The labour supply, part 2

In the previous post, I noted that the deviation of the empirical data from the model P:NGDP→LS becomes large over the long time series from the BOE. I also noticed that it seems like this deviation prevented us from defining a "natural rate" of unemployment in the model P:NGDP→U unless we account for the effect. So starting from P:NGDP→LS for the US we have this fit to the price level:

And here is the ratio of the blue line to the green line:

So if we account for this in the model P:NGDP→U by dividing
$$
P = \frac{1}{\kappa_{U}} \frac{NGDP}{U}
$$
by the factor
$$
\frac{1}{P} \frac{1}{\kappa_{LS}} \frac{NGDP}{LS}
$$
Using the parameter $\kappa_{LS}$ derived from the previous fit and then fitting on the parameter $\kappa_{U}$ we get the following best fit curve to the price level:

If we then use the price level (green curve) to represent the "natural rate" of unemployment (i.e. what the blue curve $\sim NGDP/U$ should be), we can show this plot of the unemployment rate (black) and the "natural rate" (blue dashed):

If you go through the algebra, it follows that the "natural rate" is given by
$$
u^{*} = \frac{\kappa_{LS}}{\kappa_{U}} \frac{LS}{L}
$$

where $L$ is the civilian labor force and $LS$ is the total number of people employed. The latter term is effectively the "employment rate" and is typically $\simeq 1$.

We can do a similar manipulation for the UK data resulting this fit of the model P:NGDP→U to the price level:

Which is an improvement over the naive result here. The natural rate can then be derived using the green in the previous graph as we did for the US data above:

This result has a remarkably constant "natural rate". Too remarkable -- I imagine the data for the total number of employed, the total number of unemployed and the unemployment rate are all based off of the same measure. However, the important result is that $u^{*} \simeq \kappa_{LS}/\kappa_{U}$.