Existence of financial equilibria with restricted participation

Abstract

We consider a two-date model of a financial exchange economy with finitely many agents having nonordered preferences and portfolio constraints. There is a market for physical commodities at any state today or tomorrow and financial transfers across time and across states are allowed by means of finitely many nominal assets or numéraire assets. We prove a general existence result of equilibria for such a financial exchange economy in which portfolios are defined by linear constraints, extending the framework of linear equality constraints by Balasko et al. (1990), and the existence results in the unconstrained case by Cass (1984, 2006), Werner (1985), Duffie (1987), and Geanakoplos and Polemarchakis (1986). Our main result is a consequence of an auxiliary result, also of interest for itself, in which agents’ portfolio constraints are defined by general closed convex sets and the financial structure is assumed to satisfy a “nonredundancy-type” assumption, weaker than the ones in Radner (1972) and Siconolfi (1989).

Introduction

Since the seminal paper by Radner (1972) proving the existence of equilibria in a financial exchange economy with bounded portfolio sets, and the non-existence issue raised by Hart (1975), Duffie and Shafer, 1985, Duffie and Shafer, 1986 showed a generic existence result with real assets. An extensive body of literature built upon their argument, see e.g. Geanakopolos and Shafer (1990), Hirsch et al. (1990), Husseini et al. (1990) and Bich and Cornet, 2004, Bich and Cornet, forthcoming. Subsequently, the literature on the existence problem paid particular attention to incomplete asset markets with only nominal assets or only numéraire assets; this was considered either in the case of unconstrained agents’ portfolio holdings, e.g. Cass, 1984, Cass, 2006, Werner (1985), or Duffie (1987) for nominal assets and Geanakoplos and Polemarchakis (1986) for numéraire assets, as well as when agents’ participation to financial markets might be restricted.

With restricted participation, in addition to the budget constraint, each agent $i$ faces exogenous portfolio constraints $z_i\in Z_i\subset R^J$, where $J$ denotes the (finite) number of assets in the economy. The presence of such portfolio constraints is a natural cause of market incompleteness and allows to capture a wide range of imperfections in the financial markets, such as short selling constraints, collateral requirements, and more generally institutional constraints. Elsinger and Summer (2001) give an extensive discussion of these institutional constraints and how to model them in a general financial framework. The existence problem had recently a growing interest since the first work by Siconolfi (1989), and Cass, 1984, Cass, 2006. Linear equality constraints are considered by Balasko et al. (1990) with nominal assets, and by Polemarchakis and Siconolfi (1997) with real assets. More recently, the case of portfolio sets $Z_i$ which are closed, convex subsets containing zero as in Siconolfi (1989) is considered by Angeloni and Cornet (2006) for real assets and by Martins-da-Rocha and Triki, 2005, Hahn and Won, 2007, and Cornet and Gopalan (forthcoming) in the nominal case.

This paper considers a two-date stochastic model ($t=0$ and $t=1$) of a financial exchange economy with finitely many states of nature, one of which is revealed at $t=1$. There is a market for finitely many physical goods at every state today or tomorrow and financial transfers across time and across states are allowed by means of finitely many assets. There are finitely many agents with nonordered preferences and portfolio constraints described by closed, convex subsets containing zero. Our contribution is twofold. First, when financial assets are nominal or numéraire, we provide a general existence result of equilibria (Theorem 1). Apart from standard assumptions on the consumption side (preferences and endowments), we assume that portfolio restrictions are defined by linear inequality constraints. This extends the framework of linear equality constraints considered by Balasko et al. (1990), and the standard model of unconstrained portfolios. Our existence result generalizes previous work by Cass, 1984, Werner, 1985, Duffie, 1987, and Geanakoplos and Polemarchakis (1986).

Our second contribution provides an auxiliary result (Theorem 2) which is the key tool in the proof of Theorem 1. In this auxiliary result, we make an additional “nonredundancy-type” (or “reduced form”) assumption (F3 in the text) on the financial side. In the case of nominal assets and no restrictions on portfolio trades ($Z_i=R^J$ for all $i$), Assumption F3 is equivalent to the fact that the payoff matrix $V$ has no redundant assets, that is, rank $V=J$ or equivalently ker $V=\{0\}$. In this case there is a priori no loss of generality in assuming that there are no redundant assets, otherwise, by deleting the redundant columns we obtain a “reduced” financial economy, whose equilibria yield equilibria in the original one. However, as mentioned in Balasko et al. (1990), one significant source of restricted participation is financial intermediation which typically involves redundancy. So there is no a priori grounds for the standard Full Rank Assumption in the presence of restricted participation, which therefore will be superseded by Assumption F3.

In the case of linear equality portfolio constraints (i.e., the $Z_i$’s are vector spaces), Balasko et al. (1990) show how to transform the agents’ financial opportunities to obtain a “reduced” financial economy in which each agent’s portfolio choice is a subspace having the same dimension as the wealth space it generates, that is, $Z_i\cap \text{ker}V=0$ for all i (Siconolfi, 1989); moreover, every equilibrium in the “reduced” economy leads to an equilibrium in the original economy. In this paper, we extend the analysis to the case of linear inequality constraints with nominal or numéraire assets. We show how to “reduce” the financial structure to obtain a new financial structure satisfying Assumption F3, a weaker condition than Siconolfi (1989)’s, keeping the correspondence between the equilibria; moreover, Assumption F3 coincides with Siconolfi’s when the $Z_i$’s are linear subspaces. Finally, we mention the companion papers Aouani and Cornet, 2008a, Aouani and Cornet, 2008b and the paper by Hahn and Won (2007), which study the more general case of closed convex portfolio sets.

The paper is organized as follows. In Section 2, we describe the financial exchange economy, we state our main existence result (Theorem 1) in the case of nominal or numéraire assets, and we state the auxiliary result (Theorem 2) under the additional Assumption F3. This section also provides examples under which Assumption F3 and the Financial Survival Assumption are satisfied. In Section 2.6, the proof of our main result (Theorem 1) is given as a consequence of Theorem 2 by “reducing” the initial economy into a new economy satisfying Assumption F3 whose equilibria yield equilibria in the original one. Finally, we also discuss the relationship with the existence results in the literature by Radner, 1972, Siconolfi, 1989, Cass, 1984, Cass, 2006, Werner, 1985, Duffie, 1987 and Geanakoplos and Polemarchakis (1986). Section 3 is devoted to the proof of the auxiliary result (Theorem 2). The Appendix gathers the proofs of some lemmata used in the proofs of Theorem 1, Theorem 2.