Marginal Revenue Derived from Elasticity

I'm studying for my second Managerial Economics quiz next week when I was reviewing something our professor showed us in class which I thought was absolutely brilliant. We had previously talked about elasticity as being defined as percent change in quantity demanded over percent change in price. Now, you can intuitively already begin to tell that there is probably some relationship between elasticity and maximum revenue (using calculus, it would be related to the idea of finding the value of a curve at dy/dx = 0).

It's basically the idea that you total revenue is maximized at elasticity = -1 where any change would either decrease the price or quantity demanded more than it's counterpart resulting in a decline in total revenue. The only way to not have unit elasticity is if the numerator and denominator were not equal implying one was shrinking faster than the other was growing.

What I thought was brilliant was the mathematical proof he used to describe the relationship by focusing on marginal revenue. At first I got incredibly confused with his notation: He writes "Price as a function of quantity demanded Qd" as P(Qd) where I misunderstood that as meaning "Price multiplied by quantity demanded" (which would actually give you revenue, not price). It's hard to "type formulas" in text boxes rather than use software like Equation Editor in Office.

Anyways, he described the concept of profit maximizing by assuming a constant or target marginal cost (MC) and tried to derive a formula for marginal revenue which incorporated elasticity. For simplicity (and clarity) I'll define my notation:

P0 - Old price
P1 - New price
Q - Quantity demanded at old price
Note: New quantity is (Q + 1)

Total revenue is price x quantity, so for two different prices P1 and P0, the difference in total revenue (marginal revenue or for an increase in quantity by 1 unit) is:

= Total Revenue at one more unit - Total Revenue at original quantity
= P1 * (Q+1) - P0 * Q
= P1Q + P1 - P0Q
= P1 + Q(P1 - P0)
= P1(P0/P0) + Q(P1 - P0)(P0/P0)
= P0 [P1/P0 + Q(P1 - P0)/P0]
Now using calculus like assumptions (the line where the prof states the lines are "approximately the same") you can assume for a "small" increase in quantity (Q only increases by 1), that P1 - P0 is the slope (because the change in unit is only 1 unit increase of Q) and that P1 is approximately the same as P0 or P1/P0 is approximately 1 to arrive at
= P (1 + (1/E))

Where E is elasticity as defined in our previous neat little math trick.

So if you know your target MC, to maximize profit, you set MC = MR
= P (1 + (1/E))

But wait... there's more! I was just looking at this formula and realized something... This looks an awful lot like a growth rate when doing financial modeling! Like revenue growth year over year (or like NPV using EBITDA with a growth rate and a discount rate):
Revenue Next Year = Revenue This Year * (1 + Year over year growth rate)

or more generically:

New Value = Old Value * (1 + % rate of change)

Recall that E = P/mQ therefore 1/E = mQ/P or [% change in price] / [% change in quantity demanded] (acting as a sort of elasticity correction factor).

In the same way:
Marginal Revenue = Revenue from one additional unit plus change Total Revenue due to elasticity
Note that Price is approximately (calculus like assumption) equal to the revenue from one additional unit so
MR = P + P/E
And the P/E term acts as a compensation term in total revenue using the elasticity of an additional unit (or the % change in price for a % change in quantity) multiplied by the current price.

This goes back to the original idea that elasticity and marginal revenue "feel" like they should be related.