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

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 $$t$$).

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 $$\vec{x}_0$$.

• 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. Nov 21 at 13:41
• @NicholasTorres Yes, I believe that the standard definition requires that this hold for all input vectors x. Nov 21 at 14:28

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

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

Or, more recently, (2020)