The inverse function theorem and the monetary policy instrument

I'm not very good at math. So treat this post accordingly.

[Ooops! As Kevin points out. I meant the *inverse* function theorem. I did say I was no good at math. I've edited it now. Can't find the strikeout function.].

The basic idea here seems rather obvious. But I don't remember anyone making this connection before. Between the inverse function theorem and the choice of monetary policy instrument.

There's something in math called the Inverse Function Theorem. It says that if Y is a function of X then, under certain conditions, there must exist some inverse function such that X is a function of Y. If Y=F(X), then X=F^-1(Y).

IIRC, those "certain conditions" are continuity and monotonicity. If you could draw that function on a sheet of graph paper without lifting your pen from the paper (continuity), and if the curve always sloped up or always sloped down (monotonicity), then those conditions were satisfied. If you can find Y from X, you can find X from Y. But if the graph is U-shaped, and so non-monotonic, the inverse function theorem does not hold. You can find Y from X, but each value of Y gives two possibles values for X, so you can't find X from Y.

I used to drive a farm tractor that (almost) obeyed those certain conditions for the inverse function theorem to hold. It had a hand throttle and a foot throttle, and those two throttle linkages were connected up somewhere underneath. If you moved one throttle the other moved. So you could control the speed either with your hand or with your foot. It didn't really matter which policy instrument you used.

I said "almost", because those certain conditions didn't exactly hold. If you had the hand throttle set at half-speed, you could use the foot throttle to make it go anywhere from half-speed to full-speed; but you couldn't go below half-speed just by lifting your foot. You had to move the hand throttle back to do that. The failure of the inverse function theorem meant there were some things you could do with the hand throttle that you couldn't do with the foot throttle. The linkage between the two throttles wasn't monotonic everywhere.

It's easy to think of mechanisms where the inverse function theorem is not true. A crank is an example. The pistons go up and down, but the motor could be running either forward or backwards. The only reason cars go the right way is because we use a DC electric starter motor to start them the right way, and then they keep on going the same way. But Farley Mowat once had a boat which had a different starting mechanism for the engine, and half the time it would start and go backwards.

Economies are like tractors, and the central bank is like the driver. Everything is linked up under the hood, and if the central bank moves one lever other levers will move too. Change the interest rate, and the stock of money and the exchange rate will adjust too. Change the stock of money, and the interest rate and exchange rate will adjust too. Change the exchange rate, and the rate of interest and stock of money will adjust too.

If the inverse function theorem held true, it wouldn't matter which lever the Bank held onto. And it wouldn't matter which lever economists assumed the bank held onto. And it wouldn't matter if the lever the bank thought it was holding was the same lever economists thought the bank was holding. But if the inverse function theorem didn't hold true, it would matter which lever the bank was holding.

Economies are more complicated than tractors. The speed of the economy doesn't just depend on the current position of any policy lever, but on the past and expected future settings as well. And the different levers are not connected to each other in any simple immediate way, where the current position of one lever depends only on the current position of the other lever. The whole time-path of one lever depends on the whole time-path of the other. Each lever's positions is a function of time (and other things), and those two functions are related to each other.

Under certain conditions, if we know how one lever moves as a function of time, we can figure out how the other lever will move as a function of time, and vice versa. In other words, under certain conditions a meta inverse function theorem will hold true, so that if the first function is a meta-function of the second function, the second function will be some meta-function of the first function.

I don't know what those "certain conditions" are. But I think they are unlikely to hold true.

A model which exhibits the neutrality of money is one example where they don't hold true. Start in one equilibrium time-path. Now consider a second time-path where all nominal variables are doubled compared to the first, but the time-path of all interest rates is the same, and that is also an equilibrium time-path. You can figure out the time-path for interest rates from the time-path of prices, but you can't figure out the time-path of prices from the time-path of interest rates.

You don't need to believe in the exact literal truth of monetary neutrality to see that this example poses a problem for the inverse function theorem. The same time-path for nominal interest rates could be both a far too loose monetary policy -- compatible with ever-rising inflation, as well as a far too tight monetary policy -- compatible with ever-falling inflation.

Monetary policy instruments aren't always all equivalent. You can sometimes do things with one lever that you can't do with another, even if all the levers are connected up underneath. I could make my tractor go faster with the foot throttle, but couldn't slow it down. I needed the the hand throttle for that. It's the opposite problem with the nominal interest rate, at the zero lower bound. The bank can't speed the economy up. The inverse function theorem doesn't always work.

Off canoeing for a few days.

Comments

[Kevin Donoghue]

Nick, that's the Inverse Function Theorem you're thinking of.

[William Peterson]

I think there's something wrong with the monetary neutrality argument at the end of this post. There are two steady state paths with the same nominal interest rate, but different values for the monetary stock and prices: both have the same rate of monetary expansion, the same inflation rate, and the same values for all real variables. So the function from money stock to nominal interest rates is many->one, and we cannot find an inverse function. But if you compare two paths with different monetary expansion rates, they will have different inflation rates, different values for real variables, and different nominal interest rates: if we assume super-neutrality (inflation perfectly anticipated, no effect of inflation on real variables) then there is a one->one relationship (the Fisher equation) between nominal interest rates and monetary expansion (or inflation).

[Nick Rowe]

Kevin: Ooops! I said I was no good at math. Premature something or other. What's that word for forgetting the name of things?

Changed it everywhere.

William: yes, there's neutrality, and super-neutrality. I was talking about neutrality. The Fisher equation says that higher inflation means higher nominal interest rates, but it can't be reversed. If you raise the nominal interest rate, as a policy lever, that doesn't mean inflation will be higher. Remember the Kocherlakota controversy?

[Adam P]

"If you raise the nominal interest rate, as a policy lever, that doesn't mean inflation will be higher."

Under certain conditions, very much like the ones we have right now, James Tobin would disagree with that. (Maybe that's what Kocherlakota meant, though if so he could have said something!)

[Nick Rowe]

Adam: Was that the Tobin and Buiter paper, for the very long run? To get the budget back in balance? I vaguely remember that paper. But wasn't the stability of that comparative static result questionable?

"(Maybe that's what Kocherlakota meant, though if so he could have said something!)"

Thoughts along that line have been running through my mind ever since.

[Patrick]

Is this the paper?

http://cowles.econ.yale.edu/P/cp/p04a/p0437a.pdf

[Adam P]

I don't think it was that paper, it's this standard debt-deflation result where the AD curve can slope the wrong way. I just recently saw it in a book from Tobin (a set of lectures).

He draws an IS-LM diagram in (p,r) space instead of (Y,r) space. Then, due to the effect of deflation on debtors you get the IS curve sloping the upwards, so on the IS curve higher p goes with higher r. He draws the LM curve as usual.

The upward sloping IS curve I'm fine with. However, he then depicts an expansion in the money supply, M increases, which shifts the LM curve in the usual way. The result, (which he explicitly states so I can't be misreading the diagram), is that higher M implies lower P, lower r and *lower Y*. (This part I don't understand).

[Adam P]

As far as I can tell, the idea is that higher M initially increases P, and reduces r. The higher P causes debtors to spend less, lower r causes creditors to spend more but the change in debtor spending is bigger so net spending declines.

The decline in aggregate expenditure lowers Y, causing debtors and creditors to spend less and p to fall by more than its initial rise.

[Mark A. Sadowski]

These sort of analogies always make my head spin.

I find it easier to think about it this way:

James Bullard (2010), "Seven Faces of 'The Peril'" http://research.stlouisfed.org/econ/bullard/pdf/SevenFacesFinalJul28.pdf

"In this paper I discuss the possibility that the U.S. economy may become enmeshed in a Japanese-style, deflationary outcome within the next several years. To frame the discussion, I rely on an analysis that emphasizes two possible long-run outcomes (steady states) for the economy, one which is consistent with monetary policy as it has typically been implemented in the U.S. in recent years, and one which is consistent with the low nominal interest rate, defĺationary regime observed in Japan during the same period.... When the line describing the Taylor-type policy rule crosses the Fisher relation, we say there is a steady state at which the policymaker no longer wishes to raise or lower the policy rate, and, simultaneously, the private sector expects the current rate of ináation to prevail in the future.... In the right-hand side of the Figure, short-term nominal interest rates are adjusted up and down in order to keep inflation low and stable.... as we move to the left... the two lines cross again, creating a second steady state..... The policy rate cannot be lowered below zero, and there is no reason to increase the policy rate since well, inflation is already "too low." This logic seems to have kept Japan locked into the low nominal interest rate steady state. Benhabib, et al., sometimes call this the "unintended" steady state..."

In short, there are two stable equilibriums, one good and one bad. As long as we're in areas where the vector fields lead us to the good equilibrium, using the interest rate as the policy instrument is fine. It's when we fall into areas where the vector fields lead to the bad equilibrium that we need to adopt other policy instruments to help get us out. And in the final analysis, since using the interest rate as the sole policy instrument always has the possibility of leading to the bad equilibrium, why not just shift to targeting the price level or the nominal GDP level?

The problem with Kocherlakota's analysis was that he failed to consider the dynamics involved.

Brad DeLong has gone on in length on Bullard's paper and makes some criticisms which take this model a little further but do not essentially change the fundamental conclusion:

http://delong.typepad.com/sdj/2010/08/extremely-rough-a-note-on-bullards-interpretation-in-his-seven-faces-of-the-peril-paper-of-benhabib-et-al.html

[Adam P]

Mark, the Tobin thing has nothing at all to do with what Bullard was talking about, you've misunderstood.

[Mark A. Sadowski]

Adam, I'm not familiar with the Tobin thing. Which paper is that?

However I think Bullard paper is relevant to this post and the Kocherlakota contoversy. In fact DeLong used it himself in discussing Steven Williamson's response to it.

[Jim Rootham]

To interject a math nitpick.

There are no continuous functions in economics. Money is not continuous.

[Adam P]

Mark, sorry I misunderstood you.

[Determinant]

On the domain of what? Sure money can be continuous, we just need to define the domain correctly.

We never have negative money, so X >= 0.

Just to be annoying, why are we surprised when economic functions exhibit the kind of degenerate behaviour we spend First Year Calculus examining? That's half the point of the course, to teach to you to be careful when composing and analyzing your functions. The other point is to get you read for second-year calculus where we examine Laplace Transformations, Differential Equations and multi-dimensional Calculus.

[DavidN]

‘There are no continuous functions in economics’ - I don’t think this is true. Someone can correct me if I’m wrong but I recall economists assume continuity of preferences in choice theory as one example.

[DavidN]

Also money is continuous for x>=0 it’s just that if you want change for example $15.234578432579803425079234 it’s rounded.

[RSJ]

The independent variable is interest rates. The dependent variable is the money supply.

The central bank cannot control the money supply -- it can only control the quantity of bank reserves.

By controlling bank reserves, it controls interest rates, and from that the money supply adjusts, based not only on interest rates, but the other factors that would encourage or discourage credit creation.

So this is a great example of understanding the importance of the inverse function theorem. Belief that control of bank reserves somehow enables you to control the money supply requires a kooky multiplier theory (e.g. M = Reserves/C, which is a 1-1 map). We can easily disprove such a relation by observing that for a given level of the money supply there are many different levels of bank reserves.

Therefore the inverse function theorem tells you that you cannot convert the dependent variable (money supply) into an independent variable under the central bank's control. Therefore the central bank is not "driving" the economy, or steering it. It is setting the short rate, and the money supply will be whatever the private sector demands at that short rate.

[himaginary]

"You can find Y from X, but each value of Y gives two possibles values for X, so you can't find X from Y."

Sometimes you can't even find Y from X, when hysteresis steps in.

Another analogy I'm recently obsessed is phase transition, where increase in X (added heat) doesn't cause change in Y (temperature) until increase in X suffices a certain value (latent heat). (I roughly sketched this analogy in my recent posts.) In economy, X can be thought of as base money, and Y as money stock.

I think these analogy cast some insight into ketchup theory which I brought up here.

[Old Ari]

You cannot run a gasoline engine backwards. Once you have made a grievous error of fact , the rest of your article is suspect.

[Nick Rowe]

Adam: on Tobin. That's a different paper than the one I was thinking of. There's presumably some sort of AS curve in there somewhere, implicitly or explicitly. And that the stability of his equilibrium (and hence the usefulness of his comparative statics results) would depend on the relative slopes of the various curves, as well as his assumptions about what adjusts to what, out of equilibrium.

Mark: yes, I think the Bullard paper is talking about the same sort of thing. At a very abstract level, draw a U-shaped function in {Y, X} space. For a given X there is only one Y, but for a given Y there are two possible X's. If Y is the nominal interest rate, and X is inflation, that's what Bullard is saying.

Jim: there are no coins smaller than one cent. So, in that sense, money may not be continuous, but that probably doesn't matter much at the sorts of scales we are talking about.

Determinant: "Just to be annoying, why are we surprised when economic functions exhibit the kind of degenerate behaviour we spend First Year Calculus examining?"

That's not "annoying". That's the sort of question I was trying to get my head around!

RSJ: "The independent variable is interest rates. The dependent variable is the money supply."

That's just proof by re-iterated assertion. Remember axiom's comment on my last post? Saying that the Fed can only control the price of gold by controlling the rate of interest? And yet Roosevelt nearly doubled the price of gold while doing next to nothing to short term safe interest rates.

himaginary: My mind keeps obsessing over the ketchup theory too. "Don't worry, QE2 can't cause too much inflation. It if does, we will simply reduce the money supply again." Thereby making the increased money supply expected to be temporary, and therefore making it much less stimulative.

Old Ari: Are you sure? Doesn't it depend on how the intake and exhaust valve timing is set? Perhaps I shouldn't rely on Farley Mowat as my authority (he wasn't that accurate on wolves, I hear), but I don't want to test the theory on my MX6, by trying to push start it backwards, in forward gear.

[RSJ]

NIck,

"And yet Roosevelt nearly doubled the price of gold while doing next to nothing to short term safe interest rates."

Sure, if the government buys something, its price will go up. But that does not mean the general price level goes up, just because the price of gold does.

At that time, this was forex devaluation, since international payments were settled in gold. The effect of a forex devaluation is to try to stimulate exports, and this may affect the price level, but price movements in gold (or currencies) do not correspond in a simple way to inflation. Particularly when other nations are also trying to devalue, or if other nations effect trade barriers that reduce the effect of the currency devaluation. In any case, this was forex pegging not monetary policy, which I suppose is why you said "Roosevelt" instead of "The Fed".

The argument that the CB controls interest rates instead of money is simple. When the bank buys a bond from households, households turn around and buy another bond from another sector. The "someone else" is typically government or the financial sector, but it could also be the foreign sector. In the last round of QE, the government was selling more bonds to households than households were selling to the CB. The quantity of deposits held by households decreased slightly.

So household deposits are at the level demanded at the given interest rate. What changes is the interest rate, but the change in interest rates as a result of quantity interventions is slight. Bonds prices are going to be driven more by arbitrage against the short rate than by quantity movements. Therefore when the CB announces a change to the short rate, that is a powerful influence on the price of bonds, which translates to a more muted change in the demand for deposits, which then causes the quantity of deposits to be equal to the (new) amount demanded.

But when the CB tries to buy bonds directly, without changing the short rate, then it is fighting against itself. The price of bonds will move only slightly, and the move in household deposits will be imperceptible.

No amount of handwaving or philosophizing can give the CB powers that it does not have.

[RSJ]

FYI, I made some diagrams to illustrate this point here

[Jim Rootham]

@Determinant Not having negative money is not relevant to money being continuous. Continuity is about having breaks in a set, not what the range is.

@Nick you are probably right that currency granularity is not critically significant. However, that's not the only argument.

Prices are not continuous with respect to time. To a significant degree. If the price of something is P at time t and P' at time t', with t'>t, it is NOT true that there is a price P" at time t" such that t < t" < t' and P > P" > P'. In fact, the failure of this relation on a large scale is characteristic of a bubble bursting (I think you could probably use that as a definition of a bubble).

[RSJ]

K,
If you can't borrow against future labor income or future social security income (it is illegal to do the latter), or equivalently, if you are charged different interest rates for borrowing against different types of future income, and if these different forms of income have different probability distributions, then you cannot just "add them up" when solving your intertemporal problem. Of course, you can add them up in some philosophical sense, but not in any sense that is relevant to your budget constraint or your consumption problem.

[RSJ]

Or a better way to think about this is that what you are really saying is that the value function is what counts.

The value function takes present wealth, future income, future taxes, future benefits, etc. and weighs them somehow in terms of consumption possibilities.

You seem to think that the value function is the identity map -- i.e. that wealth *is* the value function, so that if something gives value, it must be counted as wealth.

But in order to estimate the value function, we measure the inputs -- market wealth, expected future income, expected future benefits, etc. We understand that ultimately we are after the value function, but muddying the definitions of the inputs doesn't help, particularly if the effect of a change in income volatility on the value function is different than the effect of a change in the volatility of the interest rate. Here, it's good not to try to glob together all the inputs, particularly when you are trying to measure a specific input -- e.g. market wealth in the case of Frances' problem.