<p>Solution: Your approach is right but it wont work in this situation as there are certain avoidable fixed costs associated with your cost function.</p>
<p>we are given the demand function as, <span class="math-container">$p^d(q)=100-q$</span> (i.e., <span class="math-container">$q^d(p)=100-p$</span>)and the cost function as
<span class="math-container">$c_i(q_i)= (10+2q_i)\mathbb{I}_{(q_i \in (0,\infty))}$</span> which tells us that there are avoidable fixed costs of 10.</p>
<p>Now we can setup the firm's optimization problem,
<span class="math-container">$$\max_{\{q_i\}} \hspace{4mm} pq_i-(10+2q_i) \mathbb{I}_{(q_i \in (0,\infty))}$$</span></p>
<p>if we solve the above problem, we will get the supply as a function of price(p) for firm i,</p>
<p><span class="math-container">$$q_i^s(p)= \cases{\{0\}, &amp; \text{$p \le 2$}\\ \infty, &amp; \text{otherwise} } $$</span></p>
<p>Notice that the markets wont clear for any price <span class="math-container">$p \le 2$</span> as the firms wont be willing to produce anything and for price <span class="math-container">$p &gt;2$</span>,the firms are ready to produce <span class="math-container">$+\infty$</span> amounts of quantity, Thus even in this case, the market wont clear.</p>
<p>Hence, There doesn't exist a competitive equilibrium.</p>

CC BY-SA 4.0
2024-04-17T08:28:17.850Z
The cost functions precise definition makes it clear that these are quasi-fixed costs, not fixed costs, which would occur even when $q=0$.
Giskard
CC BY-SA 4.0
2024-04-17T08:35:17.673Z
Yeah you are right. I should have used the term "avoidable fixed costs", so do you also mean the same thing by quasi-fixed costs?
SGP
CC BY-SA 4.0
2024-04-17T08:40:19.840Z
Yes. There is one more occurence "*which tells us that there are **fixed costs** of 10*"
Giskard
CC BY-SA 4.0
2024-04-17T08:46:05.927Z
Yeah thanks for your inputs @Giskard
SGP
CC BY-SA 4.0
2024-04-17T05:06:52.823Z
2024-04-17T08:45:01.487Z
2024-04-17T08:45:01.487Z
SGP
<p>Consider a market demand function p = 100 − q, where p is
market price and q is aggregate demand. There is a large number
of firms with identical cost functions</p>
<pre><code>ci(qi) = 10 + 2qi, if qi &gt; 0
         0, otherwise
</code></pre>
<p>Solving the cost function, I am getting that Marginal Cost is equal to 2. For a competitive equilibrium price, we set Price equal to Marginal Cost. So, price = 2. But the answer says that competitive equilibrium price is not well defined.</p>
<p>Could anyone please explain this to me? Why is it not well defined?</p>

CC BY-SA 4.0
2024-04-16T18:17:56.633Z
Competitive Equilibrium does not exist in this case. This is because at $p=2$, firm will choose to produce $0$ and that is not equal to the demand which is $98$.
Amit
CC BY-SA 4.0
2024-04-16T18:24:29.537Z
Can you please explain why would the firm choose to produce 0 units at price = 2?
Shreya
CC BY-SA 4.0
2024-04-16T20:01:23.910Z
@Shreya because by producing $98$ (or any non zero quantity), it would have a profit of $-10$, which is worse than having a profit of $0$ which is achieved by producing $0$.
Nicolas Torres
<competitive-equilibrium><perfect-competition>
Why is the competitive equilibrium price not well defined in this question?
Shreya
# Why is the competitive equilibrium price not well defined in this question?

(Asked by: username_0 on 2024-04-16)

Consider a market demand function p = 100 − q, where p is
market price and q is aggregate demand. There is a large number
of firms with identical cost functions
ci(qi) = 10 + 2qi, if qi > 0
         0, otherwise

Solving the cost function, I am getting that Marginal Cost is equal to 2. For a competitive equilibrium price, we set Price equal to Marginal Cost. So, price = 2. But the answer says that competitive equilibrium price is not well defined.
Could anyone please explain this to me? Why is it not well defined?

**Tags:** competitive-equilibrium, perfect-competition

> **Comments:**
> **username_1**: Competitive Equilibrium does not exist in this case. This is because at $p=2$, firm will choose to produce $0$ and that is not equal to the demand which is $98$.  
> **username_0**: Can you please explain why would the firm choose to produce 0 units at price = 2?  
> **username_2**: @username_0 because by producing $98$ (or any non zero quantity), it would have a profit of $-10$, which is worse than having a profit of $0$ which is achieved by producing $0$.  

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## 1 answers

### Answer by username_3 on 2024-04-17. Score: 2 | Accepted answer

Solution: Your approach is right but it wont work in this situation as there are certain avoidable fixed costs associated with your cost function.

we are given the demand function as, $p^d(q)=100-q$ (i.e., $q^d(p)=100-p$)and the cost function as
$c\_i(q\_i)= (10+2q\_i)\mathbb{I}\_{(q\_i \in (0,\infty))}$ which tells us that there are avoidable fixed costs of 10.

Now we can setup the firm's optimization problem,
$$\max\_{\{q\_i\}} \hspace{4mm} pq\_i-(10+2q\_i) \mathbb{I}\_{(q\_i \in (0,\infty))}$$

if we solve the above problem, we will get the supply as a function of price(p) for firm i,

$$q\_i^s(p)= \cases{\{0\}, & \text{$p \le 2$}\\ \infty, & \text{otherwise} } $$

Notice that the markets wont clear for any price $p \le 2$ as the firms wont be willing to produce anything and for price $p >2$,the firms are ready to produce $+\infty$ amounts of quantity, Thus even in this case, the market wont clear.

Hence, There doesn't exist a competitive equilibrium.

> **Comments:**
> **username_4**: The cost functions precise definition makes it clear that these are quasi-fixed costs, not fixed costs, which would occur even when $q=0$.  
> **username_3**: Yeah you are right. I should have used the term "avoidable fixed costs", so do you also mean the same thing by quasi-fixed costs?  
> **username_4**: Yes. There is one more occurence "*which tells us that there are **fixed costs** of 10*"  
> **username_3**: Yeah thanks for your inputs @username_4  

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