tag:blogger.com,1999:blog-6837159629100463303.post4367577349688380193..comments2023-06-18T01:25:08.748-07:00Comments on Information Transfer Economics: When is an intertemporal budget constraint a true budget constraint?Jason Smithhttp://www.blogger.com/profile/12680061127040420047noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-6837159629100463303.post-3590200017269182532015-10-16T15:39:40.421-07:002015-10-16T15:39:40.421-07:00Jason, I love how much mileage you've gotten o...Jason, I love how much mileage you've gotten out of that simplex diagram.Tom Brownhttp://www.google.comnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-36872069401502226762015-10-16T12:02:34.128-07:002015-10-16T12:02:34.128-07:00Jason, thanks as always for the detailed reply.
...Jason, thanks as always for the detailed reply. <br /><br />I read through the wikipedia pages you linked to, and I have to admit it mostly brought back memories of not doing very well in statistical mechanics as an MIT undergrad... hopefully I will figure all of this stuff out someday.<br /><br />> There are no states in the interior!<br /><br />Yes, understood. It doesn't matter what probability distribution across the state space you use as d grows large, you'll wind up spending most of your money most of the time for any reasonable distribution.<br />Kenneth Dudahttps://www.blogger.com/profile/10593455504357461005noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-18162371334194830212015-10-16T10:25:33.262-07:002015-10-16T10:25:33.262-07:00Hi Ken,
It's not really off-topic. Treating e...Hi Ken,<br /><br />It's not really off-topic. Treating each point as equally likely is the principle of indifference, and is basically equivalent to several different formulations of the fundamental postulate of statistical mechanics:<br /><br /><a href="https://en.wikipedia.org/wiki/Statistical_mechanics#Fundamental_postulate" rel="nofollow">https://en.wikipedia.org/wiki/Statistical_mechanics#Fundamental_postulate</a><br /><br />The uniform distribution is the maximum entropy distribution assuming you have a maximum value (our budget constraint), however there are other maximum entropy distributions (that depend on your constraint) ... including the partition function, which maximizes entropy given a constraint on the average value. We could say the intertemporal constraint is the average of a large system of agents instead of it being the a strict boundary. That actually might be more realistic because banks can create money through loans, so the budget isn't really a hard wall.<br /><br />That would probably be the optimal way to deal with this problem and it would still lead to occupying many of the states near the surface (because there are more of them that are indistinguishable), but the distribution of points would look more like a Maxwell-Boltzmann distribution with some tail that exceeds the budget wall.<br /><br />The primary reason for the uniform distribution (other than the principle of indifference) is that it is easy to work with. I like to think of it as the Einstein model vs the Debye model:<br /><br /><a href="https://en.wikipedia.org/wiki/Einstein_solid" rel="nofollow">https://en.wikipedia.org/wiki/Einstein_solid</a><br /><br />The Einstein model gets the basic physics right (freezing out of phonon degrees of freedom), but the Debye model is more accurate about how the phonons freeze out vs temperature.<br /><br />Also, the volume of the simplex (the area under the d-1 dimensional budget constraint) is 1/d! so for d >> 1, 1/d! ~ 0. There are no states in the interior!Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-52063881229279173572015-10-16T07:04:08.388-07:002015-10-16T07:04:08.388-07:00Jason, apologies for the off-topic comment.
I fin...Jason, apologies for the off-topic comment.<br /><br />I finally made myself work through the expected value of total spending for a randomly-behaving agent subject to a budget constraint spending on k commodities. I worked out the volume of a unit right k-simplex, which is (as you know) the space of feasible points where each point is a k-vector comprising the amount spent on each of the k commodities assuming the sum of the money spent is less than or equal to 1. The result I got is simply 2^(-1/k), which means in one dimension, you spend half your money on average; two dimensions, 71%; and by the time you get to 50 or so dimensions, you're probably spending 99% of your money.<br /><br />Assuming I did this analysis right, it raises in my mind a question: why do we assume, a priori, that each point within the simplex is equally likely? I know people who basically spend their money on apparently random things until they have none left. This behavior is still nice and random (to me at least), uniformly distributed across the simplex surface furthest from the origin, rather than uniformly distributed across the simplex's full volume as I think you are assuming.<br /><br />Maybe it makes no difference and the ITM gives you the same result with either consumer model. But I'm still curious.<br /><br />Thanks,<br /> -Ken<br /><br />Kenneth Dudahttps://www.blogger.com/profile/10593455504357461005noreply@blogger.com