# Perfect competition profits question

Market is supplied by: 50 competitive companies all of them have relatively low costs given by an equation $C_l(q)=350+2q+q^2$ and by n companies of higher costs given by an equation $C_h(q)=400+2q+q^2$. Market demand is given by $Q=250-10P$. If none of the companies with higher costs will gain positive profits, how large is n? What is the profit of companies with low costs?

I have no idea how to solve this question, i do not know how to start.

It says

If none of the companies with higher costs will gain positive profits

I assume this means they're making 0 profits, while strictly speaking it could mean that they make negative profits and leave the market.

• So with 0 profits, this means that average costs = price for the high-cost firms.

First, you need to derive the equilibrium price, using that, for the equilibrium price, demand = supply.

$$\text{total demand} = Q(p) = S(p) = \text{total supply}$$

A firm's supply is the solution to a maximization problem:

$$q_h(p) = \arg\max_q (p-C_h(q))q \\ q_l(p) = \arg\max_q (p-C_l(q))q$$

Then, total supply given price $p$ and amount of high-cost firms $n$ is

$$S(p, n) = 50\cdot q_l(p) + n \cdot q_h(p)$$

Then, our equilibrium condition is simply

$$S(p, n) = Q(p)$$

This is under defined, as we have one equation and two variables $(p, n)$ to solve for. But wait, we know more! 0 profits for the high-cost firms means

$$p = C_h(q)$$

To get rid of the additional variable $q$, just use the solution to the firm's problem:

$$p = C_h(q_h(p))$$

Now we have two equations and two variables, you are all set!

$$S(p, n) = Q(p) \\ p = C_h(q_h(p))$$

• do not get this maximization problem could you explain this further? why not just calculate derivative of cost function? – Krowskir Apr 9 '15 at 18:07
• @Krowskir that's the general formula, perhaps there is a different one, IO is not my strength. The problem is: Taking prices as given, maximize profits $(p-c(q))*q$ – FooBar Apr 9 '15 at 18:50