Problem $1$:

Consider the following GDP equation, consumption function, exports function. imports function and investment function and use them to answer the following questions

$Y_t=C_t+I_t+G_t+E{x}_t-I{M}_t$

$\frac{I_t}{\frac{?}{\text{b}\text{a}\text{r}}(Y{)}_t}={}_i\frac{-}{b}ar(b)(R_t-(\frac{?}{\text{b}\text{a}\text{r}}(r)))$;$\frac{,}{b}ar(b)>0$

$\frac{C_t}{\frac{?}{\text{b}\text{a}\text{r}}(Y{)}_t}={}_c+\stackrel{}{x}tilde(Y{)}_t$;

$G=\frac{{}_g}{b}ar(Y{)}_t$

$Ex=0$

$IM=0$

$($a$)$ Derive the IS curve as a relation between short run output tilde$(Y{)}_t$ and the real interest rate gap $R_t\frac{-}{b}ar(r).$

$($b$)$ Find the government spending multiplier $(\frac{d(tilde(Y))}{d{}_g})$ holding the real interest rate constant.

$($c$)$ Is the multiplier derived in b$)$ greater than one or smaller than one? Briefly provide intuition for your answer along with mathematical reasoning.

$($d$)$ Consider the following MP curve where monetary policy and real interest rates now depends on the short$-$run output.

$R_t\frac{-}{b}ar(r)\frac{?}{b}=ar(n)tilde(Y{)}_t$

Derive the AD curve.

$($e$)$ Find the government spending multiplier $(\frac{d(tilde(Y))}{d{}_g}).$ If $\frac{?}{\text{b}\text{a}\text{r}}(n)$ is positive, is the multiplier higher or lower than in part c$).$

$1$

$($f$)$ Suppose that in the Great Recession, monetary policy is constrained and cannot respond to fiscal policy. In other words, replace the MP curve by

$R_t=0$

Is that an argument in favor or against the use of fiscal policy to stimulate the economy?