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.     

Theorems on the core of an economy with infinitely many commodities and consumers

It is known that the classical theorems of Grodal [Grodal, B., 1972. A second remark on the core of an atomless economy. Econometrica 40, 581–583] and Schmeidler [Schmeidler, D., 1972. A remark on the core of an atomless economy. Econometrica 40, 579–580] on the veto power of small coalitions in finite dimensional, atomless economies can be extended (with some minor modifications) to include the case of countably many commodities. This paper presents a further extension of these results to include the case of uncountably many commodities. We also extend Vind’s [Vind, K., 1972. A third remark on the core of an atomless economy. Econometrica 40, 585–586] classical theorem on the veto power of big coalitions in finite dimensional, atomless economies to include the case of an arbitrary number of commodities. In another result, we show that in the coalitional economy defined by an atomless individualistic model, core–Walras equivalence holds even if the commodity space is non-separable. The above-mentioned results are also valid for a differential information economy with a finite state space. We also extend Kannai’s [Kannai, Y., 1970. Continuity properties of the core of a market. Econometrica 38, 791–815] theorem on the continuity of the core of a finite dimensional, large economy to include the case of an arbitrary number of commodities. All of our results are applications of a lemma, that we prove here, about the set of aggregate alternatives available to a coalition. Throughout the paper, the commodity space is assumed to be an ordered Banach space which has an interior point in its positive cone.     

On Vind’s theorem for an economy with atoms and infinitely many commodities

We extend Vind’s classical theorem on the measure of blocking coalitions valid in finite dimensional atomless economies (see Vind (1972)), to include the possibility of infinitely many commodities as well as the presence of atoms. The commodity space is assumed to be an ordered Banach space which has possibly the empty positive cone. The lack of interior points is compensated by an additional assumption of a cone of arbitrage that allows us to use Lyapunov’s convexity theorem in its weak form. The measure space of agents involves both negligible and non negligible traders. The extension is proved in the general class of Aubin coalitions for which a suitable version of Grodal’s result (Grodal (1972)) is also formulated. Our results wish to point out the relevance of cone conditions dealing with blocking coalitions of arbitrary measure or weight.     

Existence of equilibria with infinitely many commodities: Banach lattices reconsidered

The purpose of this paper is twofold. The first aim is to present an extension of the results on the existence of Walrasian equilibrium to the infinite dimensional setting. The result depends on two crucial assumptions. These are the compactness of the collection of feasible allocations and the non-emptiness of the interior of the production set. The proof is a direct generalization of Bewley's (1972) proof for the L_∞ case. The second purpose of this paper is to show that the recent result of Mas-Colell (1986) on the existence of equilibrium for exchange economies on Banach lattices can be obtained through an argument based on the result outlined above. That is, exchange economies on Banach lattices with ‘uniformly proper’ preferences behave as though they were production economies in which the production sets have non-empty interior.     

New conditions for the existence of Radner equilibrium with infinitely many states

In atomless differential information economies, equilibria are known not to exist prevalently even when agents are risk averse expected utility maximizers. The notion of prevalence involves essentially picking an economy at random. In this paper, however, we establish existence results with economically meaningful assumptions on the information structure. We obtain existence when agents have independent information, and also when the total endowment of the economy is common knowledge.     

A note on strategies for bandit problems with infinitely many arms

A bandit problem consisting of a sequence of n choices (n) from a number of infinitely many Bernoulli arms is considered. The parameters of Bernoulli arms are independent and identically distributed random variables from a common distribution F on the interval [0,1] and F is continuous with F(0)=0 and F(1)=1. The goal is to investigate the asymptotic expected failure rates of k-failure strategies, and obtain a lower bound for the expected failure proportion over all strategies presented in Berry et al. (1997). We show that the asymptotic expected failure rates of k-failure strategies when 0<b1 and a lower bound can be evaluated if the limit of the ratio F(1)–F(t) versus (1–t)^b exists as t1^– for some b>0.     

On the asymptotic optimality of the LIML estimator with possibly many instruments

We consider the estimation of the coefficients of a linear structural equation in a simultaneous equation system when there are many instrumental variables. We derive some asymptotic properties of the limited information maximum likelihood (LIML) estimator when the number of instruments is large; some of these results are new as well as old, and we relate them to results in some recent studies. We have found that the variance of the limiting distribution of the LIML estimator and its modifications often attain the asymptotic lower bound when the number of instruments is large and the disturbance terms are not necessarily normally distributed, that is, for the micro-econometric models of some cases recently called many instruments and many weak instruments.     

Fractional cointegration and price discovery in Canadian commodities

Using as a unifying theme commodities important to the Canadian economy, recently developed tools are applied to studying price discovery in the spot and futures markets. For each commodity the fractionally cointegrated vector autoregression (FCVAR) model of Johansen and Neilsen is estimated and tested against the special case of the conventional cointegrated vector autoregression (CVAR). These models characterize the fundamental value of a commodity as the common stochastic trend shared by its cointegrated spot and futures prices, and so price discovery can be analyzed using the permanent-transitory decomposition of Gonzalo and Granger. Model forecasts are evaluated and compared using a distributional result due to Clark and West. The generalization to fractional cointegration is found to be statistically significant. However the economic significance of this generalization—in terms of forecast accuracy and the profitability of mean–variance dynamic trading strategies—is more fragile than may have been appreciated.     

Core equivalence theorem: countably many types of agents and commodities in $\vec{L}^{1}(\mu)$

Abstract  We prove a core-Walras equivalence result for a finitely additive confederate economy with commodity space L¹(μ) and a measurable bounded map of extremely desirable commodities: when the map of extremely desirable commodities is simply bounded the properness of preferences is no longer equivalent to the existence of just one extremely desirable commodity as assumed in the countably additive model by Rustichini and Yannelis. Mathematics Subject Classification: 91B50, 91B54; 28B20, 46A22, 28A25 Journal of Economic Literature Classification: D51; C02     

A methodology for comparing predictions from many models to few data

A common problem in data analysis occurs when one has many models to compare to a single or just a few data sets. For example, a researcher may conduct an experiment in which subjects respond by choosing one category from a small set of categories. The data set then consists of the frequencies with which the categories occur. Many substantive models may yield predictions of these frequencies, so that the researcher is faced with the problem of comparing the data to many a priori equally attractive theoretical predictions. This paper proposes a method for the simultaneous study of the predictions and data. The method improves on the standard approach to judging goodness-of-fit by treating the predictions as rows in a two (or higher) way contingency table. Log linear models for the probabilities that subjects respond in specific ways are used to determine how the predictions compare to the data and to rank the predictions in terms of their accuracy.     

The effects of economic news on commodity prices

We assess how commodity prices respond to macroeconomic news and show that commodities have been relatively insensitive to such news over daily frequencies between 1997 and 2009 compared to other financial assets and major exchange rates. Where commodity prices are influenced by news, there is a pro-cyclical bias and these sensitivities have risen as commodities have become increasingly financialized. However, models based on news still do a relatively poor job of forecasting commodity prices at daily frequencies. We also find some asymmetries in how commodity prices respond to news, most notably for gold, which alone among commodities acts as a safe-haven when “bad” economic news emerges.     

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.     

Positioning of commodities using the Kraljic Portfolio Matrix

This article proposes a more objective methodology to classify and position commodities (works and services) in the Kraljic Portfolio Matrix (KPM) in a continuous scale around the two dimensions of supply risk and profit impact—the two dimensions which mainly influence the choice of a firm's purchasing strategy. Fuzzy multi-attribute scoring is used to assign performance scores to different commodities on supply risk as well as profit impact attributes. With a multidimensional scaling approach the commodities are placed in the KPM. An application of the proposed approach is attempted on commodities procured by the Rural Development Department of a state government in India.     

Comparison of Input–Output,Input–Output + Econometric and Computable General Equilibrium Impact Models at the Regional Level

Economic impact studies are a common practice—indeed, a necessary prerequisite in many cases of project development—in Australia. Although input–output (IO) is still regarded as the ‘bread-and-butter’ model for these types of study, attention in recent years has turned towards more sophisticated models, the main contenders being integrated IO + econometric and computable general equilibrium models. All these models, which are often promoted as substitutes, exhibit characteristics which are theoretically and empirically appealing, yet questions have been raised with respect to the different approaches, with apparently little awareness at the practitioner level as to the extent of these differences. This paper compares the three models and demonstrates that the differences can be quite substantial, even when the models rely on the same database and are subjected to the same impact scenario.     

Core equivalence with differentiated commodities

This paper presents improved core equivalence results for atomless economies with differentiated commodities in the framework of Ostroy and Zame (1994). Commodity bundles are elements of the space $M\left(K\right)$ of signed Borel measures on a compact space $K$ of commodity characteristics. Ostroy and Zame provide two sufficient conditions for core equivalence: It is sufficient that markets are “physically thick”, so that there are many suppliers of every commodity, or that markets are “economically thick”, so that consumers are sufficiently willing to substitute commodities with a similar composition for each other. The sufficient conditions in Ostroy and Zame (1994) all imply that there are “many more agents than commodities”, an idea of Aumann that was formalized and discussed in Tourky and Yannelis (2001) and Greinecker and Podczeck (2016). We generalize the framework in Ostroy and Zame (1994) and weaken their sufficient conditions to not imply the presence of “many more agents than commodities”. In particular, we drop the requirement that $K$ is metrizable from the basic model, the requirement of an uniform bound on endowments from the condition of “physically thick markets”, and the requirement that preferences are weak${}^1$-continuous from the condition of “economically thick markets”. Core equivalence still holds, showing that “many more agents than commodities” are not needed for core equivalence in models of commodity differentiation.     

On finite sample properties of alternative estimators of coefficients in a structural equation with many instruments

We compare four different estimation methods for the coefficients of a linear structural equation with instrumental variables. As the classical methods we consider the limited information maximum likelihood (LIML) estimator and the two-stage least squares (TSLS) estimator, and as the semi-parametric estimation methods we consider the maximum empirical likelihood (MEL) estimator and the generalized method of moments (GMM) (or the estimating equation) estimator. Tables and figures of the distribution functions of four estimators are given for enough values of the parameters to cover most linear models of interest and we include some heteroscedastic cases and nonlinear cases. We have found that the LIML estimator has good performance in terms of the bounded loss functions and probabilities when the number of instruments is large, that is, the micro-econometric models with “many instruments” in the terminology of recent econometric literature.     

Bivariate garch estimation of the optimal commodity futures Hedge

Six different commodities are examined using daily data over two futures contract periods. Cash and futures prices for all six commodities are found to be well described as martingales with near-integrated GARCH innovations. Bivariate GARCH models of cash and futures prices are estimated for the same six commodities. The optimal hedge ratio (OHR) is then calculated as a ratio of the conditional covariance between cash and futures to the conditional variance of futures. The estimated OHRs reveal that the standard assumption of a time-invariant OHR is inappropriate. For each commodity the estimated OHR path appears non-stationary, which has important implications for hedging strategies.     

On cores and indivisibility

An economic model of trading in commodities that are inherently indivisible, like houses, is investigated from a game-theoretic point of view. The concepts of balanced game and core are developed, and a general theorem of Scarf's is applied to prove that the market in question has a nonempty core, that is, at least one outcome that no subset of traders can improve upon. A number of examples are discussed, and the final section reviews a series of other models involving indivisible commodities, with references to the literature.     

Existence of equilibria in economies with externalities and non-convexities in an infinite-dimensional commodity space

We prove an equilibrium existence theorem for economies with externalities, general types of non-convexities in the production sector, and infinitely many commodities. The consumption sets, the preferences of the consumers, and the production possibilities are represented by set-valued mappings to take into account the external effects. The firms set their prices according to general pricing rules which are supposed to have bounded losses and may depend upon the actions of the other economic agents. The commodity space is L_∞(M,M,μ), the space of all μ-essentially bounded M-measurable functions on M.As for our existence result, we consider the framework of Bewley (1972). However, there are four major problems in using this technique. To overcome two of these difficulties, we impose strong lower hemi-continuity assumptions upon the economies. The remaining problems are removed when the finite economies are large enough.Our model encompasses previous works on the existence of general equilibria when there are externalities and non-convexities but the commodity space is finite dimensional and those on general equilibria in non-convex economies with infinitely many commodities when no external effect is taken into account.     

Supply-constrained fixed price equilibria in monetary economies

In this paper we consider an economy with restrictions on the relative prices of non-money commodities. The non-money commodities are partitioned into two groups, index makers and price following commodities. Then two cases are considered, namely that the relative prices of the index makers are fixed, respectively free. In both cases the money prices are flexible, whereas the relative prices of the price following commodities are tied to the prices of the index makers. The existence of a supply-constrained equilibrium is proved with (i) no rationing on the money commodities (stores of value), and (ii) at least one non-money commodity is not rationed. If prices of the index makers are fixed the result strengthens a theorem of Dehez and Drèze, if the prices are free a theorem of Kurz is strengthened. This paper is not only concerned with these existence results, but also with the question whether supply-constrained equilibria should appear more frequently than demand-constrained equilibria.