I am trying to retrive the law of supply and demand from game theory. I don't understand the result.
Suppose we have a probability $p$ to sell a good at price $q$. I can calculate $p$ as the fraction of people $N(t)$ who will buy the good at the price $q$ over the whole number of people who can buy $N_0$ (the size of market) as
$$ p=\frac{N(q)}{N_0} $$
if the probability $p$ depends only on price difference, then
$$N(q)=N_0 e^{-\frac{q}{\theta}}$$
where $\theta$ is a parameter and rapresent the elasticity of the demand. So we have that
$$ p=e^{-\frac{q}{\theta}} $$
The expected value of selling a good at price $q$ is
$$ EV=qe^{-\frac{q}{\theta}} -c $$ where $c$ is the cost of good.
Now I differentiate respect $q$ and solve the condition $\frac{\partial EV}{\partial q}=0$ and I find that
$$ q=\theta $$
I can understand that the optimal price to sell is not related to the cost of good, but why I have no relation with the $N_0$? Why the optimal price does not depend of the size of market $N_0$?