Noah Smith states that there is no “cycle” to the business cycle. (link) My response on twitter was, “…there can be definable limits without a cycle.”

Can we define the limit of a business cycle?

I have a model that describes what a limit to the business cycle might look like in terms of a TFUR (total factor utilization rate). (link)  The TFUR is a measure of utilizing both labor and capital…

TFUR = capacity utilization*(1 – unemployment rate)

The model and its equation are…

Production = PT
Effective Demand Consumption line, EDCL = PT(aT/L)(1 -(1 – 1/a)(T/L))

P = Productive capacity of the economy
T = TFUR
(PT = real GDP ouput)
L = Limit function upon T. (80% in graph) (This variable will be developed below)
a = coefficient to cross lines at L.

In the graph, the blue line is production. As utilization of labor and capital increase, production increases. The orange line represents the effective demand function to determine the consumption limit upon a business cycle. The point where the maximum of the effective-demand consumption line crosses the production line gives the natural limit of output. If production goes beyond the crossing point of the maximum, effective demand goes lower than production, which inhibits production. Production then returns to the stable equilibrium at the maximum of effective demand.

So, what variables in the economy will help us determine the effective demand limit function (L)? First let's look at the TFUR through the decades.

We can see cycles where the TFUR rises and then falls. I have marked the peaks of those cycles to show the limits of business cycles. These limits have been falling over the decades to lower combined utilization rates of labor and capital (TFUR). The task now is to find variables that trace a limit line on top of those business cycle peaks.

I normally use labor share to represent effective demand. (Non-farm business sector, link for data) Let's plot the labor share index (LS) against the TFUR with the following equation…