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.