Friday, July 4, 2014

How good is the price level function approximation?


In this post I defined the partition function approach and noted that the "ansatz" (i.e. fancy guess)

$$
\text{(1) } P = p_{0} \frac{1}{\kappa (NGDP, M0)} \left( \frac{M0}{m_{0}}\right)^{1/\kappa (NGDP, M0) - 1}
$$

seemed to be a pretty good approximation to

$$
\text{(2) } P = p_{0} \langle a (m/m_{0})^{a - 1} \rangle
$$

which I intend to explore further in this post. First, I needed to see how the expected valueof $NGDP \sim \langle m^{a} \rangle$ (in 100 random markets again) worked against the empirical data. In the following plot I show the equation (black) along side the data (blue) in both log scale and linear scales:


This was a two parameter fit: and overall normalization of $NGDP$ and the relative normalization of $m$ so that 

$$ NGDP = n_{0} \langle (m/m_{0})^{a} \rangle $$

This fit was then used in the price level ansatz equation (1) and compared with the numerical evaluation of the expectation value equation (2). What I am doing here is trying to figure out how well the functional form (1) approximates the "true" solution (2). It turns out it fits pretty well:


The ansatz (blue dashed), which was motivated through some squishy arguments [1], is a pretty good approximation to the exact solution (black), both of which fit pretty well to the data (green), again shown in log and linear scales. This means that the approach to macroeconomics taken on this blog has some pretty solid grounding.

[1] Equation (1) uses the definition of the information transfer index as counting the number of symbols and posits that the number of symbols in the demand is proportional to NGDP, while the number of symbols in the supply is proportional to the money supply. Additionally, there is an assumption that changes in the information transfer index are slow (compared to changes in the size of the economy or the money supply) so it can be taken out the integral.

4 comments:

  1. Jason, I'm surprised to see an exponent of kappa-1 rather than 1/kappa-1 in equation 1.

    ReplyDelete
  2. Jason, an O/T aside here: regarding science and what you're doing here, do you think that the way you are going about developing this theory lends itself to establishing a clear criterion for falsification. In other words, can you say something along the lines of:

    Hypothesis X states if A then B, thus if A and not B we know X is false.

    The reason I'm wondering this is because it seems like you might be able to formulate such a statement, or the equivalent. Maybe I'm wrong though... you tell me! On most of the econ blogs I look at, I never see something like that. I never see anyone spell out clearly under what circumstances their theory can be falsified. Is that asking too much?

    ReplyDelete
    Replies
    1. If there start to be significant deviations from the model calculations (without a plausible "hyperinflation episode"), then I'd say it would be falsified. That's why I'm keeping my eye on Japan:

      http://informationtransfereconomics.blogspot.com/2014/07/more-on-japanese-inflation.html

      Additionally, if there are countries out there that can't be fit to the model then there might be an issue.

      I suspect the interest rate models might fail, too.

      In general, if I have to keep re-fitting parameters, then there's probably something wrong.

      That's the macro stuff. The underlying information transfer framework is effectively equivalent to supply and demand, so you'd have to falsify supply and demand to falsify that part.

      Delete

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.