There is considerable empirical evidence that both individual consumers and larger entities such as countries bear more idiosyncratic risk than is consistent with complete and frictionless Arrow-Debreu markets. Evidence at the level of the individual consumer is discussed, for example, in Hayashi [1985] and Zeldes [1989], who show that individual consumption is poorly correlated with aggregate consumption. Evidence at the international level is discussed, for example, in Backus, Kehoe, and Kydland [1992], who point out the low correlation between consumption levels across countries.
That individuals bear idiosyncratic risk can be captured by many departures from the Arrow-Debreu framework. Three important examples of such models are incomplete market models, where there are not enough securities to insure against all events; models of liquidity constraints in which individual consumers are assumed unable to borrow as much as they would like in loan markets; and models of adverse selection and moral hazard. Incomplete market models are discussed by Radner [1972], Hart [1975], and Duffie and Shafer [1985], for example. Examples of models of liquidity constraints can be found in Bewley [1980], Dumas [1980], Townsend [1980], Scheinkman and Weiss [1986], Abel [1990], Kehoe, Levine, and Woodford [1992] and Heaton and Lucas [1997]. Models of liquidity constraints typically involve incomplete markets, as not only are there short sales constraints on securities, but securities are limited in number as well. These papers have largely focused on the computation of special types of equilibria in economies where the outside asset is a fiat money of no intrinsic value. In these equilibria shocks have long term consequences. We show that this is also the case in the incomplete market model considered in this paper.
Models of adverse selection and moral hazard, with the notable exception of Prescott and Townsend [1984], are not ordinarily general equilibrium models, so fall outside the scope of this paper, but the interested reader should consult Green [1987] who shows some of the links between asset market models and models of adverse selection.
Models with incomplete markets and/or liquidity constraints typically have the properties that in equilibrium individuals bear idiosyncratic risk, and interest rates are lower than subjective discount rates. There is also a fourth model that shares these properties: a model with individually rational debt constraints. Here the setup differs from that of Arrow-Debreu only in the assumption that a portion of the endowment is inalienable and cannot be seized if a consumer goes bankrupt.
This model has been studied by Schechtman and Escurdero [1977], Manuelli [1986], Marcet and Marimon [1992], and Kehoe and Levine [1993]. Kocherlakota [1996], and Albuquerque and Hopenhyn [1999]. It has been applied to the study of existing asset markets by Kehoe and Perri [1998], Krueger and Perri [1998], and Alvarez and Jermann [2000]. It is worth noting that there are two distinct models of debt constraints: those in which traders can be excluded from spot markets, or those, as in Kehoe and Levine [1993] where they cannot. The latter possibility leads to a failure of the welfare theorems, and is conceptually more like the incomplete market model. In the single good model studied here, and widely used in applications, including the papers cited above, however, there is no spot market, and as a result the welfare theorems hold.
In contrast to the general equilibrium approach employed in this paper, models with debt constraints can also be analyzed using the tools of optimal contracts. See, for example, Kocherlakota [1996] and Albuquerque and Hopenhyn [1999].
This paper directly compares the debt constrained model to the incomplete markets/liquidity constrained model in the same physical environment in which consumers alternate either deterministically or randomly between having high and low endowments. The bottom line is that the debt constrained model, largely because it involves a much smaller departure from the Arrow-Debreu framework, leads to a vastly simpler and more tractable model of equilibrium in the stochastic case, but nevertheless incorporates the main features of equilibrium idiosyncratic risk bearing and interest rates lower than subject discount rates.
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