Believe those who are seeking the truth. Doubt those who find it. Andre Gide

## Wednesday, March 31, 2021

### A Natural Rate of Interest

This post was motivated by a conversation with Eric Lonergan. It began with a simple question: what should be the interest rate paid on reserves? I answered that according to theories I'm familiar with, reserves should earn the "natural" rate of interest, which I defined as the sum of population and productivity growth. So, assuming 2% "real" growth and 2% inflation, reserves (and government debt more generally) should be yielding around 4%.

I think it's fair to say most people did not find my answer very satisfying. So I thought I'd take a moment to explain how I arrived at it. I want to do so in the context of a model economy. Let me describe the model first. We can discuss its limitations and possible extensions later on.

Consider an economy where people live for two periods; they are "young" and then become "old." Let N(t) denote the population of young at date t. Assume that the population grows at (gross) rate n; that is, N(t) = nN(t-1). In this "overlapping generations" model, the population at date t is given by N(t)+N(t-1).

Individuals in this economy generate y units of perishable output (goods and services) when they are young. I'm going to treat y as fixed over time. This implies that the RGDP at date t is given by N(t)y and that the RGDP grows at rate n over time (there is no productivity growth). In what follows, I label n the "natural" rate of interest.

Suppose that people only value consumption when they are old. This poses an interesting economic problem. The young can produce goods that the old value, but the old have no way of paying for these goods. Private credit markets don't work here.

The cooperative solution is very simple: the young should "gift" their goods y to the old. If everyone followed this cooperative protocol, then the young of generation t would consume (in their old age) c(t+1) = N(t+1)y/N(t) = ny.

That is, by following this protocol, it's as if the young "deposit" their income y in a savings account that generates a (gross) real yield equal to n, the "natural" rate of interest.

Since private competitive markets cannot be expected to implement this socially-desirable outcome, what other mechanisms might be employed? In small communities, reciprocal gift exchange seems to work quite well. In the present context, the young look after their parents, expecting their children to return the favor, and so on.

Larger communities need to rely on other mechanisms. In the present context, a PAYGO social security system that taxes the young y and pays the old ny would do the trick.

The same outcome could be achieved through monetary exchange. Suppose the government lets all individuals open a central bank money account. The government creates (out of thin air) M dollars and credit the accounts of the "initial old" with M/N(0) dollars. Assume that M is kept constant over time. The old are expected to spend these dollars on a competitive spot market, where goods exchange for dollars at price p(t).

The equilibrium price-level is easy to derive in this example. At any date t, we have N(t-1) old people collectively holding M dollars. These M dollars will be spent (the old have no reason not to) on the goods available for sale, N(t)y. The market-clearing condition here is M = p(t)N(t)y at every date t. Because both M and y are constant, and because population N(t) is growing at rate n, it follows that the equilibrium price-level p(t) must be falling at rate n.

So, if we interpret M as "reserves" in this model economy, then reserves yielding the natural rate of interest would be consistent with economic efficiency. If reserves yield zero nominal interest rate, then efficiency requires some deflation. But the same outcome is possible if reserves were to yield a nominal interest rate n in a zero-inflation rate regime.

This result continues to hold for more general preferences. Suppose that people care about consumption when young and old. Then the young will only want to consume a fraction of their income. That fraction will depend on (among other things) the real rate of return they expect on their retirement savings. As it turns out, the so-called "Golden Rule" allocation requires that money/bonds yield the natural rate of interest.

Is this a good place to start thinking about Eric's question? It may be a good place to start, but we don't want to stop here. The analytical framework above is "bare bones." Among other things, my analysis implicitly assumes that there is no difference between reserves and government treasury securities. Does this matter? If it does, the reasons need to be spelled out. What modifications to the simple model above would imply that to meet a given social objective, it is desirable to have reserves yield less than treasury securities? In reality there is duration risk. But why does the Treasury issue bonds with different maturities in the first place? Moreover, why are these bonds purposefully rendered illiquid (for example, by discouraging the Fed from monetizing the entire bond issue or, at least, from setting up standing purchase facilities?). The answers to these questions are not immediately obvious to me. But they may be to you!

## Thursday, March 4, 2021

### Is it time for some unpleasant monetarist arithmetic?

The title of this post alludes to a paper written by Tom Sargent and Neil Wallace 40 years ago "Some Unpleasant Monetarist Arithmetic." The startling conclusion of this paper is that a central bank (limited to interest rate policy and/or open market operations) does not have unilateral control over the long-run rate of inflation. The result is made all the more powerful by the fact that it relies mostly on arithmetic and only minimally on theory.

So, what's the basic idea? First, begin with the fact that monetary and fiscal policy are inextricably linked via a consolidated government budget constraint. This implies that monetary policy will have fiscal consequences. In particular, interest-rate policy affects the interest expense associated with rolling over any given amount of government debt. The question is how the fiscal authority intends to finance interest expense. There are two basic ways it can to do this: (1) use primary surplus (increase taxes and/or cut spending); (2) issue debt. The first option is associated with what economists call a Ricardian fiscal policy; the second option is associated with a Non-Ricardian fiscal policy.

A central bank has no control over (1) or (2); these are determined by the tax and spend decisions made by the fiscal authority. Suppose the fiscal authority chooses (2). If so, then what is accomplished by increasing the policy rate? Ceteris paribus, it increases the rate at which nominal debt is issued. This is arithmetic. If nominal debt is money (and it is), then this must lead to higher inflation, not lower inflation.

The corollary here is that a central bank has no unilateral control over the long-run rate of inflation. A central bank may be permitted to choose a long-run inflation target, but only with the blessing of the fiscal authority. Canadians know this: the Bank of Canada and the Government of Canada meet every five years to review their joint inflation-control agreement. In other jurisdictions, the central bank simply assumes that fiscal policy will be conducted in a "responsible" manner (i.e., a manner that will not un-anchor long-run inflation expectations).

If the primary deficit is managed in a manner to anchor long-run inflation, then the central bank is left free to use interest rate policy for the purpose of stabilizing shocks to aggregate demand. To stabilize the inflation rate around target requires, in this context, that the central bank raises its policy rate aggressively against above-target inflation. (Hopefully, even the threat of such a response keeps inflation close to target. This is the so-called Taylor principle.)

The question monetary policymakers might want to mull over these days is whether this standard policy prescription is appropriate in an environment where the traditional fiscal support for inflation-targeting seems to be waning? I want to be clear here: I am not saying this is happening today or that it will happen in the future. I'm saying that it might happen and that if it does, monetary policy makers should have a contingency plan in place. What should this contingency plan look like?

So, to take a concrete example, suppose that the tax and spend decisions coming from Congress imply an elevated primary deficit for the foreseeable future. Perhaps there's been a "regime change" in thinking that transcends political parties so, no matter who controls Congress, the expectation is for elevated primary deficits for as long as we can see. (I am not suggesting this is good or bad, I'm just saying suppose.)

Next, suppose the economy is humming along at or near what anyone would call "full employment." And then suppose inflation rises to 3, 4, 5% or higher and stays there with no sign of ever wanting to return to the Fed's official 2% long-run inflation target. What should the FOMC do in this hypothetical scenario? (Again, consider this as the type of thought-experiment that is necessary to form a contingency plan -- I do not mean to suggest that this scenario is likely, only that it is possible--and not in a Dumb and Dumber way).

The monetary policy advice coming from a model like Sargent and Wallace (1981) might suggest something like this: For as long as Congress remains in a regime of high primary deficits

[1] Keep policy rate low, or even lower it, if possible; and
[2] Announce a temporarily higher inflation target (consistent with the new fiscal regime).

Recommendation [1] comes from Unpleasant Monetarist Arithmetic. Increasing the interest rate in this fiscal regime will only lead to higher inflation. Lowering the interest rate has the opposite effect. If recommendation [2] is not adopted, the monetary authority would have to explain (after every meeting) why it is missing its 2% inflation target. They might, of course, just say it's "temporary," but this would wear thin after a few years.

This advice is based on the assumption that everyone knows there's been a change in fiscal regime and that it will be persistent. What if no one is really sure of regime change or, if regime change, how long it might last?

The prudent thing to do in this more realistic case is to hedge your bets. In terms of [1], one might recommend raising the policy rate, but not by as much as would normally be done given the observed inflationary pressure. In terms of [2], one could probably get away with maintaining the long-run inflation target at 2% and legitimately explaining away deviations from target as "transitory."

Probably the last thing monetary policy should do under these circumstances is to raise the policy rate aggressively against inflationary pressure (as recommended by the Taylor principle). The Taylor rule works fine under a Ricardian fiscal policy. But it may backfire un a Non-Ricardian fiscal policy--this was the whole point of Sargent and Wallace (1981).

Tightening monetary policy might have the effect of bringing inflation down temporarily (this is consistent with the Sargent and Wallace model). But in reality (and in some models, like here and here), this would come at the cost of economic recession.

I can see no rationale for creating a recession to bring inflation down temporarily. But there may be a political-economy rationale for the threat of such a policy. That is, a Congress that does not trust future Congresses may want to create an independent (but accountable) central bank to pursue a low-inflation mandate and to do whatever it can with interest rate policy to achieve that mandate, even at the cost of recession. Future Congresses are in this manner obliged to behave in a Ricardian manner (so only temporary deficits permitted), which has the effect of anchoring long-run inflation.

Well, maybe. But I can't help but think of Dr. Strangelove here.

PS. I recently gave a talk on this that you can view here beginning at the 3 hour 12 minute mark.