Fun with numbers

At BLS, after some pruning, we read:

What are "unit labor costs"?

Unit labor costs are calculated by dividing total labor compensation by real output...
That is, unit labor costs = total labor compensation / real output...

Unit Labor Cost (ULC) is the ratio of inflating labor cost to inflation-adjusted output. Here's the graph:

Graph #1: Unit Labor Cost for Non Farm Business (ULCNFB)

That's "total labor compensation" (in current dollars) divided by "real output" (in constant dollars). The labor number has inflation in it, and the output number does not. The arithmetic distorts the blue line upward along a path similar to inflation:

Graph #2: The GDP Deflator (red) as a Measure of Prices, and Unit Labor Cost (blue)

The blue line (unit labor cost) is similar to the red line (prices going up).

The ULC is an inflating labor number divided by inflation-adjusted GDP. If we multiply ULC by inflation-adjusted GDP, we get the actual (inflating) labor number back. Then we can divide it by the actual (inflating) GDP number to see the true relation:

Graph #3 Shows the True Relation: Total Labor Compensation Relative to GDP, Falling Since 1961

Now the blue line goes down, not up. And all we did was stop having inflation in only *ONE* of the two number series. When we have inflation in the labor number (see Graph #1), the labor number goes up like prices going up (see graph #2). When we have inflation in BOTH number series, the labor number goes DOWN. And all we did was stop doing sneaky things with price numbers.

The first two graphs leave inflation in the labor number and take it out of the GDP number, forcing the blue line to go up like the red line. The third graph leaves inflation in both the labor number and the GDP number, the way things are in the real world. It shows that labor costs have been falling for a long time.

But you probably knew that already.