$2.(30$ marks$)$ The desired consumption Cd$,$ investment, Id$,$ and the demand for monetary assets, Md$,$ for the closed economy of Hilarius can be represented by the following equations:

Cd $=100+0\mathrm{.}6$Y Id $=801000$r

Md

P $=0\mathrm{.}3$Y$0\mathrm{.}1($r$+$e$)$

where Y denotes GDP$,$ r denotes the real interest rate and P is the domestic price level. Government spending is given by G $=50,$ the money supply is M $=100$ and expected inflation is e $=0\mathrm{.}02.$

Initially, assume that the full employment level of output is Y $=500$

$($a$)$ Derive the equation for the IS and LM curves for this economy $,$ expressing each of them as r as a function of Y$.$

$($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 Y $=400.$

$($d$)$ Starting from the initial general equilibrium situation, what are the short run equilibrium values of output and the real interest rate?

$1$

$($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 $($due to supply$-$chain disruptions$)$ by increasing the money supply to M $=120.$ What will be the resulting output and the price level in general equilibrium?