I've got a following graphic, where green line is sale.
From economics classes, I think marginal return from media advertising can be specified as a first derivative (actually, partial derivative but here we consider special case -function of one variable).
- I've been wondering if those three marked points (threshold, optimal and saturation) can be specified in terms of derivatives and optimal points of them?
- If not in general, does it help if I assume that my sale has a arcus tangens shape?
- Also can rate of decline in marginal return second be described as second derivative?
I agree that you need to define "uniform decrease" and "begins to taper off". But I think I understand qualitatively what you mean. I try to answer. Abbreviations:
FD = first derivative ; SD=Second derivative ; MR=marginal return ; RDMR = rate of decline of marginal return
Your RDMR is the FD of the MR. (Related to your third question.) In the case shown, it should be a negative number reaching a minimum (highly negative = fast decline) at the optimal point (see below).
Threshold: FD of RDMR is almost constant. How you define "almost constant" is up to you. Hint: if the FD of something is almost constant, then its SD is almost zero.
Optimal: FD of RDMR = zero and SD of RDMR = positive.
Saturation: A tipping point generally is the point where the SD vanishes. From a calculus point of view, that would be the point where the SD of the RDMR vanishes after the optimal point. On the other hand, the term "saturation" implies that something stays almost zero. Indeed, from a practical point of view, you may want to define your own criterion, involving also the FD or related.
If you make the approximation you mention (arc) you will get analytical expressions.
I hope this makes sense!