This paper defines a notion of an equilibrium and a pseudo-equilibrium for infinite horizon economies with incomplete asset markets. This definition generalizes the usual ones for finite horizon economies with incomplete markets and for infinite horizon economies with complete markets. We establish the existence of a pseudo-equilibrium when assets are short-lived and denominated in general commodity bundles; we obtain a true equilibrium when assets are denominated solely in a single numeraire commodity, or in units of account. It seems to us that the notion of an equilibrium we define is a natural and compelling one; as evidence, we show that our notion actually coincides with several other apparently quite distinct notions of an equilibrium.
The crucial issue that divides the infinite horizon setting from the finite horizon setting is the nature of debt constraints. In the finite horizon setting the constraint that there be no debt following the terminal date, together with the budget constraint, imply limits on debt at earlier dates. In the infinite horizon setting these terminal debt constraints and the implied debt constraints at earlier dates are absent If no additional debt constraints were imposed, then an equilibrium could not possibly exist: all traders would attempt to finance unbounded levels ot consumption by unbounded levels of borrowing. When markets are complete, such Ponzi schemes may be ruled out by the simple requirement that debt at each date/event never exceeds the current value of future endowments; this is frequently called a solvency requirement.
Completeness of markets guarantees that solvency is an unambiguous requirement. Moreover, in the presence of appropriate assumptions about preferences and endowments, it is sufficient to guarantee that an equilibrium exists (see Bewley 1972, for instance). However, when markets are incomplete, solvency is no longer an unambiguous requirement. When markets are incomplete, marginal rates of substitution for different traders may not be equal at equilibrium; as a consequence, traders may not agree on current value prices.
In the complete markets setting, an alternative formulation of the solvency requirement is that, at each node, almost all the debt can be repaid in finite time; this latter formulation has the advantage that it makes perfect sense in the incomplete markets setting as well. 2 We say that such debt constraints are finitely effective This condition expresses the same intuition as the usual solvency condition and rules out Ponzi schemes. We show that it also meets the basic consistency test of sufficing for the existence of what we term a finitely effective equilibrium (with appropriate assumptions on preferences and endowments). A finitely effective equilibrium reduces to the usual notions of equilibrium in the infinite horizon setting with complete markets and in the finite horizon setting with incomplete markets.
Finitely effective debt constraints are not the only debt constraints that will rule out Ponzi schemes. In a sense, however, they are the only debt constraints that are compatible with equilibrium and with the minimal ability to borrow and lend that we expect in our model. To make this assertion precise, we identify a broad class of debt constraints and show that whenever one of these more general debt constraints is compatible with an equilibrium, it necessarily reduces to the finitely effective constraints.
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