# 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.