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!

1. Let's make this a market.
First, it is not ergodic because an entry into the market is either a buyer or seller, and old people are never sellers. And the youngsters partition their lives, being seller sometimes and buyer sometime.

We need a auctioneer, so make this a Walmart checkout counter line up and add in Sam Walton. We are going with random independent arrivals.

Sam Walton will engineer a precise ratio of clerks (sellers) to customers. He will walk the line changing items per basket so no customer is third in line and no clerk is second. He opens and closes counters as needed.

In this case, the ratio of clerks to customers is a constant, within known, small bound. From number theory having nothing to do with price, that constant is known. y, which is really gains to scale, is determined by the number of checkout counters. If gains to scale are fixed, the number active counters is fixed, the flow of good is within some know error bound and the monetary flow equations work. Same Walton can then calculate and get a 1/ny yield on the dollar. But no one is doing time averaging, there is no Godot counting time. And best of all, this model meets the Lucas criteria, it is closed and dots connected, the only thing we added was am Walton walking the line. Further, the ratio of clerks stocking the floor to customers is the same as at the checkout cuunter. From here can can set the floor size and number of counters if we know market size.

1. This model is exactly the asynchronous savings and loan model which fintech has automated. But there are no time bets. And Sam Walton is replaced by an automated pit boss which matches deposits and loans, in advance. The matching error is bounded and know by all parties, ex ante, a provable guarantee. All traders are bots, the S/L ratios for each bot set to maximize returns and all bots get round robin access to the trade book (checkout counter congestion map). Interest swaps are made from loans to deposits, as needed to keep matching within bounded error. Coinbase uses this model. It is the basic reductionist model of banking. Betting term length is an insurance function as Paul Samuelson and Cochran discovered, the solution is indeterminate when betting terms and insurance is needed.

This model is the foundation of our two new nobel prize winners, I am certain. This model is adapted and works to understand our government as a value added channel, it explains the interaction between House and Senate. It is the basis of Krugman agglomeration theory. It scales from small corner stores to the big box retailers. It explains fiscal recession cycles, covers Roger Farmer and shocks to our beliefs. Handles Hayek just fine. And so on. Works for atomic orbitals. Predicts double entry accounting. Add a dimension and we can handle quarks.

2. Not sure my comment is germane, but here goes.

Throughout much of the world, there is "forced" savings going on, as in Germany, China, really the entire Asian Pacific.

Add on aging populations, that save.

Add on required insurance for nearly everything, that effectively becomes large pools of savings.

So, what is "natural" and what is not, and what is the "natural" rate of interest?

If savings is greater than good investment opportunities, then we should see negative real rates of interest.

3. I’ll take a stab.

Government buys ½ of y from (credits the accounts of) the young. It gives the goods to the old but debits the public account, not the accounts of the old, thus incurring a public debt of ½ y.

The young spend their \$ throughout their lives, i.e. on both their own produce and that of the future young, to smoothen consumption until they pass away.

Because they do not know whether the future y which they will be able to command with their savings will rise or fall, this inter-generational deal is subject to uncertainty.

In an arrangement with only one type of government asset, everyone who has an account with a credit balance is subject to that risk.

The government can, however, choose to offer two types of account. One which says: if you give up goods when you’re young, you are entitled to receive goods at a ratio of 1:1 whenever you like. And one which says: if you’re patient, we will also pass future surpluses (or deficits) of y on to you.

So, normatively, you have a 0 inflation deal for the holders of ‘M’, while the delta accrues to the holders of government debt. From that, you can derive a ‘natural’ rate of interest on M equal to inflation, should the government, for whichever reason, not be able (or willing?) to fulfill its promise of keeping \$ prices constant.

I suppose one could construct an analogous, normative story for any other trajectory of prices, but I think price stability is the default setting.

My intuition is that this is a story of productivity gains / losses more than of changes in population size but I'm no good at math.

The question of duration only makes sense, I’d say, if the goods being bought by the public can actually be defined as investment goods. In the current situation, that might be public health, for example. Assuming there is a social benefit to keeping businesses afloat while telling people to stay at home, the question is, over which period do these benefits accrue / does the debt pay off itself (in real terms). It can be derived directly from the golden rule of government spending.