Asset Demands of Heterogeneous Consumers with Uninsurable Idiosyncratic Risk

We examine asset market equilibrium in an intertemporal economic model with individual and aggregate uncertainty and where the asset market is incomplete. Modigliani-Miller leverage irrelevance holds, even when consumers face borrowing constraints, because individual firms cannot alter the equilibrium portfolio of securities available to consumers. We show that households demand less risky portfolios as their financial wealth increases because a given asymmetry in asset holdings imparts more variability to income when wealth is high. Finally, we confirm previous results that endogenous rates of time preference, uninsurable idiosyncratic risk and household borrowing constraints produce a very low risk-free real interest rate. Version: August 20, 1999 Asset Demands of Heterogeneous Consumers with Uninsurable Idiosyncratic Risk Models used to analyze household saving and portfolio allocation, corporate financial policy and equilibrium asset prices within an intertemporal setting often assume that there is no production, no uncertainty or that optimizing agents are identical (for example, see Uzawa (1968), Becker (1980) and Lucas and Stokey (1982)). There is considerable doubt, however, whether representative agent models can account for the evidence on consumption and saving, asset prices and portfolio allocations (for example, see Deaton (1991), Cochrane and Hansen (1992) and Aiyagari (1994)). Some authors have argued that uninsurable income risk and household borrowing constraints might enable the models to account for some of the evidence. Since these modifications make asset markets incomplete for investors, however, their effects have usually been examined in endowment economies. We examine the effects of uninsurable income risk and household borrowing constraints within a general equilibrium production economy. Even though the capital market is incomplete for investors, Modigliani-Miller leverage irrelevance holds in our model. Although household choices result in a unique aggregate debt-equity ratio, the financial policies of individual firms do not affect their market values because no single firm can alter the set of securities available to households.1 The uninsurable income shocks lead to differences in household wealth. Since we assume all households have the same constant relative risk averse utility function, we might expect households to choose portfolios with a level of risk that was independent of wealth. We find, however, that households desire less risky portfolios as their financial wealth increases. Household behavior depends on total wealth, which includes the (state independent) capitalized value of expected labor income. As financial wealth increases, it becomes a larger proportion of total wealth. Households with an indirect utility of total wealth function that has the same concavity as their utility of consumption function would make the proportionate changes in consumption and total wealth the same. They therefore would reduce the riskiness of their portfolios of financial assets as financial wealth increases. We nevertheless find that even the wealthiest households choose end of period financial wealth that is more variable across aggregate states than is consumption. The capitalized value of expected labor income is constant across states. A given proportional variation in total wealth across states therefore requires more than a proportional variation in financial wealth. We confirm the findings of Heaton and Lucas (1992) that households use financial assets to selfinsure against income shocks. They smooth consumption by saving when there is a good shock to their income and dissaving when there is a bad shock to it.2 In fact, households smooth consumption more against idiosyncratic risk than against aggregate uncertainty. Since idiosyncratic risk is fully 1. This is the basis for Fisher separation holding in our model. When firms face short-selling constraints, the capital market may be incomplete for both households and firms, and Fisher separation can fail. The objective function of the firm then depends on the intertemporal consumption preferences of its shareholders. 2. Household saving thus mitigates the welfare effects of the lack of formal insurance. Dixit (1987, 1989) argues that asymmetric information is likely to restrict the availability of private insurance against individual risk. The same problems do not arise for aggregate uncertainty because it can be observed. Asset Demands in Incomplete Markets diversifiable across households, changes in wealth to insure against idiosyncratic risk are offsetting in aggregate. In contrast, the attempt to save against aggregate income shocks affects net saving and therefore alters the equilibrium asset prices in a way that tends to discourage self-insurance. Our results also have implications for the equity premium puzzle (Mehra and Prescott (1985)). We confirm evidence in previous studies that uninsured idiosyncratic income risk in the presence of credit or borrowing constraints (Aiyagari (1994), Aiyagari and Gertler (1991) and Hartley (1994)), and non-separable utility (Constantinides3 (1990) and Ferson and Constantinides (1991)) reduce the risk-free interest rate. For plausible values of risk aversion and income uncertainty, however, they do not greatly increase the risk premium on equity.4 There are other reasons for relaxing the assumption of time separability in the utility function. Becker (1980) argues that when households have additive utility with different constant rates of time preference, in a long-run steady state all the capital will be owned by the household with the lowest rate of time preference. If more than one household shares this low rate of time preference then the distribution of capital across these households will be indeterminate. We confirm this result in our model. Household borrowing constraints can eliminate this problem (Heaton and Lucas (1992), Aiyagari (1994) and Hartley (1994)). Another approach, however, is to allow time-interdependencies in household utility functions while retaining a constant rate of time preference (Ryder and Heal (1973), and Constantinides (1990)). Alternatively, the rate of time preference can be endogenized (Uzawa (1968), Epstein (1987), Epstein and Hynes (1991) and Shi and Epstein (1993)).5 We assume consumers become more impatient as wealth increases. This approach draws directly from Epstein and Hynes (1983), who consider a class of utility functions (in continuous time) where the rate of time preference depends positively on an index of future consumption. These functions are weakly additively separable because the marginal rates of substitution depend only on current and future, but not past, consumption. We use wealth as the index of future consumption because it greatly simplifies calculation of the household value function. The indirect utility of wealth function becomes more concave when the rate of time preference is an increasing function of wealth. The increased concavity helps produce an equilibrium cross-sectional wealth distribution when households do not face borrowing constraints. The increased concavity also 3. Constantinides relaxes time separability by introducing habit persistence. Other modifications to time discounting have also been proposed as explanations for the asset pricing puzzles. For example, Benninga and Protopapadakis (1990) allow the household discount rate to exceed unity. Kocherlakota (1990) shows that, if consumption grows, positive interest rates may exist in infinite horizon growth economies where individuals have discount factors larger than one. 4. The results in Hartley (1994) suggest that including banks in the economy might yield different results. 5. Shi and Epstein (1993) compare the effects of habit formation when it enters the utility function directly versus when it makes the rate of time preference endogenous.