# Conditions on turnover as a function of number of sales to be a concave function

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?