How are returns to scale of a non homogeneous production function defined?

Question

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 $t$).

Answer 1

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

Answer 2

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 $\vec{x}_0$.