# Economic interpretation of Returns to Scale

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!

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.

• 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 Nov 30 '20 at 10:02
• @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 Nov 30 '20 at 10:42
• @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 Nov 30 '20 at 10:45
• 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: youtube.com/watch?v=ftmvsahQ6Wo. See starting from 28:30 – Duo Nov 30 '20 at 10:47
• @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 Nov 30 '20 at 11:03

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, 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.

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.

• +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 Nov 30 '20 at 15:06
• 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 Dec 2 '20 at 6:51
• 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 Dec 2 '20 at 6:55