Despite the strong competition, no merchant/supplier will publicly advertise his lowest possible price because that would be incredibly short-sighted. They advertise the lowest price they can offer OR the highest price sufficient to undercut the competition, whichever of these is higher. So it's effectively 1 cent below the second cheapest offer for merchants who can go (sometimes much) lower.
Therefore, it's reasonable to expect them to accept lower priced offers made to them directly in some cases.
In cases where there is no direct competition (i.e. authors who are the sole providers of said product), prices are strategically chosen (not dictated by competition) and therefore direct sales at lower prices are much more likely.
(Amazon will never know the lower price, I presume)
The author will just delete your email, why would he spend one minute on confirming the deal for a lousy 9 dollar, he doesn't care and he already has his lowest price on Amazon. If he agrees with you, then Amazon will also want a lower price (and then it will cost him a lot of money)