« Mathematicus, Bernardus Silvestris | Main | The beast is rearing its ugly head »

October 01, 2004

Comments

Feed You can follow this conversation by subscribing to the comment feed for this post.

Bernard Guerrero

Let me see if I've got the transition matrix right:

a=Stagnation
b=Implosion
c=Plateau
d=Boom
e=Technological Leadership

a b c d e
a 0.8850 0.1000 0.0105 0.0045 0.0000
b 0.4000 0.6000 0.0000 0.0000 0.0000
c 0.1000 0.0000 0.8000 0.1000 0.0000
d 0.0000 0.0000 0.1000 0.9000 0.0000
e 0.0000 0.0000 0.0000 0.0000 1.0000

I'll run some Markov Chain stuff and see what there is to see.

Any chance you can modify to fill in the transition probabilities to absorbing state 'e', which were dropped off of the simplified version?

Bernard

Gareth Wilson

"In Pritchett's model, if you start off as a country in economic stagnation, you have an 88.5% chance of remaining stagnant the next year, a 10% chance to implode even further, a 1.05% chance to hit a plateau of moderate growth, and a 0.45% chance to boom."

Perhaps it's just the sleep deprivation talking, but I immediately thought of the board game version of this. "Tiger: The Exciting Game of Macroeconomic Development". The dice would be odd.

Carlos

Bernard, that's right. You might find this utility useful.

The problem is, in Pritchett's model, that a boom to a leadership transition depends on cumulative factors, which violate Markov assumptions. So there's no good way to model it in a small matrix, though there is a crude way to do it in a large sparse one. Pritchett explicitly calls this his "toy collection" model, so it's probably not best to let it bear too much load.

Here's a theoretical question to chew on: why is the matrix so strongly asymmetric, especially in the bad states?

Gareth, you might use four dice, one for each state. Three regular, one weighted. Or a modified Snakes and Ladders; I believe William Easterly has made that analogy.

C.

Cosma

While there are few things I like better than a nice phenomenological Markov model, I hve my doubts about this one. It sounds like Pritchett is treating each country's history as an independently generated sample-path, with a common generating process. Even at this level of coarse-graining, though, it seems to me that this isn't a good approximation, since growth states are spatially correlated. A better approach would be a Markov random field, done on a graph (nodes = countries, edges = trading links, perhaps) and not a simple lattice. This only brings us closer to the board game, of course. If someone wants to make it, I can guarantee you at least one sale; it'd be the perfect gift for my father.

Bernard Guerrero

Bummer, looks like it stabilizes at approximately 63%-16%-9%-12% for states 'a' through 'd' respectively. I worked up a little MATLAB script to produce a track of transitions and plotted 'em. If anybody wants to see the .xls or .m, please let me know.

Bernard Guerrero

"This only brings us closer to the board game, of course. If someone wants to make it, I can guarantee you at least one sale; it'd be the perfect gift for my father."

Hmmnnn. Gotta think about how to make it gamier. Or, er, something like that...

Cosma

With any finite Markov chain, with probability 1 the trajectory enters a strongly connected component of the chain and then stays there forever. Within each component, there's a stable long-run distribution over states, so every finite chain eventually settles down to one such distribution; the only question is which component it gets trapped in. Here there's only one component, so that doesn't arise. If you find the eigenvectors of the transition matrix, you'll see that the distribution Bernard got is the eigenvector for the largest eigenvalue, which is 1.

Carlos

Bernard, you wussed out. I did it like General Anna in The Pushcart War, by hand. Hey, it's only 4 by 4: (1 + .25 + .15 + .195)x = 1 .

Cosma, good to hear from you! Yeah, growth in the real world is spatially correlated, but less strongly than it appears at first glance, and especially in the post-WWII era, when transport and other costs have dropped so much.

The classic example is Botswana, which fifty years ago was a landlocked, artificial region on the map, filled with Iron Age cattle herders, and neighbor to the most predatory state on the continent; and today is as rich as 1975 Wyoming, while most of the region is still stagnant, or worse.

William Easterly brought up a similar point a few years back: if a model of random growth gives the same overall distribution of fast growing and stagnant countries as the real world without introducing geographical factors, should we then include them?

It's odd, because we know -- or at least we think we know -- that they are there.

So I am not sure that the lack of geography is that big of a damper on the model. I think a larger distortion comes from the mix of "grain" sizes of the real world data set. Are Mauritius and the People's Republic of China really the same sort of economic entity?

In the board game, I'd use a map, and overlay circles of colored transparent plastic to denote areal effects.

C.

Bernard Guerrero

"Bernard, you wussed out. I did it like General Anna in The Pushcart War, by hand. Hey, it's only 4 by 4: (1 + .25 + .15 + .195)x = 1 ."

Ah, well, you know the old saying. When you've got an automated matrix algebra hammer lying around, everything starts to look like a nail. Besides, I got to see the time-path, such as it was.

The comments to this entry are closed.