# 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.

What does “capital utilization” mean, and how is it measured?

I first misread it as “capacity utilization” which caused me all sorts of confusion.

JzB

Your model neglects raw materials prices. Given that raw material prices, especially energy, have been seeing significant price increases over the last decade, that seems odd.

Do you have some data set to test this model against?

Jazz,

Capital utilization and capacity utilization are the same. It’s just two different ways to say the same thing. I should just stick with one, but in my mind I use both terms. Sorry for the confusion. Run74441 tells me I need to be consistent with my terms. He is right.

Jay,

I am preparing a video with data since the crisis.

On the prices of raw materials, utilization of labor and capital is a separate issue. Yes, there are effects as the economy moves up the curve, but the limit of utilization is what the model above is showing. Raw material prices, or housing prices or even consumer credit don’t change those limits.

I am looking forward to the video, Mr Lambert!

I hope you will offer some guide on how this model should

be applied…

Thank you, Sir.

Hans,

The video is already done and scheduled to be published here tomorrow morning.

Capacity U has changed over the last 50 years. If it is at 80-81, that is a sign of economic boom.

The general structure of the economy has changed over the last 50 years. You just can’t seem to grasp that.

John,

Agreed… the general structure of the economy has changed. I boil that down to lower labor share, which is suppressing capacity utilization through the years.

I would not say as a rule that a cap u of 80-81 is a sign of economic boom. That was the experience years ago. It depends on the context of other factors. And booms can be relative.

Mr Lambert, thank you I look forward to it…

The first thing that jumps out at me is that the equation is labor / capital symmetric which implies that the measures of capacity and labor utilization have to be normalized.

If a and b are zero, then the equation is just linear with a factor of MFP. The problem I see with this is that there is no penalty for substituting capital for labor or vice versa. You would imagine profits to be optimal with a particular mix. You’d have to add m more machines and w more workers in some suitable ratio to maximize profitability. You’d expect the profit curve to have something that looked like ax^n+by^n front and center with n > 2, depending on how important you think having the right mix is.

You say you want to subtract out the “diminishing returns from production and demand”. Are you trying to capture the decreasing returns as one tries to build work force or capitalization up towards 100% as labor recruitment and capital acquisition get harder? Your b term seems to behave something like this, cutting profits more and more as the markets get tighter. I guess I am arguing that the b term is definitely a good idea and is almost certainly positive and shouldn’t be too small.

It’s an interesting approach, but I’m not sure how the equation lines up with your model.

I’m also not sure of what “as capital utilization falls and employment rises” means. My naive interpretation is that companies are hiring more people, but either buying equipment and not unpacking it or turning off existing equipment. That is obviously wrong. Is the idea that they’ve bought too much machinery and hired too many people and at some point get caught overextended?

I’ll be the first to suggest that I am missing something here, probably my brain. I’ve been doing all too much calculus lately so I’m probably burned out.

Hi Kaleberg,

I have changed the equation and the coefficients. See posts from today the 18th.

There is just one coefficient now, that is a function of labor share.

And if you look at the most recent post,

http://angrybearblog.strategydemo.com/2013/10/update-to-cobra-equation.html

you will see that for 7 years, as employment increased, capital utilization decreased. The new form of the equation shows how that can happen.

The trick now is explaining how the equation works. Why would capital be used less and labor be used more when the economy is at profit maximization?