The news is no stranger to third derivatives, although it doesn’t sneak in very often – we’ve mentioned before the October 1996 issue of the Notices of the AMS in which Hugo Rossi wrote, “In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection.”
Well, just like a blog that doesn’t post for so long that you figure it’s basically dead, and then all of a sudden out of nowhere WHOA there’s a new post (Hi everybody!), the third derivative made a new appearance recently. On the appropriately palindromic- (in the US) date of March 10, 2013, Paul Krugman wrote in the Opinion pages of the New York Times:
People still talk as if the deficit were exploding, as if the United States budget were on an unsustainable path; in fact, the deficit is falling more rapidly than it has for generations, it is already down to sustainable levels, and it is too small given the state of the economy.
Did you catch that? The line about the deficit falling more rapidly than it has been? Let’s take a closer look:
Assume that the National Debt at year t is the original function: D(t). This is positive, since we have debt.
Then the Deficit is the derivative, D'(t). It’s also positive, because the National Debt is increasing.
If the Deficit is falling, that means that the Deficit is a decreasing function, so the derivative of D'(t) – that is, D”(t) – is negative. That would mean the Debt is increasing, but concave down.
But the quote said the Deficit is falling more rapidly than is has been: the derivative of the Deficit is getting more negative, so to speak. In other words, the Deficit itself is decreasing and concave down, which means that D”'(t) is negative.
And so we have a third derivative! Welcome back old friend!