Below are three different demand curves (i) - (iii), which depend on advertising (A).
(i) Q(P,A) = A $\times$ ($\alpha$ - $\beta$P), where $\alpha$, $\beta$ > 0
(ii) Q(P, A) = $\alpha$ + A - $\beta$P
(iii) Q(P,A) = $\alpha$ - $\beta$(B - A)P
For each of the cases, what happens to the optimal price as advertising increases? How does your answer depend on the behavior of the cost function C(Q)?
Assuming constant marginal cost of $c$ and that the firm is a monopoly:
$$\max (\pi(\text{A}, \text{P})) = (\text{P} - c)\text{Q} - \text{A}$$
WRT (i),
$$\max (\pi(\text{A}, \text{P})) = \text{A}\alpha\text{P} - \text{A}\beta\text{P}^2 - \text{A}\alpha c - \text{A}\beta\text{P}c - \text{A}$$
For that, we calculate
$$\frac{d\pi}{d\text{P}} = \text{A}\alpha - 2\text{A}\beta\text{P} - \text{A}\beta c = 0$$
and
$$\frac{d\pi}{d\text{A}} = \alpha\text{P} - \beta\text{P}^2 - \alpha c - \beta \text{P}c - 1 = 0$$
We get to
$$\frac{e\text{A}}{-e\text{P}} = \frac{\alpha\text{P} - \beta\text{P}^2 - \alpha c - \beta\text{P}c - 1}{-\text{A}\alpha + 2\text{A}\beta\text{P} + \text{A}\beta c}$$
Any continued guidance would be greatly appreciated