By random chance, I happened to run into a law professor starting her first year here at Tulane. By an even stranger coincidence, it turns out she’s the law professor I first encountered through another weblog, and who I tried to help with a data analysis problem she had for a paper she was writing. I never felt completely satisfied with the answers I ended up giving, and I’ve turned the problem over in my mind from time to time since then. I’d like to toss out my thoughts on the problem here.
What we’re interested in is a way of comparing changes in CEO compensation with changes in a company’s profits, year to year. Originally, this was presented in terms of percent change, but I think the problem is more naturally stated in terms of a multiplier. That is, if one year the CEO has a salary 
, and the next year the salary is 
, then we’re concerned with the multiplier 
. Similarly, if the profit goes from 
to 
, we’re concerned with the multiplier 
.
Now consider two companies with the same profit multiplier. The company with the larger CEO profit multiplier is, in some sense, favoring the CEO more than the other one is. Thus we want to spit out a number for the first company that’s smaller than that for the second, to reflect that this is worse for the company. Similarly, given two companies with the same value of 
, we want the higher value of 
to correspond to a higher number — it’s “better for the company” to make more profit.
At a first glance, an obvious solution would be to consider the function 
. Indeed, as goes up this value will go up. But a funny thing can happen for changes in the value. If a profit becomes a loss — if is negative — then an increase in will actually make 
larger. Sure, the absolute value becomes smaller, but now the overall value is negative so we’re actually going up as absolute value goes down. We don’t want to just replace with 
, because that would say that a company doubling or halving its profit (at a fixed value of ) is the same situation.
The problem is essentially one of coordinate systems. Imagine laying out a piece of graph paper with CEO compensation multiplier increasing to the right, while profit multipliers increase towards the top. Then we’re working in the right half-plane, because we’ll assume that a CEO won’t keep working at all if he suddenly makes no money, or has to pay money to keep his job — is always strictly positive.
If we plot a company as a pair 
on this graph, then our guess tells us the slope of the line that passes through the origin and this point. Thus all companies on the same line get the same slope, and the lines through the origin give a collection of coordinate lines for some new coordinate system. Under the regular rectangular coordinates, the coordinate lines are the straight horizontal (constant ) and vertical (constant ) lines. But now we want to find a new system of coordinates on the half-plane so that one of the coordinates is our “how good is this situation for the company?” function.
So we’ve got one function, and we’ve got its constant curves on the plane. Now let’s throw in another function: 
. Notice that we can recover the original and functions by
The curves of constant 
are hyperbolæ, crossing over the curves (lines) of constant , so we get a coordinate system. But notice that something still seems weird when 
. This is going to be the crack that lets us break this problem open. Notice that when we must have 
. That is, the weirdness happens along the -axis. This is also where the hyperbolæ all have a common asymptote. There seems to be a natural weakness in our coordinates along that line, but in the first quadrant — 
— and in the fourth quadrant — 
— everything’s fine.
What we’re going to do is break the coordinate system in half here. We’ll use the coordinates in one order above the line and in the other order below the line. That is, define a new coordinate 
so that if then the curves of constant are the curves of constant , but if the curves of constant are the curves of constant . This will be our comparison function, and we can similarly define a function 
for the other coordinate.
We need to sew these values all together somehow. As an attempt, let’s try to make run between 
and 
in the first quadrant. That is, the line of infinite slope gets value , while the line of zero slope gets value . A natural choice here is to use the arctangent of the slope — which runs from 
to 
here — and adjust it by dividing by 
and subtracting . All together this means that if we set 
.
Now for the negative values of we need to start the values associated to the hyperbolæ at and let them run off to 
as we move out from the origin. Here we can just use 
, which does the trick.
Now the coordinate is continuous as we cross the line , but it’s not differentiable there. My puzzle to you is:
Can you come up with a smooth(er) coordinate system that models this “good for the company” function? Is there a better choice of how to sew the two quadrants together that does it? And does the fact that this looks a teensy bit like complex analysis mean anything, or is it just a coincidence?
