Optimal Pricing with Advertising

Question

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

Answer 1

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 $\partial P / \partial A$