## Wednesday, September 4, 2013

### The Fisher equation and information transfer

David Glasner talks about the Fisher equation breaking down:
"In other words, if r + dP/dt < 0, where r is the real rate of interest and dP/dt is the expected rate of inflation, then r < -dP/dt. But since i, the nominal rate of interest, cannot be less than zero, the Fisher equation does not hold, and must be replaced by the Fisher inequality i > r + dP/dt."
The Fisher equation represents a "minimal coupling" between the price level and the interest rate, which would be implemented in the information transfer framework ($i : NGDP \rightarrow MB$) as:

$$\text{(1) } i = r + \frac{dP}{dt} = \frac{1}{\kappa}\frac{NGDP}{MB}$$
Glasner refers to it as a pathology (that applies to our current circumstances in the US since 2008):
"Perhaps one way to think about the nature of the pathology is that the Fisher equation has been replaced by the Fisher inequality, a world in which changes in inflation expectations are reflected in changes in real interest rates instead of changes in nominal rates ... "
And I'd agree that it is a pathology. In particular, it seems to be a specific pathology where the market defined by Eq. (1) breaks down (in the thermodynamic analogy, zero price means a zero pressure system -- no atoms, no pressure). The market no longer transfers information that is detected by the interest rate.