Interest on Reserves Again

I did some math about interest on reserves and the money multiplier.

If=

c  = A - B*id

id = ir - K

r  = Z + G* ir

and:

mm = 1+ c/c+r

then:

dmm/dir  < 0   if:

G  >  (1-r)/(1+c) B

where c = currency deposit ratio,  r = reserve deposit ratio, id = interest rate on deposits, ir = interest rate on reserves.

If the responsiveness of the  reserve ratio to interest on reserves is sufficiently high that it is no less than 1 minus the reserve ratio divided by 1 plus the currency deposit ratio times the responsiveness of the demand for deposits to the interest rate on deposits, then an increase in the interest rate on reserves is contractionary.

In other words, if the responsiveness of the reserve ratio to the interest rate on reserves is greater than the responsiveness of the currency deposit ratio to the interest rate on deposits, an increase in the interest rate on reserves is contractionary.   And it can be somewhat less.

I am a bit worried about the result in that if the currency deposit ratio falls too zero, this result must be wrong.    The mm = 1/r or 1/(Z+G*ir) .    The money multiplier is obviously negatively related to the interest rate on reserves.    (I guess I should check again and make sure the 1 is in the denominator.)

However, I still stand by my verbal argument that for the expansionary scenario to hold, the banks were not maximizing profit and should have increased the interest rate on deposits and expanded loans.   I don't think that the money multiplier result above is inconsistent with that being true.

In a world with no interest on deposits, or else, a world where banking is with banknotes, and even more so when usury laws create a shortage of bank loans, then something like the money multiplier is a  fine framing.    Of course, there is no interest rate on deposits to impact the the currency deposit ratio.   In the result above, B = 0 and interest on reserves is contractionary..

When banks can charge competitive interest rates on  loans and deposits, then there remains an element of truth in the money multiplier approach.   I don't doubt that a change in preferences leading to a reduced currency deposit ratio would be expansionary.   But if banks pay higher interest on deposits in order to attract currency to deposit at  the central bank, it is hard to see why they would lend that money out.  On the contrary, they would be contracting lending as well to increase reserve holdings.   And, of course, the increase in the interest rate on deposits is contractionary in and of itself, along with the decrease in the quantity of money.

One final note:  If holding reserves is considered a cost of operating a bank, which is certainly true, to a degree, and especially with required reserves, then paying interest on reserves could result in lower interest on loans and higher interest on deposits both.   This tendency would be for bank balance sheets to be larger, but it is unlikely to be expansionary--that is, create an excess supply of money.