## Tuesday, April 4, 2017

### Productivity, growth, and Verdoorn's law

... or rather, We know nothing about productivity, growth, or Verdoorn's law.

Reading Chris Dillow's post on wages and productivity (and especially the accompanying graph) inspired me to take at look at UK output and productivity data. I also wanted to bring in some insights from my previous post about the major transitions in our macroeconomies obscuring the relationships between macro variables.

First, let's discuss Verdoorn's law which originally said that productivity rises with the square root of output. This is actually a very simple information equilibrium relationship RGDP ⇄ P with information transfer index k = 2. Rather straightforwardly, this implies:

log P ~ 0.5 log RGDP

or

P ~ √RGDP

Since RGDP = NGDP/DEF (where DEF is the GDP deflator), we can look at the long run dynamic equilibrium for NGDP and DEF and find what the dynamic equilibrium for RGDP is. Here are the results (solid lines) for NGDP and DEF followed by RGDP:

So we'll use this dynamic equilibrium to fit to the productivity data. The actual function I fit was

log P = a log NGDP - b log DEF + c

But the best fit parameters found that a = 0.57 and b = 0.47 (and c = 3.3) which is pretty close to what would be expected for Verdoorn's law (a = b = 0.5). So does that mean Verdoorn's law is true? And if so, what of the post-financial crisis productivity drop in the UK? In order to try and understand the data, I also posited a dynamic equilibrium model for productivity. The two models are shown in the following graph of the UK productivity index:

Where we have data, these are not too different from each other. However outside of the data, they are very different. First, let's look at where we have data. We can see the fit Verdoorn's law (equation above, blue dashed below) is pretty closely approximated by √RGDP (gray dashed). The vertical lines represent the centers of the shocks to the NGDP and DEF data above. However the dynamic equilibrium model works just as well and also closely follows √RGDP within the noise of the data.

The difference between these two models is mostly outside the region where we have data:

The dynamic equilibrium model hints at a long process of a productivity growth surge across the 20th century (centered at 1949.7) and that "normal" productivity growth is zero. The Verdoorn's law frame sees the productivity data as part of a more recent surge associated with the high inflation of the 70s. Going forward, the dynamic equilibrium frame sees continued falling productivity growth while the Verdoorn's law frame sees constant, but lower, productivity growth.

The recent fall in productivity growth is part of the dynamic equilibrium picture. In the Verdoorn's law picture, it is anomalously low (a productivity "puzzle").

Which is correct?

The data doesn't tell us! The dynamic equilibrium view is definitely more speculative (we are estimating a peaked function where we don't have any data from the peak). The high correlation between productivity growth and RGDP growth definitely lends credence to the Verdoorn's law picture, but that correlation seems to be fading with the more recent data. We could imagine the dynamic equilibrium model with the stochastic innovation term coming from the same source as the one for the RGDP data.

Overall, we have the same problem I pointed out in the previous post: our macroeconomic data comes from periods that are still strongly affected by major shocks or transitions. It is really hard to say what is "normal" or what is a "normal" correlation. Is log P ~ 0.5 log RGDP? Or is log P ~ constant + 20th century industrialization? Since the only quality data comes from a limited post-war time period, we can barely start to address these questions scientifically.