Competitive equilibria in economies with multiple indivisible and multiple divisible commodities

In this paper we consider a general equilibrium model with a finite number of divisible and a finite number of indivisible commodities. In models with indivisibilities it is typically assumed that there is only one divisible good, which serves as money. The presence of money in the model is used to transfer the value of certain amounts of indivisible goods. For such economies with only one divisible commodity Danilov et al. showed the existence of a general equilibrium if the individual demands and supplies belong to a same class of discrete convexity. For economies with multiple divisible goods and money van der Laan et al. proved existence of a general equilibrium if the divisible goods are produced out of money using a linear production technology and no other producers are present in the model.     

Market economies with many commodities

In this survey problems from the consideration of infinite dimensional commodity spaces in the Arrow-Debreu-McKenzie model are discussed. It has become clear in the last couple of decades that in order to address real policy questions, economic models that are both stochastic and dynamic have to be employed. These models lead naturally to infinite dimensional commodity spaces.

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Riassunto In questo lavoro si discutono i problemi che si presentano quando, in connessione all'esistenza e all'efficienza di equilibri Walrasiani nel modello di Arrow-Debreu-McKenzie, si ammette che lo spazio dei beni sia di dimensione non finita. è diventato chiaro nei due ultimi decenni che per affrontare questioni reali è necessario usare modelli sia dinamici che stocastici e ciò conduce in modo naturale alla considerazione di spazi infinito-dimensionali. Si rimane principalmente confinati alle economie di puro scambio al solo scopo di semplificare la trattazione.

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This paper has been presented atWorkshop in Economic Theory, Venice-June 2–7, 1997 organized by Istituto Veneto di Scienze, Lettere ed Arti and atXXI Convegno AMASES, Rome-September 10–13, 1997. It is an expanded version of our paper [5] that appeared inJournal of Economic Theory, vol. 74, n. 1, 1997, pp. 62–105. We are grateful to Karl Shell and Academic Press for permission to include that portion of the material here. The research of C. D. Aliprantis was supported in part by the NATO Collaborative Research Grant #941059.     

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.     

Finitely additive economies with free extremely desirable commodities

Continuing the investigation on finitely additive economies with infinite dimensional commodity space, we state a core-walras individualistic equivalence for an economy where preferences admit a summable map of extremely desirable commodities. The main result extends previous equivalences obtained both in the countably additive and in the finitely additive setting. Extra assumptions on the model are discussed via a pair of examples.     

Equilibrium with coordination

The obvious equilibrium concepts in the simplest institutions for transferring ownership of commodities—bilateral exchange—are neither Nash equilibria nor cooperative equilibria. To study such equilibria as special cases of equilibria of a social system it is necessary to introduce coordination. Two or more agents coordinate their actions, if, when they consider an alternative to a state, they take as given—for agents with whom they coordinate—the alternative state. If there is no coordination we obtain Nash equilibrium as a special case. If there is complete coordination we obtain optimality as a special case. The main result is an existence theorem for a social system with coordination. This theorem is then applied to prove existence of exchange equilibria in an economy with bilateral exchange.     

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.     

Equilibria in markets with a Riesz space of commodities

We present a new proof of the existence of competitive equilibrium for an economy with an infinite dimensional space of commodities.     

Hedonic prices and composite commodities

In this paper we conjecture and, to an extent, prove that recently noted restrictions required for the logical coherence and empirical relevance of hedonic price models make these models no more general than traditional housing services models. In particular, intra-urban variation in hedonic prices may not be substantively related to market equilibration at all, and therefore is not evidence for the existence of housing sub-markets. Moreover, in the case of jointly produced housing characteristics, the hedonic price models are found to be less general than the traditional homogeneous housing services models.     

论商品质量特性

准确地评价、检验和监督商品的质量，有赖于对质量特征的正确理解和把握。由于消费的需求是多种多样的，所以反映商品质量的特性也是多种多样的，商品质量的适用性就是建立在质量特性基础之上的。     

Volatility transmission across currencies and commodities with US uncertainty measures

This paper uses the Multi-chain Markov Switching model (MCMS) conditioned on US uncertainty measures (VIX, VIX-oil and FSI) to examine the patterns of volatility transmission across the resource, major and safe haven currencies The results with and without the uncertainty variables generally identify three patterns of volatility transmission: interdependence, spillover and comovement. They reveal the dominance of interdependence over spillovers and comovements when the uncertainty variables are excluded, highlighting the significance of mutual reciprocity of individual market shocks over common shocks across the selected assets. Within portfolios of a two-variable framework (two variables representing two minimum variance portfolios (à la Markowitz), containing a weighted combination of the currencies and of the commodities, respectively), we find interdependence between the two portfolios with and without the VIX, a spillover from commodities to currencies in the case when the FSI is included and independence between the two portfolios in the case when the oil-VIX is accounted for. The implications of the results are important for the portfolio managers in selecting portfolios’ components during high oil volatility periods.     

Resource allocation and public commodities

This paper characterizes Pareto efficient allocations by means of prices within a public-goods economy. The underlying notion of public commodities is a generalization covering private and pure public ones as extreme cases. The model uses as measure space of agents.     

Local public goods as indivisible commodities

A sharper version of Ellickson's (1979) result on the existence of an approximate Tichout equilibrium is shown to be true under much weaker assumptions. It is also shown that there exists a second-best policy regarding the bundle of public goods each local government offers to a resident. The existence result can also be seen as an extension of a result of Anderson, Khan and Rashid (1982) to economies with production. In this context, the main technical difficulty in that the approximate equilibrium allocation does not lie in the attainable and standard techniques of truncation are not adequate.     

Forecasting realized volatility of agricultural commodities

We forecast the realized and median realized volatility of agricultural commodities using variants of the heterogeneous autoregressive (HAR) model. We obtain tick-by-tick data on five widely-traded agricultural commodities (corn, rough rice, soybeans, sugar, and wheat) from the CME/ICE. Real out-of-sample forecasts are produced for between 1 and 66 days ahead. Our in-sample analysis shows that the variants of the HAR model which decompose volatility measures into their continuous path and jump components and incorporate leverage effects offer better fitting in the predictive regressions. However, we demonstrate convincingly that such HAR extensions do not offer any superior predictive ability in their out-of-sample results, since none of these extensions produce significantly better forecasts than the simple HAR model. Our results remain robust even when we evaluate them in a Value-at-Risk framework. Thus, there is no benefit from including more complexity, related to the volatility decomposition or relative transformations of the volatility, in the forecasting models.     

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.     

Learning cycles in Bertrand competition with differentiated commodities and competing learning rules

This paper stresses the importance of heterogeneity in learning. We consider a Bertrand oligopoly with firms using either least squares learning or gradient learning for determining the price. We demonstrate that convergence properties of the rules are strongly affected by heterogeneity. In particular, gradient learning may become unstable as the number of gradient learners increases. Endogenous choice between the learning rules may induce cyclical switching. Stable gradient learning gives higher average profit than least squares learning, making firms switch to gradient learning. This can destabilize gradient learning which, because of decreasing profits, makes firms switch back to least squares learning.     

Unconventional monetary policy and financialization of commodities

Our paper has two stages of analysis. First of all, we examine whether volatility spillover between US equity and commodity markets has significantly changed with the heavy influx of index traders in commodity derivatives markets, which is a phenomenon referred to as financialization. Given that previous findings show institutional traders enter into commodity markets at high liquidity episodes, in the second stage of our analysis, we investigate the particular impact of US quantitative easing policy on spillover between commodity and US stocks. Our results indicate that during financialization period, spillover from stocks to commodities have significantly increased for almost all commodities. More importantly, we show that quantitative easing is one of the underlying reasons for increasing volatility spillover between markets. Including interest rate, currency factors or default spread does not diminish the explicit role of quantitative easing on spillovers.     

Equilibrium turnpike theory with time-separable utility

The purpose of this paper is to show that under reasonably general conditions intertemporal competitive equilibrium has a turnpike property. The model is general because it permits (1) time-variant, time-separable utility functions, (2) heterogeneous rates of discount across consumers, and (3) matching flatness in utilities and production possibilities. The results rely on bounded marginal utility for all consumers and aggregate stationarity of utilities of the set of most patient consumers. Under these assumptions, the neighborhood turnpike result holds with (1) and (2) because of the eventual unimportance of the impatient consumer. Matching flatness requires the use of a two-sided Liapounov function and the growth theoretic methods of McKenzie.     

Competitive General Equilibrium with network externalities

We identify sufficient conditions for existence of competitive equilibrium with network externalities and indivisibilities. Such combination of externalities and indivisibilities is present in many goods and services with network effects, and it makes existence of equilibrium non-trivial. We provide an existence theorem in a model with a measure space of consumers. Key conditions for existence are anonymity of network effects and dispersion in the economy’s income distribution.     

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.