Edgeworth and the Newsvendor

Several months ago I received a new paper by Mukherjee, Brown and Rusmevichientong on
empirical Bayes methods for the "newsvendor problem."  It is a matter of considerable personal
embarrassment that I hadn't ever heard about the newsvendor problem.  In its simplest form we
have a newspaper vendor (remember them?) who must decide how many papers of various sorts
he should stock.  He sacrifices profits if he stocks too many or too few, the loss is linear in both
directions, but asymmetric.  He knows the distribution, F, of the (random) demand for each
paper, and of course his solution is to stock the a/(a+b) quantile of F, where a is the marginal
cost of stocking too few papers, and b is the marginal cost of stock too many.  I learned this
bit of basic decision theory wisdom from Tom Ferguson's wonderful textbook in a course from
Bruce Hill eons ago, and I was aware that it appears in Raiffa and Schlaifer's text as well, but
I was surprised to learn that there was an extensive operations research literature going back
at least to the seminal paper of Arrow, Harris and Marschak in 1951.  Why I was surprised,
since this is a rather fundamental problem in inventory policy, is another question, but I won't
delve into that. *  Instead, I decided to look into the earlier history of this idea and began to see
quite a lot of references to Edgeworth's (1888) paper "The mathematical theory of banking."
More embarrassment ensued when I realized that I'd never read this paper.  Reading it revealed
a very clear formulation of the newsvendor problem applied to banking, unfortunately Edgeworth's
solution seemed a little murky.  I've tried to untangle all this in a couple of pages below, but as
you will see, it resists a fully satisfactory untangling.

Edgeworth.pdf

* I would like to stipulate that the simple static model underlying the newsvendor problem
is only a small part of the accomplishment of the AHM paper that introduced dynamic sS
rules as well.