Problem $3$:

Consider the following equations

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

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

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

$Ex=0$

$IM=0$

Instead of the textbook investment function, consider the following "accelerator" investment function where investment rises with short run outptut:

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

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

$2$

$(b)$ Find the government spending multiplier $(\frac{dY}{d{}_g})$ holding the real interest rate constant. How does $an$ increase $in$ the parameter $\frac{?}{\text{b}\text{a}\text{r}}(i{)}_y$ affect the multiplier? Explain briefly the mechanism.