# Growth Accounting

In growth accounting the production function is defined as follows:

$$Y= AF(K,L)$$

That means output $Y$ is dependent upon capital $K$, labour $L$ and technology $A$. Now to produce something, in addition to capital and labour, you also need raw material and energy resources and even land. I do not understand why this production function does not include these three primary categories of input. Can someone please explain. A simple example could be very helpful.

-
Where did you find that version of the production function? All production functions that I have seen look like Y = A*F(L,K,H,N) where H is human capital or knowledge and N is natural resources. –  Mathematician Dec 7 at 18:47
This production function is found in Mankiw's books on macroeconomics. This is also found in some journal articles where others variables can be found from or derived from publicly available data and the value of A (Technology) is calculated as the residual which balances the equation. In other words technology is the endogenous variable. Just to be sure this production function is modelling the economy at the aggregate level. Y is the output of the economy as a whole. –  Rain Dec 7 at 22:29
That's odd. My equation, Y = A*F(L,K,H,N), is also from Mankiw (Principles of Macroeconomics 6th Edition). –  Mathematician Dec 7 at 23:34
Well, precisely this is from page 260 of mankiw's book (eigth edition). it's an appendix which explains the growth accounting methodology. if you look in index for growth accounting, maybe you may find it? –  Rain Dec 8 at 0:29

Including additional inputs into a production function is a simple exercise. For example, in Acemoglu (2009):

You can include any input, but you should specify how it behaves (presumably different from $L$ and $K$).

-

Land is included inside capital. You should not think of land only as the fields for agricultural production but as any physical space needed for productive processes. Therefore, land could be the terrain where a factory is placed, a block of offices etc. Similarly, do not think that land is a fixed factor. Although the dimensions of a country are given, abandoned terrains are available for the construction of new factories and buildings, forests can be cut, residential spaces can be turned to commercial etc. It is true that all this could be added as additional factors, but that will not lead to any improvement.

With respect to energy and raw materials, notice that they are very special factors different than capital and labor. For instance, in a given year the maximum quantity of capital and labor that you can use is difficult to modify, you are constrained. However, this will change over time through capital accumulation, employment transitions etc. Changes in these two factors will determine the variation of GDP over time. However, one may think that at any point in time all energy and raw materials needed for the productive process are immediately available, they do not accumulate but are consumed and disappear, and therefore they do not play a role in determining the evolution of production over time (it is true that natural resources are a stock that deteriorate and decrease with their use, but in this framework, it is like if they were infinitely available).

Productivity can make land more productive, increasing for example the tons of grain produced by ha. This would be captured in A. Nonetheless, if technologies allow a more efficient used of resources, this would have no impact on the production function because as we saw, they do not enter as factors.

To conclude, the production function $Y=F(K,L)$ is a simplification of reality, but it is useful to the extent that it captures part of the most crucial factors that determine production and its evolution over time. It is not meant to be perfect, simply to help us understanding growth, in fact, differences in capital and labor alone can explain more than 80% of the cross-country differences of GDPpc and its growth rate. Given that the pressure on resources it is becoming more and more important and it is likely to constrained future production possibilities (if not today), it would be sensible to take this into account, and many economists do it with more sophisticated production functions.

-
"Land is included inside capital": Conceptually it can, but you should check what is included in the "Capital" data series that is actually used in "growth accounting". Check for example sdw.ecb.europa.eu/browseExplanation.do?node=2120798 –  Alecos Papadopoulos Dec 7 at 16:13

The standard growth model treads all production as equal, it is assumed that you can transform capital, consumption and all other commodities without friction.

If this is not satisfying, think about them as omitted variables. They will not cause bias in the estimators for $\log K$, $\log L$. They could bias the estimates of $\log A$, which growth regressions treat as a residual.

Personally, I don't think that there's too much to learn from growth regressions anyways. They're mostly eyeballing econometrics where we hope to learn something from correlations without having real random variation.

But apparently I'm not the majority. Some people got a lot of publications by just regressing anything on anything, or even running 2 millions of regressions and hoping to find significance somewhere.

-