Suppose I have a customer base of size 100. An arbitrary customer has a private valuation for my product, which shall be represented by the random variable $X$. (Suppose $X$ takes on values between $0$ and $1$.) Now I set a selling price p. Then I will sell
w(p)=100*P(X>p)
products. My question is under what conditions on X (apart from finite support) my turnover as a function of total number of sales is concave. Is there any resource on this question?
Example: in the basic case where $X$ is uniform, we have that $w^{-1}(S)=1-S/100$. I.e. to have sales $S=25$, we need to set a price $p=w^{-1}(25)=0.75$. Now turnover (number of sales times selling price per product) as a function of the number of sales is $S*w^{-1}(S)$, which is a concave function as can be seen easily.
Now what about other distributions of X? Is this a known problem? Is it easy to specify a bit more general sufficient condition for turnover to be concave in total number of sales?
