<p>You ask about the <strong>MP</strong> curve. Let start with the IS:</p>

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

<p>Your simplified Taylor rule is:</p>

<p>$$R^{CB}_t = \bar{r} + \bar{n}\tilde{Y}_t $$</p>

<p>And the <em>effective</em> real interest rate is:</p>

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

<p>Thus, the MP is:</p>

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

<p>This is upward sloping, because $\bar{n}&gt;0$.</p>

<p>(Notice that the investment decision by firms depend on the difference between the <em>effective</em> 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 $R^{CB}_t$).</p>

<p>This result holds even before we mention the nominal interest rate. Since the relationship between real and nominal interest rate is linear and <strong>positive</strong>, 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.</p>

CC BY-SA 3.0
2017-02-22T12:36:59.630Z
2017-02-22T12:36:59.630Z
luchonacho
<p>I get it now </p>

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

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

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

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

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

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

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

<p>Hence 
$$i_t = R_t + \pi_t+R_t-\bar r$$
$$i_t = R_t + \pi_t+n \tilde Y $$</p>

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

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

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

CC BY-SA 3.0
2016-10-21T00:14:52.733Z
2016-10-22T02:35:55.460Z
2016-10-22T02:35:55.460Z
Mark
<p>Ok so we're given a monetary policy rule where</p>

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

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

<p>Where $\bar p $ is risk premium $\bar n$ is a parameter and $R^{CB}_t$ is short run interest rate.</p>

<p>The IS curve is of course $ \tilde Y = \bar a - b (R_t - \bar r)$</p>

<p>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.</p>

<p>Also i thought by rearranging that $R_t - \bar r=\bar m (\pi_t - \bar \pi) =\bar n \tilde Y + \bar p  $</p>

<p>Which mean IS curve is  $\tilde Y = \bar a - b(\bar n \tilde Y + \bar p )$
With mp curve </p>

<p>I'm a bit confused</p>

<p>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?</p>

<p>Oh yeah i haven't consider the fishing equation which i presume calculates the MP curve
$$i_t = R_t + \pi_t = r+\pi_t +m(\pi_t - \bar \pi) $$</p>

<p>Then if $R_t = R^{CB}_t+\bar p  $
$$i_t =R^{CB}_t+\bar p + \pi_t = r+\pi_t +m(\pi_t - \bar \pi) $$</p>

<p>But i want to make it look like this 
$$i_t =R^{CB}_t+\bar p + \pi_t + \bar n \tilde Y \space \text{With an upward sloping demand curve} $$</p>

<p>I'M quite confused</p>

<p><a href="https://i.sstatic.net/7kiEH.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/7kiEH.png" alt="enter image description here"></a></p>

<hr>

<p>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 </p>

<p><a href="https://i.sstatic.net/7kiEH.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/7kiEH.png" alt="enter image description here"></a></p>

<macroeconomics><monetary-policy><interest-rate><central-banking>
How would the risk premium and a simplified taylor rule change the MP curve?
Mark
# How would the risk premium and a simplified taylor rule change the MP curve?

(Asked by: username_0 on 2016-10-20)

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

[![enter image description here](https://i.sstatic.net/7kiEH.png)](https://i.sstatic.net/7kiEH.png)

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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](https://i.sstatic.net/7kiEH.png)](https://i.sstatic.net/7kiEH.png)

**Tags:** macroeconomics, monetary-policy, interest-rate, central-banking


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## 2 answers

### Answer by username_1 on 2017-02-22. Score: 1

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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### Answer by username_0 on 2016-10-21. Score: 1

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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