<p>The answers to these questions is &quot;no, yes, no&quot;.</p>
<p>Note that if you have increasing RTS like in your example and fixed prices, it is likely that the firm can always increase their profits by doubling input use, thus there won't be a profit maximizing output.</p>

CC BY-SA 4.0
2022-10-09T20:03:12.743Z
Yes, there won't be any profit maximizing solution. So in that case, how does the firm pick which quantity and inputs to choose? Can you please explain this in your answer? I learnt that a competitive firm has zero profit in the long run, so this seems contradictory or at least, confusing.
Rick_Morty
CC BY-SA 4.0
2022-10-09T23:08:39.830Z
There is no positive answer to that question, which is why they usually assume CRS or DRS.
Giskard
CC BY-SA 4.0
2022-10-09T19:57:57.437Z
2022-10-09T19:57:57.437Z
Giskard
<p>If the price of the output good is positive, there can be no profit-maximizing profit plan for the production function <span class="math-container">$q(l,k) = l^2k$</span>, which means that this production function is not compatible with perfect competition.</p>
<p>However, it is in principle possible to have a production function with increasing returns to scale under perfect competition- provided producing positive amounts is never optimal. Here is an example: Let the output price be <span class="math-container">$p=1$</span>, the input price be <span class="math-container">$w=1$</span> and the production function be given by <span class="math-container">$q(l)=l-\log(1+l)$</span>. This production function is strictly increasing and strictly convex, which means it also has increasing returns to scale. But the slope is always smaller than <span class="math-container">$1$</span>, so the unique optimal production plan is to produce zero output using zero input.</p>
<p>More importantly, a production function needs to have neither decreasing, constant, or increasing returns to scale if one takes these notions to be global.</p>

CC BY-SA 4.0
2022-10-09T22:02:43.527Z
2022-10-09T22:02:43.527Z
Michael Greinecker
<p>In a perfectly competitive market, is it necessary for the production functions to have decreasing returns to scale? Can it have a constant or increasing returns to scale production function like <span class="math-container">$q(l,k) = l^2k$</span>? Further, is homogeneity of the production function a necessity?</p>

<microeconomics>
Returns to scale in perfect competition
Rick_Morty
# Returns to scale in perfect competition

(Asked by: username_0 on 2022-10-09)

In a perfectly competitive market, is it necessary for the production functions to have decreasing returns to scale? Can it have a constant or increasing returns to scale production function like $q(l,k) = l^2k$? Further, is homogeneity of the production function a necessity?

**Tags:** microeconomics

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## 2 answers

### Answer by username_2 on 2022-10-09. Score: 4 | Accepted answer

If the price of the output good is positive, there can be no profit-maximizing profit plan for the production function $q(l,k) = l^2k$, which means that this production function is not compatible with perfect competition.

However, it is in principle possible to have a production function with increasing returns to scale under perfect competition- provided producing positive amounts is never optimal. Here is an example: Let the output price be $p=1$, the input price be $w=1$ and the production function be given by $q(l)=l-\log(1+l)$. This production function is strictly increasing and strictly convex, which means it also has increasing returns to scale. But the slope is always smaller than $1$, so the unique optimal production plan is to produce zero output using zero input.

More importantly, a production function needs to have neither decreasing, constant, or increasing returns to scale if one takes these notions to be global.


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### Answer by username_1 on 2022-10-09. Score: 2

The answers to these questions is "no, yes, no".

Note that if you have increasing RTS like in your example and fixed prices, it is likely that the firm can always increase their profits by doubling input use, thus there won't be a profit maximizing output.

> **Comments:**
> **username_0**: Yes, there won't be any profit maximizing solution. So in that case, how does the firm pick which quantity and inputs to choose? Can you please explain this in your answer? I learnt that a competitive firm has zero profit in the long run, so this seems contradictory or at least, confusing.  
> **username_1**: There is no positive answer to that question, which is why they usually assume CRS or DRS.  

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