It turns out that while the Fed is legally permitted to pay interest on reserves that is less than other money market rates or to pay the same rate, it cannot pay more. David Pearson, commenting on David Glasner's blog quoted the low:
Balances maintained at a Federal Reserve bank by or on behalf of a depository institution may receive earnings to be paid by the Federal Reserve bank at least once each calendar quarter, at a rate or rates not to exceed the general level of short-term interest rates.
Steven Williamson noted this as well, and suggested that the Fed needs to break the law to "do its job." Williamson claims that it must be that T-bills are more liquid than reserves. Still, it is hard to believe that the Fed's job is to pay interest on reserves at a level that reflects the liquidity value of T-bills relative to reserves. With Williamson, it is hard to tell.
I would argue that if the Fed is going to use the interest rate paid on reserves as a policy instrument to restrain nominal expenditure on output, then it must sometimes lead the market. If banks are holding the relevant money market instrument, then it would seem that raising the rate paid on reserves would cause banks to sell the money market instruments, lowering their prices and raising their yields. But if the banks sell off all of them, then that avenue for adjustment no longer exists. What happens instead is that the higher rate paid on reserves pulls up the rate that banks pay on deposits, so that bank depositors also shift out of the relevant money market instrument and into deposits.
It is also possible that when banks respond to higher rates paid on reserves by raising bank loan rates, some borrowers might sell commercial paper rather than utilize bank loans. That would tend to raise the rates on money market instruments too. This would also involve the Fed "leading" the market.
However, I don't favor the use of the interest rate on reserves as a policy instrument. If there is an excess supply of base money, the Fed should reduce the quantity through open market sales. I don't think that the Fed should be using the composition of its asset portfolio to try to direct credit. (Keep nominal GDP on the target growth path and let the market allocate credit.)
More importantly, the Fed doesn't need to be restraining nominal expenditures on output now. Quite the contrary. If the Fed were to use the interest rate on reserves as a policy instrument, it would be "leading" the market lower. And there is nothing in the law that prohibits that--at least until the interest rate hits zero.
Could the Fed "get away" with charging the banks fees based upon the size of their reserve balances without some kind of authorization by Congress? Probably, if they could point to some financial cost to them when banks hold reserve balances. For example, if the yields on T-bills are so low, the Fed is earning next to nothing, or perhaps, literally nothing, or perhaps, receiving a negative yield like everyone else, then any operating costs would have to be covered. The Fed has to keep track of the banks' balances, collect on T-bills and reinvest the funds, and so on. The Fed could buy longer term Treasuries, but that involves a risk. The banks should share in that potential cost.
Of course, if the interest rate on reserve balances falls so low that the result is a currency drain, with either banks accumulating vault cash or the nonbanking public filing safes or stuffing mattresses, then it is too low. Here again, the Fed would need to "lead" the market by "paying" sufficient interest on reserves, regardless of whether market rates are lower, to avoid the cost of printing currency.