I am stuck on the following question:
Consider a profit maximizing monopoly. The demand for the monopoly's product is given by $Q=\ln(a-bP)$ and its cost function is $C(Q)=ce^{Q}$, where the parameters $a$,$b$, and $c$ are all positive. Let $Q^{*}$ be the monopoly's optimal (profit-maximizing) output.
Derive an expression for $\frac{\partial Q^{*}}{\partial a} $ and determine its sign.
I know that profit ($\pi$) is $\pi=R(Q)-C(Q)$ (revenue less costs) where $R(Q)=PQ$. I need to find the max of $\pi(Q)$. First I need to write the demand function in terms of $Q$:
$e^{Q}=a-bP$ $P=\frac{a-e^{Q}}{b} $
Now I can derive the revenue function,
$R=PQ=Q\left(\frac{a-e^{Q}}{b}\right) $
and the profit function:
$\pi(Q)=Q\left(\frac{a-e^{Q}}{b}\right)-ce^{Q}$
I now need to find where
\pi'(Q)=0
$\frac{a-e^{Q}(bc+Q+1)}{b} =0$
I am unable to find the zeros. Perhaps I have not taken the right steps. Once I have found $Q^{*}$ (which is just the maximum point of $\pi (Q)$) I understand that I just take the partial derivative. Thank you for any help.