<p>It says</p>

<blockquote>
  <p>If none of the companies with higher costs will gain positive profits</p>
</blockquote>

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

<ul>
<li>So with 0 profits, this means that <em>average costs</em> = <em>price</em> for the high-cost firms.</li>
</ul>

<p>First, you need to derive the equilibrium price, using that, for the equilibrium price, <em>demand</em> = <em>supply</em>.</p>

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

<p>A firm's supply is the solution to a maximization problem:</p>

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

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

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

<p>Then, our equilibrium condition is simply</p>

<p>$$ S(p, n) = Q(p) $$</p>

<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>

<p>$$ p = C_h(q)$$</p>

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

<p>$$ p = C_h(q_h(p))$$</p>

<p>Now we have two equations and two variables, you are all set!</p>

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

CC BY-SA 3.0
2015-04-09T18:07:13.547Z
do not get this maximization problem could you explain this further? why not just calculate derivative of cost function?
Krowskir
CC BY-SA 3.0
2015-04-09T18:50:01.047Z
@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
CC BY-SA 3.0
2015-04-09T13:37:27.653Z
2015-04-09T13:37:27.653Z
FooBar
<p>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?</p>

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

<microeconomics><competition><cost>
Perfect competition profits question
Krowskir
# Perfect competition profits question

(Asked by: username_0 on 2015-04-09)

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.

**Tags:** microeconomics, competition, cost


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## 1 answers

### Answer by username_1 on 2015-04-09. Score: 1 | Accepted answer

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!

$$ S(p, n) = Q(p) \\
p = C\_h(q\_h(p))$$

> **Comments:**
> **username_0**: do not get this maximization problem could you explain this further? why not just calculate derivative of cost function?  
> **username_1**: @username_0 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 $(p-c(q))*q$  

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