# If $F$ is MHR (log-concave), can the monopoly reserve price exceed the mean?

Let $$V\sim F$$ be some positive valued random variable such that $$F$$ has a monotone hazard rate (alternatively $$F$$ is log-concave). Let $$p = \operatorname{argmax}q(1-F(q))$$. We call $$p$$ the monopoly reserve price, when $$F$$ is MHR it's well known that $$p$$ is unique.

I'm interested the relationship between the monopoly price and the mean, specifically I want to know if $$p \le \Bbb E[V]$$, or else be given an MHR distribution such that $$p > \Bbb E[V]$$. The best bounds I've seen in the literature come from Lemma 3.4 of Giannakopoulos et al. (2021)1 which shows that $$\frac{p}{\mu} \le \frac{e}{e-1} = 1.58$$ by loosely using a result of Barlow et al. (1964)2.

For the canonical MHR distributions like exponential, uniform, etc. $$p$$ is either equal to or less than $$\mu$$ so I believe if the result fails is should take some sort of construction.

References

 Giannakopoulos, Y., Poças, D., Zhu, K. (2021). Optimal Pricing For MHR and $$\lambda$$-Regular Distributions. ACM Transactions on Economics and Computation. 9(1).

 Barlow, R. E., Marshall, A. W. (1964). Bounds for distributions with monotone hazard rate, i. Ann. Math. Stat. 35(3):1234-1257.

• You might have better luck on Mathematics StackExchange. – prubin Mar 1 at 23:03
• Note: as outlined in the paper the upper bound of $e/(e-1)$ can also be constructed from Chawla et al. (2010). – TheSimpliFire Mar 2 at 9:50