Arithmetic and Integrity

These are the relations between GDP, inflation-adjusted GDP, and prices:

That is a fact. So, what does the arithmetic tell us? It says

1. If you divide "nominal" GDP by "real" GDP, you get "prices".
2. If you divide "nominal" GDP by "prices", you get "real" GDP. And
3. If you multiply "real" GDP by "prices" you get "nominal" GDP.

The three series of numbers considered here are more closely tied than braided hair. You can always convert nominal GDP values to real, by dividing prices out. You can always convert real GDP values to nominal, by multiplying prices in.

Again: The relations between these series is such that if you divide NGDP by RGDP it gives you prices; if you divide NGDP by prices you get RGDP; and if you multiply RGDP by prices you get NGDP. These are the relations, and nothing can be done about it.


On Sunday I wrote:

Unit Labor Cost is Employee Comp multiplied by the price level...

They take numbers like Employee Compensation going down relative to GDP. They times it by prices to make the numbers go up. They say Look, look! Labor costs are going up! And they claim that rising labor costs are pushing prices up.

I don't know which employee cost data is used to figure Unit Labor Cost. I know I'm in the ballpark because the lines on the graph were very close. But that's not the point. The point is, they multiply the employee cost numbers by the price level to get the Unit Labor Cost numbers.

It is a point I've made before. In mine of 6 October 2012, I wrote:

Kaminska seems to think the graph shows labor cost. How does she describe it? "The labour cost attached to the production of one unit". Oh right, right: "One unit".

As I showed the other day, the Unit Labor Cost plot is almost identical to the price level plot:

Graph #2: Unit Labor Cost (blue) and the price level (red)

The similarity between ULC and the price level is so remarkable as to inspire disbelief. And well it should, for the red line is used to calculate the blue line. To calculate Unit Labor Cost, labor costs are multiplied by prices.

And then the graph is used to claim that Labor cost makes prices go up.

The first comment on that old post disputes my view:

I'm afraid you've got everything upside down. Unit labour costs are exactly as described, the cost of one unit of output. Normally when we say this, we mean nominal unit costs. Since labour costs account for most costs, as labour costs go up so do prices. It is therefore not remotely surprising that the labour cost line and the price line move together.

Labour costs are NOT multiplied by prices. They are a nominal variable which at least partially drive prices.

There is nothing about multiplying by prices involved. Nominal gdp does go up as prices go up or or as real gdp, which is nominal gdp divided by prices, goes up. It is the nominal gdp which is the directly observed measure.

The similarity between the two lines is "so remarkable as to inspire disbelief," I said. My anonymous commenter's view is that the similarity is not remotely surprising because "labour costs account for most costs". I had to laugh at the difference of opinion. I do think that someone with enough knowledge of math and economics could determine with certainty which view is closer to the truth. This one has an answer.

Meanwhile, I stand by my view. The two lines are inordinately, unjustifiably similar. More similar than can be accounted by labor's share of cost which, unlike prices, has been falling for 30 years. The similarity is artificial.

The similarity is created when labor costs are multiplied by prices.

That brings me to the second point in the comment: "There is nothing about multiplying by prices involved," my anonymous friend writes. "Nominal gdp does go up as prices go up or or as real gdp, which is nominal gdp divided by prices, goes up."

Real GDP is Nominal GDP divided by prices, he says. He's right about that. So if we are dividing something by Real GDP, we can instead divide by "Nominal GDP divided by Prices" and get the same result. The same result, and better transparency:

But we're dividing by a fraction here. Do you remember how to do that? "Invert and multiply." To divide by a fraction, invert the fraction and multiply. We can do that:

The fraction "Nominal GDP over Prices" becomes "Prices over Nominal GDP", and the "divided by" symbol gets replaced by a multiply. But now that we're multiplying, we can rearrange the calculation a bit more:

Now it is obvious that we are dividing our original number by Nominal GDP and multiplying by prices. This is the arithmetic: You can divide something by Real GDP, or you can divide it by Nominal GDP -- actual GDP -- and multiply by prices. Either way you get the same answer.

Either way, you get the same answer.

For figuring unit labor costs, the commenter says, "Labour costs are NOT multiplied by prices ... There is nothing about multiplying by prices involved." But Unit Labor Cost is total labor compensation divided by real GDP. And dividing by "real GDP" gives the same answer as dividing by actual GDP and multiplying by prices.


Actual GDP is GDP at the prices we actually paid to buy it. They call it "nominal".


If Real GDP was "the directly observed measure" then there would be nothing wrong with the Unit Labor Cost calc. But that's not the case. Real GDP is an artificial measure, created by stripping price changes out of actual GDP.

Or if they didn't multiply prices into labor cost, and compare the resulting numbers to prices (and discover an unbelievable similarity) then what they are doing might be okay. But the arithmetic is not okay, because they multiply prices into labor cost and create the similarity they pretend to discover.

When they choose to divide labor costs by "real" GDP rather than actual GDP, they are choosing to multiply labor costs by prices. So doing, they create the appearance of similarity between Unit Labor Cost and prices. To use this artifice as evidence that labor costs have been pushing prices up is an abomination.