Optimal Consumption with a Stochastic Income Stream and Two Interest Rates

Abstract

An individual's choice between consumption and investment is very complex and simplifications must be introduced in every model building. A common simplification is the elimination of uncertainty from the model by replacing random variables by their expected values, since in general the deterministic case is much simpler than its probabilistic counterpart. Examples of such random variables are future labor incomes and investment returns. Another type of simplification is the elimination of variables or opportunities from the model. For example, borrowing may be excluded from the individual's financial opportunities or it may be assumed that there exists only one interest rate, common to borrowers and lenders. A combination of the above attitudes has led to an important result obtained by Mirman [8] and Leland [4] in two period models, and Miller [6] and Mendelson and Amihud [5] in a multiperiod model. They have proved that in the case of one interest rate and additive utility functions, concavity and positive third derivative of the utility functions imply an increase in consumption when going from a random labor income case to its deterministic counterpart. In [5] and [7] this result is extended, and it is shown that consumption also increases when a random sequence of labor income is replaced by a less risky sequence. The importance of the above results is twofold; they supply qualitative knowledge on the impact of increased uncertainty in future labor income on the consumer's decisions, and they give an upper bound on the immediate consumption in probabilistic models. These results can be explained as follows: The assumptions imply convexity of the marginal utility functions which tend therefore to increase when randomness is introduced or increased. Therefore the expected marginal utility of future consumption (related to present savings) increases. This leads to an increased preference for future consumption at the expense of immediate consumption.