# 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 $$t$$).

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)

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 $$\vec{x}$$.
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 $$\vec{x}_0$$.