A wide range of empirical applications rely on linear approximations to dynamic Euler equations. Among the most notable of these is the large and growing literature on precautionary saving that examines how consumption growth and saving behavior are affected by uncertainty and prudence. Linear approximations to Euler equations imply a linear relationship between expected consumption growth and uncertainty in consumption growth, with a slope coefficient that is a function of the coefficient of relative prudence. This literature has produced puzzling results: estimates of the coefficient of relative prudence (and the coefficient of relative risk aversion) from linear regressions of consumption growth on uncertainty in consumption growth imply estimates of prudence and risk aversion that are unrealistically low. Using numerical solutions to a fairly standard intertemporal optimization problem, our results show that the actual relationship between expected consumption growth and uncertainty in consumption growth differs substantially from the relationship implied by a linear approximation. We also present Monte Carlo evidence that shows that the instrumental-variables methods that are commonly used to estimate the parameters correct some, but not all, of the approximation bias.

This paper studies the optimal consumption behavior of individuals who face borrowing limitations that vary stochastically with their income. This framework is motivated by new empirical evidence that I document in U.S. aggregate data: predictable growth in consumer credit is significantly related to consumption growth, a finding that is inconsistent with existing models of consumer behavior. The time-varying liquidity constraint model considered here correctly predicts two key properties of the U.S. aggregate data: the correlation of consumption growth with predictable credit growth documented in this paper, and the well-known correlation between consumption growth and predictable income growth that has been documented extensively elsewhere.