How does the central bank control interest rate?

How does the central bank control interest rate? Anyone who is slightly familiar with history knows that it is notoriously hard to control price. For example, many governments have dictated a price floor for gasoline, the result would always be a queue in the gas station. Central banks are not even able to dictate interest rates. They have to do it trough market by buying or selling Treasury bills. But a hedge fund can buy or sell Treasury bill as well, and it can be larger than the Fed (post crisis Fed is definitely larger). Benjamin Friedman argues that Fed's action can influence the interest rate because it changes the amount of bank reserves, of which Fed monopolizes the supply. However, reserve requirement is becoming less relevant. (Many countries have no requirement for any banks. In the U.S., shadow banks do not have reserve requirement. Many types of commercial bank liability also does not have reserve requirement.) Now, how do central banks control interest rates if it is not able to control the price of gasoline?

It seems that there is a fundamental difference between gasoline and money. Central banks cannot create gasoline out of nothing, but they can create money. Bank reserve, the liability of central bank to commercial banks, is the so called high power money. Neglecting other inside money, central bank can change money supply by borrowing more or less. If central bank wants to borrow more, it can just buy real assets from commercial banks with bank reserves, which the central bank can costlessly add; if the central bank wants to borrow less, it can just sell the real assets and repay the liability. But does the change of quantity of money always link to interest rate?

Let us illustrate how it works. Suppose now the market equilibrium interest is 5%, and the Fed wants 10%. So it announces that "The interest rate should be 10%". It starts to sell Treasuries to suck up bank reserves. We assume that there is no shortage of reserves, or no required reserves, or Treasury bills are so liquid which functions essentially as reserves. In this world, there would be no effect if the Fed merely changes the composition of Treasury and bank reserves. In fact, now bank reserves start to pay interest, it is hard to conceive any intrinsic difference between reserves and Treasuries. The Fed has to increases the interest on reserves, which set a floor for interest rate, since no one would lend lower if they can deposit to central bank and get 10%. It is also the ceiling, since reserve is not special, so no one would borrow reserve higher than 10%, which is already higher than the original 5%.

But wait, wouldn't a interest rate higher than equilibrium creates over-supply of credit? Does central bank need to absorb all the extra credit? Let us start with the simplest world where there is only real assets called apple, but no money. Now the central bank promises a 10% return if you lend your apple to someone. 100 people want to lend at this rate, only 60 want to borrow (each has one apple), so the central bank has to borrow the extra 40. So the central bank get the 40 apples, and issue 40 IOUs to lenders. The central bank can collect apples only through tax. Now the central bank promises a 0% return for the apple. 100 people want to borrow, but only 60 want to lend, so the central bank has to lend the extra 40. How? The central bank need to get apple by tax.

Important conclusion: in a world without money, the only way that the central bank can control the interest rate without causing shortage or excess supply is fiscal policy.

Now suppose households have money in their utility function. Central bank liability is money. It can exchange apples with money which can be costlessly supplied. Now suppose 10% case where the central bank has to accommodative excessive supply for credit. The central bank borrowed the 40 apples, and issue 40 IOUs to lenders. Money supply increases. But wait, should high interest rate be associated with low money supply?

There is a flaw in the above argument. The market for money should also cleared by interest rate. 10% definitely causes over supply of credit in terms of money in this market. So central bank has to accommodative the excess 40 supply of money, by issuing 40 IOUs. But, IOU is money, so central bank is replacing IOUs by IOUs, effectively doing nothing!

The contradictory result comes from the fact that we do not have Treasuries here. It seems essential to have an asset, which is an IOU issued by the government/central bank, but differs in liquidity from money. And more importantly, the liquidity difference matters for interest rate.

Can liquidity really affect interest rate? The amount of liquidity should only affect liquidity premium, i.e. the difference between return on liquid liability claims and illiquid liability claims. Central bank can affect the composition of liquid and illiquid claims, which will affects liquidity premium. But can it affect the return of illiquid liability claims, which is the true interest rate? In another words, does change of liquidity composition change the demand and supply for credit? How can one price (the interest rate) clear both money market and credit market?

Another way to formulate the question is: Suppose the liquid claims pay zero interest. If the central bank replace all the illiquid claims by liquid claims, can it bring down the interest rate (return on illiquid claim) to zero? Maybe not. There can be excessive credit demand at very low interest rate.

So we have come back to the claim that I made above:"It seems essential to have an asset, which is an IOU issued by the government/central bank, but differs in liquidity from money." In a world where there is no special thing about bank reserve as against Treasury, (e.g. no reserve requirement), the central bank can  still control interest rate. It is not only theoretical possible, it also works in reality. New Zealand has no reserve requirement, but the bank of New Zealand can still control interest rate. How does it work? The central bank can just announce a rate, at which it is willing to borrow and lend at any amount. Since the central bank can always costly create the liability (by printing money) or eliminate the liability (selling assets, if not enough, by taxation), the central bank can always achieve the interest rate target. Therefore, to reconcile with the discussion between money and Treasury, changing liquidity premium is just one way to change interest rate, the central bank can always use other ways to change interest rate, (e.g. simply dictating the rate).

However, it is still not clear how central bank can change the rate that people borrow and lend apples by printing money. Do we have to go back to the moneyless model where the rate control is effectively through the fiscal policy? If so, how to map the printing money action to fiscal policy?

The fundamental misconception here is the distinction between nominal and real interest rate. The rate that people borrow and lend apples is real interest rate. The rate that people/central bank borrow and lend dollars is nominal interest rate. The central bank can control nominal rate, but it does not mean that it can also affect real rate. Continue our thought experiment. We add money in the apple economy, but people do not really need money. We call this the "parallel economy" since there is no interaction between nominal and real. The central bank can do what it wants on the returns on the money. But the real interest rate on apple is still 5%. An empirical analogy is Bit-coin. It can assign whatever interest on the Bit-coin, but it would not affect the interest rate of the economy since (almost) no one is using it.

So the real question is: when central bank changes nominal interest rate, does it change the real interest rate? To answer this question, we first note that the difference is inflation. In the parallel economy, the real rate is always constant. An increase of nominal rate mechanically means that the inflation rate will go up (quite unintuitively if one believes high nominal rate means monetary tightening and disinflation). In an economy where inflation is rigid, an increase nominal rate will lead to an increase of real rate. This is basically what New Keynesian framework is: central bank controls nominal rate directly, and inflation is sticky. They do not need to specify how the interest rate target is achieved, they do not pay any attention to the quantity of money or the size of balance sheet of the central bank, because it is almost tautological that central bank can control nominal rate. What they do is to model why inflation is sticky.

Now the question left is that when the central bank tries to control nominal interest rate, will it end up with a gigantic balance sheet? In the parallel economy, since money has no use, central bank's action will have no effect on the size of balance sheet (in terms of apples), since it is constant zero. In the other extreme, where inflation is fixed, the central bank's action causes real rate to vary, but how?

Suppose the price of apple is fixed, so inflation is zero. Now central bank raises the nominal rate to 10%, which means the real rate also becomes 10%. More people want to lend than borrow. So central bank accept the excess credit supply, either in terms of money or apple. In a second scenario, the central bank reduces the nominal rate to 0%. So the central bank has to accommodate the excess demand, either in terms of money or apple. If the central bank lends apple out, it has to tax apple in the first place. This thought experiment seems to show that the central bank's action, if it really changes the term that people borrow and lend the real good, will cause substantial variation of the size of central bank balance sheets, and very likely involves fiscal policy.

This points to the curious feature of New Keynesian model, in which the size of central bank balance sheet or fiscal policy does not show up explicitly. We need to resort to fiscal theory to address this mystery.