I don’t have time to do posts on what I’d really like to talk about, the new Krugman and Wells article or Russ Roberts’ piece on banking. So I’ll defer those until after my trip to Oxford, and instead do a short fun piece on Zipf’s Law. Well, fun for nerds like me, who find descriptive statistics to be endlessly fascinating. (Not the other kind of statistics.)
Mankiw recently linked to an Edward Glaeser article on Zipf’s Law, which reminded me of a table of skyscraper statistics. It may be that everything I say is already widely known, and fully explained. If so I can count on my very smart commenters to point that out. For those who don’t know, Zipf’s Law says that in a ranking of entities by size, the second on the list will be about 1/2 the size of the first, the third will be 1/3 the size of the first, the 10th largest will be one tenth the size of the first, etc. A good example is the population of US cities:
Rank↓ City↓ State↓ Population↓
1 New York New York 8,363,710
2 Los Angeles California 3,833,995
3 Chicago Illinois 2,853,114
4 Houston Texas 2,242,193
5 Phoenix Arizona 1,567,924
6 Philadelphia Pennsylvania 1,540,351
7 San Antonio Texas 1,351,305
8 Dallas Texas 1,279,910
9 San Diego California 1,279,329
10 San Jose California 948,279
I find this kind of spooky, as these cities grew spontaneously. Note that if you look at this Wikipedia list you will find that the big cities are actually a bit too small when compared to cities ranked, 20, 30, etc. The Glaeser article shows that metro populations have the same problem–big cities are a bit too small.
Does this work for other countries? My hunch is that it works for smaller cities in most countries, but the bigger cities are often much less perfectly representative of Zipf’s Law. Germany, for instance, has no dominant NYC-type city, but rather a half dozen cities with about 2 million people. This may reflect the fact that Germany was once a collection of independent states. To make this point another way, I don’t think that Zipf’s Law even comes close to working at the world level. I suppose Tokyo is the largest metro area (30-35 million?), but there must be at least a dozen metro areas of at least 16 million. If not, there soon will be.
I got to thinking about Zipf’s Law when I ran across this data for cities with the most skyscrapers:
1 84922 Hong Kong
2 35811 New York (inc Jersey City, Fort Lee, Guttenburg)
3 19670 Tokyo
4 18129 Shanghai
5 16426 Chicago
6 15262 Dubai
7 13375 Bangkok
8 10368 Guangzhou
9 8849 Chongqing
10 7923 Shenzhen
11 7697 Singapore
12 7674 Kuala Lumpur
13 7195 Seoul
14 6598 Manila
15 6053 Toronto (inc. Mississauga)
16 5590 Jakarta
17 5473 Osaka
18 5371 Beijing
19 4909 Miami (inc Miami Beach)
20 4903 Nanjing
21 4812 Houston (inc Pasadena)
22 4693 Sydney (inc. N. Sydney, Chatswood, Bondi, St. Leonards)
23 4411 Moscow
24 3978 São Paulo
25 3842 Los Angeles (inc Burbank, El Segundo, W Hollywood)
26 3626 Melbourne
27 3564 Atlanta (inc Vinnings, North Atlanta)
28 3435 San Francisco
29 3274 Panama City
30 2959 Wuhan
50 1993 Taipei
90 823 Ankara
This is actually far better than the US city population example, as it holds pretty well throughout the entire 1 to 90 range. Indeed the top of the list looks almost spookily like the US population data.
At this point you might be thinking; so what? After all, skyscraper intensity is presumably correlated with population. But that’s exactly the problem. At the world level, metro area populations don’t even come close to following Zipf’s law, at least for the largest metro areas. Also notice that Hong Kong is not among the top 30 world metro areas, and Dubai is not in the top 100. On the other hand the list includes only three cities from Europe and Latin America, which have lots of big metro areas. Nor does it seem related to level of development. Poor Asian cities often have many more skyscrapers than Japanese cities of equal size. (Compare Osaka to the big cities in China and SE Asia.
And finally, is this just a fluke, or is there some underlying reason for the pattern? Skyscrapers are being built at a furious rate in Asia, especially China. I’m pretty sure that the nice Zipf’s Law pattern for the top 10 will break down in a few decades. Enjoy it while it lasts. Any thoughts would be appreciated. Apologies if I have merely reproduced what is already widely known.
BTW, I tried to model the failure of Zipf’s Law for world metro populations by considering a model of the world as a limitless plain, where a combination of pre-modern transport constraints, language regions, and nationalism, created lots of similar size countries, each having major capital cities of roughly equal size. As the number of countries approached infinity, I’d expect Zipf’s law to do very poorly. Unfortunately, although Zipf’s law doesn’t work for world metro pops, it does work for world country populations. And in a hundred years it will probably work even better for country pops, as the largest countries are expected to have:
India: 1.5 billion
China: 750 million
USA: 500 million
Part 2. More on NGDP targeting.
I’ve argued that once you start thinking in terms of NGDP, it’s hard to avoid evaluating monetary policy in terms of changes in NGDP, or M*V. And once you start using NGDP as an indicator of policy, it’s hard to avoid the next logical step, which is that the Fed should target NGDP. Matt Yglesias has been recently using NGDP as evidence of the need for more stimulus. Now he has taken the plunge, and gone for NGDP targeting:
It’s probably worth observing that the dual mandate is arguably conceptually incoherent. In normal times, the Fed only uses one policy instrument so it can’t really be targeting two things. The main practical upshot of the dual mandate is that it’s impossible to say for sure whether or not the Fed is meeting it. If you gave the Fed a single clear mandate—keep M*V growing at a steady rate of approximately such-and-such then Congress and the President could specifically say whether or not the Fed was executing its mission.
It’s good to have left-of-center pundits on board. As I have said, this idea should appeal to liberals who worry that inflation targeting gives too little attention to unemployment, but are also knowledgeable enough to realize that the Fed can only hit one target at a time.
3. Immigration and housing
Ryan Avent recently cited my earlier post linking the 2007 crackdown on immigration and the housing bust, and then added the following comments:
I think you want to be careful about assigning too much causation to this factor, but demand is demand, and a negative shock to expected growth in housing demand from immigrants certainly wouldn’t have helped matters in 2007. Here in the Washington area Prince William County, in the Virginia suburbs, adopted particularly draconian immigration-status check rules during the late stages of the bubble, and it subsequently experienced some of the largest declines in real estate values.
Along these lines, here‘s Richard Green:
“My colleague Dowell Myers points out that for the housing market in the US to remain healthy, we must “cultivate new immigrant residents.” Arizona’s new law, which would require immigrants (legal or otherwise) to “carry papers” creates what I would consider to be an atmosphere of hostility to immigrants–all immigrants. I am also awaiting the spectacle of a police officer demanding the “papers” of a native-born Latino.
In any event, people have a propensity to go where they feel welcome, and avoid places where they are not. Hostility to immigrants in general and Latinos in particular seems to be a political loser in California, so Arizona’s policies may lead to higher demand for houses in California.”
I buy what he’s selling. And consider that Phoenix home values have declined 52% from their peak, are still off on a year-over-year basis, and declined in both January and February of this year. As Mr Sumner put it, now might not be the optimal moment to send out a signal to property markets that Hispanic immigration is about to slow sharply.
I agree, and would add that I didn’t mean to suggest that the immigration slowdown caused the entire housing bust, I think the bubble/bust was mostly due to previous errors by private lenders, F&F, moral hazard, and then later in 2008 by tight money. But immigration probably played a non-trivial role in southwestern markets.