No Matter What the Red Line Does

The black line is prices.

Multiply the red line by prices, and you get the blue line.

The blue line is similar to the black line. Similar to prices.

"Of course it is," you say. "That's what happens when you multiply by prices."

Well yes, of course it is. But dividing something by "real output" is just another way to multiply by prices. Surreptitiously. It is this multiplying by prices, in my first graph, that makes Federal Spending relative to output look similar to prices. It is this that makes labor cost look similar to prices. And it is this that makes Milton Friedman's "money relative to output" look similar to prices.

1. The red line is Federal spending as a percent of GDP. The blue has prices factored in:

Graph 1: The blue line is very similar to prices

2. The red line is total labor cost as a percent of GDP (Index 2005=100). The blue has prices factored in:

Graph #2: The blue line is very similar to prices

3. The red line is M2 money as a percent of GDP. The blue has prices factored in:

Graph #3: The blue line is similar to prices

Of the three graphs, it appears that Friedman's "money relative to output" shows the least similarity to prices. Friedman's evidence is the weakest of the three.

Of course, it isn't evidence at all, really. It is only the surreptitious factoring-in of prices.

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

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.

The Lie

What's Wrong with this Picture?

Graph #1: Federal Spending Relative to Output

Hint: It uses "real output" in the denominator.


My sense of humor is a little off from the rest of the world. I'd best not leave it at that. Federal spending was not really so small, in the early years shown. Nor did it increase so rapidly in the years since.

Graph #1 is not the honest picture of "Federal Spending Relative to GDP". The honest picture shows Federal spending increasing, but nowhere near as much as on Graph #1. The honest picture is the red line in Graph #2, below:

Graph #2: The Blue Line is the Red Line with Prices Multiplied In

Graph #2 shows the same data as Graph #1 (blue, again) plus the honest picture (red). The red line does not use the "cheat" that I've been talking about for the past week. The blue line uses the cheat.

The cheat is to use "real output" in the denominator. It makes the resulting numbers increase more rapidly than honest numbers, in a pattern that mimics inflation. That's what makes the blue line so up-sloping on these graphs, and so similar to the trend of prices:

Graph 3: The Black Line is Prices. The Blue Line is the Cheat

The same cheat displayed in these three graphs is typically used in "Money relative to output" graphs and in "Unit Labor Cost" graphs. In all three cases, what seems to be a shocking increase is really just an outrageous lie.

Fake Output

They add up all the stuff we produce in a year, at the prices we paid to buy it, and they call it "nominal" output.

Then they take all the price changes out of that number and of GDP numbers for other years, so it is like there was no inflation. And they call these numbers "real" output.

This thing they call "real" output is useful for looking at changes in the volume of production, as opposed to changes in prices.

It is also useful for faking evidence to support bad arguments.

From now on, let's not call it "real output". Let's call it fake output.

Symmetry in Deception: Similarities between "Unit Labor Cost" and "Money Relative to Output"

Consider the mathematical deception entangled in the "Unit Labor Cost" (ULC) calculation, and in the comparison of "money relative to output" (MRTO) to inflation. Both follow the same pattern:

1. Divide GDP by a price index to get a fraction called "real" GDP.
2. Use this fraction as the denominator of another fraction.
3. Compare the result to inflation, and find similarity.

The MRTO uses this formula: (M2) / (GDP/Prices)
The ULC uses this formula: (Total Labor Cost) / (GDP/Prices)

In both cases we divide by the fraction (GDP/Prices). The schoolboy's rule "To divide by a fraction, invert and multiply" tells us that:

The MRTO calculation in fact is (M2 / GDP) * Prices
The ULC calculation in fact is (Total Labor Cost / GDP) * Prices

In both cases a ratio is multiplied by prices, and the resulting numbers climb upward on a path comparable to the path of prices. (For "Prices", both use the GDP Deflator.) Without multiplying by Prices, neither ratio is similar to the trend of prices.

Milton Friedman's "Money relative to output" graphs compare the quantity of money to the "real GDP" ratio, show similarity to inflation, and are used as evidence that printing money causes inflation.

The "Unit Labor Cost" calculation compares Total Labor Cost to the "real GDP" ratio, shows similarity to inflation, and is used as evidence that rising labor costs are the cause of inflation.

These are fraudulent uses of arithmetic, and are not acceptable.

"This mathematical deception is guaranteed to make labor costs look as if they are increasing on a path similar to inflation."

I said it, but I didn't show you. So, here it is:

Similarity: Unit Labor Cost and the Implicit Price Deflator
(Click graph for the FRED source page)

The Unit Labor Cost series: Suspiciously similar to the Price Deflator, don't you think?

The Uses of Fake GDP (2): Unit Labor Cost

I looked at "Unit Labor Cost" the other day. According to yourdictionary.com, ULC is calculated as "total labor costs (including benefits)" divided by "real output".

Wikipedia has a problem with that. Under Compensation of Employees (CE) it says:

The main criticisms made of the accounting concept of CE are that it can make workers' incomes look larger than they truly are, and that the main components of CE are not separately itemised in the accounts. What CE really contains is not made explicit.

Often economists confuse CE with the total wage bill of a country, which is false. They might use CE to strike a quick "wages-profits ratio" or calculate unit labor costs, without realising what they are really doing. CE is not equal to gross wages, or real disposable income of workers, nor - strictly speaking - total labour costs.

I dunno. The OECD's definition pretty much agrees with yourdictionary.com:

Unit labour costs (ULC) measure the average cost of labour per unit of output and are calculated as the ratio of total labour costs to real output.

OECD seems not to define "total labour cost" but does define Labour Cost:

Labour cost is defined as the total expenditure borne by employers in order to employ workers, a concept which has been adopted in the Community framework and complies broadly with the international definition of the International Conference of Labour Statisticians (Geneva, 1966).

Again, it seems to agree with yourdictionary.com. And it seems to be "the total" labor cost. I think I have to dismiss the objections of the Wikipedia article.


Now about that calculation: Add up all the costs you can associate with labor, and divide it by GDP. Oh -- no, that's wrong. Divide it by real GDP.

So they take a cost number and an output number, make them into a ratio, and then reduce the denominator to account for inflation. The value of the ratio must therefore be skewed upward on an inflation-like path. That is a blatant falsification of the facts.

Why bother to divide inflation out of the denominator? Why not just multiply the ratio by inflation directly? You'd get the same result.


The Wikipedia article has all sorts of complaints about the calculation. But apart from economists' confusion, the main complaint seems to be that CE "can make workers' incomes look larger than they truly are".

Sheesh. If I try to get a job that pays $10 an hour, but the employer has to pay a total of, say, $25 an hour if they hire me, then $25 is the number that determines whether anyone gets hired. If you're concerned about employment -- or unemployment -- the total labor cost number is the number you have to watch.

The Wikipedia article is pretty bad. But it's almost right about one thing. Something is made to look larger than it really is. But it's not total labor cost. It is the ULC, the Unit Labor Cost, that is made to look falsely large. But this is not accomplished by counting the costs associated with labor.

It is accomplished by stripping inflation out of the denominator of the Unit Labor Cost calculation. This mathematical deception is guaranteed to make labor costs look as if they are increasing on a path similar to inflation. When the deception is not used, it is easy to see that Unit Labor Cost is falling and has been falling for half a century.

Velocity and Other Funny Things

Two kids behind a fence

When I go to FRED and type velocity in the search box, this is what I get:

Sorted by popularity (default), M2 Velocity is first on the list.

This, then, is "Velocity":

Graph #1: The Velocity of M2 Money

Velocity is a measure of how fast money moves. Something like that. A measure of how often the average dollar is spent. But not just any dollar, and not just any spending. It is a measure of how often M2 money is used for final spending.

M2 money is like the tall kid behind the fence, but just the part you can see. Funny thing is, the short kid is M1 money, which is the money people spend. M2 money is the money we have in savings and the money we spend, all added together. The Velocity graph -- which supposedly shows how fast the average dollar is spent -- is based largely on money we don't even spend. On savings. Meanwhile, behind the fence, from the eyeballs down, is other money -- money not included in the Velocity calculation, whether we spend it or not.

The standard calculation for how often the average dollar is spent assumes that money in the spending stream and money in savings are spent alike -- but only a portion of that money is used in to figure Velocity. The rest is behind a fence.

Come to think of it, the other number, the spending used in the Velocity calculation is also behind a fence. The standard calculation uses final spending, but excludes intermediate spending behind some other fence.

And that, ladies and gentlemen, is Velocity. The ratio of two numbers behind fences.

Economists depend on it.


If you take the Velocity graph and invert it, it looks like this:

Graph #2: Inverted Velocity

No no, that's not right. It looks like this:

Graph #2: Inverted Velocity

To invert Velocity, divide the value 1 by the Velocity number.

Funny thing. Velocity is GDP divided by M2. Velocity is a fraction. When we divide 1 by Velocity we are dividing by a fraction. For me, this brings back memories of grade school math: To divide by a fraction, invert and multiply.

To calculate 1 divided by V, calculate 1 multiplied by 1/V.

But maybe it makes more sense this way: V is the fraction GDP/M2. To divide by the fraction GDP/M2, invert and multiply. So to calculate 1/(GDP/M2), figure it as 1 multiplied by (M2/GDP). The answer is M2/GDP.

Oh yeah, the funny thing: M2/GDP is the same as "the quantity of money divided by output." Perhaps you've heard of that one. Milton Friedman made it famous.

Friedman wrote:

"Changes in the quantity of money have important, and broadly predictable, economic effects. Long-period changes in the quantity of money relative to output determine the secular behavior of prices."

- The Monetary Studies of the National Bureau (PDF), page 20

Definitive, don't you think? But when you compare prices (the red line) to M2/GDP, it's hard to see any similarity at all:

Graph #3: Money Relative to Output, and Prices

Isn't that funny?

That's Final!

"Final spending" is a technical term. "Final" as opposed to "intermediate" spending. "Intermediate" spending is separated from "final" to avoid double-counting income.

If you go to the Mall and buy something for $10, that $10 is final spending. But when the store bought the thing, and when the shipper shipped it, and when the packer packed it, and when the manufacturer manufactured it, and when the engineer engineered it, and when the entrepreneur thought it up, none of the spending on it they did then was final spending.

Actually, each of them spent something on it, and each of them made something on it. If you take what they made on it, and subtract out what they spent on it, what's left is their income. And if you take all their income for the work on the thing you bought, and add up all that income, you get $10 of income. Exactly the same as the $10 you spent, the "final" spending.

"Real" output is calculated, not measured

Real GDP cannot be measured. It must be calculated.

I found a great statement to that effect from Statistics Canada. It's got a technical tone to it, but the meaning is clear:

Growth in the gross domestic product (GDP) or any other nominal value aggregate can be decomposed into two elements: a "price effect", or the part of the growth linked to inflation, and a "volume effect", which covers the change in quantities, quality and composition of the aggregate. The volume effect is presented in the National Accounts by what is referred to as the "real" series (such as the real GDP).
...

Statistics Canada uses the chain Fisher index as a measure of real GDP. Following the same sequence that we used with Equation (4), chaining Equation (6) gives us:

(7)

This is the formula used as the basis of the calculations of real GDP at both the national and provincial levels.

In case you were wondering.