Can Interest on Reserves be Expansionary?

Josh Hendrickson has written some posts critical of New Keynesian macroeconomics over the last few months.   In one post, he questioned whether increased interest on reserves is really contractionary.   Now, his  broader point is that New Keynesian models are problematic because they try to do monetary economics without money.   I certainly agree that this is a problem.  However, I find it difficult to believe that increasing the interest rate paid on one portion of base money does not increase  the demand for base money.

Hendrickson's analysis is that a higher interest rate on reserves raises the opportunity cost of holding currency while reducing the opportunity cost of holding reserves.    The currency/deposit ratio would fall while the excess reserve ratio would rise.    A lower currency deposit ratio increases the money multiple while a higher excess reserve ratio decreases the money multiplier.  The net effect is ambiguous and so is the effect on broader monetary aggregates.   The impact on spending on output and inflation is therefore also ambiguous.   It is possible that higher interest on reserves could be inflationary, deflationary, or have no effect.

I don't believe it.

First, suppose the increase in the interest rate on reserves is expansionary.    The higher interest rate on reserves will motivate banks to increase the interest rate on loans and the interest rate on deposits.   The higher interest rate on loans results in a lower quantity of loans demanded and so a smaller quantity of bank deposits while the increase in the interest rate on deposits results in an increase in the demand for deposits.   This is contractionary.

However, the higher interest rate on deposits creates an incentive to reduce currency holdings by depositing currency into banks.  

The deposit of currency into banks increases the quantity of bank reserves.   This causes the banks to expand lending, which they do by lowering the interest rate on loans.   The additional lending increases the quantity  of deposits.   The lower earnings on the bank's asset portfolios lead banks to lower the interest rate on deposits.   This is expansionary.

Which effect is larger?

Well, it seems to me that the expansionary scenario is going to require that the interest rate on deposits and loans both be lower than their initial values.   But if the interest rate on deposits is lower than its initial value, then there would be no incentive to deposit currency into the banks.   Looks like a contradiction.

Now, let's consider a corner solution.   Once the interest rate on reserves is so high that all of the currency has been deposited, then clearly any further increase in the interest rate on reserves cannot possibly result in a deposit of currency and an expansion in the quantity of money through that avenue.   All that is left is a decrease in bank lending and so a decrease in the quantity of deposits which is the same thing as money after all the currency is deposited.   While a higher interest rate on reserves would still lead to a higher interest rate on deposits, that would increase the demand for money, reinforcing the contractionary impact.

Of course,  the section of Woodford cited by Hendrickson describes a "cashless" payments system, and so would be just such a corner solution.  I would note that if all hand-to-hand currency is private, redeemable banknotes, then all base money is reserves and so an increase in the interest rate on reserves is unambiguously contractionary.

Since we don't live in a world with privatized currency or where all currency has been deposited into banks, that corner solution is of little practical significance except to show that even if there is some range over which an increase in the interest rate on reserves is expansionary or at least not contractionary, a sufficiently large increase in the interest rate on reserves must be contractionary.

Now, consider a scenario where banks hold 100% reserves.   An increase in the interest rate on reserves has no impact on bank loans, because there are none.   But it still results in  higher interest rate on deposits.   While this would attract more deposits of currency, the result would leave the total quantity of money unchanged.   However, the higher interest rate on deposits implies an increase in the demand for money.   This is contractionary.

Suppose there were a monopoly bank.   The interest rate on reserves increases.   The interest rate on reserves is greater than the interest rate on deposits.   The bank raises the interest rate paid on deposits to attract more deposits of currency so that the currency can be in turn deposited at the central bank to earn interest.    Is it sensible to say that the bank would then take this extra currency and lend it out?   There was no increase in the interest rate that it can earn on loans.  It intended to raise the interest rate on deposits to obtain currency to deposit at the central bank to earn interest on reserves.  

Now, with  competitive banking system, we can image that each bank increases the interest rate it pays on deposits to attract deposits from other banks.   Each bank attempts to increase its reserve holdings and so the amount it can earn on reserves.    This is different from the monopoly bank that can only be raising interest rates on deposits to attract deposits of currency.   Still, even with a competitive system, one reason why they would be raising interest rates on deposits would be to attract more currency deposits.   So why would the banks, to some degree seeking to obtain currency to be deposited at the central bank and earn interest on reserves, end up expanding loans instead?

What must be going on for there to be an expansionary scenario?   The banks receive all of these deposits of currency, which they deposit at the central bank and are now earning interest on their reserves.  They then notice that they could make even more interest by lending the funds out.   And then as they lend the money out, the money multiplier process expands the total quantity of deposits.   If the increase in the quantity of deposits is greater than the increase in the demand to hold them due to the higher interest rate on deposits, then the result is expansionary.

But then, why didn't the banks already increase the interest rate on deposits to obtain market share and attract currency deposits in order to make these added loans?   If they were already maximizing profit before, then there is no reason to expect them to expand loans at all.  Quite the contrary, they will contract lending to hold more reserves.  

Suppose there are two types of banks, one set is 100% reserve and the other is no reserve.   Both issue checkable deposts, and the zero reserve banks lend out the funds.   The interest rate on reserves increases, and the 100% reserve banks pay higher interest on deposits.  Currency is deposited in the 100% reserve banks.   The quantity of money is unchanged, but because the interest rate on deposits has increased, the demand for money is higher and so this is contractionary.

Now, suppose the no reserve banks must increase their deposit interest rates to match those paid by the 100% reserve banks.   With higher costs, they raise the interest rate charged for loans, the quantity of loans demanded falls, and as old loans are repaid and borrowers checking accounts are debited, the quantity of deposits falls.   This is contractionary.   Higher interest rates paid on deposits and a smaller quantity of deposits.

But these higher interest rates on deposits attracts currency deposits for the no reserve banks too.  Doesn't this result in more lending and an expansion in the quantity of money?   Of course, that is inconsistent with the raising interest rates on loans due to the higher costs.   If it would be profitable for the zero reserve banks to expand lending in response to an influx of currency deposits due to a higher interest rate on deposits, then they would have already increased the interest rate on deposits and lowered the interest rate on loans, and expanded their balance sheets.

It just doesn't add up.