tag:blogger.com,1999:blog-6837159629100463303.post6556026216860217339..comments2023-06-18T01:25:08.748-07:00Comments on Information Transfer Economics: More on stock-flow modelsJason Smithhttp://www.blogger.com/profile/12680061127040420047noreply@blogger.comBlogger137125tag:blogger.com,1999:blog-6837159629100463303.post-13382995393590027442016-03-17T10:39:54.245-07:002016-03-17T10:39:54.245-07:00... another way to say that is Γ = 0.5 approximate...... another way to say that is Γ = 0.5 approximately halves the sample period of the system, but we keep calling each sample period "1 period" instead of "half a period" so it appears to approximately double the time constant instead.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-19525059387112524752016-03-16T22:25:51.122-07:002016-03-16T22:25:51.122-07:00Thinking about this:
(1) ΔH = Γ*(G - T)
In terms...Thinking about this:<br /><br />(1) ΔH = Γ*(G - T)<br /><br />In terms of my own formulation, where I have for the dynamics of H and measurement for T:<br />(2) H[n+1] = A*H[n] + B*G[n+1]<br />(3) T[n+1] = CT*H[n] + DT*G[n+1]<br /><br />Where <br />A = 1-α2θ/(1-α1(1-θ))<br />B = 1-θ/(1-α1(1-θ))<br />CT = α2θ/(1-α1(1-θ))<br />DT = θ/(1-α1(1-θ))<br /><br />Substituting (3) into (1) indeed produces (2) when Γ=1.<br /><br />Then Γ ≠ 1 is equivalent to multiplying both α2 and B by Γ, which is commentator A H's approximate prescription for moving to a new sample period Ts2 (in this case Γ times the old sample period Ts1, i.e Ts2 = Γ*Ts1) with the same steady state for H, except in this case without changing the time steps on the plot. So without changing the time steps on the plot Γ = 0.5 should approximately double the apparent time constant Tc (AKA "adjustment time"), using the approximation exp(-t/Tc) ≈ 1 - t/Tc.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-14463613385241254792016-03-16T13:37:32.918-07:002016-03-16T13:37:32.918-07:00BTW, making some assumptions (continuous compoundi...BTW, making some assumptions (continuous compounding and all variables=0 at t=0, and restricted average gov spending functions per period (G)), I did finally make a <a href="http://banking-discussion.blogspot.com/2016/03/sim6-updated-sim-to-preserve-time.html" rel="nofollow">version of SIM (SIM6)</a> that satisfies ALL G&L's equations (I think) (including gamma=1) at all times, matches their results and is sample-period invariant. I do it by adjusting alpha1 AND aphla2 for changes in the sample period. Adjusting theta wasn't necessary. I describe the adjustment procedure <a href="http://www.bondeconomics.com/2016/03/models-are-not-frequency-invariant.html?showComment=1458155418614#c7106654837245534400" rel="nofollow">here</a>.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-24479330202583894512016-03-12T18:45:21.292-08:002016-03-12T18:45:21.292-08:00Just for laughs, I did another version of SIM (SIM...Just for laughs, <a href="http://banking-discussion.blogspot.com/2016/03/sim4.html" rel="nofollow">I did another version of SIM (SIM4) (just the expressions)</a>: the goal being to match G&L at Ts=1, stay sample period invariant, always satisfy ΔH = G - T, and to allow for a wider class of government spending functions.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-21484393456326672672016-03-12T16:15:38.456-08:002016-03-12T16:15:38.456-08:00Bill, OK, I fleshed out the case where g' = ca...Bill, OK, I fleshed out the case where g' = can be expressed as a Taylor series + Dirac deltas, where g represents the integral of all government spending. <a href="http://banking-discussion.blogspot.com/2016/03/sim4.html" rel="nofollow">I called it SIM4</a>, and it's just the math. I think it's right. ... as if you care! LolTom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-29457509580460999602016-03-11T12:54:50.072-08:002016-03-11T12:54:50.072-08:00Bill, here's an updated pair of block diagrams...Bill, <a href="https://2.bp.blogspot.com/-dt8OsQDCHMQ/VuMuJLwHcNI/AAAAAAAAAwg/mzX10nF1hY0HuO--EZ_PMAmSu0k_h4o8A/s1600/SIM_discrete_and_continuous_time.png" rel="nofollow">here's an updated pair of block diagrams</a> with the discrete and continuous time versions side by side. In general X[n] = x(t=n*Ts), and x' = dx/dt.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-37053804029666505752016-03-11T11:01:51.634-08:002016-03-11T11:01:51.634-08:00Bill, also, I discuss in a comment here (intended ...Bill, also, I discuss in a <a href="http://banking-discussion.blogspot.com/2016/03/answer-for-henry-1.html?showComment=1457648477325#c4285117314032871480" rel="nofollow">comment here</a> (intended for Henry) and the one below it how to transform that block diagram into one for an equivalent continuous time system with an identical time constant. As long as you can express g(t) for t>=0 (representing the integral of all government spending from -inf to t) as a Taylor expansion about t=0 with a finite number of terms, it's quite easy to construct an equivalent discrete time system (i.e. one producing the exact same answers as the continuous case at all the sample times). Doing so, however, requires expanding the scalar B into a row vector, one element per Taylor expansion term of g at t=0. Of course that may not be a very useful exercise!<br /><br />Anyway, I'm sure Jason is sick of this, so leave a comment for me anywhere on that blog if you want.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-74165326083568017962016-03-11T10:44:01.911-08:002016-03-11T10:44:01.911-08:00Bill, you lost me there. In my formulation Y,T,YD ...Bill, you lost me there. In my formulation Y,T,YD and C are outputs of a system who's dynamics are completely determined by a feedback loop with H:<br /><br />H[n+1] = A*H[n]<br /><br />with an exogenous input G added:<br /><br />H[n+1] = A*H[n] + B*G[n+1]<br /><br />A(Ts=1) = 1 - θ∙α2/(1 - α1 + θ∙α1)<br />B(Ts=1) = 1 - θ/(1 - α1 + θ∙α1)<br /><br />A and B are *ONLY* functions of the sample period (Ts=1 period) and the parameters α1, α2 and θ, as I indicate in <a href="https://2.bp.blogspot.com/-Wl8jGPivS3Y/VuG-88lrQoI/AAAAAAAAAvk/X8DtHDc7N5M/s640/SIM2.png" rel="nofollow">the block diagram here</a>. The system dynamics (namely a time constant Tc in this case) are completely determined by that feedback loop. I discuss how in <a href="http://informationtransfereconomics.blogspot.com/2016/03/moderation-and-comment-policy.html?showComment=1457699940155#c3748551702782109055" rel="nofollow">this comment</a>.<br /><br />As for Y, T, YD and C: they're all just scaled versions of H offset by a scaled version of input G (again see my block diagram). All the scaling parameters for these outputs are ONLY functions of α1, α2 and θ. I give what they are (written out as text) in the table in the lower left hand portion of <a href="http://banking-discussion.blogspot.com/2016/03/sim.html" rel="nofollow">the spreadsheet on this page</a>.<br /><br />I'm not sure what you mean by a "within period equilibrium." But it's easy to see what the steady state ("across time period?") equilibrium is: just remove the sample indices:<br /><br />Hss = A*Hss + B*Gss<br /><br />Assuming G's steady state, measured as dollars/(initial period Ts) is Gss, and H's steady state is Hss. The answer is<br /><br />Hss = B*Gss/(1-A) <br /><br />I can also calculate Yss, Tss, YDss and Css as functions of Hss, Gss, α1, α2 and θ.<br /><br />And as I discuss in that comment I link to above, the time constant (Tc) at Ts=1 is:<br /><br />Tc = -Ts/log(1-θ*α2/(1-α1+θ*α1)) = 5.9861 periods<br /><br />Regardless of G or the initial value of H.<br /><br />Can you tell me how you arrived at<br /><br />Y∗ = G/(1 − α1 · (1 − θ))<br /><br />?<br /><br />You can leave <a href="http://banking-discussion.blogspot.com/2016/03/sim.html" rel="nofollow">a comment for me directly here</a> if you want to give Jason a break from chatter about this problem. :DTom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-3296573869475059242016-03-11T08:35:42.023-08:002016-03-11T08:35:42.023-08:00Bill, just go back to the previous post and toggle...Bill, just go back to the previous post and toggle between the last two pairs of graphs until you get it.<br /><br />It's more of a velocity multiplier, but if you can't get past the idea that the equation is ambiguous (or rather has implicit assumptions that there's no reason to make in the model so should be left as a parameter) then it's going to be difficult.<br /><br />I replied to Greg on the following post with an example that might help.Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-53787464089698422722016-03-11T08:32:59.719-08:002016-03-11T08:32:59.719-08:00Tom, there are two equilibrations in their SIM mod...Tom, there are two equilibrations in their SIM model. One occurs across time periods, one occurs within them. The within period equilbrium value of GDP is given by this equation:<br /><br />Y∗ = G/(1 − α1 · (1 − θ))<br /><br />What happens after within period equilibrium is reached? Apparently nothing, until the next time period. <br /><br />The model does not work if the time period is too short for within period equilibration, so there is a minimum time period. It doesn't make much sense to have an extended period within which nothing happens after equilibration, either. So I think that there is an implied time period in the model, we just don't know what it is. ;)<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-15016530517807240812016-03-11T08:27:50.912-08:002016-03-11T08:27:50.912-08:00Bill
It's that the definition of H is ambiguo...Bill<br /><br />It's that the definition of H is ambiguous numerically, so it could have a multiplier or not. That equation does not define H to be equal to government debt. It could be. But it doesn't have to.<br /><br />You can interpret the degree of freedom allowed by that ambiguity as a multiplier.<br /><br />Another way to say it is that the toy model doesn't define the situation it says it does.<br /><br />What G&L need is a mechanism that says there's no multiplier -- since the equations they wrote down are ambiguous.<br />Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-58444898456808877202016-03-11T08:00:25.730-08:002016-03-11T08:00:25.730-08:00Jason, in the toy economy with no private banks, h...Jason, in the toy economy with no private banks, how do you get a multiplier for government debt? Please spell out how that happens. Thanks. :)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-84674848561031294362016-03-10T19:16:24.830-08:002016-03-10T19:16:24.830-08:00Re: model dynamics, it's the feedback loop in ...Re: model dynamics, it's the feedback loop in the upper left of <a href="https://2.bp.blogspot.com/-Wl8jGPivS3Y/VuG-88lrQoI/AAAAAAAAAvk/X8DtHDc7N5M/s1600/SIM2.png" rel="nofollow">this block diagram of SIM</a> that creates those. And a big component of that loop is A, which is dependent on alpha1 and alpha2 (our two behavioral parameters). The tax rate theta you could consider the government's behavior parameter I guess. The way they're put together I suppose results from the "accounting." The structure of A is interesting too: you can see in it G&L's implicit approximation to <a href="https://2.bp.blogspot.com/-xaA8ZF_ElRY/VuHdUBzjhtI/AAAAAAAAAwI/FoyuvmevuuA/s1600/approx_exp.png" rel="nofollow">exp(a*t) as 1 + a*t</a>. a <= 0 for a stable system.<br /><br />1 - alpha2*(other terms).<br /><br />Thus when Ramanan says to adjust alpha2 for different sample periods (for example divide it by N if you divide the sample period by N). He's trying to keep the time constant approximately the same in their approximation to exp(a*t).Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-29110238882533491342016-03-10T18:13:09.488-08:002016-03-10T18:13:09.488-08:00Yes, and the way you interpret the G&L model i...Yes, and the way you interpret the G&L model is that if you use the equation:<br /><br />ΔH = G - T<br /><br />it it actually ambiguous as to what H really is -- the meaning of H becomes dependent on the time step. Adding the time scale τ = Δt/ Γ to the equation gives us<br /><br />ΔH = (Δt/τ)(G - T)<br /><br />nails down what H is -- effectively choosing the multiplier for government debt (= "money") and what you decide to call H (= "high powered money"). If you choose the time scale to effectively follow the path of G - T, then (and only then) does H mean what G&L say it means.<br /><br />Otherwise, it's a bit like a "<a href="https://en.wikipedia.org/wiki/Gauge_fixing" rel="nofollow">gauge</a>" in physics. I can add an arbitrary constant to H because<br /><br />ΔH = H - H(-1) = H + C - (H(-1) + C) = ΔH<br /><br />It is this ambiguity that makes the definition of money ambiguous -- there's no definition of money I know where it doesn't matter the level (H), only changes (ΔH).<br /><br />I understand what G&L want to do, but this equation doesn't do it. Really, they want H = D where D is government debt -- the integral of government spending minus taxes.<br /><br />Using the equation H = D however, removes all dynamics from the model! What remains are just a series of linear transformations. Given a value of G, you could solve for everything else up to an overall scale (i.e. definition of a dollar).<br /><br />PS -- that brings up another thing: accounting has an overall scale degree of freedom. If I multiply every dollar by 100, the accounting all still works out. If the G&L model was all just accounting, there'd be no way to determine the overall level of anything!Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-2312824042046503522016-03-10T13:41:28.205-08:002016-03-10T13:41:28.205-08:00Bill, last comment on this: one (narrow) way inter...Bill, last comment on this: one (narrow) way interpret the issue is that G&L implicitly make the following approximation:<br /><br />exp(a*t) ≈ 1 + a*t<br /><br />Which only holds when |a*t| << 1Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-18325630943002506382016-03-10T10:41:30.457-08:002016-03-10T10:41:30.457-08:00Bill, one way to look at SIM here.Bill, <a href="https://2.bp.blogspot.com/-Wl8jGPivS3Y/VuG-88lrQoI/AAAAAAAAAvk/X8DtHDc7N5M/s640/SIM2.png" rel="nofollow">one way to look at SIM here</a>.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-9856114111356887012016-03-10T08:30:25.906-08:002016-03-10T08:30:25.906-08:00Bill, measuring time in years in my example, you f...Bill, measuring time in years in my example, you find a parameter A such that exp(A*t) passes through any of the points produced by those compounding cases. Then the time constant Tc = 1/A for each system.<br /><br />2% per year, compounded 1 time / year: Tc = 50.498 years<br /><br />2% per year, compounded 4 time / year: Tc = 50.125 years<br /><br />2% per year, compounded continuously: Tc = 50 years<br /><br />What about for 200%?<br /><br />1 time / year, Tc = 0.189 years (9.8 weeks)<br /><br />continuously, Tc = 0.005 years (44 hours)<br /><br />0.189/0.005 = a factor of 37.7!Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-51547303661757041012016-03-09T23:07:00.550-08:002016-03-09T23:07:00.550-08:00Bill, given the annual interest rate r percent per...Bill, given the annual interest rate r percent per year, compounded every T years (consider T <= 1), then after 1 year, the actual annual rate realized (rr) is:<br /><br />rr = ((1 + r*T/100)^(1/T) - 1)*100 percent<br /><br />lim T -> 0, rr = (exp(r/100)-1)*100 percent<br /><br />So for r = 2 percent<br /><br />T = 1 gives rr = 2<br />T = 1/4 gives rr = 2.015 (quarterly)<br />T = 1/365 gives rr = 2.02 (daily)<br />And as T -> 0, rr -> 2.02 (continuously)<br /><br />Not a huge change. But for r = 200 percent, a different story:<br /><br />T = 1 gives rr = 200<br />T = 1/4 gives rr = 406 (quarterly)<br />T = 1/365 gives rr = 635 (daily)<br />And as T -> 0, rr -> 639 (continuously)<br /><br />So it can make a substantial difference. G&L's SIM model (for example) only assumes the compounding is done at the sample times, so clearly the answer they get after a year depends on how many sample times per year. Whether it's a big effect or not depends on the rates involved.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-82804793390320107272016-03-09T21:01:18.763-08:002016-03-09T21:01:18.763-08:00Thanks, guys! :)Thanks, guys! :)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-68046033536727864342016-03-09T20:59:05.655-08:002016-03-09T20:59:05.655-08:00Jason:
"If the government prints up ¥100,000...Jason:<br /><br />"If the government prints up ¥100,000 that is held by banks, our net holdings of deposits (M1) go up (to say ¥120,000) because the banks make loans on that ¥100,000."<br /><br />Right, in the real world. :) However, in the toy economy of their simple model there are no banks and no loans, only cash:<br /><br />"We shall eventually cover both types of money creation and destruction. But we have reluctantly come to the conclusion that it is impossible to deploy a really simple model of a complete monetary economy in which inside and outside money both make their appearance at the outset. We have therefore decided to start by constructing and studying a hypothetical economy in which there is **no private money at all**, that is, a world where there are **no banks**, where producers **need not borrow** to produce, and hence a world where there are no interest payments." (p. 57) Emphasis mine.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-39893075501591575602016-03-09T18:13:57.981-08:002016-03-09T18:13:57.981-08:00Jamie,
Actually, one of the results in the inform...Jamie,<br /><br />Actually, one of the results in the information transfer model is that the money multiplier (ie. information transfer index for M2 to M0 with M0 being physical currency) <a href="http://informationtransfereconomics.blogspot.com/2016/02/money-money-money.html" rel="nofollow">falls over time</a>. Base reserves never seemed to have any effect except on short term interest rates.<br /><br />Bill,<br /><br />That isn't how macro works. If the government prints up ¥100,000 that is held by banks, our net holdings of deposits (M1) go up (to say ¥120,000) because the banks make loans on that ¥100,000.<br /><br />In your example, if you deposit ¥100,000 in the bank, that bank will make loans against it (say ¥20,000), crediting another customer -- not necessarily making the loan to you.<br /><br />However, you can actually sort of do what you say: if you make a big enough deposit, a bank could give you a line of credit on top of your bank account. They don't usually do it in the form of a loan where they deposit money in your account right away, but functionally it is similar.Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-42707188863650936562016-03-09T18:02:10.085-08:002016-03-09T18:02:10.085-08:00Bill, yes a 2% continuously compounded annual inte...Bill, yes a 2% continuously compounded annual interest rate is invariant to the time period. (The 2% annual interest rate is a scale itself.)<br /><br />In the model, if you change the time scale <b>observable things in the real world change</b>. This is not desirable in any model. If I decide to measure time in weeks rather than months, it shouldn't change how fast people adjust to government spending shocks.<br /><br />In the model discussed above, if you change from weeks to months, the time it takes people to adjust to a government spending shock changes by a factor of 50. Since it is just a re-labeling, that doesn't make any sense. There are time series measured with months and with quarters on FRED -- the two series aren't different in any way besides resolution:<br /><br /><a href="https://research.stlouisfed.org/fred2/graph/?g=3KoM" rel="nofollow">https://research.stlouisfed.org/fred2/graph/?g=3KoM</a><br /><br />Check it out; except for resolution, GDP is basically the same if you measure GDP in years or quarters. In the model above, one of the measures would be multiplied by 4 if you changed from quarters to years. And that's in addition to the adjustment for the seasonally adjusted annual rate.Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-5307511166220535502016-03-09T17:49:19.714-08:002016-03-09T17:49:19.714-08:00Peiya,
That isn't how it is set up in the mod...Peiya,<br /><br />That isn't how it is set up in the model under consideration (from Godley and Lavoie). In the model, government debt is basically cash.Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-45963248611696502042016-03-09T13:30:39.189-08:002016-03-09T13:30:39.189-08:00Actually, I think I satisfy all three here.Actually, I think I satisfy all three <a href="http://banking-discussion.blogspot.com/2016/03/sim3.html" rel="nofollow">here</a>.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-89143632433616869652016-03-09T12:03:09.130-08:002016-03-09T12:03:09.130-08:00Hi, Tom. :)
I am not sure what you mean by being ...Hi, Tom. :)<br /><br />I am not sure what you mean by being invariant to the sample period. For instance, do you consider something like an interest rate of 2% per year to be invariant to the sample period, even though you have to adjust it to the period? If not, I don't see why invariance to the sample period is desirable.Anonymousnoreply@blogger.com