Existence of Equilibria in Financial Markets With Restricted Participation

Investors facing restrictions on the portfolios that they can trade, is more of a norm than an exception. We consider a model in which investors’ portfolio sets are constrained. As in Balasko, Cass and Siconolfi (1990) these constraints are exogenously given (possibly arising due to some institutional reasons). Moreover, we consider very general restrictions on portfolio sets as in Siconolfi (1986), where each agent’s portfolios set is assumed to be convex and containing zero. In two date (one period) models without restrictions on portfolio sets, the existence issue has been extensively studied. Cass (1984) and Werner (1985, 1989) showed existence with nominal assets. Duffie and Shafer (1985) showed a generic existence result with real assets. This second approach has been extensively used as surveyed in Magill and Shafer [14]. This paper primarily examines the characterization of equilibrium asset prices with arbitrage free asset prices, in a multiperiod model when investors face such general portfolio restrictions. In the absence of such portfolio constraints the approach initiated by Cass (1984), has been extensively used to characterize equilibrium asset prices with arbitrage free asset prices. See Cass (1984), Duffie (1987) and Florenzano and Gourdel (1994). Moreover this approach is also useful in showing the existence of an equilibrium. See Magill and Shafer (1991), Florenzano and Gourdel (1994), Magill and Quinzii (1996) and Angeloni and Cornet (2006) among others. Another approach to prove existence in a differentiable economy is to show existence in a numeraire asset economy and infer the existence in the nominal asset economy (See Villanacci et al. 2002 and Magill and Quinzii 1996). In the approach of Cass (1984), one agent is assumed to be unrestricted and hence behaves as in an Arrow-Debreu world. This assumption breaks the symmetry of the problem and thus the proof is not symmetric with respect to the agents. Hahn and Won (2003) are able to avoid this Cass approach, albeit with monotonic preferences and a more involved notion of ‘projective’ arbitrage. In a recent working paper, DaRocha and Triki (2005) have been able to show this characterization in a symmetric manner. We follow this approach, but with a more general notion of absence of market arbitrage and a more general compatibility condition.