<ol>
<li>That portion of answer about decreasing price is not generally correct as a is both in numerator and denominator and you don't know what <span class="math-container">$c$</span> is. You can answer the question about what happens to <span class="math-container">$p$</span> as <span class="math-container">$a$</span> becomes arbitrary high using limits. Taking your solution for optimum price and taking limit as <span class="math-container">$a$</span> goes to infinity gives:</li>
</ol>
<p><span class="math-container">$$\lim_{a \to \infty} \left(\frac{A+ac}{2a}\right)=\frac{c}{2}$$</span></p>
<p>(here we had to use L'Hôpital's rule as the limit was in indeterminate form of <span class="math-container">$\infty/\infty$</span>). So if <span class="math-container">$a$</span> is arbitrarily high the price will be half of <span class="math-container">$c$</span></p>
<ol start="2">
<li><p>You do not mention the whole problem is so it is hard to be precise here (you say <span class="math-container">$c$</span> is cost but <span class="math-container">$c$</span> is usually <em>marginal</em> costs in these problems), regardless optimum price can be below costs that would just mean that firm would eventually exit the market and stop providing the service. That is because the demand is simply too much decreased by even small increases in price.</p>
<p>This is essentially situation where people would want some goods or service but only if price is so low that firms are not willing to provide, so the market won't exist. This can happen in reality. You probably had at least once in your life thought that it would be nice if there would be movie/game/book/some service etc that caters to some niche taste but you see that nobody is providing it. Well one of the reasons why nobody is providing it might be that there is no demand at the price where it would actually be economically viable to provide the good or service.</p>
</li>
</ol>

CC BY-SA 4.0
2020-12-10T14:47:29.753Z
Thanks you very much - I get what you're saying.
Matt
CC BY-SA 4.0
2020-12-10T14:41:15.187Z
2020-12-10T14:41:15.187Z
1muflon1
<p>Given the demand function,</p>
<p><span class="math-container">$$D(p)=A-ap$$</span></p>
<p>I've found the optimal price,</p>
<p><span class="math-container">$$p=\frac{A+ac}{2a}$$</span></p>
<p>Where <span class="math-container">$c$</span> is cost and <span class="math-container">$A,a &gt;0$</span>.</p>
<p>My question is how is the optimal price is dependent of <span class="math-container">$a$</span> (1) - what will happen with the optimal price when <span class="math-container">$a$</span> is arbitrarily high? (2)</p>
<p><strong>(1): My answer</strong> is that the optimal price will decrease with an increasing <span class="math-container">$a$</span>. But what happens when <span class="math-container">$a$</span> is arbitrarily high?</p>
<p><strong>(2): My guess</strong> is that the optimal price will near <span class="math-container">$\frac{c}{2}$</span>, but I can't put  a word on why this happens. Why would the firm take an optimal price that is lower than the cost?</p>
<p>It doesn't seem very intuitive for me.</p>

<microeconomics><optimization><demand>
The optimal price for a demand curve with a steep slope
Matt
# The optimal price for a demand curve with a steep slope

(Asked by: username_0 on 2020-12-10)

Given the demand function,

$$D(p)=A-ap$$

I've found the optimal price,

$$p=\frac{A+ac}{2a}$$

Where $c$ is cost and $A,a >0$.

My question is how is the optimal price is dependent of $a$ (1) - what will happen with the optimal price when $a$ is arbitrarily high? (2)

**(1): My answer** is that the optimal price will decrease with an increasing $a$. But what happens when $a$ is arbitrarily high?

**(2): My guess** is that the optimal price will near $\frac{c}{2}$, but I can't put a word on why this happens. Why would the firm take an optimal price that is lower than the cost?

It doesn't seem very intuitive for me.

**Tags:** microeconomics, optimization, demand

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## 1 answers

### Answer by username_1 on 2020-12-10. Score: 0 | Accepted answer

1. That portion of answer about decreasing price is not generally correct as a is both in numerator and denominator and you don't know what $c$ is. You can answer the question about what happens to $p$ as $a$ becomes arbitrary high using limits. Taking your solution for optimum price and taking limit as $a$ goes to infinity gives:

$$\lim\_{a \to \infty} \left(\frac{A+ac}{2a}\right)=\frac{c}{2}$$

(here we had to use L'Hôpital's rule as the limit was in indeterminate form of $\infty/\infty$). So if $a$ is arbitrarily high the price will be half of $c$

2. You do not mention the whole problem is so it is hard to be precise here (you say $c$ is cost but $c$ is usually *marginal* costs in these problems), regardless optimum price can be below costs that would just mean that firm would eventually exit the market and stop providing the service. That is because the demand is simply too much decreased by even small increases in price.

   This is essentially situation where people would want some goods or service but only if price is so low that firms are not willing to provide, so the market won't exist. This can happen in reality. You probably had at least once in your life thought that it would be nice if there would be movie/game/book/some service etc that caters to some niche taste but you see that nobody is providing it. Well one of the reasons why nobody is providing it might be that there is no demand at the price where it would actually be economically viable to provide the good or service.

> **Comments:**
> **username_0**: Thanks you very much - I get what you're saying.  

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