Arbitrage and equilibrium in economies with infinitely many commodities

An ‘arbitrage opportunity’ for a class of agents is a commodity bundle that will increase the utility of any of the agents and that has non-positive price. The non-existence of ‘arbitrage opportunities’ is necessary and sufficient for the existence of an economic equilibrium. A bundle is ‘priced by arbitrage’ if there is a unique price that it can command without causing an ‘arbitrage opportunity’ to exist. For economies that have infinitely many commodities, appropriate notions of ‘arbitrage opportunities’ and ‘bundles priced by arbitrage’ depend on the continuity of agents’ preferences. This paper develops these notions, thereby providing a foundation for recent work in financial theory concerning arbitrage in continuous-time models of securities markets.     

A note on the equilibrium theory of economies with asymmetric information

In this note, we give an equilibrium existence theorem for exchange economies with asymmetric information and with an infinite dimensional commodity space. In our model, we assume that preferences are represented by well behaved utility functions, the positive cone has a non empty interior and the individual rational utility set is compact. Our result complements the corresponding one in Podczeck and Yannelis (2008), in the sense that is applicable to commodity spaces in which the order intervals are (possibly) not compact with respect to any Hausdorff linear topology.     

Consumer surplus analysis under uncertainty: A general equilibrium perspective

This paper derives an exact form of partial equilibrium efficiency measure under uncertainty which is consistent with expected utility maximization in a general equilibrium situation with ex-post spot markets for many goods and asset markets which are in general incomplete.We consider that the good under consideration tends to be negligibly small compared to the entire set of commodity characteristics which is assumed to be a continuum, and look into the limit property of preferences over state-contingent consumption of the good and state-contingent income transfer associated to it. We show that the limit preference exhibits risk neutrality, not only that it exhibits no income effect, meaning that the two conditions are tied together. We also show that the marginal rate of substitution between extra income transfers at different states of the world converges to the ratio between the Lagrange multipliers associated to those states. When the asset markets are complete such ratios are equalized between consumers, but it is not the case in general when the asset markets are incomplete. This means that using the aggregate expected consumer surplus as the welfare measure will be in general inconsistent with individuals’ expected utility maximization in the general equilibrium environment or with ex-ante Pareto efficiency.     

Perturbed utility and general equilibrium analysis

We study general equilibrium theory of complete markets in an otherwise standard economy with each household having an additive perturbed utility function. Since this function represents a type of stochastic choice theory, the equilibrium of the corresponding economy is defined to be a price vector that makes its mean expected demand equal its mean endowment. We begin with a study of the economic meaning of this notion, by showing that at any given price vector, there always exists an economy with deterministic utilities whose mean demand is just the mean expected demand of our economy with additive perturbed utilities. We then show the existence of equilibrium, its Pareto inefficiency, and the upper hemi-continuity of the equilibrium set correspondence. Specializing to the case of regular economies, we finally demonstrate that almost every economy is regular and the equilibrium set correspondence in this regular case is continuous and locally constant.     

Uniqueness of competitive equilibrium with solvency constraints under gross-substitution

Under a gross substitution assumption, we prove existence and uniqueness of competitive equilibrium for an infinite-horizon exchange economy with limited commitment and complete financial markets. Risk-sharing is limited as only a part of the private endowment can be used as collateral to secure debt. The unique equilibrium is Markovian with respect to a minimal state space consisting of exogenous shocks and Negishi’s welfare weights. We represent equilibrium dynamics via a monotone operator acting on entire wealth distribution functions. We construct a fixed point of this operator generating a lower and an upper orbit and proving coincidence of accumulation points.     

Stationary Markovian equilibrium in altruistic stochastic OLG models with limited commitment

We introduce a new class of infinite horizon altruistic stochastic OLG models with capital and labor, but without commitment between the generations. Under mild regularity conditions, for economies with either bounded or unbounded state spaces, continuous monotone Markov perfect Nash equilibrium (henceforth MPNE) are shown to exist, and form an antichain. Further, for each such MPNE, we can also construct a corresponding stationary Markovian equilibrium invariant distribution. We then show for many versions of our economies found in applied work in macroeconomics, unique MPNE exist relative to the space of bounded measurable functions. We also relate all of our results to those obtained by promised utility/continuation methods based upon the work of Abreu et al. (1990). As our results are constructive, we can provide characterizations of numerical methods for approximating MPNE, and we construct error bounds. Finally, we provide a series of examples to show the potential applications and limitations of our results.     

General competitive equilibrium of the spatial economy : Two Teasers

The local indivisibility of space complicates general equilibrium analysis. Mazzoleni and Montesano (1984, p. 286), suggest that the usual reasoning of the non-spatial analyses of competitive equilibrium can be performed in the context of a commodity space that contains subsets of land. This space, however, lacks a linear structure, so it resists application of the usual reasoning. Two examples will show that Mazzoleni and Montesano's existence results are flawed.     

Piecewise linear exchange equilibrium

It is known that if a pure exchange model in which all consumers have linear utility functions has an equilibrium it has one which is rational in the initial data. Examples show on the other hand that this is no longer true in general if utility functions are merely piecewise linear. In this note we observe that for the case of two traders the rationality result carries over and we sketch a finite procedure for calculating the equilibrium.     

On the convergence to the Cournot equilibrium in a productive asset oligopoly

We derive a feedback equilibrium of a dynamic Cournot game where production requires exploitation of a renewable asset. As in the classical Cournot model, quantity-setting firms compete in the same market for a given homogeneous good. We show that, when the asset stock grows sufficiently fast, the unique globally asymptotically stable steady state of the dynamic Cournot game corresponds to the static Cournot solution. Initial differences between firms’ production rates due to asymmetric allocations of asset stocks tend to disappear over time. When instead the asset stock grows slowly, the system does not converge to any stationary point. We also show that, within the class of linear feedback equilibrium strategies, besides the couple of strategies that stabilizes the states for every possible initial conditions, there exists another couple which is more efficient, in that it leads to higher stationary equilibrium profits for both firms, closer to the collusive outcome. Finally, we show that, as the discount rate approaches zero, there exist multiple linear feedback equilibrium strategies that induce a price trajectory that converges asymptotically to a price which is above the static Cournot equilibrium price.     

Nash equilibria in a model of multiproduct price competition: an assignment problem

We study the market interaction of a finite number of single-product firms and a representative buyer, where the buyer consumes bundles of these goods. The buyers’ value function determines their willingness to pay for subsets of goods. We show that Nash-equilibrium outcomes are solutions of the linear relaxation of an integer programming assignment problem and that they always exist. The Nash-equilibrium price set is characterized by the Pareto frontier of the associated dual problem’s projection on the firms’ price vectors. We identify the Nash-equilibrium prices for monotonic buyers’ value functions and, more importantly, we show that some central solution concepts in cooperative game theory are (subgame perfect) equilibrium prices of our strategic game.     

Increasing the price variation in a repeated cross section

Insufficient price variation seriously hampers many applications of consumer demand models. This paper examines the empirical performance of a potential remedy for this problem that was suggested by [Lewbel, A., 1989. Identification and estimation of equivalence scales under weak separability. Review of Economic Studies 56, 311–316], the construction of individual specific price indices for bundles of goods. These individual specific price indices allow for a population with heterogeneity in preferences for goods within a given bundle of goods. We confine ourselves to heterogeneous Cobb Douglas within bundle preferences, while between bundles, we allow for several parametric and even general nonparametric specifications. In a variety of settings, we show that such prices produce superior empirical results than the ones obtained through the traditional practice of using aggregate price indices. Our empirical analysis is based on the British Family Expenditure Survey data, and uses several categories of food. Both in parametric as well as nonparametric models, we obtain higher precision of estimates for parameters or functions, as well as economically more plausible results.     

Stackelberg equilibrium with multiple firms and setup costs

I provide conditions that guarantee that a Stackelberg game with a setup cost and an integer number of identical leaders and followers has an equilibrium in pure strategies. The main feature of the game is that when the marginal follower leaves the market the price jumps up, so that a leader’s payoff is neither continuous nor quasiconcave. To show existence I check that a leader’s value function satisfies the following single crossing condition: When the other leaders produce more the leader never accommodates entry of more followers. If demand is strictly logconcave, and if marginal costs are both non decreasing and not flatter than average costs, then a Stackelberg equilibrium exists. Besides showing existence I characterize the equilibrium set and provide a number of results that contribute to the applied literature. As the number of leaders increases, leaders produce more and eventually they deter entry. Leaders produce more than the Cournot best reply, but they may underinvest in entry deterrence. As the number of followers increases, leaders become more aggressive. When this number is large, if leaders can produce the limit quantity and at the same time have market power, then they deter entry.     

A new look at static spatial price equilibrium models

In this paper we propose a new view and more general formulations of static single commodity spatial equilibrium models, that lead to simpler and more efficient algorithms than those previously employed for solving models of this type. The proposed formulations incorporate general transportation networks and may be extended to allow for multiple commodities. Solution algorithms are suggested for the problem with multiple commodities, when there exists an equivalent optimization problem. We show that the multiple commodity problem may be cast in the form of a variational inequality, when there does not exist an equivalent optimization problem and propose algorithms to solve this version of the problem as well.     

A computable market equilibrium model with markets for transferable discharge permits

This paper presents a mathematical programming model for numerically analyzing the impacts of a transferable discharge permit (TDP) system on resource allocation and the distribution of income in a competitive equilibrium setting. Key assumptions of the model are linear household commodity demands, Leontief production functions requiring both primary and produced factors, fixed primary factor supplies and linear pollution dispersion processes. A basic model is presented with discussion of how it may be modified to examine alternative TDP configurations. In addition, there is also a discussion of relationships between key primal and dual variables and of issues involved in actual numerical implementation of the model.     

Minimaximin results with applications to economic equilibrium

Abstract theorems are proved that can be used to establish the existence of a free disposal equilibrium in some infinite dimensional models of the economy. The results generalize those previously proved in that they apply to concave–convex functions rather than bilinear functions and excess demand correspondence dedicated to Professor Heinz König rather than excess demand functions.     

Equilibria in a class of aggregative location games

Consider a multimarket oligopoly, where firms have a single license that allows them to supply exactly one market out of a given set of markets. How does the restriction to supply only one market influence the existence of equilibria in the game? To answer this question, we study a general class of aggregative location games where a strategy of a player is to choose simultaneously both a location out of a finite set and a non-negative quantity out of a compact interval. The utility of each player is assumed to depend solely on the chosen location, the chosen quantity, and the aggregated quantity of all other players on the chosen location. We show that each game in this class possesses a pure Nash equilibrium whenever the players’ utility functions satisfy the assumptions negative externality, decreasing marginal utility, continuity, and Location–Symmetry. We also provide examples exhibiting that, if one of the assumptions is violated, a pure Nash equilibrium may fail to exist.     

Second‐Price Auctions with Different Participation Costs

This paper studies equilibria of second‐price auctions in independent private value environments with different participation costs. Two types of equilibria are identified: monotonic equilibria in which a bidder with a lower participation cost results in a lower cutoff for submitting a bid, and nonmonotonic equilibria in which a lower participation cost results in a higher cutoff. We show that there always exists a monotonic equilibrium, and further, that the monotonic equilibrium is unique for either concave distribution functions or strictly convex distribution functions with nonincreasing reverse hazard rates. There exist nonmonotonic equilibria when the distribution functions are strictly convex and the difference of the participation costs is sufficiently small. We also provide comparative static analysis and study the limiting properties of equilibria when the difference in bidders’ participation costs approaches zero.     

Equilibria with coordination failures

This paper extends the literature on equilibria with coordination failures to arbitrary convex sets of admissible prices. This makes it possible to address coordination failures for cases with price indexation or more general price linkages between commodities. We introduce a new equilibrium concept, called quantity constrained equilibrium (QCE), giving a unified treatment to all cases considered in the literature so far. At a QCE the expected trade opportunities on supply and demand are completely determined by a rationing vector satisfying that the prevailing price system maximizes the value of the rationing vector within the set of admissible prices. When the set of admissible prices is compact, we show the existence of a connected set of QCEs. This set connects two trivial no-trade equilibria, one with completely pessimistic expectations concerning supply opportunities and one with completely pessimistic expectations concerning demand opportunities. Moreover, the set contains for every commodity a generalized Drèze equilibrium, being a QCE at which for that commodity no binding trade opportunities on both supply and demand are expected, and also a generalized supply-constrained equilibrium at which no binding constraints on demand opportunities are expected and for at least one commodity also not on supply. We apply this main result to several special cases, and also discuss the case of an unbounded set of admissible prices.     

Two definitions of correlated equilibrium

Correlated equilibrium constitutes one of the basic solution concepts for static games with complete information. Actually two variants of correlated equilibrium are in circulation and have been used interchangeably in the literature. Besides the original notion due to Aumann (1974), there exists a simplified definition typically called canonical correlated equilibrium or correlated equilibrium distribution. It is known that the original and the canonical version of correlated equilibrium are equivalent from an ex-ante perspective. However, we show that they are actually distinct – both doxastically as well as behaviourally – from an interim perspective. An elucidation of this difference emerges in the reasoning realm: while Aumann’s correlated equilibrium can be epistemically characterized by common belief in rationality and a common prior, canonical correlated equilibrium additionally requires the condition of one-theory-per-choice. Consequently, the application of correlated equilibrium requires a careful choice of the appropriate variant.     

Wealth concerns and equilibrium

Most properties of the classical general equilibrium model without externalities fail to extend to the wildest forms of consumption externalities. The recent interest for wealth concerns, a kind of externality associated with herding behavior and other-regarding preferences, motivates a study of the general equilibrium exchange model with those externalities. The diffeomorphism of the equilibrium manifold with a Euclidean space, the smoothness and properness of the natural projection and its non-zero degrees are shown to hold true for endowment spaces with variable total resources. Other properties of the classical exchange model without externalities are fragile in the sense that they do not resist the introduction of wealth concerns even in models where consumers preferences are represented by the simplest forms of utility functions like the log-linear (or Cobb–Douglas) functions. The most notable fragile properties are the uniqueness and regularity of equilibrium at equilibrium allocations and the stability of no-trade equilibria.