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.     

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.     

On top responsiveness and strict core stability

Alcalde and Revilla [Journal of Mathematical Economics 40 (2004) 869–887] introduce a top responsiveness condition on players’ preferences in hedonic games and show that it guarantees the existence of a core stable partition. In the present note we strengthen this observation by proving that under top responsiveness even the strict core is non-empty.     

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.     

The coordinate-wise core for multiple-type housing markets is second-best incentive compatible

We consider the generalization of Shapley and Scarf’s (1974) [Shapley, L., Scarf’s, H., 1974. On cores and indivisibility. Journal of Mathematical Economics 1, 23–37.] model of trading indivisible objects (houses) to so-called multiple-type housing markets. We show that the prominent solution for these markets, the coordinate-wise core rule, is second-best incentive compatible.     

Equilibrium theory with a measure space of possibly satiated consumers

Walras equilibria may not exist when consumers’ preferences are possibly satiated. To overcome this difficulty, several extended notions of equilibria have been proposed and all reduce to Walras equilibria under nonsatiation and free disposal. This includes the notions of equilibria with slack (also called dividend equilibria) as by Drèze and Müller [J. Economic Theory 23 (1980) 131], Makarov [J. Mathematical Economics 8 (1981) 87], Aumann and Drèze [Econometrica 54 (1986) 1271], Mas-Colell [Equilibrium theory with possibly satiated preferences, in: Majumdar, M. (Ed.), Proceedings of the Essays in Honour of David Gale on Equilibrium and Dynamics, Macmillan, London, pp. 201–213], monetary equilibria as by Kajii [J. Mathematical Economics 25 (1996) 75], or weak equilibria as by Polemarchakis and Siconolfi [J. Mathematical Economics 22 (1993) 85], which are all defined when there are finitely many consumers. This includes also the notion of free disposal equilibrium, when markets clear in a weak sense, allowing free disposal. Our paper considers an economy with a measure space of consumers and provides a general existence result of equilibria for the various existing notions. This result extends in particular the result by Hildenbrand [Econometrica 38 (1970) 608] on the existence of Walras equilibria.     

An axiomatization of the inner core using appropriate reduced games

I adapt a reduction process introduced by Serrano and Volij [Serrano, R., Volij, O., 1998. Axiomatization of neoclassical concepts for Economies. Journal of Mathematical Economics 30, 87–108] so that the reduced games of convex-valued games are convex-valued. I use the corresponding consistency property and its converse to axiomatize the inner core for games that are convex-valued, non-level and smooth.     

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.     

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.     

Purification of Bayesian–Nash equilibria in large games with compact type and action spaces

We present a purification result for incomplete information games with a large but finite number of players that allows compact metric spaces for both actions and types. We then compare our framework and findings to the early purification theorems of Rashid (1983. Equilibrium points of non-atomic games: asymptotic results. Economics Letters 12, 7–10), Cartwright and Wooders (2002 On equilibrium in pure strategies in games with many players. University of Warwick Working Paper 686 (revised 2005)), Kalai (2004. Large robust games. Econometrica 72, 1631–1665) and Wooders, Cartwright and Selten (2006. Behavioral conformity in games with many players. Games and Economic Behavior 57, 347–360). Our proofs are elementary and rely on the Shapley–Folkman theorem.     

Discrete fixed point theorem reconsidered

The aim of this note is to indicate an example that demonstrates the incorrectness of Iimura’s discrete fixed point theorem [Iimura, T., 2003. A discrete fixed point theorem and its applications. Journal of Mathematical Economics 39, 725–742] and to present a corrected statement using the concept of integrally convex sets.     

Core-stable rings in second price auctions with common values

In a common value auction in which the information partitions of the bidders are connected, all rings are core-stable. More precisely, the ex ante expected utilities of rings, at the (noncooperative) sophisticated equilibrium proposed by Einy et al. [Einy, E., Haimanko, O., Orzach, R., Sela, A., 2002. Dominance solvability of second-prices auctions with differential information. Journal of Mathematical Economics 37, 247–258], describe a cooperative games in characteristic function form, in spite of the underlying strategic externalities. A ring is core-stable if the core of this characteristic function is not empty. Furthermore, every ring can implement its sophisticated equilibrium strategy by means of an incentive compatible mechanism. An example shows that, if the bidders’ information partitions are not connected, rings may no longer be core-stable.     

Discrete fixed point theorem reconsidered

The aim of this note is to indicate an example that demonstrates the incorrectness of Iimura’s discrete fixed point theorem [Iimura, T., 2003. A discrete fixed point theorem and its applications. Journal of Mathematical Economics 39, 725–742] and to present a corrected statement using the concept of integrally convex sets.     

Representation of transferable utility games by coalition production economies

We prove that, by the method of construction of a coalition production economy due to Sun et al. [Sun, N., Trockel, W., Yang, Z., 2008. Competitive outcomes and endogenous coalition formation in an n$n$-person game. Journal of Mathematical Economics 44, 853–860], every transferable utility (TU) game can be generated by a coalition production economy. Namely, for every TU game, we can construct a coalition production economy that generates the given game. We briefly discuss the relationship between the core of a given TU game and the set of Walrasian payoff vectors for the induced coalition production economy.     

Instability and fluctuations in intertemporal equilibrium models: Presentation

This paper introduces the special issue of the Journal of Mathematical Economics on Instability and Fluctuations in Intertemporal Equilibrium Models.     

A note on the closed-form solution to the Lucas–Uzawa model with externality

Ruiz-Tamarit [2008. The closed-form solution for a family of four-dimension nonlinear MHDS. Journal of Economic Dynamics and Control 32, 1000–1014] provides a closed-form solution to the two-sector model of endogenous growth with externalities. He assumes that the coefficient of the relative risk aversion is equal to the physical capital share, but this assumption is empirically and theoretically implausible. This note uses the result of Boucekkine and Ruiz-Tamarit [2008. Special functions for the study of economic dynamics: the case of the Lucas–Uzawa model. Journal of Mathematical Economics 44, 33–54] and derives a closed-form solution without setting the parametric assumption. The solution path is expressed in terms of the Gauss hypergeometric functions.     

Finite sample multivariate structural change tests with application to energy demand models

This paper considers finite sample motivated structural change tests in the multivariate linear regression model with application to energy demand models, in which case commonly used structural change tests remain asymptotic. As in Dufour and Kiviet [1996. Exact tests for structural change in first-order dynamic models. Journal of Econometrics 70, 39–68], we account for intervening nuisance parameters through a two-stage maximized Monte Carlo test procedure. Our contributions can be classified into five categories: (i) we extend tests for which a finite-sample theory has been supplied for Gaussian distributions to the non-Gaussian context; (ii) we show that Bai et al. [1998. Testing and dating common breaks in multi-variate time series. The Review of Economic Studies 65 (3), 395–432] test severely over-rejects and propose exact variants of this test; (iii) we consider predictive break test approaches which generalize tests in Dufour [1980. Dummy variables and predictive tests for structural change. Economics Letters 6, 241–247] and Dufour and Kiviet [1996. Exact tests for structural change in first-order dynamic models. Journal of Econometrics 70, 39–68]; (iv) we propose exact (non-Bonferonni based) extensions of the multivariate outliers test from Wilks [1963. Multivariate statistical outliers. Sankhya Series A 25, 407–426] to models with covariates; (v) we apply these tests to the energy demand system analyzed by Arsenault et al. [1995. A total energy demand model of Québec: forecasting properties. Energy Economics 17 (2), 163–171]. For two out of the six industrial sectors analyzed over the 1962–2000 period, break and further goodness-of-fit and diagnostic tests allow to identify (and correct) specification problems arising from historical regulatory changes or (possibly random) industry-specific effects. The procedures we propose have potential useful applications in statistics, econometrics and finance (e.g. event studies).     

The Beveridge–Nelson decomposition in retrospect and prospect

Beveridge and Nelson [Beveridge, Stephen, Nelson, Charles R., 1981. A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘business cycle’. Journal of Monetary Economics 7, 151–174] proposed that the long-run forecast is a measure of trend for time series such as GDP that do not follow a deterministic path in the long run. They showed that if the series is stationary in first differences, then the estimated trend is a random walk with drift that accounts for growth, and the cycle is stationary. In contrast to linear de-trending, the smoother of Hodrick and Prescott (1981) and Hodrick and Prescott [Hodrick, Robert, Prescott, Edward C., 1997. Post-war US business cycles: An empirical investigation. Journal of Money Credit and Banking 29 (1), 1–16] and the unobserved components model of Harvey, [Harvey, A.C., 1985. Trends and cycles in macroeconomic time series. Journal of Business and Economic Statistics 3, 216–227]. Watson [Watson, Mark W., 1986. Univariate detrending methods with stochastic trends Journal of Monetary Economics 18, 49–75] and Clark [Clark, Peter K., 1987. The cyclical component of US economic activity. The Quarterly Journal of Economics 102 (4), 797–814], the BN decomposition attributes most variation in GDP to trend shocks while the cycles are short and brief. Since each is an estimate of the transitory part of GDP that will die out, it seems natural to compare cycle measures by their ability to forecast future growth. The results presented here suggest that cycle measures contain little if any information beyond the short-term momentum captured by BN.     

Characterization of an extended Walrasian concept for open economies

The consistency axiom, which is quite prominent in the framework of concepts of cooperative games, can be used to characterize concepts for economies as well. Dagan [Dagan, N., 1996b. Consistency and the Walrasian allocations correspondence. A revised version of Economics Working Paper No. 151, Universitat Pompeu Fabra, Barcelona, Spain.] and van den Nouweland et al. [van den Nouweland, A., Peleg, B., Tijs, S., 1996. Axiomatic characterization of the Walras correspondence for generalized economies. Journal of Mathematical Economics 25, 355–372.] used the setting of open economies to give axiomatic characterizations of an extension of the Walras correspondence. Here, we will characterize the proportional Walrasian concept, an extension which was proposed by Thomson [Thomson, W., 1992. Consistency in exchange economies. Mimeo, Department of Economics, University of Rochester, Rochester, NY.] and which is non-empty on a much bigger class of open economies. Apart from consistency properties and other frequently used axioms, we also employ an axiom on distribution.     

Coalition structure values in differential information economies: is unity a strength?

The coalition structure (CS) value, introduced by Owen [G. Owen, Values of games with a priori unions, in: Essays in Mathematical Economics and Game Theory, Springer Verlag, Berlin, 1977] and Hart and Kurz [Econometrica 51 (1983) 1047], generalizes the Shapley value to social situations where coalitions form for the purpose of bargaining. This paper introduces the CS value to economies with differential information. We show that the private CS values exists and is Bayesian incentive compatible. Moreover, we construct examples that go against the intuitive viewpoint that “unity is strength”. In particular, we consider a three-person economy in which two agents bargain as a unit against the third agent. We show that bargaining as a unit is advantageous if and only if information is complete. This result sheds new light on bargaining under differential information.