Inconsequential arbitrage

We introduce the concept of inconsequential arbitrage and, in the context of a model allowing short-sales and half-lines in indifference surfaces, prove that inconsequential arbitrage is sufficient for existence of equilibrium. Moreover, with a slightly stronger condition of nonsatiation than that required for existence of equilibrium and with a mild uniformity condition on arbitrage opportunities, we show that inconsequential arbitrage, the existence of a Pareto optimal allocation, and compactness of the set of utility possibilities are equivalent. Thus, when all equilibria are Pareto optimal — for example, when local nonsatiation holds — inconsequential arbitrage is necessary and sufficient for existence of an equilibrium. By further strengthening our nonsatiation condition, we obtain a second welfare theorem for exchange economies allowing short sales.Finally, we compare inconsequential arbitrage to the conditions limiting arbitrage of Hart [Hart, O.D., 1974. J. Econ. Theory 9, 293–311], Werner [Werner, J., 1987. Econometrica 55, abs1403–1418], Dana et al. [Dana, R.A., Le Van, C., Magnien, F., 1999. J. Econ. Theory 87, 169–193] and Allouch [Allouch, N., 1999. Equilibrium and no market arbitrage. CERMSEM, Universite de Paris I]. For example, we show that the condition of Hart (translated to a general equilibrium setting) and the condition of werner are equivalent. We then show that the Hart/Werner conditions imply inconsequential arbitrage. To highlight the extent to which we extend Hart and Werner, we construct an example of an exchange economy in which inconsequential arbitrage holds (and is necessary and sufficient for existence), while the Hart/Werner conditions do not hold.     

No unbounded arbitrage,weak no market arbitrage and no arbitrage price system conditions; Equivalent conditions

Page and Wooders [Page Jr., F.H., Wooders, M., 1996. A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium. Economics Letters 52, 153–162] prove that the no unbounded arbitrage (NUBA), a special case of a condition in Page [Page, F.H., 1987. On equilibrium in Hart’s securities exchange model. Journal of Economic Theory 41, 392–404], is equivalent to the existence of a no arbitrage price system (NAPS) when no agent has non-null useless vectors. Allouch et al. [Allouch, N., Le Van, C., Page F.H., 2002. The geometry of arbitrage and the existence of competitive equilibrium. Journal of Mathematical Economics 38, 373–391] extend the NAPS introduced by Werner [Werner, J., 1987. Arbitrage and the existence of competitive equilibrium. Econometrica 55, 1403–1418] and show that this condition is equivalent to the weak no market arbitrage (WNMA) of Hart [Hart, O., 1974. On the existence of an equilibrium in a securities model. Journal of Economic Theory 9, 293–311]. They mention that this result implies the one given by Page and Wooders [Page Jr., F.H., Wooders, M., 1996. A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium. Economics Letters 52, 153–162]. In this note, we show that all these conditions are equivalent.     

Tiebout's tale in spatial economies: Entrepreneurship, self-selection, and efficiency

This paper establishes the existence and efficiency of equilibrium in a local public goods economy with spatial structures by formalizing Hamilton's [Hamilton, B.W., 1975. Zoning and property taxation in a system of local governments Urban Studies 12, 205–211] elaboration of Tiebout's [Tiebout, C., 1956. A pure theory of local public expenditures. Journal of Political Economy 64, 416–424] tale. We use a well-known equilibrium concept from Rothschild and Stiglitz [Rothschild, M., Stiglitz, J.E., 1976. Equilibrium in competitive insurance markets: an essay on the economics of imperfect information. Quarterly Journal of Economics 40, 629–649] in a market with asymmetric information, and show that Hamilton's zoning policy plays an essential role in proving the existence and efficiency of equilibrium. We use an idealized large economy following Ellickson, Grodal, Scotchmer and Zame [Ellickson, B., Grodal, B., Scotchmer, S., Zame, W.R., 1999. Clubs and the market, Econometrica 67, 1185–1217] and Allouch, Conley and Wooders [Allouch, N., Conley, J.P., Wooders, M.H., The Tiebout Hypothesis: On the Existence of Pareto Efficient Competitive Equilibria, (2004), mimeograph]. Our theorem is directly applicable to the existence and efficiency of a discrete spatial approximation of mono- or multi-centric city equilibria in an urban economy with commuting time costs, even if we allow the existence of multiple qualities of (collective) residences, when externalities due to traffic congestion are not present.     

Precise computation of a competitive equilibrium of the discrete land market model

This paper provides a finite algorithm to compute a competitive equilibrium of the Alonso-type discrete land market model (known as bid-rent equilibrium) without assuming any condition on the spatial configuration such as monocentricity. The existence of the finite algorithm implies that the discrete land market model is sufficiently tractable for computer simulation analysis (sensitivity analysis) even if the spatial configuration is variable. Specifically, we can evaluate numerically the effects of new routes in a transportation network upon equilibrium rents and commuting mode.     

Equilibria without the survival assumption

It is well known that an equilibrium in the Arrow–Debreu model may fail to exist if a very restrictive condition called the survival assumption is not satisfied. We study two approaches that allow for the relaxation of this condition. Danilov and Sotskov [Danilov, V.I., Sotskov, A.I., 1990. A generalized economic equilibrium. Journal of Mathematical Economics 19, 341–356], and Florig [Florig, M., 2001. Hierarchic competitive equilibria. Journal of Mathematical Economics 35, 515–546] developed a concept of a generalized equilibrium based on a notion of hierarchic prices. Marakulin [Marakulin, V., 1988. An equilibrium with nonstandard prices and its properties in mathematical models of economy. Discussion Paper No. 18. Institute of Mathematics, Siberian Branch of the USSR Academy of Sciences, Novosibirsk, 51 pp. (in Russian); Marakulin, V., 1990. Equilibrium with nonstandard prices in exchange economies. In: Quandt, R., Triska, D. (Eds.), Optimal Decisions in Market and Planned Economies. Westview Press, London, pp. 268–282] proposed a concept of an equilibrium with non-standard prices. In this paper, we establish the equivalence between non-standard and hierarchic equilibria. Furthermore, we show that for any specified system of dividends the set of such equilibria is generically finite. As a consequence, we have generic finiteness of Mas-Colell’s equilibria with slack, uniform dividend equilibria, and other special cases of our concept.     

Equilibria in large economies with a separable Banach commodity space and non-ordered preferences

The purpose of this paper is to provide an existence result of equilibria for economies with a measure space of agents, a non-trivial production sector and an infinite dimensional commodity space. The commodity space is modeled by an ordered separable Banach space whose positive cone has a non-empty interior. The discretization approach proposed in this paper, allows us to extend the existence results in Khan and Yannelis [Equilibrium in markets with a continuum of agents and commodities. In: Khan, M.A., Yannelis, N.C. (Eds.), Equilibrium Theory in Infinite Dimensional Spaces. Springer, Berlin, 1991] and Podczeck [Economic Theory 9 (1997) 585] to economies with a non-trivial production sector and with possibly non-ordered but convex preferences as well as partially ordered (possibly incomplete) but non-convex preferences.     

Technology choice in an evolutionary oligopoly game

In this paper, we propose and analyze a two-stage oligopoly game in which firms first simultaneously choose production technologies and in the second stage simultaneously choose production quantities. After characterizing the Nash equilibrium of the game, we cast our static model in a dynamic setting exploring the stability properties of the market equilibrium in two different cases: (i) exogenously distributed technologies and Cournot adjustments and (ii) endogenously distributed technologies in an infinite population game with Cournot–Nash equilibrium outputs. The main aim of the paper is that of extending the results about Cournot oligopoly stability in an evolutionary setting of heterogeneous decreasing returns-to-scale technologies. We show how the interplay between production decisions and R&D decisions can generate endogenous market fluctuations leading to complex dynamic phenomena.     

Virtual Nash implementation with admissible support

A social choice correspondence (SCC) is virtually implementable if it is ?$?$-close (in the probability simplex) to some (exactly) implementable correspondence [Abreu, D., Sen, A., 1991. Virtual Implementation in Nash Equilibrium. Econometrica 59, 997–1021] proved that, without restriction on the set of alternatives receiving strictly positive probability at equilibrium, every SCC is virtually implementable in Nash Equilibrium. We study virtual implementation when the supports of equilibrium lotteries are restricted. We provide a necessary and sufficient condition, imposing joint restrictions on SCCs and admissible supports. Next, we discuss how to construct supports, and we underline an important difficulty. Finally, we study virtual implementation when the support is restricted to the efficient or individually rational alternatives.     

Equilibrium theory with unbounded consumption sets and non-ordered preferences : Part I. Non-satiation

A new condition is introduced for the existence of equilibrium for an economy where preferences need not be transitive or complete and the consumption set of each agent need not be bounded from below. The new condition allows us to extend the literature in two ways. First, the result of the paper can cover the case where the utility set for individually rational allocations may not be compact. As illustrated in Page et al. [Page Jr., F.H., Wooders, M.H., Monteiro, P.K., 2000. Inconsequential arbitrage. Journal of Mathematical Economics 34, 439–469], the no arbitrage conditions do not apply to an economy with a non-compact utility set. Second, we generalize the arbitrage-based equilibrium theory to the case of non-transitive preferences.     

From mind to market: A global,dynamic analysis of R&D

Existing models of R&D are not easily reconciled with four observable aspects of R&D: initial technologies (ideas) need to be developed further, only a minority of initial ideas are successfully brought to the market, production and process innovations take place simultaneously (whereby, initially, there is no production at all), and process innovations are implemented for technologies that are destined to leave the market. We present a detailed bifurcation analysis for a dynamic model of R&D that captures these observations in one, unifying framework. As we provide a global analysis, we do not limit initial technologies to carry marginal costs that are below the choke price. We show that there always exists a critical value of initial marginal cost above which the firm does not initiate any (R&D) activity; the path to the saddle-point steady state is never globally optimal. We also sketch some tentative policy implications of our analysis.     

Existence of competitive equilibrium under financial constraints and increasing returns

This paper studies a ‘factor cost in advance’ model with increasing returns in production. In the model both partial equilibrium and general equilibrium may exist since working capital of firms limit their input demand. We provide a sufficient condition for the existence of partial equilibrium of a firm operating on a non-convex choice set. Furthermore we establish the existence and uniqueness of competitive equilibrium in the special case of logarithmic utility.     

Correlated equilibria in homogeneous good Bertrand competition

We show that there is a unique correlated equilibrium, identical to the unique Nash equilibrium, in the classic Bertrand oligopoly model with homogeneous goods and identical marginal costs. This provides a theoretical underpinning for the so-called “Bertrand paradox” as well as its most general formulation to date. Our proof generalizes to asymmetric marginal costs and arbitrarily many players in the following way: The market price cannot be higher than the second lowest marginal cost in any correlated equilibrium.     

On pure-strategy Nash equilibria in price–quantity games

This paper examines the existence and characteristics of pure-strategy Nash equilibria in oligopoly models in which firms simultaneously set prices and quantities. Existence of a pure-strategy equilibrium is proved for a class of price–quantity games. If the demand function is continuous, then the equilibrium outcome is similar to that of a price-only model. With discontinuous demand and limited spillover, there are rationing equilibria in which combined production falls short of market demand. Moreover, there might again be an equilibrium reflecting the outcome of a price game. Competition in price and quantity thus yields Bertrand outcomes under a variety of market conditions.     

Efficiency of markets for heterogeneous,indivisible housing

A rental housing market with a finite number of traders is presented. Tenants trade money to landlords for the use of an indivisible, heterogeneous house. Equilibrium is defined as the outcome of a competitive bidding arrangement among agents, the bid rent approach. Using this bidding arrangement, the two fundamental welfare theorems are shown to apply to this market. The second welfare theorem is verified via an algorithm which constructs equilibrium prices for Pareto-efficient allocations.     

Unbounded exchange economies with satiation: How far can we go?

We unify and generalize the existence results in Werner [Werner, J., 1987. Arbitrage and the existence of competitive equilibrium. Econometrica 55 (6), 1403–1418], Dana et al. [Dana, R.-A., Le Van, C., Magnien, F., 1999. On the different notions of arbitrage and existence of equilibrium. Journal of Economic Theory 87 (1), 169–193], Allouch et al. [Allouch, N., Le Van, C., Page Jr., F.H., 2006. Arbitrage and equilibrium in unbounded exchange economies with satiation. Journal of Mathematical Economics 42 (6), 661–674], Allouch and Le Van [Allouch, N., Le Van, C., 2008. Erratum to “Walras and dividends equilibrium with possibly satiated consumers”. Journal of Mathematical Economics 45 (3–4), 320–328]. We also show that, in terms of weakening the set of assumptions, we cannot go too far.     

Existence of a competitive equilibrium in the Lucas (1988) model without physical capital

This paper considers an endogenous growth model with human capital accumulation. It gives sufficient conditions and a necessary condition for the existence of a unique competitive equilibrium with externalities. These conditions are more stringent than those which prevail for the existence of an equilibrium defined as the solution to a fixed-point problem.     

Rational expectations equilibrium and the strategic choice of costly information

This paper studies costly information acquisition in one-good production economies when agents acquire private information and prices transmit information. Before asset markets open, agents choose the quality of their private information. After this information stage, agents trade assets in sequentially complete markets taking into account their private information and the information revealed by equilibrium prices (rational expectations equilibrium, (Radner, R., 1979. Rational expectations equilibrium: generic existence and the information revealed by prices, Econometrica 47, 655–678.)). An overall equilibrium in asset and information market is defined as a Nash equilibrium of the information game in which agents’ actions are information choices and their utility payoffs are the ex-ante expected utilities of the corresponding rationale expectations equilibrium. This paper shows that for a generic set of economies parameterized by endowments and productivity shocks, an overall equilibrium in information and asset market (a Nash equilibrium of the induced information game) with costly information acquisition and fully-revealing prices exists. In other words, informational efficiency is in general consistent with costly information acquisition.     

Monotonic matching in search equilibrium

It is well-known from Becker’s [Becker, G., 1973. A theory of marriage, Part I. Journal of Political Economy 81, 813–846] analysis of the neoclassical marriage market that matching is positive assortative if agent-types are complements in household production. Shimer and Smith [Shimer, R., Smith, L., 2000. Assortative matching and search. Econometrica 68, 343–369] have shown that this result does not generally extend to markets with search frictions, and they provide additional conditions on the production function that guarantee assortative matching in these settings. Here, the relationship between the Shimer–Smith restrictions and equilibrium matching pattern is considered, and alternative proofs of their results on assortative matching are provided.     

The spatial price equilibrium problem with path variables

Most of the recent literature dealing with the solution of the spatial price equilibrium model has dealt with the arc variable formulation; that is, the situation where the flows on the paths connecting an origin-destination pair in a network with trans-shipment nodes are not explicitly considered. This paper presents a clarification of the main assumption underlying this arc formulation and presents an argument for the need for using a model with path variables in certain applications. After this discussion, a new solution algorithm for the model with path variables is stated and compared against the linear approximation algorithm first proposed by Florian and Los [1] for this problem. Numerical examples are presented which illustrate the features of this new algorithm.