By popular request (well, David Andolfatto asked, but I think he's right to ask) I'm drawing a picture to illustrate my previous post on the distinction between the interest rate Rb you get paid for *lending* money and the interest rate Rm you get paid for *holding* money.

But first I need to take a short detour back to Micro 1000, where we show the effect of taxes on a supply and demand diagram.

A tax of $t per apple bought-and-sold creates a wedge between the price the buyer pays Pb and the price the seller gets Ps. Pb-Ps=t. Quantity demanded depends on Pb, and quantity supplied depends on Ps. There are three ways we can draw the supply and demand diagram. We can put Pb on the vertical axis and shift the supply curve vertically up by t. We can put Ps on the vertical axis and shift the demand curve vertically down by t. We can put both Pb and Ps on the vertical axis, and stick a vertical wedge of height t between the supply and demand curves to the left of the point where they cross. And we get the same answer either way.

OK, back to macro 2000. I now have two interest rates, Rb and Rm, and the spread between them Rb-Rm. Plus I have the added complication of the distinction between nominal and real interest rates, with the difference between them being the (expected) inflation rate.

This,* I think*, is the simplest way to draw the ISLM diagram.

I will put the *real* interest rate for *lending* money on the vertical axis. So Rb denotes the real interest rate on "bonds". If we make the standard assumption that desired saving and investment depend on the real interest rate on "bonds", that means the IS curve does not shift around when Rm or expected inflation changes. So the IS curve is simple textbook.

**The opportunity cost of holding money is the spread between the interest paid on lending money (i.e paid on holding "bonds") and the interest paid on holding money.** Since we are talking about the difference between two interest rates, it does not matter if both are nominal or both are real (but we can't have one of each, obviously). So let Rm be the

**real**interest rate on holding money, so Rb-Rm is the spread between them.

The equation for the LM curve is: M/P = L(Y,Rb-Rm), where L_{Y} > 0 and L_{R} < 0. M is the stock of money, and P is the price level. This is just like the textbook, except the textbook assumes Rm=0%.

Draw the LM curve showing combinations of {Y,Rb} such that money supply = money demand, holding constant M, P, and Rm. It slopes up, just like the simple textbook one does.

Let the nominal rate of interest paid on holding money be Im, and the (expected) rate of inflation be p (lower case). So Rm=Im-p.

If the bank that issues the money holds Im constant, an increase in expected inflation from 0% to 2% reduces Rm by 2%, which shifts the LM curve vertically down by 2%.

If the bank that issues the money cuts Im from 0% to minus 2%, holding expected inflation constant, that also shifts the LM curve vertically down by the same 2%.

Both (expected) inflation and negative interest rates on holding money have exactly the same consequences for the LM curve, because both work by reducing Rm by the same amount, and so shift the Aggregate Demand curve (not shown) rightwards by the same amount from Y^ to Y*. And both reduce Rb by the same amount (which is less than 2%).

Alternatively, if the bank that issues the money always adjusts Rm to maintain a constant spread between Rb and Rm, the LM curve will be vertical (for a given M and P). Given M, P, and the Rb-Rm spread, there is only one Y at which money demand equals money supply. That means that shifts in the IS curve have no effect on Aggregate Demand.

If the bank increases the money supply M, holding Rm constant, the LM curve shifts right, just like in the textbook. This means the bank (if it is the central bank) has two monetary policy instruments, not just one like in the simple textbook model. (Plus, in a more complete model where the price level is endogenous, the bank can change expected inflation by making announcements about how it will change M or Rm in future.)

Nick,

OK, I think I follow now. I'll have to think about the assumption that the IS curve is unaffected by the interest rate spread. In my OLG model, the object I would identify as an IS curve does depend on the spread. In my model, capital spending is constrained by the need to hold cash reserves. Lowering Rm lowers the opportunity cost of holding reserves and stimulates capital spending.

But if IS curve is invariant to the spread as you assume, then OK. I suppose one could use this as a rationale for negative nominal interest rates (central bank deposit rates)?

Posted by: David Andolfatto | February 25, 2016 at 01:38 PM

David: the standard textbook assumption is that desired saving and investment depend only on (real) Rb, and not on expected inflation, and hence implicitly not on (real) Rm. But you could make a transactions costs argument that they depend on the spread too.

Yes, this is the simplest rationale for negative rates on CB deposits. (But this little model assumes there is only one type of money.)

Posted by: Nick Rowe | February 25, 2016 at 02:06 PM

Has this any relation to the proposal of Olivier Blanchard of two interest rate to de IS-LM model? I'd rather prefer a private versus public interest rate, because they tend to diverge significantly in circumstances as Today. The spread between the two have risen in recent times, and is a good indicator of coming recessions, as you can see grafting the 10 years public private Baa bonds spreads. In fact, the spread has nor return to normal historical level since de GR. Perhaps taking Rb as Baa bond and Rm as Treasury bond?

Posted by: Miguel Navascues | February 25, 2016 at 02:29 PM

For example, https://research.stlouisfed.org/fred2/graph/fredgraph.png?hires=1&g=3A1O

Posted by: Miguel Navascues | February 25, 2016 at 02:33 PM

Miguel: related, but different. There are many different interest rates, and aggregate saving and investment will be a function of some sort of weighted average of those interest rates. So what I call Rb is a composite. Changes in spreads between that Rb and the interest rate on safe bonds will also change the wedge.

Posted by: Nick Rowe | February 25, 2016 at 02:52 PM

Perhaps this is not the right place for this but I want to lay it out. Feel free to move if want to somewhere else. Thanks.

Negative rates are alluring because they get rid of the ZLB and keep everything else constant in a conventional macro account. It is of course more than of simply passing interest that many of the advocates and practitioners of NIRP either have a heterodox view about the anchor for the future price level or want to leave it up to someone else to define how this anchor should be arrived at.

Let's start with the second case. Switzerland strikes me as pretty axiomatic. The SNB has an ambiguous inflation target that functions in practice by keeping their interest rates below that of others--i.e. a fixed interest rate peg with the important practical qualifier that it is the spread to some other monetary zone that is functionally targeted. Notice right away how this modifies the normal sense in which interest rate pegs are thought to be unstable. It is not the price level/monetary base pair that is ambiguous but rater the relative deflation/nominal effective exchange rate pairs required to keep the real effective exchange rate stationary over time. The SNB would prefer a slightly positive domestic inflation rate coexisting with a modest increase in its nominal exchange rate over the long run. The problem is obvious. Its pretty easy to get knocked into outright deflation, since you don't control the anchor, if spillovers from abroad start to hurt instead of help. I think the above fairly sums up how the SNB thinks about monetary policy, but don't take my word for it, take Governor Jordan's word.

https://www.snb.ch/en/mmr/speeches/id/ref_20150828_tjn/source/ref_20150828_tjn.en.pdf

The critical point with regards to negative interest rates is as follows. Advocates of NIRP say it is a means of returning to higher interest rates in the future. The market seems to disagree with this line of reasoning, and I think it is important to try and understand why. (The following analysis will not work for all cases--like for example countries where negative rates are explicitly presented as a means of returning to the inflation target from below and therefore should lead to higher rates in the future, ceteris paribus. These cases are obviously important, and in my mind, it is even more of a challenge to NIRP theory that these countries as well have seen dramatic falls in their long term bond yields post the introduction of NIRP)

I think one potential explanation for why NIRP has so far precipitated a fall in long-term bond yields is that NIRP actually does convey new information in an unstable way about where the long-dated nominal anchor for the economy is likely to lie. Of course it does not help that advocates of NIRP explicitly advocate that the inflation target should be changed to zero--at a time when the market already doubts the viability of the current target no less!! But let's work through the reasoning. The U.S. and the Eurozone and Japan have at least stated inflation targets of basically 2%. Switzerland has an inflation target of strictly less than 2%. I randomly chose the CHFUSD exchange rate over the last 40years and calculated that the the CHF appreciated against the Dollar by about 2.3% per year. As one would expect over such a long period--for two almost equally rich countries across the whole sample--Swiss inflation was 2% per year lower than U.S. inflation on average. So far, so good.

Now Lets iterate forward the NIRP solution. Assume that the U.S. adopts NIRP and moves to a zero percent inflation target as some suggest (by design this will not have a permanent impact as the vector of real rates is assumed unchanged--a very tenuous assumption give the prevailing nominal structure in the economy but lets go with it). With 10yr inflation expectations currently at 1.4%, we might assume that nominal bond yields in the U.S. would fall by 140bps all else equal. In order to offset immediate appreciation pressure against the Franc, lets assume that the SNB cuts by the same amount--140bps in the target rate. Now my contention is that it is perfectly rational in our simple setup to assume that term rates in Switzerland would fall by an equal amount--this is in fact guaranteed by the adjustments needed to tie down the REER path if the nominal exchange rate is not allowed to adjust on Day 1 (Remember that the indeterminacy in the Swiss example lies along the path of the NEER and the inflation differential. This reaction function would also be consistent with the SNB modus operandi as laid out by Jordan--cut relative to the leader in order to keep the CHF forward rate below the CHF spot rate--so that the spot rate doesn't converge painfully to the forward rate in one fell swoop).

Swiss 10yr rates today are -45 bps, so we would expect them to go to -185bps. Because we have completely de-nominalized the economy by assumption, everyone will like this outcome just as much as the prior equlibirum. Yes the tax on cash will go up, but so will the value of its long term purchasing power in expectation (the tax on cash has to go up to make the deeply negative policy rate functional). This can be seen either as the domestic purchasing power increasing due to deflation, or perhaps more clearly that CHF will buy more dollars worth of stuff every year by precisely the same amount that CHF deposit accounts are taxed (an ad hoc result of assuming the U.S. inflation rate is precisely zero).

Finally, I want to highlight a different aspect of this example. Swiss 30yr Bonds currently yield 15bps. Lets assume that they fall to 0bps. Then a permanent increase in the monetary base executed by buying 30yr bonds would not be distortionary (assuming equal utility of both assets, etc...) even if the central bank is assumed to always (30yrs more accurately) treat reserves like cash. This is because sellers would willingly trade reserves for bonds for cash and vice versa even if they expected those reserves always to be compensated at zero like cash. Contrast this with the normal case where such an asset swap would only be willingly entered into if it was thought to be temporary. Notice that in the conventional account, if this asset swap is NOT temporary then the monetary authority loses control of the future interest rate through permanent super-abundance of reserves. Of course it might be unwillingly entered into because of the regulatory requirement to hold an arbitrary quantity of reserves even if those reserves are strictly dominated in return by equally useful instruments.

Advocates of NIRP think that this example represents an oddity or a peculiarity: namely, the specific monetary policy where the returns to cash and bonds are equal both in nominal and real terms even without a transaction value for money. In my opinion, I cant stress enough how much of a defeat this is. It is not an oddity at all. It represents the success of turning the medium of exchange and account into an equally dominant store of value. And it does so with no tricks or oddities whatsoever. In fact, I think it is almost certain that before this experiment is over, very long-dated bonds WILL TRADE below 0%. And the deeper the NIRP the greater the likelihood.

addendum: This whole comment assumes that the CB can solve the cash storage incentive at deeply negative rates. Thus we impose no lower limit on the short-term policy rate by virtue of the short-term 0% yield to cash. Instead, cash and government bonds are substitutes as stores of value because the government runs a zero interest rate policy on average forever by choice. We have chosen Switzerland to show that this situation does not have to arrive from the institutional characteristic of cash whatsoever, and might arise as part of normal practice quite soon.

Also, this has little to do with the optimal money growth rule because we still leave unanswered the question of what actually does tie down the long-term price level in A domestic economy. Our example only ties down the relative price level through open macro, and assumes away the policy suite which ties down the U.S. price level in the far future. But htis is precisely the point, the expectations of the future price level SHOULD be tied down by money being strictly dominated in return by other assets. To ties it down by going in reverse is a terrible mistake. that will be a follow up comment using Japan as the problem situation.

Posted by: T. | February 25, 2016 at 05:10 PM

I understand. But I suspect that the IS-LM prove that real investment and saving are equal in spite of some level of unemployment. So, the monetary problem -that money policy cannot by itself restablish full employment- is not completely incorporated.

Posted by: Miguel Navascues | February 26, 2016 at 03:31 AM

Nick,

Effectively a move to a negative Rm on money that you hold increases the net interest that you pay (Rb - Rm) on an existing loan - yes? You could achieve the same effect if all borrowers paid a floating interest rate on long term loans.

I am not sure Aggregate Demand increases as the spread between Rb and Rm increases. Taking on a floating rate long term loan entails more risk for the borrower. And so, how would risk appetite affect the demand for credit?

Posted by: Frank Restly | February 26, 2016 at 08:16 AM

Frank: M/P=L(Y,Rb-Rm) means the LM curve is vertical if Rb-Rm is held constant.

As I told you on the last post, you lack the prerequisite to talk about this stuff. So stop. Go and read an intermediate textbook.

Posted by: Nick Rowe | February 26, 2016 at 10:17 AM

Nick, I tried to find the logical of your new very fascinating IS-LM. Nevertheless, I thing that Rb is no so easy to move as in your example in some circumstances. Particularly if the estimation of risk change. Only at constant risk premia the monetary policy is effective. http://www.miguelnavascues.com/2016/02/una-nueva-is-lm.html.

Posted by: Miguel Navascues | February 27, 2016 at 03:05 PM

I use a wedge too, based off an article I read in JEE several years ago as well as some of your posts. I draw LM using nominal interest rates on safe assets (such as US government bonds) and IS versus real interest rates on riskier assets (such as corporate bonds and mortgage rates). The wedge then combines a risk premium as well as expected inflation.

I like adding in the idea of Rb versus Rm too.

Posted by: primedprimate | March 05, 2016 at 08:12 PM

primed: "The wedge then combines a risk premium as well as expected inflation."

I like doing it that way too. But for some reason (it's too early in the morning for me to remember why) I convinced myself the wedge method wouldn't work as well for Rb versus Rm. Ah! (sips tea) I think it's because Md (and hence the LM curve) is a function of that very wedge, so we can't hold that wedge constant when we draw the standard LM curve. We have to hold Rm constant instead.

Posted by: Nick Rowe | March 06, 2016 at 06:08 AM