tag:blogger.com,1999:blog-6837159629100463303.post9013855393473632238..comments2023-06-18T01:25:08.748-07:00Comments on Information Transfer Economics: The importance of transversality conditions (more on the Ramsey model)Jason Smithhttp://www.blogger.com/profile/12680061127040420047noreply@blogger.comBlogger34125tag:blogger.com,1999:blog-6837159629100463303.post-41559239740681195682015-07-15T13:36:27.014-07:002015-07-15T13:36:27.014-07:00Hi Bill,
Sorry for the delay in answering your q...Hi Bill, <br /><br />Sorry for the delay in answering your question (I actually thought I did, but somehow the comment didn't appear), but the short answer is yes. The average of paths that start near a point on the saddle path will on average have a starting point on the saddle path and will therefore on average follow the saddle path (the difference will be small).Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-13981491610024372642015-07-01T15:29:28.458-07:002015-07-01T15:29:28.458-07:00Thanks, Nick.
Crusoe is still going to want to ha...Thanks, Nick.<br /><br />Crusoe is still going to want to have no savings if he dies on day omega, which means that he is going to want to have infinitesimal savings for the day before day omega, and so on. Homo sapiens might view that as a model for overconsumption.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-70258224866096111672015-07-01T06:53:27.063-07:002015-07-01T06:53:27.063-07:00Thanks Bill.
If Robinson Crusoe has diminishing M...Thanks Bill.<br /><br />If Robinson Crusoe has diminishing Marginal Utility of consumption d2U/dC2 < 0, which is what we normally assume, then he won't want to wait till the last day. He will smooth his consumption over time (to some extent).<br /><br />Here's how we handle uncertain lifetime:<br /><br />If Robinson Crusoe had a probability of death p(t) per period, then the Euler equation becomes C(t)/C(t+1) = (1+rho)/((1-p(t))(1+r)<br /><br />(That's assuming U=log(C), so dU/dC = 1/C )<br /><br />One way to approach the transversality condition: assume there's a constant risk of death p per period up to time T, but a cutoff at date T, so p(T)=1 if he's still alive at that point. Then take the limit as T approaches infinity.Nick Rowehttps://www.blogger.com/profile/04982579343160429422noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-31237546831961764312015-07-01T02:13:45.789-07:002015-07-01T02:13:45.789-07:00Yes, E. T. Jaynes. He was talking about probabilit...Yes, E. T. Jaynes. He was talking about probabilities, but I think that the idea applies in general. There are any number of probabilities related to infinities where people disagree. Keynes mentions some in his discussion of the principle of indifference in his book on probability. Jaynes realized that taking finite models to the limit could let you prove that a certain probability was correct in the limit. If the same infinite case could be reached in the limit in a different way, then a different probability would be valid in that case. <br /><br />Certain paradoxes, such as two envelopes paradox, do not have finite models. For instance, if there are only two possibilities for the amounts of money in the envelopes, keep the envelope if it has the higher amount, otherwise switch envelopes. With Jaynes's approach the paradox is simply incoherent.<br /><br />In the infinite Robinson Crusoe case Crusoe wants to eat everything on the last day, but does not know when the last day will be. If you simply allow Crusoe to eat everything on the last day of the finite cases, with no other restrictions, then the last day hardly matters. OTOH, if the next to last day needs to be like the last day, and the day before needs to be like it, etc., you can reason backwards. But won't that give you a higher rate of consumption, by comparison with the anything goes model? Froelich machet das Haus den Esser: er leert es.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-85440479397036354072015-06-30T17:59:27.356-07:002015-06-30T17:59:27.356-07:00Thanks Jason.
I did take a look at your roundabou...Thanks Jason.<br /><br />I did take a look at your roundabout post. Couldn't get my head around it, but that's probably my fault.Nick Rowehttps://www.blogger.com/profile/04982579343160429422noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-82411049611092759322015-06-30T17:30:38.858-07:002015-06-30T17:30:38.858-07:00I also toyed around with your Wicksellian roundabo...I also toyed around with your Wicksellian roundabout idea as a MaxEnt problem afterwards:<br /><br /><a href="http://informationtransfereconomics.blogspot.com/2015/03/the-wicksellian-roundabout-and-entropy.html" rel="nofollow">http://informationtransfereconomics.blogspot.com/2015/03/the-wicksellian-roundabout-and-entropy.html</a>Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-79493663302191183962015-06-30T17:23:58.313-07:002015-06-30T17:23:58.313-07:00Nick,
This is ET Jaynes of maximum entropy fame ....Nick,<br /><br />This is ET Jaynes of maximum entropy fame ...<br /><br /><a href="https://en.wikipedia.org/wiki/Principle_of_maximum_entropy" rel="nofollow">https://en.wikipedia.org/wiki/Principle_of_maximum_entropy</a><br /><br />He took a wild swing at how economics might look as an entropy maximization problem:<br /><br /><a href="http://informationtransfereconomics.blogspot.com/2015/02/jaynes-on-entropy-in-economics.html" rel="nofollow">http://informationtransfereconomics.blogspot.com/2015/02/jaynes-on-entropy-in-economics.html</a>Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-9452640887750469492015-06-30T17:17:08.599-07:002015-06-30T17:17:08.599-07:00Bill: "My impression is that the transversali...Bill: "My impression is that the transversality condition is in line with Jaynes's recommendation to derive infinite cases by taking the limit of finite cases."<br /><br />Sounds sensible. Dumb question: who is Jaynes?Nick Rowehttps://www.blogger.com/profile/04982579343160429422noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-89200144706941597282015-06-30T16:54:27.363-07:002015-06-30T16:54:27.363-07:00Jason: Yep. The output = consumption + investment ...Jason: Yep. The output = consumption + investment thing is a true accounting identity (in this model, though it could also be reinterpreted as an equilibrium condition, in other models, just to confuse things!)Nick Rowehttps://www.blogger.com/profile/04982579343160429422noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-71856136771929522122015-06-30T16:35:40.082-07:002015-06-30T16:35:40.082-07:00... my co-worker says 1996 was when the HRGs were ...... my co-worker says 1996 was when the HRGs were first used. They did have one failure, but it was in the supporting electronics (a bad component). And he accidentally broke his desktop one some time ago...Tom Brownhttp://www.google.comnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-69375613761703156342015-06-30T15:06:07.230-07:002015-06-30T15:06:07.230-07:00Nick,
I stand corrected -- only one of the RCK e...Nick, <br /><br />I stand corrected -- only one of the RCK equations is an accounting identity (for k' = production - consumption - depreciation = change in capital). The Euler equation (for c') is a necessary condition for utility maximization/optimality.Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-58204137860034383642015-06-30T14:09:02.394-07:002015-06-30T14:09:02.394-07:00Thanks Jason. Goleta: yeah, I went to UCSB and nev...Thanks Jason. Goleta: yeah, I went to UCSB and never left town. I really enjoy it here (although I do prefer next door in Santa Barbara... one of the few moderately sized cities in California (~90k) where you can take a pleasant, sensory-appealing walk, nearly any time of the day or year, to nearly everywhere you might need to go. Too bad we don't have any fresh water left. )c:Tom Brownhttp://www.google.comnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-72792457789821420332015-06-30T13:39:08.293-07:002015-06-30T13:39:08.293-07:00Jason: re your update: I think you understand it r...Jason: re your update: I think you understand it right.<br /><br />" In that case, however, it seems agents optimize once and then live feet up, mind in neutral."<br /><br />Well, I would say they make their optimal plan once, and then just carry out that plan. Like a driver who plots his route only once, at the beginning of the journey. He only changes his plan if new information arrives, but even her you could say he has only one *contingent* plan, that says what he will do conditional on all possible news.<br /><br />But if he did re-examne his optimal plan, halfway along the route, he would continue on as before. See Bellman's Principle https://en.wikipedia.org/wiki/Bellman_equation (Though there are cases where optimal plans are "time-inconsistent", like when you make a promise and want to renege on it later.)<br /><br />" The RCK equations are more an accounting identity..."<br /><br />Oh no they aren't. They are a *necessary* (not sufficient) condition for optimality. An accounting identity is like "number of my sons + number of my daughters = number of my children."Nick Rowehttps://www.blogger.com/profile/04982579343160429422noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-65298043103583839362015-06-30T12:31:40.535-07:002015-06-30T12:31:40.535-07:00Hi Nick,
I added an update above with what I thin...Hi Nick,<br /><br />I added an update above with what I think you are talking about. In that case, however, it seems agents optimize once and then live feet up, mind in neutral. The RCK equations are more an accounting identity (all the 2000 paths satisfy it), while the transversality conditions are doing the optimizing (selecting which path).<br /><br />Srini,<br /><br />Thanks for the reference -- I will look into it.<br /><br />Tom,<br /><br />I agree -- Robinson Crusoe has to spend valuable drinking time to determine not only his location but the location of the saddle path every time he is subject to shocks.<br /><br />Bill,<br /><br />History is probably important, but not so much for this model -- once you've moved to a given path, your previous path doesn't matter much anymore.Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-33125579137503100382015-06-30T11:36:22.868-07:002015-06-30T11:36:22.868-07:00Wiki is more authoritative than me on this subject...Wiki is more authoritative than me on this subject: https://en.wikipedia.org/wiki/Turnpike_theoryNick Rowehttps://www.blogger.com/profile/04982579343160429422noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-39017755499064726042015-06-30T11:30:21.723-07:002015-06-30T11:30:21.723-07:00Now assume r = K^-0.5 (or something like that), an...Now assume r = K^-0.5 (or something like that), and take the limit as T approaches infinity. As T gets larger and larger, Robinson Crusoe will spend an increasingly large fraction of his life increasingly close to the stationary state where C(t)=C(t+1) and K = rho^-2. <br /><br />This is in line with what Bill says below.<br /><br />I think there's something called a "turnpike theorem" that's related to this. But I never understood the math. On a very long journey, starting at K(0) being whatever its initial value is, and ending at K(T)=0, you spend a very long time driving very close to the turnpike in the middle of your journey. At T = infinity, you keep driving closer and closer to the turnpike.Nick Rowehttps://www.blogger.com/profile/04982579343160429422noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-74774045095782925352015-06-30T10:31:14.363-07:002015-06-30T10:31:14.363-07:00Please ignore obvious typo in the last paragraph. ...Please ignore obvious typo in the last paragraph. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-58301096662604348782015-06-30T10:28:57.079-07:002015-06-30T10:28:57.079-07:00I think that is a more recent gyroscope design; th...I think that is a more recent gyroscope design; the story I am thinking of involves MIT/Draper labs and is summarized here:<br /><br /><a href="https://en.wikipedia.org/wiki/Inertial_navigation_system#Guidance_in_human_spaceflight" rel="nofollow">https://en.wikipedia.org/wiki/Inertial_navigation_system#Guidance_in_human_spaceflight</a><br /><br />I've been down to Goleta -- once did a work trip down there. Beautiful area.Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-74628980559292433752015-06-30T10:28:20.295-07:002015-06-30T10:28:20.295-07:00Thanks, Nick.Thanks, Nick.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-676652351519964752015-06-30T10:25:36.510-07:002015-06-30T10:25:36.510-07:00My impression is that the transversality condition...My impression is that the transversality condition is in line with Jaynes's recommendation to derive infinite cases by taking the limit of finite cases. Since all of the finite cases end with the complete consumption of capital, that should happen in the limit, as well. That is not jiggery-pokery. <br /><br />Whether that is a realistic condition is another question. I am reminded of this line from "Von Armen B. B." by Bertolt Brecht, <br /><br />"Froelich machet das Haus den Esser: er leert es."<br /><br />Gleeful the house makes the eater: he empties it. Homo economicus, probably; real humans, probably not.<br /><br />Wait. The transversality condition is more than that, isn't it? The central planner adapts by increasing consumption when the people are saving too much. How she does that is unclear. Is this a well defined model?<br /><br />Also, under a random consumption model I suppose that on average half of what is available is consumed in the last period, instead of half of it. If the only "transversality condition" is that everything is consumed in the last period, then that may well not matter in the limit. But if under some incompletely specified conditions the central planner burns part of the crops or something, in order to guarantee that everything is consumed in the last period, then that may be a significant difference. Now, the transversality condition implies that there is a limit to what a person can consume in one period. Perhaps setting a top limit to capital would work in the random model.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-65296136780271280852015-06-30T09:56:21.345-07:002015-06-30T09:56:21.345-07:00Srini: for simplicity, assume U=log(C), and time p...Srini: for simplicity, assume U=log(C), and time preference rho, and a single agent.<br /><br />We know that C(t)/C(t+1) = (1+rho)/(1+r(t)) is a *necessary* condition for optimisation, but it is not sufficient. For example, with a finite-lived agent, who dies at time T, we also know that K(T)=0 is a necessary condition for optimisation. The (implicit) constraint is that K(t) >= 0 for all t.<br /><br />Tom and Bill: those are different models. For Tom's model we have to maximise E(U), and specify the probability distribution, and how new information arrives. For Bill's model we need to introduce some cost of change, like a different utility function, so current utility depends not just on current consumption but on past consumption too (rum is habit-forming).Nick Rowehttps://www.blogger.com/profile/04982579343160429422noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-77194481203672339702015-06-30T09:26:36.432-07:002015-06-30T09:26:36.432-07:00Hemispherical resonator gyroscope? I have a co-wor...Hemispherical resonator gyroscope? I have a co-worker with one on his bookshelf (made into a trophy of sorts) from a previous job where he worked on that. Actually a few people here were involved with that project... now relocated from here (Goleta) down to Woodland Hills (where by brother works, by chance). My brother told me last week they sell about 12 of them a year, but they have something like 30 million hours w/o a failure, so if that's what you're after you don't have too many other choices. He says the proximity of the freeway forced some expensive modifications to the facility (because of vibrations). The Wikipedia article wasn't that helpful (in terms of dates of service), but I can ask my co-workers... Tom Brownhttp://www.google.comnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-13629056462585357442015-06-30T09:04:47.387-07:002015-06-30T09:04:47.387-07:00Nick,
I understand that the RCK model is an optim...Nick,<br /><br />I understand that the RCK model is an optimizing model. All paths, including non saddle path ones, result from an optimization set up. So, I don't think your argument is correct.<br /><br />One can think of the transversality conditions as intertemporal constraints. However, they are not as innocuous as generally assumed. Lance Taylor has a good discussion on this topic.<br /><br /> Srininoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-69412114895912140092015-06-30T07:18:29.926-07:002015-06-30T07:18:29.926-07:00BTW, what did the people on Pitcairn Island do?BTW, what did the people on Pitcairn Island do?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-28243513943075111912015-06-30T07:05:21.952-07:002015-06-30T07:05:21.952-07:00Not so sure that people do not have something akin...Not so sure that people do not have something akin to momentum. After all, they have drives, they have wishes, they have plans. Drives like hunger and thirst have short cycles, while plans tend not to be cyclical.<br /><br />As for history, I think that humans are more affected by that than pendulums.Anonymousnoreply@blogger.com