On the Concavity of the Consumption Function with Liquidity Constraints

Carroll and Kimball (1996) prove that the consumption function is concave if infinitely-lived risk-averse households have a utility function which exhibits Hyperbolic Absolute Risk Aversion (HARA), face income uncertainty, and are prudent. However, the empirical evidence is inconclusive about the importance of income uncertainty for households. On the other hand, empirical results suggest that liquidity and liquidity constraints are important determinants of household behavior. In this paper, I prove that the consumption function is strictly concave in wealth for infinitelylived risk-averse households with a utility function of the HARA class if there exists a liquidity constraint which binds for some level of wealth. This result is independent of prudence. Furthermore, the introduction of a liquidity constraint always reduces consumption and increases the marginal propensity to consume out of wealth.