Verifying gross substitutability

Summary. In labor market models as well as in exchange economies with indivisible goods gross substitutability is used as a property to guarantee the existence of competitive equilibria. This paper develops an easy way to check gross substitutability for utility functions concerning a finite set of indivisible goods (or employees) and money. Concavity is one of the conditions that has to be satisfied. Only one other, but similar, type of relation must be checked to guarantee gross substitutability. Received: 21 August 2000; revised version: 28 November 2001     

Existence and Welfare Properties of Equilibrium in an Exchange Economy with Multiple Divisible and Indivisible Commodities and Linear Production Technologies

In this paper we consider a class of economies with a finite number of divisible commodities, linear production technologies, and indivisible goods and a finite number of agents. This class contains several well-known economies with indivisible goods and money as special cases. It is shown that if the utility functions are continuous on the divisible commodities and are weakly monotonic both on one of the divisible commodities and on all the indivisible commodities, if each agent initially owns a sufficient amount of one of the divisible commodities, and if a “no production without input”-like assumption on the production sector holds, then there exists a competitive equilibrium for any economy in this class. The usual convexity assumption is not needed here. Furthermore, by imposing strong monotonicity on one of the divisible commodities we show that any competitive equilibrium is in the core of the economy and therefore the first theorem of welfare also holds. We further obtain a second welfare theorem stating that under some conditions a Pareto efficient allocation can be sustained by a competitive equilibrium allocation for some well-chosen redistribution of the total initial endowments. Journal of Economic Literature Classification Numbers: D4, D46, D5, D51, D6, D61.     

Nonprofits and public good provision: A contest based on compromises

Two competing nonprofits with ideologically distinct missions compete for donor funding to provide an indivisible public good in a population with heterogeneous preferences. This paper examines the extent to which (average) public values are undermined and nonprofits’ ideology compromised in a contractual game in which the right to provide the public good is the outcome of competition between nonprofits. We also scrutinize the roles of (i) cooperative versus competitive contracting, (ii) multiple public goods, (iii) enforceability of actions and (iv) observability of nonprofit costs in determining the equilibrium terms of the contract. In each case, the intensity of the ideological divide between the donor and the nonprofits jointly impact the degree to which compromises are made in terms of both the public's and nonprofit's missions, and the ability on the part of the donor to reap double (cost-saving and strategic) financial gains.     

An expository note on individual risk without aggregate uncertainty

In economies where agents bear some risk, the analysis would often be facilitated by the assumption that the risks are i.i.d. and disappear in the aggregate. A formal appeal to the law of large numbers requires the consideration of a sequence of finite economies. In an economy with a continuum of agents there are countable families of sets for which it is impossible for a law of large numbers to be valid. With countably many agents and a finitely additive measure, independence is compatible with the law of large numbers.     

The exact law of large numbers via Fubini extension and characterization of insurable risks

A Fubini extension is formally introduced as a probability space that extends the usual product probability space and retains the Fubini property. Simple measure-theoretic methods are applied to this framework to obtain various versions of the exact law of large numbers and their converses for a continuum of random variables or stochastic processes. A model for a large economy with individual risks is developed; and insurable risks are characterized by essential pairwise independence. The usual continuum product based on the Kolmogorov construction together with the Lebesgue measure as well as the usual finitely additive measure-theoretic framework is shown further to be not suitable for modeling individual risks. Measurable processes with essentially pairwise independent random variables that have any given variety of distributions exist in a rich product probability space that can also be constructed by extending the usual continuum product.     

Indivisibilities,lotteries, and sunspot equilibria

Summary We analyze economies with indivisible commodities. There are two reasons for doing so. First, we extend and provide some new insights into sunspot equilibrium theory. Finite competitive economies with perfect markets and convex consumption sets do not allow sunspot equilibria; these same economies with nonconvex consumption sets do, and they have several properties that can never arise in convex environments. Second, we provide a reinterpretation of the employment lotteries used in contract theory and in macroeconomic models with indivisible labor. We show how socially optimal employment lotteries can be decentralized as competitive equilibria without lotteries once sunspots are introduced.We thank Kenneth Arrow, Aditya Goenka, Ed Green, Jeremy Greenwood, Walter Heller, Steve Matthews, Herve Moulin, Roger Meyerson, Jim Peck, Patrick Kehoe, Ramon Marimon, Ed Prescott, Richard Rogerson, Nancy Stokey and Raghu Sundaram for their comments. We also thank participants in seminars at Northwestern, Yale, USC, Cornell, Barcelona, Madrid, Santander, and the Canadian Economics Association annual meetings in Victoria. We are grateful to the National Science Foundation (through grants SES-8606944 and SES-8821225), the Center for Analytic Economics, the Thorne Fund, and the University of Pennsylvania Research Foundation for research support. The views expressed here are those of the authors, and not necessarily those of the Federal Reserve System or the Federal Reserve Bank of Minneapolis.     

Existence and optimality of oligopoly equilibria in linear exchange economies

Summary. We consider a linear exchange economy and its successive replicas. We study the notion of Cournot-Walras equilibrium in which the consumers use the quantities of commodities put on the market as strategic variables. We prove that, generically, if the number of replications is large enough but finite, the competitive behaviour is an oligopoly equilibrium. Then, under a mild condition, which may be interpreted in terms of market regulation and/or market activity, we show that any sequence of oligopoly equilibria of successive replica economies converges to the Walrasian outcome and furthermore that every oligopoly equilibrium of large, but finite, replica is Pareto optimal. Consequently, under the same assumptions on the fundamentals of the economy, one has an asymptotic result on the convergence of oligopoly equilibria to the Walras equilibrium together with a generic existence result for the Cournot-Walras. Received: June 20, 2002; revised version: November 20, 2002 RID="*" ID="*" Part of this paper was written while the second author was visiting the Universidad de Vigo. The support of the department of mathematics is gratefully acknowledged. Correspondence to: J.M. Bonnisseau     

Are incentives against economic justice?

We consider the problem of fairly allocating a social endowment of indivisible goods and money when the domain of admissible preferences contains, but is not restricted to, quasi-linear preferences. We analyze the manipulability of the Generalized Money Rawlsian Fair (GMRF) solutions. (i) We show that the Nash and strong Nash equilibrium correspondences of the “preference revelation game form” associated with each GMRF solution coincide with the no-envy solution (in equilibrium, efficiency is preserved according to agents' true preferences). (ii) A corollary is that the GMRF solutions “naturally implement” the no-envy solution in Nash and strong Nash equilibria.     

Liquid markets and competition

A market is liquid if no individual's actions have a big effect on the prices of goods traded in that market. Perfectly competitive markets are therefore perfectly liquid. It is well known that market liquidity can be achieved by increasing the number of traders so that individual trades are small compared to total trades. We show that even when there are only few traders, market liquidity can be achieved through large short sales in which net trades are small relative to gross trades. In particular, for a natural variant of the market game which permits unlimited short sales, we show that there is always a Nash equilibrium allocation arbitrarily close to a competitive equilibrium allocation. Of course, not all NE are near competitive. Only the large-short-sales NE are nearly liquid and hence close to CE.     

Gross substitutes and complements: A simple generalization

We extend the gross substitutes and complements framework (Sun and Yang, 2006). Competitive equilibrium with indivisible goods exists under significantly weaker, intuitive and interpretable conditions. A generalized dynamic double-track procedure (Sun and Yang, 2008, 2009) finds the competitive equilibrium outcome.     

Bargaining, coalitions and competition

We study a Gale-like matching model in a large exchange economy, in which trade takes place through non-cooperative bargaining in coalitions of finite size. Under essentially the same conditions of core equivalence, we show that the strategic equilibrium outcomes of our model coincide with the Walrasian allocations of the economy. Our method of proof makes use of the theory of the core. With respect to previous work, our positive implementation result applies to a substantially larger class of economies: the model relaxes differentiability and convexity of preferences, and also admits an arbitrary number of divisible and indivisible goods.     

On the set of Pareto efficient allocations in economies with public goods

Summary We examine the set of Pareto-efficient allocations in economies with public goods. We show that even if preferences are continuous and strongly monotonic, it need not coincide with the set of weakly efficient allocations. We then study topological properties of the Pareto set. We show that it is neither connected nor closed in allocation space. Furthermore, if the public goods are local, the image of the Pareto set in utility space need not be closed or connected. We provide two independent sufficient conditions for the closedness of the Pareto set. The results are directly applicable to private goods economies with joint production. Our results should be of interest for general equilibrium and mechanism design theory; where for example, the properties of the efficient set are important for proving the existence of an equilibrium and for the study of the properties of monotone-path social choice correspondences.We thank Hideo Konishi, Tomoichi Shinotsuka, Nicholas Yannelis and an anonymous referee for valuable comments on previous drafts of this paper.     

Uncertainty and insurance in strategic market games

Summary. For perfectly competitive economies under uncertainty, there is a well-known equivalence between a formulation with contingent goods and one with state-specific securities followed by spot markets for goods. In this paper, I examine whether this equivalence carries over to a particular form of imperfect competition. Specifically, I look at three Shapley-Shubik strategic market games: one with contingent commodities, one with Arrow securities traded under imperfect competition and one with Arrow securities traded under perfect competition. First I compare the feasibility constraints of these three games. Then I compare their equilibrium sets. As in Peck and Shell (1989), the only common equilibria between the first and the second game are those which involve no transfer of income across states. However, if the securities markets are competitive, then the set of equilibria of the contingent commodities game and the securities game coincide. Received: June 16, 1997; revised version: April 30, 1998     

SPECIALIZATION AND TRADE: A GAME THEORETICAL APPROACH

Abstract.  In this paper we consider the problem of specialization and trade for large economies with a continuum of ex ante identical individuals and with a finite number of goods. Different from the classical treatment, we adopt a game theoretical approach. Therefore in our models the prices of traded goods are endogenously formulated according to the bidding strategies of the producer-consumers. Furthermore, we assume that in the beginning individuals randomly choose their professions. As a result, with a short-run Nash equilibrium different types of professionals may have different utility levels; while through a dynamic process, a long-run Nash equilibrium with utility equalization is reached. Besides, we also attempt to provide a new algorithm for the computation of general equilibrium models in the Yang-Ng framework.     

Search cost,trading strategies and optimal market structure

The theory of Walrasian equilibrium yields a set of prices at which the aggregate competitive demand for each commodity equals its aggregate competitive supply. However, even at equilibrium prices the theory of competitive equilibrium does not explicitly offer explanation regarding the manner in which trades are actually executed. This paper considers a model where trade takes place in a decentralized fashion and examines in a dynamic game-theoretic framework, the role of social institution of money and markets in facilitating exchange. The steady state Nash equilibrium derived in the paper demonstrates how, depending on the level of transaction costs associated with a market setup (synonymously, trading posts to exchange possible pairs of goods) appropriate monetary trade emerges, which like a hub and spoke network (Starr and Stinchcombe, 1999) makes some markets non-functioning and in equilibrium only the markets having trade through the medium of exchange continue to exist. However, despite the obvious advantages of a market setup in reducing search costs, pure random search for a complementary trading partner (as considered by Ostroy and Starr, 1974; Kiyotaki and Wright , 1989; and others) prevails in many economies, especially, in many developing economies. This paper models this feature of developing economies by introducing differences in transaction costs across agents and shows why sustainable equilibria might exist exhibiting random search for certain commodities even in the presence of established markets.     

Endogenous, Efficient Long-Run Cyclical Unemployment, Endogenous Long-Run Growth, and Division of Labor

The paper uses a dynamic equilibrium model to explain the concurrence of economic growth and business cycles. Introducing durable goods into a model with ex-ante identical consumer–producers and economies of specialized learning-by-doing, the author shows that when job-shifting costs, economies of specialized learning-by-doing, and trading efficiency are sufficiently large, a dynamic equilibrium with business cycles and unemployment might be Pareto superior to noncyclical growth patterns. Long-run cyclical unemployment still exists when the credit market is imperfect. Extending the model into overlapping generations framework, the author shows that complete division of labor with business cycles would still occur in equilibrium.     

Indivisible labor implies chaos

Summary. We study a simple infinite horizon model with indivisible labor. We characterize the optimal plans under the assumptions that and that , where R is the gross interest rate and is the discount factor. We show that under those assumptions, optimal plans are almost always asymptotically nonperiodic unless initial wealth is excessively small or large. Received: May 25, 1999; revised version: June 24, 1999     

Growth in economies with non convexities: sunspots and lottery equilibria

Summary. We investigate the relation between lotteries and sunspot allocations in a dynamic economy where the utility functions are not concave. In an intertemporal competitive economy, the household consumption set is identified with the set of lotteries, while in the intertemporal sunspot economy it is the set of measurable allocations in the given probability space of sunspots. Sunspot intertemporal equilibria whenever they exist are efficient, independently of the sunspot space specification. If feasibility is, at each point in time, a restriction over the average value of the lotteries, competitive equilibrium prices are linear in basic commodities and intertemporal sunspot and competitive equilibria are equivalent. Two models have this feature: Large economies and economies with semi-linear technologies. We provide examples showing that in general, intertemporal competitive equilibrium prices are non-linear in basic commodities and, hence, intertemporal sunspot equilibria do not exist. The competitive static equilibrium allocations are stationary, intertemporal equilibrium allocations, but the static sunspot equilibria need not to be stationary, intertemporal sunspot equilibria. We construct examples of non-convex economies with indeterminate and Pareto ranked static sunspot equilibrium allocations associated to distinct specifications of the sunspot probability space.Received: 25 August 2003, Revised: 16 March 2004, JEL Classification Numbers: D84, D90.Correspondence to: Paolo SiconolfiWe thank Herakles Polemarchakis for helpful conversations on the topic. The research of Aldo Rustichini was supported by the NSF grant NSF/SES-0136556.     

Determinacy of competitive equilibria in economies with many commodities

This paper provides a framework for establishing the determinacy of equilibria in general equilibrium models with infinitely many commodities and a finite number of consumers and producers. This paper defines a notion of regular economy for such models and gives sufficient conditions on the excess savings equations characterizing equilibria under which regular economies have a finite number of equilibria, each of which is locally stable with respect to perturbations in exogenous parameters, and under which regular economies are generic. This paper also defines two notions of concavity, called uniform concavity and weighted uniform concavity, which generalize standard finite-dimensional notions of differential concavity to an infinite-dimensional setting by prohibiting goods from becoming perfect substitutes asymptotically. For the case of economies in which there are countably many commodities, such as discrete time models or markets with countably many assets, results in this paper show that equilibria are generically determinate as long as utility functions and production sets are uniformly concave or weighted uniformly concave. Received: November 7, 1996; revised version: March 13, 1998     

A DYNAMIC ENTRY AND PRICE GAME WITH CAPACITY INDIVISIBILITY

Strategic market interaction is here modelled as a two‐stage game in which potential entrants choose capacities and next active firms compete in prices. Due to capital indivisibility, the capacity choice is made from a finite grid and there are economies of scale. In the simplest version of the model with a single production technique, the equilibrium turns out to depend on the ratio between the level of total output at the long‐run competitive equilibrium and the firm's minimum efficient scale: if that ratio is sufficiently large (the market is sufficiently ‘large’), then the competitive price emerges at a subgame‐perfect equilibrium of the capacity and price game; if not, then the firms randomize in prices on the equilibrium path. The role of the market size for the competitive outcome is shown to be even more important if there are several available production techniques.