A simple method for computing equilibria when asset markets are incomplete

The problem of computing equilibria for general equilibrium models with incomplete real asset markets, or GEI models for the sake of brevity, is reconsidered. It is shown here that the rank-dropping behavior of the asset return matrix could be dealt with in rather a simple fashion: We first compute its singular value decomposition, and then, through this decomposition, construct, by the introduction of a homotopy parameter, a new matrix such that it has constant rank before a desired equilibrium is reached. By adjunction of this idea to the homotopy method, a simpler constructive proof is obtained for the generic existence of GEI equilibria. For the purpose of computing these equilibria, from this constructive proof is then derived a path-following algorithm whose performance is finally demonstrated by means of three numerical examples.     

Inefficiency of equilibria with incomplete markets

As is well known, equilibria with incomplete markets are generically Pareto inefficient. In this paper, we demonstrate the leading role of a budget constraint in the occurrence of Pareto inefficiency of equilibria with incomplete markets. Specifically, on the basis of the classical two-period one-good pure exchange model we prove that so long as a budget constraint is met for all agents, equilibria with incomplete markets are generically Pareto inefficient in initial endowments and utility functions regardless of the optimization behavior of each agent. All we require of utility functions is a very weak hypothesis called current monotonicity. A simple unified method applicable to both a real asset case and a nominal asset case is presented, so that our claim is proved in both cases.     

Minimax price bounds in incomplete markets

This paper develops an approach to tighten the bounds on asset prices in an incomplete market by combining no-arbitrage pricing and preference-based pricing, and the approach is applied to a call option in the absence of dynamic rebalancing. With the no-arbitrage pricing, it is straightforward to obtain the initial bounds, which are too wide to be of practical uses. By accepting that a representative agent exhibits risk aversion from a benchmark pricing kernel, it is possible to narrow the bounds considerably. Using the unbiased minimax deviation implicit in the parameters, one can restrict further the set of reasonable values on assets in incomplete markets.     

Taxes and money in incomplete financial markets

Abstract We analyze a model with incomplete financial markets, where money is needed to pay taxes. Equilibria exist, are typically regular and not Pareto optimal. Moreover, generically, there exists a redistribution of money among households which leads to a Pareto superior equilibrium. The intervention occurs only in the first period and it does not require either closing markets or upper bounds on the number of households. Mathematics Subject Classification (2000): 91B50 Journal of Economic Literature Classification: D52, D60, E50, H20     

Robust nonexistence of equilibrium with incomplete markets

We construct a pure exchange economy with spot and real security markets for which there does not exist a competitive equilibrium. Moreover, we show that the problem of nonexistence is robust to small perturbations of the endowments of the consumers. The result is driven by a lack of strict convexity of preferences.     

Game contingent claims in complete and incomplete markets

A game contingent claim is a contract which enables both the buyer and the seller to terminate it before maturity. For complete markets Kifer [Finance and Stochastics 4 (2000) 443–463] shows a connection to a (zero-sum) Dynkin game whose value is the unique no-arbitrage price of the claim. But, for incomplete markets one needs a more general approach. We interpret the contract as a generalized non-zero-sum stopping game. For the complete case this leads to the same results as in Kifer [Finance and Stochastics 4 (2000) 443–463]. For the general case we show the existence of an equilibrium point under the condition that both the seller and the buyer have an exponential utility function. For other utility functions such a point need not exist in the context of incomplete markets.     

Notes on pareto improvement in incomplete financial markets

these notes describe some recent developments in the analysis of the possibilities for Pareto improvement when, in a competitive environment, financial markets are incomplete. The basic framework is a general methodology for investigating the welfare effects of policy or institutional changes in equilibrium models, when these can be characterized in terms of perturbing the solutions to a system, of smooth equations. Two particular scenarios are discussed in detail: First, the situation where a central government can intervene in the form of wealth taxes and subsidies, and second, the situation where the financial institutions can be modified by increasing the number of available instruments.

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Sommario Queste osservazioni descrivono alcuni sviluppi recenti dell'analisi del miglioramento paretiano quando, in un contesto competitivo, i mercati finanziari sono incompleti. La struttura di riferimento è la metodologia generale per indagare gli effetti sul benessere sociale di cambiamenti politici o istituzionali nei modelli di equilibrio, quando questi possono essere caratterizzati come una perturbazione delle soluzioni di un sistema di equazioni. Due scenari particolari vengono discussi in dettaglio: nel primo, un governo centrale può intervenire attraverso la tassazione sulla ricchezza e le sovvenzioni; nel secondo, le istituzioni finanziarie possono essere modificate mediante l'aumento degli strumenti disponibili.

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While working in this general area I have benefitted substantially from interaction and collaboration with several of my, sometime Penn colleagues or students, Alex Citanna, Atsushi Kajii, Marcos Lisboa and Antonio Villanacci. Conversations at various points with Ronel Elul, John Geanakoplos, Heracles Polemarchakis and especially, Paolo Siconolfi have also been extremely helpful and illuminating. None of them, however, is responsible for my biases and idiosyncrasies. These notes were prepared for a presentation at the AMASES Conference held in Pugnochiuso, Italy during September 25–28, 1995.     

Excess demand function around critical prices in incomplete markets

We show that the aggregate excess demand function in an economy with incomplete real asset markets can be characterized by Walras’ law, homogeneity, and continuity around critical prices that cause one-dimensional drop of the dimension of the budget set.     

Determination of general equilibrium with incomplete markets and default penalties

This paper is concerned with the existence and computation of general equilibrium with incomplete asset markets and default. Due to the incompleteness of asset markets, the excess demand functions are typically not continuous at prices and delivery rates for which the assets have redundant nominal deliveries. This discontinuity results in a serious problem for the existence and computation of general equilibrium. We show that this problem can be resolved by replacing the nominal delivery matrix with a constant-rank one and restricting the macro variables in a subset of the domains. With this approach, the economies with incomplete markets and default penalties can be analyzed with differentiable homotopy techniques, and thus in the same framework as standard general equilibrium models. As a by-product, the existence of equilibrium is ensured for generic economies. Several computational examples demonstrate the effectiveness of the algorithm and show some quantitative features of equilibria in the model with default penalties.     

Equilibrium and arbitrage in incomplete asset markets with fixed prices

At arbitrary prices of commodities and assets, fix-price equilibria exist under weak assumptions: endowments need not satisfy an interiority condition, utility functions need only satisfy a very weak monotonicity requirement, and the asset return matrix allows for redundant assets. Prices of assets may permit arbitrage. At equilibrium, though restricted through endogenously determined trading constraints, arbitrage possibilities may persist; in an example, an individual holds an arbitrage portfolio.     

Adaptive learning and distributional dynamics in an incomplete markets model

Recent research shows that several DSGE models provide a closer fit to the data under adaptive learning. This paper extends this research by introducing adaptive learning in the model of Krusell and Smith (1998) with uninsurable idiosyncratic risks and aggregate uncertainty. A first contribution of this paper establishes that the equilibrium of this framework is stable under least-squares learning. The second contribution consists of showing that bounded rationality enhances the ability of this model to match the distribution of income in the US. Learning increases significantly the Gini coefficients because of the opposite effects on consumption of the capital-rich and of the capital-poor agent. The third contribution is an empirical exercise that shows that learning can account for increases in the income Gini coefficient of up to 25% in a period of 28 years. Overall, these findings suggest that adaptive learning has important distributional repercussions in this class of models.     

Production in incomplete markets: Expectations matter for political stability

In the present paper we study voting-based corporate control in a general equilibrium model with incomplete financial markets. Since voting takes place in a multi-dimensional setting, super-majority rules are needed to ensure existence of equilibrium. In a linear–quadratic setup we show that the endogenization of voting weights (given by portfolio holdings) can give rise to – through self-fulfilling expectations – dramatical political instability, i.e. Condorcet cycles of length two even for very high majority rules.     

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.     

Bounds for the utility-indifference prices of non-traded assets in incomplete markets

Abstract We consider a special class of financial models with both traded and non-traded assets and show that the utility indifference (bid) price of a contingent claim on a non-traded asset is bounded above by the expectation under the minimal martingale measure. This bound also represents the marginal bid price for the claim. The key conclusion is that the bound and the marginal bid price are independent of both the utility function and initial wealth of the agent. Thus all utility maximising agents charge the same marginal price for the claim. This conclusion is in some sense the opposite conclusion to that of Hubalek and Schachermayer (2001), who show that any price is consistent with some equivalent martingale measure. Mathematics Subject Classification (2000): 91B28, 91B16, 60J70 Journal of Economic Literature Classification: G13     

Comparison of solutions to the incomplete markets model with aggregate uncertainty

This paper compares numerical solutions to the model of Krusell and Smith [1998. Income and wealth heterogeneity in the macroeconomy. Journal of Political Economy 106, 867–896] generated by different algorithms. The algorithms have very similar implications for the correlations between different variables. Larger differences are observed for (i) the unconditional means and standard deviations of individual variables, (ii) the behavior of individual agents during particularly bad times, (iii) the volatility of the per capita capital stock, and (iv) the behavior of the higher-order moments of the cross-sectional distribution. For example, the two algorithms that differ the most from each other generate individual consumption series that have an average (maximum) difference of 1.63% (11.4%).     

Solving the incomplete markets model with aggregate uncertainty by backward induction

This paper describes a method to solve models with a continuum of agents, incomplete markets and aggregate uncertainty. I use backward induction on a finite grid of points in the aggregate state space. The aggregate state includes a small number of statistics (moments) of the cross-sectional distribution of capital. For any given set of moments, agents use a specific cross-sectional distribution, called “proxy distribution”, to compute the equilibrium. Information from the steady state distribution as well as from simulations can be used to chose a suitable proxy distribution.     

Slack in incomplete markets with nominal assets: A symmetric proof

In this paper, we provide an equilibrium analysis in the framework of incomplete markets where some agents’ preferences are possibly satiated at some state of the nature. We will consider nominal assets with exogenously fixed asset prices. We extend the notion of equilibrium with slack – introduced by Drèze and Müller [Drèze, J., Müller, H., 1980. Optimality properties of rationing schemes. Journal of Economic Theory 23, 150–159] in a fixed price setting – to the GEI framework.     

Solving the incomplete markets model with aggregate uncertainty using explicit aggregation

We propose a method to solve models with heterogeneous agents and aggregate uncertainty. The law of motion describing aggregate behavior is obtained by explicitly aggregating the individual policy rule. The algorithm is simpler and faster than existing algorithms that rely on parameterization of the cross-sectional distribution and/or a computationally intensive simulation step. Explicit aggregation establishes a link between the individual policy rule and the set of necessary aggregate state variables, an insight that can be helpful in determining what state variables to include in other algorithms as well.     

A remark on rational expectation equilibria with incomplete markets and real assets

In this note, I established the existence, for a generic set of endowments, of a fully revealing rational expectation equilibrium (REE) in an economy characterized by incomplete markets and real assets.     

Equilibrium in incomplete markets: II : Generic existence in stochastic economies

We model multiperiod securities markets with differential information. A price system that admits no free lunches is related to martingales when agents have rational expectations. We introduce the concept of resolution time, and show that a better informed agent and a less informed agent must agree on the resolution times of commonly marketed events if they have rational expectations and if there are no free lunches. It then follows that if all elementary events are marketed for a less informed agent then any price system that admits no free lunches to a better informed agent must eliminate any private information asymmetry between the two. We provide an example of a dynamically fully revealing price system that is arbitrage free and yields elementarily complete markets.