<p>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 $\partial P / \partial A$</p>

<p>We can think of p in the Foc as being a function of all the other parameters: $A\alpha-2A\beta P(A,\alpha,\beta,c)-A\beta c=0$</p>

<p>Now differentiate this wrt A: $\alpha-2\beta P(A,\alpha,\beta,c) -2A\beta (\partial P / \partial A)-\beta c=0$</p>

<p>Now you can solve for $\partial P / \partial A$</p>

CC BY-SA 3.0
2016-09-11T20:07:25.887Z
2016-09-11T20:07:25.887Z
VCG
<blockquote>
<p>Below are three different demand curves (i) - (iii), which depend on advertising (A).</p>
<p>(i) Q(P,A) = A <span class="math-container">$\times$</span> (<span class="math-container">$\alpha$</span> - <span class="math-container">$\beta$</span>P), where <span class="math-container">$\alpha$</span>, <span class="math-container">$\beta$</span> &gt; 0</p>
<p>(ii) Q(P, A) = <span class="math-container">$\alpha$</span> + A - <span class="math-container">$\beta$</span>P</p>
<p>(iii) Q(P,A) = <span class="math-container">$\alpha$</span> - <span class="math-container">$\beta$</span>(B - A)P</p>
<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)?</p>
</blockquote>
<p>Assuming constant marginal cost of <span class="math-container">$c$</span> and that the firm is a monopoly:</p>
<p><span class="math-container">$$\max (\pi(\text{A}, \text{P})) = (\text{P} - c)\text{Q} - \text{A}$$</span></p>
<p>WRT (i),</p>
<p><span class="math-container">$$\max (\pi(\text{A}, \text{P})) = \text{A}\alpha\text{P} - \text{A}\beta\text{P}^2 - \text{A}\alpha c - \text{A}\beta\text{P}c - \text{A}$$</span></p>
<p>For that, we calculate</p>
<p><span class="math-container">$$\frac{d\pi}{d\text{P}} = \text{A}\alpha - 2\text{A}\beta\text{P} - \text{A}\beta c = 0$$</span></p>
<p>and</p>
<p><span class="math-container">$$\frac{d\pi}{d\text{A}} = \alpha\text{P} - \beta\text{P}^2 - \alpha c - \beta \text{P}c - 1 = 0$$</span></p>
<p>We get to</p>
<p><span class="math-container">$$\frac{e\text{A}}{-e\text{P}} = \frac{\alpha\text{P} - \beta\text{P}^2 - \alpha c - \beta\text{P}c - 1}{-\text{A}\alpha + 2\text{A}\beta\text{P} + \text{A}\beta c}$$</span></p>
<p>Any continued guidance would be greatly appreciated</p>

CC BY-SA 3.0
2016-09-11T19:17:22.253Z
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
CC BY-SA 3.0
2016-09-11T19:51:29.493Z
@VCG - I edited with my line of thought and I am still stuck. I would greatly appreciate any other guidance you could offer.
Anonymous
CC BY-SA 3.0
2016-09-12T06:38:54.087Z
@Anonymous, could you edit your post and explicitely add what is your question? Essentially what you write here in the comment.
clem steredenn
<microeconomics><optimization><advertising>
Optimal Pricing with Advertising
Anonymous
# Optimal Pricing with Advertising

(Asked by: username_0 on 2016-09-11)

> 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

**Tags:** microeconomics, optimization, advertising
> **Comments:**
> **username_1**: 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.  
> **username_0**: @VCG - I edited with my line of thought and I am still stuck. I would greatly appreciate any other guidance you could offer.  
> **username_2**: @Anonymous, could you edit your post and explicitely add what is your question? Essentially what you write here in the comment.  

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## 1 answers

### Answer by username_1 on 2016-09-11. Score: 1 | Accepted answer

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 $\partial P / \partial A$


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