I need a reference post for the diminishing effect of monetary expansion in the US (and the negative effect in Japan). The motivation of the model is here. The following graphs represent fits to the US (and Japanese) CPI (core for the US and so-called 'core-core' for Japan). In both cases we use the currency component of the monetary base (referred to as the 'base'). Here is the model for the US:
We can see the reduced sensitivity to monetary expansion if we look at the derivative of the price level with respect to the monetary base:
Another way to see the same effect is to look at the exponent of MB in the model equation (log P ~ β log MB), shown in this plot, color coded for larger values of β (red) and smaller values (blue):
In the inset graph, I plotted P vs MB corresponding to the earliest (1960, red) and latest (2014, blue) points on the graph. The slope is lower for the blue line, which corresponds to the right side of the second graph above and higher for the red line which corresponds to the left side.
And finally, here is the plot of the fit to the price level for Japan:
The key driver in the diminishing effect of monetary expansion is the size of the monetary base relative to the size of the economy (measured by NGDP) -- when the base gets large relative to the economy (and the economy gets large), then the same percentage increase in the monetary base leads to a smaller increase in the price level with the larger monetary base. This overall effect can be seen in this graph of the price level vesus the monetary base for several countries:
The diminishing effect of monetary expansion is seen as the curve bends downward as you move to the right. An additional point is that just because a country finds itself on the right side of the graph above, doesn't mean it is doomed to remain there. The US actually had a very large monetary base relative to the size of its economy during the period just before and after the Great Depression. A bout of accelerating inflation during/after WWII caused NGDP to grow much faster than the base, leading to an overall reduction in the ratio of MB/NGDP.
I think in your symbolic solution (integration) of the differential equation you treat k as constant. I think that works if k is a constant, but fails if k is variable in time or S or D that solution would not be correct.ReplyDelete
"(5) dAD/AD=1/κ dM/M
<<< Step not shown:
∫dAD′/AD′=∫1/κ dM′M´ <-- k can only come out of integration if k is constant
if k is a funtion of time, k(t), then k is not constant and you could non do this. Then the simbolic solution would not be exactly correct.
Using equation (4) again, we have
(6) P=1/k (AD0/M0) (M/M0) raised to the power of 1/κ−1"
But, in your fit in figure 1, K has GDP(t) term which would make it a driving function of your differential equation and k not constant.
Do you have some fitting situation that can closely fit any thing? Is 1/k like v and its variation is neglegable to the variation in the other varaiables?
Thanks for your comment.Delete
There is an assumption that k is slowly varying with respect to M and AD and can be pulled out the integral (treated as constant, with negligible variation over the time series).
The assumption is specifically stated at this link:
This assumption is justified empirically, e.g. here:
The assumption is justified theoretically, e.g. here:
I also numerically integrated the exact differential equation:
In all, the assumption that k is slowly varying with respect to AD and M turns out to be a very good approximation.