Let a demandfunction be defined as $D(p)=B-bp$, where $b,B>0$. A firm has some production cost, $c$, and can set the price $p$ under the constrain given by the Demand.
- What is the optimal price?
- Is the priceelasticity higher or lower than $1$ (absolute value)
- How does price depend on $b$?
How would one do such a maximization problem?
My attempt
The elasticity would be given by
$\frac{p}{q}\cdot(-b)$ since $D'(p)=-b$
Then the problem is \begin{equation*} max_p(p-c)D'(p) \end{equation*}
which then follows that
\begin{equation*} D(p)+(p-c)D'(p)=0 \Leftrightarrow \end{equation*}
\begin{equation*} D(p)\left ( 1+(p-c)\frac{\epsilon}{p} \right )=0\Leftrightarrow \end{equation*}
and then the optimal price $p^*$ \begin{equation*} p^*=\frac{\epsilon}{1+\epsilon}c=\frac{\left | \epsilon \right |}{\left | \epsilon \right |-1}c=\frac{\left | \frac{-bp}{q} \right |}{\left | \frac{-bp}{q} \right |-1}c=\frac{c\left | b \right |\left | p \right |}{\left | b\right |\left |p \right |-\left | q \right |} \end{equation*}
and then we notice that
\begin{equation*} \left | \epsilon \right |>1 \end{equation*}
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I feel like this is not correct nor the right approach. Help appreciated.