Continuing in this series (here, here and here), I found Cobb and Douglas's original paper from 1928 [pdf] where their least squares fit gives them the function:
And they get a pretty good result:
Also, Noah Smith writes today:
Yes, in a Solow model you can tie capital K to observable things like structures and machines and vehicles. But you'll be left with a big residual, A.
Now if we use the information equilibrium model:
NGDP=AKαLβ
And use the "economic potential" (see also here):
NGDP=TS+X+Y+...
NGDP≈(c/κ+ξ+η+...)NGDP
So that ...
NGDP≈(c/κ+ξ+η+...)AKαLβ
=(Ac/κ+Aξ+Aη+...)KαLβ
=(Ac/κ⏟residual productivity+Aξ+Aη+...⏟measurable output)KαLβ
or
=(Ac/κ⏟entropy+Aξ+Aη+...⏟real output)KαLβ
So that we say
NGDP≈(ATS+A0)KαLβ
Noah's statement is essentially that we expect a number the size of A0, but it turns out it is large (i.e. the size of ATS+A0) and ATS is this large residual (or the whole term is the large residual). In this description, the Cobb Douglas production function works because the entropy term is approximately proportional to output: TS≈(c/κ)NGDP.
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