<p>The standard definition is that a production function <span class="math-container">$F(\vec{x})$</span> has constant, increasing, or decreasing returns to scale if <span class="math-container">$F(a\vec{x})$</span> is equal to, greater than, or less than <span class="math-container">$a F(\vec{x})$</span> (respectively) for <em>all</em> positive numbers <span class="math-container">$a$</span> and input vectors <span class="math-container">$\vec{x}$</span>.</p>
<p>It follows that any homogeneous production function has well-defined returns to scale determined by its degree of homogeneity. But a generic non-homogeneous production function will not necessarily have well-defined returns to scale.</p>
<p>Although this isn't standard usage as far as I know, there may be some way to define the &quot;local returns to scale&quot; at a given set of inputs <span class="math-container">$\vec{x}_0$</span> by considering how dilating <span class="math-container">$\vec{x}_0$</span> by an infinitesimal amount <span class="math-container">$1+\epsilon$</span> (with <span class="math-container">$0 &lt; \epsilon \ll 1$</span>) changes the value of <span class="math-container">$F$</span> near the point <span class="math-container">$\vec{x}_0$</span>.</p>

CC BY-SA 4.0
2022-11-21T13:41:43.470Z
Thank you for answering. I guess you mean the condition has to hold for ALL input vectors x? Interesting that you brought the concept of "local returns to scale". This means my cubic would have increasing local returns to scale beyond a certain point.
Nicolas Torres
CC BY-SA 4.0
2022-11-21T14:28:08.703Z
@NicholasTorres Yes, I believe that the standard definition requires that this hold for all input vectors x.
tparker
CC BY-SA 4.0
2022-11-21T00:03:58.783Z
2022-11-21T14:29:18.747Z
2022-11-21T14:29:18.747Z
tparker
<p>I find interesting the proposal of Nicolas Torres of a way, through <span class="math-container">$R$</span>, of measuring returns to scale. But, actually, the problem is that it cannot be a global measure, as can change from point to point of the production function.
But, also the usual  definition cannot give a ‘global’ measure of returns to scale for functions that are not homogeneous.
So,  tparker proposes a ‘local’ measure of returns to scale.</p>
<p>Actually, the suggestion of tparker of a 'local' measure of returns to scale seems similar to a directional derivative, along a direction such that  all  factors increase in the same proportion, that is, which leaves constant the ratios of the products. But  how to compare this derivative with <span class="math-container">$t f(K,L,...)$</span> to establish whether there are constant or increasing or decreasing return to scale?</p>
<p>However, a similar proposal of a local measure of returns to scale exists in the literature.</p>
<p>A local measure of returns to scale is the <em>elasticity of scale</em>, which was introduced by Frisch, and it is a measure of the percentage increase of output due to a unitary percentage increase of all inputs. Formally it is defined as:</p>
<p><span class="math-container">$$\epsilon=\frac{df(\lambda x)}{d \lambda} \frac{\lambda}{f(\lambda x)}$$</span></p>
<p>calculated for <span class="math-container">$\lambda=1$</span>, where <span class="math-container">$\lambda$</span> is a positive parameter.</p>
<p>See the following article of Wikipedia (I’m sorry the following  article is not in English, but I can’t find an equivalent article of Wikipedia in English.)</p>
<p><a href="https://it.wikipedia.org/wiki/Elasticit%C3%A0_di_scala" rel="nofollow noreferrer">https://it.wikipedia.org/wiki/Elasticit%C3%A0_di_scala</a></p>
<p>And there is  a literature about this subject, see for instance:</p>
<p><a href="https://www.jstor.org/stable/40751123" rel="nofollow noreferrer">https://www.jstor.org/stable/40751123</a></p>
<p>Or, more recently,  (2020)</p>
<p><a href="https://link.springer.com/referenceworkentry/10.1007/978-981-10-3450-3_23-1" rel="nofollow noreferrer">https://link.springer.com/referenceworkentry/10.1007/978-981-10-3450-3_23-1</a></p>

CC BY-SA 4.0
2022-11-21T15:19:36.033Z
2022-11-21T15:19:36.033Z
BakerStreet
<p>Most of the production functions encountered in Intermediate Microeconomics are homogeneous (Cobb-Douglas, perfect substitutes, perfect complements).</p>
<p>So their returns to scale are easy to get, comparing their degree of homogeneity to <span class="math-container">$1$</span>.</p>
<p>However, how would you define the returns to scale of non-homogeneous functions such as this one?</p>
<p><span class="math-container">$f(L,K) = \alpha K^2 L^2 - \beta K^3 L^3, \alpha &gt;0, \beta &gt; 0$</span></p>
<p>What I did was define the “returns to scale function”, for <span class="math-container">$t&gt;1$</span>:</p>
<p><span class="math-container">$R(L,K;t) = f(tL,tK) - t f(L,K)$</span>, which would give the returns to scale according to these cases:</p>
<ul>
<li><span class="math-container">$R&gt;0 \implies $</span>Increasing returns to scale</li>
<li><span class="math-container">$R=0 \implies $</span> Constant returns to scale</li>
<li><span class="math-container">$R&lt;0 \implies $</span> Decreasing returns to scale</li>
</ul>
<p>For this function we would get</p>
<p><span class="math-container">$R(L,K;t) = t^4 \alpha K^2 L^2 - t^6 \beta K^3 L^3 - t \alpha K^2 L^2 + t \beta K^3 L^3$</span></p>
<p>Factoring <span class="math-container">$t K^2 L^2$</span>,</p>
<p><span class="math-container">$R(L,K;t) = t K^2 L^2 (t^3 \alpha - t^5 \beta K L - \alpha + \beta K L)$</span></p>
<p>Factoring <span class="math-container">$\beta KL$</span> from the second and fourth terms we get,</p>
<p><span class="math-container">$R(L,K;t) = t K^2 L^2 (t^3 \alpha - \alpha + (1-t^5) \beta K L)$</span></p>
<p>Since <span class="math-container">$t K^2 L^2$</span> can’t be negative, the sign of <span class="math-container">$R$</span> is the same sign as</p>
<p><span class="math-container">$S(L,K;t):=t^3 \alpha - \alpha + (1-t^5) \beta K L$</span></p>
<p>For <span class="math-container">$KL &gt; \frac{\alpha}{\beta}$</span>, since <span class="math-container">$t&gt;1 \implies 1-t^5 &lt;0$</span>,</p>
<p><span class="math-container">$S(L,K;t) &lt; t^3 \alpha - \alpha + (1-t^5) \alpha$</span></p>
<p>Factoring <span class="math-container">$\alpha$</span>,</p>
<p><span class="math-container">$S(L,K;t) &lt; \alpha (t^3 - t^5)$</span></p>
<p>Since <span class="math-container">$\alpha &gt; 0$</span> and <span class="math-container">$t&gt;1$</span>, the above is negative.</p>
<p>This implies that for <span class="math-container">$KL &gt; \frac{\alpha}{\beta}$</span>, we have that <span class="math-container">$S&lt;0$</span> and hence <span class="math-container">$R&lt;0$</span>.</p>
<p>Therefore, the production function <span class="math-container">$f$</span> has (should have) decreasing returns to scale?</p>
<p>I tried to do something similar with <span class="math-container">$g(L,K) = \exp{(KL)}$</span> and <span class="math-container">$h(L,K) = \log{(KL)}$</span> but couldn’t get uniform lower bounds for <span class="math-container">$K,L$</span> (that don’t depend on <span class="math-container">$t$</span>).</p>

<microeconomics><production-function><producer-theory>
How are returns to scale of a non homogeneous production function defined?
Nicolas Torres
# How are returns to scale of a non homogeneous production function defined?

(Asked by: username_0 on 2022-11-20)

Most of the production functions encountered in Intermediate Microeconomics are homogeneous (Cobb-Douglas, perfect substitutes, perfect complements).

So their returns to scale are easy to get, comparing their degree of homogeneity to $1$.

However, how would you define the returns to scale of non-homogeneous functions such as this one?

$f(L,K) = \alpha K^2 L^2 - \beta K^3 L^3, \alpha >0, \beta > 0$

What I did was define the “returns to scale function”, for $t>1$:

$R(L,K;t) = f(tL,tK) - t f(L,K)$, which would give the returns to scale according to these cases:

* $R>0 \implies $Increasing returns to scale
* $R=0 \implies $ Constant returns to scale
* $R<0 \implies $ Decreasing returns to scale

For this function we would get

$R(L,K;t) = t^4 \alpha K^2 L^2 - t^6 \beta K^3 L^3 - t \alpha K^2 L^2 + t \beta K^3 L^3$

Factoring $t K^2 L^2$,

$R(L,K;t) = t K^2 L^2 (t^3 \alpha - t^5 \beta K L - \alpha + \beta K L)$

Factoring $\beta KL$ from the second and fourth terms we get,

$R(L,K;t) = t K^2 L^2 (t^3 \alpha - \alpha + (1-t^5) \beta K L)$

Since $t K^2 L^2$ can’t be negative, the sign of $R$ is the same sign as

$S(L,K;t):=t^3 \alpha - \alpha + (1-t^5) \beta K L$

For $KL > \frac{\alpha}{\beta}$, since $t>1 \implies 1-t^5 <0$,

$S(L,K;t) < t^3 \alpha - \alpha + (1-t^5) \alpha$

Factoring $\alpha$,

$S(L,K;t) < \alpha (t^3 - t^5)$

Since $\alpha > 0$ and $t>1$, the above is negative.

This implies that for $KL > \frac{\alpha}{\beta}$, we have that $S<0$ and hence $R<0$.

Therefore, the production function $f$ has (should have) decreasing returns to scale?

I tried to do something similar with $g(L,K) = \exp{(KL)}$ and $h(L,K) = \log{(KL)}$ but couldn’t get uniform lower bounds for $K,L$ (that don’t depend on $t$).

**Tags:** microeconomics, production-function, producer-theory

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## 2 answers

### Answer by username_2 on 2022-11-21. Score: 3 | Accepted answer

I find interesting the proposal of Nicolas Torres of a way, through $R$, of measuring returns to scale. But, actually, the problem is that it cannot be a global measure, as can change from point to point of the production function.
But, also the usual definition cannot give a ‘global’ measure of returns to scale for functions that are not homogeneous.
So, tparker proposes a ‘local’ measure of returns to scale.

Actually, the suggestion of tparker of a 'local' measure of returns to scale seems similar to a directional derivative, along a direction such that all factors increase in the same proportion, that is, which leaves constant the ratios of the products. But how to compare this derivative with $t f(K,L,...)$ to establish whether there are constant or increasing or decreasing return to scale?

However, a similar proposal of a local measure of returns to scale exists in the literature.

A local measure of returns to scale is the *elasticity of scale*, which was introduced by Frisch, and it is a measure of the percentage increase of output due to a unitary percentage increase of all inputs. Formally it is defined as:

$$\epsilon=\frac{df(\lambda x)}{d \lambda} \frac{\lambda}{f(\lambda x)}$$

calculated for $\lambda=1$, where $\lambda$ is a positive parameter.

See the following article of Wikipedia (I’m sorry the following article is not in English, but I can’t find an equivalent article of Wikipedia in English.)

<https://it.wikipedia.org/wiki/Elasticit%C3%A0_di_scala>

And there is a literature about this subject, see for instance:

<https://www.jstor.org/stable/40751123>

Or, more recently, (2020)

<https://link.springer.com/referenceworkentry/10.1007/978-981-10-3450-3_23-1>


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### Answer by username_1 on 2022-11-21. Score: 4

The standard definition is that a production function $F(\vec{x})$ has constant, increasing, or decreasing returns to scale if $F(a\vec{x})$ is equal to, greater than, or less than $a F(\vec{x})$ (respectively) for *all* positive numbers $a$ and input vectors $\vec{x}$.

It follows that any homogeneous production function has well-defined returns to scale determined by its degree of homogeneity. But a generic non-homogeneous production function will not necessarily have well-defined returns to scale.

Although this isn't standard usage as far as I know, there may be some way to define the "local returns to scale" at a given set of inputs $\vec{x}\_0$ by considering how dilating $\vec{x}\_0$ by an infinitesimal amount $1+\epsilon$ (with $0 < \epsilon \ll 1$) changes the value of $F$ near the point $\vec{x}\_0$.

> **Comments:**
> **username_0**: Thank you for answering. I guess you mean the condition has to hold for ALL input vectors x? Interesting that you brought the concept of "local returns to scale". This means my cubic would have increasing local returns to scale beyond a certain point.  
> **username_1**: @NicholasTorres Yes, I believe that the standard definition requires that this hold for all input vectors x.  

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