Estimating GDP per worker via acres per worker data In History class today we discussed the problem of getting GDP data for countries BEFORE those data where collected. Since GDP wasn't collected during most of the period that economic historians are interested in, historians have to estimate this information.

Let Q = the quantity produced or the GDP or the total income for a country, L = the number of workers in the country, K = the capital (i.e. machines), Z = the amount of worked land in the country and A = the productivity of labor, capital and land taken together (A is called the total factor productivity). A is just a single measure for the productivity of all the inputs to the economy. Also, let lower case letters represent the 'per worker' value of that variable. For example, q = Q / L.

First, a common production function used is called the Cobbs-Douglas function, Q = A L^a K^b Z^c where a+b+c = 1. The 'a' is the share of the total income that labor gets (i.e. total wages divided by the total income). 'b' is the share for capital and 'c' the share for land (i.e. rent). You can see why these need to add up to 1.

This function is relatively easy to get your head around and more complex production functions share some interesting economic properties with it. Economists tend to like to use it.

If a+b+c = 1, then a = 1 - b - c. So, Q = A L^a K^b Z^c = A L^(1 - b - c) K^b Z^c. Just substituting the 'a' out.

Also, if you're interested in per worker numbers than the production looks like this:
Q/L = (A L^(1 - b - c) K^b Z^c)/L
= A (L^(1 - b - c)/L) K^b Z^c
= (A K^b Z^c) / (L^( b+ c))
= (A K^b Z^c )/ (L^bL^ c)
= A (K/L)^b (Z/L)^c

Thus, q = A k^b z^c. In other words, the GDP per worker is a function of the capital per worker and the land per worker.

If you assume that capital always gets the same percentage (j) of total outcome over time (which I suspect is empirically true), then q = A (jq)^b z^c.

A fancy trick that economist like to make to get the percentage change in something is to take the log and then the derivative. Taking the log: ln(q) = ln(A (jq)^b z^c) = ln A + b ln (jq) + c ln z. (Using the laws of logarithms.)


Differentiating q'/q = A'/A + (b j q'/jq) + c z'/z = A'/A + b q'/q + c z'/z. Here, "x'/x" just means the percent change in the variable x.

But we can bring all the q's on one side... q'/q - b q'/q = q'/q (1 - b) = A'/A + c z'/z. This means q'/q = (A'/A + c z'/z)/(1 - b).

Now we got it. The change in GDP per worker as a function of the change in total factor productivity, the shares of income for capital and land and the change in land per worker.

Presumably, these data are easy to get... Just don't ask me where to get total factor productivity data, though.  ¶ 10:02 PM