Tedious Math On Economic Growth And Return On Capital

This is from the tail of a post written in response to a Paul Krugman column.  The subject is "growth" versus "Return on Capital" in an economy.

GDP = Output = Workers x (Output/Worker);

Output/Worker = Productivity

Profit = Output x Income Share to Capital = (Workers x Productivity) x Inc Sh to Cap

Capital = Workers x (Capital/Worker)

Higher growth comes from adding workers (workforce growth), or increasing output per worker (productivity growth).

SO: Return on Capital =

[Workers x Productivity x Inc Sh to Cap] / Workers x (Capital/Worker)

Is it obvious that increasing growth by adding workers to this mix does not increase the return on capital?  Since "Workers" appear in the numerator and the denominator, they cancel.  This relects the notion that increasing the scale of an operation does not change the return on capital (unless something is *not* scaling the same way).

The implicit modeling assumption is that new workers do the same tasks as the current ones, with the same average capital usage.  If new workers need substantially less capital, then the overall capital/output ratio could fall.  But that is not the way it was modeled by Dean Baker, who started us down this road with his Feldstein letter.

Productivity is trickier (and my verbal formulation may be off a bit). Basically, as Output per Worker rises, the Capital team captures some of the benefit.  (Normally, I should add, productivity is associated with "capital deepening", where it takes more capital to make workers more productive.  I am setting that aside here, since it makes it harder for productivity to turn growth into a higher return on capital).

So, in this formula, if the old Return on Capital was 5%, and the increase in productivity is 1.7%, the new Return on Capital will be (5.0%) x (1.017) = 5.085%.  Keep that up for 16 years with no capital deepening, and the Return on Capital will rise to 6.5%.  Patient capital, indeed.  And remember, we are assuming that, down in the denominator, (Capital/Worker) is not changing.

Suppose we increase Productivity so that it grows by 2.7% per year.  Now, the Return on Capital will grow from 5% to 6.5% in just 10 years.  And when will it stop growing?  Ahh, we have a bit of a glitch here - without some more elaborate assumptions about the relationship of capital to labor and output, the Return on Capital will just keep going up.  However, any reasonable adjustment I make will slow the (already slow) improvement in the Return on Capital, so I stand by my point that the most plausible place to look for an improvement in the Return on Capital is not growth, but the Income Share to Capital.  Increase the Income Share to Capital, and that can solve the Return on Capital problem right away.

As modeled here, higher growth achieved by adding workers does nothing for the return on capital; higher growth through higher productivity reduces the time it takes the return on capital to grow to the target, with an unlikely assumption about capital deepening and the total ratio of (Capital/GDP).  So yes, growth can solve the Return on Capital problem eventually, depending on what one assumes about the relative capital deployed (as I noted originally).