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The payment on reserve process proposed by Robert Hall and Ricardo Reis is a way of remunerating reserves which would give the central bank better control over the price level.
The basic intuition is that given the central bank offer reserves that promised an indexed payment instead of an interest rate, for $1 of reserve the bank could receive a payment (x) tomorrow that is indexed to the price level then (p’). Therefore:
Nominal payment on reserves = (1+x)(p’)
The inflation rate of the next period is p’ divided by price level this period (p), such that:
Real payment on reserves = (1+x)(p’)(p/p’)
= (1+x) (p)
The no-arbitrage condition dictates that the real payment on should be equal to the real interest rate of the risk-free assets (1+r), so that:
(1+x)(p) = (1+r)
p = (1+r) / (1+x)
The central bank can then use this equation to ensure the price level is equal to the targeted price level. For a given estimate of the safe real rate, if the price level was running below the target, the central bank could lower the payment on reserves, so to raise prices. The mechanism can be interpreted as follow: when the real payment on reserves is lower than the safe real rate, banks would move the money out of reserves and invest in alternative assets. As the real value of reserves is the inverse of the price level, such that the price must rise.
The one above is the base case for payment on reserves process. In their research paper “Achieving Price Stability by Manipulating the Central Bank’s Payment on Reserves“, Hall and Reis demonstrated that with certain alterations, the payment on reserves process can be applied not just to indexed nominal payment on reserves, but also on simple nominal payment on reserves with a very similar intuition. The authors also considered how certain fractions like segmentations of the financial markets, mismeasurement of the real interest rate and nominal rigidities affect the payment on reserves process. They showed that the payment on reserves rule still works under those fractions.