# Finding long run equilibrium price, quantity and number of firms with a linear average cost function

I've been been brushing up on my micoreocnomics lately and I came across a question in Perloff that looked really simple, but for some reason I am struggling to answer:

Assume we are in the long run and each firm in a competitive market has a cost function $$C = q^2$$ and market demand is $$Q = 24 - p$$.

Determine the long-run equilibrium price, quantity per firm, market quantity and number of firms.

Attempt

In the long run, profits are $$0$$, therefor firms should enter until $$MC = AC$$.

$$MC = 2Q$$ and $$AC = Q$$. Setting them equal to another we get:

$$2Q = Q$$, which implies $$Q = 0$$.

However this result seems incorrect to me, could someone please point me in the right direction. I feel like I am getting this weird result because my AC curve is linear whereas it is usually U shaped.

Thanks

• Your analysis is not wrong but I guess you just need to answer in terms of the number of firms (n). That is you can base the equilibrium quantity, price quantity and profit in terms of n. If each firms produces a tiny amount and there are enough of them to satisfy the quantity demanded it might work. – DJJ Mar 23 '19 at 17:13
• The key to the problem I guess is to find the maket supply in terms of the number of firms. – DJJ Mar 23 '19 at 17:16
• @DJJ Could you please elaborate? I don't see how any firms would be in the market, given that a firm is at equilibrium point when q = 0? – Joseph Mar 23 '19 at 17:29
• The supply curve of a firm is derived by p = MC, that is $$q_i = \frac{p}{2}$$. if we assume that all firms are identical that make a market supply of $$Q_s = \frac{np}{2}$$ , where n is the number of firms. Then you can find the equilibrium price, quantity and profit in terms of n. – DJJ Mar 23 '19 at 17:37
• In the long run to have q=0, there must be an infinit amount of firms as well. – DJJ Mar 23 '19 at 17:43