Constant Returns To Scale

We will use the assumption of constant returns to scale to analyze all quantatites in per-worker terms. First, let's build up some intuition for the idea of constant returns to scale.

We consider that the economy has $K=4$ units of capital and $L=2$ units of labour. Hence output $Y$ equals:

$Y = F(4, 2) = 4^{0.3} 2^{0.7} = 2.46$

This economy is represented by point A on the graph. Constant returns to scale predicts that if we increase both $K$ and $L$ by some constant $z$, we will also increase $Y$ by that same constant:

$2.46 z = F(4 z, 2 z)$

We can verify that this equality holds by varying $z$:

We can see graphically that for the given $z$, output equals 2.46, or $2.46 z$. Hence we have shown that scaling the inputs by some constant $z$ has the effect of scaling $Y$ by the same constant.

Geometrically, the economy moves along a straight line that goes through the origin as we scale it's inputs. This is equivalent to saying that the production function is homogenous of degree 1.