<p><strong>Constant returns to scale</strong> mean that:</p>
<p><span class="math-container">$$  F(aK,aL)=aF(K,L) $$</span></p>
<p>or in plain English if you increase the number of inputs (here capital <span class="math-container">$K$</span> and labor <span class="math-container">$L$</span>) output will increase by the same factor. So for example if you double the inputs you also double the output  <span class="math-container">$F(2K,2L)=2F(K,L)$</span>.</p>
<p>This means that an economy can always scale its production by the same proportion by which it scales its resources. If you want to double production just double amount of labor and capital.</p>
<p><strong>Increasing returns to scale</strong> mean that:</p>
<p><span class="math-container">$$ F(aK,aL)&gt;aF(K,L)$$</span></p>
<p>or in plain English, if you increase inputs by some factor output increases by more than that factor. For example, you might double the number of capital and labor you use but quadruple the output you actually get.</p>
<p>This means that the output will increase by larger proportion than the resources you put in. So if your goal would be doubling the production you would be fine just with less than double of resources and be able to do that.</p>
<p><strong>Decreasing returns to scale</strong> mean that:</p>
<p><span class="math-container">$$ F(aK,aL)&lt;aF(K,L)$$</span></p>
<p>or in plain English, if you increase the number of inputs by a factor the output will increase by some smaller factors. For example you might double the inputs but output would increase by just factor of 1.5.</p>
<p>This simply means production always increases by smaller proportion than the proportion by which you double the resources. Here if your goal would be to double the output you would need to increase resources by more than a double.</p>

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2020-11-30T10:02:38.693Z
Thanks for the answer! I am trying the understand the underlying concept behind returns to scale. So, if the production function is increasing returns to scale (IRS), aside from implying that output increases by more than the proportional change in all inputs, what this means in terms of economic interpretation? Is there any underlying insight behind increasing returns to scale? For example, can we say that for the IRS case the economy experiences significant technological progress or production inputs are efficiently used, etc.?
Duo
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2020-11-30T10:42:01.087Z
@Duo but that is the insight I mean the insight literally is that with constant returns to scale for some F(L) if you double hours worked also your product doubles. For example, if you are writer if you double hours writing you write twice as many books. For decreasing returns to scale if you double hours writing you will get less then twice as many books and for increasing returns to scale the implication is that you get more than twice as many books.
1muflon1
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2020-11-30T10:45:48.287Z
@Duo if your question is what further implications there are like for example what are implications of constant returns on economic growth etc then in that case actually your question is broad and it should be closed as needing more focus. Returns to scale will have different implications in macro, micro and for policy. Such question would require several books to answer and our rule here is that too broad questions are off topic. If that is your Q consider asking separate questions on this on separate subtopic e.g what is their implication for growth  in Solow model can be 1 separate Q
1muflon1
CC BY-SA 4.0
2020-11-30T10:47:26.387Z
If Jack and Rose are writers, both are doubling their working hours. In this case, Jack write twice as many books, but Rose get less than twice as many books. What does this mean? Does it imply that Jack is more talented, gritty or more productive, or uses its time more efficiently, etc.? Here I found some explanation on firm level: https://www.youtube.com/watch?v=ftmvsahQ6Wo. See starting from 28:30
Duo
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2020-11-30T10:53:33.527Z
@Duo that means Jack has constant returns to scale and Rose has decreasing returns to scale. It does mean that Jack is never getting tired and no matter how much he writes always when he doubles his work his number of books increases by the same proportion. Rose here would be getting ‘tired’ after some time where first hour of her work would be more productive than the second and second more than the third. On average Jack would be more productive but that would be because his productivity does not decrease as his work amount increases
1muflon1
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2020-11-30T10:58:34.310Z
About Jack and Rose, I think in the same way, as you presented. And what about the whole economies, if we substitute Jack and Rose with Germany and Poland? Can we say that Germany uses its inputs more efficiently than Poland, or Germany experiences more technological progress than Poland?
Duo
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2020-11-30T11:03:18.957Z
@Duo no technological progress is not directly related to returns to scale even a function with absolutely no technology can have constant, decreasing and increasing returns to scale. You can say exactly the same thing about Jack and Rose - in this case Germany would be *on average* more productive because it’s productivity does not diminish as it utilizes more resources, while Poland would be *on average* less productive, because as it utilizes more resources it’s productivity declines
1muflon1
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2020-12-02T09:59:42.647Z
Thanks for the comment; I was thinking about it a while. You say that Poland is on average less productive, because as it utilizes more resources it's productivity declines. How this can happen, if productivity does not depend on the amount of used factors? If Poland uses $L_0=10$ and $K_0=25$, then productivity doesn't change by doubling inputs: $L_1=20$ and $K_1=50$. In other words, $Y_t=A_t F(K_t, L_t)$ production function's $A_t$ term can't be influenced by scaling inputs, i.e.  $A_t$ won't be changed in $\delta Y_t=A_t F(\gamma K_t,\gamma L_t)$, where $\delta=\gamma$ if prod. fun. is CRS.
Duo
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2020-12-02T10:25:57.160Z
@Duo productivity is defined as number of output per units of input - for example measure of productivity would be GDP/hours worked. $A_t$ parameter is not productivity per se but technology parameter. This technology parameter is sometimes referred to as total factor productivity because technology affects productivity but it is not synonymous with it. For example, you can have constant returns of scale production functions $Y=AL$ where the productivity is always constant $Y/L=A$ and then you can have production function with exactly the same technology $A$ but diminishing returns to scale
1muflon1
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2020-12-02T10:27:28.887Z
such as $Y= A \sqrt{ L}$ where the productivity would be given by $Y/L = A/\sqrt{L}$ and clearly in this case you can see that even with the exactly same stock of technology the productivity here will be lower when our use of labor $L$ increases.
1muflon1
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2020-12-02T10:54:56.527Z
Thanks for the comment! For the CRS production function $Y=AL$, the productivity is always constant, since scaling input doesn't change productivity, i.e. $A=Y/L$ and after scaling by $\gamma$ we have $A=\gamma Y/ \gamma L=Y/L$. But this is true for DRS too. That is, from $Y=A \sqrt L$ we have $A=Y/\sqrt L$ and after scaling input by $\gamma$ we have $A=\frac{\sqrt \gamma Y}{\sqrt {\gamma L}}=\frac{Y}{\sqrt L} $; again scaling doesn't change anything in terms of productivity ($A$)
Duo
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2020-12-02T11:34:06.903Z
@Duo no it does not hold to DRS that calculation is not correct. In CRS we have: $F(L) =Y=AL \implies F(\gamma L) = \gamma F(L) $ thus $\gamma Y  = \gamma AL$ . In case of DRS this does not hold  since $F(L) =Y=A \sqrt{L} \implies F( \gamma L) \neq \gamma F(L) $ In fact here if we scale $L$ by gamma production will increase only by $\sqrt{\gamma}$ since $F(\gamma L) = A \sqrt{\gamma L} = \sqrt{\gamma} A \sqrt{L} = \sqrt{\gamma} F(L) = \sqrt{\gamma} Y $ which means that if you increase $L$ by $\gamma$, $Y$ increases only by $\sqrt{\gamma}$ so productivity is $\frac{\sqrt{\gamma}}{\gamma L}$
1muflon1
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2020-12-02T11:44:46.307Z
I don't know what is wrong in my comment. Anyways, you wrote that holds $F(\gamma L)=A \sqrt {\gamma L}=\sqrt \gamma A \sqrt L= \sqrt \gamma F(L)=\sqrt \gamma Y$! If $\sqrt \gamma A \sqrt L= \sqrt \gamma Y$, then holds $A=\frac{Y}{\sqrt L}$ ($\sqrt \gamma \neq 0$). The productivity $A$ is the same before and after scaling production function.
Duo
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2020-12-02T11:49:10.797Z
@Duo but A is technology not productivity per se. Ok forget for a second about A because I think that creates the confusion here. Let us assume that A=1 which would be equivalent to having no extra advantage thanks to technology. In that case if $Y=L$ productivity of labor will be exactly 1 $Y/L =1$ this should make sense because that just says that 1 unit of labor produces one unit of output. Importantly here this productivity will be constant. In case of $Y = \sqrt{L}$ the productivity will be $Y/L = \frac{1}{\sqrt{L}}$ here note that mathematically $\frac{\sqrt{L}}{L} =\frac{1}{\sqrt{L}}$.
1muflon1
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2020-12-02T11:52:25.383Z
in the case of $Y= \sqrt{L}$ the productivity will depend on what $L$ is. When $L=1$ also productivity or $Y/L$ will be 1 (same as with CRS), but when $L=4$ productivity of  of labor will be $Y/L = 1/2$, if $L=9$ productivity will be $Y/L=1/3$. You will see that as you add more labor productivity will decline in DRS case.
1muflon1
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2020-12-02T12:05:41.660Z
Thank you very much for the thorough comments! For the DRS $Y=A \sqrt L$ (where $A$ is technology measuring the productivity) we have that **by definition** technology is defined as $A=\frac{Y}{\sqrt L}$. After scaling prod. function by $\gamma$ the technology is equal to $A=\frac{ \sqrt \gamma Y}{\sqrt {\gamma L}}=\frac{Y}{\sqrt L}$.  In your comments, technology is not constant because you change the definition of technology; instead of $A=\frac{Y}{\sqrt L}$ definition, you use $\frac{A}{\sqrt L}=\frac{Y}{L}$. Why you change the definition? It is given by specific form of the prod function.
Duo
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2020-12-02T12:36:42.237Z
@Duo But this is to my best understanding that would not be the right interpretation.  Definitional relationships are given by $\equiv$ not by $=$. These are similar but not same $=$ just means that one variable value will equal to the other, but not that it is by definition the other value. Technology is not defined by the production function - like you cant say that just because $D=S$ at equilibrium price demand is by definition supply. In fact the distinction between productivity and technology is good example of distinction between definition and equal relationship.
1muflon1
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2020-12-02T12:42:13.103Z
By definition productivity $(P)$ is output over input so by definition $P \equiv \frac{F(x)}{x}$, where $Y \equiv F$.  You see this is true definition because it holds no matter what production function is. Whether $Y =A L $, $Y=\sqrt{L}$ or $Y=  BL^2$  by definition productivity will be $P\equiv \frac{Y}{L}$ (since here we have only $L$ as an input). But of course productivity will be in each case equal to different expression. In first case $P = A$ in second $P = \frac{1}{\sqrt{L}} $ in third case $P = BL$.
1muflon1
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2020-12-02T12:43:32.200Z
I got your point! Thanks a lot! Btw, a small question; I don't want to post a separate question on this. Could you please give me a reference(s) where production function and its properties are thoroughly discussed. I found some sources, where it is either superficial or non-rigorous. Thanks!
Duo
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2020-12-02T12:47:23.780Z
@Duo depends on what production function you are talking about. Production function $F = AL$ will have different properties than $F= A + L$. Usually production functions are discussed in quite large degree of detail in microeconomic textbooks. Varian Microeconomic Aanalysis has whole chapter on production functions (chapter 1). However, note Varian calls production functions technology functions - this is because the whole $F(x)$ could be also viewed as a technology in itself. In macro literature (where I come from) technology is usually reserved just for the $A$ parameter.
1muflon1
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2020-12-02T12:57:17.273Z
Thanks, I'll check Varian's book!
Duo
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2020-12-02T12:58:46.737Z
@Duo you are welcome. Also btw this will be definitely in any micro book - if you already own some you can start there instead of getting/borrowing Varian - but this being said Varian has very nice and detailed treatment.
1muflon1
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2020-12-02T13:16:50.410Z
Thanks a lot; I'll check several book, including Varian!
Duo
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2020-11-30T09:36:53.780Z
2020-11-30T09:43:40.763Z
2020-11-30T09:43:40.763Z
1muflon1
<p>Since the OP has labeled their question &quot;Economic Interpretation of Returns to Scale&quot;:</p>
<p>The economist mostly responsible for spreading the use of mathematical rigor in economics, <a href="https://en.wikipedia.org/wiki/Paul_Samuelson" rel="nofollow noreferrer">Paul Samuelson</a>, who also never lost sight of the essence of economics, writes as follows in <strong>&quot;Foundations of Economic Analysis&quot; (1947, Enlarged Edition 1983), p. 84</strong> (emphasis mine):</p>
<blockquote>
<p>The problem of homogeneity of the production function is one about
which much controversy has raged. It has long been held on
philosophical grounds that product must be a homogeneous function of
the first order of all the variables, and if this is not so, it must
be either because of &quot;indivisibility&quot; or because not all &quot;factors&quot;
have been taken into account.</p>
<p>With regard to the first point, it is clear that labeling the absence
of homogeneity as due to indivisibility changes nothing and merely
affirms by the implication that &quot;indivisibility&quot; does exist, the
absence of homogeneity.</p>
<p>With respect to the second point, we may reverse the Aristotelian
dictum and affirm that anything that must be true self-evidently
(&quot;philosophically&quot;), intuitively -i.e. by conventional definition of the terms involved- that <em><strong>such a principle can have no empirical
content</strong></em>. <em><strong>It is a scientifically meaningless assertion that doubling
all factors must double product</strong></em>. This is not because we do not have
the power to perform such an experiment; such an objection is of
course irrelevant. <em><strong>Rather the statement is meaningless because it
could never be refuted, in the sense that no hypothetically
conceivable experiment could ever controvert the principle enunciated.
This is so because if product did not double, one could always
conclude that some factor was &quot;scarce</strong></em>&quot;.</p>
</blockquote>
<p>&quot;Principles that have no empirical content&quot; can be seen, and used, as <em>methodological / modeling choices</em>.</p>
<p>For example declaring that a production function <span class="math-container">$Q = F(K,L)$</span> has constant returns to scale (homogeneous of first order/degree, or &quot;linearly homogeneous&quot;), is equivalent to declare that we understand the symbols <span class="math-container">$K$</span> and <span class="math-container">$L$</span> as capturing <em>all forces affecting the level of output.</em></p>
<p>Samuelson goes on to write</p>
<blockquote>
<p>...the expression &quot;factor of production&quot;... has been used in at least
two senses... First it has been used to denote broad composite
quantities such as &quot;labor, land and capital&quot;. On the other hand, it
has been used to denote any aspect of the environment which has any
influence on production. I suggest that only &quot;inputs&quot; be explicitly
included in the production function, and that this term be confined to
denote measurable quantitative economic goods or services... So
defined the production function need not be homogeneous of the first
order.</p>
</blockquote>
<p>The profession has implemented his suggestion <em>exactly</em>, and this is why we see models with decreasing or increasing returns to scale. But in light of the above wise comments, we must keep in mind that, say, a production function with increasing returns to scale does <em>not</em> reflect some real-world impressive technology that can more than double output if we only double inputs. It is just a methodological / modelling choice we have made, following Samuelson's suggestion.</p>
<p>A way to <em>justify</em> Samuelson's suggestion as regards what to include in the &quot;production function&quot; construct, is that he essentially suggests <em>to include those output-affecting factors related to which the firm has a strong degree of control and decision making over their quantities employed</em>.</p>
<hr />
<p>Responding to comments by the OP</p>
<blockquote>
<p>Thanks for the answer! Summarizing Samuelson's points I understand the
following: if we are able to capture all forces affecting production,
than the production function is homogenous of first order. However,
since we are not able to measure all inputs/factors, then it is
reasonable to assume that the production function is not linearly
homogenous. If so, then I assume that the production function is never
linearly homogenous, because K and L are measurable, but A (TFP) is
never measurable.</p>
<p>Aside from that, from your answer I assume that there is no strong
ground to find any economic interpretation of the returns to scale,
since it is merely methodological/modeling choices, right?</p>
</blockquote>
<p>Don't confuse data availability on a factor with symbolic representation of a factor. So one could in principle say that <span class="math-container">$A(TFP)$</span> are &quot;all other factors&quot; and so we have constant returns to scale, but then <span class="math-container">$A(TFP)$</span> would become a catch-all concept devoid of any economic meaning.</p>
<p>Returns to scale are indeed a methodological/modeling choice so they do not have &quot;intrinsic&quot; economic interpretation. But they affect the results of the model. As such, they are <em>effective</em>, because they indirectly reflect the existence (or not) of factors that will not enter a specific model quantitatively.</p>
<p>So choosing diminishing/constant/increasing returns to scale will reflect the knowledge/belief of the researcher as to whether there is some <em>important</em> factor that affects output and stays out of the model. So in a sense the represent an aspect of the model that has this indirect economic meaning.</p>

CC BY-SA 4.0
2020-11-30T15:06:53.013Z
+1 Excellent answer, especially the technology part is important from my experience students often confuse returns to scale with technology but of course technology in most production function is orthogonal on what the returns to scale are.
1muflon1
CC BY-SA 4.0
2020-12-02T06:51:10.300Z
Thanks for the answer! Summarizing Samuelson's points I understand the following: if we are able to capture all forces affecting production, than the production function is homogenous of first order. However, since we are not able to measure all inputs/factors, then it is reasonable to assume that the production function is not linearly homogenous. If so, then I assume that the production function is never linearly homogenous, because $K$ and $L$ are measurable, but $A$ ($TFP$) is never measurable.
Duo
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2020-12-02T06:55:35.967Z
Aside from that, from your answer I assume that there is no strong ground to find any economic interpretation of the returns to scale, since it is merely methodological/modeling choices, right?
Duo
CC BY-SA 4.0
2020-11-30T13:44:50.273Z
2020-12-02T10:12:08.137Z
2020-12-02T10:12:08.137Z
Alecos Papadopoulos
<blockquote>
<p>There are three possible types of returns to scale: increasing returns to scale, constant returns to scale, and diminishing (or decreasing) returns to scale. If output increases by the same proportional change as all inputs change then there are constant returns to scale (CRS). If output increases by less than the proportional change in all inputs, there are decreasing returns to scale (DRS). If output increases by more than the proportional change in all inputs, there are increasing returns to scale (IRS).</p>
</blockquote>
<p>What are the economic interpretation of constant returns to scale, decreasing returns to scale, increasing returns to scale? In other words, what means and how it can be interpreted if the economy has a specific type of returns to scale?</p>
<p>Thanks!</p>

<macroeconomics><production-function>
Economic interpretation of Returns to Scale
Duo
# Economic interpretation of Returns to Scale

(Asked by: username_0 on 2020-11-30)

> There are three possible types of returns to scale: increasing returns to scale, constant returns to scale, and diminishing (or decreasing) returns to scale. If output increases by the same proportional change as all inputs change then there are constant returns to scale (CRS). If output increases by less than the proportional change in all inputs, there are decreasing returns to scale (DRS). If output increases by more than the proportional change in all inputs, there are increasing returns to scale (IRS).

What are the economic interpretation of constant returns to scale, decreasing returns to scale, increasing returns to scale? In other words, what means and how it can be interpreted if the economy has a specific type of returns to scale?

Thanks!

**Tags:** macroeconomics, production-function

---

## 2 answers

### Answer by username_1 on 2020-11-30. Score: 2 | Accepted answer

**Constant returns to scale** mean that:

$$ F(aK,aL)=aF(K,L) $$

or in plain English if you increase the number of inputs (here capital $K$ and labor $L$) output will increase by the same factor. So for example if you double the inputs you also double the output $F(2K,2L)=2F(K,L)$.

This means that an economy can always scale its production by the same proportion by which it scales its resources. If you want to double production just double amount of labor and capital.

**Increasing returns to scale** mean that:

$$ F(aK,aL)>aF(K,L)$$

or in plain English, if you increase inputs by some factor output increases by more than that factor. For example, you might double the number of capital and labor you use but quadruple the output you actually get.

This means that the output will increase by larger proportion than the resources you put in. So if your goal would be doubling the production you would be fine just with less than double of resources and be able to do that.

**Decreasing returns to scale** mean that:

$$ F(aK,aL)

or in plain English, if you increase the number of inputs by a factor the output will increase by some smaller factors. For example you might double the inputs but output would increase by just factor of 1.5.

This simply means production always increases by smaller proportion than the proportion by which you double the resources. Here if your goal would be to double the output you would need to increase resources by more than a double.

> **Comments:**
> **username_0**: Thanks for the answer! I am trying the understand the underlying concept behind returns to scale. So, if the production function is increasing returns to scale (IRS), aside from implying that output increases by more than the proportional change in all inputs, what this means in terms of economic interpretation? Is there any underlying insight behind increasing returns to scale? For example, can we say that for the IRS case the economy experiences significant technological progress or production inputs are efficiently used, etc.?  
> **username_1**: @Duo but that is the insight I mean the insight literally is that with constant returns to scale for some F(L) if you double hours worked also your product doubles. For example, if you are writer if you double hours writing you write twice as many books. For decreasing returns to scale if you double hours writing you will get less then twice as many books and for increasing returns to scale the implication is that you get more than twice as many books.  
> **username_1**: @Duo if your question is what further implications there are like for example what are implications of constant returns on economic growth etc then in that case actually your question is broad and it should be closed as needing more focus. Returns to scale will have different implications in macro, micro and for policy. Such question would require several books to answer and our rule here is that too broad questions are off topic. If that is your Q consider asking separate questions on this on separate subtopic e.g what is their implication for growth  in Solow model can be 1 separate Q  
> **username_0**: If Jack and Rose are writers, both are doubling their working hours. In this case, Jack write twice as many books, but Rose get less than twice as many books. What does this mean? Does it imply that Jack is more talented, gritty or more productive, or uses its time more efficiently, etc.? Here I found some explanation on firm level: https://www.youtube.com/watch?v=ftmvsahQ6Wo. See starting from 28:30  
> **username_1**: @Duo that means Jack has constant returns to scale and Rose has decreasing returns to scale. It does mean that Jack is never getting tired and no matter how much he writes always when he doubles his work his number of books increases by the same proportion. Rose here would be getting ‘tired’ after some time where first hour of her work would be more productive than the second and second more than the third. On average Jack would be more productive but that would be because his productivity does not decrease as his work amount increases  
> **username_0**: About Jack and Rose, I think in the same way, as you presented. And what about the whole economies, if we substitute Jack and Rose with Germany and Poland? Can we say that Germany uses its inputs more efficiently than Poland, or Germany experiences more technological progress than Poland?  
> **username_1**: @Duo no technological progress is not directly related to returns to scale even a function with absolutely no technology can have constant, decreasing and increasing returns to scale. You can say exactly the same thing about Jack and Rose - in this case Germany would be *on average* more productive because it’s productivity does not diminish as it utilizes more resources, while Poland would be *on average* less productive, because as it utilizes more resources it’s productivity declines  
> **username_0**: Thanks for the comment; I was thinking about it a while. You say that Poland is on average less productive, because as it utilizes more resources it's productivity declines. How this can happen, if productivity does not depend on the amount of used factors? If Poland uses $L_0=10$ and $K_0=25$, then productivity doesn't change by doubling inputs: $L_1=20$ and $K_1=50$. In other words, $Y_t=A_t F(K_t, L_t)$ production function's $A_t$ term can't be influenced by scaling inputs, i.e.  $A_t$ won't be changed in $\delta Y_t=A_t F(\gamma K_t,\gamma L_t)$, where $\delta=\gamma$ if prod. fun. is CRS.  
> **username_1**: @Duo productivity is defined as number of output per units of input - for example measure of productivity would be GDP/hours worked. $A_t$ parameter is not productivity per se but technology parameter. This technology parameter is sometimes referred to as total factor productivity because technology affects productivity but it is not synonymous with it. For example, you can have constant returns of scale production functions $Y=AL$ where the productivity is always constant $Y/L=A$ and then you can have production function with exactly the same technology $A$ but diminishing returns to scale  
> **username_1**: such as $Y= A \sqrt{ L}$ where the productivity would be given by $Y/L = A/\sqrt{L}$ and clearly in this case you can see that even with the exactly same stock of technology the productivity here will be lower when our use of labor $L$ increases.  
> **username_0**: Thanks for the comment! For the CRS production function $Y=AL$, the productivity is always constant, since scaling input doesn't change productivity, i.e. $A=Y/L$ and after scaling by $\gamma$ we have $A=\gamma Y/ \gamma L=Y/L$. But this is true for DRS too. That is, from $Y=A \sqrt L$ we have $A=Y/\sqrt L$ and after scaling input by $\gamma$ we have $A=\frac{\sqrt \gamma Y}{\sqrt {\gamma L}}=\frac{Y}{\sqrt L} $; again scaling doesn't change anything in terms of productivity ($A$)  
> **username_1**: @Duo no it does not hold to DRS that calculation is not correct. In CRS we have: $F(L) =Y=AL \implies F(\gamma L) = \gamma F(L) $ thus $\gamma Y  = \gamma AL$ . In case of DRS this does not hold  since $F(L) =Y=A \sqrt{L} \implies F( \gamma L) \neq \gamma F(L) $ In fact here if we scale $L$ by gamma production will increase only by $\sqrt{\gamma}$ since $F(\gamma L) = A \sqrt{\gamma L} = \sqrt{\gamma} A \sqrt{L} = \sqrt{\gamma} F(L) = \sqrt{\gamma} Y $ which means that if you increase $L$ by $\gamma$, $Y$ increases only by $\sqrt{\gamma}$ so productivity is $\frac{\sqrt{\gamma}}{\gamma L}$  
> **username_0**: I don't know what is wrong in my comment. Anyways, you wrote that holds $F(\gamma L)=A \sqrt {\gamma L}=\sqrt \gamma A \sqrt L= \sqrt \gamma F(L)=\sqrt \gamma Y$! If $\sqrt \gamma A \sqrt L= \sqrt \gamma Y$, then holds $A=\frac{Y}{\sqrt L}$ ($\sqrt \gamma \neq 0$). The productivity $A$ is the same before and after scaling production function.  
> **username_1**: @Duo but A is technology not productivity per se. Ok forget for a second about A because I think that creates the confusion here. Let us assume that A=1 which would be equivalent to having no extra advantage thanks to technology. In that case if $Y=L$ productivity of labor will be exactly 1 $Y/L =1$ this should make sense because that just says that 1 unit of labor produces one unit of output. Importantly here this productivity will be constant. In case of $Y = \sqrt{L}$ the productivity will be $Y/L = \frac{1}{\sqrt{L}}$ here note that mathematically $\frac{\sqrt{L}}{L} =\frac{1}{\sqrt{L}}$.  
> **username_1**: in the case of $Y= \sqrt{L}$ the productivity will depend on what $L$ is. When $L=1$ also productivity or $Y/L$ will be 1 (same as with CRS), but when $L=4$ productivity of  of labor will be $Y/L = 1/2$, if $L=9$ productivity will be $Y/L=1/3$. You will see that as you add more labor productivity will decline in DRS case.  
> **username_0**: Thank you very much for the thorough comments! For the DRS $Y=A \sqrt L$ (where $A$ is technology measuring the productivity) we have that **by definition** technology is defined as $A=\frac{Y}{\sqrt L}$. After scaling prod. function by $\gamma$ the technology is equal to $A=\frac{ \sqrt \gamma Y}{\sqrt {\gamma L}}=\frac{Y}{\sqrt L}$.  In your comments, technology is not constant because you change the definition of technology; instead of $A=\frac{Y}{\sqrt L}$ definition, you use $\frac{A}{\sqrt L}=\frac{Y}{L}$. Why you change the definition? It is given by specific form of the prod function.  
> **username_1**: @Duo But this is to my best understanding that would not be the right interpretation.  Definitional relationships are given by $\equiv$ not by $=$. These are similar but not same $=$ just means that one variable value will equal to the other, but not that it is by definition the other value. Technology is not defined by the production function - like you cant say that just because $D=S$ at equilibrium price demand is by definition supply. In fact the distinction between productivity and technology is good example of distinction between definition and equal relationship.  
> **username_1**: By definition productivity $(P)$ is output over input so by definition $P \equiv \frac{F(x)}{x}$, where $Y \equiv F$.  You see this is true definition because it holds no matter what production function is. Whether $Y =A L $, $Y=\sqrt{L}$ or $Y=  BL^2$  by definition productivity will be $P\equiv \frac{Y}{L}$ (since here we have only $L$ as an input). But of course productivity will be in each case equal to different expression. In first case $P = A$ in second $P = \frac{1}{\sqrt{L}} $ in third case $P = BL$.  
> **username_0**: I got your point! Thanks a lot! Btw, a small question; I don't want to post a separate question on this. Could you please give me a reference(s) where production function and its properties are thoroughly discussed. I found some sources, where it is either superficial or non-rigorous. Thanks!  
> **username_1**: @Duo depends on what production function you are talking about. Production function $F = AL$ will have different properties than $F= A + L$. Usually production functions are discussed in quite large degree of detail in microeconomic textbooks. Varian Microeconomic Aanalysis has whole chapter on production functions (chapter 1). However, note Varian calls production functions technology functions - this is because the whole $F(x)$ could be also viewed as a technology in itself. In macro literature (where I come from) technology is usually reserved just for the $A$ parameter.  
> **username_0**: Thanks, I'll check Varian's book!  
> **username_1**: @Duo you are welcome. Also btw this will be definitely in any micro book - if you already own some you can start there instead of getting/borrowing Varian - but this being said Varian has very nice and detailed treatment.  
> **username_0**: Thanks a lot; I'll check several book, including Varian!  

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### Answer by username_2 on 2020-11-30. Score: 3

Since the OP has labeled their question "Economic Interpretation of Returns to Scale":

The economist mostly responsible for spreading the use of mathematical rigor in economics, [Paul Samuelson](https://en.wikipedia.org/wiki/Paul_Samuelson), who also never lost sight of the essence of economics, writes as follows in **"Foundations of Economic Analysis" (1947, Enlarged Edition 1983), p. 84** (emphasis mine):

> The problem of homogeneity of the production function is one about
> which much controversy has raged. It has long been held on
> philosophical grounds that product must be a homogeneous function of
> the first order of all the variables, and if this is not so, it must
> be either because of "indivisibility" or because not all "factors"
> have been taken into account.
>
> With regard to the first point, it is clear that labeling the absence
> of homogeneity as due to indivisibility changes nothing and merely
> affirms by the implication that "indivisibility" does exist, the
> absence of homogeneity.
>
> With respect to the second point, we may reverse the Aristotelian
> dictum and affirm that anything that must be true self-evidently
> ("philosophically"), intuitively -i.e. by conventional definition of the terms involved- that ***such a principle can have no empirical
> content***. ***It is a scientifically meaningless assertion that doubling
> all factors must double product***. This is not because we do not have
> the power to perform such an experiment; such an objection is of
> course irrelevant. ***Rather the statement is meaningless because it
> could never be refuted, in the sense that no hypothetically
> conceivable experiment could ever controvert the principle enunciated.
> This is so because if product did not double, one could always
> conclude that some factor was "scarce***".

"Principles that have no empirical content" can be seen, and used, as *methodological / modeling choices*.

For example declaring that a production function $Q = F(K,L)$ has constant returns to scale (homogeneous of first order/degree, or "linearly homogeneous"), is equivalent to declare that we understand the symbols $K$ and $L$ as capturing *all forces affecting the level of output.*

Samuelson goes on to write

> ...the expression "factor of production"... has been used in at least
> two senses... First it has been used to denote broad composite
> quantities such as "labor, land and capital". On the other hand, it
> has been used to denote any aspect of the environment which has any
> influence on production. I suggest that only "inputs" be explicitly
> included in the production function, and that this term be confined to
> denote measurable quantitative economic goods or services... So
> defined the production function need not be homogeneous of the first
> order.

The profession has implemented his suggestion *exactly*, and this is why we see models with decreasing or increasing returns to scale. But in light of the above wise comments, we must keep in mind that, say, a production function with increasing returns to scale does *not* reflect some real-world impressive technology that can more than double output if we only double inputs. It is just a methodological / modelling choice we have made, following Samuelson's suggestion.

A way to *justify* Samuelson's suggestion as regards what to include in the "production function" construct, is that he essentially suggests *to include those output-affecting factors related to which the firm has a strong degree of control and decision making over their quantities employed*.

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Responding to comments by the OP

> Thanks for the answer! Summarizing Samuelson's points I understand the
> following: if we are able to capture all forces affecting production,
> than the production function is homogenous of first order. However,
> since we are not able to measure all inputs/factors, then it is
> reasonable to assume that the production function is not linearly
> homogenous. If so, then I assume that the production function is never
> linearly homogenous, because K and L are measurable, but A (TFP) is
> never measurable.
>
> Aside from that, from your answer I assume that there is no strong
> ground to find any economic interpretation of the returns to scale,
> since it is merely methodological/modeling choices, right?

Don't confuse data availability on a factor with symbolic representation of a factor. So one could in principle say that $A(TFP)$ are "all other factors" and so we have constant returns to scale, but then $A(TFP)$ would become a catch-all concept devoid of any economic meaning.

Returns to scale are indeed a methodological/modeling choice so they do not have "intrinsic" economic interpretation. But they affect the results of the model. As such, they are *effective*, because they indirectly reflect the existence (or not) of factors that will not enter a specific model quantitatively.

So choosing diminishing/constant/increasing returns to scale will reflect the knowledge/belief of the researcher as to whether there is some *important* factor that affects output and stays out of the model. So in a sense the represent an aspect of the model that has this indirect economic meaning.

> **Comments:**
> **username_1**: +1 Excellent answer, especially the technology part is important from my experience students often confuse returns to scale with technology but of course technology in most production function is orthogonal on what the returns to scale are.  
> **username_0**: Thanks for the answer! Summarizing Samuelson's points I understand the following: if we are able to capture all forces affecting production, than the production function is homogenous of first order. However, since we are not able to measure all inputs/factors, then it is reasonable to assume that the production function is not linearly homogenous. If so, then I assume that the production function is never linearly homogenous, because $K$ and $L$ are measurable, but $A$ ($TFP$) is never measurable.  
> **username_0**: Aside from that, from your answer I assume that there is no strong ground to find any economic interpretation of the returns to scale, since it is merely methodological/modeling choices, right?  

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