# Some thoughts on (negative) interest rates and inflation from a micro perspective, Part 2

*the next part is [not up yet]; here a related post on the **Iron Bank of Braavos*

In my previous post, I have started to look at how one might look at inflation from a micro perspective, and how symmetries can help in this respect. I’ll continue with this topic in this post here.

#### Review of Part 1

In Part 1 I have previously defined an *exchangeable *as anything of value that can be exchanged against something else, including assets, consumables, and services. The *exchange ratio* between to exchangeables is defined as the amount of one you need to get another, and I have shown that under certain market efficiency conditions (eg no friction, no arbitrage) bilateral exchange ratios *Rij* can be derived from a price unilateral price via the relationship

*Rij = Pi / Pj*

This price however is not unique, as only the exchange ratios *Rij *are observable, and those exchange ratios are invariant under scaling the *Pi* with a common factor.

I have then introduced a world where markets organise themselves periodically, and that introduces a *money* token as a physical manifestation of price (one unit of money is deemed to have unity price). However, this money is only issued during one period and redeemed for real assets at the end, and we have shown that in this environment there is no need for monetary stability over time — the value of the money token can be redefined at every new trading session, and whilst this introduces notional inflation or deflation in the quoted price levels, this effect is entirely spurious.

#### A monetarily connected multi period model

Above I described a model that introduced money tokens in order to facilitate trade during each trading session, but where those tokens were newly reissued at each and every session, a situation that I might pompously refer to as monetarily disconnected.

Now I want to look at a situation where monetary tokens can be retained between trading sessions. Detailed assumptions are as follows

- monetary tokens are electronic and issued by a central bank that allow for a frictionless and instantaneous transfer amongst market participants
- the central bank operates a cost-less pawn-broking facility where participants can obtain monetary tokens by pledging exchangeables, so liquidity is never a problem (solvency might be of course)
- there is neither an overdraft facility and nor an organised unsecured credit market; there might or might not be market participants who lend each other tokens or exchangeables
- because of highly liquid markets and some overcollateralisation on the pawn-broking facilities there is absolutely no risk of those facilities ever defaulting, hence the fact that it is cost-less does not imply a subsidy
- markets are frictionless and non-arbitrageable (implying that all exchange ratios are determined by prices), trading is continuous, and prices are variable over time

Other than having assumed away a lot of the friction and inefficiencies from the system, this looks very close to a world where notional central bank policy rates — borrowing and lending — are predictably held at zero.

**Positive nominal interest rates**

Now we introduce a twist: the central bank continuously increases all monetary token balances (positive and negative) by a certain fixed (and predictably fixed) annualised percentage (the so called — drumroll — interest rate). So if we assume this percentage is 10% then 100 tokens deposited at the beginning of the year will become 110 of the end of the year, and the same will apply to tokens owed under the pawn-broking facility. For simplicity we assume that interest is paid continuously, with continuous compounding.

In this world it is common knowledge — everybody knows, and everybody knows that everybody knows etc — that a certain amount of tokens received at time *T* will correspond to a different well known amount of tokens received at time *T+dT*. We can also assume that the marginal participant does not face financing constraints, because everyone who is solvent is liquid by virtue of the pawn-broking facility. **Therefore rational actors will simply adjust prices to exactly track the well known increase in tokens, and as a consequence in the real world strictly nothing changes: all exchange ratios between exchangeables remain the same, and the difference in exchange ratios between exchangeables and monetary tokens exactly compensates for their expansion.**

En passant, and because I might need to later, and want to introduce a mathematical trick (akin to ‘1990 dollars’) that is sometimes helpful: I define a *time-T-token* as a token issued at time *T* and I assume that it does not change over time. However, before it can be used at a time *t > T *it needs to be converted into a *time-t-token*, and the conversation ratio is simply the interest compounding factor from time *T* to time *t*. Whatever the evolution of our ‘spot’ money, that prices as expressed to exchange ratios in terms of time-T-tokens will be exactly those that one would obtain in a world where monetary tokens do not accrue interest.

#### Negative and weird nominal interest rates

Of course nothing in the above analysis actually relied upon the interest rate being positive — all arguments would be analogous in the case where 100 tokens transform to not 110 tokens but to 90 tokens at the end of the year. We have already seen that in the scenarios considered, positive (nominal) interest rates are not different from zero (nominal) interest rates, and the same holds of course for negative (nominal) interest rates.

In fact, we can go a step further. Implicitly we have assumed the well known continuous compounding function *CF(t,T)=exp(r(t-T)) *when compounding a token obtained at time *T* to time *t > T*. In fact, any positive function *c(t)* can be used as a compounding function with the following definition

*CF(t,T) := c(t) / c(T)*

In periods where *c(t)* is increasing we have positive (nominal) interest rates, where it is decreasing we have negative (nominal) interest, and where it is flat, (nominal) interest rates are zero. As an aside: *c(t)=exp(t)* yields our original compounding function.

#### Conclusion

I’ll leave it here for today, as I’ll be actually off for the airport soon, and I just want to write a short conclusion of the analysis so far.

We have looked at a world where people trade exchangeables, and we have found that in order to being able to efficiently do so one probably wants to introduce permanent monetary tokens related to the price levels from which exchange ratios are derived. We found that the system of price levels is not fully defined even if the entire set of exchange ratios is known, due to a symmetry where the prices of all exchangeables are scaled by the same number. Introducing a monetary token and tying it to unity price breaks the symmetry, and within the system prices are now unique.

However, when we are looking at the evolution of the system over time we find that the scaling symmetry comes back: we looked at a world with positive nominal interest rates, then one with negative nominal interest rates and one where interest rates deterministically change over time defined by an arbitrary positive function *c(t)*, and we have found that all of those worlds were behaving exactly like the zero-interest world as for as exchange ratios (which is all that really matter!) are concerned.

Or, to put it the other way round: nominal interest rates on their own do not matter, even if they are zero or negative. What matters is… *(to be continued)*

*the next part is [not up yet]; here a related post on the **Iron Bank of Braavos*