Suppose there is an increase in the demand for T-bills. Their prices rise, and their yields fall, clearing the market. However, suppose the increase in demand is so great that that market clearing yield is negative. What happens when T-bills hit the zero nominal bound?
If the zero nominal bound were a ceiling on the prices of the T-bills at their face values, then the result would be a shortage of T-bills. What do the frustrated T-bill buyers do with the funds they had planned to use to purchase the T-bills? If they just choose to hold the money, the indirect consequence of this "price ceiling" on T-bills is to create an excess demand for money. The excess demand for T-bills has spilled over to money and has become an excess demand for money.
However, the zero-nominal bound on T-bills or any other nonmonetary asset isn't like a price ceiling at all. While mathematical economists might see it as a side constraint (R greater than or equal to zero,) everyone else says, "if the interest rate is zero, why not just stuff money under the mattress?" The entire rationale for the zero nominal bound is that rather than lend by holding assets with a negative nominal yield, people will just hold money. The zero nominal bound exists because an excess demand for a nonmonetary asset at a yield of zero will shift to an excess demand for money.
But doesn't an excess demand for money lead to lines at the money store? No it won't, because money is the medium of exchange. People regularly receive money in payment for goods and resources and can build money holdings by refraining from spending. The result is an inability to complete habitual sales of goods and services. An excess demand for money creates the illusion that everyone doesn't want to purchase as many goods and services. Or worse, the illusion that too many goods and services have been produced.
If those frustrated by an inability to sell reduce production, the lower output results in a lower real income. Money being a normal good, the demand to hold money falls to meet the given supply. The excess demand for T-bills at the zero nominal bound leads to a decrease in real income and output.
Why is monetary policy ineffective at the zero nominal bound?
What does monetary policy mean?
If monetary policy means open market operations using T-bills, and T-bills are at the zero nominal bound, then purchasing T-bills with newly created money will not directly clear up the excess demand for money. The nominal yield on T-bills is at zero and the excess demand for T-bills has shifted to an excess demand for money. The central bank buys T-bills, increasing the demand for them and creating a new excess demand for T-bills. This excess demand for T-bills will be shifted into an excess demand for money. The central bank has created new money to purchase the T-bills, and so this increased quantity of money accommodates the excess demand for money. However, the net result of the open market operations is an increase in the demand for money that is matched by an increase in the quantity of money.
If this occurs before real income falls, the initial excess demand for money remains. If it occurs after real income falls, no excess supply of money is generated. Real income remains depressed. There appears to be some kind of trap.
If monetary policy isn't defined in terms of open market operations in terms of one particular asset, then monetary policy isn't ineffective when T-bills have a nominal interest rate of zero. The reason for ineffective monetary policy is simple. If there is an excess demand for some asset, and it is shifted into an excess demand for money, then if the central bank attempts to accommodate the increased demand for money using open market purchases of the particular asset that was in excess demand to start with, the policy will be ineffective.
The central bank simply needs to increase the quantity of money by purchasing some asset that is not in excess demand. In other words, it must purchase some asset whose yield is not at the zero nominal bound. If the central bank generally buys T-bills, and the yield on one maturity is driven to zero, then buying more of that maturity is pointless. The central bank should switch to another maturity. Presumably, the central bank will need to purchase bonds of progressively greater terms to maturity or progressively greater credit risk until the quantity of money rises to meet the demand to hold money. If real income has already been depressed by reduced expenditures due to the excess demand for money, the central bank must increase the quantity of money enough to match how much money would be demanded if real income were equal to potential income.
Suppose monetary policy is implemented using "helicopter drop," with money being created and given away. The liquidity trap no longer applies even if T-bills are at the zero nominal bound. The excess demand for money may still exist because an excess demand for T-bills spilled over to an excess demand for money. But when new money is created, the central bank is no longer demanding more T-bills, creating a greater excess demand for T-bills, spilling over to a greater demand for money, matching the added quantity of money. The newly created money can accommodate the added demand to hold money.
Further, if real income has already been depressed, reducing the demand for money to match the existing quantity of money, there is no reason to assume that the additional money will not be spent on current output. If the additional money were all spent on T-bills, then this would create an excess demand for T-bills. Given that they are at the zero negative bound, it would spill over to a demand for money, and the demand for money would rise to match additional quantity of money. However, with the helicopter drop scenario, the notion that people will spend all of the new money they have been given on zero-yield T-bills seems strained.
Rather than a helicopter drop, suppose the real quantity of money increases through a proportional decrease in all prices, including the prices of resources, like labor. Does the liquidity trap apply? Again, suppose the initial problem is caused by an excess demand for T-bills. The nominal yield hits zero, and any remaining excess demand for T-bills spills over into an excess demand for money. Prices (including wages) all drop to a level so that the real quantity of money matches this additional demand. There is no longer an excess demand for money.
Why no liquidity trap? The central bank has not purchased T-bills, increasing the demand for them, causing a further spillover into the demand for money, and so matching the increase in the real quantity of money with an increase in the demand for money. On the contrary, the decrease in all prices and wages raises the real quantity of T-bills. (As well as the real tax burden of paying the debt.)
Having bonds, such as T-bills, hit the zero nominal bound can easily lead to an excess demand for money and depressed output, employment, and real income. Open market operations using the very T-bills in excess demand at a zero nominal yield will not fix the problem. However, open market operations using other assets, even government bonds, that are not at the zero nominal bound can still be effective, as can helicopter drops of money or an increase in the real quantity of money created by a lower price level.