A couple weeks ago, I found the work of Anwar Shaikh. Briefly, he has developed unique and better models to describe the economy, so that there is no need to use rational expectations, utility functions, monopoly power, and much more. He describes the economy as always developing around the central aspects of* Real Competition* and profitability.

He has numerous original equations and models.

He researches effective demand, as I do, so there is common ground. Our work shares similar factors used to determine effective demand. So I see areas where our work can blend toward better models.

This post is concerned with… Is there an equation to describe the profit maximization that takes place as the economy hits the effective demand limit?

In Shaikh’s book, *Capitalism: Competiton, Conflict and Crisis*, he writes about profit rates, wage share, unemployment and capacity utilization. These are factors that I use in my own work of effective demand. I was hoping to see an equation in his book to determine the aggregate profit rate of business as it is related to an effective demand limit, but he does not seem to have the effective demand limit nailed down yet. So he presents in a general way the idea that profit rates peak at an effective demand limit…

“So it becomes crucial to ascertain the conditions under which a rise in the real wage causes the normal rate of profit to fall.”

He is describing the rise in real wages,or labor share, to such a point as to make the aggregate profit rate fall. This would describe the effective demand limit.

So I set out to find an equation…

First, a brief explanation of effective demand as I see it.

Of aggregate national income, labor receives its labor share, or wage share as Anwar Shaikh calls it. Capitalists will then optimize the utilization of their capital to make profits off of the labor share. In the aggregate, capital does not want to make profits off of its own capital utilization. That would make profit rates decline. Labor share becomes the measure by which capital **in the aggregate** can determine its maximum profit rate.

Think of it this way… In general…

Labor share > utilization of capital … in order to insure increasing profit rates.

The portion of income from production that goes to labor must be greater than the portion of capital used for production. If utilization rates of capital in the aggregate supersede the effective labor share rate, then capital must consume its own income for production and capital profits will decline by eating themselves.

But not all of labor works… A certain % of labor is employed. So the true proportional share going to labor is found by…

Labor share/employment

Employment = (1 – unemployment)

Anwar Shaikh uses this measure of employment, (1 – unemployment), quite a bit in his work as I do.

From my work, the effective demand limit that labor share establishes upon utilization rates of capital is found in the central tendency of labor share within the cycles of capacity utilization. Anwar Shaikh is a proponent of central tendencies.

The equation from the central tendency to show the effective demand limit upon utilization rates of capital is…

Effective labor share = Labor share * 0.76

This is similar to Anwar Shaikh’s concept of a normal capacity utilization rate around which capacity utilization gravitates. I can offer Anwar Shaikh my equation using labor share to determine this normal rate.

So now the ideas come together in this equation.

Effective labor share/employment = Proportional effective labor share/100% employment

Let’s say for example that effective labor share is 75% and employment 95%. Then the left side of the equation says the 95% employed workers are receiving proportionally 79% labor share. The labor share is given to only 95% of the labor force, not the whole labor force. If 100% of the labor force was employed, they would receive 79% of the income at the current production.

Proportional effective labor share = 79%

This 79% now becomes the limit threshold upon capital utilization. If capacity utilization were to rise above this limit, profit rates fpr capital income would start to fall. Here is a graph from my work to show that profit rates peak when capacity utilization is hitting the Proportional effective labor share limit. (1967 to 2015, years left out on x-axis.)

Ok now… Can the same variables be used to estimate the profit rate itself? Yes…

Here is the equation which is built upon my cobra equation…

Corporate profit rate = b * (2*(cap + emp + T^{2}*els) – 3*(T^{2} + els))/2*els

cap = capacity utilization

emp = (1 – unemployment rate)

T = cap * emp

els = effective labor share

b = coefficient, 0.352 for United States

The equation uses just one simple variable coefficient (b). It is designed to give decreasing returns to utilization at the effective demand limit. Here is a graph of annual data since 1969… (data on profit rate)

The lines have a 93.6% correlation. They have the same average of 7.9%. The standard deviation of the difference between the two lines is just 0.5%. It is a good fit.

Here is what the equation looks like in 3-dimensional space with an effective labor share of 75%. It is like a cobra. (link to graph site)

The x and y axes are for capacity utilization and employment from 0% to 100%. The z-vertical has a range from 0% to 10% profit rate in the picture. Profit rates climb upwards on the back of the cobra increasing profits as they go.

So as Anwar Shaikh wrote…

“So it becomes crucial to ascertain the conditions under which a rise in the real wage causes the normal rate of profit to fall.”

The equation describes the conditions. It shows that as labor share rises, or as real wages rise faster than productivity, the aggregate profit rate falls. That is what we have seen since 2014. Labor share has risen 4% since profit rates peaked. It is a common dynamic at the end of a business cycle.

and Yes… we are heading toward an economic contraction… the dynamics of the equation above are pointing in that direction.