### Total Factor Productivity does NOT measure what you think it does

The conventional discussion of Total Factor Productivity (TFP) starts from a production function. For that reason most people when they hear about the problems with TFP, think that it refers to aggregation problems (which on top many confuse with the capital debates, which did involve a critique of the production function, but was about more than that).

The problem is that TFP is NOT productivity at all. So start from a Cobb-Douglas production function (it doesn’t really matter what production function for our argument):
Log-linearizing and deriving with respect to time you obtain:
Where alpha is the share of wages in national income. Solow suggested that the residual (dA/A) must be the combined effect of the increase in productivity of all factors of production (TFP).

Now consider the income identity that by definition tells us that net income (net of taxes) is composed by the remuneration of capital and labor. We have:
There is no need here to be concerned with aggregation questions. This is an identity and the data reflects NOT a theoretical construct, like the production function, but a national accounting identity.
The total derivative of the identity would be:
Diving by Y, and with some manipulation, knowing that r=R/K, we get:
Note that the first two terms can be re-written using the above notation for the shares of profits (R) and wages (wN) in total income. We have:
Note any similarity with the growth accounting equation? The Solow residual is basically a weighted average of the rate of growth of profits and wages. This is what is being measured, since the data actually comes from the identity, not an imaginary production function. Even if you had a production function, you still would be measuring the identity.

Jesus Felipe and John McCombie have done extensive work, building on the work of Franklin Fisher, Herbert Simon, Anwar Shaikh and others. The use of TFP, when the evident measure of productivity should be labor productivity, only leads to additional confusion in mainstream analysis of economic growth (the other big problem being the supply constraint and the notion of a natural rate).

PS: There are several papers that one should read on (subscrirequired), but I suggest the following one (subscription required) by Felipe and McCombie that shows the kind of confusion that using TFP leads to when applied to the interpretation of the East Asian growth experience.

1. Where does the change in the rates of return come from? It seems like you need the theory to explain that. Of course, Solow is not complete.

1. Certainly you need theories. However, what you have above is NOT a theory, is the measurement of an identity. It is simply a fact, that if you measure the change of income over time, it would be a combination of the change in capital, the change in labor and a weighted average of the change in wages and profits. So Solow or any other model that is based on a production function cannot be measured, since when you try to measure the production function you DO measure the identity.

I discussed in several posts my views on distribution, which are based on the old classical political economy authors. Class conflict goes into the determination of one of the distributive variables (real wages or rate of profit) exogenously. Sraffa suggested that the rate of profit was determined exogenously by the central bank's interest rate policy, for example.

2. The Cobb-Douglas form of the Solow growth model is a theory of growth. Of course, it is. It is not a very complete theory, but it is a theory. It can be measured, though not well, just like we cannot get good measurements of dK or r.

Your accounting identity is not a theory (at least not an interesting one) by itself. It is an identity, a tautology. It is important, especially when checking whether a theory makes sense, since it must hold by definition. It does not give us much insight.

Most economists want to go deeper. They want an explanation of *why* the rate of return changes. Technology is one option. I'm glad that in your comment you offered two other possible explanations. Those could be used to rebuff the Solow explanation. Your accounting identity does not, by itself, show that A is not the term economists claim it is.

3. The fact it's an accounting identifica means it's noto a theory. On the other hand, DSGE is a theory but rejected followin Popper. Mainstream is only ideology to obtain idiot-cracy.

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3. I bought Felipe and McCombie's book at the AEA, haven't had time to read past chapter 1 at the moment (and the concluding chapter :D ) but from what I can tell is a well researched book from two very knowledgeable people who have been working on this for a long time. As they say, of the aggregate production function: not even wrong!

4. n = N/Y and k = K/Y, right? And is there a typo in the last equation? shouldn't it be new, not ndr?

1. Yes, on both counts.

5. This post is correct and useful but I don't think it will convince anyone in the mainstream. Once we have conceded that it is possible to measure K independent of r, then it is possible to decompose growth in output into the portion due to changes in the quantity of factor inputs (the first two terms in your final equation) and the portion independent of growth in factor inputs (the second two terms). By definition, the latter portion implies an improvement in the production technology, where "technology" includes anything that affects maximum output for a given set of inputs. So a mainstream economist would look at this and say, "Sure, of course -- and kdr + ndw is just another way of writing the Solow residual!" The dispute isn't over the accounting, it's over the interpretation.

Not hypothetical, by the way -- I was talking about this post with someone at a top mainstream department and that is just what he said...

1. Hi JW:
No need to tell me recalcitrant mainstremers will not be converted, to logic (capital debates?). Ask Jesus how many times he has presented this. Mind you, it is about accounting. There is no possibility that some average rate of growth of profits and wages, which is by definition what the residual measures, is anything but about income distribution. No connection whatsoever with technology or productivity. Wages may grow (fall) and profits fall (grow) with no significant or clear relation to increases in productivity. Whatever the guy you talked to said to convince himself that he can still say something that makes no sense, is a question for therapy, not science.

2. NK, what is your take on this piece: "Total Factor Productivity An Ecological-Economic Critique" by Paul Burkett - http://oae.sagepub.com/content/19/2/171.short (subscription required)

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