I share Matt's bafflement at the argument that a flat tax code is significantly simpler than a progressive one. In fact back in my Plastic days I commented on this topic. I've always liked the idea of using very small-width or even infinitesimal tax brackets, because as a physicist I find discontinuous jumps unsettling. I have this utopian vision of some economist working out a model for an optimal tax code that could then be perfectly implemented with a continuous differential marginal tax rate. Unfortunately, what would actually happen is that you'd give Congress an infinite number of parameters to adjust, so you'd soon have all kinds of horrible delta functions littering the tax rate, and Senate debates over the value of the 8th Taylor coefficient, so we're probably better off with relatively wide tax brackets.

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8th coefficient? Wow. We're talking, like, squiggles and jiggles and wiggles all over. (Well, not necessarily, but you know what I mean).

We tend to avoid polynomial approximations, except in special cases where we have "absurdly fine" data, in which case a 3rd order polynomial gives us machine precision. I've always found it strange and interesting how often 3rd order is the "right" order for many practical applications... Then again, if we count the number of 0th or 1st order approximations floating around that aren't really presented as such, and then just compare to second order approximations, it's far less of a deal.

Posted by: Lemming | March 7, 2005 6:42 PMInteresting, I didn't think about that. It occurs to me that polynomials are probably a terrible set of basis functions for a progressive tax code. I wonder what a more useful set would be? Probably the overall marginal tax rate starts at 0 and monotonically increases, but levels off at some maximum rate, which puts some constraints on the basis functions. I imagine you and/or Mason could tell me...

Posted by: Arcane Gazebo | March 7, 2005 8:07 PMReally, there are far too few constraints to define a good basis here. All we really have is:

Where f(x) is the dollar amount you pay in taxes for an income of x,

f(x) is O(x)

lim (f(x)/x, x->+0) = 0 (You did say rate, depending on how you meant it you might mean f(x) instead of f(x)/x)

df(x)/dx >= 0

Not a whole lot to go on.

Of course, if there was enough to go on, I still wouldn't likely know.

Posted by: Lemming | March 8, 2005 1:52 AMYou can get a "good" basis even without constraints, but it could require a lot of trial and error.

If you want to find someone with a good intuition for what type of basis should be used on a given set of data, I'd ask the signal processing people first... personally, we could expand in Bessel functions (or parabolic cylinder functions, Mathieu functions, etc.) just because I'd love to see special functions in the tax code. But that's just me...

In terms of continuous versus discrete, how about Congressmen arguing over an epsilon-delta proof... that would amuse me greatly.

If you want to see long expansions with tons of Taylor coefficients, take a look at the celestial mechanics literature (especially in problems where they need to include all 6 orbit parameters...).

Hmmm... I think "longer expansions" goes right up there with curve fitting and group actions...

Posted by: Mason | March 8, 2005 6:06 PM