sorting invariants

Of the standard fast sorting algorithms, heap and quick are in-place but unstable, merge is stable (or can be) but take O(n) extra space. You can decorate input with the original sequencing in order to break ties and make a stable sort, but now that takes O(n) extra space. Insertion sort is stable and in-place, but takes O(n^2) time; if you insert into a tree it becomes faster, but takes space again (as you're not just copying the data, but creating two pointers per item, too.)

If I understand radix sort right, it has worst case O(n*log(n)) performance (if there are n distinct entries, they take log(n) bits to be represented, and radix is O(n*numbits), and the LSD version is stable, but I think takes extra space. MSD can be stable but takes extra space for that.

So, seems like a pattern! Except Wikipedia intimates of in-place stable mergesorts in O(n*log n * log n) -- okay, that still takes extra time -- or in just O(n log n) but with high constant values. And there's blocksort, allegedly an in-place, stable, worst case O(n log n), best case O(n) algorithm. Which sounds like the Ultimate Sorting Algorithm, doing everything you'd want. Why haven't I heard of it more? Apart from seeming pretty complicated.

is $15 minimum wage too high?

I knew that Australia had a "high" minimum wage; somehow I'd remembered it as $22/hour, but that seems totally wrong. (I think 22 is where the US would be if it had tracked productivity gains.)

As often, Wikipedia has a handy and hopefully accurate table: https://en.wikipedia.org/wiki/List_of_minimum_wages_by_country

the fun part being to sort the columns. In nominal (exchange rate) US$ terms, the highest minimum wage is yes, Australia, at $15.58/hour. Next is Luxembourg at $14.75, and a handful of countries scattered around $12. Germany drops down to $11.28, then Canada at $9.45.

In PPP terms, the highest is San Marino, at only $12.55. Luxembourg is second at $11.43, Australia third at only $11.14, and things quickly slide down from there.

Also interesting is the annual minimum wage as a %age of GDP/capita. There's some variation -- Luxembourg is only 26%, Argentina 75%! -- but 40-50% is common, with the US low at 27.6%. 40% would mean $10.50/hour, 50% would mean $13.13/hour. $15 would mean an annual wage of 57% of GDP/capita, which would put us alongside San Marino and New Zealand.

(Side note: many poor countries allegedly have a minimum wage that is a *multiple* of the GDP/capita.)

Note: Switzerland and the Nordic countries have no official minimum wage, relying on massive collective bargaining instead, and aren't listed. Some web pages say Denmark has an average (over sectors) minimum of US$20, with $15 at the minimum. Another says $19 for Sweden. Methodologies aren't given, so I'd guess those nominal, not PPP; still high. Swiss search results are poisoned by the referendum for a $25/hour wage (overwhelmingly rejected by the voters.)

The PPP multiplier for both Denmark and Sweden seems to be 1.3, so that $19-20 nominal would be about $15. But also note that it's not actually universal; there may be some lower wage sectors.

Also note that a lot of these countries have relatively high unemployment rates, especially among youth, while being much nicer countries to be unemployed in.

According to https://www.dol.gov/featured/minimum-wage/chart1 the highest inflation-adjusted minimum wage in the US was $10.34, in 1968; $8-9 was a more sustained plateau.

So what does this mean for US policy? Going by PPP numbers, the highly popular $15/hour is in fact notably higher than any other country's legal minimum wage; it's also pretty high as a %GDP, though not uniquely so. It may also be standard for de facto heavily unionized Nordic countries. Even Hillary's $12 is on the high end in dollars, though fairly standard as %GDP. (Note that Hillary also proposes inflation-indexing it, which might be a radical detail in itself. Possibly risky, even -- part of the macroeconomic role of inflation is to quietly depress wages when that's needed.

So I'd say that $15 for the whole country really is pushing the envelope, especially if we're not prepared for higher unemployment as a possible outcome.

EDIT: I should say, I wrote this post from a perspective of comparative macroeconomics, as in "what levels can we safely say won't mess up a modern economy." If we look at lifestyle attained, obviously one needs less money in a country with subsidized health care, college, parental leave, and with superior public transit and bus/train networks. You'll live better on $24K (PPP) in Germany than in the US because of the greater public services.

EDIT 2: Also of interest might be how many, and what sort of, people receive minimum wage, for how long, in various countries. Stepping stone, or permanent underclass?

housing supply and demand

Recent thought about equilibrium effects on housing:

If many people actually *prefer* denser areas, or the amenities that result from them, then shifting the supply curve right (by loosening zoning codes) leads to more housing and lower rates, but may be followed by the demand curve shifting right because the area is more desirable due to higher density, leading to more housing and higher rates. And then shifting again, until the demand curve stops shifting or you run into a steeper part of the supply curve (whether due to looser but still extant legal limits, or sheer physical capacity) so that price increases outweigh increased amenities in desirability. (Or you simply run out of ability to add people.)

So it's true that free market housing won't necessarily lower prices for good; OTOH, it does allow more people to live in a place they'd prefer, which is still good.