Problem $1.(45$ Points$).$

Consider the New Classical framework developed in class with a constant real return

$R_t-=\frac{1}{}>1$

Suppose the monetary authority sets the nominal interest rate $i_t$ in period $t$ in

response to current inflation ${}_t$ based on a Taylor rule of the form

$i_t\frac{?}{b}=ar()+*{}_t,>0.$

Suppose the parameters in $(2)$ satisfy $>1$ and

$(1$a$)$ State the Fisher equation and combine it with equations $(1)$ and $(2)$ to obtain a

function $G$:RlongrightarrowR which determines the inflation dynamics as ${}_{t+1}=G({}_t).$

Define and compute the fixed point $\frac{?}{\text{b}\text{a}\text{r}}()$ of $G.$

$[8$ Points$]$

$(1$b$)$ Use a diagram to analyze the long$-$run behavior of the inflation dynamics gen$-$

erated by $G.$ Determine the set of initial inflation rates ${}_0$ compatible with

a bounded inflation path $({}_t{)}_{t}0.$ Explain the consequences for the determi$-$

nacy$/$indeterminacy of monetary equilibrium in the Ricardian case.

$[10$ Points$]$

$(1$c$)$ Explain the ZLB $(\text{'}$zero lower bound'$)$ problem associated with the Taylor rule

$(2)$ and modify the rule appropriately to incorporate the ZLB$.$ Determine the

critical inflation rate ${}^{\text{c}\text{r}\text{i}\text{t}}$ at which the ZLB becomes binding. Derive the

inflation dynamics in the form ${}_{t+1}=H({}_t)$ where the function $H$ is linear

with slope $$ for ${}_t{}^{\text{c}\text{r}\text{i}\text{t}}$ and constant for

$[12$ Points$]$

$(1$d$)$ Compute all fixed points $($steady states$)$ of $H.$ Show that $H$ has two fixed

points under our parameter restrictions and

$>1.$

$[8$ Points under our parameter restrictions and

$>1.$

$[8$ Points$]$