<p>It looks like your question is implying all the firms are identical, so they have the same production function.</p>
<p>Then each firm must provide the same amount, because they are the same internally (in their production function) and face the same price in perfect competition (so face the same external environment too).</p>
<p>Then we can denote each <span class="math-container">$q_i$</span> simply as <span class="math-container">$q$</span> because they’re all the same.</p>
<p>Suppose you have N firms.</p>
<p>Then <span class="math-container">$Q = \Sigma^N q = q + q + q$</span> (N times) <span class="math-container">$= N*q$</span>.</p>
<p>Use that to solve for <span class="math-container">$q$</span>.</p>

CC BY-SA 4.0
2022-10-08T16:09:19.560Z
How do you know that the firms have the same production function? I know the marginal costs are equal, but that doesn't mean the production functions will be the same I suppose. Let me ask something: **Without further information about the production capabilities, can we say that $Q = \frac{\alpha - c}{\beta}$ will be reached? Can the total supply go above that? Or below that?**
Rick_Morty
CC BY-SA 4.0
2022-10-08T17:27:07.317Z
Q must be reached in equilibrium, because in equilibrium the market clears and demand = supply. While I don’t know for sure that all the firms are identical, that is usually assumed in these types of questions at this level, especially if the marginal costs are equal.
BB King
CC BY-SA 4.0
2022-10-08T18:31:53.447Z
Alright, one final question: Since the profits will be zero, how do you know that each will produce $Q/n$ units of output? For them, producing any unit will be the same (since profit is always $0$), so what guarantees that each will produce the exact same amount?
Rick_Morty
CC BY-SA 4.0
2022-10-08T14:44:52.457Z
2022-10-08T14:44:52.457Z
BB King
<p>Suppose the market demand is <span class="math-container">$P(Q) = \alpha - \beta(Q)$</span> where <span class="math-container">$Q = \sum q_1$</span>. Variable <span class="math-container">$q_i$</span> denotes the output of the <span class="math-container">$i$</span>th firm and <span class="math-container">$Q$</span> is the total output. The marginal cost for each firm is <span class="math-container">$c$</span>.</p>
<p><strong>What's the quantity that each firm will set for themselves in a perfectly competitive market?</strong></p>
<p>I reckon the (market) price will be <span class="math-container">$c$</span> which means the overall market size will be given by <span class="math-container">$Q = \frac{\alpha - c}{\beta}$</span>. What next?</p>

CC BY-SA 4.0
2022-10-19T21:16:14.273Z
"What next?", indeed - a great question! Show us your attempt (all the way to the answer) for problems that appear to be homework.
RegressForward
CC BY-SA 4.0
2022-10-20T07:41:13.563Z
@RegressForward The first bit cracked me up but no, I didn't mean it that way. It isn't a homework; I just wanted to see if the firms produce all of $Q = (\alpha-c) / \beta$ regardless of their cost functions and productions. The current answer assumes the same cost function and nothing about the production function, so it back when I posted it, it didn't clear my doubt. Now I have learnt that the production function and cost functions mean the same (since we can derive one from the other) and hence, identical cost functions will mean identical production functions.
Rick_Morty
CC BY-SA 4.0
2022-10-20T07:43:44.200Z
@RegressForward As for attaining the full quantity $Q = (\alpha - c) / \beta$, the firms will do again do so by definition of equilibrium. This the current answerer mentions as a comment if you see. That said, I agree that I should have been more clear with what I wanted as an answer.
Rick_Morty
<microeconomics><markets><perfect-competition>
Perfect competition allocations
Rick_Morty
# Perfect competition allocations

(Asked by: username_0 on 2022-10-07)

Suppose the market demand is $P(Q) = \alpha - \beta(Q)$ where $Q = \sum q\_1$. Variable $q\_i$ denotes the output of the $i$th firm and $Q$ is the total output. The marginal cost for each firm is $c$.

**What's the quantity that each firm will set for themselves in a perfectly competitive market?**

I reckon the (market) price will be $c$ which means the overall market size will be given by $Q = \frac{\alpha - c}{\beta}$. What next?

**Tags:** microeconomics, markets, perfect-competition

> **Comments:**
> **username_1**: "What next?", indeed - a great question! Show us your attempt (all the way to the answer) for problems that appear to be homework.  
> **username_0**: @username_1 The first bit cracked me up but no, I didn't mean it that way. It isn't a homework; I just wanted to see if the firms produce all of $Q = (\alpha-c) / \beta$ regardless of their cost functions and productions. The current answer assumes the same cost function and nothing about the production function, so it back when I posted it, it didn't clear my doubt. Now I have learnt that the production function and cost functions mean the same (since we can derive one from the other) and hence, identical cost functions will mean identical production functions.  
> **username_0**: @username_1 As for attaining the full quantity $Q = (\alpha - c) / \beta$, the firms will do again do so by definition of equilibrium. This the current answerer mentions as a comment if you see. That said, I agree that I should have been more clear with what I wanted as an answer.  

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## 1 answers

### Answer by username_2 on 2022-10-08. Score: 4 | Accepted answer

It looks like your question is implying all the firms are identical, so they have the same production function.

Then each firm must provide the same amount, because they are the same internally (in their production function) and face the same price in perfect competition (so face the same external environment too).

Then we can denote each $q\_i$ simply as $q$ because they’re all the same.

Suppose you have N firms.

Then $Q = \Sigma^N q = q + q + q$ (N times) $= N\*q$.

Use that to solve for $q$.

> **Comments:**
> **username_0**: How do you know that the firms have the same production function? I know the marginal costs are equal, but that doesn't mean the production functions will be the same I suppose. Let me ask something: **Without further information about the production capabilities, can we say that $Q = \frac{\alpha - c}{\beta}$ will be reached? Can the total supply go above that? Or below that?**  
> **username_2**: Q must be reached in equilibrium, because in equilibrium the market clears and demand = supply. While I don’t know for sure that all the firms are identical, that is usually assumed in these types of questions at this level, especially if the marginal costs are equal.  
> **username_0**: Alright, one final question: Since the profits will be zero, how do you know that each will produce $Q/n$ units of output? For them, producing any unit will be the same (since profit is always $0$), so what guarantees that each will produce the exact same amount?  

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