Conditional Impatience and Concavity of Consumption Functions

Concave consumption functions imply a marginal propensity to consume that falls with wealth. I characterize the utility functions that guarantee this property in finite-horizon optimal saving problems with stochastic discounting, returns, income, and borrowing limits. Under conditional impatience—the conditional expected discounted gross return does not exceed one—consumption functions are always concave if and only if inverse absolute prudence, $-u^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}{u}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}$, is concave. When no conditional-impatience restriction is imposed, hyperbolic absolute risk aversion (HARA) is necessary and sufficient for uniform concavity. Thus conditional impatience permits declining marginal propensities to consume for a preference class strictly larger than HARA.