Existence of a competitive equilibrium when all goods are indivisible

This paper investigates an economy where all consumption goods are indivisible at the individual level, but perfectly divisible at the overall level of the economy. In order to facilitate trading of goods, we introduce a perfectly divisible parameter that does not enter into consumer preferences — fiat money. When consumption goods are indivisible, a Walras equilibrium does not necessarily exist. We introduce the rationing equilibrium concept and prove its existence. Unlike the standard Arrow–Debreu model, fiat money can always have a strictly positive price at the rationing equilibrium. In our set up, if the initial endowment of fiat money is dispersed, then a rationing equilibrium is a Walras equilibrium. This result implies the existence of a dividend equilibrium or a Walras equilibrium with slack.     

Competitive equilibria in economies with multiple indivisible and multiple divisible commodities

In this paper we consider a general equilibrium model with a finite number of divisible and a finite number of indivisible commodities. In models with indivisibilities it is typically assumed that there is only one divisible good, which serves as money. The presence of money in the model is used to transfer the value of certain amounts of indivisible goods. For such economies with only one divisible commodity Danilov et al. showed the existence of a general equilibrium if the individual demands and supplies belong to a same class of discrete convexity. For economies with multiple divisible goods and money van der Laan et al. proved existence of a general equilibrium if the divisible goods are produced out of money using a linear production technology and no other producers are present in the model.     

A strong incompatibility between efficiency and equity in non-convex economies

In allocation problems of perfectly divisible goods, we study the equity property of ‘no-domination', according to which no agent can receive strictly more of all goods than any other agent. We prove that no-domination is incompatible with Pareto efficiency, as soon as preferences are allowed to be non-convex.     

Existence and description of housing market equilibrium

A rental housing market with finite numbers of non-identical consumers and indivisible housing units, each composed of a vector of attributes, is studied. A partial equilibrium, open city analysis is presented in which all other commodities are perfectly divisible and elastically supplied on national markets. Sufficient conditions for the existence of an equilibrium are rigorously established using a fixed point argument that is based on a bidding arrangement between agents. It is then shown that, although the set of equilibria is not a singleton, all equilibria are similar enough to ensure a strong resemblance between open and closed cities.     

Erratum to “Do pure indivisibilities prevent core equivalence? Core equivalence theorem in an atomless economy with purely indivisible commodities only” [J. Math. Econ. 41 (4–5) (2005) 571–601]

We consider a pure exchange economy with a continuum of agents and finitely many indivisible commodities. Every commodity can be consumed only in integer amounts. Thus, agents’ preferences are locally satiated and no commodity bundle has necessarily local cheaper points. We introduce a core which is an intermediate concept between the strong core and the weak core. In our economy, this core is the most natural concept in the sense that it coincides with the set of all exactly feasible Walras allocations.     

Characterization of the Walrasian equilibria of the assignment model

We study the assignment model where a collection of indivisible goods are sold to a set of buyers who want to buy at most one good. We characterize the extreme and interior points of the set of Walrasian equilibrium price vectors for this model. Our characterizations are in terms of demand sets of buyers. Using these characterizations, we also give a unique characterization of the minimum and the maximum Walrasian equilibrium price vectors. Also, necessary and sufficient conditions are given under which the interior of the set of Walrasian equilibrium price vectors is non-empty. Several of the results are derived by interpreting Walrasian equilibrium price vectors as potential functions of an appropriate directed graph.     

Slack in incomplete markets with nominal assets: A symmetric proof

In this paper, we provide an equilibrium analysis in the framework of incomplete markets where some agents’ preferences are possibly satiated at some state of the nature. We will consider nominal assets with exogenously fixed asset prices. We extend the notion of equilibrium with slack – introduced by Drèze and Müller [Drèze, J., Müller, H., 1980. Optimality properties of rationing schemes. Journal of Economic Theory 23, 150–159] in a fixed price setting – to the GEI framework.     

Natural oligopolies with exogenous sunk costs: A non-Suttonian result

In a spatial competition model with exogenous fixed costs and divisible goods, we obtain non-Suttonian results. When the economy is infinitely replicated, the number of firms does go to infinity but, as consumers’ income goes to infinity, the equilibrium number of firms tends toward a finite value. This occurs because the global demand to each firm becomes in the limit infinitely sensitive to price differentials since they give then rise to infinitely large differences in purchase expenditure.     

Double implementation of the Lindahl equilibrium by a continuous mechanism

We consider a class of economies with public goods that have the following properties: (i) The preferences of the agents are convex, interior, and strictly increasing. (ii) The technology for production of public goods is a closed convex cone that satisfies free disposal and an additional mild assumption. No assumptions are made on continuity, completeness or transitivity of preferences. We provide a continuous and feasible mechanism that implements the Lindahl equilibrium by Nash equilibria, and has the following property: For every economy in our class every Nash equilibrium of the game induced by the mechanism is a strong Nash equilibrium.     

A family production overlapping generations economy

This paper provides a theoretical analysis of an overlapping generations economy in which production decisions and input–output allocations are all carried out at the family level. I consider a single class of output allocation schemes and various degrees of knowledge about the production technology. Under complete knowledge, I show that a family organizational structure in which everyone receives his marginal contribution to output, invests less in physical capital than under a perfectly competitive equilibrium environment. Under incomplete knowledge, I analyze and compare how beliefs about the input–output relationship affect the physical capital accumulation dynamics and the long-run standards of living.     

Voluntary contributions to multiple public goods in a production economy with widespread externalities

In a production economy, multiple public goods are produced by firms in competitive markets, and provided by the government together with contributions from consumers. There are widespread externalities: all consumers’ consumption and contributions and all firms’ production enter into utility functions. Public goods can be imperfect substitutes or complements, and they can be local public goods or club goods. Zero bounds that require consumers to make nonnegative contributions complicate the differentiable approach. Applying the transversality theorem for smooth economies in a regular parameterization, we obtain the existence of equilibrium in such an economy, and generically equilibria are regular and locally unique.     

Strategyproof exchange of indivisible goods

The exchange of heterogeneous indivisible goods is considered among agents with responsive preferences, who may be endowed with several goods. We introduce the segmented trading cycle rules which allow the agents to trade goods, one-for-one, subject to market segmentation. In particular, the market is divided into segments such that each agent in each segment may trade at most one good. The segmented trading cycle rules are characterized by four axioms: strategyproofness, nonbossiness, trade sovereignty, and strong individual rationality.     

Probabilistic assignments of identical indivisible objects and uniform probabilistic rules

We consider a probabilistic approach to the problem of assigning k indivisible identical objects to a set of agents with single-peaked preferences. Using the ordinal extension of preferences we characterize the class of uniform probabilistic rules by Pareto efficiency, strategy-proofness, and no-envy. We also show that in this characterization no-envy cannot be replaced by anonymity. When agents are strictly risk averse von Neumann-Morgenstern utility maximizer, then we reduce the problem of assigning k identical objects to a problem of allocating the amount k of an infinitely divisible commodity.     

Equilibrium in a mixed economy of arrow-debreu type

The functioning of the current NIS economies is severely affected by regulatory interferencies with supply and demand for many commodities. In particular, the presence of dual markets is characteristic for these economies. On the first market, typically the “low-price sector” of the economy, prices are fixed and the allocation of goods is determined by rationing schemes and governmental orders. On the second market, flexible prices resulting from the market mechanism coordinate demand and supply. It is allowed that any surplus of good purchased on the first market can be sold on the second market at the then prevailing market prices.     

Fair allocation in a general model with indivisible goods

In this paper we study the problem of fair allocation in economies with indivisible goods, dropping the usual restriction that one agent receives at most one indivisible good. We show that most of the results obtained in the literature do not hold when the aforementioned restriction is dropped. Received: 13 July 1995 / Accepted: 2 December 1997     

Indivisible commodities and an equivalence theorem on the strong core

We consider a pure exchange economy with finitely many indivisible commodities that are available only in integer quantities. We prove that in such an economy with a sufficiently large number of agents, but finitely many agents, the strong core coincides with the set of expenditure-minimizing Walrasian allocations. Because of the indivisibility, the preference maximization does not imply the expenditure minimization. An expenditure-minimizing Walrasian equilibrium is a state where, under some price vector, all agents satisfy both the preference maximization and the expenditure minimization.     

Arbitrage and equilibrium in unbounded exchange economies with satiation

In his seminal paper on arbitrage and competitive equilibrium in unbounded exchange economies, Werner (1987) proved the existence of a competitive equilibrium, under a price no-arbitrage condition, without assuming either local or global nonsatiation. Werner’s existence result contrasts sharply with classical existence results for bounded exchange economies which require, at minimum, global nonsatiation at rational allocations. Why do unbounded exchange economies admit existence without local or global nonsatiation? This question is the focus of our paper. First, we show that in unbounded exchange economies, even if some agents’ preferences are satiated, the absence of arbitrage is sufficient for the existence of competitive equilibria, as long as each agent who is satiated has a nonempty set of useful net trades– that is, as long as agents’ preferences satisfy weak nonsatiation. Second, we provide a new approach to proving existence in unbounded exchange economies. The key step in our new approach is to transform the original economy to an economy satisfying global nonsatiation such that all equilibria of the transformed economy are equilibria of the original economy. What our approach makes clear is that it is precisely the condition of weak nonsatiation – a condition considerably weaker than local or global nonsatiation – that makes possible this transformation.     

On second-price auctions and imperfect competition

Consider two sellers each of whom has one unit of an indivisible good and two buyers each of whom is interested in buying one unit. The sellers simultaneously set reserve prices and use second-price auctions as rationing device. An equilibrium in pure strategies where each seller has a regular customer is characterized. The result is applied in order to demonstrate that not allowing sellers to use second-price auctions may enhance total surplus.     

A graph theoretic approach to markets for indivisible goods

Many important markets, such as the labor market and the housing market, involve goods that are both indivisible and of budgetary significance. We introduce new graph theoretic objects ideally suited to analyzing such markets. We show that the minimum equilibrium price is characterized by a certain optimization problem on these graph theoretic objects.