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

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

CC BY-SA 4.0
2019-07-08T14:07:44.260Z
2019-07-08T14:07:44.260Z
Regio
<p>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
<span class="math-container">\begin{equation}
\max_{p} \ p[1-F(p)]
\end{equation}</span>
when marginal costs are 0 and where <span class="math-container">$F$</span> is the CDF of the consumers' valuations. 
This yields the solution
<span class="math-container">\begin{equation}
p^*=\frac{1-F(p^*)}{f(p^*)}.
\end{equation}</span>
<span class="math-container">$\textbf{Question}$</span>: How would one write the value function (optimal profit function) of the monopolist in this case, given that the solution is implicitly (?) defined?</p>

<p>Can one just say that it is
<span class="math-container">\begin{equation}
\text{optimal profit function}=\frac{[1-F(p^*)]^2}{f(p^*)}
\end{equation}</span> 
or is this incorrect?</p>

<p>Many thanks.</p>

<microeconomics><game-theory><monopoly><profit-maximization><producer-theory>
Value Function For Durable-Good Monopolist with General Distribution
Hal_Incandenza
# Value Function For Durable-Good Monopolist with General Distribution

(Asked by: username_0 on 2019-07-06)

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.

**Tags:** microeconomics, game-theory, monopoly, profit-maximization, producer-theory


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## 1 answers

### Answer by username_1 on 2019-07-08. Score: 1 | Accepted answer

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.


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