Wednesday, August 10, 2016

A Little Confusion About Inflation

It may help to clear up a little confusion about inflation. Inflation is basically dimensionless— price over price— and does not have dimension of 1/time as suggested by Nick Rowe. Rather, inflation is an exchange rate. Instead of converting between currencies of different countries (say) at a single moment, inflation (and interest rates!) convert between different times the same nominal currency. That is, one may convert today’s dollars in today’s euros; likewise, one may convert yesterday’s dollars into today’s dollars. An inflation rate of 2 percent over the past year means that what cost \$100 last year today costs \$102. Thus \$100 last year is equivalent to \$102 today.

Where folks get a bit mixed up is, hey, that’s 2 percent per year. Doesn‘t that put time in the denominator? No. All we are saying is that we add 2 percent (compounded) in each period— in this case one year. For example, 10 percent inflation over 5 years is not 10/5=2 percent per year, because if we added to prices 2 percent per year for 10 years, we would wind up with a price level about 10.4 percent higher. ”Per year" describes the frequency of compounding at the specified rate.

But we still divide by time to compute the rate, right? We say $$ 1+\pi=\exp{\frac{\ln{\!\left({P_f}/{P_i}\right)}}{t_f-t_i}} $$ do we not? No, we do not. This is obvious shorthand. (Obvious because the dimensions do not work out.) More carefully, we may write $$ 1+\pi=\exp{\!\left[\Delta\frac{\ln{\!\left({P_f}/{P_i}\right)}}{t_f-t_i}\right]} $$ where $\Delta$ is the period of time between compoundings. Equivalently, $$ 1+\pi=\exp{\frac{\ln{\!\left({P_f}/{P_i}\right)}}{n}} $$ where $n$ is the (dimensionless) number of compoundings. That is, “year” specifies $\Delta$— with units of time.