$$ Q^d = \text{NGDP}(t) $$

And

$$ Q^s = \text{MB}(t) $$

As in the previous few posts, and using the equations

$$

P = \frac{1}{\kappa} \left( \frac{Q^d_\text{ref}}{Q^s_\text{ref}} \right) \left( \frac{Q^s}{Q^s_\text{ref}} \right)^{1/\kappa-1}

$$

And

$$ \frac{1}{\kappa} = \frac{\log Q^d}{\log Q^s} $$

I graphed the empirical data from FRED (in green, with the identification $P = \text{CPI}$) and the model fit (blue) along with the the previous equations as a function of $Q^s = \text{MB}$ and $Q^d = \text{NGDP}$ (the 2D surface). One can imagine the blue line as the best fit to the green curve constrained to the surface. Time goes along the curves from 1960 to 2013. The dashed gray line is the local maximum of the surface in the $Q^s$ dimension for a fixed $Q^d$ (i.e. it follows the ridgeline). I show two different perspectives on the surface (the flat region is simply the edge of the graphed region and not significant):

In the following pair, I've zoomed in on the time series from 1990 to 2013

I am going to leave my musings in the comment section, if I end up having anything novel to say.

Part of the reason I put this plot together is that I hoped it would illustrate a counterintuitive effect in the model: increases in the money supply can be deflationary in the short run. It depends on the NGDP and $\kappa$ at the time.

ReplyDeleteAnd it does! You can see the bursts of QE pushing the time series data over the ridge, resulting in a decrease in the price level.

Of course the empirical data doesn't do this -- it just goes across at a constant level before rising again.

Note that NDGP can be thought of as a time variable along the curve.

ReplyDeleteAlso note the data was fit to the entire time series so as to minimize the integrated difference between the CPI and model result.

ReplyDeleteDoes this model show that the quantitative easing hasn't really done much of anything? If we are near the local maximum/ridgeline of the surface, the gradient is approximately zero and large changes in $Q^s$ are going to have little effect.

ReplyDeleteThe location of the ridgeline is where $Q^s \sim \exp \sqrt{\log Q^d}$ for large $Q^d$. Luckily for the QTM model, this runs off to infinity so if $Q^s$ is over the ridge, slow to zero growth in the MB will push us back to the other side eventually.

There is a much better fit in this post:

ReplyDeletehttp://informationtransfereconomics.blogspot.com/2013/07/predicting-inflation-again.html