Monday, 23 December 2013

Determinancy of Rates and Quantities

Monetarists such as Nick Rowe and Scott Sumner are sceptical about the use of interest rates as a monetary policy tool.  As I understand it, the argument goes something like this.  If monetary policy involves targeting some variable with a $ value, such as the money supply or the exchange rate, then the value of this variable in, say, ten years time will provide a solid peg for the price level at that time.  We might not be able to predict the exact ratio of one to the other, but that ratio must be somehow determinate.  On the other hand, knowing the interest rate in ten years time tells us nothing.  Any price level could be consistent with a given interest rate.

I don't like this argument because it seems to ignore the necessary accounting implications of the path that takes us from today to the future time.  I want to illustrate this with a simple model of a closed service economy. 

In this economy, there is a single consumer service produced in quantity c at price p.  There is a government sector which imposes a fixed per head tax and spends on government services incurring a fixed deficit D.  The only assets are claims on the government being money, M, and an amount b of single period bonds.  The bonds are issued at price q and redeemed at 1. 

The government budget constraint is

b.q + M = M-1 + b-1 + D

The marginal benefit of holding money for liquidity purposes is a function L of the ratio of the money supply to nominal consumption.  In equilibrium, this is equal to the nominal yield on bonds:

L( M / p.c ) = 1 / q -1

Finally, households allocate their resources, being net income (p.c + D) and financial assets (b-1 + M-1), between consumption (p.c) and the new level of asset holdings (b.q + M).  We assume that they do this according to some preference rule, which satisfies the following function V for the new level of wealth*:  

b.q + M = V( p.c , D, b-1+M-1 )

For each period, these three equations determine p.c, b, and then either M or q.  We can either set M and get q, or set q and get M.

This means that if we know all the values of q from now until time t and we know the starting values of M and b, then we know all the values of M from now till time t.  Alternatively, we could note that knowing the value of M in ten years time is insufficient to determine p.c.  To determine p.c, we need to also know the value of M-1 and b-1, and once we know the value of those, then knowing q is as good as knowing M.

So it is absolutely correct that we need a nominal value to provide our peg.  But knowledge of one current nominal value is insufficient.  We also need to know preceding values.  And once we know preceding values, we have our peg and we no longer need to know the current value.  An interest rate will suffice.

Knowledge of the starting nominal values and the subsequent history of rates gives us as much information as the starting values and the subsequent history of the money supply.

* I have not included the real interest rate in this function, to avoid getting into issues which would greatly complicate the analysis.  I do not believe it is relevant to what I am saying here.

[Edit.  This post was prompted by an exchange with Nick Rowe here.)


  1. Am I right in understanding that in your model
    L(M/p.c)= (1/q)-1
    means that supposing the liquidity preference L is fixed, then the central bank can increase prices by keeping interest rates fixed (perhaps at zero) whilst doubling the amount of money M?
    Basically isn't that just the sort of idea that Scott Summer and Nick Rowe have?
    I mean Scott Summer and Nick Rowe definitely allow for central bank intervention in their view of the world don't they? It is the core of what they are about.
    Am I right in understanding that your model has no scope for variations in central bank balance sheet size like that?

    I guess a crucial problem with the Scott Summer view is that liquidity preference L isn't fixed in real life. So changing M or q or whatever is often simply accommodated by shifts in L.

    My impression is that QE perhaps actually "works" by making it very hard for the central bank to subsequently raise interest rates after the central bank has amassed a huge portfolio of long duration low yield assets. The central bank then no-longer has the capability to gather back all of the money it initially spent to accumulate its portfolio because it wouldn't be able to sell off its assets for as much as it initially paid for them. The central bank's portfolio of low yielding assets also would not provide it with the funds sufficient to pay interest on reserves above what its own portfolio received.
    Basically QE is a bridge burning exercise that reassures the market that it will be impossible for interest rates to be raised much. The market then takes on debt and prices long duration assets in a way that it would be fearful of if their was any danger of a rise in interest rates.

    1. There are all sorts of problems with the Sumner view, but that's a different story.

      This model is intended to be really simple. It's largely set out in monetarist terms, but it's structurally very similar to Godley and Lavoie's model PC. The point is that it is not possible to change M whilst keeping q constant. Nick Rowe seems to think it is, but I think that's only possible in trivial models where the only financial asset is money and in those models, there is no role for the (nominal) interest rate.

      This, to me, is at the core of money non-neutrality, but it seems to be brushed under the carpet sometimes when monetarists appeal to models without these complications.

      I don't think I'd agree with your analysis of QE there. I don't think it practically restricts future actions of the central bank at all.

    2. Isn't it possible to change M when q is at the zero bound and have q remain at the zero bound?

      Do you have any links or anything that explain how the central bank can increase interest rates when there is a glut of reserves and it has a huge portfolio of long duration, low yielding assets? The Fed web site says they would sell certificates of deposit as a way to sequester bank reserves but why would banks buy those at an elevated price any more than why would anyone buy back long duration bonds at an elevated price?

    3. You're right. There may be a range (at the ZLB say) say where changes in M will not change q. That's slightly different to my point, as the authorities still cannot choose both M and q. I'd have to think about what it would do in the model.

      In the real world, there are more assets and so more degrees of freedom for the monetary authorities. The easiest way for the Fed to raise (short) rates, whilst keeping the existing level of reserves would be to increase the IOER rate.

    4. But doesn't the central bank need to fund the interest on excess reserves from the money that the central bank receives from its own portfolio? So if say the central bank has a portfolio of $1T that overall pays it at 2% and there is $1T of excess reserves, then it won't be able to fund IOER at say 5%. Finding the money for that would require a transfer of funds from the treasury to the central bank and that would be a totally of the wall innovation requiring politicians to give the go-ahead wouldn't it?

      Basically I thought the central bank is allowed to create more bank reserves to buy assets but it doesn't have the facility to create more bank reserves simply to pay out as interest.

      Obviously it is all just rules but isn't our money no more than a set of rules.

    5. If the central bank is subject to such a budgetary constraint, then it will place a limit on the range within which in can manipulate rates. As you say, it is just rules though, and such a constraint doesn't achieve anything else.

    6. But it has put a ceiling on what rates could be over the coming periods and the more QE there is and the longer the duration of the assets in the Fed's (or BoE) portfolio the longer lasting and lower that ceiling becomes. Say I want to max out and borrow lots of money for some purchase or enterprise and I know that I will be bankrupted if rates were to go to 5% in three years time (unless inflation wiped out my debt). With enough QE done, I can be confident that that is no longer a danger so I can feel emboldened to go ahead.

      I'm not sure I really understand what you mean by, "such a constraint doesn't achieve anything else". I guess it boils down to whether it is considered appropriate to have a central bank at all rather than simply having a genuine printing press.

    7. Asset prices in general are pretty much a function of expected interest rates aren't they? If interest rates go up then prices of perpetual bonds drop. So QE does genuinely support asset prices in a rational way unless I'm in a total muddle.

      Personally though I think supporting asset prices should be the bottom not the top of priorities for our economy though.

    8. The point I'm making here is a quite theoretical one, about whether quantity or rate targets are equally deterministic. If the central bank was constrained as you describe, that would come down to the rules about how it was managed rather than any theoretical constraint. I'm not saying that doesn't matter, but I don't think it can be used as an argument against the theoretical position, particularly as that's not the point Rowe and Sumner are making.

      (Also worth noting that steps are often taken to avoid these type of constraints biting. For example, the Treasury indemnifies the BoE against any loss on the APF portfolio).

      Yes, QE does support asset prices. This may be partly through expectations of future short rates, but not necessarily so (see for example my post

      I don't think supporting asset prices is seen as the end in itself. Certainly with later QE, rising asset prices is just seen as part of the transmission mechanism to higher GDP. Nevertheless, QE does seem to me to produce a lot of adverse redistribution for a small amount of GDP boost.

    9. To be honest, I'm still totally Flummoxed by the theoretical idea that quantities can have an effect divorced from any rationalization such as QE influencing future rates or anything (such as I wondered about).
      I don't see how any given person can be influenced by a quantity that she has no way of sensing and that doesn't effect her directly. The quantity people such as Scott Summner say that they are talking about total quantities irrespective of distribution. So the shipping containers of hundred dollar bills that the late Muammer Gaddafi supposedly has buried in the Sahara exert that "quantity power" just as much as any other money?????

  2. Your first paragraph is correct. Except this isn't just something that monetarists like me and Scott say. This goes all the way back to Wicksell.

    I'm not 100% sure, but this is what I *think* is going on in your model.

    Suppose actual and expected inflation is 0%, and suppose the central bank sets a nominal interest rate equal to the natural rate. The economy is in equilibrium, but it's a metastable equilibrium like a ball resting on a flat table. The ball is the price level. Nothing makes it roll way, but nothing prevents it rolling away.

    Suppose the price level did fall. That would increase the real value of the existing stock of government bonds. That would increase people's wealth, and they would demand more goods, which would cause the price level to rise again, back to the original equilibrium. Therefore the equilibrium is stable after all.

    I think that above paragraph describes what is going on in your model (at least roughly). Short version: since monetary policy is not doing what it needs to do to pin down the price level, fiscal policy needs to pin down the price level instead. Inflation targeting by the fiscal authority, rather than by the central bank. It can be done (as long as you don't have Ricardian Equivalence, of course) but would it be a good way to do it? Not if the fiscal authority has other targets of its own. Because of Tinbergen's Rule. You are leaving one instrument (monetary policy) unemployed, when instruments are scarce, because you have more targets than instruments.

    1. Nick,

      Thanks for your comments there. I find your posts very good for making me think about things like this.

      What you describe is pretty much what's going on. There's a few points worth noting. In terms of what the natural rate of interest is here, it has to be a level that is consistent with the assumed deficit, given the government budget constraint, in order for financial assets to grow at a rate consistent with NGDP growth. That is more important here than any inter-temporal substitution.

      I don't think I'd go so far as to suggest that fiscal policy alone was used for inflation targeting. But it can't be ignored. Money might be providing one peg, but if fiscal policy is providing a different peg, that's going to matter.

      In any event, the key point I'm trying to make is that the peg is there anyway. So we don't lose anything by following a rates policy, even if fiscal policy is targeted at some other objective.

    2. Yep: I would state it this way: fiscal policy affects the natural rate of interest, and so affects the rate of interest the central bank needs to set to keep inflation on target. But if expectations of inflation are not well-anchored by a credible inflation target, or if the ZLB becomes a binding constraint, monetary policy by setting interest rates alone won't do the job.

    3. Agree the ZLB imposes a limit on the ability to use rates as the policy instrument (at least in one direction). As to the anchoring of inflation expectations, I'd say this may or may be achieved by the fiscal stance, depending on how it is cast.

  3. I write to confirm that I am reading correctly.

    When you say "The bonds are issued at price q and redeemed at 1. ", your meaning is that the interest rate has been predetermined so the term q is some fraction, maybe 0.95 for an interest rate near 5%? ( I am assuming an ideal case where on Jan. 1, we know last period M,b and D. Then gov needs b.q + M (with M estimated) to pay coming bills. for the base year. )

    My problem with my own assumption is that the liquidity equation then becomes always negative. This the result of 1/q-1 when q is always a number less than zero.

    Perhaps I fail to understand a basic precondition.

  4. Roger Sparks, if I understand you right, could this be a mix up between
    (1/q)-1 and 1/(q-1)?
    My understanding was that when Nick wrote 1/q-1 , it meant (1/q)-1
    whilst what you are saying gives me the impression that you might have thought it meant 1/(q-1).

    It is a very long time since I had a maths lesson but my dim recollection is that the convention is that 1/q-1 means (1/q)-1.

    Using your example of q being 0.95, 1/0.95-1=0.0526 when just plugging it in like that into my calculator.

  5. Stone.
    Thanks. I think you are correct. My mistake was accepting 1 / q - 1 for 1/(q-1). I should have thought 1/q - 1, with the lack or existence of spaces between symbols controlling the operational sequence.

    With that linkage between equations explained, I think I can accept all the equations. Together, they seem to provide a tight relationship between government and private sector wealth.

    I expect to continue thinking about these three equations. Money supply (an important component in my thinking) is present here but in a little different context. Banks are also here in the sense that they are private sector. Banks as money supply creators are not included here except as may be merged invisibly in the term p.c.

    At first glance, these equations all seem to pass a dynamic test as well as the static test.

    Weird, these three equations remind me, just a little, of the three electromagnetic equations that described electromagnetic theory until Maxwell added the fourth equation. The fourth equation tied the earlier three equations together and made a complete theory. Is there a fourth equation yet-to-be-found?

    1. Roger,

      stone is right that I mean (1/q)-1. I have to apologise, because reading this now I realise that I have could have expressed this in terms of interest rates rather than issue prices and that probably would have been clearer. It's because I originally drew this up differently where it made more sense to do it that way and then I changed it.

      As I mentioned to stone, this is structurally very similar to Model PC from ch. 4 of Godley and Lavoie. Main differences: I have assumed lump sum taxation and I have consolidated government and central bank. So, yes, banks don't really appear in this.

    2. Nick,

      Thinking of the government constraint equation, I did find it quite hard to understand. However, the beauty of writing as you did, is that b can transition to b-previous-period unchanged. I thought that was a very perceptive contribution. (I acknowledge that it does make the liquidity equation much more difficult to understand, but considering all three equations, I think this presentation is neat and complete in describing the monetary side of wealth.)

      (The three equations seem to tie seamlessly with my Government Provided Money Supply description [found at] . That perception is a source of comfort for me.)

  6. Nick Rowe,

    I have been bothered by your comment
    "Suppose the price level did fall. That would increase the real value of the existing stock of government bonds. That would increase people's wealth, and they would demand more goods, which would cause the price level to rise again, back to the original equilibrium. Therefore the equilibrium is stable after all.".

    I think there is more to the story.

    Let us assume that the reason for the average price level fall is a technological improvement in one product. Yes, consumers would have more to spend on other goods and presumably the GDP would recover and return to the same level. This is based on spendable income remaining the same.

    That said, the actual point of equilibrium would have shifted. The mix of goods has changed and presumably the consumer has an improved life with more goods at same level of spendable income. That is the rest of the story. The actual (or physical) equilibrium has shifted even though the equilibrium as measured by GDP may be unchanged.

    Distribution changes between sector groups are not reflected in GDP measurements.

    1. Roger: You are doing a comparative statics experiment, where some exogenous change caused prices to fall. I am doing a stability experiment, where I ask what would happen if prices fall, to see if there is any force pushing them back up again to where they were originally. The "Hand of God" forces the price level down initially in my experiment, then disappears.

  7. What is the monetarist rebuttal of this?:
    "The monetary aggregate that the Federal Reserve directly controls (monetary base) is actually quite negatively correlated with future inflation. "

    1. Money demand is a negative function of nominal interest rates, which are a positive function of expected future inflation, which is positively correlated with actual future inflation. Standard monetarist (plus non-monetarist) economics.

      Plus, Google "Milton Friedman's Thermostat".

    2. Nick Rowe, thanks for the tip about Milton Friedman's Thermostat (I've got to your post about that). I actually just this minute had a go posting about QE wondering whether QE actually might increase the preference to hold cash as a store of wealth.

    3. Nick Rowe, to be honest your posts about Milton Friedman's Thermostat make the point so convincingly that I'm sort of embarrassed that the point wasn't glaringly obvious to me from the start. My only excuse is, it's not easy being stupid :)

      Is it fair to say though that the negative correlation indicates either that central banks are not aggressive enough (as Scott Sumner would argue I guess) or perhaps that monetary policy doesn't always work terribly well?