Beginning with “How Subordinating Paper Currency to Electronic Money Can End Recessions and End Inflation,” and continuing in the many posts collected in my electronic money sub-blog, I have often written of how a crawling-peg exchange rate between paper currency and electronic money can eliminate the zero lower bound. In this post, I want to give more detail about how to determine the appropriate exchange rate.  Also, in “A Minimalist Implementation of Electronic Money,” I wrote of a “deposit charge” for paper currency deposited with the central bank. One minus the deposit charge is the effective exchange rate there. That is, $1 of paper currency would, in effect, be worth $(1-deposit charge) of electronic money, since that is what it would become once deposited. So determining the appropriate exchange rate at which paper currency can be exchanged for electronic money also determines the “deposit charge” (and ideally, the equal withdrawal discount) in the minimalist implementation of electronic money. 

The way to determine the appropriate exchange rate between paper currency and electronic money is to first decide what the effective interest rate on paper currency should be. The current paper currency policies all around the world amount to the choice to have the nominal interest rate on paper currency be equal to zero all the time. (In this post, I will always be talking about the effective interest rate on paper currency before storage costs.) This policy decision (or non-decision) to keep the interest rate on paper currency equal to zero is what creates the zero lower bound. 

In order to eliminate the zero lower bound as a lower bound, the interest rate on paper currency needs to be chosen somewhere below the desired policy interest rate (the fed funds rate in the US, the bank rate in the UK, the repo rate in Sweden, etc.) In an electronic money system, the monetary policy committee would decide on a paper currency interest rate at the same time it chose the policy interest rate. Once the paper currency interest rate is decided upon, that paper currency interest determines how the exchange rate between paper currency and electronic money evolves. 

The math is exactly the math needed to see what $1 turns in to if it earns at each moment the variable interest rate in the graph. Of course the graph above is only an example. What matters is the paper currency interest rate the central bank decides on.

I will give the math for compound interest with a variable interest rate below, but in this context, the basic idea is that to make the paper currency interest rate negative, the exchange rate between paper currency and electronic money has to make a paper dollar worth gradually less and less compared to an electronic dollar. On the other hand, to make the paper currency interest rate positive, the exchange rate between paper currency and electronic money has to make a paper dollar worth gradually more and more compared to an electronic dollar.  

To give the technical description of how the exchange rate is determined from the math of compound interest with a variable interest rate, start from a moment when the paper dollar (or euro, or yen or pound, …) is at par relative to the electronic dollar. An obvious moment when paper currency is at par is the moment electronic money is introduced. Draw the graph of the desired paper currency interest rate as in the graph above. Figure out the area between the x-axis and the curve of the desired paper currency interest rate, with area below the axis counting as negative and area above the axis counting as positive. In calculus, this area is called an integral. (This integral needs to be calculated with 1% per year being represented as .01/year, 2% and .02/year, and so on; or if not you will need to divide the integral by 100 to get the right number. Also, notice that -3% per year is only -0.25% per month, or -0.75% per quarter. I like to think of the % sign as just another name for .01, with a hint that some sort of proportion and maybe some kind of compounding is going on.) The integral of the paper currency interest rate from when electronic money is introduced is then the natural logarithm of the appropriate exchange rate–the value of a paper dollar in terms of electronic dollars. (I give an introduction to natural logarithms in my post  “The Logarithmic Harmony of Percent Changes and Growth Rates.”) Equivalently, if you calculate the integral (with 1% treated as another name for .01) and then use the exp key (or e to the x power key) on a calculator on that number, you will get the appropriate exchange rate. 

Another way to describe things is that, to a good approximation, for exchange rates not too far from par, the integral gives the size of the deposit charge in the minimalist implementation of electronic money that determines the effective exchange rate, except that a negative integral corresponds to a positive deposit charge. 

In the graph above, the integral is shown shaded in over a period of time when the intended interest rate on paper currency is always negative. That will yield an exchange rate in which a paper dollar is worth less than an electronic dollar. 

In the graph below, the integral is shown shaded in over a longer period of time over which the intended interest rate on paper currency is first negative, then positive. With the area below the axis counted as negative and the area above the axis counted as positive, the overall integral is getting close to zero again, so the exchange rate is quickly coming back up towards par. (The earlier moment in time when the paper currency hit zero on its way from negative territory to positive territory was the moment when the value of a paper dollar relative to an electronic dollar was the lowest. After that the exchange rate is going up again.)

The most important point is this: the zero lower bound is a policy choice. From the perspective I am taking, the zero lower bound arises when governments choose to have a paper currency interest rate equal to zero all the time, in order to keep paper currency at par. The world has suffered a great deal in the past few years from paper currency interest rates too high to allow full economic recovery. In our current environment, a paper currency interest rate of zero is too high in many countries. In countries that are still below the natural level of output, let’s lower paper currency interest rates, along with policy interest rates, the interest rate on reserves, and the discount rate at which central banks lend.