Value Function For Durable-Good Monopolist with General Distribution

Question

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.

Answer 1

You probably want to say that the optimal profit is $\pi=\frac{[1-F(p^*)]^2}{f(p^*)}$ where $F(x)$ is a given probability distribution, $f(x)$ is its density and $p^*$ satisfies $p^*=\frac{1-F(p^*)}{f(p^*)}$.

If you are dealing with a general $F$ function, you may want to state the conditions that guarantee that $p^*$ exists and it is unique. If $F$ is uniform or some other well-known distribution, however, you should probably find in closed form the value of $p^*$ and give an actual number as optimal profit.