Noah Smith states that there is no cycle to the business cycle. (link) I responded to him on twitter saying, "...there can be definable limits without a cycle."

Can we define the limit of a business cycle?

The business cycle implies limits within which the economy fluctuates. When the economy reaches its limit, a recession eventually occurs and we call it a business cycle.

I write here on this blog about the effective demand limit upon the business cycle. Does my equation for the effective demand limit behave as a limit? In order to find out, we first need a model to describe limits. This post will lay out a model for a limit upon the economy. In a following post, I will see if my effective demand equation behaves as expected according to this model of a limit.

So how can a limit be modeled?

I developed an equation to model a limit from predator-prey dynamics in Population Ecology (link) and the work of Samuel Bowles on Lenders and Borrowers under Inequality (link, pages 42-50).

I view the limit in a business cycle working upon the utilization rates of labor and capital, namely the TFUR (total factor utilization rate). The TFUR is...

TFUR = capacity utilization*(1 - unemployment rate)

I have presented this model for the limit of the business cycle before... (link)

This graph shows the limit as a straight horizontal line. But the interplay of a limit upon production is more complex when we look at the model for population dynamics between predator and prey. (link)

This model shows that at times production rate of the prey is higher than that of the predator. The result is that we see 3 states of equilibrium. The black dots represent stable equilibrium. The white dot represents an unstable equilibrium. However, we see that the population of the prey is limited at some point.

Now I develop the model for a "2-equilibrium state" limit upon the utilization of labor and capital (TFUR).

I start with the basic equation for the upsloping line in the first graph above...

Production = P*T

P = Productive capacity of the economy

T = TFUR

Then I add the basic dynamic for establishing a consumption limit upon production.

Consumption limit = PT*T (1-T)

T(1-T) combines the upward pressure of production (T) and the downward pressure upon production (1-T). Here is a graph of the equation...

Now I modify the equation to determine a more precise limit.

Consumption limit = PT(T/L)(1-(T/L))

L = the determined limit upon T.

Let's say that the limit occurs at a TFUR of 80%. (L = 80%)

We can see that the equation crashes at a TFUR of 80%. But we want the equation to cross the production line (blue) at 80%. So we have to make one more modification.

Consumption limit = PT(aT/L)(1-(1-1/a)(T/L))

a = coefficient to raise line.

There is more to this variable a. We can use it to change the form of the line. But I will not cover that in this post. Here is the graph of this equation with a = 3. The line itself reaches its maximum at L = 80% when a = 3.



What do we have now? We have a line that represents the consumption of the production, similar to a predator-prey relationship in nature. At times consumption is stronger than production, which leads an expansion of production through higher utilization rates of labor and capital. At times, production is stronger than consumption, which would act to suppress production.

In this light, we see 2 states of equilibrium where consumption equals production.

The stable equilibrium (red dot) represents the natural limit upon production. If the economy tries to surpass this level, production will be inhibited with suppressed utilization rates of labor and capital.

The unstable equilibrium (blue dot) represents a critical point for an economy. If during a recession, the economy was to fall below this state, it would in theory totally collapse all the way to zero. Fortunately as an economy falls into a recession, there are consumption powers pushing it back to the stable equilibrium at the natural level of production.

In the following post, I will explore if the basic equation for the effective demand limit is in aggreement with this model.