An Analytic Solution to a Stochastic Consumption/Saving Problem with Liquidity Constraints

Abstract

A substantial amount of recent work focusses on the importance of liquidity constraints and their effect on consumption behavior [5]. Liquidity constraints play an important role in determining the path of consumption over time. When income is uncertain and individuals are unable to borrow, they will take precautions against being caught short of income in the future. This precautionary saving has been used as an explanation for several consumption puzzles [7] and as a channel for fiscal policy to influence the economy [1]. Liquidity constraints are also important in the analysis of government policy. For example, the effectiveness of fiscal policy depends on the consumers' ability to smooth their path of consumption when faced with a change in the path of income as evidenced in the controversy over Ricardian Equivalence [2]. Finally, liquidity constraints have been offered as a reason permanent income models are rejected by excess sensitivity tests [4; 6]. If individuals cannot adjust current consumption to new information about future consumption, then consumption changes more than expected when the adjustment in income arrives. Past income helps predict future consumption since it affects the likelihood of being liquidity constrained in the future. For these reasons, it is important to understand the mechanisms through which liquidity constraints influence individual's consumption-saving decisions. Despite the existence of a large body of empirical work, analytic solutions have not yet been obtained for consumption models that include liquidity constraints. Instead, solutions are characterized using indirect methods such as numerical techniques or other approximations. This paper derives an analytic solution to a 3-period consumption/savings model with uncertain income and liquidity constraints, and examines the economic implications of the solution for consumer behavior. The results show that liquidity constraints induce precautionary saving and excess sensitivity on the part of an optimizing individual. Further, these two phenomena occur jointly since both arise from the consumer's desire to avoid being liquidity constrained in the future. The analytic solution also allows us to examine how consumption-saving behavior changes with variations in model parameters and to characterize the threshold level of wealth where the liquidity constraint becomes irrelevant.