The word liquidity trap is somewhat ambiguous, so, for the sake of clarity, the definition I will use in the post is as follows: a liquidity trap is an extended period of time during which the nominal interest rate is roughly equal to zero.

Given this definition and the Fisher relation, it becomes clear that a liquidity trap is simply a period of deficient inflation expectations.

$$ (1) \: i_t = \rho + E_t \pi_{t+1} $$

The nominal interest rate, $ i_t $, is low because expected inflation over the next period, $ E_t \pi_{t+1} $, is low. When confronted with this situation, a central bank like the Bank of Japan, the Federal Reserve, or the Bank on England may be tempted to affect a one-off increase in the size of the monetary base and call it "quantitative easing". Unfortunately, this will have next to no effect on the price level.

Take an example economy where the central bank has complete control over nominal spending and real GDP is constant:

$$ (2) \: M_t = P_t\: y $$

By doing some algebra, we can see that expected inflation in this economy is a function of the expected size of the money supply next period and the price level this period:

$$ (3) \: E_t \pi_{t+1} = \frac{E_t M_{t+1}}{P_t\: y} - 1 $$

If the central bank sets the money supply, $ M_t $, to grow at a constant trend rate, but be subject to a bit of discretion every period so that the money supply evolves like this:

$$ (4) \: M_t = \phi M_{t-1} + v_t $$

then we can simplify expected inflation to only being a function of $ \phi $.

$$ (3a) \: E_t \pi_{t+1} = \phi - 1 $$

This shows that

*the only way***for monetary policy to increase expected inflation in this economy is to increase the***trend rate of growth*of the money supply. In other words, quantitative easing would have no effect on the nominal interest rate in this model.
Governments may also want to engage in fiscal stimulus during a liquidity trap in order to improve economic conditions (not modeled here) or to increase expected inflation. If they do this correctly, it can work.

Consider a small change to equation 2. Now real GDP consists of only government spending (having government spending and private spending would yield the same result but involve annoying amounts of algebra) which can vary through time.

$$ (2a) \: M_t = P_t\: g_t $$

Expected inflation can now be written as a function of the expected money supply, the current price level, and the expected level of government spending:

$$ (3b) \: E_t \pi_{t+1} = \frac{E_t M_{t+1}}{P_t\: E_t g_{t+1}} - 1 $$

If we add a growth rule for government spending so that government spending grows at some rate $\theta_t$ every period so that government spending evolves as such:

$$ (6)\: g_t = \theta_t\: g_{t-1} $$

then expected inflation can again be simplified to an increasing function of the money supply growth rate, $ \phi $, and a decreasing function of the government spending growth rate, $ \theta_t $.

$$ (3c) \: E_t \pi_{t+1} = \frac{\phi}{E_t \theta_{t+1}} - 1 $$

In order to increase inflation expectations, the government needs to

*reduce*the expected growth rate of government spending. You may be wondering how this is at all consistent with me saying that fiscal stimulus can cause expected inflation to increase in this model. There is a relatively simple explanation.
There are two ways for the government to reduce $ E_t \theta_{t+1} $. They can either decrease $ E_t g_{t+1} $ while holding $ g_t $ constant or that can increase $ g_t $ while holding $ E_t G_{t+1} $ constant. In this way,

*current*stimulus with the promise of*future*austerity will cause the necessary increase in expected inflation.
Of course, the government could just choose to decrease the trend rate of growth of government spending, but that would annoy all the Keynesian's too much.

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