## Saturday, May 16, 2015

### Cobb and Douglas didn't have changing TFP, and Is TFP entropy?

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:

$$P = 1.01 L^{3/4} C^{1/4}$$

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.

$$NGDP = A \; K^{\alpha} \; L^{\beta}$$

$$NGDP = TS + X + Y + ...$$

$$NGDP \approx (c/\kappa + \xi + \eta + ... ) NGDP$$

So that ...

$$NGDP \approx (c/\kappa + \xi + \eta + ... ) A \; K^{\alpha} \; L^{\beta}$$

$$= (A c/\kappa + A \xi + A \eta + ... ) \; K^{\alpha} \; L^{\beta}$$

$$= (\underbrace{A c/\kappa}_{\text{residual productivity}} + \underbrace{A \xi + A \eta + ...}_{\text{measurable output}}) \; K^{\alpha} \; L^{\beta}$$

or

$$= (\underbrace{A c/\kappa}_{\text{entropy}} + \underbrace{A \xi + A \eta + ...}_{\text{real output}}) \; K^{\alpha} \; L^{\beta}$$

So that we say

$$NGDP \approx (A_{TS} + A_{0}) \; K^{\alpha} \; L^{\beta}$$

Noah's statement is essentially that we expect a number the size of $A_{0}$, but it turns out it is large (i.e. the size of $A_{TS} + A_{0}$) and $A_{TS}$ 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: $TS \approx (c/\kappa) NGDP$.