omorka | The perils of being a math teacher

moontyger posted a Link of the Day, in the comments of which the Laffer Curve was referenced.

Now, I've just started a unit on polynomials, and we just covered the review of all the function stuff they had back in Algebra II, so when someone mentions that the Laffer Curve has only two points - they said "zero and 100," but they really mean (0, 0) and (1.00, 0), where x is the tax rate and y is the tax revenue - I started coming up with questions:

Does the Curve have, as commonly assumed by economists and neocons, a single maximum? Or does it have multiple local maxima? If there are multiple local extrema, how would we know whether we were at the abolute maximum point rather than a local maximum? How many parameters are we talking about?

Is this thing polynomial or rational? Is there an asymptote anywhere? A flattening followed by an asymptotic fall-off? Is it even really a function?

Is it differentiable? Again, how many parameters are we talking about? There might be a whole family of potential Laffer Curves, and we might not be able to tell which one we're on. This might be a three-dimensional function in two variables, x and t, of which a given Laffer Curve - dare I say a curve in the Laffer family - is merely a cross-section at some time t?

What if it's sensitively dependent on initial conditions? What if it's *gasp* fractal?

Is it even continuous?

Then, as all mathematicians find their hopes of finding a new result dashed early in their careers by the realization that Gauss proved their particular result at the age of 17 or some such, I discover that Martin Gardner beat me to it.

Darn. :-/