Wednesday, March 26, 2014

The sign of monetary policy, part II

(This blog post uses mathjax to show equations. You should see pretty equations, not ugly LaTex code.)

The ECB is in the news today. They want some inflation, yet the overnight rate is already zero. They're talking about negative interest rates, which leads to a great lunchroom discussion about bags of euros wandering around Europe.  All very interesting.

Yet it brings to mind a heretical thought I explored in an earlier blog post: What if we have the sign wrong on the effect of monetary policy? Could it be that to get more inflation, our central banks should raise rates not lower them? (Leave aside whether you think more inflation is good, which I don't. But suppose you want it, how do you get it?)

It's not as crazy as it sounds.

We know in the long run that higher inflation must come with higher nominal interest rate. Nominal rate = real rate plus expected inflation. Tradition says though that you temporarily steer the wrong way. First lower the nominal rate, then inflation picks up, then deftly raise the nominal rate to match inflation. If you instead raise rates and then just sit there waiting for inflation to catch up all sorts of unstable things happen.

But maybe not. Here is a simple and complete model of the "wrong" sign.

At the end of each period \(t-1\) the government issues \( B_{t-1} \) face value of bonds. In the morning of period \(t\), the government redeems the bonds for newly printed cash. At the end of period \(t\), the government soaks up the cash by selling new bonds \(B_t\) and with lump sum taxes net of transfers \(S_t\). The real interest rate is \(r\) and the price level at time t is \(P_t\). The real value of government debt is then the present value of future primary surpluses,

\[ \frac{B_{t-1}}{P_t} = E_t \sum_{j=0}^{\infty} \frac{1}{(1+r)^j} S_{t+j}. \]

(You can derive this from just watching the flow of money,

\[ B_{t-1} = P_t S_t + Q_t B_{t}; \ Q_t = E_t \frac{1}{1+r} \frac{P_t}{P_{t+1}} \]

where \(Q_t\) is the nominal bond price. Divide by \(P_t\) and iterate forward.)

Now, taking expected and unexpected values of the bond valuation equation

\[ \frac{B_{t-1}}{P_{t-1}} E_{t-1}\frac{P_{t-1}}{P_t} = E_{t-1} \sum_{j=0}^{\infty} \frac{1}{(1+r)^j} S_{t+j} (1) \]

\[ \frac{B_{t-1}}{P_{t-1}} [E_{t}-E_{t-1}] \frac{P_{t-1}}{P_t} = [E_t-E_{t-1}] \sum_{j=0}^{\infty} \frac{1}{(1+r)^j} S_{t+j} (2) \]

(1): By changing the nominal quantity of debt, with no change in fiscal policy \( {S_t}\), the government can freely pick expected inflation. This is like a share split. Doubling debt with no change in surpluses must raise the same revenue, so cut bond prices in half.  It also means the same surplus is divided among twice as many bonds next period, so causing the inflation.

(2): Once debt \(B_{t-1}\) is predetermined,  unexpected fiscal shocks translate one for one to unexpected inflation.

In practice, my little model government adopts an inflation target. This is an agreement between "Treasury" and "Fed," binding both. To the "Treasury," it's a commitment to equation (2): You won't give us any surplus surprises. You will raise as much surplus \( {S_t} \) as needed to validate the inflation target.

The "Fed" figures out what it thinks the real rate is, and announces a nominal rate, supplying as much debt as anyone wants at that rate -- but not touching fiscal policy \( {S_t} \).  By fixing the nominal rate, and thus fixing expected (inverse) inflation, (1) describes the amount of debt \( B_{t-1} \) that will be sold at this auction. (Equation 1 sounds a little warning, however. That might take a lot of debt! To change the price level 5%, the government has to issue 5% more debt, or about a trillion dollars.)

In this model, to raise (expected) inflation, the Fed and Treasury agree to a higher inflation target, and then the Fed raises rates.

This isn't that deep. Again, we've known about \(i_t = r_t + E_t \pi_{t+1} \) for a long time. But this fills in the determinacy and dynamics question. Yes, if the government just fixes \(i_t\), once \(r_t\) sorts itself out, then inflation must follow.

Ok, I left out stickiness, short runs, and so forth. But this seems (to me) like a pretty compelling simple long-run model of interest rate and inflation targeting, and it at least spells out a mechanism by which raising nominal rates and waiting for the inflation to happen will not be completely destabilizing.

Here is some history. I plotted the change from a year ago of inflation, together with the  3 month treasury rate. You should mentally shift the inflation rate to the right a year, as interest rates are associated with future, not past inflation, but I couldn't get Fred to do that. Once you do, you see pretty much my story. Higher interest rates lead to higher inflation. And the history since 1982 has been slowly lower interest rates leading to slowly lower inflation. Of course you can say that higher interest rates anticipate higher inflation. But there's precious little evidence for the opposite story, that higher interest rates lower inflation and vice versa.

Well, except 1980-1982. There are some short term dynamics, but if you're worried about decades of no inflation like Japan, maybe you shouldn't be thinking about vigorous short run dynamics.

More deeply,  we are, and will remain, in a brave new world, where the mechanism for short-run dynamics may have changed completely.  We are living the Friedman Rule -- $2.5 trillion or so of excess reserves, and interest rate = 0 mean that money and bonds are the same thing.

Here's a conventional reserve demand picture. We're out at the right edge. The conventional mechanism would have the Fed unwind $2.45 trillion of open market operations, until the reserve demand curve wants a larger interest rate, as illustrated by "really?"

Everything I hear out of the Fed says they won't do that.  We will stay satiated in liquidity, we will stay on the horizontal axis of the money demand curve, we won't go back to rationing reserves. Instead, they'll just raise the whole graph by paying more interest on reserves.

Living the Friedman optimal quantity of money is good. But who is to say any theory or experience based on the old mechanism will still apply to dynamics? 1980 was arguably a strong move on the left side of the graph, creating all sorts of monetary havoc. Raising the whole graph and leaving it there, with no rationing of liquidity whatsoever, is a completely different experiment.

As before, I view this just an intriguing possibility, not settled theory, and I'm using today's news to think out loud.

Some credit (without blame if you think this is all nuts):  Lars Svensson motivated this thought at a conference a while ago, while I was expounding on the fiscal theory. Lars pointedely asked why I thought inflation targeting countries had done so well. Well, I think this is the answer: The inflation target binds the Treasury as much as it does the the central bank. Then together they slowly lower rates to lower inflation, the slowly part to tiptoe over shortrun dynamics.


  1. John,

    You are ssssoooo close.

    "Could it be that to get more inflation, our central banks should raise rates not lower them?"

    No, to get higher inflation the fiscal authority should raise the risk free rate of return it offers on it's securities (bonds). In essence the fiscal authority offers an income (interest payments from tax revenue) for no productive effort.

    1. In my little world fed and treasury rates are the same, so this is how it works.

    2. Why would a primary dealer bank (a profit maximizing firm) borrow from the Fed (or another bank) at a rate and lend to Treasury at the same rate (0% spread)?

    3. It would borrow from you and lend to either Fed (reserves) or Treasuries, at the same rate.

    4. Why would I lend to a bank at a rate less than what I could get from Treasuries?

      Also, Fed reserves are lent by the Fed to banks, not the other way around - unless you are talking about interest on reserves.

      If that is what you are talking about, then your equation makes no sense at all. Federal government borrows at market rate and receives above market rate for interest payments on reserves that it deposits with the Fed.

      The whole point of interest on reserves is to dissuade banks from lending. If Fed offers above market rate for interest on reserves, then member banks will hold reserves rather than lending out the money.

      However, the primary dealer arrangement ensures that the federal government can always borrow. And so the Fed could try to offer an interest rate on reserves above Treasuries, however, the federal government would do one of two things:

      1. Find a new group of central bankers
      2. Deposit borrowed money with federal reserve and earn the interest

  2. interesting idea.

    the ceteris paribuses are worth considering though:

    what if the fed raised interest on reserves, but at the same time soaked up 99% of these reserves in exchange for 10-year bonds? what if private parties were able to swap this all back into instantaneous debt? what if a large foreign country decided to dollarize? etc.

  3. Shouldn't the flow of money equation read:

    B = Nominal Quantity of bonds
    Q = Nominal Price of bonds
    P = Price Level
    S = Lump Sum Taxes Net of Transfers

    B(t-1) * Q(t-1) = B(t) * Q(t) + P(t) * S(t)

    This makes the assumption that nominal tax revenue is improved by inflation (it may not be - it depends on tax structure).


    B(t) = [ B(t-1) * Q(t-1) - P(t) * S(t) ] / Q(t)

    1. These are one period bonds so the price of the B(t) bonds is one.

  4. "The real value of government debt is then the present value of future primary surpluses"

    That does not seem right. I have seen you assert the same proposition in the past. If the economy is growing at a rate faster than the risk free interest rate then the "primary surplus" can be negative but investors will still confident that they will be paid (because the total debt can be a declining share of GDP). The bonds will have value but your formula says they won't have value.

    1. If the economy will grow forever at a rate faster than the real interest rate on government bonds (not quite "risk free") then indeed, a claim on tax rate times gdp has an infinite present value. All our problems are over, borrow as much as we want, consume as much as we want, take the day off, cut taxes to epsilon, rising tax revenues will eventually pay it all off. I don't think the 'if" holds.

  5. Professor Cochrane - like you my background was in physics and math. You became an economist and I became a lawyer. One of the things they taught me in physics was to do "gedanken" experiments and to test theories (and formulas) against limiting cases. You assert a universal formula for the value of the public bonds. I posit a limiting case where your formula breaks down. When I did my physics degree, that would have suggested your formula is wrong.

    The other problem I see with your formula (again as a limiting case) is that it assumes that primary surpluses will be handed over to the bond holders. If there was a surplus that was greater than the amount due to the bond holders, why would the government over pay?

    As for your "counter example". There is no doubt that achieving a growth greater than the risk free rate "indefinitely" would require good policy and good conduct - but so what? Your formula is supposed to be true for all circumstances.

    My recollection is that you have written papers built on a foundation of that valuation formula and I think (with all respect) that the formula is wrong.

    1. Absalon, like you, I have difficulty wrapping my head around the Fiscal Theory of the Price Level. However, I think your questions are misguided.

      Your first point is as much of a problem for any asset pricing equation as it is for the equation Cochrane referenced. Is it a problem for asset pricing that shares of a non-dividend paying company are valued positively? Of course not. The value is simply determined by the future payoff. The same logic exists for government debt.

      Your second problem: you are correct that there would be no reason for the government to give any windfall to the bondholders, but the point of the FTPL is that future primary surpluses are a payoff to money holders. This either takes the form of reduced tax liability (in a monetary regime) or lower inflation (in a fiscal regime).

  6. John, do you view the VAR literature and similar on the price puzzle (including this by you as broadly supportive of the above? It certainly looks to be very powerfully in favour of your hypothesis

    1. Good point. Thanks for reminding me. And you made my day -- that anyone has even seen these notes is heartwarming.

    2. Thanks - and they were great. Sorry to bother, but are there/could you point me towards any convincing resolutions of the price puzzle? Commodity prices don't seem to cut it.

      More generally, the whole back-and-forth across the blogosphere (Stephen Williamson is also very involved) in the sign of monetary policy seems to ultimately boil down to empirics. Ultimately you can build an at least moderately convincing DSGE model for either side quite easily. Given that the price puzzle seems to come out of most plausible identification strategies even by NK authors (including Christiano, Eichenbaum and Evans IIRC), it seems to be quite a good cudgel to beat the standard NK model with.

  7. I may have misunderstood, but I didn't think the ECB wanted to raise inflation just for its own sake; I thought the concern was that a prolonged period of low inflation makes deleveraging difficult. Wouldn't raising nominal rates and waiting (hoping) for inflation to follow actually hinder deleveraging?

    1. Nick,

      Deleveraging is a choice of the borrower (fiscal authority / private borrower), not the lender (central bank). Deleveraging is easy as long as the fiscal authority is on board with deleveraging.

    2. OK, but why do you think they want to raise inflation, and does (temporarily) raising real interest rates actually help achieve that end objective?

    3. Nick,

      Because they (the ECB) are not in control of fiscal policy and the ECB is a bank who like any bank wants to make loans (increased leverage).

      The ECB would rather have higher inflation and more loans than zero / negative inflation and fewer / no loans. The ECB (like any other enterprise) has it's own interests at heart.

  8. This is all straight out of cloud cukoo land, and for the following reasons.

    1. Interest rate adjustments have little effect on investment spending. Here are two recent studies, but I could quote more.

    2. Interest rate adjustments have no effect on credit card rates. See:

    3. Even if interest rate cuts do increase borrowing and investment, there is no more sense in imparting stimulus just via investment than there would be in imparting it just via car sales and restaurant meals.

    4. As for negative rates, they can lead to negative output, i.e. wealth destroying rather than wealth producing types of economic activity. Here’s a simple example. If the rate was minus 4%, it would pay me to buy 100 houses, knock down two of them, and a year later sell the remaining 98. Assuming no other costs, and assuming the price of the houses in real terms remained the same, I’d make 2% on the “transaction”.

    1. Your links are great, and I read them as really bad news for the conventional view that raising Fed rates raises other interest rates which lowers investment demand which feeds through a Phillips curve and finally lowers inflation. I would think that means "we agree" but you cloud cuckoo comment suggests not.

  9. What if the CB simply ignored interest rates and engaged in OMOs? Do they really have to care about when interest rates are "expansionary" or "contractionary"? They can keep printing money and buying things until they buy all the assets in the world, but surely inflation will increase before then. And yes, this is the result of reading Nick Rowe, so I never understood why interest rates are supposed to be so important!

  10. John: "In this model, to raise (expected) inflation, the Fed and Treasury agree to a higher inflation target, and then the Fed raises rates."

    But that is not controversial. If you raise the inflation target, and if doing so raises expected inflation, the Fed would need to raise the nominal rate as a consequence.

    That is quite different from saying that if actual and expected inflation are below target, the Fed should raise the nominal rate to help it hit the target.

    1. Nick,

      Why would the central bank "need" to raise the nominal rate as a consequence?

    2. To keep the real interest rate constant.

    3. Why do you believe that a central bank could simultaneously hit an inflation target and a real interest rate target?

  11. "But that is not controversial. If you raise the inflation target, and if doing so raises expected inflation, the Fed would need to raise the nominal rate as a consequence.

    That is quite different from saying that if actual and expected inflation are below target, the Fed should raise the nominal rate to help it hit the target."

    Exactly! A good attempt at trying to be contrarian, Dr. Cochrane, but one that ultimately fails.

  12. If the Fed has a single mandate, that of price stability, then the current regime (IOER, QE and low interest rates) makes sense, according to Cochrane.
    In fact, in terms of price stability this is the most successful Fed in the postwar era.
    So, should the Fed stay the course, maintain the regime with steady QE at, say $25 billion a month? Given the Cochrane model, what happens in the long run?

  13. Add on:

    If the current regime (IOER, QE and low interest rates) does result in low inflation, and the Fed continues with QE set a lower steady state--say $25 billion a month--can the Fed eliminate from taxpayer federal taxpayer liabilities about $3.6 trillion in next 10 years? In addition to the roughly $2 trillion in Treasuries it has already monetized?

    This would require the Fed to maintain a portfolio of about, say, $5-6 trillion in Treasuries, and to freshen it, buy buying new Treasuries as older ones matured.

    If this works---and keep inflation low---is it the solution to Dr. Cochrane's worries about federal debt burdens? Why or why not?

    If at some point is does lead to a surge in spending, and then inflation, when would that be? And why does that not happen now?

  14. I think the article ignores asset price inflation. We may have had a long period of ever declining inflation corresponding with ever declining but we had a big inflation in nearly all kind of assets.

    There is no visible effect of QE on the CPI, only QE1 showed a limited effect on long term interest rates but a huge effect on stock market prices.

    An increase of interest rates by the ECB would strenghten the Euro further, invrease borrowing costs for governments.

    What is needed is a better understanding how asset price inflation can be transformed into consumer price and wage inflation.


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