The Calculation of TFP

One of the problems with economic models is that they seem to be based on imaginary conditions. So, when I found a PDF titled "Creating a Macroeconomic Model Using Real Economic Data" I judged that book by its cover (so to speak). I liked it right away.

That PDF is SolowProjectNotes.pdf, from the archives of Professor Rodney Smith at the University of Minnesota.

The PDF shows a calculation for Total Factor Productivity. For me that's useful, because I've been wondering for a while now how they calculate TFP. Hey, don't be intimidated by this:

See the equal sign? All that big mess of stuff on the right side of the equal sign is equal to TFP at some time t. I know this because I see TFP_t on the left side of the equal sign.

I know. I know. It's not the stuff on the left side of the equal sign that's intimidating. It's all the stuff on the right. But parts of that are familiar. Take the first part:

"Y" is output. "t" is time. Y_t is output at some particular time t.

Y_t-1 is output at the time just before time t. For example: If "t" is the year 1995, then "t-1" is 1994. And if you take output (or Y) for the year 1995 and subtract the output for 1994, then you get the change in output from 1994 to 1995. That's what this is:

Then if you divide that "change" number by Y_t-1, the output for 1994, you get the percent change number. I guess you'd have to multiply by 100 and stick a percent sign after the number to make it a percent, but that's just formatting.

And then if you look at the equation again, you've got that "Y minus Y divided by Y" thing there, percent change in Y. And you've got two other things just like it, except one of them has K instead of Y, and the other has L  instead of Y:

So you've got "percent change" calculations three times -- once for Y, which is output... once for K, which is capital... and once for L, which is labor. I didn't make those up, K is for capital and all that. It's in the PDF somewhere. But it's pretty easy to remember.

I guess C is for "consumption" so you can't use C for capital. That'd be my guess.

Going back to the complete equation again...

we've got TFP (for some year "t") equals the percent change in output, minus something times the percent change in capital, minus one-minus-something times the percent change in labor.

It's kind of nice that everything we've figured out so far is "percent change" numbers. Because subtracting "percent change" from "percent change" seems pretty reasonable. It's not like trying to subtract number-of-employees from gross-domestic-product. (You can't do that!)

The only other things in that equation are the "something" and the "one-minus-something". That's how "labor share" and "capital share" are distributed. I'm not sure why it's reasonable to do that the way they do. But it seems like it might be. Anyway, the "something" is capital share, and the "1" is 100% of what is shared, and 100% minus capital share is labor share. So there ya go.

In broad stroke then, TFP is what's left when you take output and subtract the contribution of capital and labor from it. As the formula is written, it is figuring the "percent change" in TFP, because it is subtracting percent change values from percent change values and what's left must also be percent change values.

But GDP is GDP, whether you look at it in billions of dollars or as percent change values. And the same is true of TFP.


In broad stroke, TFP is what's left when you subtract from output the contribution of capital and labor. That's not me talking. It's the SolowProjectNotes.

Anjaree says it too: "TFP growth rate = growth rate of output - growth rate of factors of production."

At Researchgate, John Ryding of RDQ Economics says it too: "total factor productivity ... is the growth of output that cannot be explained by the growth of the inputs (labor, capital etc.)"

Even the Bank of England says it, if not in so many words. In the PDF titled Measuring total factor productivity for the United Kingdom, in the appendix, they show "The formula used to calculate TFP growth":

Now, this doesn't look much like the equation from the SolowProjectNotes. But there are key similarities. On the right side of the equal sign we start with Y -- output. We subtract some messy thing that ends with K -- capital. And we subtract some other messy thing that ends with L -- labor. Total Factor Productivity is calculated by taking output and subtracting, very precisely and carefully, some number arising from capital and some number arising from labor. This is exactly the same as in the SolowProjectNotes.

Total Factor Productivity is the contribution to economic growth that's left over, after we subtract the contributions to growth that we understand -- the contributions from labor and capital. Total Factor Productivity is the discrepancy. It is the contribution to growth that we do not understand.

Don't you want to open that black box and see what's in there? I do.