Trade-off between equity and efficiency in a general economy with indivisible goods

In most of the recent literature on fair allocation in economies with indivisible goods and a single infinitely divisible good, it is assumed that each agent can consume at most one indivisible good. In this paper, we show that if this assumption is dropped, there do not necessarily exist envy-free and Pareto efficient allocations. However, envy-free allocations still exist and so do Pareto efficient allocations. Hence, a trade-off between equity and efficiency arises.     

The equity-efficiency trade-off reconsidered

Envy-free and Pareto optimal allocations never exist in large economies with production, if there is enough variation with respect to preferences and innate abilities of the agents. If distributional equity is given priority only second best envy-free allocations may be considered. Those allocations are characterised in finite and continuum economies. In continuum economies envy-free allocations are type-egalitarian, i.e. agents with the same preferences but different abilities obtain the same consumption-labour bundle. Regarding implementation envy-free allocations can only be implemented by a tax depending on labour and income.I would like to thank two anonymous referees and the editor in charge of this paper for very helpful comments.     

An algorithm for envy-free allocations in an economy with indivisible objects and money

This paper studies envy-free allocations for economies with indivisible objects, quasi-linear utility functions, and an amount of money. We give a polynomially bounded algorithm for finding envy-free allocations. Connectedness of envy-graphs, which are used in the algorithm, characterizes the extreme points of the polytopes of sidepayments corresponding with envy-free allocations. Received: 22 October 1997/Accepted: 19 January 1999     

Welfare-domination under preference-replacement: A survey and open questions

The objective of this paper is to describe various applications of a requirement of solidarity pertaining to situations in which the preferences of some of the agents may change. It says that the welfares of all agents whose preferences are fixed should be affected in the same direction: they should all weakly gain, or they should all weakly lose. We show how this condition, which we name “welfare-domination under preference-replacement”, can help in evaluating allocation rules. We discuss it in several contexts: private good allocation in classical economies, public good decision, binary choice with quasi-linear preferences, economies with indivisible goods, economies with single-peaked preferences, both in the private good case and in the public good case, and economies with time. For some of these models the implications of the property are well understood. For others, we state a number of open problems. Received: 2 January 1997/Accepted: 26 February 1998     

Natural implementation in public goods economies

In this paper, we identify necessary and sufficient conditions for social choice correspondences to be Nash implemented by “natural” mechanisms in economies with arbitrary numbers of private and public goods. We find that when there exist only one public good and an arbitrary number of private goods, the Lindahl correspondence is implementable by a natural quantity mechanism in which each agent announces his own consumption bundle of private goods and input vectors for public goods. However, regardless of the numbers of private and public goods, the Pareto correspondence is not implementable even by any natural price ⁿ -quantity mechanism in which each agent reports a price vector of private goods and all agents’ personalized price vectors of public goods, in addition to his own consumption bundle of private goods and input vectors for public goods.     

A simple procedure for finding equitable allocations of indivisible goods

The paper investigates how far a particular procedure, called the “descending demand procedure,” can take us in finding equitable allocations of indivisible goods. Both interpersonal and intrapersonal criteria of equitability are considered. It is shown that the procedure generally fares well on an interpersonal criterion of “balancedness”; specifically, the resulting allocations are Pareto-optimal and maximize the well-being of the worst-off individual. As a criterion of intrapersonal equitability, the property of envy-freeness is considered. To accommodate envy-freeness, a modification of the basic procedure is suggested. With two individuals, the modified procedure is shown to select the envy-free allocations that are balanced, i.e. the allocations that maximize the well-being of the worse-off individual among all envy-free allocations. Received: 3 March 2000/Accepted: 27 November 2000     

Double implementation of Lindahl allocations by a pure mechanism

This paper considers the issue of designing mechanisms whose Nash allocations and strong Nash allocations coincide with Lindahl allocations for public goods economies when coalition patterns, preferences, and endowments are unknown to the designer. It will be noted that the mechanism presented here is feasible and continuous, and the implementation result is obtained without defining an artificial preference profile on prices announced by individuals. In addition, unlike most existing Nash-implementing mechanisms which need to distinguish the case of two agents from that of three or more agents, this paper provides a unified mechanism which is irrespective of the number of agents. Received: 19 August 1997/Accepted: 16 November 1998     

Linear cost share equilibria and the veto power of the grand coalition

We consider pure exchange economies with finitely many private goods involving the choice of a public project. We discuss core-equivalence results in the general framework of non-Euclidean representation of the collective goods. We define a contribution scheme to capture the fraction of the total cost of providing the project that each blocking coalition is expected to cover. We show that for each given contribution scheme defined over the wider class of Aubin coalitions, the resulting core is equivalent to the corresponding linear cost share equilibria. We also characterize linear cost share equilibria in terms of the veto power of the grand coalition. It turns out that linear cost share equilibria are exactly those allocations that cannot be blocked by the grand coalition with reference to auxiliary economies with the same space of agents and modified initial endowments and cost functions. Unlike the Aubin-type equivalence and results presented in Diamantaras and Gilles (Soc Choice Welf 15:121–139, 1998), this characterization does not depend on a particular contribution scheme.     

A derivation of the money Rawlsian solution

We study the set of envy-free allocations for economies with indivisible objects and quasi-linear utility functions. We characterize the minimal amount of money necessary for its nonemptiness when negative distributions of money are not allowed. We also find that, when this is precisely the available amount of money, there is a unique way to combine objects and money such that these bundles may form an envy-free allocation. Based on this property, we describe a solution to the envy-free selection problem following a pseudo-egalitarian criterion. This solution coincides with the Money Rawlsian Solution proposed by Alkan et al. (1991).I am indebted to I. Gilboa for his valuable suggestions and his patience during the elaboration of the final version. I also wish to thank S. Barbera for his guidance in an earlier version and M. Boldrin, H. Moulin, Z. Neeman, W. Thomson and the referees for their comments. Financial support from FPU-MEC (Spain) is gratefully acknowledged.Northwestern University.     

Linear cost sharing in economies with non-Samuelsonian public goods: Core equivalence

 We consider an economy with non-Samuelsonian public goods and we focus on linear cost sharing. In a linear cost sharing equilibrium all agents in the economy optimize given a certain fixed cost share to be contributed towards the provision of public goods in the economy. Hence, each agent pays a certain fraction of the total establishment costs of public goods and these cost shares are common knowledge. We show that for a certain fixed contribution scheme the resulting linear cost share equilibria are equivalent to corresponding core allocations, in which the core is based on the integral of the individual cost shares. We also show that there is no equivalence of the Foley core with cost share equilibria, even in well-behaved large economies. Received: 16 August 1995/Accepted: 29 July 1996     

On fairness and welfare analysis under uncertainty

This paper investigates the role of fairness, uncertainty, and a “veil of ignorance” in efficient resource allocation. It focuses on the choice of private and public goods, the method of financing, as well as the choice of information available for public decision-making. A fair-equivalent and Pareto efficient allocation is presented using a maximin criterion defined in terms of individual willingness-to-pay. The paper investigates the role of information in public decision making in terms of its implications for both efficiency and fairness. While better information typically generates improved efficiency, it can also contribute to unfair allocations. The effects of asymmetric information are discussed. Received: 4 May 1998/Accepted: 28 February 2002 We would like to thank two anonymous referees for very useful comments on an earlier draft of the paper. The research was supported in part by a Hatch grant from the University of Wisconsin, Madison.     

Existence of efficient envy-free allocations of a heterogeneous divisible commodity with nonadditive utilities

This paper studies the existence of Pareto optimal, envy-free allocations of a heterogeneous, divisible commodity for a finite number of individuals. We model the commodity as a measurable space and make no convexity assumptions on the preferences of individuals. We show that if the utility function of each individual is uniformly continuous and strictly monotonic with respect to set inclusion, and if the partition matrix range of the utility functions is closed, a Pareto optimal envy-free partition exists. This result follows from the existence of Pareto optimal envy-free allocations in an extended version of the original allocation problem.     

The relation between monotonicity and strategy-proofness

The Muller–Satterthwaite Theorem (J Econ Theory 14:412–418, 1977) establishes the equivalence between Maskin monotonicity and strategy-proofness, two cornerstone conditions for the decentralization of social choice rules. We consider a general model that covers public goods economies as in Muller–Satterthwaite (J Econ Theory 14:412–418, 1977) as well as private goods economies. For private goods economies, we use a weaker condition than Maskin monotonicity that we call unilateral monotonicity. We introduce two easy-to-check preference domain conditions which separately guarantee that (i) unilateral/Maskin monotonicity implies strategy-proofness (Theorem 1) and (ii) strategy-proofness implies unilateral/Maskin monotonicity (Theorem 2). We introduce and discuss various classical single-peaked preference domains and show which of the domain conditions they satisfy (see Propositions 1 and 2 and an overview in Table 1). As a by-product of our analysis, we obtain some extensions of the Muller–Satterthwaite Theorem as summarized in Theorem 3. We also discuss some new “Muller–Satterthwaite preference domains” (e.g., Proposition 3).     

On the behavioural relevance of optional and mandatory impure public goods

Impure public goods combine a private good with a public good. Often, impure public goods have a charitable or ethical dimension, giving ethically motivated consumers a convenient option to contribute to public goods through the marketplace (in addition to direct donations). Impure public goods could potentially promote ethical giving or alternatively hinder charitable behaviour. We implement an economics experiment with a between-subject design to test the behavioural relevance of impure public goods with only a token (i.e. small) contribution to a public good. Contributions to the public good are negatively affected by the presence of impure public goods with token contributions. We explore one mechanism to offset this negative impact by making the token impure public good mandatory. We observe higher average contributions and several positive impacts on charitable behaviour, which supports the claim that this mechanism can potentially offset the negative impact of impure public goods.     

On increased risk and the voluntary provision of public goods

This paper examines the impacts of increased risk on the voluntary provision of public goods. I strengthen the view that uncertainty ameliorates free riding even given negative non-Nash conjectures. This result holds since the change in the perceived free riding (i.e., the magnitude of the conjecture) drives voluntary contribution to public good supply. The result has two main implications. First, the traditional model of the rational self-interested individual can explain voluntary contributions. Second, a public policy to encourage volunteerism would maintain private information of private contributions to the supply of public goods. However, risk as a public policy instrument is limited by ethical concerns and behavioral responses to risk.U.S. Environmental Protection Agency provided support for this research. Garey Durden, Todd Sandler, Fred Sterbenz, Robert Sugden, and Joe Kerkvliet provided useful comments. I owe tanks to Peter Bohm, Hans Wijkander, and Charles Mason for encouraging my work in this area. Two referees provided detailed suggestions that greatly improved the paper.     

Identical preferences lower bound solution and consistency in economies with indivisible goods

We consider the problem of allocating a finite set of indivisible goods and a single infinitely divisible good among a group of agents, and we study a solution, called the Identical Preferences Lower Bound solution, in the presence of consistency properties. This solution is not consistent. We prove that its maximal consistent subsolution is the No-envy solution. Our main result is that the minimal consistent extension of the intersection of the Identical Preferences Lower Bound solution with the Pareto solution is the Pareto solution. This result remains true in the restricted domain when all the indivisible goods are identical, but not when there is a unique indivisible good.This paper was developed during my stay at Rochester University in the summer of 1992. I would like to express my special thanks to Professor William Thomson for all his help and advice. Iam also grateful to my supervisor Luis Corchón, to Koichi Tadenuma and to the anonymous referees for their helpful comments. The remaining errors are my exclusive responsibility. Financial support from the DGCYT under project PB 91-0756 and the Instituto Valenciano de Investigaciones Económicas are gratefully acknowledged.     

Equity and efficiency in non-convex economies

The objective of this paper is to consider the following question. Does the presence of increasing returns introduce a fundamental trade-off between equity and efficiency objectives? We show that if the no-envy notion of Foley (1967) is taken as the equity criterion and Pareto optimality as the efficiency criterion, then the answer is yes; there exist economies with increasing returns and well-behaved preferences (and no agent-specific inputs) in which there do not exist any envy-free and Pareto optimal allocations. We also propose a weakening of the no-envy criterion and prove that this weaker equity notion is compatible with Pareto optimality in general non-convex economies.     

Nash implementation of Lindahl allocations

We define a class of Nash based mechanisms that implement Lindahl allocations in standard public good economies. As those that first proved Nash implementation of Lindahl allocations, defined by Hurwicz (1979) and Walker (1981), the mechanisms provided here are continuous, feasible, and independent of agents' characteristics. In economies with two agents (e.g., bilateral monopolies, duopolies, principal-agent models), our mechanisms are also applicable, in contrast to the well-behaved mechanisms in the literature, and they satisfy favourable stability properties.     

Money-egalitarian-equivalent and gain-maximin allocations of indivisible items with monetary compensation

Suppose that a certain quantity M of money and a finite number of indivisible items are to be distributed among n people, all of whom have equal claims on the whole. Different allocations are presented using various criteria of fairness in the special case where each player's utility function is additively separable. An allocation is “money-egalitarian-equivalent” (MEE) if each player's monetary valuation of his or her bundle is a fixed constant. We show that there is an essentially unique allocation that is MEE and Pareto-optimal; it is also envy-free. Alternatively, the “gain” of a player may be defined as the difference between how the player evaluates his bundle and an exact nth part of the whole according to his numerical evaluation of the whole. A “gain-maximin” criterion would maximize the minimum gain obtained by any player. We show that Knaster's procedure finds an allocation which is optimal under the gain-maximin criterion. That allocation is not necessarily envy-free, so we also find the envy-free allocation that is optimal under the gain-maximin criterion among all envy-free allocations. It turns out that, even though there exist allocations that are simultaneously envy-free and Pareto-optimal, this optimal allocation may fail to be Pareto-optimal, and it may also violate monotonicity criteria. Received: 30 September 1996/Accepted: 6 March 2002 The author would like to thank Professor William Thomson for a discussion on this subject; and he would like to thank the anonymous referees, who made many substantive suggestions for improving this paper – shortening it, streamlining the arguments, improving the terminology, making further ties with the literature, and improving the exposition.     

The possibility of efficient provision of a public good in voluntary participation games

In this study, we provide the conditions for efficient provision of a public good in a participation game in which a non-negative integer number of units of the public good can be provided. In the case in which at most one unit of the public good can be provided, we provide refinements of Nash equilibria at which agents choose only a Nash equilibrium with an efficient allocation and provide sufficient conditions for cost-sharing rules that guarantee the existence of a Nash equilibrium with an efficient allocation. In the case of a multi-unit public good, we provide a necessary and sufficient condition for the existence of a Nash equilibrium with an efficient allocation and prove that Nash equilibria are less likely to support efficient allocations if the participation of many agents is needed for efficient provision of the public good in the case of identical agents. I would like to thank Koichi Tadenuma, Yukihiro Nishimura, Toshiyuki Fujita, John Weymark, Tatsuyoshi Saijo, Takehiko Yamato, Toshiji Miyakawa, Motohiro Sato, Takashi Shimizu, Nobue Suzuki, and Dirk T. G. Rübbelke for helpful comments and suggestions. I also thank the anonymous referees for detailed comments and suggestions. This research was supported by the Japanese Economic Research Foundation and the Grand-in-Aid for Young Scientists (Start-up) from the Japan Society for the Promotion of Science. Any remaining errors are my own.