It is known that with a unit mass of consumers, each of whom has a value distributed between 0 and 1, one can think of the monopolist solving \begin{equation} \max_{p} \ p[1-F(p)] \end{equation} when marginal costs are 0 and where $F$ is the CDF of the consumers' valuations. This yields the solution \begin{equation} p^*=\frac{1-F(p^*)}{f(p^*)}. \end{equation} $\textbf{Question}$: How would one write the value function (optimal profit function) of the monopolist in this case, given that the solution is implicitly (?) defined?
Can one just say that it is \begin{equation} \text{optimal profit function}=\frac{[1-F(p^*)]^2}{f(p^*)} \end{equation} or is this incorrect?
Many thanks.