Switching to Per-Worker Quantities
Constant returns to scale allow us to analyze the economy relative to the size of the labour force (or in other words, at the per-worker level). We set in the production function as follows:
Hence, the amount of output per worker depends only on the amount of capital per worker and is independant of the size of the economy. This is a consequence of constant returns to scale; the number of workers in the economy does not affect the relationship between capital per worker and output per worker.
In our example economy with four units of capital and two workers, we have . The vertical axis now represents output per worker . Let's see what happens when change capital per worker:
We have obtained the per-worker production function: the relationship between and .