# A LIQUIDITY TRAP IN AN AS/AD FRAMEWORK (PART II)

Welcome back. Last time we saw three important curves:

IS: $\tilde{Y}_t = \bar{a} - \bar{b}(R_t - \bar{r})$
MP: $R_t = \bar{r} + \bar{m} (\pi_t-\bar{\pi})$
AS: $\pi_t = \pi_{t-1} + \bar{v}\tilde{Y}_t + \bar{o}$

However, this MP curve is misspecified.  In order to fix it, we need to consider two additional relationships:
ZLB: $i_t \ge 0$
Fisher: $i_t = R_t + \pi_t$

The first equation, the Zero Lower Bound (ZLB), says that nominal interest rates cannot go below zero. This relationship holds in real life, because if banks charged a negative interest rate people would not put their money in the bank. Instead, they would just hold cash in their house, or buy nonperishable goods that they expected to keep their value, like gold.

The second equation, the Fisher equation says that the nominal interest rate, $i_t$ is equal to the sum of the real interest rate and inflation. If we substitute this equation into the first, we have:
$R_t + \pi_t \ge 0 \\ R_t \ge - \pi_t$.
In words, the central bank cannot ever lower the real interest rate below the negative rate of inflation. Again, this is because the nominal interest rate can’t go below zero.

Therefore we see that the central bank must use a piecewise function to conduct monetary policy:
MP: $R_t = \bar{r} + \bar{m}(\pi_t-\bar{\pi})$  if  $\bar{r} + \bar{m}(\pi_t-\bar{\pi}) \ge -\pi_t$
$R_t = -\pi_t$   otherwise.

Plugging the MP curve into the IS curve will give us the new AD curve, which will also be a piecewise function:
AD: $\tilde{Y}_t = \bar{a} - \bar{b} \bar{m} (\pi_t - \bar{\pi})$  if  $\pi_t \ge \frac{\bar{m}\bar{\pi}-\bar{r}}{1+\bar{m}}$
$\tilde{Y}_t = \bar{a} + \bar{b} \bar{r} + \bar{b} \pi_t$  otherwise.

This is a remarkable result! If inflation falls low enough, so that the policy rule gets stuck at the lower bound, then the slope of the AD curve switches sign, and turns positive. Looking at this result graphically, we have:

Next time I will show how demand shocks (changes in $\bar{a}$) will shift the AD curve.  First, I will do a simple example using the AD curve without the ZLB to show what happens during “normal” recessions.  Next, I will re-introduce the AD curve I just derived, and show that if the reduction in demand is large enough, the economy can get stuck in a deflationary spiral.