The structure of the set of sunspot equilibria in economies with incomplete financial markets

Summary This paper considers the set of equilibria of two-period, sunspot economies withS purely extrinsic states of nature in the second period andI assets with linearly independent nominal payoffs. The span of the payoff matrix contains the vector [1, ... , 1] (i.e., inside money). The set of economies is described in terms of (sunspot-invariant) utility functions. IfS>I> 0, there is an open, dense set of economies such that, given a vector of no arbitrage asset prices, the set of equilibrium allocations contains a smooth manifold of dimensionS—I. Such a manifold contains at least one nonsunspot equilibrium (and at most a finite number of such equilibria).The paper was written while I was a visitor at C.O.R.E., Universitè Catholique de Louvain, with the financial support of a S.P.E.S. fellowship. I would like to thank D. Cass, H. Polemarchakis and P. Siconolfi for their helpful comments.     

The set of sunspot equilibria in economies with incomplete financial markets: variable asset prices

Summary. I consider the set of equilibria of two-period economies with S extrinsic states of nature in the second period and I assets with linearly independent nominal payoffs. Asset prices are variable. If the number of agents is greater than (S-I), the payoff matrix is in general position and S 2I, the set of equilibrium allocations generically (in utility function space) contains a smooth manifold of dimension (S-1). Moreover, the map from states o f nature to equilibrium allocations (restricted to this manifold) is one-to-one at each equilibrium. Received: February 23, 1998; revised version: June 1, 2000     

Stochastic equilibria with incomplete financial markets

We demonstrate the existence of equilibria with incomplete financial markets for stochastic economies whose information structure is given by an event tree, restricting attention to purely financial securities, those paying in units of account (e.g., “dollars”). Financial markets may be incomplete: some consumption streams may be impossible to obtain by any trading strategy. Securities may be individually precluded from trade at arbitrary states and dates. Sufficient conditions for the existence of stochastic equilibria are: continuous, convex, strictly monotonic preferences and strictly positive aggregate endowments. These conditions are weakened. A corollary states that any regime of security prices precluding arbitrage can be embedded in an equilibrium.     

Equilibria in incomplete assets economies with infinite dimensional spot markets

The paper studies the two period incomplete markets model where assets are claims on state contingent commodity bundles and there are no bounds on portfolio trading. The important results on the existence of equilibrium in this model assume that there is a finite number of commodities traded in each spot market and that preferences are given by smooth utility functions. With these assumptions an equilibrium exists outside an “exceptional” set of assets structures and initial endowments. The present paper extends these results by allowing for general infinite dimensional commodity spaces in each spot market. These include all the important commodity spaces studied in the literature on the existence of Walrasian equilibrium—in each spot market the consumption sets are the positive cone of an arbitrary locally solid Riesz space or of an ordered topological vector space with order unit or of a locally solid Riesz space with quasi-interior point. The paper establishes that even with our very general commodity spaces there exists an equilibrium for a “very” dense set of assets structures. Our approach is in the main convex analytic and the results do not require that preferences be smooth or complete or transitive. The concepts and techniques studied in this paper have important finite as well as infinite dimensional applications. This paper has benefited from the comments of Martine Quinzii, Wayne Shafer, Manuel Santos and Yeneng Sun. The research of C. D. Aliprantis is supported by the NSF Grants SES-0128039, DMS-0437210, and ACI-0325846. The research of R. Tourky is funded by the Australian Research Council Grant A00103450.     

Sunspots,indeterminacy and Pareto inefficiency in economies with incomplete markets

Summary. We consider two periods economies with both intrinsic and extrinsic uncertainty. Asset markets are incomplete in the certainty economy. If assets are nominal, there are enough commodities and the number of agents is greater than two and smaller than the total number of states of nature tomorrow (minus one), then a sunspot-invariant equilibrium is generically Pareto dominated by some sunspot equilibria. When assets are real, and there are enough commodities, if there are sunspot equilibria, there are sunspot equilibria Pareto dominating sunspot-invariant equilibria under the same restriction on the number of agents (and stronger restrictions on the number of commodities).Received: 20 October 2003, Revised: 1 April 2004, JEL Classification Numbers: D52.I wish to thank Paolo Siconolfi for helpful suggestions and comments. I aknowledge the financial support of M.I.U.R. and the kind hospitality of C.C.D.R. in Summer 2003.     

Computing equilibria in finance economies with incomplete markets and transaction costs

Summary. Transaction costs on financial markets may have important consequences for volumes of trade, asset pricing, and welfare. This paper introduces an algorithm for the computation of equilibria in the general equilibrium model with incomplete asset markets and transaction costs. We show that economies with transaction costs can be analyzed with differentiable homotopy techniques and thus in the same framework as frictionless economies despite the existence of non-differentiabilities of agents asset demand functions and the existence of locally non-unique equilibria. We introduce an equilibrium selection concept into the computation of economic equilibria that picks out a specific equilibrium in the presence of a continuum of equilibria.Received: 2 December 2002, Revised: 15 November 2004, JEL Classification Numbers: C61, C62, C63, C68, D52, D58, G11, G12. Correspondence to: P. Jean-Jacques HeringsThis research started when Jean-Jacques Herings enjoyed the generous hospitality of the Cowles Foundation for Research in Economics at Yale University. His research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences and a grant of the Netherlands Organisation for Scientific Research. We thank audiences at Stanford University, UC San Diego, and Venice for discussions on the subject. We are very grateful to an anonymous referee for very helpful comments on an earlier draft.     

Pareto improving financial innovation in incomplete markets

Summary. In this paper we develop a differential technique for investigating the welfare effects of financial innovation in incomplete markets. Utilizing this technique, and after parametrizing the standard competitive, pure-exchange economy by both endowments and utility functions, we establish the following (weakly) generic property: Let S be the number of states, I be the number of assets and H be the number of households, and consider a particular financial equilibrium. Then, provided that the degree of market incompleteness is sufficiently larger than the extent of household heterogeneity, S−I≥2H−1 [resp. S−I≥H+1], there is an open set of single assets [resp. pairs of assets] whose introduction can make every household better off (and, symmetrically, an open set of single assets [resp. pairs of assets] whose introduction can make them all worse off ). We also devise a very simple nonparametric procedure for reducing extensive household heterogeneity to manageable size, a procedure which not only makes our restrictions on market incompleteness more palatable, but could also prove to be quite useful in other applications involving smooth analysis. Received: August 14, 1995; revised version: April 14, 1997     

Real indeterminacy in incomplete financial market economies without aggregate risk

Summary We analyze an exchange economy with incomplete financial markets and assets whose returns are fixed in units of account. Moreover, we assume absence of aggregate risk, i.e., that individual preferences and total resources are constrained to be invariant across different states of the world. In this framework we show that the set of (commodity) price-endowment equilibria is diffeomorphic to a Euclidean space. We then exploit this global parameterization to prove that the set of equilibrium allocations associated with each endowment in a generic set contains a smooth manifold, whose dimension is equal to the number of missing assets.     

Interaction of formal and informal financial markets in quasi-emerging market economies

The primary objective of this paper is to investigate the interaction of formal and informal financial markets and their impact on economic activity in quasi-emerging market economies. Using a four-sector dynamic stochastic general equilibrium model with asymmetric information in the formal financial sector, we come up with three fundamental findings. First, we demonstrate that formal and informal financial sector loans are complementary in the aggregate, suggesting that an increase in the use of formal financial sector credit creates additional productive capacity that requires more informal financial sector credit to maintain equilibrium. Second, it is shown that interest rates in the formal and informal financial sectors do not always change together in the same direction. We demonstrate that in some instances, interest rates in the two sectors change in diametrically opposed directions with the implication that the informal financial sector may frustrate monetary policy, the extent of which depends on the size of the informal financial sector. Thus, the larger the size of the informal financial sector the lower the likely impact of monetary policy on economic activity. Third, the model shows that the risk factor (probability of success) for both high and low risk borrowers plays an important role in determining the magnitude by which macroeconomic indicators respond to shocks.     

Sunspots and incomplete financial markets: The general case

Summary This paper examines the effects of extrinsic uncertainty or sunspots on competitive equilibrium when financial markets are incomplete. For the canonical two-period, pure-exchange model with bonds (or so-called nominal assets, yielding fixed overall returns specified in units of account, and including pure inside money), the following result is established: Generically in endowments, if there areS sunspot states in the second period, but only 0<I<S distinct types of bonds, then — corresponding to the inherent deficiency in the financial markets — sunspots will generateD=S–I dimensions of consumption allocation or real (as well as spot price or nominal) indeterminacy.     

The inflationary effects of stochastic resource revenues in resource-rich economies with less well-developed financial markets

This article develops a simple Dynamic Stochastic General Equilibrium (DSGE) model to illustrate how economies that face restrictions in their ability to alter both government spending and taxation in the short run and cannot borrow easily (perhaps because of incomplete internal capital markets) can find external fluctuations in resource revenues producing unexpected variations in their internal money supply and ultimately in their inflation rate. The main channels for these effects run through the government budget and through the country's balance of payments position. The model is calibrated to illustrate the case of Iran.     

Equity home bias,incomplete financial markets,and nominal rigidities

Abstract This paper analyzes optimal portfolio decisions in a monetary open‐economy framework. It is found that market completeness and the specific form of nominal rigidities, namely, nominal price vs. nominal wage rigidities, matter for justifying the observed structure of equity holdings. When markets are complete, sticky prices generate a negative correlation between the non‐diversifiable labour income and the profit of domestic firms with respect to the productivity shocks, which explains why households invest little abroad. In contrast, when markets are incomplete, rigidities in goods prices result in a counterfactual ‘super home bias’, because domestic equities provide a good hedge against not only the labour income risk but also the relative price risk. Wage rigidities, however, have the opposite effect. Therefore, nominal rigidities in both goods prices and wage rates are needed to address the empirical composition of gross equity positions under incomplete markets.     

A social welfare function characterizing competitive equilibria of incomplete financial markets

A classic characterization of competitive equilibria views them as feasible allocations maximizing a weighted sum of utilities. It has been applied to establish fundamental properties of the equilibrium notion, such as existence, determinacy, and computability. However, it fails for economies with missing financial markets.We give such a characterization for economies with a single commodity and missing financial markets, by an amended social welfare function. Its parameters capture both the relative importance of households welfare, through the classic welfare weights, as well as the disagreements among them as to the value of the missing markets.As a by-product, we identify the dimension of the set of interior equilibrium allocations.     

Indeterminacy in incomplete market economies

Summary It is shown that in a two-period economy with a continuum of states and real assets, the following holds: (1) if the asset structure is complete, then generically the number of equilibria is finite; (2) if there are a finite number of real assets (this can approximate completeness arbitrarily close) then, for a nonempty open set of economies, there are a continuum of distinct equilibria. Asymptotic versions (on the number of states and on the number of assets) of the result are also given. It is argued, therefore, that incompleteness, by itself, may be a leading source of indeterminacy.I want to thank R. A. Dara, J. Geanakoplos and the audience of a larger number of presentations (at Brown, Stanford, Columbia, Harvard, Paris,...) for useful comments.     

Multiplier effects in economies with missing risk markets

The paper analyses the impact of pecuniary externalities in a two-sector economy with an incomplete market structure. Agents in each sector choose their proportion of risky investment. Sector specific risks are assumed to be perfectly negatively correlated. It is shown that the economy is more volatile if risk markets do not exist. With a complete set of risk markets, shocks in one sector will be dampened on the aggregate level. In contrast, when risk markets are absent, pecuniary externalities arising from higher risky investment in one sector can create feedback effects in the other sector. When agents are sufficiently risk averse (their coefficient of relative risk aversion being greater than one), an individually optimal response to the increased riskiness of the price distribution will result in an even riskier price distribution: an increase in risky activity in one sector will lead to an increase in risky activity in the other sector, and this gives multiplier effects.Part of this work has been done while I was visiting assistant professor at the University of Western Ontario, London, Canada.I am grateful to Hans-Werner Sinn for helpful comments. The suggestions of the two referees have contributed to improve the paper considerably.     

Endogenous trading constraints with incomplete asset markets

This paper endogenizes the borrowing constraints on capital in a production economy with incomplete markets. We find that these limits get looser with income, a property that is consistent with US data on credit limits. The framework with endogenous limits is then used to study the effects of a revenue neutral tax reform that eliminates capital income taxes. Our results illustrate that it is very important to take into account the effects of tax policies on the limits. Throughout the transition, these effects can be big enough to change the overall conclusion about the desirability of a tax reform.     

On constrained optimal allocations with incomplete markets

Summary We provide an example of an economy with incomplete asset markets in which there is no constrained optimal allocation.I would like to thank Ramon Marimon for many helpful discussions, and to an anonymous referee for useful comments.     

Complete and incomplete markets with short-sale constraints

Summary. This paper argues that the introduction of a short-sale constraint in the Arrow-Radner framework invalidates standard definitions of complete and incomplete markets. Two threshold values with familiar properties arise in this constrained set-up. If short sales are not allowed on some security, then financial markets will be incomplete in the standard sense. Beyond a particular level of the short-sale bound, financial markets are “complete”, since the short-sale constraint is not effective. For intermediate bounds the distinction between complete and incomplete financial markets is blurred. Although some technical definitions hold, agents can not fully transfer wealth among states. These intermediate cases, called “technically incomplete markets”, exhibit interesting welfare properties. For instance, the resulting equilibrium allocations may not be Pareto-dominated by those of the non-restricted complete markets equilibrium. Received: November 28, 2000; / revised version: November 9, 2001