I've looked for any mathematical model that shows headphone quality as a function of the price of the headphones, but have not found anything yet. Of course, "quality" is quite subjective, which could be a reason I'm not finding much, but from my research (i.e. Google) it seems that it's pretty much accepted that initially, as more money is invested in a pair of headphones, the quality begins to increase fairly rapidly, and then around $150 is the point of diminishing returns.
My goal is to find this point of diminishing returns by finding the second derivative of the function and solving for zero, which would give me the inflection point, or the point of diminishing returns.
My guess would be a logistics growth model of the form $$f(x) = \frac{a}{1+b e^{-k x}}, k>0$$ where $a$ is the maximum quality and $\frac{a}{1+b}$ is the inital value for a pair of headphones costing $0.
The function would probably need to be adjusted where I set a pair of headphones costing $c$ dollars would give evaluate to $f(c) = 1$ and then $f(x)$ would give the quality as more of a "percentage" of quality compared to the one costing $c$ dollars. I could also start by setting $150 as the point of diminishing returns, which would mean
$$f''(150)=0$$
The problem with that is I want to find what the point of diminishing returns is, so this doesn't really address my motivation for doing this.
I don't have any data that I have found to try doing logistic regression, and I am not too knowledgeable in regards to headphone quality. Ideally someone else has already modeled this so I can use that as a starting point.