Are Open Market Operations wrong-signed when nominal rates are negative? (No)

I'm going to start out with a silly unrealistic thought-experiment. Then I'm going to say why I think my silly unrealistic thought-experiment might matter.

A central bank issue two types of money: $1 notes; and $100 notes. The two types of money are not perfect substitutes: the $1 notes are convenient for small purchases; the $100 notes are convenient for large purchases. The central bank is willing to exchange $1 notes for $100 notes at a fixed exchange rate of 100 to 1. There are no other denominations, no other banks, and no other forms of money.

The $1 notes and $100 notes pay the bearer a nominal rate of interest that is chosen daily by the central bank. Those two rates of interest can be the same or different, and can be positive or negative.

1. Suppose the central bank sets the same rate of interest on both notes. What happens if the central bank cuts that rate of interest, holding the total $ value of the stock of notes constant? That's an easy question to answer. The opportunity cost of holding money increases, so the demand for money falls, the Velocity of circulation increases so MV rises, so Aggregate Demand increases. It's an "expansionary" monetary policy. It doesn't matter whether the interest rate is positive or negative.
2. Suppose the central bank always sets the rate of interest on $1 notes at 0%. What happens if the central bank cuts the rate of interest on $100 notes, holding the total $ value of the stock of notes constant? That's a slightly harder question to answer, but I think the answer is qualitatively the same as in the previous question above. It's still an expansionary policy, because the opportunity cost of holding money has increased, on average, so the demand for money has decreased, though it's less expansionary than the first policy. But the answer is complicated by the fact that some $100 notes will be exchanged for $1 notes, so the composition of the money supply changes, even though the total stays constant.
3. Suppose the central bank wants to change the composition of the money supply, so there are more $1 notes and fewer $100 notes in public hands, holding the total $ value of the stock of notes constant. And it adjusts the rate of interest on $100 notes, holding constant the rate of interest on $1 notes, to hit that new target for the composition of the money supply? The answer to question 3 is the same as the answer to question 2 above. It's the same question, except the exogenous and endogenous variables have been swapped in the central bank's targeting perspective. If the central bank buys $100 notes and sells $1 notes, and adjusts the rate of interest on $100 notes so the exchange rate stays at 100 to 1, it's an expansionary policy, even though the total money stock stays the same. Because the rate of interest on $100 notes will fall.

The above thought-experiment was a metaphor. The "$1 notes" in my thought-experiment stand for all central bank money (currency and reserves). The "$100 notes" in my thought-experiment stand for Treasury bills. My policy experiment 3 above stands for an Open Market Operation where the central bank buys Treasury bills.

And if my thought-experiment metaphor is correct, Open Market Operations still have the "right" sign, even if nominal interest rates on Tbills are negative, and even if Tbills are used as money, and even if those negative nominal interest rates on Tbills indicate that Tbills are more liquid (more money-like) than the money (currency and reserves) issued by the central bank.

Which might matter again soon, if pessimistic forecasts turn out to be right.

[Liquidity Preference is a bad theory of "the" rate of interest, but a good theory of interest rate differentials (people prefer more liquid to less liquid assets, other things equal). Silvio Gesell was right about interest on money being what matters for Aggregate Demand.]

Comments

[Market Fiscalist]

If people have a high demand to hold money and are not bothered if they hold it in $1 or $100 format then options 2 and 3 may be ineffective. Large negative rates on $100 bills will tend to cause people to switch to $1 format but not necessarily spend any more.

Option 1 would seem more likely to work at least in the short run. Reducing rates from -10% to -20% would definitely cause the demand for money to decrease. But it feels wrong and that it would likely have unexpected (or perhaps expected?) bad side effects especially if the nominal value of currency (or bonds+currency if applied to more realistic scenario) in circulation is held steady.

[Market Fiscalist]

And an interesting complexity is that if the total money supply stays constant then when rates are positive the CB has to levy taxes to fund the interest payments or redistribute the revenue if rates are negative.

[Nick Rowe]

MF: "If people have a high demand to hold money and are not bothered if they hold it in $1 or $100 format then options 2 and 3 may be ineffective."

If $1 and $100 notes were perfect substitutes, the CB would have to set the same rate of interest on both notes, otherwise people would hold only those notes that paid the higher interest rate, and swap all the lower rate notes at the central bank. (We get a corner solution for 2, and 3 is ineffective, as you say.) That's why I assumed they were imperfect substitutes.

"And an interesting complexity is that if the total money supply stays constant then when rates are positive the CB has to levy taxes to fund the interest payments or redistribute the revenue if rates are negative."

Or, to say the same thing differently, the CB's seigniorage profits are lower, so it gives less profits to the government that owns the CB, which needs to have higher taxes or lower spending.

[Market Fiscalist]

'That's why I assumed they were imperfect substitutes.'

Missed that - thanks for the clarification.

[Alex Godofsky]

The effectiveness of (3) is contingent on it actually achieving the outcome (2), viz. lower interest rates. If QE doesn't move rates, or moves them very little, then it has no effect in this model.

[Nick Rowe]

Alex: I think that's right. If $1 and $100 notes were perfect substitutes, then the CB could change the composition without affecting rates or anything.
(Keep in mind though, that rates also depend on expected future NGDP, so a successful OMO will have two effects on rates, going in opposite directions. The "model" in this post ignores those expectations.)

[Judge Glock]

Is the other necessary assumption here that the central bank would let the exchange rates between the two types of denominations fluctuate? I get that the two types of bills are not perfect substitutes in the marketplace, but aren't they by definition perfect substitutes at the central bank, which severely limits the ability of the bank to manipulate the differential interest rates or size of the alternate denomination stocks? It seems to me like the distinction between your example and the typical reserves and T-bills example, is that the dollar signs in front of the the $1 and $100 means the central bank has to keep them steady or the market will set the unit of the account as one or the other. If, as in your third example, the central bank reduces the stock of $100s through purchases, but then sets interest rates to keep the exchange rate between the two the same, the two effects would perfectly offset, and there would be no expansion or contraction. The bank would have some minor wiggle room depending on how far the banks' offices were from the marketplace, but the "gold points" or "denomination points" spread would be pretty thin.

(The example put me in mind of Sergeant's and Velde's penny-in-advance and dollar-in-advance model for coinage in middle ages, in which the increased relative liquidity of the silver penny led to a gradual appreciation of the gold dollar as a way for equalizing returns, which eventually led to a corner solution in which the pennies fell below the value necessary to coin them and only dollars existed. But in that case the central bank equivalents, or mints, could not arbitrarily determine how much gold or silver was brought in to the mint, so they could not completely control exchange rates )