As a recovery gets underway after a recession, wouldn't it be nice to have an equation that would forecast the peak of the profit rate cycle? Well, below is one.

Profits are a function of labor share. If labor share rises, the profit rate will decline.

Profits are also a function of optimizing the utilization of capital or labor. Profits will peak when the utilization of either capital or labor is optimized. In the United States, the optimization of capital always takes precedent over the optimization of labor.

With that in mind, here is the equation...

Corporate profit rate = a*((c + e - (d - s)*c^{2*}e^{2})/s + b

c = capacity utilization

e = (1 - unemployment rate)

s = effective labor share, (labor share index * 0.76)

a, b and d are coefficients determined by regressions on real data.

This equation can be used to forecast the profit rate cycle. All you need are the coefficients a, b and d... an estimate of where labor share will be through the biz cycle... and a way to estimate when capacity utilization will be optimized.

### Let's see how this down...

First the regression results to determine a, b and d. Quarterly data from 1967 to 2016 is used. (link to data for Corporate Profit rate). The part of the equation highlighted in blue above is regressed against quarterly corporate profit rates varying the variable d within the blue section.

Adjusted R square is 82.4%. P-values very small. According to this best regression fit...

a = 0.3370

b = -0.5140

d = 1.453

The Adjusted R square is over 87% by using data from 1998 to present. Yet, this is a workable equation for this post.

Corporate profit rate = 0.3370*((c + e - (1.453 - s)*c^{2*}e^{2})/s - 0.5140

### So how would you use this equation?

Apart from estimating what the actual aggregate profit rate might be, it would be better to estimate the peak of the profit rate cycle to guide such things as investing and monetary policy.

When the derivative of this equation is equal to zero, you have reached the top of the profit rate cycle. Here is the derivative in terms of capacity utilization. (Capacity utilization is used instead of employment since capital, and not labor, is always optimized by corporations.)

Derivative in terms of c = 0.3370*(1 - (1.453 - s) * 2*c*e^{2})/s

You can see from this graph that the derivative reliably reaches its limit near zero. It may not have reached its zero limit this biz cycle due to OPEC lowering the price of oil a bit before.

### Let's forecast the profit rate cycle for the next biz cycle.

Let's assume that effective labor share will equal 78%. What about employment? Well, capital will optimize when unemployment is between 4.5% and 6.2%. Let's use 5.5% unemployment as the place where capital utilization will optimize. (Note: Capital optimized when unemployment was at 6.1% in this biz cycle.)

0 = 0.3370*(1 - (1.453 - 0.78) * 2*c*0.945^{2})/0.78

Solve for c...

c = 83.2%

So when capacity utilization gets close to 83%, assuming unemployment and effective labor share are on target, profit rates will be peaking.

As well, the Fed should have already started raising the Fed rate by that time because they really do not want to raise it much afterward when the profit rate is declining.

As the next recession fades and the recovery starts, we will get a better picture of where labor share, capacity utilization and unemployment are trending.