## 3 dimensional equation gives another view of the path to a recession

I am working on a new equation and want to pass it along for feedback. The equation is 3 dimensional and simulates the effective demand limit upon the utilization of labor and capital. Here is the equation…

Measure of profitability = MFP * ((x+y) – a * (x^2 * y^2) – b * (x^2 + y^2))

MFP = Marginal factor productivity
x = capital utilization rate
y = employment rate
a = coefficient of labor share to establish effective demand limit on capacity utilization. Note: If coefficient b is zero, then a can be used alone.  a = 0.88*s^2 – 2.31*s + 1.95 … (s = effective labor share, for example as 80%, 0.80).
b = coefficient as 1/6 to 1/10 of a. This coefficient may be zero thus taking out the last part of the equation, but it is added for exploration.

The equation gives a measure of profitability for utilizing more or less labor and capital. Basically the equation takes a measure of total utilization of labor and capital, (x+y), then subtracts out diminishing returns from production and demand based on labor share.

As I am just beginning to develop this equation, I present in the light of exploration. What is it saying? What does it show? What am I seeing so far? Yet, I want to present the equation because it is showing something interesting.

I have prepared a video explaining what is interesting.
The first graphic you will see in the video shows numbers representing the surface of a 3 dimensional curve. The numbers rise as profitability increases and vice versa.
What is interesting is that the plot in the graphic is showing a different path to a recession. The numbers show that as the economy recovers from a recession it will employ more capital and labor. Then, a limit is reached for capital utilization. But then the economy has a profit incentive to employ more labor, but as it does so, capital will be dis-employed. So as employment is pushed below a natural level as companies reach for profit, the decrease in capital utilization triggers a recession.

Here is the video to explain…

A recession may simply be a reach for more profits that causes a contraction in capital utilization. The mechanism would be something like… as capital utilization falls and employment rises, production is held in check and inflation begins to appear from increased money in the hands of labor. Then the Fed reacts to control inflation and then both capital and labor will start to contract in a cascading dynamic.

The economy recovers along the original path until capital utilization reaches its maximum. Then labor is employed in an effort to reach for more profits and a capital contraction is induced. Eventually the path to more profits ends in a contraction.

The contraction of capital and then labor is actually seen in historic data before a recession. This equation presents a path of profitability to explain how that might happen.

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