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

Published in *Journal of Mathematical Economics*, 2021

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.)