A natural rate of unemployment in the Cobra equation

This post continues the exploration into the Cobra equation, which measures the profitability of utilizing labor and capital as a function of labor share. Now it looks as though the Cobra equation is implying a natural rate of unemployment.

The simplified Cobra equation is now…

Measure of profitability in the aggregate = (x + y) – ax2y2

x = capital utilization rate
y = employment rate
a = coefficient for labor share to establish profit maximization.
Coefficient “a” = (els)2 – 2.474*(els) + 2.0 … (els = effective labor share, for example 80% as 0.80). (Note: I have stream-lined the equation for the coefficient “a”. No change from the previous version over the effective range of the economy.)

The Cobra equation is in 3 dimensions, but I have made a 2 dimensional graph of it with an Excel spreadsheet.

Cobra 1
The orange line marks the profit maximization for utilizing labor and capital in the aggregate.

The equation for profit maximization (orange line) is simply the derivative of the profitability measure above with respect to x then solved for y, employment…

Profit max = (1/(2*a*x))0.5

The blue line is the effective demand limit, which has the equation…

Effective demand limit = (els)/x
x = capital utilization
els = effective labor share (This is determined by multiplying Labor Share index for the business sector by 0.766… 2009 as base year.)

As the economy utilizes more labor and capital in the expansion phase of the business cycle, it eventually reaches one of the two lines and stops. Then the economy will either rise up the line employing more labor or it will contract into a recession.

It appears so far in the data that the economy gravitates to the crossing point of the orange and blue lines. The blue dots mark the crossing points as changes in labor share shift the lines. (Labor share is increasing from left to right.)

The blue dots imply a natural rate of unemployment, because the economy gravitates to there in equilibrium. The natural rate of unemployment implied in the graph stabilizes at higher rates of labor share (to the right). Then as labor share falls to 70%, the equilibrium crossing point drops implying a higher natural rate of unemployment. Then as labor share falls below 70%, the equilibrium point reverses direction and starts rising implying a lower natural rate of unemployment.

I have prepared a video about this. The video starts out with a recap of the equation. (Sorry that I mixed up employment and unemployment a couple of times in the video.)