It's of course well-known that a finite horizon presents problems in monetary economies. If the economy has a known final date, then no one wants to hold money (an intrinsically worthless object) at the final date. So, working backward, money will be valueless at all dates. Not very useful if we want to do monetary economics. There are fixes, though. The standard one, used extensively of course, is to just assume an infinite horizon. This solves other problems too - even in non-monetary models an infinite horizon can make life easy for the modeler. Another fix is to suppose that the world ends with some fixed probability on the next date. That works much like infinite horizon, and is tractable. As well, one could assume a finite horizon in a monetary model, and suppose that the government does something on the final date to give money value. For example, money is required to pay your taxes, or the government in some other fashion retires the money stock in exchange for goods. That approach is not so tractable, and is not used much (off the top of my head, I can't think of a reference, but maybe you know one).

In Narayana's finite horizon economy, there is a further set of issues which come up, as this is not a monetary economy in the usual sense. It's a cashless Woodford NK economy. In the infinite-horizon version, prices are all quoted in terms of a numeraire, called money. In this world, anyone can trade one unit of numeraire for

*1+i(t)*units of the money in the next period, where

*i(t)*is the nominal interest rate, and the central bank has the power to set

*i(t)*each period. No money balances are held in this world, and the central bank avoids plenty of messy stuff that it has to do in the real world - open market operations, etc. Mike Woodford took great pains in his book to convince us that this setup makes sense.

Narayana then faces a couple of problems in working with a finite-horizon version of the standard NK model. The first is that the New Keynesian Phillips curve in the infinite horizon reduced form is derived from Calvo pricing in a stationary environment. But in Narayana's setup, when firms make pricing decisions, they are going to do it differently depending on how close they are to the terminal date. That problem is never addressed in Narayana's note. The second problem he does address, which is that we need to determine the inflation rate at the terminal date. Narayana assumes that the final-period inflation rate is fixed by fiscal policy. You might think that sounds OK, if you're accustomed to thinking about standard monetary models - in that context, fiscal policy can indeed solve the problem. But here, it's hard to see what it is that the fiscal authority does to fix the inflation rate in the final period, as the model is cashless. Seemingly, this works just like monetary policy does - if the central bank can dictate the nominal interest rate, I guess the fiscal authority can dictate the inflation rate. Of course, you might then ask why the fiscal authority can't just dictate a whole sequence of inflation rates for all time, and be done with it.

Narayana takes this model and considers a policy experiment. The central bank fixes the nominal interest rate for all time. What happens? With an infinite horizon, this is straightforward. Given Narayana's parameter restrictions there are many equilibria, but all equilibrium paths converge in the limit to a steady state in which the real interest rate is equal to the "natural rate," and the inflation rate is equal to the nominal interest rate minus the natural rate - in the long run there is a one-for-one Fisher effect. Narayana calls this the "neo-Fisherian hypothesis," but it's not a hypothesis, it's just a property of a wide class of models that macroeconomists work with. Neo-Fisherism is actually more accurately stated as: "Excuse me people, I think you might have the sign wrong." I expand on that in this post. Indeed, a serious neo-Fisherian is going to worry not only about Fisher effects, but about short-run liquidity effects, persistent movements in the real interest rate, and the long-run effects of monetary policy on the real interest rate.

Now, consider the finite horizon. If the fiscal authority happened to choose the Fisherian steady state inflation rate in the final date, then the equilibrium would be that steady state, for all time. However, since the steady state is stable solving forward, it's unstable solving backward, so given a final-period inflation rate away from the steady state, the inflation rate is further from the steady state the further in the future the final date is. In equation (5) of Narayana's note, a higher nominal interest rate then implies that inflation is lower at every date. Equation (5) also tells us that the fixed final-period inflation rate - that is fixed ("anchored" as Narayana says) inflation expectations in the last period - has a larger and larger effect the further we are from the final period. That is, anticipated inflation very far in the future has an enormous effect on inflation today. If you think that raises questions about how seriously we want to take this, then I agree with you.

Another experiment which Narayana does not look at, is to ask what happens if the central bank follows a standard linear Taylor rule. Suppose, in typical fashion, that (using Narayana's notation)Notation is defined as follows: I've left the output gap term out of the Taylor rule, for convenience.

In the infinite horizon case, if we assume the Taylor principle (alpha > 1), then there are two steady states - the zero lower bound (ZLB) and the desired steady state in which the central bank achieves its inflation target. There are many equilibria converging to the ZLB. Both steady states are Fisherian, in the sense that the real interest rate in the steady state is the natural rate, and the inflation rate is equal to the nominal interest rate minus the natural rate. With 0 < alpha < 1, the desired steady state is the unique steady state, and there are many equilibria, all of which converge in the limit to the steady state.

Now, consider the finite horizon. As with Narayana's analysis, suppose the fiscal authority fixes the inflation rate at the final date. Then, under the Taylor principle, if we solve backward from the terminal date, we get closer to the desired inflation rate. The fiscal authority, presumably not in tune with the central bank, can set a terminal inflation rate that differs from the central bank's target inflation rate, but the further in the future is the terminal date, the less that matters for the first-period inflation rate. Indeed, as the time horizon increases, the initial inflation rate gets closer and closer to the target inflation rate. Thus, for a long horizon, this economy is Fisherian to an extreme at the first date. Indeed, under the Taylor principle, and with a long horizon, if the central bank increases the target inflation rate, then it increases the nominal interest rate roughly one-for-one at the initial date, and the actual inflation rate increases at the first date, roughly one-for-one. So, it would be incorrect to say that this finite-horizon economy does not have Fisherian properties.

If the central bank does not adhere to the Taylor principle in the finite horizon economy, then things can go badly. If the horizon is long and the fiscal authority chooses a terminal inflation rate below the central bank's target rate, then the nominal interest rate will stay at the zero lower bound for some time before it increases until the terminal date - inflation is always below the target.

So, Narayana's note didn't give me cause to worry about infinite horizon monetary models, or about neo-Fisherism. His finite horizon framework has some problems, but in spite of those problems it actually has some Fisherian properties.

"But here, it's hard to see what it is that the fiscal authority does to fix the inflation rate in the final period, as the model is cashless."

ReplyDeleteCorrect me if I'm wrong, but Woodford's cashless system still has a medium of account, some sort of asset or security that is used to represent the economy's pricing unit. This isn't a very liquid asset, not enough to be called a medium of exchange or cash, but it's still something. Why can't the fiscal authority just buy that security back and retire it?

There is something - we could call it money - and the supply of that stuff is zero in a Woodford cashless model. I may be missing something, but when I go to the store to buy a toothbrush, and there are none there, I can't buy a toothbrush.

DeleteDear SW, one possible fix to work in a well specified monetary environment and solve backward induction would be to use an OLG structure and set up a stationary equilibrium . This could turn helpful to teach monetary environment, esp CIA, to bachelor students for instance (!)

ReplyDeleteBest,

No, OLG is infinite horizon. The agents in the model are finite lived, but the economy runs forever.

Delete