In the preceding post, I showed a line calculated by subtracting potential real GDP from effective demand. It is a measure of how far effective demand is over the center line of the business cycle. I am now going to plot this value against the UT index since 1967. Here is the graph...

The lines fan out from a y-intercept. Actually the earliest contractions are down low and then with each succeeding contraction, the slopes increase. The most recent contraction, the crisis, is over the others. The most recent data, 4Q-2012, is at a UT index of 2.82% and a value of $669 billions for effective demand over potential real GDP.

I want to start zooming in on the y-intercept.

You can see the lines don't cross the y-intercept cleanly, but there is a tight pattern there. Above you can see the most recent line coming at a high angle. Let's look even closer...

There is a lot of action going on around the y-intercept. Let's see if we can make sense of it. First of all, let's recall the equation for effective demand, ED...

ED = real GDP * (els/cu(1-u))

els = effective labor share ... cu = capacity utilization ... u = unemployment rate.

Now, when UT is equal to zero, (els/cu(1-u)) = 1.

Next, we look at the equation for potential real GDP, Y*...

Y* = real GDP - $3000 billion * (cu - els)/els

In the graph above, when UT = 0, effective demand - potential real GDP = $150 to $190 billion. Or approximately $170 billion..

Therefore at UT = 0, ED - Y* = real GDP - (real GDP - $3000 billion * (cu - els)/els) = $170 billion

Simplify, $3000 billion * (cu - els)/els) = $170 billion

cu = (1 + 170/3000)els = 1.0567 * els

Recall from above, (els/cu(1-u)) = 1 and also, 1.0567 * els/cu = 1,

therefore when UT = 0, (els/cu(1-u)) = 1.0567 * els/cu

Simplify for unemployment**, u =0.0567/1.0567 = 5.4% **

5.4% is the natural rate of unemployment at $170 billion. When the line crossed the y-intercept around $110 billion in the 60's, the natural rate of unemployment was 0.0367/1.0367 = 3.5%. (110/3000 = 0.0367).

In order for the natural rate of unemployment to have risen to 7.0%, the line must now cross at $225 billion. It looks to be close to that trajectory in the first 2 graphs above. We will see.