## Sunday, November 3, 2013

### 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 $u^{*} \simeq \kappa_{LS}/\kappa_{U}$.

1. A few notes:

--In the second equation, all of the variables -- especially the P -- represent the empirical price level data.

--In the last graph, there are two sudden falls. A plot of the approximate labor force = total number of unemployed/unemployment rate (measures available in the BOE data) shows a smooth curve with two large spikes that correspond to extremely low unemployment rates (as low as 0.14%) during WWI and WWII ... the effective civilian labor force became suddenly much larger.

--It is interesting that the "natural rate" $u^{*} \sim 1 - u$ where $u$ is the unemployment rate with $e = 1 - u$ where $e = LS/L$ being the employment rate mentioned in the post. This means that the "natural rate" drops during recessions and calls into question the identification with either an average or the "natural rate".

1. The first item should read:

--In the second equation, all of the variables -- especially the P -- represent empirical data.

(Except the $\kappa_{LS}$, which is a fit parameter.)

2. Oh well, it might be nice for a quite well educated person to be able to understand this and you could try if being understood by almost any person is at all important to you.

3. I get it, the point was not to be understood. At least I know now.

1. I'm sorry; I don't intend to be exclusionary with jargon and math, it's just the interesting part about what I am writing about is technical. Academic economists tend to look down on drawing supply and demand diagrams, and I've been using this blog to show that they are actually much better than they realize for reasons that come from information theory. I'd refer you to Paul Krugman for someone who is much better at being non-technical than I am and who has a similar idea he tries to push when he discusses IS-LM and supply and demand diagrams:

http://krugman.blogs.nytimes.com/2011/10/09/is-lmentary/

I believe you came here though a link here (since this post is an older post that I just linked to yesterday):

which discusses a graph where the employment population ratio seems to suggest employment is back to "normal" ... my only point with respect to Mark Thoma's post was that the fourth graph down from the top shows that unemployment is still elevated.

2. Here is the data I am referencing (all set up with a model):

http://research.stlouisfed.org/fred2/graph/?g=scJ

4. Excellent, I really am grateful for the careful explanation since I wanted to understand what you have been doing. The explanation allows me to work back over and understand what you were initially getting at. Well done.

5. With the explanation added, along with the data reference, I both understand and consider this analysis significant.