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

• So now you want to do a comparative static: take the derivative of the FOC with respect to A but knowing that $p=p(A)$ so you have to differentiate p with respect to A. Then solve for the partial of p wrt A and you have your answer. I can write it up if you give it a shot and get stuck. – VCG Sep 11 '16 at 19:17
• @VCG - I edited with my line of thought and I am still stuck. I would greatly appreciate any other guidance you could offer. – Anonymous Sep 11 '16 at 19:51
• @Anonymous, could you edit your post and explicitely add what is your question? Essentially what you write here in the comment. – clem steredenn Sep 12 '16 at 6:38

Given how the problem is set up, I think they want you to take A as given when you choose price. If that is the case, then you can find the comparative static $\partial P / \partial A$
We can think of p in the Foc as being a function of all the other parameters: $A\alpha-2A\beta P(A,\alpha,\beta,c)-A\beta c=0$
Now differentiate this wrt A: $\alpha-2\beta P(A,\alpha,\beta,c) -2A\beta (\partial P / \partial A)-\beta c=0$
Now you can solve for $\partial P / \partial A$