The following set of equations describe an economy:

$C=15,000+0\mathrm{.}5(Y-T)-50,000r$

$I^P=10,000-25,000r$

$G=8,000$

$Nx=2,000$

$T=8,200$

$Y**=46,800$

a$.$ Find a numerical equation relating planned aggregate expenditure to output and to the real interest rate.

PAE $=,r,$

b$.$ At what value should the Fed set the real interest rate to eliminate any output gap? $($Hint$.$ Set output $Y$ equal to the value of potential output given above in the equation you found in part a$.$ Then solve for the real interest rate that also sets planned aggregate expenditure equal to potential output.$)$

Instructions: Enter your response as a whole number.

Real rate of interest:

$\%$