Economic interpretation of Returns to Scale

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.

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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.