tag:blogger.com,1999:blog-5908830827135060852.post7954065514631237442..comments2024-08-05T02:11:03.545-04:00Comments on Bond Economics: The Horrifying Mathematics Of Infinitesimal AgentsBrian Romanchukhttp://www.blogger.com/profile/02699198289421951151noreply@blogger.comBlogger22125tag:blogger.com,1999:blog-5908830827135060852.post-49534974923156847382017-10-14T04:06:23.583-04:002017-10-14T04:06:23.583-04:00This comment has been removed by a blog administrator.srikanthhttps://www.blogger.com/profile/06260681586506848278noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-35173176735098180112017-06-24T00:26:05.514-04:002017-06-24T00:26:05.514-04:00Wow you're not very smart. SAD!Wow you're not very smart. SAD!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-13940191671401543282017-06-16T09:10:22.989-04:002017-06-16T09:10:22.989-04:00That seems to capture my understanding. Aumann has...That seems to capture my understanding. Aumann has a discussion of how competition is supposed to work; you create coalitions (?) of firms with non-zero measure, and it is not supposed to be possible for them to increase their profits by a different choice of working hours (at the given wage and price).<br /><br />The problem then becomes: how can the coalition change working hours, without implying that households change their optimising solution? There may be no supply of labour (or too much) at the new hours demanded.<br /><br />My guess is that they are just ignoring the coupling between the sectors, and pretending that we have two completely independent optimisations inside the same model. That is certainly consistent with the proofs seen in places like Gali, where first order conditions are derived for each sector independently.<br /><br />You'd think that for a "workhorse model" that is used by multiple central banks, someone would be bothered to write out the formal mathematics somewhere, and that paper would be referenced.Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-10769508878744134262017-06-16T07:46:50.913-04:002017-06-16T07:46:50.913-04:00For what it is worth, I have outlined in more deta...For what it is worth, I have outlined in more detail than anybody probably cares about my understanding this subject: http://robertvienneau.blogspot.com/2017/06/perfect-competition-with-uncountable.htmlRobert Vienneauhttps://www.blogger.com/profile/00872510108133281526noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-60160095464856126682017-06-12T14:52:01.677-04:002017-06-12T14:52:01.677-04:00Here are two papers directly addressing
Markets ...Here are two papers directly addressing <br /><br />Markets with a Continuum of Traders - Aumann 1964 (12 pages):<br />http://econweb.ucsd.edu/~rstarr/201/Aumann%20Econometrica%201964.pdf<br /><br />Continuum in Economics: Realism of Assumptions in Economic Theory (22 pages):<br />http://www2.gcc.edu/dept/econ/ASSC/Papers2007/infinities_in_economics.pdf<br /><br />I recall some applied math of distribution functions (which I think by definition always span the interval 0,1 and have value = 1) from a course on semiconductor theory:<br /><br />https://ecee.colorado.edu/~bart/book/distrib.htm<br /><br />It seems to me the economists are simply attempting to argue that the methods developed in the field of statistical mechanics somehow also apply to economics? Personally I can't follow the math models as described in most of the economic papers in the way I could follow the application of distribution functions to large systems of discrete energy states or large systems of particles as a reasonable model in a given context.Joe Leotehttps://www.blogger.com/profile/01292763300917387201noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-42874256681286922582017-06-02T14:14:42.214-04:002017-06-02T14:14:42.214-04:00The first two pages of this five page paper discus...The first two pages of this five page paper discuss the stochastic integral:<br /><br />Diffusions and the Weiner Process:<br />http://www.stat.cmu.edu/~cshalizi/754/notes/lecture-17.pdf<br /><br />The final paragraph on the second page reads "It turns out that, properly constructed, this sort of integral, and so this sort of stochastic differential equation (SDE), make sense even when dY/dt does not make sense as any sort of ordinary derivative, so the the more usual way of writing an SDE is (shows Eq. 17.5) even though this seems to invoke infinitesimals, which don't exist (footnote 3)."<br /><br /><br />This 24 page paper is a good introduction to the theory of stochastic methods with discussion of simulation of Brownian motion per the subject of Einstein's paper in 1905:<br /><br />Stochastic Simulation and Monte Carlo Methods:<br />http://arachne.it.uu.se/edu/course/homepage/bervet2/MCkompendium/mc.pdf<br /><br />This 30 page paper says it lays out the standard New Keynesian model based on Calvo (1983) staggered price setting:<br /><br />The Simple New Keynesian Model:<br />http://www3.nd.edu/~esims1/new_keynesian_model.pdf<br /><br />Even if the math is applied consistently based on well-developed theorems in the context of economics the basic assumptions are much less likely to correspond with actual parameters of social systems. Joe Leotehttps://www.blogger.com/profile/01292763300917387201noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-36123904810450316292017-06-01T21:31:52.041-04:002017-06-01T21:31:52.041-04:00All aggregates are generated by integrating over [...All aggregates are generated by integrating over [0,1]. The function f is just the "infinitesimal" profits, and needs to be integrated to be brought into the right "units",<br />(The infinitesimal variables can be thought of as density per unit of "volume", and the intervals you are integrating over are the volumes. The technical term from this volume is the measure. The problem is that a single agent has a measure of zero, so it does not matter what the density of the infinitesimal function is.) <br /><br />I was given a reference on the "continuum" of agents. The way in which it is supposed to work is that it is not really an optimisation for a single agent (as I somewhat suspected). This means that my comments about integration from i to i are not in the spirit of what they do, but it raises other issues. The actual mathematics of what they are supposed to be doing bears no resemblance to what they write out in their papers. <br /><br />Profits going to zero is another one of those maintream ideas that sound good until you think about accounting identities... Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-13733802502421393422017-06-01T21:18:39.578-04:002017-06-01T21:18:39.578-04:00Ah yes, Robert Vienneau's blog. I used to read...Ah yes, Robert Vienneau's blog. I used to read it, and I thought he stopped writing (or I lost the bookmark when I changed tablets). Thanks for the reminder. (I read econospeak, but did not remember that article you linked.)<br /><br />I am not hugely optimistic (I gave up earlier on the subject). But I want to write a paper on the issue with Alexander Douglas. We are both outsiders to economics, and we need to be as open as possible.Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-3231486374647833172017-06-01T17:27:56.314-04:002017-06-01T17:27:56.314-04:0016 pages of Lecture Notes on Monetary Economics:
...16 pages of Lecture Notes on Monetary Economics:<br /><br />http://web.mit.edu/14.461/www/part1/slides4.1.pdf<br /><br />There is a discussion of optimal price setting on page 9 asserting that firms choose a variable P*(sub_t) to maximize profits subject to a sequence of demand constraints. Unfortunately many symbols used in the equations do not appear to have a verbal description within the reference.Joe Leotehttps://www.blogger.com/profile/01292763300917387201noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-6543792801144256582017-06-01T17:14:47.928-04:002017-06-01T17:14:47.928-04:00Dear Brian,
This is a good exercise (i.e. trying ...Dear Brian,<br /><br />This is a good exercise (i.e. trying to interpret DSGE models) but I ended up being confused.<br />Even though I suppose you're getting the maths right, I am not sure about the interpretation:<br /><br />You have already defined the profit function f.<br />How do we end up with the integral s(i) for the profit of the i-th agent and not simply f(i) ? Is s(i) = f(i) and thus f(i) = 0 ?<br /><br />On a side-note, micro theory says that under perfect competition the long-run profit for a firm is zero. Not sure if the whole framework is such by design, in order to comply with this, but seems consistent in this manner.<br /><br />SteliosCrossoverhttps://www.blogger.com/profile/18104929127001564162noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-67838422659462410072017-06-01T17:04:36.981-04:002017-06-01T17:04:36.981-04:00«However, it seems that it should be possible to w...«<i>However, it seems that it should be possible to write down the mathematics, so that it is possible to discuss the model.</i>»<br /><br />Ah such optimism :-). For "internal consistency" (which is what matters to becoming and remaining and Economist) the "technicalities" matter only inasmuch they support the "valid" conclusions. Or else the long forgotten "Cambriges Capital Controversy" would have had an impact. So your quest to «<i>discuss the model</i>» is quite futile. BTW here is an interesting and very related post on the issue of making-up assumptions about infinities in Economics here:<br /><br />http://econospeak.blogspot.co.uk/2015/11/what-has-not-been-said-about-later.html<br /><br />There is another mathematically trained non-Economist who has tried to make sense of Economics models and was "disappointed" by the "difficulties" in doing so, his blog is at:<br /><br />http://robertvienneau.blogspot.com/<br />Blissexnoreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-79603178070554572017-06-01T15:35:26.599-04:002017-06-01T15:35:26.599-04:00(You usually need to write [1, infinity); we do no...(You usually need to write [1, infinity); we do not include infinity in the set. Yay for finicky math!)<br /><br />The technical problem is that we cannot conceive of the set [0,1] as the limit of taking a large number of firms. From the point of proofs, the structure for taking the limit of a sequence versus the continuum is completely different. <br /><br />The defense of the mainstream approach is simplicity, and so the fact that it is not really a limit of a finite number of firms might be considered a triviality. However, the structure of the oprimisation problem is different in each case. The "proofs" I have seen using Calvo pricing are all technically incorrect, since they did not specify the proper optimisation structure. The excuse is that the proofs are a shorthand for the real problem structure. However, if the actual problem statement is obscured, how does one validate proof validity? Mathematics is supposed to eliminate ambiguity, not create it.Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-62965860224520746122017-06-01T15:25:49.590-04:002017-06-01T15:25:49.590-04:00There's a hierarchy of inifinities. With a cou...There's a hierarchy of inifinities. With a countable set, it can be represented as the limit of a finite sequence.<br /><br />Uncountable - we cannot define such a such that covers all of the points.<br /><br />I have read (I am unsure about this, but it seems plausible) that the set of all numbers that we can represent in the set [0,1] has measure 0; the full set has measure 1. In other words, all of the "weight" in the real line consists of numbers that we cannot represent.<br /><br />The real line is one of those Lovecraft-ian concepts that we do not want to think about too much.Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-25201597354291768962017-06-01T14:08:49.187-04:002017-06-01T14:08:49.187-04:00Brian,
Forget my last request. I'll have you ...Brian,<br /><br />Forget my last request. I'll have you bogged done in explaining things you probably can't be bothered explaining - no doubt I will have other questions. Your post is very technical and it seems you are looking to engage with people having a technical appreciation of the issues raised.<br /><br />It seems to me that the set [0,1] and [1,infinity] are both infinite sets and equally uncountable. And I can't see the relevance of "countability" anyway. The index is just a way of naming an element within a set (isn't it?), so what's in a name?<br /><br />HenryAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-41208957287387429362017-06-01T13:44:47.251-04:002017-06-01T13:44:47.251-04:00Brian,
Please forgive me if I say some stupid thi...Brian,<br /><br />Please forgive me if I say some stupid things - I am sure I have misconstrued your entire argument. So I will jump in the deep end with a question.<br /><br />How is an infinite set countable?<br /><br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-30514742030048680612017-06-01T13:22:50.876-04:002017-06-01T13:22:50.876-04:00To underline, that's the worst case interpreta...To underline, that's the worst case interpretation. There could be a reference somewhere that cleans up my definitional concerns.<br /><br />But if the issues are not cleared up, the situation is like the following mathematical model:<br />1) x is an element of the real line;<br />2) x=0;<br />3) x=10.<br /><br />No such solution x exists, so it tells us nothing. It's just a long-winded way of saying "empty set"; and there's an infinite number of ways of doing that.<br /><br />(If that assertion bothers any mainstream economists that somehow reads this, feel free to comment!) Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-62211613409014815222017-06-01T12:43:48.943-04:002017-06-01T12:43:48.943-04:00"DSGE "models" are not actually mod..."DSGE "models" are not actually models...."<br /><br /><br />Brian,<br /><br />By this do you mean that there is no cogent logical structure to them?<br /><br />HenryAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-44545600312656292242017-06-01T12:35:25.949-04:002017-06-01T12:35:25.949-04:00Einstein ponders the problem of correspondence bet...Einstein ponders the problem of correspondence between a math model and reality in this essay:<br /><br />Geometry and Experience<br />http://pascal.iseg.utl.pt/~ncrato/Math/Einstein.htm<br /><br />I learned the definition of a point as an axiom of geometry and the definition of a particle as a point-mass as a simplifying assumption in physics to eliminate the hard math and solve problems as a first-case approximation. I recall reading an article where Paul Krugman says an economist must have a talent for making simplifying assumptions that are reasonable approximations. So if the math applied is axiomatic and consistent one must attack the reasonableness of the assumptions and if the math is axiomatic and inconsistent with the axioms then of course it is bad math.Joe Leotehttps://www.blogger.com/profile/01292763300917387201noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-22114449876619354452017-06-01T09:01:03.187-04:002017-06-01T09:01:03.187-04:00One could see that conclusions might be pre-ordain...One could see that conclusions might be pre-ordained for ideological reasons; that's an old critique. (I believe that the word "ideology" was invented as part of one critique.) However, it seems that it should be possible to write down the mathematics, so that it is possible to discuss the model. It's not a realism complaint, it's a definitional complaint. <br /><br />If it's not real mathematics (I do not know whether this is the case; it is still possible), that pretty much ends all debates about realism: DSGE "models" are not actually models, so there's no question of realism. But if they incorporate real mathematics, we can then analyse the model and see how it compares to reality. Theoretically, it should be easy to determine whether the mathematics is valid; in practice, it seems to involve chasing through a "vast literature" and finding that each reference does not cover what I want.Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-45154378474956476302017-06-01T07:36:49.525-04:002017-06-01T07:36:49.525-04:00The formal mathematics is a smokescreen for the em...The formal mathematics is a smokescreen for the embedded political beliefs in the structure. The outcomes of all the debates lead inevitably to that conclusion.<br /><br />It is like criticising Catholic Dogma. There is no way you're ever going to convince a catholic that it is wrong, no matter how good the argument. They first have to stop being a catholic. <br /><br />Or you don't bother talking to catholics, bypass them, and talk to other people instead about the undue influence of catholics in society. There are historical precedents for the process :-)<br /><br /><br /><br />NeilWhttps://www.blogger.com/profile/11565959939525324309noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-88677837195725187162017-05-31T18:23:41.847-04:002017-05-31T18:23:41.847-04:00As an outside observer, I would not argue against ...As an outside observer, I would not argue against that view.<br /><br />The problem is that I want to be able to explain economic debates to my readers, and it's hard to explain mathematical models that appear to be formally incoherent, but mainstream economists say they are fine. I doubt that anything I write will get anywhere near a "top journal," but since I already have a Ph.D. in a real academic field, I hardly care. <br /><br />It's not even a question of realism, it is a question of writing down the actual optimisation problem that we are supposed to be solving. Considering that I bill myself as an applied mathematician, saying that the mathematics makes no sense to me is very awkward. (Once again, it may be that the formal mathematics makes sense, but the publishing standards in economics are sloppy to say the least. A great deal of formal mathematics is dropped from discussion, when that formal mathematics is obviously critical.)Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-21846881406437521312017-05-31T16:57:28.332-04:002017-05-31T16:57:28.332-04:00There are many problems with economic models based...There are many problems with economic models based on infinities and infinitesimals, never mind the continuum, in general equilibrium and DSGE models derived from them, for example about infinitesimal probabilities.<br /><br />But all these "technical" problems don't matter to the Economics profession: what matters is whether a model is "validated", which means it supports JB Clark's three fables, in particular the one that the distribution of income, absent government intervention, is (Pareto-) optimal and exactly determined by merit, that is productivity.<br /><br />Economic theories or models that are not validated by their support for the "three fables" are regarded by the Economists, and their sponsors, as communist, and we all know that "communism never worked and never will". Therefore such invalid models cannot possibly work, and "top journals" will not welcome papers on them, unless cleverly disguised, and their proponents or adopters will not become Economists.<br />PhD students in "top departments" of Economics or Business rapidly understand which approaches are "internally consistent" and how much "internal consistency" matters, and how little "technicalities" matter, as long as "internal consistency" is achieved.Blissexnoreply@blogger.com