I am exploring a new equation to describe Effective Demand. The equation is not perfect... yet, but it is showing some things to look at.

Here is the equation...

measure of profitability = l * (c + e) - (c^2 + e^2 - c * e)

l = effective labor share

c = capacity utilization

e = employment rate (94% for example which means 6% unemployment)

The equation has two parts, a demand part and a production part.

The demand part...

l * (c + e) ... This part says that for the utilization rate of labor and capital, there is a portion determined by effective labor share that is paid out for demand of production.

The production part...

(c^2 + e^2 - c * e) ... This part is subtracted from the demand part. It is an expression to give a measure of the combined utilization of labor and capital. It is different than just a straight average.

The equation seeks to give a measure of the potential profitability for business in the aggregate.

Here is a little video showing what the equation looks like graphed out. (web site source for #D graph)

The graph shows that capacity utilization and employment will reach their maximums at 100%, where the effective labor share constraint is.

As the effective labor share declines, the maximum point of the graph moves with it.

When I apply this equation to actual data since the crisis. I get some graphs.

Graph #1

This graph shows contour lines from the 3d graph as the economy recovers from the crisis. The red dots show where the economy was at periods of 6 months. The first dot up to the left was low on the contour and showed there were returns to production available. Over time, the red dots are closer to the maximums of the contours meaning less profit potential available.

The measure of profitability has been dropping since the crisis by this is not normal. Usually the economy just moves up the contour but this time is different because labor share has been dropping, which shifts the lines down. Here is what the graph would look like if labor share had not fallen since the crisis.

Graph #2

As you can see, the measure of profitability would have continued upward on a normal path. Yet, the fall in labor share has hurt aggregate demand and thus profitability, thus many people are complaining about a sub-par economy. Graph #1 would say that the problem has to do with falling labor share of income.

Graph #2 is saying that if we factor out the effects of labor share, the utilization of capital has been recovering on a normal course.

In graph #1, as the utlization rate of capital increases, the profitability of the economy rises. So a business has an incentive to use capital for production. However, business must hire labor to run much of the capital, not always. Yet, there is a disencentive to hire labor as seen in the next graph.

Graph #3

The curve is sloping downward here, because on the surface of the 3D graph, the economy would be on the downsloping part over the topside. So increases in utilization of capital rise on the surface, whereas increases in employment slide down the surface. Be that as it may, I will flow with the logic of the moment and see where it leads.

As unemployment falls, the measure of profitability falls too according to the downward sloping curve. So as the economy recovers, it is profitable to put capital back to work. But it is not profitable to put labor back to work. However, if labor is not put back to work, and the labor share falls, the contours lines will fall even faster, thus lowering the measure of profitability even more.

Here is what graph #3 would look like if labor share hadn't fallen.

Graph #4

You may wonder why these lines shifted upward, but stayed steady in the graph for capacity utilization. The reason is that capital was employed much faster than labor in the early stages of the recovery. Whereas capital was re-employed at a normal rate, employment lagged behind. The result would have been an increase in potential profitability even for employment. The contours of employment would have reached a point of equilibrium with capacity utilization, as seen by the lines staying steady at a higher level of potential profitability.

So, even though profitability declines with employment, the general tendency if labor share stays constant is for the contour lines of employment to shift upwards as the economy recovers. Thus there are exogenous factors that make employment profitable. But the fall in labor share negated those factors.

Nevertheless, graph #3 shows that labor share fell after the crisis bringing down profitability.

We may wonder why profits are so high while these graphs are showing that profitability has declined. It is no wonder. The drop in labor share has pushed profits up at the expense of employment. It is like the mystery of a monopoly. How can a monopoly make more profits by producing less? It's a similar situation here.

OK... just wanted to show this equation as I am working on it.