Wednesday, March 16, 2016

Scott Sumner writes down another information equilibrium model

NGDP = MB*(Base velocity), where V is positively related to nominal interest rates.
Hey, that's a pretty good model. Where have I seen this before ...

Let's write down a pair of information equilibrium relationships


i ⇄ V

So that in general equilibrium we have:


log i = α log V

Or more compactly:

log i = α log NGDP/MB + β 

with α = 3.6 and β α log k  = 9.1. Here's a graph:

This has the added bonuses of using numerical values, showing explicit functional forms, and being fit to actual data (in this case from 2000 to the present).

Quantitative analysis: ask for it by name!


  1. Meanwhile, Sumner makes an econ 101 mistake that no one seems to notice (or get aggravated by) but me:

    Also, I still stand by my similar point in a similar post that Sumner's model is not an information equilibrium model; he's literal just written down the money demand curve in IS-LM, so his money demand curve is just as IE as IS-LM (not your version) is.

    1. You're right -- that next sentence from Sumner isn't something you can do with an IE model ...

      "Thus if you cut interest rates without increasing the money supply, then V falls and policy becomes more contractionary."

      In the IE version, you can't really cause velocity to change without changing either money demand or money supply (or both).

      I am actually working on a post about what happens if you try to move the price in an IE model. In an ideal gas, you can change the pressure, but only be changing energy (temperature or number of particles = demand) or volume (= supply).

      But I still need to think it through a bit.

    2. Every time I read, "thus if you cut interest rates without increasing the money supply," I feel a strong urge to vomit. Sumner is literally saying "thus if you decrease NGDP, NGDP will fall," but no one seems to call him out for arguing from this reasoning -- the injustice is unbearable.

    3. John, why can't you understand what Scott is doing?

      Of *course* a sophisticated economist like you understands that if you do things that cause interest rates to fall but don't increase the money supply that necessarily means a drop in V, hence in NGDP. But tons of idiots out there don't get it. They've been trained by mediamacro to think that lower interest rates will "stimulate the economy" and mean that money is "loose". "Easy money" and "irresponsible money printing" and all that. Those are the people Scott is fighting against, not you. He's trying to get them to see what you already see, which is it's M and V (and their product) that count, and interest rates are an epiphenomenon and the wrong thing to target. When sailing a ship, you should be targeting the destination, not the angle of the steering wheel, especially when the wind and currents can change rapidly.

      What "injustice" are you even talking about? It sounds like you agree with Scott, so much so that you think his point is obvious.


      Kenneth Duda
      Menlo Park, CA

    4. Sorry, just a quick follow-up.

      I am deeply confused by your and Jason's incessant criticism of Scott.

      Scott is not trying to create a new theoretical framework for all of economics. He's trying to figure out how to improve monetary policy. It seems to me that NGDP level targeting guided by a prediction market would work dramatically better than today's muddle.

      Do you disagree? Do you think what the Fed did from 2007 to 2015 was better than what simple market-forecast-based NGDP level targeting would have done? Because what I see is millions of people needlessly unemployed for years because of monetary policy failure that would have been impossible under NGDP level targeting, and I see super-smart economists like you opposing the few voices I hear for better policy, and it seems very sad.


      Kenneth Duda
      Menlo Park, CA

    5. I wasn't really criticizing Scott in the post above. Whenever Scott writes down any equations, he tends to write down information equilibrium models (and much of what he says can be captured by them ... Especially on interest rates).

      My question is: if you like Sumner's theory, why wouldn't you like information equilibrium? It can get the same results but has the added benefit of being quantitative!

      Sure, they're not all the best models built with the IE framework, but many of them are pretty good. Part of the original intention behind the effort was to build Sumner's model. I was going to send the original draft paper to him back in 2012 before I started the blog.

      Posts like the one above are basically questions like: do you like model A ot model B that is the same as model A but is quantitative?

      Why does no one want to have their cake and eat it too?

    6. I love the part of information equilibrium that I understand, which is modeling agent behavior as essentially random subject to constraints e.g budget constraint, which seems much smarter than the hyper intelligent utility maximizing representative agent stuff, and then showing the system tends towards maximum entropy just like in statistical thermodynamics. ... Brilliant! I hope someday this framing is part of standard undergrad econ.

      I've made a modest effort to understand the details, but I get lost quickly. Like when supply and demand are to be in information equilibrium... I don't even know what the units of "supply" are. I've gathered that the IT framework is not a model, but rather a tool for creating models. But that just makes it one level more abstract, where it was already too abstract for me to follow.

      When I retire from Arista, maybe I'll try to figure this out, and also why Selgin thinks free banking leads to stable aggregate demand, which is another great mystery of the intelligent part of the econ blogosphere. :-)

    7. Regarding whether I like Sumner's theory... I'm not sure he has one in the sense you use the word. If you mean a bunch of practical beliefs, the belief that broad nominal aggregates are a better gauge of what matters in the macroeconomy than inflation or interest rates are, then yes, I do really like his ideas.

    8. I can answer the question about units.

      Short answer: they don't really matter.

      Long answer: Since

      dD/dS = k D/S

      I can take S → α S and get the same equation:

      dD/d(αS) = k D/(αS)

      (1/α) dD/dS = (1/α) k D/S

      dD/dS = k D/S

      But if you want concreteness, you can think of S measured in widgets or widgets*(price at some reference time). Since solving the differential equation introduces the reference points Dref and Sref so that

      D/Dref = (S/Sref)^k

      As long as the units of S and Sref (and D and Dref) are the same, the units don't matter (this is a consequence of that symmetry transformation above).

    9. "Do you think what the Fed did from 2007 to 2015 was better than what simple market-forecast-based NGDP level targeting would have done?"

      No, because I believe in liquidity traps. Otherwise, even conceding that liquidity traps don't exist (which is a strange position to have given that each episode of QE was consistent with a tiny change in the FFR but a huge drop in velocity - which either means that the marginal effect of the money supply on NGDP becomes really really small at high levels of the money supply or that Krugman's version of the liquidity trap is basically right), the only theoretical reason that NGDPLT would be better than inflation targeting and/or a price level target is if nominal wage rigidity is a lot more important than everyone currently thinks it is. I can't think of a specific paper, but I'm fairly certain the reason NK focuses so much on sticky prices is that it's really hard to find sticky wages in the data (i.e., countercyclical real wages). Given the lack of empirical evidence for sticky wages, NGDPLT shouldn't be any better than a price level or inflation target.

    10. Jason, thanks as always for your patience. It helps me to know that S and D are measured in widgets. Are D and S things you can measure? Does it make sense to argue that if 1000 widgets were sold in Q1 2016, that means S=D=1000? (It can't be that simple, because then D=S as an identity.)

      Also, once you have:

      D/Dref = (S/Sref)^k

      What do you do with this relationship? Is this a constraint that holds over all time periods, so that you can determine or estimate S if you know D somehow at some given point in time?



      What do you think of David Beckworth's proposal of an automatic fiscal backstop?

      Do you think that approach would have significantly reduced the pain of the great recession?

    11. Sorry, David's article is a little long and contains a bunch of things I already know you don't buy. The critical passage I am hoping you will react to is this one:

      > How to Make the Fix Credible
      > As noted above, a NGDP growth path target should create its own
      > self-fulling expectations of stable demand growth. One way to
      > reinforce this tendency and insure against central bank incompetence
      > is to have the U.S. Treasury Department provide an automatic backstop
      > for the spending target. This would make the system foolproof.
      > The way it would work is that once a year the Treasury Department
      > would check to see if the Fed was keeping total dollar spending on
      > target. If it fell below target, the Treasury Department would
      > automatically deposit bonds at the Fed and send the new money created
      > by those deposits directly to households. It would continue to do so
      > until spending got back up to its targeted growth path.
      > If total dollar spending were above target, the Treasury Department
      > would again deposit bonds at the Fed. But this time the Fed would be
      > required sell the bonds to the public, which would take money out of
      > circulation. The Treasury Department and the Fed would continue doing
      > this until spending fell back down to its targeted growth path.
      > ...
      > For conservatives there would be far less need for discretionary
      > fiscal policy to respond to the business cycle. That would mean less
      > growth of government debt. It would also mean that monetary and fiscal
      > policy would be become far more predictable and rule-like.
      > For liberals there would be far less human suffering since large
      > cyclical swings in unemployment would disappear. It would also mean
      > more stable wage growth since a total dollar spending target would, in
      > effect, also be stabilizing the growth of total dollar income.

    12. Hi Ken,

      D = S would apply if the price was always 1. From the equation

      P ≡ dD/dS = k D/S

      The derivative is the "detector of information flow", which is defined to be the abstract price (more like the price level than the true price because of the units discussion above).

      You have

      D = (1/k) P(S) S

      With P = Dref (k/Sref) (S/Sref)^(k-1) ... which is just dD/dS. If k = 1, then P = k Dref/Sref = constant.

      So you can see the units of D don't really have to be same as the units of S ... they can be, if P = 1. In the most general case, demand D is proportional to P × S ... which is say price level times aggregate supply ... if you measure aggregate supply in widgets times nominal price at some reference time, then D = NGDP.

      The equation D/Dref = (S/Sref)^k holds as a statistical average, just like PV^γ = constant does in thermodynamics (isentropic process). In thermodynamics you have 10^23 particles and the statistical error is tiny. In economics, you only have thousands, millions or billions of "particles" (widgets, dollars, etc) and the error is larger. So measures like inflation or nominal output will jump around.

      But yes, you can use D/Dref = (S/Sref)^k as a function to determine either D or S given the other. In the IE model, this becomes the equation of exchange if you think of supply as the money supply (M) and D as NGDP = PY ...

      PY/(p0 y0) = (M/m0)^k
      PY = [c M^(k-1)] M ≡ V M

      where c is the combination of constants and "zero" instead of "ref". Just search on equation of exchange or quantity theory of money on this blog for more about that. That equation can was also used for the interest rate model shown above with the additional assumption that velocity is in information equilibrium with the interest rate.

      D/Dref = (S/Sref)^k is the "general equilibrium" solution. There are two "partial equilibrium" solutions where you assume either S moves faster than D or D moves faster than S. These are supply and demand curves, and k is related to the price elasticities (slopes).

      There's more on the basic mechanics in the paper -- this is just a summary of the first couple sections:

    13. Regarding my criticism of Scott, the reason I got so mad at him is that he suggested something that is impossible: a central bank cannot, in reality, control both the money supply and the nominal interest rate, so Sumner is implicitly assuming that the money demand curve shifts left when the nominal interest rate falls.

      This would be fine if he had said "if you lower interest rates while people start wanting to hold more money, the monetary base will not change and NGDP will fall." But Sumner doesn't say this, he insinuates that it is possible for 1) the Fed to cut the nominal interest rate and 2) the Fed to force the money supply to remain constant. This is indeed a possible equilibrium, but if the Fed does 1, then if 2 happens it was not caused by the Fed.

      Alternatively, if Sumner said "if you cut interest rates by lowering expectations of future money supply growth, then NGDP will fall" he would have only partially been wrong, and only then because his model is static and therefore shouldn't be used for dynamic analysis.

    14. Thanks John. I guess I understand, but I think you misunderstand Scott. I think he meant *if* those things happened. He was not asserting that the Fed could engineer both to happen simultaneously.

      Jason, thanks as always. I need to read your response to me one more time. I had never imagined that D could be NGDP while S is the money supply! I had been thinking of them too literally as supply and demand in econ 101 e.g. quantity of widgets that would be demanded/supplied as a function of market price.


    15. I don't care if Scott didn't mean to say what he did. What he did say was horrendous and typical of him; he may be fine assuming NGDP is an instrument of the central bank, but I am not and no one should let this slide, specifically because of his liberal use of unstated assumptions.

  2. Or even if you believe that money demand is indeterminate at the ZLB and therefore monetary policy is ineffective, hopefully you also realize that your beliefs might be wrong, and that it's at least possible that the expectations channel is sufficient to drive enough marginal extra spending even at the ZLB to escape the trap, and shouldn't you support that policy even if your preferred models tell you that it will neither hurt nor help?

  3. Because maybe we are operating outside of the scope conditions for your model. It's not really possible that money demand is indeterminate? Surely each economic actor must have some preference. What would it even mean in reality (i.e., outside of a model) if demand was "indeterminate"?

    It reminds me of hanging a picture by a wire. In theory, if you have a wire pulled tight between two nails in a picture, and then try to hang the picture by that wire, the tension in the wire is indeterminate and the wire snaps. In reality, the wire stretches a bit, and is no longer perfectly straight, and the tension quickly drops to something manageable. Maybe it's the same with NGDP targeting at the ZLB.


Comments are welcome. Please see the Moderation and comment policy.

Also, try to avoid the use of dollar signs as they interfere with my setup of mathjax. I left it set up that way because I think this is funny for an economics blog. You can use € or £ instead.

Note: Only a member of this blog may post a comment.