## Thursday, December 4, 2014

### The information transfer Solow growth model is remarkably accurate

I wanted to jump in (uninvited) on this conversation (e.g. here, here) about total factor productivity (TFP, aka phlogiston) using some previous work I've done with the information transfer model and the Solow growth model.

In using the Solow growth model, economists assume the Cobb-Douglas form

$$Y = A K^{\alpha} L^{\beta}$$

as well as assume $\alpha + \beta = 1$ to enforce constant returns to scale and then assign the remaining variation to $A$, the Solow residual aka TFP.

The conversation linked at the top of this post is then about why TFP seems to have slowed down (e.g. the Great Stagnation or whatever your model is). This is all a bit funny to me because it is effectively asking why the fudge factor is going away (if TFP is constant then $A$ is just a constant in the formula above).

Well, my intended contribution was to say "Hey, what if $\alpha$ and $\beta$ are changing?". In the information transfer model, the Solow growth model follows from a little bit of algebra and gives us

$$Y = A K^{1/\kappa_{1}} L^{1/\kappa_{2}}$$

where $\kappa_{i}$ are the information transfer indices in the markets $p_{1} : Y \rightarrow K$ and $p_{2} : Y \rightarrow L$.

The first step in looking at changing $\kappa$ is to look at constant $\kappa$ (and constant $A$). That threw me off my original intent because, well ... because the information transfer Solow growth model with constant TFP and constant $\kappa_{i}$ is a perfect fit:

It's so good, I had to make sure I wasn't implicitly using GDP data to fit GDP data. Even the derivative (NGDP growth) is basically a perfect model (for economics):

Of course, in the information transfer model $\kappa_{1}$ and $\kappa_{2}$ have no a priori relationship to each other and in fact we have

$$\frac{1}{\kappa_{1}} + \frac{1}{\kappa_{2}} = 1.25$$

or individually $\kappa_{1} = 1.18$ and $\kappa_{2} = 2.50$. So there aren't "constant returns to scale".

In the information transfer model, this is not a big worry. The two numbers represent the relative information entropy in the widgets represented by each input (dollars of capital and number of employees, respectively) relative to the widgets represented in the output (dollars of NGDP) -- why should those things add up to one in any combination? That is to say the values of $\kappa$ above simply say there are fewer indistinguishable types of jobs than there are indistinguishable types of capital investments so adding a dollar of capital adds a lot more entropy (unknown ways in which it could be allocated) than adding an employee. A dollar of capital is interchangeable for a lot of different things (computers, airplanes, paper) whereas a teacher or an engineer tend to go into teaching and engineering slots**. Adding the former adds more entropy, and entropy means economic growth.

PS Added 12/5/2014 12pm PST:  The results above use nominal capital and nominal GDP rather than the usual real capital and real output (RGDP). The results with 'real' (not nominal) values don't work as well. I am becoming increasingly convinced that "real" quantities may not be very meaningful.

** This is highly speculative, but it lends itself to a strange interpretation of salaries. Economists may make more money than sociologists because they are interchangeable among a larger class of jobs; CEO's may make the largest amount of money because they are interchangeable among e.g. every level of management and probably most of the entry level positions. A less specific job description (higher information entropy in filling that job) corresponds with a bigger contribution to NGDP and hence a higher salary.

1. I like your highly speculative interpretation of relative worth of professions. Given that this is somewhat accurate, how to explain skilled trades (i.e. physicians) who specialize for a very specific, highly skilled, and highly regulated trade, yet are compensated (relatively) well?

1. Actually in checking the definitive source for all information (wikipedia), turns out there is a modification of the Solow growth model that adds another term for "human capital". Perhaps this would be a partial answer to my previous question- highly skilled individuals can also claim a chunk of entropic salary?

http://en.wikipedia.org/wiki/Solow%E2%80%93Swan_model

2. Ha! Maybe wikipedia is more of an absolute reference point?

Yes, depending on how the Solow model is used, TFP includes both training and knowledge as well as technology improvements. However in the ITM-Solow model, all of those factors are basically constant since 'A' doesn't change.

I should probably extend the footnote idea a bit -- it ignores supply and demand and costs to entry like education and licensing. Physicians both spend a lot on education and are limited in supply by the AMA in the US.

You'd imagine the market, given CEO salaries are so high in the US, would move to produce more CEOs to bring down their salaries. This happens in other fields -- you get more computer programmers if salaries are high and fewer when salaries are low. But CEO salaries are consistently high, which may reflect an equilibrium price, not supply or demand shocks. The equilibrium price might be the 'entropic salary' and differences from that are based on supply and demand.

It's fun to think about, but still pretty speculative.

3. I think that the market just might tend to produce more CEOs, except for the monopolistic power of large corporations. So-called "free" markets tend to monopoly, as was evident by the late 19th century, hence anti-trust legislation. OC, today anti-trust laws have almost been eviscerated.

4. Hi Bill,

My thinking now is that maybe this mechanism is more at play ...

http://informationtransfereconomics.blogspot.com/2015/04/solving-dark-matter-problem.html

There is definitely some non-economic forces going on as the people who sit on corporate boards are frequently CEOs of other companies or in general boosters of the CEOs they vote for ...

2. Jason, can this be used to forecast growth? I saw your footnote there about real vs nominal. Too bad. Allan Gregory discusses "Why real?" in his latest post.

1. Inasmuch as you think you can extrapolate capital and labor supply.

That doesn't really explain "why real" with an answer besides "well, theorists at one time thought to use real growth in their models, but it is still under investigation ... so forecasters use real growth".

???!

3. This does seem like a remarkable fit. Do you suppose that some(one in particular) might say it's just a "spurious correlation?"

1. No because it also fits first differences (derivative).

Comments are welcome. Please see the Moderation and comment policy.

Also, try to avoid the use of dollar signs as they interfere with my setup of mathjax. I left it set up that way because I think this is funny for an economics blog. You can use € or £ instead.