As I've argued here, conventional money demand models suggest that the price level becomes indeterminate at the zero lower bound and monetary expansion can not do anything to change inflation. In a recent conversation with Scott Sumner, Scott pointed to Paul Krugman's 1998 paper about this issue. Krugman suggests in his paper that only current monetary expansions are useless, but commitments to larger money supplies in the future (or, as Scott would probably like me to say, commitments that the current monetary expansion will be permanent) can both alleviate the liquidity trap and raise the current price level.
So, in line with Krugman's model, let's assume that there is a representative household that maximizes the utility function
$$(1)\: U = \sum^\infty_{t=0}\beta^t\left(u(c_t)\right) $$
where $\beta$ is the household's discount factor and $u(c_t)$ is the utility that the household gains from its consumption, $c_t$, in period $t$. The household is endowed without output $y$ every period and participates in an asset market where it trades one period government bonds and government money. The household's budget constraint is
$$(2)\: M_{t-1} + (1 + i_{t-1}) B_{t-1} + P_t y = P_t c_t + B_t + M_t + T_t $$
where $M_t$ is the money supply, $B_t$ is the household's holding of government bonds, $i_t$ is the nominal interest rate that government bonds pay, $P_t$ is the price level, and $T_t$ is the lump sum tax from the government. The household also faces a cash-in-advance constraint; it must finance its consumption with government cash. This constraint takes the form
$$(3)\: M_t \geq P_t c_t $$
Notice the fact that this is an inequality constraint. The household can hold as much money as it wants, but must at minimum have enough cash on hand to pay for its consumption. The household maximizes $1$ subject to $2$ and $3$ which yields the following first order conditions:
$$(4a)\: M_t = P_t y\: \mbox{if}\: i_t > 0$$
$$(4b)\: M_t \geq P_t y\: \mbox{if}\: i_t = 0$$
$$(5)\: 1 + i_t = \frac{1}{\beta}\frac{P_{t+1}}{P_t}$$
If, like Krugman did, we assume hat next period's price level is constant, we can draw a nice diagram with $4a$, $4b$, and $5$:
The solid blue line is the curve from $5$, the dotted blue line marks the zero lower bound, and the black lines represent the money supply. Normally, the central bank is in complete control of the price level and can move it around by moving the money supply around. But, because the cash-in-advance constraint does not bind at the zero lower bound, increases in the money supply at the zero lower bound will not be immediately spent by the household. This means that, given a constant future price level, the central bank can only push the price level up until it hits the zero lower bound. After that, no amount of current monetary expansion can increase the current price level.
Of course, all that was exactly in line with Krugman. Here's where it gets interesting, though. Krugman assumes in his paper that the central bank has control of the future price level the entire time and can easily increase the future money supply to end the liquidity trap. If we drop the assumption that the cash-in-advance constraint must bind in the next period, can the monetary expansion, regardless of permanence be effective? In order to escape the liquidity trap, the central bank needs to make the household expect that the price level next period will be higher than the price level this period (this would shift the solid blue curve in the graph to the right).
I'm having a lot of trouble wrapping my head around it, but I think that everything hinges on expectations. The cash-in-advance constraint will only bind in the next period if the price level two periods ahead is expected to be higher than the price level next period and so on, ad infinitum. This means that the central bank can only exit the liquidity trap if the household subjectively expects inflation to be greater than the rate of time preference (the inverse of the discount factor subtracted by one) in the future. This is independent of the path of the money supply; not only is there no equilibrium for the price level in the static analysis at the zero lower bound, there is no equilibrium for the entire path of the price level once the zero lower bound has been reached.
The alternative is to reduce the current money supply until the cash-in-advance constraint binds once again; basically to cause a bunch of deflation now instead of in the future. The problem with this is that prices are sticky and a massive monetary contraction would cause a recession.
