## 13 February 2011

### Which comes first, the nominal or real record high?

A quick puzzle: say that gas prices hit a record high in nominal dollars. And say they also, around the same time, hit a record high in real (inflation-adjusted) dollars. Which comes first?

Say the nominal price is f(t) and the real price is h(t) = f(t)g(t), where g is monotone decreasing. Then h'(t) = f'(t) g(t) + f(t) g'(t). So if the nominal price is at a maximum at time T, then f'(T) = 0 and so h'(T) = f(T) g'(T). f(T) is positive because it's a price and g'(T) is negative by assumption. So h'(T) is negative and at the time of the nominal high, the real price is already decreasing. The real high comes first, and there's a short period in which the real price is decreasing but the nominal price is still increasing. This makes sense - a nominally constant price is decreasing in real dollars.

This is brought to you by an example from Geoff Nunberg's book Talking Right, on how the Right has distorted political language in the United States in an effort to marginalize the Left. At the particular point I'm commenting on, argues that the right likes to proclaim "record highs" in gas prices which are basically always in nominal dollars and therefore make the gas price rise look worse than it is. My argument, however, says nothing about that - taking derivatives makes this a local problem, and "record highs" (nominal or real) are global maxima.

Tim Dierks said...

If prices are at a record high at any particular moment, there's no guarantee that the derivative is zero at that time (there's a discontinuity because we don't have any prices of the future).

You're correct that if there's a recent pair of local maxima on nominal & real prices and prices & currency values and their derivatives are continuous that the nominal price will have a local maxima that is after the local maxima of the real price.

However, if instead of looking for maxima, we look for the time that the previous record is broken (which is what most people mean by "hit a record high"), then the nominal price will surpass the previously highest nominal price before the real price will surpass the previously highest real price.

Michael Lugo said...

Tim, you're right that the assumption of continuity is unwarranted. And for that matter, the assumption that the inflation rate is always positive is unwarranted.

Joseph said...

Speaking as a wingnut, I thought we were more likely to complain about a decreasing value of the dollar in general (also known as inflation) than about specific manifestations of it such as gasoline prices.

dp1080 said...

This article reminds me of the mathematics behind the economic fallacy: "cutting taxes always increases revenue for the government."

Have you heard of this argument?