the last section (XI) presents the more general relations between the various structural properties of inequality measures.¹     

Saddle points of Hamiltonian systems in convex Lagrange problems having a nonzero discount rate

Problems are studied in which an integral of the form ∫₀^+∞L(k(t),k(t))e^?ptdt is minimized over a class of arcs k: [0, +∞) → Rⁿ. It is assumed that L is a convex function on Rⁿ × Rⁿ and that the discount rate ? is positive. Optimality conditions are expressed in terms of a perturbed Hamiltonian differential system involving a Hamiltonian function H(k, q) which is concave in k and convex in q, but not necessarily differentiable. Conditions are given ensuring that, for ? sufficiently small, the system has a stationary point, in a neighborhood of which one has classical “saddle point” behavior. The optimal arcs of interest then correspond to the solutions of the system which tend to the stationary point as t → +∞. These results are motivated by questions in theoretical economics and extend previous work of the author for the case ? = 0. The case ? < 0 is also covered in part.     

Price characterization of stable allocations in exchange economies with externalities

A pure exchange economy where the consumers have utility functions U_i(v₁(x₁),…, v_m(x_m)) for i = 1,…, m and where x_j is the consumption of consumer j, is studied. U_i may be nonincreasing or nondecreasing in v_j for j ≠ i. i is said to be nonbenevolent or nonmalevolent towards j, accordingly.An allocation is stable if no coalition can redistribute what it receives in the allocation to get an allocation which is preferred, given the consumptions of the consumers in the complementary coalition. Results concerning the relation among the Paretooptimal, stable and equilibrium allocations (under different definitions of equilibrium) are given. In particular, it turns out that in case every consumer is non-benevolent towards every other consumer, the classical results, concerning the relation between Paretooptimal allocations and equilibrium allocations, can be generalized in a satisfactory way.     

Min, Max, and Sum

This paper provides characterization theorems for preferences that can be represented by U(x₁, …, x_n)=min{x_k}, U(x₁, …, x_n)=max{x_k}, U(x₁, …, x_n)=∑ u(x_k), or combinations of these functionals. The main assumption is partial separability, where changing a common component of two vectors does not reverse strict preferences, but may turn strict preferences into indifference. We discuss applications of our results to social choice. Journal of Economic Literature Classification Numbers: C0, D1, D6.     

Resource egalitarianism with a dash of efficiency

We study the problem of defining inequality-averse social orderings over allocations of commodities when individuals have different preferences. We formulate a notion of egalitarianism based on the axiom that any dominance between consumption bundles should be reduced. This Dominance Aversion requirement is compatible with Consensus, a version of the Pareto principle saying that an allocation y is better than x whenever everybody finds that everyone?s bundle at y is better than at x. We characterize a family of multidimensional leximin orderings satisfying Dominance Aversion and Consensus.     

Topological upper limits of mixed Nash equilibria

Our purpose in this article is to prove that given any integer n ≥ 2 and any non-empty compact Polish spaces S ₁, ..., S _n , if for any u ∈ C( S ₁ × ... × S _n , R) ⁿ , we denote by MNE(u) the set of mixed Nash equilibria of (S ₁, ..., S _n , u), then MNE(u) is a non-empty compact subset of P(S ₁) × ... × P(S _n ) and if u ^k → u in C(S ₁ × ... × S _n , R) ⁿ as k → ∞, then lim sup _{k → ∞} MNE (u ^k ) MNE(u). The author would like to thank the referee for offering critical comments on this paper.     

Choice and individual welfare

We propose an abstract method of systematically assigning a “rational” ranking to non-rationalizable choice data. Our main idea is that any method of ascribing welfare to an individual as a function of choice is subjective, and depends on the economist undertaking the analysis. We provide a simple example of the type of exercise we propose. Namely, we define an individual welfare functional as a mapping from stochastic choice functions into weak orders. A stochastic choice function (or choice distribution) gives the empirical frequency of choices for any possible opportunity set (framing factors may also be incorporated into the model). We require that for any two alternatives x and y, if our individual welfare functional recommends x over y given two distinct choice distributions, then it also recommends x over y for any mixture of the two choice distributions. Together with some mild technical requirements, such an individual welfare functional must weight every opportunity set and assign a utility to each alternative x which is the sum across all opportunity sets of the weighted probability of x being chosen from the set. It therefore requires us to have a “prior view” about how important or representative a choice of x at a given situation is.     

On choosing which game to play when ignorant of the rules

This paper suggests a theory of choice among strategic situations when the rules of play are not properly specified. We take the view that a “strategic situation” is adequately described by a TU game since it specifies what is feasible for each coalition but is silent on the procedures that are used to allocate the surplus. We model the choice problem facing a decision maker (DM) as having to choose from finitely many “actions”. The known “consequence” of the ith action is a coalition from game f _i over a fixed set of players \(N_i\cup\{d\}\) (where d stands for the DM). Axioms are imposed on her choice as the list of consequences (f ₁,..., f _m ) from the m actions varies. We characterize choice rules that are based on marginal contributions of the DM in general and on the Shapley Value in particular.     

Some results on “an income fluctuation problem”

A consumer at each period, given the income available, y, has to decide how much to consume and save. If he consumes c ? 0 units he gets u(c) units of satisfaction or utility, and if x = y ? c ? 0 is the amount saved then the available income in the next period is rx + ω^k, where ω^k is a random variable, and r is an interest factor that is assumed to be known with certainty. Infinite time horizon problems are considered, and it is shown that if 0 < δr < 1, where 0 < δ < 1 is a discount factor, then the limiting policy is optimal. Questions about the behavior of the stock level, such as boundness, are considered, and an example is given that shows that the stock level might converge almost surely to infinity. Finally an economic explanation is given.     

On the informational efficiency of the competitive resource allocation process

The purpose of this paper is to formalize the competitive process as a parametric process, and then prove the minimality of its message space among the message spaces for a broad class of parametric processes that includes the class of processes considered by Hurwicz (in “Studies in Resource Allocation Processes” (K. J. Arrow and L. Hurwicz, Eds.), pp. 413–423, Cambridge Univ. Press, Cambridge, 1977), Mount and Reiter (J. Econ. Theory6 (1974), 161–192), and Osana (J. Econ. Theory17 (1978), 66–78). The proof of this result turns crucially on the “asymmetry property” (which is stronger than the well-known “uniqueness property” of Hurwicz) and on an injectiveness lemma which is applicable to parametric processes.     

Overlapping expectations and Hart's conditions for equilibrium in a securities model

Hart (J. Econ. Theory9 (1974), 293–311) gave conditions for equilibrium to exist in a securities model where each agent undertakes asset transactions to maximize expected utility of wealth. These conditions rule out agents wanting to undertake unbounded balanced transactions to reach a Pareto superior allocation given their expectations. With mild extra assumptions to make agents unwilling to risk incurring unbounded losses on their portfolios, Hart's conditions become equivalent to an assumption of “overlapping expectations,” which is comparable to a much weaker form of Green's “common expectations” (Econometrica41 (1973), 1103–1124).     

Optimal control problems from second-order difference equations

We examine solutions x?l_∞(Rⁿ) of the equations $\text{F}{}_{\mathrm{t}}\text{(}\text{x}{}_{\mathrm{t?1}}\text{,}\text{x}{}_{\mathrm{t}}\text{,}\text{x}{}_{\mathrm{t+1}}\text{) = 0}$ and derive conditions for the existence of objective functions G_t so that $\text{x}$ solves max_x?l∞ΣG_t(x_t, x_t+1). We specialize the conditions to time autonomous equations and apply them to a competitive industry in temporary equilibrium. The objective function in the example is of the cost-benefit form: consumer surplus minus opportunity cost of labor minus industry-wide cost of investment.     

Temporal risk and the nature of induced preferences

It has been known since the work of H. Markowitz (“Portfolio Selection: Efficient Diversification of Investments,” Yale Univ. Press, 1959) and J. Mossin (Amer. Econ. Rev.59 (1969), 172–174) that even an individual whose underlying preferences satisfy the von Neumann-Morgenstern axioms will not choose over delayed (i.e., “temporal”) risky prospects in a manner which can be modelled as expected utility maximizing. Since most economically important instances of risk taking (insurance, real investment, agriculture, career training) involve delayed as opposed to immediately resolved risk, the standard use of expected utility theory to model such decisions must be questioned. In this paper the technique of “generalized expected utility analysis” (M. J. Machina, Econometrica50 (1982), 277–323) and the theory of support functions (R. T. Rockafellar, “Convex Analysis,” Princeton Univ. Press, 1970) are applied to exactly model and hence determine the nature of preferences over temporal risky prospects.     

The Phillips Curve and Optimal Policy in a Structural Signal Extraction Model

We study a monetary economy subject to “signal extraction” problems, and investigate within that framework the positive and normative aspects of monetary policy. As in Lucas (1972, Journal of Economic Theory,4, 103–124; 1973, American Economic Review, 63, 326–334), imperfect signal perception generates macroeconomic correlations similar to those found in the “Phillips curve” literature. Moving to normative aspects, we find that, when aggregate shocks are present, traditional nonactivist policies do not permit reaching the first best, and that an intelligent activist policy always leads to better outcomes. The specific characteristics and effectiveness of this optimal policy also depend crucially on the problem of signal extraction. Journal of Economic Literature Classification Number: E5.     

Market Entry Regulation and International Competition*

We analyze a non‐cooperative two‐country game where each government decides whether to allow free market entry of firms or to regulate market access. We show that a Pareto‐efficient allocation may result in equilibrium. In particular, if the cost difference between home and foreign production is “significant,” production will be located in the cost‐efficient country exclusively; and if this cost difference is even “substantial,” the induced allocation is also Pareto efficient. Only if the cost difference is “insignificant,” production may take place in both countries and the allocation is inefficient.     

A leximin characterization of strategy-proof and non-resolute social choice procedures

Summary. We characterize strategy-proof social choice procedures when choice sets need not be singletons. Sets are compared by leximin. For a strategy-proof rule g, there is a positive integer k such that either (i) the choice sets g(r) for all profiles r have the same cardinality k and there is an individual i such that g(r) is the set of alternatives that are the k highest ranking in i's preference ordering, or (ii) all sets of cardinality 1 to k are chosen and there is a coalition L of cardinality k such that g(r) is the union of the tops for the individuals in L. There do not exist any strategy-proof rules such that the choice sets are all of cardinality to k where . Received: November 8, 1999; revised version: September 18, 2001     

Some comparative-statics results for nonlinear input-output models of a multisectored economy,and related results for nonlinear price-demand relations

For a nonlinear version of the linear input-output model of Leontief, we consider the effect, on the vector x of total gross outputs, of changes in the components of the final-demand vector c. For example, it is shown that, for a certain important class of models, the “large-change elasticity” of any component of x, with respect to any component of c, is less than or equal to unity. Some similar results are proved concerning properties of nonlinear price-demand relations for the case in which all goods are weak gross substitutes. Unlike earlier work in this general area, we do not introduce hypotheses concerning the indecomposability or homogeneity of certain mappings. For that reason, the results are valid for the price-demand relation for a single firm, or for any collection of firms.     

Super-modularity: Applications to convex games and to the greedy algorithm for LP

L. S. Shapley (Internat. J. Game Theory1 (1971), 11–26) showed that if a game is convex, then all the marginal worth vectors are members of the core. J. Edmonds (in “Combinatorial Structures and Their Applications” (R. Guy et al., Eds.), pp. 69–87, Gordon & Breach, New York, 1970) considered a class of linear programming problems for which the greedy algorithm works. The present paper unifies these studies, and establishes the converse of each theorem.     

Null or nullifying players: The difference between the Shapley value and equal division solutions

A famous solution for cooperative transferable utility games is the Shapley value. Most axiomatic characterizations of this value use some axiom related to null players, i.e. players who contribute zero to any coalition. We show that replacing null players with nullifying players characterizes the equal division solution distributing the worth of the ‘grand coalition’ equally among all players. A player is nullifying if every coalition containing this player earns zero worth. Using invariance we provide similar characterizations of the equal surplus division solution assigning to every player its own worth, and distributing the remaining surplus equally among all players.     

When do market games have transferable utility?

A pure exchange economy generates a “market game” in which the allocations achievable by any coalition are determined by the initial endowments of its members. Subject to certain regularity conditions, it is shown that for a market game it is possible to find utility representations for each consumer so that the game can be treated as a game with transferable utility if and only if indirect utility of all consumers can be represented in the Gorman polar form. This is the class for which aggregate demand behaves as if it were the demand of a single consumer.