The optimal price for a demand curve with a steep slope

Question

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.

Answer 1

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.