# Absolute Value of Total Factor Productivity in an Aggregate Cobb-Douglas Production Function

A standard formulation of the Cobb-Douglas production function (e.g. here) is: $$Y=AK^{\alpha}L^{\beta}$$ Have there been estimates, at aggregate level for any large economy, of the absolute value of total factor productivity $A$ (at a defined date, and for defined units of measurement of output $Y$, capital $K$ and labour $L$?). Even a rough order of magnitude would be of interest.

From searching the web, there have been many studies that estimate changes in total factor productivity over time, but I could not find any that focused on its absolute magnitude.

• The units of $A$ would be rather strange, as they would be the units of $\dfrac{Y}{K^\alpha L^\beta}$ and there is no reason to suppose $\alpha$ and $\beta$ are integers or even rational numbers – Henry Nov 25 '16 at 20:57

Applying a regression model

$$\ln Y = \ln A + a\ln K + b\ln L$$

on a standard macroeconomic time-series data set from a country would immediately provide an estimate for $A$, essentially an average over the sample.

The problem is that experience has shown us that we don't have $A$ but $A_t$, i.e. it is time-varying. As such, estimates of $A$ as though it was a constant seem rather meaningless or even misleading.

More-over, in a time-varying coefficient framework, what is important is the change, rather than the absolute value, since it is essentially a mark-up index on the production function, and not something that can be meaningfully measured in absolute units. In this case, we take $A_0=1$ (the first period in the sample) and we track the percentage change over the sample.

Henry is right. The units of $A$ depends on how you are defining the variables $Y$, $K$ and $L$. You could only define an "absolute value" of $A$ is you have strong reason to believe that your definitions for these variables is the "correct" one (how they are computed, their units, etc). This is not trivial. For example, notice that GDP is, by definition, a nominal variable (that is how the data is recorded/constructed). Real GDP is a derivation from nominal GDP, based on some assumed price index. Therefore, the definition of $A$ will be affected by the choice of such index, which is not trivial.

Additionally, such "absolute value" of $A$ might be incorrect because the production function could be wrong. For example, the model does not include human capital, or energy, or land, etc. Or the functional form could be incorrect (CES?), or its parameters might vary over time.

But even more severe, an aggregate production function (and therefore $A$) might not even exist. This is the old-but-still-relevant issue of the impossibility of aggregation.