Ok so we're given a monetary policy rule where
$$R^{CB}_t-\bar r = \bar n \tilde Y \space where \space \bar n>0 $$
and $$R_t = R^{CB}_t+\bar p $$
Where $\bar p $ is risk premium $\bar n$ is a parameter and $R^{CB}_t$ is short run interest rate.
The IS curve is of course $ \tilde Y = \bar a - b (R_t - \bar r)$
is $\bar p = 0$ is assume the IS-MP looks the same graphically except the MP curve is now determined by $R_t = R^{CB}_t+\bar p $ so rather than just being the target interest rate of the central bank it's a sum of the target rate and risk premium so if $\bar p > 0$ it would distort monetary policy. As it would require higher targets to get the desire interest rate or inflation.
Also i thought by rearranging that $R_t - \bar r=\bar m (\pi_t - \bar \pi) =\bar n \tilde Y + \bar p $
Which mean IS curve is $\tilde Y = \bar a - b(\bar n \tilde Y + \bar p )$ With mp curve
I'm a bit confused
I know that the whole point of taylor rule is to counteract shock in the economy with with fiscal stimulus hence $\bar n \tilde Y $ the risk premium is solely a distortion of the MP curve so maybe i should have $\bar p $ in the is curve?
Oh yeah i haven't consider the fishing equation which i presume calculates the MP curve $$i_t = R_t + \pi_t = r+\pi_t +m(\pi_t - \bar \pi) $$
Then if $R_t = R^{CB}_t+\bar p $ $$i_t =R^{CB}_t+\bar p + \pi_t = r+\pi_t +m(\pi_t - \bar \pi) $$
But i want to make it look like this $$i_t =R^{CB}_t+\bar p + \pi_t + \bar n \tilde Y \space \text{With an upward sloping demand curve} $$
I'M quite confused
Edit: So i got something like this but i'm not so sure i don't like that i have a $-\bar n \tilde Y$ it should be positive