How would the risk premium and a simplified taylor rule change the MP curve? Ok so we're given a monetary policy rule where

$$R^{CB}_t-\bar r = \bar n \tilde Y \space  where \space \bar n&gt;0  $$ 

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 $\bar p &gt; 0$ 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
I get it now 

Under traditional monetary policy 
MP Curve is set by the central bank and follow the fisher equation
$$i_t = R_t + \pi_t$$

But traditionally the taylor asks that when the central bank considers monetary policy that they consider short run output and inflationary gap.
$$i_t = R_t + \pi_t+\bar m(\pi_t-\bar \pi)+\bar n \tilde Y_t$$
Where the monetary policy rule is
$$R_t-\bar r = m(\pi_t-\bar \pi)+n \tilde Y_t  $$
Now 
$$R^{CB}_t-\bar r = \bar n \tilde Y \space  where \space \bar n&gt;0  $$ 

and 
$$R_t = R^{CB}_t+\bar p $$

I assume that in the original taylor rule
$$R_t-\bar r = m(\pi_t-\bar \pi)+n \tilde Y_t  $$

Hence in the original taylor rule
$$i_t = R_t + \pi_t+\bar m(\pi_t-\bar \pi)+\bar n \tilde Y_t$$
$$i_t = R_t + \pi_t+R_t-\bar r$$

But in this simplified taylor rule $R^{CB}_t-\bar r=n $
$$R^{CB}_t-\bar r = \bar n \tilde Y  $$

To which i assume also that 
$$R_t-\bar r = \bar n \tilde Y = m(\pi_t-\bar \pi) $$

Hence 
$$i_t = R_t + \pi_t+R_t-\bar r$$
$$i_t = R_t + \pi_t+n \tilde Y $$

If $R_t = R^{CB}_t+\bar p$

$$i_t = R_t + \pi_t+n \tilde Y $$ $$i_t =R^{CB}_t+\bar p + \pi_t+n \tilde Y $$

Hence the $ n \tilde Y $ component of the simplified taylor rule makes the MP curve upward sloping.