Perfect competition: Finding short run equilibrium price?

Question

I am trying to self teach myself some Economics, I am using an old textbook given to me by a friend, which does not contain an answer key. I have run into the following problem and I was wondering if anyone could guide me (I don't want the answers, I just want to understand the method).

The firms in a perfectly competitive industry have a cost function given by:

$c(w_{1},w_{2},Y)= Y^{2} (w_{1}, w_{2})^{\frac{1}{2}} + 8$

Where $(w_{1},w_{2})=(4,25)$

The market demand in this industry is $D(p)=40-p$

The number of firms in this industry are $30$, I'm not sure if this ties in with this half of the question or not.

Anyways, what I have done so far is computed the total cost function:

$TC= 10Y^{2} +8$

From this point, how can I proceed onwards to find the short run equilibrium price?

I would appreciate it if anyone could offer some advice.

Thanks.

Answer 1

These kinds of questions depend heavily on the market structure that you assume. Most importantly, when firms have (some degree of) market power, they usually take the demand function as given and try to solve something of the kind

$$ \max_p D(p)\cdot (p-c(D(p)))$$

However here, when there is no market power, firms take the price as given and solve for the optimal quantity of production $q$:

$$ \max_q q\cdot (p-c(Q))$$

Then, we get the supply of any single firm, given the price $S(p)$ from solving this problem. To clear the perfectly competitive market, we remember that there are $30$ firms and solve:

$$p: D(p) - 30\cdot S(p) = 0$$

Answer 2

I believe you will need a Demand and Supply Curve to find Equilibrium Price, i.e. the intersection of both.