<p>Yes the 'standard story' you reference is the correct intuition here. Lower nominal interest rate encourage consumption which puts pressure on both output and prices.</p>
<p>Household can actually save and borrow in the model. However, because it is a representative household model <em>in equilibrium</em> there would not be any borrowing - the interest rate should just adjust in order to make agent's indifferent between consuming today and in the future - but the intertemporal consumption choice still depends on <span class="math-container">$i$</span> with higher <span class="math-container">$i$</span> incentivizing  future consumption at the expense of the present one.</p>
<p>Also, even if in the description provided in question this is not shown the Euler equation has its micro foundation in model where households are allowed to save and borrow - otherwise the interest rate would not be part of the Euler equation in the first place, and I am not sure how it would even be modeled if there would be no link between present and future. For example, see the treatments of IS-LM models in lets say Woodford (2003). Interest and Prices. You will see that in dynamic models the Euler equation is typically derived from model of household behavior where households have access to saving/borrowing.</p>

CC BY-SA 4.0
2020-11-09T12:46:05.030Z
I understand that the lack of borrowing is an equilibrium outcome, and that if $i$ was set in the market it would adjust. I'm more interested in the case where $i$ is a policy instrument set by the central bank.
John
CC BY-SA 4.0
2020-11-09T12:47:23.690Z
@John but that was exactly my point. In equilibrium where $i$ would freely adjust there would be no borrowing but when monetary policy changes $i$ then you can have intertemporal substitutions of consumption
1muflon1
CC BY-SA 4.0
2020-11-09T12:49:30.017Z
If all the agents are identical, as in a representative agent model, how does that happen? At the new $i$, every agent wants to borrow, which means no one can. Who do they borrow from?
John
CC BY-SA 4.0
2020-11-09T12:55:01.473Z
@John it is not necessarily through borrowing but they can also do it through saving. For example, in typical micro founded household choice model agents will have access to bonds/assets at some rate. If interest rates increase they can all simply decide to save more  to make consumption in future period larger because they get more utility that way. What exactly would happen depends on how exactly you model the microfoundations. If you show me the household optimization problem I can give you more detail but there must be some access to to saving
1muflon1
CC BY-SA 4.0
2020-11-09T12:55:37.143Z
or saving/borrowing because otherwise $i$ should not be in the Euler equation. If interest  lower interest rate means that people save less and hence consume more in current period that still expands output. However, in the above you do not explicitly show what is happening but we can infer something like that must be happening depending on your Euler equation. For example, have a look at representative household consumption models in Romer Advanced Macroeconomics
1muflon1
CC BY-SA 4.0
2020-11-09T13:05:41.357Z
My understanding is that in the standard set up, agents borrow and save by trading financial assets. In the most basic set-up, they trade risk-free bonds (at least in money-less models). That bond market has to clear, so the agents can't just all decide to save/borrow more - there would be no one for them to buy/sell the bonds from/to. If the central bank sets the interest rate, some other mechanism must increase consumption and make the bond market clear. I don't have access to a copy, but I'm fairly sure this is the setup in the early chapters of Woodford (2003).
John
CC BY-SA 4.0
2020-11-09T13:06:07.130Z
See also e.g. <https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxnYXV0aWVnZ2VydHNzb258Z3g6NGJjODIyMjRiNjllYTY5Yw>
John
CC BY-SA 4.0
2020-11-09T13:31:15.583Z
@John in the model you linked to in the comments the bond market is created by the government. In that case representative households can all decide to save at the same time as long as government is willing to borrow. So if your model builds on microfoundations in that link even representative agents can still borrow and save. Also, there are life-cycle consumption models with representative agents where saving/borrowing is possible see the sources I provided above.
1muflon1
CC BY-SA 4.0
2020-11-10T02:33:29.327Z
Ok, I have clarified that individuals borrow and save via trading government bonds, and that the decrease in $i$ is not accompanied by any change in the net supply of government bonds. If it was accompanied by reduced government borrowing, that would imply either an increase in taxes or a reduction in government spending, which would offset the monetary policy change in the first place.
John
CC BY-SA 4.0
2020-11-10T02:47:52.183Z
@John but that’s not possible based on the equations in that document you linked. Just have a look at the FOCs of the consumer, especially equation 6. The bonds here are consumer choice variable - consumers decide how much they purchase of them. If the number of bonds would be fixed as you say then the FOC given by equation 6 is simply wrong. Government must respect its inter-temporal budget but it can borrow in first period and tax in the second. If you would fix the number of bonds then you would not get the Euler equation with interest rate anymore.
1muflon1
CC BY-SA 4.0
2020-11-10T06:39:28.350Z
I don't think that is correct at all. For a given $i$, the Euler equation determines the *demand* for bonds. But the government fixes the *supply* of bonds. In the link you refer to, the Euler equation in (7) is derived before any fiscal policy is specified at all. If $i$ is decreased, the Euler equation must continue to hold no matter what the government does with the supply of bonds (as long as the intertemporal govt budget holds). It is possible to derive the equations even if the government balances its budget in each period and there is no net supply of bonds.
John
CC BY-SA 4.0
2020-11-10T06:45:10.677Z
It seems like you're saying that if $i$ falls, that means consumers will borrow from the government to increase their consumption. For the government to lend to consumers, it would have to increases taxes, which would reduce consumption. The Euler equation would still not be satisfied.
John
CC BY-SA 4.0
2020-11-10T07:16:57.820Z
I'm late to the party, I know. The intuition here is simple. Interest rates operate on intertemporal choice. Higher rates mean a higher return on savings. So, higher rates encourage saving. Reducing rates reduces the incentive to save. The tradeoff between spending today and saving in order to consume a bit more tomorrow is driven by the consumption Euler equation. Interest doesn't adjust to satisfy this equation. Intemporal choices do.
123
CC BY-SA 4.0
2020-11-10T07:19:52.713Z
Also, I want to highlight the absurdity of basing a model of aggregate dynamics on the choices of a representative household. We are studying market activity with a model that nullifies the necessity of a market to coordinate behavior. It is absurd. Thankfully, the profession is finally moving away from the absolute absurdity of rational expectations. Such a dumb assumption.
123
CC BY-SA 4.0
2020-11-10T10:09:54.660Z
@John 1. Neither document says that once consumer chooses optimal C  the b is fixed - if that would be true then this would have to be turned into two stage game. -you can also see that nowhere in the model c_t and c_t+1 are fixed so they can change. 2. fiscal spending might be fixed but Gov still can chose how to fund it. Government does not need to run balanced budget in every single period completely valid example is where all taxes are only in period 2 and all borrowing in period 1.
1muflon1
CC BY-SA 4.0
2020-11-09T11:41:08.997Z
2020-11-09T11:54:40.993Z
2020-11-09T11:54:40.993Z
1muflon1
<p>I am trying to understand the intuition behind the effect of an interest rate change in a simple New Keynesian model (or really in any sticky price model).</p>
<p>As a simple example, I take (roughly) the model in <a href="https://docs.google.com/viewer?a=v&amp;pid=sites&amp;srcid=ZGVmYXVsdGRvbWFpbnxnYXV0aWVnZ2VydHNzb258Z3g6MTA1YTg1NTljNjE4ZjJhNg" rel="nofollow noreferrer">https://docs.google.com/viewer?a=v&amp;pid=sites&amp;srcid=ZGVmYXVsdGRvbWFpbnxnYXV0aWVnZ2VydHNzb258Z3g6MTA1YTg1NTljNjE4ZjJhNg</a>. In this model, a fraction <span class="math-container">$\gamma$</span> of firms sets their price optimally for that period, and the remaining firms index their price to last period. I have the following (log-linearised) equations:</p>
<ol>
<li>Euler equation: <span class="math-container">$\hat{c}_t = E_t\hat{c}_{t+1} - \frac{1}{\sigma}\left(\hat{i}_t - E_t\hat{\pi}_{t+1}\right)$</span></li>
<li>Labour supply: <span class="math-container">$\hat{n}_t =\frac{1}{\phi}\left(\hat{w}_t - \hat{p}_t - \sigma \hat{c}_t\right)$</span></li>
<li>Optimal pricing: <span class="math-container">$\hat{p}_t^*-\hat{p}_t = \left(\hat{w}_t-\hat{p}_t\right) - \hat{a}_t$</span></li>
<li>Inflation: <span class="math-container">$\hat{\pi}_t = \frac{\gamma}{1-\gamma}\left(\hat{p}_t^* - \hat{p}_t\right)$</span></li>
<li>Output: <span class="math-container">$\hat{y}_t = \hat{a}_t + \hat{n}_t$</span></li>
<li>Goods market clearing: <span class="math-container">$\hat{y}_t = \hat{c}_t$</span></li>
</ol>
<p>There is some redundancy because I am looking for intuition. The solution to this system is
<span class="math-container">$$
x_t = E_t x_{t+1} - \frac{1}{\sigma}\left(\hat{i}_t - E_t\hat{\pi}_{t+1} - r_t^n\right) \\
\hat{\pi}_t = \kappa x_t
$$</span>
where <span class="math-container">$x_t = \hat{y}_t - \hat{y}_t^n$</span> is the output gap, <span class="math-container">$\hat{y}_t^n = \frac{1+\phi}{\sigma + \phi}\hat{a}_t$</span> is the natural level of output, <span class="math-container">$r_t^n = -\sigma\left(\hat{y}_t - E_t\hat{y}_t^n\right)$</span> is the natural rate of interest, and <span class="math-container">$\kappa = \frac{\gamma}{1-\gamma}\left(\sigma + \phi\right)$</span>.</p>
<p>Now suppose the central bank reduces the nominal interest rate. Assume also that there is no change to the net supply of government bonds (which are the vehicle for borrowing and saving). I know that the outcome will be a positive output gap and increased inflation. I am trying to understand how that occurs.</p>
<p>The standard story is that there is consumption demand by the Euler equation, so firms have to increase output, which raises their costs. The firms that can do so will raise their prices. As a consequence, there is inflation and a positive output gap.</p>
<p>If this is the correct intuition, where does the increased demand faced by firms actually come from? Individuals do want to consume more (Euler eq, 1), but they cannot all borrow and they are already at their optimal intratemporal margin (labour supply, 2). Individuals can't actually consume more until their incomes/output is higher. They won't work more unless real wages rise. Firms won't increase output, wages, or prices unless there is more demand. What is the mechanism?</p>
<p>Thanks!</p>
<p>Edit - I clarified that individuals borrow and save via trading in government bonds, and that the net supply of government bonds is held constant.</p>

<macroeconomics><inflation><new-keynesian-economics>
Intuition for effect of interest rate changes in New Keynesian model
John
# Intuition for effect of interest rate changes in New Keynesian model

(Asked by: username_0 on 2020-11-09)

I am trying to understand the intuition behind the effect of an interest rate change in a simple New Keynesian model (or really in any sticky price model).

As a simple example, I take (roughly) the model in <https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxnYXV0aWVnZ2VydHNzb258Z3g6MTA1YTg1NTljNjE4ZjJhNg>. In this model, a fraction $\gamma$ of firms sets their price optimally for that period, and the remaining firms index their price to last period. I have the following (log-linearised) equations:

1. Euler equation: $\hat{c}\_t = E\_t\hat{c}\_{t+1} - \frac{1}{\sigma}\left(\hat{i}\_t - E\_t\hat{\pi}\_{t+1}\right)$
2. Labour supply: $\hat{n}\_t =\frac{1}{\phi}\left(\hat{w}\_t - \hat{p}\_t - \sigma \hat{c}\_t\right)$
3. Optimal pricing: $\hat{p}\_t^\*-\hat{p}\_t = \left(\hat{w}\_t-\hat{p}\_t\right) - \hat{a}\_t$
4. Inflation: $\hat{\pi}\_t = \frac{\gamma}{1-\gamma}\left(\hat{p}\_t^\* - \hat{p}\_t\right)$
5. Output: $\hat{y}\_t = \hat{a}\_t + \hat{n}\_t$
6. Goods market clearing: $\hat{y}\_t = \hat{c}\_t$

There is some redundancy because I am looking for intuition. The solution to this system is
$$
x\_t = E\_t x\_{t+1} - \frac{1}{\sigma}\left(\hat{i}\_t - E\_t\hat{\pi}\_{t+1} - r\_t^n\right) \\
\hat{\pi}\_t = \kappa x\_t
$$
where $x\_t = \hat{y}\_t - \hat{y}\_t^n$ is the output gap, $\hat{y}\_t^n = \frac{1+\phi}{\sigma + \phi}\hat{a}\_t$ is the natural level of output, $r\_t^n = -\sigma\left(\hat{y}\_t - E\_t\hat{y}\_t^n\right)$ is the natural rate of interest, and $\kappa = \frac{\gamma}{1-\gamma}\left(\sigma + \phi\right)$.

Now suppose the central bank reduces the nominal interest rate. Assume also that there is no change to the net supply of government bonds (which are the vehicle for borrowing and saving). I know that the outcome will be a positive output gap and increased inflation. I am trying to understand how that occurs.

The standard story is that there is consumption demand by the Euler equation, so firms have to increase output, which raises their costs. The firms that can do so will raise their prices. As a consequence, there is inflation and a positive output gap.

If this is the correct intuition, where does the increased demand faced by firms actually come from? Individuals do want to consume more (Euler eq, 1), but they cannot all borrow and they are already at their optimal intratemporal margin (labour supply, 2). Individuals can't actually consume more until their incomes/output is higher. They won't work more unless real wages rise. Firms won't increase output, wages, or prices unless there is more demand. What is the mechanism?

Thanks!

Edit - I clarified that individuals borrow and save via trading in government bonds, and that the net supply of government bonds is held constant.

**Tags:** macroeconomics, inflation, new-keynesian-economics

---

## 1 answers

### Answer by username_1 on 2020-11-09. Score: 2

Yes the 'standard story' you reference is the correct intuition here. Lower nominal interest rate encourage consumption which puts pressure on both output and prices.

Household can actually save and borrow in the model. However, because it is a representative household model *in equilibrium* there would not be any borrowing - the interest rate should just adjust in order to make agent's indifferent between consuming today and in the future - but the intertemporal consumption choice still depends on $i$ with higher $i$ incentivizing future consumption at the expense of the present one.

Also, even if in the description provided in question this is not shown the Euler equation has its micro foundation in model where households are allowed to save and borrow - otherwise the interest rate would not be part of the Euler equation in the first place, and I am not sure how it would even be modeled if there would be no link between present and future. For example, see the treatments of IS-LM models in lets say Woodford (2003). Interest and Prices. You will see that in dynamic models the Euler equation is typically derived from model of household behavior where households have access to saving/borrowing.

> **Comments:**
> **username_0**: I understand that the lack of borrowing is an equilibrium outcome, and that if $i$ was set in the market it would adjust. I'm more interested in the case where $i$ is a policy instrument set by the central bank.  
> **username_1**: @John but that was exactly my point. In equilibrium where $i$ would freely adjust there would be no borrowing but when monetary policy changes $i$ then you can have intertemporal substitutions of consumption  
> **username_0**: If all the agents are identical, as in a representative agent model, how does that happen? At the new $i$, every agent wants to borrow, which means no one can. Who do they borrow from?  
> **username_1**: @John it is not necessarily through borrowing but they can also do it through saving. For example, in typical micro founded household choice model agents will have access to bonds/assets at some rate. If interest rates increase they can all simply decide to save more  to make consumption in future period larger because they get more utility that way. What exactly would happen depends on how exactly you model the microfoundations. If you show me the household optimization problem I can give you more detail but there must be some access to to saving  
> **username_1**: or saving/borrowing because otherwise $i$ should not be in the Euler equation. If interest  lower interest rate means that people save less and hence consume more in current period that still expands output. However, in the above you do not explicitly show what is happening but we can infer something like that must be happening depending on your Euler equation. For example, have a look at representative household consumption models in Romer Advanced Macroeconomics  
> **username_0**: My understanding is that in the standard set up, agents borrow and save by trading financial assets. In the most basic set-up, they trade risk-free bonds (at least in money-less models). That bond market has to clear, so the agents can't just all decide to save/borrow more - there would be no one for them to buy/sell the bonds from/to. If the central bank sets the interest rate, some other mechanism must increase consumption and make the bond market clear. I don't have access to a copy, but I'm fairly sure this is the setup in the early chapters of Woodford (2003).  
> **username_0**: See also e.g. <https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxnYXV0aWVnZ2VydHNzb258Z3g6NGJjODIyMjRiNjllYTY5Yw>  
> **username_1**: @John in the model you linked to in the comments the bond market is created by the government. In that case representative households can all decide to save at the same time as long as government is willing to borrow. So if your model builds on microfoundations in that link even representative agents can still borrow and save. Also, there are life-cycle consumption models with representative agents where saving/borrowing is possible see the sources I provided above.  
> **username_0**: Ok, I have clarified that individuals borrow and save via trading government bonds, and that the decrease in $i$ is not accompanied by any change in the net supply of government bonds. If it was accompanied by reduced government borrowing, that would imply either an increase in taxes or a reduction in government spending, which would offset the monetary policy change in the first place.  
> **username_1**: @John but that’s not possible based on the equations in that document you linked. Just have a look at the FOCs of the consumer, especially equation 6. The bonds here are consumer choice variable - consumers decide how much they purchase of them. If the number of bonds would be fixed as you say then the FOC given by equation 6 is simply wrong. Government must respect its inter-temporal budget but it can borrow in first period and tax in the second. If you would fix the number of bonds then you would not get the Euler equation with interest rate anymore.  
> **username_0**: I don't think that is correct at all. For a given $i$, the Euler equation determines the *demand* for bonds. But the government fixes the *supply* of bonds. In the link you refer to, the Euler equation in (7) is derived before any fiscal policy is specified at all. If $i$ is decreased, the Euler equation must continue to hold no matter what the government does with the supply of bonds (as long as the intertemporal govt budget holds). It is possible to derive the equations even if the government balances its budget in each period and there is no net supply of bonds.  
> **username_0**: It seems like you're saying that if $i$ falls, that means consumers will borrow from the government to increase their consumption. For the government to lend to consumers, it would have to increases taxes, which would reduce consumption. The Euler equation would still not be satisfied.  
> **username_2**: I'm late to the party, I know. The intuition here is simple. Interest rates operate on intertemporal choice. Higher rates mean a higher return on savings. So, higher rates encourage saving. Reducing rates reduces the incentive to save. The tradeoff between spending today and saving in order to consume a bit more tomorrow is driven by the consumption Euler equation. Interest doesn't adjust to satisfy this equation. Intemporal choices do.  
> **username_2**: Also, I want to highlight the absurdity of basing a model of aggregate dynamics on the choices of a representative household. We are studying market activity with a model that nullifies the necessity of a market to coordinate behavior. It is absurd. Thankfully, the profession is finally moving away from the absolute absurdity of rational expectations. Such a dumb assumption.  
> **username_1**: @John 1. Neither document says that once consumer chooses optimal C  the b is fixed - if that would be true then this would have to be turned into two stage game. -you can also see that nowhere in the model c_t and c_t+1 are fixed so they can change. 2. fiscal spending might be fixed but Gov still can chose how to fund it. Government does not need to run balanced budget in every single period completely valid example is where all taxes are only in period 2 and all borrowing in period 1.  

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