# What's the relationship between Output elasticity and Returns to scale?

https://en.wikipedia.org/wiki/Output_elasticity

"If the production function contains only one input, then the output elasticity is also an indicator of the degree of returns to scale. If the coefficient of output elasticity is greater than 1, then production is experiencing increasing returns to scale. If the coefficient is less than 1, then production is experiencing decreasing returns to scale. If the coefficient is 1, then production is experiencing constant returns to scale. Note that returns to scale may change as the level of production changes."

So, the above is clear for 1 input. But what if the production function has two ore more inputs? Capital, labor, materials, etc? Can anyone show the proof?

Define the production function $$Q=f(x_1,..,x_k)$$, where $$x_i$$ denotes the ith input. Next recall that the total differential of output can be written as
$$\Delta Q = \sum_{i=1}^{\ k}\frac{\partial Q}{\partial x_i}\Delta x_i$$
Now we are interested in returns to scale, which is the change in output due to the multiplication of every input with a constant. We can write: $$x_{1i}=ax_i$$, for every x where a is a constant and the subscript 1 means the new x due to the multiplication. It follows that: $$\Delta x =x_{1i}-x_i =ax_i-x_i=(a-1)x_i$$
Substitute this last result into the total differential. We obtain:
$$\Delta Q = \sum_{i=1}^{\ k}\frac{\partial Q}{\partial x_i}\ (a-1)x_i$$
which can be rewritten as
$$\Delta Q = (a-1)\sum_{i=1}^{\ k}\frac{\partial Q}{\partial x_i}\ x_i$$
Now recall that the output elasticity with respect to an input i can be written as
$$\epsilon\ = \frac{\partial Q}{\partial x_i}\ \frac{x_i}{\ Q}$$
rewrite this to the expression
$$\epsilon\ Q = \frac{\partial Q}{\partial x_i}\ x_i$$

Substitute this result into the last version of the total differential which becomes:
$$\Delta Q = (a-1)\sum_{i=1}^{\ k}\ \epsilon\ Q = (a-1)Q\sum_{i=1}^{\ k}\ \epsilon$$
Rewriting this we obtain the multiplicator of the output associated with multiplication of every input with $$a$$:
$$\frac{Q_1}{Q} = (a-1) \sum_{i=1}^{\ k} \epsilon +1$$
Where $$Q_1$$ is output ex post.

From this last expression it can be immediatly seen that constant returns to scale ($$ax \implies aQ$$) follow from the sum of elasticities being equal to 1, increasing returns to scale from it being greater than 1 and decreasing when it's smaller than 1.

The graph is plotted for a=2. In red all points which correspond to decreasing returns to scale, in green for increasing returns to scale. Further, it can be seen that
$$\sum_{i=1}^{\ k} \epsilon = 1 \implies \frac{Q_1}{Q} = a$$