👍Necessity of Hyperbolic Absolute Risk Aversion for the Concavity of Consumption Functions

The concavity of the consumption function in optimal savings problems with hyperbolic absolute risk aversion (HARA) utility is well known since Carroll & Kimball (1996). In Proposition 2.5 of Ma et al. (2020), we prove the concavity of the consumption function under some high-level assumption, and then in Remark 2.1 we note that the assumption is satisfied for the constant relative risk aversion (CRRA) utility function (which is equivalent to HARA after a translation). As I joined this project at a later stage, I don’t know the history of Proposition 2.5, but I was not happy with the high-level assumption and tried to improve it, without success.

In September 2020, I got back to this problem again, and managed to prove that the high-level assumption in Proposition 2.5 of Ma et al. (2020) was actually both necessary and sufficient for the concavity of consumption functions. After searching a variety of sources to investigate this problem further, eventually I studied Hardy, Littlewood, & Pólya (1934) Inequalities, which we cited in Ma et al. (2020). Emulating their proof technique, I proved the necessity of HARA for the concavity of consumption functions, that is, if the consumption function is concave regardless of the parameter specifications, then the utility function must be HARA. I wrote the paper in one day and submitted to AER: Insights. The paper was rejected, but the reports were good, so I submitted the paper to JME by attaching the referee reports and the decision letter. I was happily surprised that the JME editor accepted my paper as is. (As a matter of transparancy, at that point I was not a coeditor at JME.)