Arbitrage, equilibrium, and gains from trade: A counterexample

We present a counterexample to a theorem due to Chichilnisky (Bulletin of the American Mathematical Society, 1993, 29, 189–207; American Economic Review, 1994, 84, 427–434). Chichilnisky's theorem states that her condition of limited arbitrage is necessary and sufficient for the existence of an equilibrium in an economy with unbounded short sales. Our counterexample shows that the condition defined by Chichilnisky is not sufficient for existence of equilibrium. We also discuss difficulties in Chichilnisky (Economic Theory, 1995, 5, 79–107).     

Limited arbitrage is necessary and sufficient for the existence of an equilibrium

In Chichilnisky (Working Paper No. 586, 1991), Chichilnisky (Working Paper No. 650, 1992) and Chichilnisky (Economic Theory, 1995, 5, 79–108), I introduced the concept of a global cone and used it to define a condition on endowments and preferences, ‘limited arbitrage’, which I showed to be necessary and sufficient for the existence of a competitive equilibrium. In response to a comment (Monteiro et al., Journal of Mathematical Economics, 1997, 26, 000-000), I show here that the authors misunderstood my results by focussing on brief announcements which cover other areas, social choice (Chichilnisky, American Economic Review, 1994, 427–434 and algebraic topology (Chichilnisky, Bulletin of the American Mathematical Society, 1993, 29, 189–207), rather than on the publication which contains may proofs on equilibrium. The comment's example is irrelevant to my results in Chichilnisky (Economic Theory, 1995, 5, 79–108) because it starts from different conditions. Limited arbitrae is always necessary and sufficient for the existence of a competitive equilibrium, with or without short sales, with the global cones as I defined them, and exactly as proved in Chichilnisky (Economic Theory, 1995, 5, 79–108).     

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.     

Implementation in generalized matching problems

We search for (Nash) implementable solutions on a class of one-to-one matching problems which includes both the housing market (Shapley and Scarf, Journal of Mathematical Economics, 1974, 1, 23–28) and marriage problems (Gale and Shapley, American Mathematical Monthly, 1962, 69, 9–15). We show that the core correspondence is implementable. We show, furthermore, that any solution that is Pareto efficient, individually rational, and implementable is a supersolution of the core correspondence. That is, the core correspondence is the minimal solution that is Pareto efficient, individually rational, and implementable. A corollary of independent interest in the context of the housing market is that the core correspondence is the only single-valued solution that is Pareto efficient, individually rational, and implementable.     

Generic properties of the core and equilibria of pure exchange economies

We show that for an open dense set of markets with a continuum of traders the number of equilibrium allocations [which by the celebrated theorem of ^{Aumann (Econometrica. 1964, 32. 39–50)} coincide with the core allocations for such markets are finite. This presents a limiting case result that complements similar asymptotic theorems for cores of large economies proved by ^{Bewley (Econometrica 1973, 41, 425–454)}, and ^{Dierker (Journal of Mathematical Economics 1975, 2. 155–169)}. If we require that the measure on the space of agents be one with a finite number of atoms of equal weight, our reasoning recovers the classical results obtained by ^{Debreu (Econometrica. 1970. 38, 387–392)} for economies with a finite number of agents.     

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.     

The topology of fear

For many years experimental observations have raised questions about the rationality of economic agents—for example, the Allais Paradox or the Equity Premium Puzzle. The problem is a narrow notion of rationality that disregards fear. This article extends the notion of rationality with new axioms of choice under uncertainty and the decision criteria they imply (Chichilnisky, G., 1996a. An axiomatic approach to sustainable development. Social Choice andWelfare 13, 257–321; Chichilnisky, G., 2000. An axiomatic approach to choice under uncertainty with Catastrophic risks. Resource and Energy Economics; Chichilnisky, G., 2002. Catastrophical Risk. Encyclopedia of Environmetrics, vol. 1. John Wiley & Sons, Ltd., Chicester). In the absence of catastrophes, the old and the new approach coincide, and both lead to standard expected utility. A sharp difference emerges when facing rare events with important consequences, or catastrophes. Theorem 1 establishes that a classic axiom of choice under uncertainty – Arrow’s Monotone Continuity axiom, or its relatives introduced by DeGroot, Villegas, Hernstein and Milnor – postulate rational behavior that is ‘insensitive’ to rare events as defined in (Chichilnisky, G., 1996a. An axiomatic approach to sustainable development. Social Choice andWelfare 13, 257–321; Chichilnisky, G., 2000. An axiomatic approach to choice under uncertainty with Catastrophic risks. Resource and Energy Economics; Chichilnisky, G., 2002. Catastrophical Risk. Encyclopedia of Environmetrics, vol. 1. John Wiley & Sons, Ltd., Chicester). Theorem 2 replaces this axiom with another that allows extreme responses to extreme events, and characterizes the implied decision criteria as a combination of expected utility with extremal responses. Theorems 1 and 2 offer a new understanding of rationality consistent with previously unexplained observations about decisions involving rare and catastrophic events, decisions involving fear, the Equity Premium Puzzle, ‘jump diffusion’ processes and ‘heavy tails’, and it agrees with (Debreu, G., 1953. Valuation equilibrium and Pareto optimum. Proceedings of the National Academy of Sciences, 40, 588–592) formulation of market behavior and his proof of Adam Smith’s Invisible Hand theorem.     

Restricted houseswapping games

Restricted houseswapping games (RHGs) are a generalization of ‘one-sided matching games’, in which we specify a class II* of ‘allowable’ simple trading cycles. The cores of such games may be empty. Given II*, all possible closed RHGs have non-empty cores of II* is ‘strongly balanced’. Examples include the one-sided matching markets (^{Shapley and Scarf. Journal of Mathematical Economics 1974. 1. 23–37}. ^{Tijs et al., OR Spektrum 1984, 6, 119–123}; ^{Quinzii, International Journal of Game Theory 1984, 13, 41–60}) and the two-sided matching markets (^{Gale and Shapley. American Mathematical Monthly 1962. 69, 9–16}; ^{Shapley and Shubik, International Journal of Game Theory 1972, 1, 111–130}: and ^{Demange and Gale Econometrica 1985, 53, 873–888}).We then consider the subclass of RHGs in which there is no transferable resource. In this case, a weaker condition on II*, called ‘weak balancedness’, is sufficient to guarantee core non-emptiness. In addition, if II* is not weakly balanced, then there exists a preference profile such that the strict core of the resultant game is empty.Several other examples are given of II* that are (a) strongly balanced: (b) weakly balanced but not strongly balanced: and (c) not even weakly balanced.Finally, we discuss the issues of equilibrium definition, existence, and core-equilibrium allocation equivalence in RHGs.     

Arbitrage pricing theory and risk-neutral measures

Abstract We consider a market with countably many risky assets and finite factor structure, as in the “arbitrage pricing theory” of Ross (1976). We prove necessary and sufficient conditions in terms of parameters for the existence of an equivalent risk-neutral measure, i.e., a measure under which each asset return has zero expected value. We relate these conditions to a certain absence of arbitrage property of the model. Mathematics Subject Classification (2000): 91B24, 91B28 Journal of Economic Literature Classification: G10, G12     

Do house price developments spillover across euro area countries? Evidence from a global VAR

This paper empirically assesses the prospects for house price spillovers in the euro area, where co-movement in house prices across countries may be particularly relevant given a general trend with monetary union toward increasing linkages in trade, financial markets, and general economic conditions. A global VAR is estimated for three housing demand variables (real house prices, real per capita income, and the cost of borrowing, captured by a real long-term interest rate) on the basis of quarterly data for 7 euro area countries (Belgium, Germany, Ireland, Spain, France, Italy and the Netherlands), which together comprise nearly 90% of euro area GDP, over the period 1971–2009. The results suggest limited house price spillovers in the euro area, albeit with evidence of some overshooting in the first year after the shock, followed by a long run aggregate euro area impact of country-specific changes in real house prices related in part to the country’s economic weight. This contrasts with the impacts of a shock to domestic long-term interest rates, causing a permanent shift in house prices after 2–3 years. Underlying this aggregate development are rather heterogeneous house price spillovers at the country level, with a strong importance for weights – either economic or geographic – in governing their general magnitude. More generally, the impact of financing costs on house prices appears to have grown though time.     

No arbitrage conditions and liquidity

We extend the fundamental theorem of asset pricing to the case of markets with liquidity risk. Our results generalize, when the probability space is finite, those obtained by Kabanov et al. [Kabanov, Y., Stricker, C., 2001. The Harrison-Pliska arbitrage pricing theorem under transaction costs. Journal of Mathematical Economics 35, 185–196; Kabanov, Y., Rásonyi, M., Stricker, C., 2002. No-arbitrage criteria for financial markets with efficient friction. Finance and Stochastics 6, 371–382; Kabanov, Y., Rásonyi, M., Stricker, C., 2003. On the closedness of sums of convex cones in L⁰$L^0$ and the robust no-arbitrage property. Finance and Stochastics] and by Schachermayer [Schachermayer, W., 2004. The fundamental theorem of asset pricing under poportional transaction costs in finite discrete time. Mathematical Finance 14 (1), 19–48] for markets with proportional transaction costs. More precisely, we restate the notions of consistent and strictly consistent price systems and prove their equivalence to corresponding no arbitrage conditions. We express these results in an analytical form in terms of the subdifferential of the so-called liquidation function. We conclude the paper with a hedging theorem.     

<Emphasis Type="Italic">Notes and Comments:</Emphasis> Profitability in a multiple strategy market

The link between martingales and arbitrage is well-known in financial theory: arbitrage is not available if and only if there exists an equivalent measure such that the discounted prices are martingales with respect to this measure (MME). As a consequence, under MME, any previsible (non-anticipative) strategy cannot have secure (without risk) profit. Moreover, a careful reading of a bootstrap proof of the first fundamental theorem of asset pricing (see Schachermeier (1992)) underlines the fact that, if there is no possibility of arbitrage during any unit interval, then no arbitrage is allowed with any finite temporal horizon strategy. Mathematics Subject Classification (2000): Primary: 60G48; Secondary: 60G40, 60G07 Journal of Economic Literature Classification: C50, C72, D84     

Price functionals with bid–ask spreads: an axiomatic approach

In Jouini and Kallal [Jouini, E., Kallal, H., 1995. Martinagles and arbitrage in securities markets with transaction costs. Journal of Economic Theory 66 (1) 178-197], the authors characterized the absence of arbitrage opportunities for contingent claims with cash delivery in the presence of bid–ask spreads. Other authors obtained similar results for a more general definition of the contingent claims but assuming some specific price processes and transaction costs rather than bid–ask spreads in general (see for instance, Cvitanic and Karatzas [Cvitanic, J., Karatzas, I., 1996. Hedging and portfolio optimization under transaction costs: a martinangle approach. Mathematical Finance 6, 133-166]). The main difference consists of the fact that the bid–ask ratio is constant in this last reference. This assumption does not permit to encompass situations where the prices are determined by the buying and selling limit orders or by a (resp. competitive) specialist (resp. market-makers). We derive in this paper some implications from the no-arbitrage assumption on the price functionals that generalizes all the previous results in a very general setting. Indeed, under some minimal assumptions on the price functional, we prove that the prices of the contingent claims are necessarily in some minimal interval. This result opens the way to many empirical analyses.     

Truth and Robustness in Cross‐country Growth Regressions*

We re‐examine studies of cross‐country growth regressions by Levine and Renelt (American Economic Review, Vol. 82, 1992, pp. 942–963) and Sala‐i‐Martin (American Economic Review, Vol. 87, 1997a, pp. 178–183; Economics Department, Columbia, University, 1997b). In a realistic Monte Carlo experiment, their variants of Edward Leamer's extreme‐bounds analysis are compared with a cross‐sectional version of the general‐to‐specific search methodology associated with the LSE approach to econometrics. Levine and Renelt's method has low size and low power, while Sala‐i‐Martin's method has high size and high power. The general‐to‐specific methodology is shown to have a near nominal size and high power. Sala‐i‐Martin's method and the general‐to‐specific method are then applied to the actual data from Sala‐i‐Martin's original study.     

Consistency of assessments in infinite signaling games

In this paper we investigate possible ways to define consistency of assessments in infinite signaling games, i.e. signaling games in which the sets of types, messages and answers are complete, separable metric spaces. Roughly speaking, a consistency concept is called appropriate if it implies Bayesian consistency and copies the original idea of consistency in finite extensive form games as introduced by ^{Kreps and Wilson (Econometrica 1982, 50, 863–894)}. We present a particular appropriate consistency concept, which we call strong consistency and give a characterization of strongly consistent assessments. It turns out that all appropriate consistency concepts are refinements of strong consistency. Finally, we define and characterize structurally consistent assessments in infinite signaling games.     

Risk aversion in RDEU

Various notions of risk aversion can be distinguished for the class of rank-dependent expected utility (RDEU) preferences. We discuss the relationships amongst five of these, and describe simple (testable) characterizations in terms of elementary probability transformations for all but the weakest notion. The paper also provides the first complete characterization of the RDEU orderings that are risk-averse in the sense of Jewitt [Jewitt, I., 1989. Choosing between risky prospects: the characterization of comparative static results and location independent risk. Management Science 35, 60–70]. We also extend Chew et al.’s [Chew, S.H., Karni, E., Safra, Z., 1987. Risk aversion in the theory of utility with rank-dependent probabilities. Journal of Economic Theory 42, 370–381] important characterization of strong risk aversion [Rothschild, M., Stiglitz, J.E., 1970. Increasing risk: I. A definition. Journal of Economic Theory 2, 225–243] by relaxing strict monotonicity and differentiability assumptions, and allowing for discontinuities in the probability transformation function. The important special case of maximin choice falls within this relaxed RDEU class. It is shown that any strongly risk-averse RDEU order is a convex combination of maximin and another RDEU order with concave utility and continuous, concave probability transformation. Our proof of the result on strong risk aversion is also simpler (as well as more general) than that of Chew et al. [Chew, S.H., Karni, E., Safra, Z., 1987. Risk aversion in the theory of utility with rank-dependent probabilities. Journal of Economic Theory 42, 370–381].     

Lattice structure of knowledge and agreeing to disagree

This paper is a contribution to the study of the underlying mathematical structure of common-knowledge, which gives the well-known result of Aumann about the impossibility of ‘agreeing to disagree’. We present the Bayesian subjective probability model with player's belief: i.e. a triple (? %plane1D;4AF;, μ), in which i is a player. ? is a lattice in the field of sets of a state space Ω, %plane1D;4AF;, is a correspondence assigning to each state ω a filter %plane1D;4AF;(ω) in ?, and μ is a common-prior. For this model, we impose none of the important restrictions on the information structure in the Aumann-Bacharach model: axiom of knowledge K₁. axiom of transparency K₂ and axiom of wisdom K₃. We can extend both the disagreement theorem of Aumann and the agreement theorem of Geanacoplos and Polemarchakis under the assumption that each ? is an Artinian lattice.     

Reconciling the evidence of Card and Krueger (1994) and Neumark and Wascher (2000)

We employ the original Card and Krueger (American Economic Review 1994; 84: 772–793) and Neumark and Wascher (American Economic Review 2000; 90: 1362–1396) data together with the changes‐in‐changes estimator to re‐examine the evidence for the effect of minimum wages on employment. Our study reconciles the controversial positive average employment effect reported by the former study and the negative average employment effect reported by the latter study. Our main finding, which is supported by both datasets, is that the controversial result remains valid only for small fast‐food restaurants. Copyright © 2011 John Wiley & Sons, Ltd.     

Quarterly beta forecasting: An evaluation

Ever since the inception of betas as a measure of systematic risk, the forecast error in relation to this parameter has been a major concern to both academics and practitioners in finance. In order to reduce forecast error, this paper compares a series of competing models to forecast beta. Realized measures of asset return covariance and variance are computed and applied to forecast beta, following the advances in methodology of Andersen, Bollerslev, Diebold and Wu [Andersen, T. G., Bollerslev, T., Diebold, F. X., & Wu, J. (2005). A framework for exploring the macroeconomic determinants of systematic risk. American Economic Review, 95, 398–404; and Andersen, T. G., Bollerslev, T., Diebold, F. X., & Wu, J. (2006). Realized beta: Persistence and Predictability. In T. Fomby & D. Terrell (Eds.), Advances in Econometrics, vol 20B: Econometric Analysis of Economic and Financial Times Series., JAI Press, 1–40.]. This approach is compared with the constant beta model (the industry standard) and a variant, the random walk model. It is shown that an autoregressive model with two lags produces the lowest or close to the lowest error for quarterly stock beta forecasts. In general, the AR(2) model has a mean absolute forecast error half that of the constant beta model. This reduction in forecast error is a dramatic improvement over the benchmark constant model.     

Arbitrage and price revelation with asymmetric information and incomplete markets

This paper deals with the issue of arbitrage with differential information and incomplete financial markets, with a focus on information that no-arbitrage asset prices can reveal. Time and uncertainty are represented by two periods and a finite set S of states of nature, one of which will prevail at the second period. Agents may operate limited financial transfers across periods and states via finitely many nominal assets. Each agent i has a private information about which state will prevail at the second period; this information is represented by a subset S_i of S. Agents receive no wrong information in the sense that the “true state” belongs to the “pooled information” set ∩_iS_i, hence assumed to be non-empty.Our analysis is two-fold. We first extend the classical symmetric information analysis to the asymmetric setting, via a concept of no-arbitrage price. Second, we study how such no-arbitrage prices convey information to agents in a decentralized way. The main difference between the symmetric and the asymmetric settings stems from the fact that a classical no-arbitrage asset price (common to every agent) always exists in the first case, but no longer in the asymmetric one, thus allowing arbitrage opportunities. This is the main reason why agents may need to refine their information up to an information structure which precludes arbitrage.