The Fraudulent Use of Arithmetic

Just about a year ago I considered a post by Steve Randy Waldman:

I need another look at Steve Waldman's interesting layout of our economic problem:

"Prior to the 1980s, the marginal unit of CPI was purchased from wages... Prior to the 1980s, central bankers routinely had to choose between inflation or recession."

Then came the “Great Moderation”. The signal fact of the Great Moderation was that the marginal unit of CPI was purchased from asset-related wealth and consumer credit rather than from wages.

So in this view, the policy-change was a change from intermittent wage-suppression to continuous wage-suppression. It's a tidy thought, but that doesn't make it right.

I do not accept Waldman's tidy analysis. The Federal Reserve does not manage wages.

What Waldman was saying just didn't make sense to me, despite several attempts by himself and myself to make it make sense to me.

Time goes by.

My recent "Symmetry in Deception" series drawing to a close the other day, I Googled unit labor costs and turned up interfluidity » Restraining unit labor costs is a right-wing conspiracy, from half a year back.

Waldman opens thus:

In an otherwise excellent post, Matt Yglesias commits one of the deadly sins of monetary policy:

[M]y favorite indicator of inflation is “unit labor costs”… Unit labor costs are basically wages divided [by] productivity. It’s not the price of labor, in other words, but the price of labor output. If productivity is rising faster than wages, then even if wages themselves are rising unit labor costs are falling. Conversely, if wages rise faster than productivity than unit labor costs are going up...

That all sounds reasonable. But Yglesias has fallen into a trap. Unit labor costs are not “basically wages divided [by] productivity”. That’s not the right definition at all. Unit labor costs are nominal wages per unit of output. With a little bit of math, it’s easy to show that

UNIT_LABOR_COSTS = PRICE_LEVEL × LABOR_SHARE_OF_OUTPUT

An increase in unit labor costs can mean one of two things. It can reflect an increase in the price level — inflation — or it can reflect an increase in labor’s share of output.

I've omitted Waldman's "update", where he acknowledges that something like Yglesias's definition of Unit Labor Cost -- a one hour's wage version, I think -- may be correct.

Update omitted, Waldman's definition of ULC agrees with the YourDictionary definition, and with the OECD's definition. And with my definition, as created by YourDictionary and confirmed by OECD.

(Waldman refers to "nominal wages" as labor share. My accepted sources refer to total labor cost, or wages plus benefits, let us say. The confusion here is the same confusion described in Wikipedia: Compensation of employees, which I noted here. But Waldman is close enough, for now. That discrepancy is not my topic.)

Waldman writes: Unit labor costs are nominal wages per unit of output. "Nominal" means "at the prices actually paid, inflating though they be". "Output" is GDP with caveats: Often, the word "output" is used to mean "real output" or "real GDP" or "figured at prices that have been reduced to remove the effects of inflation". All too often, with the effects of inflation surreptitiously removed. Waldman points it out.

By Waldman's definition (and mine) Unit Labor Cost divides an inflating quantity by an inflation-adjusted quantity. Division by an inflation-adjusted quantity is a back-door, sneaky, deceptive way to bring inflation *INTO* the result of the calculation. Waldman points this out, too:

UNIT_LABOR_COSTS = PRICE_LEVEL × LABOR_SHARE_OF_OUTPUT

It is a very big deal.

Rule #1: Always be wary of the word "output". It is often found in bad arithmetic.


Waldman looks at his formula, notices that two things (the price level thing, and the labor share thing) go into Unit Labor Cost, and translates the math into English:

An increase in unit labor costs can mean one of two things. It can reflect an increase in the price level — inflation — or it can reflect an increase in labor’s share of output.

Yes.

But. Let me ask you this: What is Unit Labor Cost used for? Easy answer, just recall what Yglesias said, as quoted by Waldman:

[M]y favorite indicator of inflation is “unit labor costs”
Unit Labor Cost is used as an indicator of inflation.

Right.

Multiply inflation into Labor Share of Output, use the resulting numbers as an indicator of inflation, and when you see inflation in the numbers, blame Labor Share.

Again: Multiply inflation into the Labor number, feign to discover a similarity between inflation and the resulting number, and blame the Labor number for inflation!

Used in the denominator, the calculated value called "real output" -- often simply called "output" -- surreptitiously creates the appearance of similarity between inflation and the labor number. Refer to Rule #1.


Waldman's Unit Labor Cost post isn't really about ULC. Really, it is about the Fed's "direct suppression of labor’s share" -- same as his post that I reviewed a year ago. Funny thing is, in this post that made sense to me. This part sums it up:

For labor’s share to expand, either the price level must fall, or unit labor costs must rise faster than the price level. But the Fed responds aggressively to rising unit labor costs, and is committed to preventing any decrease in the price level. Under this policy regime, expansions in labor’s share are pretty difficult to come by!

Agreed.

Okay. But now that I've belatedly agreed with Steve Randy Waldman about the suppression of labor's share, now I feel free to disagree with him over ULC.

Waldman finds a bad definition of Unit Labor Cost, corrects the definition, and then evaluates it. Given that there are two factors (prices and labor's share) that go into ULC, Waldman expresses the relation clearly: "An increase in unit labor costs can mean one of two things..."

Yes, and I don't deny it. Given the formula for Unit Labor Cost, Waldman reaches the only conclusion that can reasonably be reached.

This is where I disagree with Waldman: Given the Unit Labor Cost formula, Waldman accepts that formula. I do not.

Waldman thinks of the ULC as economics to be evaluated. I see it as bad arithmetic, to be dismissed.

If the formula// no, wait.

We don't accept or reject formulas because we think they produce results we like or don't like. Rather, formulas are either valid or not, on their own merits.

If the formula is valid, we must accept it.

If the formula is not valid, we must not accept it.

The Unit Labor Cost formula is not valid -- not when ULC is used as an indicator of inflation. It is not valid because inflation is factored into the ULC numbers. On a graph, ULC appears similar to inflation because inflation is factored into the numbers. Any comparison of ULC to inflation is a fraudulent use of arithmetic.



The Unit Labor Cost formula factors the price trend into its results and allows us to discover that the results show similarity to the price trend. The logic is circular. The logic is not valid. The formula must be rejected.

Rule #2: If "real output" is in the denominator, reject first and ask questions later.