A central bank $is$ tasked with stabilizing inflation and output using the following

monetary policy rule:

$i_t=+{}_t+{}_{}({}_t-{}^{*})+{}_y(Y_t-Y^{})$

Where, $i_t=$ nominal interest rate; $=$ natural interest rate $(2\%)$; ${}_t=$ current

inflation rate; ${}^{*}=$ target inflation $(2\%)$; $Y_t=$ Actual output; $Y^P=$ Potential output;

and ${}_{}=1\mathrm{.}5,{}_y=0\mathrm{.}5$ are policy parameters.

$(a)If$ inflation $is$ currently $5\%$ and output $is2\%$ below potential, calculate the

nominal interest rate set $by$ the central bank.

$(b)$ Suppose the economy $is$ hit $by$ a supply shock that raises inflation $to8\%$ while

keeping output $at2\%$ below potential. Now determine the new interest rate.

Discuss the impact $on$ aggregate demand assuming the central bank strictly

follows the monetary policy rule.

$(c)$ Graphically show how the central bank's response shifts the aggregate demand

curve. Calculate the output change $if$ the aggregate demand equation $is$:

$Y=C-(i-)$

where $C=100$;$=2,$ while $i$ &$are$ the nominal interest rate and inflation rates,

respectively.