Thursday, July 2, 2015

Summertime


I am going to take short break from blogging (probably until August). I'll probably answer comments. For new readers, this is a good time to go through the old posts if you'd like. Keep in mind there are two major phase transitions in the blog. One happens in June of 2013, where I slowly figured out the relationship between the IT index and the macro model, starting about here:

http://informationtransfereconomics.blogspot.com/2013/06/what-role-does-information-transfer.html

The second happens around February 2014 where I figured out the monetary base doesn't produce a good inflation model:

http://informationtransfereconomics.blogspot.com/2014/02/the-role-of-central-bank-reserves-in.html

Just keep those dates in mind. Most of the posts before June 2013 don't have anything to do with anything that follows except for the basics of supply and demand.

If you'd like something to ponder during the hiatus, I mentioned in a footnote here:

http://informationtransfereconomics.blogspot.com/2015/05/what-is-economic-growth.html
[Secular stagnation] could be visualized as an economy taking longer and longer to explore (via dither) the nearby state space. A possibility is that not only is the volume of states growing, but the dimension is growing as well.
Here is a graph of the rate of change of volume for a "log(R)-sphere" -- a n-sphere of radius R with dimension n = log R:


The volume formula is:

$$
V_{\log R}(R) = \frac{\pi^{\log R/2}}{\Gamma (\frac{\log R}{2} + 1)} R^{\log R}
$$

A couple things to note. R here would be analogous to the monetary base (minus reserves, per the link above), so that log R is approximately (proportional to) time.

If we make the assumption of an approximately constant rate of innovation (like mutations in DNA), then new products/services appear at roughly a constant rate -- therefore the dimension of the economic space would also be proportional to log R (each bit of stuff we can spend money on is a new dimension in the optimization problem).

Increasing the dimension (from some dimension n to some dimension n* > n for some n) of a sphere at constant radius decreases its n-volume (the gamma function in the denominator always wins eventually). So you can imagine increasing n at some rate and increasing R (which always causes a sphere to increase in n-volumeat some rate could lead to a case where there is a decreasing growth rate of the volume.

No comments:

Post a Comment

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.