Some Fun With Money Demand

I was doing some thinking about augmenting my liquidity trap post with an interest elastic money demand function of the form:

$$ (1) \: \log M_t + i_t = \log P_t $$

Solving for inflation in this model results in

$$ (2) \: \pi_t = \Delta \log M_t + \Delta E_t \pi_{t+1} $$

(The nominal interest rate term changes to expected inflation because the real interest rate is assumed to be constant). So far, everything looks unremarkable, but solving forward yields interesting results.

$$ (3) \: \pi_t = \Delta \log M_t - E_t \sum^{\infty}_{j=0} \Delta \log M_{t + 1 + j} - E_{t-1} \sum^{\infty}_{j=0} \pi_{t + j} $$

So, what does this tell us? Well, a couple things. Namely,

1.  Inflation is dependent on three things: the growth rate of the money supply this period, the sum of all expected money growth, and the sum of all expected inflation.

2. In order for an increase in the money supply to cause inflation it must not be accompanied by a reduction in expected money growth or an increase in expected inflation.

Knowing these two things allows us to come to the conclusion that increases in the money supply that are expected to be reversed in the future will not be inflationary.

This post kind of lacks a conclusion, but I'm going to make up for it by writing a post that evaluates quantitative easing using the findings from this post and the 'How to Escape a Liquidity Trap' post.