Some Highly Paid Consultants™ have claimed that this is a good model of our price elasticity. I haven't been able to find any reference to it and the person claiming it can't tell me why it was chosen. I was thinking it might be the solution to some ODE but I haven't had any luck. Differentiating didn't seem to add any insight either. It has a nice sigmoid shape but that's about it.
$$b(p) = e^{\alpha(p-p_0)}\frac{e^{\alpha p_0} + e^{\beta + \gamma p_0}}{e^{\alpha p} + e^{\beta + \gamma p_c}}$$
Does anyone know if this is a well-known function?
A sigmoid demand function like this effectively has 3 areas; (1) a low price, high volume area; (2) a decreasing volume area in the middle, and (3) a high price low volume area. What this means is that elasticity between 2 prices that makes up the middle area. For the lower price area, demand is saturated, so lowering price does not bring in more demand. For the higher price area, demand is constrained, so there is a core group of customers who will always buy from you (although, this floor is often zero, like in the sigmoid below).
Is it legit? I have seen theoretical works that use this as one model for elasticity. I have seen (unsuccessful) attempt to fit this to data. The dirty secret is that calculating elasticity from real data is really, really hard in most cases, so a lot of people substitute a theoretical model for elasticity and pretend it's data-driven.