This is the generally accepted theory that leads to the need for output growth to maintain a steady state economy and non-increasing unemployment.
The first step is agreeing that output (Y) depends on capital and labor. Capital (K) is the stock of existing machines and plants in the economy. Ok, I can see that. To make things simpler, the theory is commonly presented in terms of output per unit labor and capital per labor.
The next step is equating investment with savings. I=S. Then establish that savings is proportional with the output. Then
Saving is done on the household level. Firms do the investment. The reason they are equal is because of the IS relation, where S=Y-C and Y=C+I. It is a little confusing about what this means on a practical level. When people buy stocks is that consumption or investment or saving? I think according to economic theory, it's consumption. Banks can lend money so that firms and people can actually consume more than they make. Could tuition be considered investment? Anyway, it seems difficult to really measure investment vs savings vs output since most businesses serve other businesses as well as people. Is all business spending considered investment? What if businesses save, too, instead of reinvesting all their profit.