Home > Macro/Monetary Theory > A Model of Endogenous Money

A Model of Endogenous Money

I’ve been working on a macro model in which the quantity of money is endogenous and I want to post an early version of it here before I do any more complaining about other people not doing anything constructive.   I am attaching it as a pdf because I couldn’t get the equations to work in wordpress.  It is sort of a cross between a blog post and a working paper, lots of equations and boring math but with some very informal discussion here and there.  Also, to put all the math in would have been very tedious, so I tried to put in as little as possible, so some steps are omitted.  It should be possible to reconstruct what I did here from what I included but it would take some effort (and probably a mathematical computing package).  I think it is possible to organize it in a simpler way but I am still working on that.

The thing you should take away from this is that monetary policy, as we are currently doing it (inflation targeting) works much the way we think it works but that I think it either leads, in a predictable way, into a “liquidity trap” and a deflationary recession in the long run or it requires ever-expanding “fiscal policy” to keep this from happening.

A Model of Endogenous Money



  1. July 16, 2014 at 12:43 pm

    Take a standard New Keynesian model: there’s an IS curve, with output a negative function of the real interest rate; a Phillips Curve, where inflation adjusts to the output gap and expected inflation; and the central bank sets the nominal rate of interest.

    If the central bank holds the nominal interest rate constant, you get an unstable equilibrium. If it sets it a little too low, given expected inflation, inflation accelerates. If it sets it a little too high, inflation falls without limit.

    To make the system stable you need the central bank to follow some sort of Taylor Principle. If inflation rises to 1% above target, the central bank needs to raise the nominal rate by more than 1%.

    But if the central bank is too slow to adjust the nominal rate, relative to the speed at which inflation and expected inflation adjust, you can still get an unstable system.

    That’s the problem with having a monetary system in which the supply of money responds passively to demand.

    I haven’t worked through it carefully enough to be sure, but I think something like that is what is happening in your model. Under perfectly flexible prices, there is always one expected rate of inflation that keeps the system from exploding, but there is nothing that would enable people to learn what that expected rate of inflation is. If they initially expect a slightly higher rate of inflation, actual inflation will be even bigger than they expected.

    • Free Radical
      July 16, 2014 at 7:20 pm


      I have to think about that a bit to fully respond but let me say one thing off hand. I did say something parenthetically about the initial setting of a rate below the “natural rate” possibly being inadvertent but I don’t really think that central banks always intended to raise inflation by raising nominal rates and being net borrowers but they accidentally set the rate too low one period. I think they try to create inflation by lowering rates and expanding lending. What I’m trying to point out is that this works in the short run but it has a sort of opposite effect in the long run which then has to be offset by even “looser” (by conventional standards) monetary policy. This is not based on the CB making errors, I am assuming that they have perfect information and that they are hitting their target at all times. It’s true that if they set what I am calling the “natural” rate in the first period and all periods thereafter, there would be no problem. There may be a parallel there with the New Keynesian model (I’m not fully convinced but have to think about it) but I think that would be entirely inconsistent with what central banks actually do.

    • Free Radical
      July 16, 2014 at 7:27 pm

      One more thing, just to be clear. I made the model “crash” in five periods but I think those five periods are analogous to about a thirty year period (In the U.S). So I’m not saying, if the Fed screws up one period, they slide into a recession a few years later. I’m saying that the nature of the system in the long run has a baked-in deflationary pressure that gradually gets stronger over time and more difficult to hold at bay.

  2. Tom Brown
    July 16, 2014 at 4:52 pm

    Nick you write:

    “If it sets it a little too high, inflation falls without limit.”

    Was that a statement of fact, or a comment about what should happen in the standard New Keynesian model or about what should happen in Mike’s model?

    In terms of what we observe in the real world, do you think there is any evidence that this hasn’t happened when the model says it should?

  3. Tom Brown
    July 16, 2014 at 4:54 pm

    Mike, in terms of writing equations in blogs, have you tried Mathjax? I noticed that’s what Jason uses, and I tried it myself and it seemed to work. I don’t know if it does everything you want or if it works on this kind of blog (it seemed to work on blogger).

    • Free Radical
      July 16, 2014 at 6:44 pm

      Good tip Tom, I will look into it, thanks!

  4. Tom Brown
    July 16, 2014 at 7:04 pm

    Mike, I know you looked at IT specifically, but can your model be adapted to look at NGDPLT? Sorry if that’s a stupid question (answered in your write up). I didn’t spend much time on the write up yet, but I got the impression you were focused on IT.

    • Free Radical
      July 18, 2014 at 3:59 am


      Sorry this comment slipped by my. It should be pretty straightforward (famous last words) to incorporate NGDP level targeting in this. I have been toying with changing the presentation from an inflation target to a price level target actually. However, the main point of NGDPLT is that it allows the CB to make up for past mistakes and follow a countercyclical monetary policy without needing to use their “discretion.” I am abstracting from these issues here. It is an entirely different issue I am trying to point out. So I think NGDPLT would work better than what we are doing for the reasons just mentioned but I still think we would go progressively farther down this path where the CB or the Central Gov. has to continually buy up more and more of the economy until they eventually have to “buy the whole world.”

  5. Tom Brown
    July 17, 2014 at 7:29 pm

    Mike, I informed Jason of your post here because your final paragraph above (and similar statements in the pdf) seem to overlap with his conclusions based on his model. He agreed:


    • Free Radical
      July 18, 2014 at 5:58 am

      BTW, I listened to that podcast. It’s a little enlightening for me to hear that the same thing on in physics (that guy from the E=MC Sucker part was basically a portrait of an “internet Austrian”). I kind of thought that economics was special in that way. After all, it seems like it would be hard to argue with the other guy’s point about the atom bomb and all that stuff. If physics is so wrong, how does everything work so well? At least with economics we actually kind of suck at designing things and predicting things (we’re not totally helpless but we certainly can’t build the economic equivalent of an atom bomb or complete an economic moon shot). I guess it’s just part of the human condition.

      The irony, of course, is that Einstein was basically a crackpot with a crazy theory too. So every once in a while the crackpots score a point on the establishment. Then those are the only ones you read about a hundred years later. So there is probably a selection bias at work on peoples’ psyches. But to be fair, Einstein, even though he was famously employed as a patent clerk when he came up with relativity, was formally trained in physics and pretty good at math.

      The math is important. I know this from experience. I have thought lots of things were intuitively obvious but I just couldn’t quite figure out the math. Then later, after trying to figure out the math, I realize I was not correct. It’s tantalizingly easy to convince yourself that some idea you have is “obvious” even though you can’t prove it is right. Maybe beating that out of you is the most important thing a formal education can do for you.

  6. W. Peden
    July 18, 2014 at 8:11 am

    “we certainly can’t build the economic equivalent of an atom bomb”

    The Great Depression and the Great Inflation seem pretty close, and arguably they were primarily driven by economic ideas i.e. the Real Bills Doctrine and Old Keynesianism.

    • Free Radical
      July 18, 2014 at 6:26 pm

      I think you are taking my comment in the opposite of the way I intended.

  7. Tom Brown
    July 18, 2014 at 2:52 pm

    Mike, I think Sumner is asking for the intuition behind your result (I gave him a link):


    But I’m not the man for the job!

    • Free Radical
      July 18, 2014 at 6:41 pm

      This also requires me to be more clear about what I mean by “monetary” and “fiscal” policy. Guess I better get on that…

      • Tom Brown
        July 18, 2014 at 8:10 pm

        Yes, you should get on it fast, since try as I might, I couldn’t resist telling all sorts of lies about you in true modern jackass fashion. 😀

      • Free Radical
        July 18, 2014 at 9:54 pm

        Haha, I don’t know if you are going to make me a star or ruin me Tom… I will try to post something brief on it today. There is a lot to say about it though so I might not get it all in.

      • Tom Brown
        July 18, 2014 at 10:21 pm

        Sumner appreciates short comments anyway… unless you reach Sadowski status (who has free reign to make unlimited and off topic comments, full of links, whenever he pleases). Well, since Sumner is laissez faire about his comments section, the same can be said of anybody (including MF), but the thing is, will Scott actually read it? BTW, Sadowski has disappeared… I haven’t seen him for a while.

  8. Tom Brown
    July 18, 2014 at 5:05 pm

    W. Peden, are you saying that the Real Bills Doctrine resulted in either the Great Depression or the Great Inflation?

  9. Tom Brown
    July 18, 2014 at 6:09 pm

    Mike, you are referring to a long term trend, correct? Would it be correct to say that your model says the effectiveness of IT monetary policy at the ZLB might diminish over the course of this long term trend? You might ask: Why would you be at the ZLB when monetary policy was effective? We could blame that on “shocks” (suppose there were a number of shock induced business cycles (recessions) over the course of the long term trend).

    • Free Radical
      July 18, 2014 at 6:39 pm

      Yes I am referring to a long term trend. But it’s not really about IT at the ZLB. I am saying that we find ourselves at the ZLB because of monetary policy. And not because of mistakes in monetary policy, or random shocks but because of the very nature of monetary policy. This is because when they induce the public to become net borrowers, they necessarily put them on a path where their monetary wealth must be decreasing over time (in order to pay the interest). But in order to keep prices going up, they have to be expanding the money supply. This means that the amount they have to get people to borrow keeps increasing and in order to get them to do this, they have to keep lowering rates. If they hit a point where they can’t lower them any more, prices will tend to crash unless they find another way to pump money into the economy (I am calling all other ways “fiscal policy” but I probably need to do a post about that to clarify what I mean since this is not entirely clear).

  10. Tom Brown
    July 18, 2014 at 8:01 pm

    Mike, thanks. I read that a couple of times, but I think I understand what you’re saying.

    Do you think your model could be expanded to say something about how long term trends affect the ease with which a CB can escape a premature visit (say due to an AD shock) to the ZLB? (not assuming here that the CB makes any mistakes)

    Nick Rowe, for example, likes to point out that the BoC easily escaped a dip in inflation (and perhaps a brief visit to the ZLB too? I’m not sure) in 2008 by resorting to a small amount of unconventional monetary policy (i.e. a permanent increase in MB via QE), and this communicated all the market needed to know to put them right back on their 2% IT in short order. Now I suspect he attributes the BoC’s success with this (in contrast to the Fed’s much larger, lengthier and less effective QE program) to the BoC’s single mindedness about perusing their 2% IT: i.e. they much more effectively communicated the proper expectations about future inflation to the Canadian market than did the Fed to the US market. Nick calls this the “Chuck Norris effect.” Chuck doesn’t have to deliver any round house kicks to clear a room: he doesn’t even have to ask twice! It happens right away because people are well aware that he means business!

    I think Jason would argue that this indeed worked fine for Canada in 2008 (as Jason’s model would predict it should have), but long term trends are changing for Canada, and it won’t be so easy to escape the ZLB next time a shock puts them there. And furthermore, this has nothing to do with expectations or effective communication from the CB (at least not in the same sense in which Nick describes it).

  11. Tom Brown
    July 18, 2014 at 8:49 pm

    Mike, you wrote above:

    “This is because when they induce the public to become net borrowers, they necessarily put them on a path where their monetary wealth must be decreasing over time (in order to pay the interest).”

    But when that interest is paid, it improves the commercial banks’ equity positions, which increases the size of bank shareholder dividend checks and bank employee bonuses, doesn’t it? What I’m saying is, don’t those interest payments get recycled right back into the “public’s” hands?

    • Free Radical
      July 18, 2014 at 10:04 pm

      Yes, at least sort of. That’s why I specifically said in the model that they don’t pay dividends. In essence, this would fall into what I refer to as “fiscal policy” at the end. The reason I keep putting that in quotes is because I mean something very different from what most people mean. I’m trying to draw a different distinction. I probably need to make up new words I guess. But be careful trying to discuss I laims about “fiscal policy” with someone like Sumner. I basically agree with what he believes about the effectiveness of “monetary policy” at the zero bound. My point is that the CB has to inject money in some way other than lending it. One way to do that is for the government to “borrow” it and spend it. Another is for the Fed to just spend it (or to drop it from a helicopter for that matter). So I’m not saying it matters whether you do this via “monetary” or “fiscal” policy. You could say that I’m using fiscal policy as an example of one way this might be done. But I’m not saying it is the actual thing the government does with the money that matters, it is their ability to expand the money supply that matters. In that respect, I think Sumner would call even that “monetary policy,” and all the caveats about monetary offset apply. Everything in my model is monetary. I’m just trying to separate two fundamentally different methods of expanding the money supply.

    • Free Radical
      July 18, 2014 at 10:08 pm

      P.S. That is also why I chose the last graph that I did in the last post, but I suspect people will be confused by that (I’m not entirely sure that is the one I want either for the record, the fact that the government maintains a perpetual buffer of securities in the market which the Fed uses to conduct policy makes things a bit fuzzy).

  12. Tom Brown
    July 18, 2014 at 10:29 pm

    I think you will get push back from Sumner about alternative methods of CB money injection, unless that means an NGDP futures market! Just yesterday he was again shooting down helicopters.

    • Free Radical
      July 18, 2014 at 10:55 pm

      Also for the record, I’m not advocating any particular form of monetary injection. And obviously NGDP futures markets and level targeting is Sumner’s baby but he has said that QE is better than nothing.

  1. October 8, 2014 at 1:40 am
  2. November 24, 2014 at 1:47 am

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