Algorithms and Sloped Tax

This piece will start off about astronomy but segue into fair taxation.  You may have noticed that I have to keep writing about astronomy because I’m expected to, but I really want to write about all kinds of other things, so I have to find stratagems to do so.

Here is one of the charts I’ve been making toward a book; it shows the curious path of Neptune in the years 1862 and 1863.  (View it larger if you can.)

At that time Neptune, advancing along its vast 165-year orbit, crossed the celestial equator into the sky’s northern hemisphere; and, as seen from Earth, the equator-crossing was a quintuple event.  This was because the retrograde loop that Neptune appears to trace as we overtake it is long enough that each year’s loop overlaps the next; and this overlapping caused Neptune to cross the equator five times.

It was while making this particular chart that I decided to add to my charting program a way of marking the opposition of a planet: the moment in the year which is the climax of the planet’s time of observability.  This would save some of the labor of adding labels by hand (automating of labor is one of the urges for programming).  I should be able to generalize this later for other kinds of event, such as Venus’s greatest elongations.

Well, these opposition labels were more conveniently written on the south side of Neptune’s path, because that path is here convex toward the south.  (Though the curvature is slight, because Neptune isn’t far from the ecliptic.)  But what about other times at which the loops are convex toward the north, because a planet is traveling on the north side of the ecliptic plane?  In anticipation of this, I had to add a bit of programming to find whether this is so.  But how is the program to know whether the curve it is drawing from date to date is convex and in which direction?  This is an example of a situation in which you have to find an algorithm, that is, a pattern of thought that you can turn into a procedure.

Algorithm is an interesting word.  I tried to write my definition of it, just given, before seeing how the Oxford English Dictionary defines it, because I’ve so often had the experience of wondering “How on earth would I define sententious or tease or countryside or affectation or or or whatever or Well! or after all?” and finding that the dictionary’s experts hit the definitional nail on the head.  For algorithm the dictionary doesn’t do a whole lot better than I did.  It’s clear what algorithm means, but not easy to put into words.

It originated from the name of Abu Ja`far Muhammad ibn Musa al-Khwarizmi, a mathematician, astronomer, and geographer who worked in Baghdad but came from Khwârazm or Khwârizm.  That is one of the words spelt in Persian with a w which, after kh, is either sounded as v or dropped out, or sort of dropped out.  The region was in earlier Hellenistic times called Chorasmia, and is the oasis where the great river called Oxus in Greek, Amu Darya in Persian, comes out into the now nearly dried-up Aral Sea, in what is now Uzbekistan.  Khwarazm was the base of an Iranian civilization and of an empire covering much of central Asia and Iran, until destroyed by Genghis Khan the Mongol in 1220.

Al-Khwarizmi flourish four centuries before that, in the early 800s.  He more or less founded algebra, and the word derives from one of his algebraic operations, which he called al-jabr, “the restoring” (of a missing piece) and included in the title of one of his treatises.  He did not invent the Hindu-Arabic system of numerals, with zero and decimal places, but his work, translated from Arabic into Latin, introduced it to Europe, where it was called algorismus.  And the Spanish and Portuguese words for “digit” are guarismo and algarismo.  And do we suspect that arithmetic is yet another corruption of al-Khwarizmi’s name?  No, it’s from Greek, arithmos, “number.”  But compounded from that and logos, “word,” is logarithm, which by delightful coincidence is the anagram of algorithm!

From meaning the improved notation for arithmetic, algorism or algorithm moved, in the 1800s, toward a new meaning: a logical insight on which you can build a set of steps which should work exactly to achieve what you want – as in a programming language.  The algorithm underlies the steps of the program.  Recently, it has spilled over into being a buzz-word for procedures such as those devised by card-sharpers and financiers to beat the system.

So, how do we get a program to know which way a moving body’s path is curving, while it is drawing it as a series of small straight lines?  Here’s my algorithm.  If it knows the coordinates not only of the present point, but of the points a day earlier and a day later, then the line from the day before to the day after passes to the left of the present point if the curve is convex to the right.  So we make it calculate the angles from day-before to day-after and from day-before to present, and if (assuming angle increases counter-clockwise) subtracting the second angle from the first gives a positive number, the curve is convex to the right.

Of course.  Some algorithms are easy to see, after you’ve seen them.

Here’s another matter requiring an algorithm.  I read in a newspaper of October 12 that the election manifesto of Britain’s Labour party proposes “a 45% tax band on those earning more than £80,000 and a 50% rate for those on more than £123,000.”

Income tax systems are structured in steps, which are called “brackets” or “bands.”  If you earn more than X1, you pay Y1 percent; if more than X2, Y2 percent…  We could refer to the borderlines proposed in the manifesto as “thresholds” 80,000/45 and 123,000/50.  (I apologize for using such pluty incomes as examples, but they are the only ones I have at hand; I don’t at the moment know what actual thresholds there are in the American and British tax systems.)

It seems unfair that someone earning a penny over one of these thresholds pays substantially more than someone earning a penny under.  So people worry about the thresholds and cheat at them.  If you fear your income has been near one of them, you don’t know, till your accountant informs you and it’s too late; or, to make sure, you may resort to some “creative accounting.”

I long wondered: instead of this system of steps, couldn’t there be a slope?

In this graph of income vs. tax rate, you can think of “income” as dollars, pounds, or “ackers” (as British soldiers used to refer to whatever dinars or rupees were used in the countries they were sent to); it’s only for illustration.

Small circles represent the two sample thresholds I read of – 45 percent tax if your income is 80,000+, 50 percent if it’s 123,000+.  If your income is in between, your tax percentage stays on 45, as shown by the stepped line.

But the dots follow a slope pinned to the two thresholds.  In a sloped system, the rate remains proportional to actual income.

I thought it would be fairly easy to find the algorithm giving this slope, and my first idea was to get the angle of the slope and use it to calculate other points along it.  That became complicated; and you wouldn’t want to use trigonometry to figure your taxes.  The algorithm had to be based on those four given figures, which we can call income 1, rate 1, income 2, and rate 2 (80,000, 123,000, 45, and 50).  Rate has to increase in proportion to the fraction of the distance between the threshold below and that above.  That’s the algorithm (not very strictly stated), and the equation resulting from it is:

rate = rate 1 + (rate 2 – rate 1) * (income – income 1) / (income 2 – income 1)

Not quite simple, but only one equation, and simple compared with trigonometry.

It could easily be built into an app; or, rates could be read off from the dots on an enhanced version of the graph.

The formula could give a whole slope, below and above the thresholds, as indicated by the continuations of the dotted line.  But rules would be added that the rate switches to zero below a certain income, and stays flat above some top level, so as to avoid becoming 100 percent.

Moreover, there would probably be other thresholds lower down, representing decisions about what is fair taxation at various levels.  A fictitious example is added to the graph, in blue.  A tax system could consist of a series of slopes from threshold to threshold.

You will notice that, since dots in the graph are lower than the steps, almost everyone in the sloped system will be paying tax at a lower rate than in the stepped system – if the slope is kept where it is.  But the whole slope would be slid to the left so as to adjust for this.

Isn’t it fair that taxes should go up without jumps?  Maybe you know of some reason why it isn’t.

A couple of footnotes about the abstractified substance the love of which is “the root of all evil” (First Epistle to Timothy, 6:10):

An addition to my list of scummionaires has to be Paul Manafort, who financed a son-of-a-dictator spending style by running a P.R. company for right-wing despots (“the torturers’ lobby”), became chairman of the Trump election campaign, and has just now been revealed as having laundered the foreign sources of his money and hiding it from taxation.

Income inequality has risen to its highest since the corca-1905 “gilded age” or age of robber barons.  Globally, there are now 1,542 dollar billionaires; in 2016 they increased their wealth by almost 20 percent, to a record total $6,000,000,000,000 (more than twice the gross domestic product of Britain).  Illionaires (under whom we could embrace those of the mil, bil, or tril kinds) are becoming somewhat fearful of a backlash of anger.  The “philanthropy” by which they try to make themselves liked consists mainly of building art galleries and buying sports teams.  The International Monetary Fund says governments had better force the top one percent to pay more tax.