tag:blogger.com,1999:blog-6837159629100463303.post1256433432669156910..comments2021-07-22T00:29:53.205-07:00Comments on Information Transfer Economics: On using Taylor expansions in economicsJason Smithhttp://www.blogger.com/profile/12680061127040420047noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-6837159629100463303.post-20000614909055552182016-10-04T10:40:34.691-07:002016-10-04T10:40:34.691-07:00Cool! I will add this to the post linked above ......Cool! I will add this to the post linked above ...<br /><br /><a href="http://informationtransfereconomics.blogspot.com/2016/10/invariance-under-inversion.html" rel="nofollow">http://informationtransfereconomics.blogspot.com/2016/10/invariance-under-inversion.html</a>Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-71200035758039775312016-10-04T02:30:05.520-07:002016-10-04T02:30:05.520-07:00I played with the equation a bit more and I think ...I played with the equation a bit more and I think I found pretty much all the symmetries. The equation is invariant under a transformation $A \mapsto f(A), B \mapsto g(B)$ if and only if the pair of functions f,g is of the form:<br /><br />f(A)= a*A^c<br />g(B)= b*A^c<br /><br />where a,b,c is any triple of constants with the condition that b and c are nonzero (otherwise you run into division by zero). I can post the computation if you want.Mnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-90903270360489346762016-10-04T01:17:46.458-07:002016-10-04T01:17:46.458-07:00What about power transformations? I think the IE e...What about power transformations? I think the IE equation is also invariant under the transformations $A \mapsto A^{\gamma}, B \mapsto B^{\gamma}$ for any nonzero $\gamma$. Example: A -> A^2, B-> B^2. Inversion is the special case where $\gamma = -1$.Mnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-46721324903397916182016-10-03T16:13:24.799-07:002016-10-03T16:13:24.799-07:00Yeah, I'm pretty sure scale invariance and inv...Yeah, I'm pretty sure scale invariance and inversion pretty much exhaust the (interesting) symmetries of the IE equation. <br /><br />Now I have to find out if the inversion symmetry has ramifications in the real world. For instance, some of the terms I proposed in the <a href="http://informationtransfereconomics.blogspot.com/2016/03/effective-information-equilibrium-theory.html" rel="nofollow">expansion here</a> (such as the constant term) are not consistent with inversion.Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-75163432182492825602016-10-03T13:58:51.360-07:002016-10-03T13:58:51.360-07:00By the way, I don't think it should be too dif...By the way, I don't think it should be too difficult to find all the symmetries of the IE relation, or at least the ones that are linear (although the one that you mention here - inversion - isn't linear). After all, it's a relatively simple equation.Mnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-53540594691728347602016-10-03T13:50:38.313-07:002016-10-03T13:50:38.313-07:00Yes, uniform distributions definitely represent a ...Yes, uniform distributions definitely represent a symmetry principle.<br /><br />Actually, I've expanded on my note about invariance under inversion into a post ...<br /><br /><a href="http://informationtransfereconomics.blogspot.com/2016/10/invariance-under-inversion.html" rel="nofollow">http://informationtransfereconomics.blogspot.com/2016/10/invariance-under-inversion.html</a>Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-51178255989972185972016-10-03T13:47:19.471-07:002016-10-03T13:47:19.471-07:00I see. Too much focus on the mathematical details ...I see. Too much focus on the mathematical details has made me oblivious to how symmetry principles underlie the physical laws you mention. :/<br /><br />Uniform distribution is also an assumption of symmetries (invariance under permutations) when I come to think of it.<br /><br />This point of view is very interesting.Mnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-37346707973024619592016-10-03T12:26:46.639-07:002016-10-03T12:26:46.639-07:00Well, in a sense the DSGE models that are log-line...Well, in a sense the DSGE models that are log-linearized are differential equations (<a href="https://en.wikipedia.org/wiki/Finite_difference_method" rel="nofollow">finite difference equations</a> that are solved numerically because they tend not to have closed form solutions) -- even the Cobb Douglas form can be written as a <a href="http://informationtransfereconomics.blogspot.com/2014/05/more-on-cobb-douglas-functions-and.html" rel="nofollow">solution to a differential equation</a>.<br /><br />Actually, marginalism (used to set up the Cobb Douglas function) is a great differential equation to use as a starting point -- and is equivalent to the IT/IE differential equation for IT index k = 1.<br /><br />In physics, the key is that the differential equations are set up using symmetry principles (Newton's laws can be expressed as time and space translation symmetries). In the case of relativity mentioned above, the kinetic energy (1/2) m v² is an expansion of p² = pᵦpᵝ, which is really just a statement that momentum p is a contravariant 4-vector and that behaves in a certain way under Lorentz symmetry transformations (and that m² is invariant). These symmetry principles are absent in economics.<br /><br />And because those symmetry principles are absent, the matrix you get for m equations in m unknowns has ~ m² parameters ... instead of a smaller number because the matrix entries are related by symmetry.<br /><br />However, I try to point out that the IE differential equation is the simplest equation relating two variables consistent with scale invariance (a (subset of) <a href="https://en.wikipedia.org/wiki/Conformal_symmetry" rel="nofollow">conformal symmetry</a>). Actually, I look at some possible generalizations of IE following this principle <a href="http://informationtransfereconomics.blogspot.com/2016/03/effective-information-equilibrium-theory.html" rel="nofollow">here</a>.<br /><br />[Interestingly, I just checked and the IE equation is also invariant under inversion of the two variables A → 1/A and B → 1/B. This may have consequences I want to explore ... ]<br /><br />It's these symmetry principles that make the functions f(x) in physics models less ad hoc, therefore controlling the coefficients of the Taylor expansion.Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-62867502053494945682016-10-03T03:28:14.768-07:002016-10-03T03:28:14.768-07:00"The function f(x) is usually ad hoc in econo..."The function f(x) is usually ad hoc in economics"<br /><br />In physics the function usually comes from solving a differential equation, doesn't it? I wonder if anyone has tried to create economic models based on differential equations, and where that got him. I guess the IE model is kinda like that.Mnoreply@blogger.com