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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

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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

enter image description here

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

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.

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You ask about the MP curve. Let start with the IS:

$$\tilde{Y}_t = \bar{a} - \bar{b}\left(R_t - \bar{r}\right)$$

Your simplified Taylor rule is:

$$R^{CB}_t = \bar{r} + \bar{n}\tilde{Y}_t $$

And the effective real interest rate is:

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

Thus, the MP is:

$$R_t = \bar{r} + \bar{p} + \bar{n}\tilde{Y}_t $$

This is upward sloping, because $\bar{n}>0$.

(Notice that the investment decision by firms depend on the difference between the effective real interest rate and the marginal product of capital $\bar{r}$. Thus, the MP curve has to be written in terms of $R_t$ and not in terms of $R^{CB}_t$).

This result holds even before we mention the nominal interest rate. Since the relationship between real and nominal interest rate is linear and positive, the Taylor rule in terms of the nominal interest rate is also upward slopping. But that is not the MP, and it is not what you seem to be asking.

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