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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.

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.

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.

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

Missed that - thanks for the clarification.

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.

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.)

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 )

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