<p>Canonical IS-LM model is a static model (i.e. it looks at an economy from birds eye view of 'end times'). Consequently, you can't really map dynamic models like modern Taylor rule equations onto IS-LM model perfectly (you could do that with New Keynesian DSGE models that are somewhat related to IS-LM model though). Taylor rule is much more modern concept, applying it to canonical version of IS-LM is like trying to apply modern Relativity to Newtonian physics. Learning canonical IS-LM can help you understand the New Keynesian models, and lot of its intuitions still hold in modern models, so didactically there is use to learning it but you can only push it so far.</p>
<p>I will try to answer your question in context of modern New Keynesian dynamic models.</p>
<p>Modern New Keynesian IS curve looks something like this (see Woodford Interest and Prices pp 301):</p>
<p><span class="math-container">$$\hat{Y}_t = \hat{Y}_{\infty} + g_t + \chi \sigma \tilde{m}_t - \sigma \sum E_t [\hat{i}_{t+j}- \pi_{t +j+1}]  $$</span></p>
<p>and the LM curve is given by something like;</p>
<p><span class="math-container">$$\hat{m}_t = \eta_1 \hat{Y}_t - \eta_2(\hat{i}_t - \hat{i}_t^m) +\epsilon_t$$</span></p>
<p>Where <span class="math-container">$Y$</span> is output, <span class="math-container">$g$</span> government spending, <span class="math-container">$m$</span> is money demand (yes here money demand enters even in IS curve not just LM curve), <span class="math-container">$i$</span> nominal interest (<span class="math-container">$i^m$</span> is nominal interest only on money not on other loans) and <span class="math-container">$\pi$</span> inflation (difference between these two will also give you the real interest rate).</p>
<p>First you will see that in this modern LM curve you do not even have money supply. Reason for that is that modern central bank generally do not have some specific targets for changes in money supply, and are also intervening by directly changing money supply very rarely. What they typically do is to set some interest rates targets (where also the Taylor rule comes from). Second you see that money demand also enters IS more directly, since when there is higher investment there is also more demand for money.</p>
<blockquote>
<p>LM shocks are automatically offset through maintaining the interest rate through adjusting the money supply, so this wouldn't require interest rates to rise as required by the Taylor rule</p>
</blockquote>
<p>In the modern version of IS-LM both shocks to LM and IS are offset by central bank through changes to nominal interest rate not by changes to money supply while keeping nominal interest constant. Model aside, central banks can control money supply only very crudely, but they can control their rates with a surgical precision. No central bank even with modern state of the art techniques and data would be able to achieve precise target of money supply being let's say 100milion, whereas it is trivially easy for them to set their policy interest rate to lets say 2% or whatever they choose.</p>
<p>Hence there is clearly a need in a modern context for some interest rate rule.  It doesn't need to be Taylor rule, there are other interest rates rules, but realistically it has to be some interest rate rule, not a money supply rule, and money supply is also certainly not completely exogenous and given by central bank (vertical) but endogenous (upward sloping in this case) and given by the conditions in financial system (since extra money is created in financial system from the original money supply).</p>
<blockquote>
<p>IS shocks would change the equilibrium interest rate, or r∗
, but with the Taylor rule it seems like we're setting interest rates via r∗
plus some factor that depends on inflation, so this doesn't make sense either.</p>
</blockquote>
<p>This is not completely true either. The Taylor rule is in this context given by; <span class="math-container">$\hat{i}_t = \hat{r}^n_t + \pi^* + \beta_1 [ E\pi_{t+1} - \pi^*]+ \beta_2 [EY_{t+1}-Y_t^n]. $</span> Where in addition to previously mentioned variables we also have <span class="math-container">$r^n$</span> which is the natural real interest rate, and <span class="math-container">$Y^n$</span> which is the natural rate of output.</p>
<p>This rule is clearly not set up in a way where interest rates are just <span class="math-container">$r^*$</span> + some factor that depends on inflation. The interest rates are set on the real interest rate plus the intensity by which central bank cares about inflation being above its target (<span class="math-container">$\beta_1$</span>), and intensity by which they care about output gap (<span class="math-container">$\beta_2$</span>). If the inflation is exactly at the target, and there is no output gap then Taylor rule boils down just to Fisher equation, <span class="math-container">$i = r-\pi$</span> which has to hold.</p>
<p>I don't see how this does not make sense. Central banks have some mandate, in real world those mandates are typically interpreted in terms of inflation targets and output gap (this is not economically speaking how it has to be, but it is how banks do operate, hence models, even though they are simplifications of reality, should reflect that to some degree). Taylor rule captures these preferences between inflation and output gap.</p>

CC BY-SA 4.0
2024-11-03T08:15:45.447Z
I guess what I'm confused about is, since we want to bring output to potential output, and this corresponds to $r_n$, why doesn't the central bank set the interest rate to what it thinks is $r_n$ rather than $r_n$ + $a$ inflation gap + $b$ output gap
User2956
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2024-11-03T11:39:56.137Z
@User2956 two reasons, 1. CB have a non-zero inflation target, so just setting it to $r$ is no starter, at best you could argue it should set it such that $i=r+ \pi$, you could only justify $i=r$ in equilibrium if CB wants 0 inflation. 2. When economy is in a stable equilibrium without output and inflation gap, Taylor rule itself gives you $i=r+ \pi$, if there is inflation gap then CB has to react and just setting $i=r$ would not change inflation when inflation is not caused by changes in $r$ which it most of the time isn't. Similarly, output gap is not really caused by changes in $r$.
1muflon1
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2024-11-03T12:05:43.010Z
1. When the output gap is zero and interest rate = $r_n$, isn't inflation equal to expected inflation, which would be $pi_e = 2%$ rather than zero? This is what is written in Blanchard's macroeconomics. 2. Isn't $r_n$ by definition the interest rate that brings the output gap to 0 and brings inflation = expected inflation. So if there is a positive output gap and inflation exceeding expected inflation, that would imply the prevailing interest rate isn't $r_n$
User2956
CC BY-SA 4.0
2024-11-03T13:10:04.407Z
@User2956 1. But that is precisely my point. 2. The word "brings" suggests that natural interest rates causes this exogenously , but that is not really the case per se. Rather, its also the other way around when inflation and output gap are zero the real interest is equal to interest rate these are all co-determined by endogenous system. CB cannot get stable inflation just by following $i=r+\pi*$, inflation for example also depends on inflation expectations with are not dependent on $r$, so there can be inflation gap even when $i=r+\pi*$ if people expect higher inflation.
1muflon1
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2024-11-03T13:53:54.367Z
Okay. So my previous understanding, which is what is presented in my textbook (Blanchard) is when there's a positive output gap at a given interest rate, it means we're not at $r_n$. Since the central bank does not precisely know $r_n$ it may slowly increase interest rates until it finds $r_n$, at which point the output gap will close. This seems to be incorrect, and suggests that output gaps are the result of $r_n$ changing, or in other words IS shifts. Could you recommend me an article or textbook at the beginner level that explains what we're doing with monetary policy.
User2956
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2024-11-03T13:58:44.723Z
Also, as far as point 1 is concerned, you're saying inflation equals expected inflation at $i=r+\pi_e$. According to Blanchard, at $i=r_n$ inflation equals expected inflation and there's a zero output gap
User2956
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2024-11-03T19:37:44.427Z
@User2956 can you give me exact citation for blanchard, because I am pretty sure you either misquoting blanchard or blanchard makes some extra assumptions you are not telling me about, I am not saying you are doing this on purpose, but without context I am not understanding what you are asking about, please give me page and book/paper where you found this. I would be surprised if Blanchard would claim output gaps are result of changes in $r$ or that at $i=r$ inflation equals expected inflation (unless he assumes its zero, otherwise this breaks the Fisher equation)
1muflon1
CC BY-SA 4.0
2024-11-04T06:50:12.310Z
Sure, Blanchard p. 181 Macroeconomics 8e. I have taken screenshots of the entire relevant section and numbered the images in order so you can easily have a look https://imgur.com/a/4zKyO3H
User2956
CC BY-SA 4.0
2024-11-04T14:26:20.150Z
@User2956 thank you. But then you are clearly misreading Blanchard. 1. Clearly Blanchard shows on all Graphs $\pi-\bar{\pi}$ for inflation, so inflation is not zero even when we hit the x axis, when you hit zero of that graph you have inflation that is just equal to the target inflation (e.g. point A' on figure 9-2). 2. I don't see anywhere in the text anything that would imply that central bank should set $i=r$ , which would require $\bar{\pi}=0$. I am really puzzled where did you got that conclusion from
1muflon1
CC BY-SA 4.0
2024-11-04T17:20:53.930Z
Ah, I think I understood why there's confusion. I was talking about setting the real rate of interest equal to $r_n$, which is the terms in which Blanchard is describing things. Clearly, this would mean a nominal interest rate $i = r_n + \pi$. Also, for the second point, Blanchard seems to be implying that the reason there's an output gap and inflation above target is that the real interest rate is not $r_n$ and the whole section I linked talks about the central bank adjusting $r$ to bring it to $r_n$ to close the output gap and bring the economy to its medium run equilibrium
User2956
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2024-11-04T17:54:19.163Z
yes that is what I was trying to say in the comments above
1muflon1
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2024-11-05T06:39:39.417Z
Okay, so the first point is clarified. But the second point that Blanchard is making is there's an output gap and inflation above target when the real interest rate $r$ isn't 
 $r_n$ and what the central bank is attempting to return the $r$ to $r_n$. This is what seems inconsistent with Taylor's rule. Instead of setting $r = r_n$ it recommends setting it to $r_n$ + $a$ inflation gap + $b$ output gap
User2956
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2024-11-05T11:32:16.900Z
@User2956 but that is not what Taylor rule says. 1. CB’s do not set $r$, CB’s can set $i$ or manipulate money supply such that $r$ changes. 2. Im Taylor rule when there is no output gap and no inflation gap $i=\pi+r^n$ which implies that $r=r^n$
1muflon1
CC BY-SA 4.0
2024-11-05T11:42:42.620Z
But when there is an output gap and inflation gap, shouldn't the central bank set $i = r_n+\pi$. The section I linked in Blanchard suggests that output and inflation gaps are the result of the real interest rate not being equal to $r_n$, which is what the central bank is attempting to do through gradually raising interest rates
User2956
CC BY-SA 4.0
2024-11-05T12:03:07.250Z
@User2956 yes, but when there is output or inflation gap that means that $r \neq r^n$ again central bank **can't set r** it can move $i$ or money supply such that LM shifts such that $r$ shifts. In order to create these shifts, CB sets lower (higher) $i$ when output is below (above) potential, or when inflation is below (above) target.
1muflon1
CC BY-SA 4.0
2024-11-09T04:15:13.413Z
But we know that expected inflation $\pi$ = 2% and that expectations are well-anchored in developed countries. So isn't it right to then set $i = r_n+2$ to set the real interest rate to $r_n$
User2956
CC BY-SA 4.0
2024-11-09T13:05:40.870Z
@User2956 only if there is no inflation gap and no output gap. Otherwise no.
1muflon1
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2024-11-12T12:17:55.290Z
Could you please recommend an article or book at the introductory level that covers this topic well. This way I can have another look at the topic. I feel like I'm missing something here
User2956
CC BY-SA 4.0
2024-11-12T16:30:16.493Z
@User2956 I think blanchard does cover it pretty well, some alternatives would be Mankiw Macroeconomics. Papers do not look at this from introductory level. If you want more modern versions of IS-LM you can read Interest and Prices by Woodford
1muflon1
CC BY-SA 4.0
2024-11-13T09:59:08.390Z
But Blanchard's description of this is essentially that the central bank, through setting $i=r_n+\pi$ sets the real interest rate to $r_n$, which closes the output and inflation gap. You said "only if there is no inflation gap and no output gap. Otherwise no"
User2956
CC BY-SA 4.0
2024-11-13T12:01:35.047Z
@User2956 Yes because if there is inflation or output gap central bank obviously has to set higher/lower interest rate to move real rates into the direction of natural rate. On Blanchard pictures you have a static model (simplified for students) Taylor rule is dynamic model as I was explaining above. In Blanchard everything happens instantaneously but that is simplification. If you have problem with understanding simplifying abstractions then you can read the advanced books I recommend where everything is expressed in terms of dynamic processes (if you have sufficient math lvl)
1muflon1
CC BY-SA 4.0
2024-11-16T08:49:50.557Z
Oh okay, so they're essentially over/undershooting what they think is $r_n+\pi$ to move the economy there faster, and as it begins moving in that direction they bring the interest rates towards $r_n+\pi$. The advanced book you're referring to here is the Woodford one or Mankiw?
User2956
CC BY-SA 4.0
2024-11-03T02:58:11.883Z
2024-11-03T02:58:11.883Z
1muflon1
<p>In terms of the IS-LM model, what is the Taylor rule attempting to do?</p>
<p>Inflation goes above target when there's an output gap. This can happen as a result of an LM shock or IS shock.</p>
<ul>
<li>LM shocks are automatically offset through maintaining the interest rate through adjusting the money supply, so this wouldn't require interest rates to rise as required by the Taylor rule</li>
<li>IS shocks would change the equilibrium interest rate, or <span class="math-container">$r^*$</span>, but with the Taylor rule it seems like we're setting interest rates via <span class="math-container">$r^*$</span> plus some factor that depends on inflation, so this doesn't make sense either.</li>
</ul>

<macroeconomics>
Taylor rule in the context of the IS-LM model
User2956
# Taylor rule in the context of the IS-LM model

(Asked by: username_0 on 2024-11-02)

In terms of the IS-LM model, what is the Taylor rule attempting to do?

Inflation goes above target when there's an output gap. This can happen as a result of an LM shock or IS shock.

* LM shocks are automatically offset through maintaining the interest rate through adjusting the money supply, so this wouldn't require interest rates to rise as required by the Taylor rule
* IS shocks would change the equilibrium interest rate, or $r^\*$, but with the Taylor rule it seems like we're setting interest rates via $r^\*$ plus some factor that depends on inflation, so this doesn't make sense either.

**Tags:** macroeconomics


---

## 1 answers

### Answer by username_1 on 2024-11-03. Score: 0

Canonical IS-LM model is a static model (i.e. it looks at an economy from birds eye view of 'end times'). Consequently, you can't really map dynamic models like modern Taylor rule equations onto IS-LM model perfectly (you could do that with New Keynesian DSGE models that are somewhat related to IS-LM model though). Taylor rule is much more modern concept, applying it to canonical version of IS-LM is like trying to apply modern Relativity to Newtonian physics. Learning canonical IS-LM can help you understand the New Keynesian models, and lot of its intuitions still hold in modern models, so didactically there is use to learning it but you can only push it so far.

I will try to answer your question in context of modern New Keynesian dynamic models.

Modern New Keynesian IS curve looks something like this (see Woodford Interest and Prices pp 301):

$$\hat{Y}\_t = \hat{Y}\_{\infty} + g\_t + \chi \sigma \tilde{m}\_t - \sigma \sum E\_t [\hat{i}\_{t+j}- \pi\_{t +j+1}] $$

and the LM curve is given by something like;

$$\hat{m}\_t = \eta\_1 \hat{Y}\_t - \eta\_2(\hat{i}\_t - \hat{i}\_t^m) +\epsilon\_t$$

Where $Y$ is output, $g$ government spending, $m$ is money demand (yes here money demand enters even in IS curve not just LM curve), $i$ nominal interest ($i^m$ is nominal interest only on money not on other loans) and $\pi$ inflation (difference between these two will also give you the real interest rate).

First you will see that in this modern LM curve you do not even have money supply. Reason for that is that modern central bank generally do not have some specific targets for changes in money supply, and are also intervening by directly changing money supply very rarely. What they typically do is to set some interest rates targets (where also the Taylor rule comes from). Second you see that money demand also enters IS more directly, since when there is higher investment there is also more demand for money.

> LM shocks are automatically offset through maintaining the interest rate through adjusting the money supply, so this wouldn't require interest rates to rise as required by the Taylor rule

In the modern version of IS-LM both shocks to LM and IS are offset by central bank through changes to nominal interest rate not by changes to money supply while keeping nominal interest constant. Model aside, central banks can control money supply only very crudely, but they can control their rates with a surgical precision. No central bank even with modern state of the art techniques and data would be able to achieve precise target of money supply being let's say 100milion, whereas it is trivially easy for them to set their policy interest rate to lets say 2% or whatever they choose.

Hence there is clearly a need in a modern context for some interest rate rule. It doesn't need to be Taylor rule, there are other interest rates rules, but realistically it has to be some interest rate rule, not a money supply rule, and money supply is also certainly not completely exogenous and given by central bank (vertical) but endogenous (upward sloping in this case) and given by the conditions in financial system (since extra money is created in financial system from the original money supply).

> IS shocks would change the equilibrium interest rate, or r∗
> , but with the Taylor rule it seems like we're setting interest rates via r∗
> plus some factor that depends on inflation, so this doesn't make sense either.

This is not completely true either. The Taylor rule is in this context given by; $\hat{i}\_t = \hat{r}^n\_t + \pi^\* + \beta\_1 [ E\pi\_{t+1} - \pi^\*]+ \beta\_2 [EY\_{t+1}-Y\_t^n]. $ Where in addition to previously mentioned variables we also have $r^n$ which is the natural real interest rate, and $Y^n$ which is the natural rate of output.

This rule is clearly not set up in a way where interest rates are just $r^\*$ + some factor that depends on inflation. The interest rates are set on the real interest rate plus the intensity by which central bank cares about inflation being above its target ($\beta\_1$), and intensity by which they care about output gap ($\beta\_2$). If the inflation is exactly at the target, and there is no output gap then Taylor rule boils down just to Fisher equation, $i = r-\pi$ which has to hold.

I don't see how this does not make sense. Central banks have some mandate, in real world those mandates are typically interpreted in terms of inflation targets and output gap (this is not economically speaking how it has to be, but it is how banks do operate, hence models, even though they are simplifications of reality, should reflect that to some degree). Taylor rule captures these preferences between inflation and output gap.

> **Comments:**
> **username_0**: I guess what I'm confused about is, since we want to bring output to potential output, and this corresponds to $r_n$, why doesn't the central bank set the interest rate to what it thinks is $r_n$ rather than $r_n$ + $a$ inflation gap + $b$ output gap  
> **username_1**: @username_0 two reasons, 1. CB have a non-zero inflation target, so just setting it to $r$ is no starter, at best you could argue it should set it such that $i=r+ \pi$, you could only justify $i=r$ in equilibrium if CB wants 0 inflation. 2. When economy is in a stable equilibrium without output and inflation gap, Taylor rule itself gives you $i=r+ \pi$, if there is inflation gap then CB has to react and just setting $i=r$ would not change inflation when inflation is not caused by changes in $r$ which it most of the time isn't. Similarly, output gap is not really caused by changes in $r$.  
> **username_0**: 1. When the output gap is zero and interest rate = $r_n$, isn't inflation equal to expected inflation, which would be $pi_e = 2%$ rather than zero? This is what is written in Blanchard's macroeconomics. 2. Isn't $r_n$ by definition the interest rate that brings the output gap to 0 and brings inflation = expected inflation. So if there is a positive output gap and inflation exceeding expected inflation, that would imply the prevailing interest rate isn't $r_n$  
> **username_1**: @username_0 1. But that is precisely my point. 2. The word "brings" suggests that natural interest rates causes this exogenously , but that is not really the case per se. Rather, its also the other way around when inflation and output gap are zero the real interest is equal to interest rate these are all co-determined by endogenous system. CB cannot get stable inflation just by following $i=r+\pi*$, inflation for example also depends on inflation expectations with are not dependent on $r$, so there can be inflation gap even when $i=r+\pi*$ if people expect higher inflation.  
> **username_0**: Okay. So my previous understanding, which is what is presented in my textbook (Blanchard) is when there's a positive output gap at a given interest rate, it means we're not at $r_n$. Since the central bank does not precisely know $r_n$ it may slowly increase interest rates until it finds $r_n$, at which point the output gap will close. This seems to be incorrect, and suggests that output gaps are the result of $r_n$ changing, or in other words IS shifts. Could you recommend me an article or textbook at the beginner level that explains what we're doing with monetary policy.  
> **username_0**: Also, as far as point 1 is concerned, you're saying inflation equals expected inflation at $i=r+\pi_e$. According to Blanchard, at $i=r_n$ inflation equals expected inflation and there's a zero output gap  
> **username_1**: @username_0 can you give me exact citation for blanchard, because I am pretty sure you either misquoting blanchard or blanchard makes some extra assumptions you are not telling me about, I am not saying you are doing this on purpose, but without context I am not understanding what you are asking about, please give me page and book/paper where you found this. I would be surprised if Blanchard would claim output gaps are result of changes in $r$ or that at $i=r$ inflation equals expected inflation (unless he assumes its zero, otherwise this breaks the Fisher equation)  
> **username_0**: Sure, Blanchard p. 181 Macroeconomics 8e. I have taken screenshots of the entire relevant section and numbered the images in order so you can easily have a look https://imgur.com/a/4zKyO3H  
> **username_1**: @username_0 thank you. But then you are clearly misreading Blanchard. 1. Clearly Blanchard shows on all Graphs $\pi-\bar{\pi}$ for inflation, so inflation is not zero even when we hit the x axis, when you hit zero of that graph you have inflation that is just equal to the target inflation (e.g. point A' on figure 9-2). 2. I don't see anywhere in the text anything that would imply that central bank should set $i=r$ , which would require $\bar{\pi}=0$. I am really puzzled where did you got that conclusion from  
> **username_0**: Ah, I think I understood why there's confusion. I was talking about setting the real rate of interest equal to $r_n$, which is the terms in which Blanchard is describing things. Clearly, this would mean a nominal interest rate $i = r_n + \pi$. Also, for the second point, Blanchard seems to be implying that the reason there's an output gap and inflation above target is that the real interest rate is not $r_n$ and the whole section I linked talks about the central bank adjusting $r$ to bring it to $r_n$ to close the output gap and bring the economy to its medium run equilibrium  
> **username_1**: yes that is what I was trying to say in the comments above  
> **username_0**: Okay, so the first point is clarified. But the second point that Blanchard is making is there's an output gap and inflation above target when the real interest rate $r$ isn't   $r_n$ and what the central bank is attempting to return the $r$ to $r_n$. This is what seems inconsistent with Taylor's rule. Instead of setting $r = r_n$ it recommends setting it to $r_n$ + $a$ inflation gap + $b$ output gap  
> **username_1**: @username_0 but that is not what Taylor rule says. 1. CB’s do not set $r$, CB’s can set $i$ or manipulate money supply such that $r$ changes. 2. Im Taylor rule when there is no output gap and no inflation gap $i=\pi+r^n$ which implies that $r=r^n$  
> **username_0**: But when there is an output gap and inflation gap, shouldn't the central bank set $i = r_n+\pi$. The section I linked in Blanchard suggests that output and inflation gaps are the result of the real interest rate not being equal to $r_n$, which is what the central bank is attempting to do through gradually raising interest rates  
> **username_1**: @username_0 yes, but when there is output or inflation gap that means that $r \neq r^n$ again central bank **can't set r** it can move $i$ or money supply such that LM shifts such that $r$ shifts. In order to create these shifts, CB sets lower (higher) $i$ when output is below (above) potential, or when inflation is below (above) target.  
> **username_0**: But we know that expected inflation $\pi$ = 2% and that expectations are well-anchored in developed countries. So isn't it right to then set $i = r_n+2$ to set the real interest rate to $r_n$  
> **username_1**: @username_0 only if there is no inflation gap and no output gap. Otherwise no.  
> **username_0**: Could you please recommend an article or book at the introductory level that covers this topic well. This way I can have another look at the topic. I feel like I'm missing something here  
> **username_1**: @username_0 I think blanchard does cover it pretty well, some alternatives would be Mankiw Macroeconomics. Papers do not look at this from introductory level. If you want more modern versions of IS-LM you can read Interest and Prices by Woodford  
> **username_0**: But Blanchard's description of this is essentially that the central bank, through setting $i=r_n+\pi$ sets the real interest rate to $r_n$, which closes the output and inflation gap. You said "only if there is no inflation gap and no output gap. Otherwise no"  
> **username_1**: @username_0 Yes because if there is inflation or output gap central bank obviously has to set higher/lower interest rate to move real rates into the direction of natural rate. On Blanchard pictures you have a static model (simplified for students) Taylor rule is dynamic model as I was explaining above. In Blanchard everything happens instantaneously but that is simplification. If you have problem with understanding simplifying abstractions then you can read the advanced books I recommend where everything is expressed in terms of dynamic processes (if you have sufficient math lvl)  
> **username_0**: Oh okay, so they're essentially over/undershooting what they think is $r_n+\pi$ to move the economy there faster, and as it begins moving in that direction they bring the interest rates towards $r_n+\pi$. The advanced book you're referring to here is the Woodford one or Mankiw?  

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