Three monetary regimes, aiming to avoid or mitigate the liquidity trap, are introduced here. They are Inflation Targeting (IT), Price Level Targeting (PLT), and Nominal GDP Targeting (NGDPT).
A brief summary is that NGDPT performs better than PT ( in terms of dealing with the liquidity trap), which in turn performs better than IT.
Before doing the analysis, we modify the model a little bit that makes the price to be “somewhat flexible”.
$$ \bar{p}_t=\frac{m_t}{y_t} $$
In that, if not in the liquidity trap, we assume \( p_t\geq \gamma \bar{p_t}\), with \(\gamma \in (0,1)\). (we previously assume \(p_t\geq \bar{p_t})\).
Therefore, our Euler equation is,
$$ u'(\hat{y})=\beta\gamma \frac{\bar{p_t}}{p_{t+1}}u'(y’) $$
Inflation Targeting
Let the inflation target be \( \frac{p_{t+1}}{p_t}=1+\pi\), then our Euler equation becomes，
$$ u'(\hat{y})=\beta \frac{1}{1+\pi}u'(y’) $$
Therefore, \( \uparrow \pi \Rightarrow \uparrow RHS \Rightarrow \uparrow LHS \Rightarrow \uparrow \hat{y}\). Increase in inflation would raise the current output.
Implication:
- The fall in current output in the crisis is less severe (as \(\uparrow \hat{y}\)).
- The economy is less likely to fall into a liquidity trap in the first place, because people know inflation in the future, so they will not likely be strucked in the liquidity trap, coz increasing demans in current time) High inflation means money loses value quickly, and thus agents are reluctant to save using cash.
Price Level Targeting
With the price level targeting, the CB aims to keep the price level on a certain path. This means if the CB fails to meet the target, it will catch up in teh later period. For example, if the price level is 100 at period \(t\), and the CB’s price level target is 2%, then the price level in period \(t+1\) should be 102. However, if the CB fails to do that in \(t+1\), then in \(t+2\) the CB should catch up and keep the price level to be 104.
In that, the CB follows \( p_{t+1}=(1+\mu)\bar{p_t}\), where \( \bar{p_t} \) is the “normal times price level”, and \(\bar{p_t}=\frac{m_t}{y_t}\).
Then, the Euler equation
$$ u'(\hat{y})=\beta \gamma \frac{\bar{p_t}}{p_{t+1}}u'(y’) $$
would become,
$$ u'(\hat{y})=\beta \gamma \frac{1}{1+\mu}u'(y’) $$
The difference between the price level target and inflation target in the Euler equation is the \( \gamma\) term. Therefore,
$$ u'(\hat{y_{IT}})=\beta \frac{1}{1+\pi}u'(y’) $$
$$ u'(\hat{y_{PLT}})=\beta \gamma \frac{1}{1+\mu}u'(y’) $$
If \(\pi=\mu\), then the RHS of the second Euler equation is smaller, and thus \(\hat{y_{PLT}} > \hat{y_{IT}}\). (easy to show in maths by assuming the isoelasicity utility function).
NGDP Targeting
With NGDPT, the CB aims to keep nominal GDP on a certain path. For example, aiming to increase NGDP by 2% per year. A failure in one period means a cathcing up in the next period.
$$\underbrace{ p_{t+1}y_{t+1}}_{nominalGDP_{t+1}}=(1+\mu)\underbrace{\bar{p_t}y_t}_{nominalGDP_t} $$
By assuming not in the liquidity trap, \(m_{t+1}=(1+\mu\)m_t\).
Then, the Euler equation
$$ u'(\hat{y})=\beta \gamma \frac{\bar{p_t}}{p_{t+1}}u'(y’) $$
becomes,
$$ u'(\hat{y})=\beta \gamma \frac{ \frac{m_t}{y_t} }{ \frac{m_{t+1}}{y’} }u'(y’) $$
$$ u'(\hat{y})=\beta \gamma \frac{ \frac{m_t}{y_t} }{ \frac{m_t(1+\mu)}{y’} }u'(y’) $$
u'(\hat{y})=\beta \gamma \frac{y’}{y_t}\frac{1}{1+\mu} u'(y’)
Since \( \frac{y’}{y_t}<1 as y'<y_t, and \gamma \in (0,1)\), so RHS is even smaller than that of the PLT Euler equation. Therefore, \(\hat{y_{NGDPT}}\) is even greater than the \(\hat{y_{PLT}}\). That implies that the cirsis would be less severe for a given \(y’\).