Question $3(65$ marks$)$

$($Open Economy IS$-$LM and Dornbusch Model$)$: Consider the following open$-$economy IS$-$LM

model, where capital is perfectly mobile across countries.

Assumptions: $1.$ the domestic and foreign price $P_t$ and $P_t^{*}$ are constant at period t$.2.$ UIP

condition: $i_t=i_t^{*}+E_t(s_{t+1})-s_t$ holds, where $s_{t+1}=ln(S_{t+1})$ and $s_t=ln(S_t),$ and $S_t$ is an

exchange rate between a Home currency and foreign currency. $3.$ Standard IS$-$LM setting, where

IS curve is given by $Y_t=C_t(Y_t-T_t)+G_t+I_t+N{x}_t(Y_t,Y_t^{**},S_t).$

a$)(5$ marks$)$ Argue how the net export $Nx$ changes with respect to $Y_t,Y_t^{*},$ and $S_t.$ Note $Y_t^{*}$

is the GDP for the foreign country. You should provide reasons for your answer.

b$)(10$ marks$)$ By substituting the UIP condition into IS curve, obtain the modified open econ$-$

omy IS curve. Is the IS curve flatter or more sloped than the closed economy IS curve? Prove

the reason for your answer.

c$)(20$ marks$)$ The central bank conducts a contractionary monetary policy through reducing

the money supply, and $E_t{S}_{t+1}$ is fixed at $S$ even after the monetary policy. i$)$ Show how the

domestic interest rate $i_t$ changes responding to the monetary policy in a IS$-$LM graph. ii$)$

How the exchange rate $S_t$ changes? iii$)$ Do we have "under" or "over" shooting of exchange

rate dynamics? iv$)$ What would be the effect of the policy on the net exports?

d$)(20$ marks$)$ Initially, $i_t=3\%,i_t^{**}=3\%,S_t=E_t{S}_{t+1}=S,$ and the money supply was reduced

permanently by $1\%.$ Moreover, this money supply decrease leads to an increase in $i_t$ by $1\%.$

The relative PPP holds in this economy in the long$-$run. Plot the nominal exchange rate

moments upon a government's monetary policy, and explain i$)$ how $S_t$ changes in the short

and long$-$run. ii$)$ how $P_t$ changes over time?

e$)(10$ marks$)$ Suppose that even after the government's monetary policy, it wants to hold the

exchange rate $S_t$ at S$.$ Provide reasons why this is not possible citing "impossible trinity".