<p>Suppose that a production set <span class="math-container">$Y \subset \mathbb R^I$</span> has the following two properties:</p>
<ol>
<li>Constant returns to scale: <span class="math-container">$\forall y \in Y, \alpha \geq 0$</span>, we have <span class="math-container">$\alpha y \in Y$</span>.</li>
<li>Separability: For some <span class="math-container">$\widetilde I \subset I$</span>, there exist sets <span class="math-container">$Y^i \subset \mathbb R^I$</span> for all <span class="math-container">$i \in \widetilde I$</span> such that (a) for all <span class="math-container">$y^i \in Y^i$</span>, <span class="math-container">$y^i_{j \neq i} \leq 0$</span> and (b) <span class="math-container">$Y = \{ \sum_{i \in \widetilde I} y^i | \forall i\in \widetilde I,  y^i \in Y^i \}$</span>.</li>
</ol>
<p>Does this imply that one may take the sets <span class="math-container">$Y^i$</span> to be constant returns to scale? Formally, do there exist sets <span class="math-container">$\widetilde Y^i \subset \mathbb R^I$</span> for all <span class="math-container">$i \in \widetilde I$</span> such that</p>
<ul>
<li>for all <span class="math-container">$y^i \in \widetilde Y^i$</span>, <span class="math-container">$y^i_{j \neq i} \leq 0$</span>,</li>
<li><span class="math-container">$Y = \{ \sum_{i \in \widetilde I} y^i | \forall i\in \widetilde I,  y^i \in \widetilde Y^i \}$</span>, and</li>
<li>for all <span class="math-container">$y^i \in \widetilde Y^i$</span> and <span class="math-container">$\alpha \geq 0$</span>, we have <span class="math-container">$\alpha y^i \in \widetilde Y^i$</span>?</li>
</ul>

CC BY-SA 4.0
2021-06-23T18:42:15.413Z
With unit elasticity firms and free entry, can't you get each firm having no meaningful returns to scale (they only operate at a single quantity or 0) but aggregate constant returns to scale? My thinking is that when you double the quantity of labor and capital you double the amount of firms and therefore double the output, but each firm does the same old thing.
BKay
CC BY-SA 4.0
2021-06-23T20:33:28.397Z
Thanks for the suggestion. I don't think this works within the formalism described above, though. If what you had in mind is that each firm is its own $i$, then the profile where you e.g. produce double of one firm's quantity is not in $Y$, the aggregate production set. If what you had in mind is that all firms that produce the same type of good should be thought of as producing the same $i$, then that would mean $Y^i$ actually is CRS.
John Sturm
<general-equilibrium><producer-theory><aggregate-production><input-output>
Does aggregate CRS imply firm-level CRS?
John Sturm
# Does aggregate CRS imply firm-level CRS?

(Asked by: username_0 on 2021-06-23)

Suppose that a production set $Y \subset \mathbb R^I$ has the following two properties:

1. Constant returns to scale: $\forall y \in Y, \alpha \geq 0$, we have $\alpha y \in Y$.
2. Separability: For some $\widetilde I \subset I$, there exist sets $Y^i \subset \mathbb R^I$ for all $i \in \widetilde I$ such that (a) for all $y^i \in Y^i$, $y^i\_{j \neq i} \leq 0$ and (b) $Y = \{ \sum\_{i \in \widetilde I} y^i | \forall i\in \widetilde I, y^i \in Y^i \}$.

Does this imply that one may take the sets $Y^i$ to be constant returns to scale? Formally, do there exist sets $\widetilde Y^i \subset \mathbb R^I$ for all $i \in \widetilde I$ such that

* for all $y^i \in \widetilde Y^i$, $y^i\_{j \neq i} \leq 0$,
* $Y = \{ \sum\_{i \in \widetilde I} y^i | \forall i\in \widetilde I, y^i \in \widetilde Y^i \}$, and
* for all $y^i \in \widetilde Y^i$ and $\alpha \geq 0$, we have $\alpha y^i \in \widetilde Y^i$?

**Tags:** general-equilibrium, producer-theory, aggregate-production, input-output
> **Comments:**
> **username_1**: With unit elasticity firms and free entry, can't you get each firm having no meaningful returns to scale (they only operate at a single quantity or 0) but aggregate constant returns to scale? My thinking is that when you double the quantity of labor and capital you double the amount of firms and therefore double the output, but each firm does the same old thing.  
> **username_0**: Thanks for the suggestion. I don't think this works within the formalism described above, though. If what you had in mind is that each firm is its own $i$, then the profile where you e.g. produce double of one firm's quantity is not in $Y$, the aggregate production set. If what you had in mind is that all firms that produce the same type of good should be thought of as producing the same $i$, then that would mean $Y^i$ actually is CRS.  

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## 0 answers

