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Saturday 2007-08-18

fed

(Disclaimer: I am not an economist, nor a political analyst, nor a statistician.)

J K Galbraith (son of the J K Galbraith and a notable economist in his own right) and two co-authors have published an analysis of the US Federal Reserve's monetary policy, purporting to give strong evidence that, since 1983,

  • the Fed's goal is to keep unemployment from getting too low, not to keep inflation from getting too high; when both unemployment rates and inflation rates are included in their model, the dependence on the former is much stronger than on the latter (and this is true robustly across multiple models);
  • the Fed does not ease its monetary policy when unemployment is very high, as you might expect if it aims to fight recessions;
  • the Fed systematically, consistently, manipulates interest rates as presidential elections approach, lowering them when the incumbent is a Republican and (to a lesser extent) raising them when the incumbent is a Democrat;
  • this last political consideration plays as big a role in determining the Fed's monetary policy as inflation and unemployment together do.

It seems to me that the first point could have a not-so-odious interpretation: if low unemployment is (or is believed to be) a good leading indicator of high inflation, reacting to low unemployment might be a better way of keeping inflation down than waiting for high inflation rates. Or reacting to some other leading indicator of inflation might look like reacting to low unemployment. However, the paper also purports to show that low unemployment is not in fact a good predictor of high inflation in the future. (Maybe it isn't one because the Fed reacts so promptly to low unemployment. But that seems like a stretch.)

Galbraith talked about this to the House Committee on Financial Services, and his comments offer some useful further insights, especially on inflation.

I wonder what a similar study of the Bank of England would find.

polyhedra

Here's a beautiful little bit of mathematics that isn't as well known as it should be. First, by way of introduction, there's a rather elementary theorem that says: if you have two polygons of equal area, then you can cut one of them up into finitely many polygonal bits and rearrange them to make the other one. Obviously you can't do this if the areas aren't equal, so we have a necessary and sufficient condition. Yay for necessary and sufficient conditions. Now, what about polyhedra?

It turns out that equal volume isn't enough. We might hope to prove this by assigning numbers to polyhedra according to some rule, so that (1) when you put some polyhedra together to make a bigger one the numbers add up and (2) polyhedra of a given volume don't all get the same number, because #1 would mean that two "equidecomposable" polyhedra have to have the same number and then #2 would mean that you can have two polyhedra of equal volume that aren't equidecomposable.

That's more or less what we do, except that what we associate to each polyhedron is a bit more complicated than a single number. The idea, crudely, is to measure how much edge the polyhedron has.

Think about what happens when you chop up a polyhedron into smaller ones. The small ones have, between them, all the same edges as the big one, plus some others (created by the chopping-up) that fit together and thereby disappear in the bigger polyhedron. What we're going to do is to find a way of counting "edginess" so that the new edges all cancel out.

There are two different ways to measure the "size" of an edge. The obvious one is its length. The less obvious one, which will be crucial here, is what's called its dihedral angle, which is the fraction of space taken up by the polyhedron near to the edge. To be more precise: imagine a very thin cylinder with the edge for its axis, and ask what fraction of that cylinder is inside the polyhedron. There may be some funny business at the ends, but as the radius of the cylinder goes to 0 the funny business does too, and the limiting fraction is the dihedral angle -- in slightly unorthodox units, because what I'm calling 1 is usually called 2π.

Let's write L@f for an edge of length L and dihedral angle ("fraction") f. I'll call this the "edginess" of the edge.

Now, we would like the total edginess not to change, or at least to change in a well controlled and well understood way, when we chop a polyhedron up. So we'll need some rules for adding edginesses. For instance, you can cut an edge into two pieces "across" its length, so we want (L+M)@f = L@f + M@f. Or you can cut it "along" its length, producing two sharper edges, so we want L@(f+g) = L@f + L@g. More subtly, you can combine these operations: split it into k pieces lengthwise and stack them "around" the edge, or split into k thinner pieces and glom them together to make an edge k times longer. So we end up wanting kL@f = L@kf for any rational number k.

The set of sums of the form L1@f1 + ... + Ln@fn, with these rules and no others, has the concise mathematical name "R ⊗Q R". The total edginess of a polyhedron, or a set of polyhedra, is such a thing. This isn't yet quite the right quantity, but we're nearly there.

So, what happens when we chop up a polyhedron into smaller polyhedra? The rules described above account for what happens to the original polyhedron's edges, but there are new edges as well. Each such edge is either internal to the original polyhedron, in which case all the space surrounding it is neatly occupied by the smaller polyhedra around it for a total edginess of the form L@1; or it's internal to a face of the original polyhedron, in which case only half of the space around it is occupied by the smaller polyhedra and we get a total edginess of the form L@1/2, which equals L/2@1 by the third of our rules. If only these were zero we'd have found a quantity that doesn't change when you chop up a polyhedron.

So, make them zero. Instead of working with the fraction f, work with "f mod Q", which just means treating two fractions as equal if they differ by a rational number. In other words, two bits of edge are equivalent if they're the same length and their angles differ by a rational amount. We're now working in "R ⊗Q (R/Q)". I'll call the quantity we have now the "reduced edginess".

So what we've found is that when you chop up a polyhedron, the total reduced edginess doesn't change. We'll be done if we can find two polyhedra of equal volume but different reduced edginess. It turns out that a cube of volume 1 has total edginess 12@1/4 = 3@1 = 0, whereas a regular tetrahedron of volume 1 has total edginess L@arctan(1/3)/2π where L is the length of one side of that regular tetrahedron (whatever that turns out to be), which isn't 0 because arctan(1/3)/2π is irrational. Therefore, those two polyhedra can't be converted into one another by chopping them into smaller polyhedra and rearranging, even though their volumes are equal.

Even better: volume and reduced edginess between them tell the whole story, as area alone does in two dimensions: if two polyhedra have the same volume and the same reduced edginess, you can convert one into the other by chopping them up.

If you allow arbitrary pieces (not just polyhedra) and believe in the Axiom of Choice, then you don't need either equal volume or equal edginess; any two polyhedra are equidecomposable. (In fact, any two bounded sets with nonempty interior are.) This is one form of the famous Banach-Tarski paradox.

Real mathematicians say "Dehn invariant" instead of "reduced edginess"; the proof is due to Max Dehn, a student of David Hilbert.