$(30$ marks$)$ The desired consumption $C^d,$ investment, $I^d,$ and the demand for monetary

assets, $M^d,$ for the closed economy $of$ Hilarius can $be$ represented $by$ the following equations:

$C^d=100+0\mathrm{.}6Y$

$I^d=80-1000r$

$\frac{M^d}{P}=0\mathrm{.}3Y-0\mathrm{.}1(r+{}^e)$

where $Y$ denotes $GDP,r$ denotes the real interest rate and $Pis$ the domestic price level.

Government spending $is$ given $byG=50,$ the money supply $isM=100$ and expected

inflation $is{}^e=0\mathrm{.}02.$

Initially, assume that the full employment level $of$ output $is\frac{?}{\text{b}\text{a}\text{r}}(Y)=500$

$(a)$ Derive the equation for the $IS$ and $LM$ curves for this economy, expressing each $of$ them

$asras$ a function $ofY.$

$(b)$ Derive the $AD$ curve for Hilarius.

$(c)$ Assuming that Hilarius $is$ initially $in$ general equilibrium, what are the price level and

real interest rate?

Now suppose that, due $to$ supply$-$chain disruptions, the full employment level $of$ output falls

$to\frac{?}{\text{b}\text{a}\text{r}}(Y)=400.$

$(d)$ Starting from the initial general equilibrium situation, what are the short run equilibrium

values $of$ output and the real interest rate? $(e)In$ the new general equilibrium, what are the values $of$ the price level and the real interest

rate?

$(f)$ Suppose that the central bank $of$ Hilarius tries $to$ offset the eventual decline $in$ output

$(dueto$ supply$-$chain disruptions$)by$ increasing the money supply $toM=120.$ What will

$be$ the resulting output and the price level $in$ general equilibrium?