Sunday, September 28, 2014

Garbage in, garbage out

Continuing yesterday's search of this site for Total Factor Productivity . . .

After the Van Zandweghe post in March 2014, my next mention of "Total Factor Productivity" came in June with Oh! I've seen this shape before.... That was my reaction when I first looked at FRED's version of TFP: Where have I seen this before?

Just a week later, in Uptrend to 1966, then stable to 1982, then uptrend again I found the matching shape. The match was SRW's Real GDP relative to the size of the labor force, at Interfluidity.

If productivity is output per worker, then maybe it makes sense that the graphs are similar. But TFP isn't "output per worker" so I'm not clear on this.

In any case, I ended that post "wondering now how they calculate TFP."

Before long I was looking at the Solow growth model.


From Roberto Chang of Rutgers:
In the original Solow model, time is continuous and the horizon is infinite. Without loss of generality assume that time is indexed by t in [0,∞).

At each point in time, there is only one final good. The good is produced via the aggregate production function:


Here Y, A, K, and L denote output, labor productivity, capital, and labor, and are functions of time (i.e. we should really write Y(t) and so on, but we omit time arguments when not needed). The product AL is called effective labor.

Once you assume the existence of the aggregate production function Y=F(K,AL), it is clear that the growth of output can be due to growth of A, K, or L.

I like that one: If we assume the program we wrote for the Holodeck accurately mirrors reality, then we can assume the results the Holodeck shows us are correct.

It always cracked me up that on Star Trek: Next Gen, when they ran into a problem they couldn't solve, they would write a computer program and let the Holodeck solve the problem. Obviously, if you cannot determine the calculation that will solve the problem, you cannot write a computer program to run that calculation for you.

On the other hand, if you know the answer you want to get, you can write the code so it gives you that answer. Then you call your code a "model" and you call yourself an economist. Roberto Chang lays this out clearly:

Once you assume the existence of the aggregate production function Y=F(K,AL), it is clear that the growth of output can [only] be due to growth of A, K, or L.


Right or wrong, the answer we get is determined by the calculation we give the computer. Still, I am fascinated by the Solow model and want to understand it.

True confession: Usually when I'm reading a PDF and I get to a formula I just completely stop. (It even happened with Mason & Jayadev's Fisher Dynamics paper.) For an old math major like me, that's embarrassing. But the 6-page SolowProjectNotes PDF gave me no trouble at all. Things are well explained:

... for each time period t, Yt is gross domestic product (GDP), Kt is the capital stock, Lt is the size of the labor force, and At is labor productivity. The parameter α is the "capital cost share" we discussed in class, δ is the rate at which capital depreciates, and s is the savings rate for a country. We said labor grew at a rate n and labor productivity grew at a rate x ...

We next discuss how to calculate/estimate the parameters in our model...

We begin with the simple calculations first.

Some of the numbers come right out of a table. And the additional explanation you want is at hand:

α is the share of production costs paid to (owners of) capital, and 1 − α is the share of production costs paid to labor. For example, for the U.S., on average 62.9 percent of GDP was paid to labor and the rest went to owners of capital: i.e., α = 0.371 and 1 − α = 0.629. For Zambia, α = 0.533 and 1 − α = 0.467.

For me, even expanding the common concept the share of production costs paid to capital to

the share of production costs paid to (owners of) capital

helps a lot. Highly recommended reading...

But remember: Back in June I wrote:

spent most of Saturday going through the PDF, gathering data and entering calculations in a spreadsheet. It's one of those things where you have to put a lot of hours into it before you can make a graph to see if you're anywhere near right. I have patience for that kind of work, but it was most of Saturday.

Good luck with it.


By the way: The A in Roberto Chang's formula and the At in the SolowGrowthModel PDF both stand for "Labor Productivity". I think Labor Productivity is related to, but not the same as, Total Factor Productivity. I can't explain the difference yet. Couple days, maybe.


Jazzbumpa said...

Labor productivity is output per labor unit per hour.

TFP is still a chimera.

According to the Wikipedia article, it doesn't even have meaningful units.

Further, the equation is

Y = A * f1(K) * f2(L)

where Y is output - which in this rendering is a product not a sum.

I can't wrap my head around that.

If it were

Y = f1(K) + f2(L) + A

you could imagine A ast an interaction of the type.

A = f3(K) * f4(L)

As is, he whole thing strikes me as nonsense, and not worth the time and effort you are devoting to it.

The Arthurian said...

Yeah, forget about the "product" version of the equation. I've seen it this way:
Y = f1(K) + f2(AL)

In English:
Output is a function of capital and a function of labor and the problem-child A.

Since we don't know what A is, maybe it should be called X. I'm interested in solving for X. But I don't see it as
X = f3(K) * f4(L)
additional functions of capital and labor. I see it as other things that may affect output either by enhancing it or by holding it back...

The debt burden, for example.