Solow production function and nominal values
Editor's note: Still cleaning out my backlog of unfinished posts. This is a question I've had and is addressed to the economists out there ... even after reading through a couple macro textbooks, I still don't have a good answer. File under questioning whether real and nominal are actually useful concepts in economics.
As noted here, the Solow growth model Cobb-Douglas production function
$$
Y(t) = A(t) K(t)^{\alpha} \; L(t)^{\beta}
$$
works really well if $A = constant$, and $\beta \neq 1 - \alpha$ and both $K$ and $Y$ represent nominal values.
One thing I have not been able to understand is that when using real values in the traditional version with $\alpha + \beta = 1$, call them $Y = P(t) y(t)$ and $K = P(t) k(t)$, you end up with a factor of the price level relative to the nominal version ...
$$
y(t) = A(t) k(t)^{\alpha} \; L(t)^{1 - \alpha}
$$
$$
P(t) y(t) = A(t) P(t) k(t)^{\alpha} \; L(t)^{1 - \alpha}
$$
$$
Y(t) = A(t) P(t)^{1 - \alpha} K(t)^{\alpha} \; L(t)^{1 - \alpha}
$$
The extra factor of $P^{1- \alpha}$ could be absorbed by the $L$, but labor is given in terms of hours or people (or 'labor units'), not wages ... or is it?
The nominal version of the equation seems to make more sense (it is a combinatorial problem with dimensionless nominal dollar units and 'nominal' labor units).
The real version leads to the mystery of the fall-off in Total Factor Productivity (the 'great stagnation'), and TFP -- i.e. phlogiston -- being the majority of economic growth.
Editor's note, added 5/13/2015: Sounds like a no-brainer, but it requires re-thinking all the other instances of real vs nominal ... as I've addressed here. This post was an initial draft for the one at that link that I abandoned and restarted.
Real and nominal may actually be an attempt by economists to adjust for entropy terms.