<p>As you already mentioned this equation can be derived from the households consumption rule.</p>
<blockquote>
<p>the equation is derived based on consumption C rather than income Y, by considering the marginal utility of comsuming right now vs saving and delaying consumption. Investment/net export is not a part of this model, so C=Y</p>
</blockquote>
<p>So lets take a look at the Euler condition for the household (where <span class="math-container">$\pi$</span> represents inflation, <span class="math-container">$\beta$</span> the discount factor and <span class="math-container">$R_t$</span> the noninal interest rate)  and remember the market clearing condition <span class="math-container">$C=Y$</span>:</p>
<p><span class="math-container">$$\begin{align}C_t^\sigma = \beta^{-1}  E \left[\frac{\pi_{t+1}}{R_t} C_{t+1}^\sigma \right]\end{align}$$</span></p>
<p>You can directely see, that if the household expects future consumption to increase, current consumption will also increase (assuming no changes in inflation and interest rates). This is called consumption smoothing. The households tries to balance consumption over different periods. A rise in nominal interest rate makes saving more attractive and thus households shifts some of the current consumption into future periods.</p>

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2021-02-01T11:06:38.687Z
So, since there are no investments, the increase in interest rate will not  affect future output negatively?
Ma Joad
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2021-02-01T11:12:08.793Z
And can we draw an IS-LM diagram in 2D? The curve is now time-dependent.
Ma Joad
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2021-02-01T19:49:25.687Z
The increase in interest rate induces the households to save more i.e. consume less in t and more in t+n. A future increase of demand (consumption) will be met by an equal increase of Y (output). So an increase in interest will have a negative effect on current output and a positive on future output.
Levittstyle
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2021-02-01T11:01:17.660Z
2021-02-01T11:01:17.660Z
Levittstyle
<p>A simple version of the new-Keynesian IS cuve is given as follows:
<span class="math-container">$$
\ln Y_{t} =\ln Y_{t+1} -\frac{1}{\theta} r_t.
$$</span>
Here <span class="math-container">$r_t$</span> is the real interest rate and <span class="math-container">$1/\theta$</span> is the cross elasticity of consuming goods at different times. On the book I am reading (<em>Advance Macroeconomics</em> by David Romer, Chapter 6), the equation is derived based on consumption <span class="math-container">$C$</span> rather than income <span class="math-container">$Y$</span>, by considering the marginal utility of comsuming right now vs saving and delaying consumption. <strong>Investment/net export is not a part of this model,</strong> so <span class="math-container">$C=Y.$</span></p>
<p>I have some doubt about this equation. The equation shows a <strong>direct</strong> relation between real interest rate <span class="math-container">$r_t$</span> and real GDP <strong>growth rate</strong> <span class="math-container">$\ln(Y_{t+1}/Y_t),$</span> which does not look quite right, as we often say that there is an inverse relation between GDP and real interest rate for an IS curve. Of course, it is the <strong>growth rate</strong> rather than the GDP <span class="math-container">$Y$</span> itself, but this direct relationship is still kind of hard to explain. <strong>Why is this the case?</strong></p>
<p>If we keep <span class="math-container">$Y_{t+1}$</span> fixed, then there is an inverse relation between <span class="math-container">$r_t$</span> and <span class="math-container">$Y_t,$</span> as expected, but it seems unrealistic to hold the future <span class="math-container">$t+1$</span> as fixed and predetermined.</p>
<p>In summary, the additional term <span class="math-container">$\ln Y_{t+1}$</span> adds in some time-dependence, which makes this model difficult to understand in terms of an IS-LM diagram, which is a diagram at a fixed time.</p>
<p>How to explain the observations I make above about the interest rates?</p>
<p><strong>Edit:</strong> It is easy to understand the above in terms of consumption <span class="math-container">$C,$</span> but it appears harder to believe <strong>when we replace <span class="math-container">$C$</span> by <span class="math-container">$Y,$</span> for the reasons above.</strong></p>

<macroeconomics><interest-rate><new-keynesian-economics><aggregate-demand>
New Keynesian IS curve: question about time dependence
Ma Joad
# New Keynesian IS curve: question about time dependence

(Asked by: username_0 on 2021-02-01)

A simple version of the new-Keynesian IS cuve is given as follows:
$$
\ln Y\_{t} =\ln Y\_{t+1} -\frac{1}{\theta} r\_t.
$$
Here $r\_t$ is the real interest rate and $1/\theta$ is the cross elasticity of consuming goods at different times. On the book I am reading (*Advance Macroeconomics* by David Romer, Chapter 6), the equation is derived based on consumption $C$ rather than income $Y$, by considering the marginal utility of comsuming right now vs saving and delaying consumption. **Investment/net export is not a part of this model,** so $C=Y.$

I have some doubt about this equation. The equation shows a **direct** relation between real interest rate $r\_t$ and real GDP **growth rate** $\ln(Y\_{t+1}/Y\_t),$ which does not look quite right, as we often say that there is an inverse relation between GDP and real interest rate for an IS curve. Of course, it is the **growth rate** rather than the GDP $Y$ itself, but this direct relationship is still kind of hard to explain. **Why is this the case?**

If we keep $Y\_{t+1}$ fixed, then there is an inverse relation between $r\_t$ and $Y\_t,$ as expected, but it seems unrealistic to hold the future $t+1$ as fixed and predetermined.

In summary, the additional term $\ln Y\_{t+1}$ adds in some time-dependence, which makes this model difficult to understand in terms of an IS-LM diagram, which is a diagram at a fixed time.

How to explain the observations I make above about the interest rates?

**Edit:** It is easy to understand the above in terms of consumption $C,$ but it appears harder to believe **when we replace $C$ by $Y,$ for the reasons above.**

**Tags:** macroeconomics, interest-rate, new-keynesian-economics, aggregate-demand

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## 1 answers

### Answer by username_1 on 2021-02-01. Score: 1

As you already mentioned this equation can be derived from the households consumption rule.

> the equation is derived based on consumption C rather than income Y, by considering the marginal utility of comsuming right now vs saving and delaying consumption. Investment/net export is not a part of this model, so C=Y

So lets take a look at the Euler condition for the household (where $\pi$ represents inflation, $\beta$ the discount factor and $R\_t$ the noninal interest rate) and remember the market clearing condition $C=Y$:

$$\begin{align}C\_t^\sigma = \beta^{-1} E \left[\frac{\pi\_{t+1}}{R\_t} C\_{t+1}^\sigma \right]\end{align}$$

You can directely see, that if the household expects future consumption to increase, current consumption will also increase (assuming no changes in inflation and interest rates). This is called consumption smoothing. The households tries to balance consumption over different periods. A rise in nominal interest rate makes saving more attractive and thus households shifts some of the current consumption into future periods.

> **Comments:**
> **username_0**: So, since there are no investments, the increase in interest rate will not  affect future output negatively?  
> **username_0**: And can we draw an IS-LM diagram in 2D? The curve is now time-dependent.  
> **username_1**: The increase in interest rate induces the households to save more i.e. consume less in t and more in t+n. A future increase of demand (consumption) will be met by an equal increase of Y (output). So an increase in interest will have a negative effect on current output and a positive on future output.  

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