<ol>
<li><p>This is not a <em>stylized fact</em>. It is a <em>result of a stylized theoretical model</em> that, if anything, assumes uncertainty away.</p>
</li>
<li><p>With uncertainty included, the &quot;Euler formula&quot; can be written (discrete time), from the point of view of the household standing at time <span class="math-container">$t$</span></p>
</li>
</ol>
<p><span class="math-container">$$u'(c_t) =\beta R_{t}\,\mathbb{E} \big[ u'(c_{t+1})\mid t\big]\implies \frac{\mathbb{E} \big[ u'(c_{t+1})\mid t\big]}{u'(c_t)} = \frac{1}{\beta R_{t}}$$</span></p>
<p>Now, how can this be described in words?</p>
<p>&quot;If returns on savings today are increased, <span class="math-container">$R_t \uparrow$</span>, I will form a consumption <em>plan</em> <span class="math-container">$\{c_t, c_{t+1}\}$</span> where the left hand ratio will be lower than before the increase in returns&quot;.</p>
<p>How can I actually make this ratio lower?</p>
<p>By decreasing my <em>current</em> consumption (leading to higher marginal utility today) and by <em>planning</em> to increase it in the future.</p>
<p>So what the formula really says, is that higher interest rates will increase savings today, with the <em>intent</em> to consume more tomorrow (which is presumably feasible due to these higher savings).</p>
<p>Meeting reality, we should check in the data</p>
<ol>
<li>Do higher interest rates increase savings?</li>
<li>Suppose they do. Do we detect this positive relation between interest rates today and consumption tomorrow?</li>
</ol>
<p>If we don't, we start thinking about what other forces left out of the core theoretical model may be important after all... and the second link the OP provides says</p>
<blockquote>
<p>Low interest rates have stimulated consumption of durable goods, but
the expansionary effect is partially dampened by households’ desire to
deleverage voluntarily.</p>
</blockquote>
<p>In English, households had debts already, and they wanted to lower their debt, something that &quot;dampened&quot; the effect predicted by the Euler equation.</p>
<p>Now, note, that that study is concerned with <em>lowering</em> interest rates, to push consumption <em>today</em>. This is the reverse effect, but comes from the same logic. &quot;If <span class="math-container">$R_t \downarrow$</span> etc.&quot;</p>

CC BY-SA 4.0
2022-08-20T21:02:31.513Z
2022-08-20T21:02:31.513Z
Alecos Papadopoulos
<p>Here is a mathematical proof of the Euler equation. Suppose the household is alive in periods <span class="math-container">$t$</span> and <span class="math-container">$t+1$</span>. We assume that the househould wants to maximise lifetime utility, which is equal to
<span class="math-container">$$u(c_{t})+\beta u(c_{t+1}),$$</span>
where the <span class="math-container">$\beta \in [0,1]$</span> term is the discount rate. The house hold has a budget constraint:
<span class="math-container">$$c_{t}+s= y_{t}, \quad c_{t+1} = R s + y_{t+1},$$</span>
where <span class="math-container">$s$</span> denotes savings, and <span class="math-container">$y_{t}$</span> denotes period <span class="math-container">$t$</span> income. We can substitute out <span class="math-container">$s$</span> to obtain the lifetime budget constraint:
<span class="math-container">$$c_{t}+R^{-1}c_{t+1} = y_{t} + R^{-1}y_{t+1}.$$</span>
Thus, the household faces the following utility maximisation problem:
<span class="math-container">\begin{align}\max_{c_{t}, c_{t+1}}u(c_{t})+\beta u(c_{t+1})\\s.t. c_{t}+R^{-1}c_{t+1} = y_{t} + R^{-1}y_{t+1}\end{align}</span>
The Lagrangian for this problem is:
<span class="math-container">$$\mathscr{L}=u(c_{t})+\beta u(c_{t+1})+\lambda (c_{t}+R^{-1}c_{t+1} - y_{t} -R^{-1}y_{t+1}).$$</span>
Assuming the utility function is `well-behaved' (increasing and concave), the solution to the maximisation problem satisfies the first-order conditions:
<span class="math-container">\begin{align}
u'(c_{t})=\lambda,\\
\beta u'(c_{t+1})=R^{-1}\lambda.
\end{align}</span>
The Euler equation comes from these first-order conditions. We simply substitute <span class="math-container">$\beta R u'(c_{t+1})= \lambda$</span> into the first to obtain:
<span class="math-container">$$u'(c_{t})=\beta R u'(c_{t+1}),$$</span>
which is equivalent to
<span class="math-container">$$\frac{u'(c_{t+1})}{u'(c_{t})}=\beta^{-1} R^{-1}.$$</span></p>
<p>The intuition isthat when <span class="math-container">$R$</span> is higher, future consumption is <span class="math-container">`</span>cheaper'; because there is a greater return on savings. Thus, the household is better off trading off some of its present consumption for future consumption.</p>

CC BY-SA 4.0
2022-08-14T00:46:21.293Z
2022-08-14T00:46:21.293Z
Louis Becker
<p><a href="https://mitsloan.mit.edu/shared/ods/documents?DocumentID=4171#:%7E:text=An%20Euler%20equation%20is%20a,variables%20along%20an%20optimal%20path" rel="nofollow noreferrer">https://mitsloan.mit.edu/shared/ods/documents?DocumentID=4171#:~:text=An%20Euler%20equation%20is%20a,variables%20along%20an%20optimal%20path</a>.</p>
<p>With respect to the intertemporal utility function for consumption delineated in the paper above, u'(c_(t+1))/u'(c_(t))=beta^(-1)*R^(-1), is there a more fundamental reasoning that goes into this equation? In other words, are we supposed to accept that lower interest rates at time t will increase consumption at time t+1, or is there a mathematical proof for this?</p>
<p>This question is in light of the fact that there was and is <a href="https://cepr.org/voxeu/columns/did-low-interest-rates-boost-households-consumption" rel="nofollow noreferrer">skepticism about this stylized fact</a>. I'm not saying it is right or wrong, but that maybe the relationship should not be a foundational bedrock of macroeconomics without more motivation.</p>

CC BY-SA 4.0
2022-08-14T07:47:39.720Z
Quoting from page 2 of your linked document: "*The mathematics was developed by Bernoulli, Euler, Lagrange and others centuries ago*" The paper also lists references for where the equation originated...
Giskard
<macroeconomics>
Intertemporal Euler Equation
Guset
# Intertemporal Euler Equation

(Asked by: username_0 on 2022-08-13)

[https://mitsloan.mit.edu/shared/ods/documents?DocumentID=4171#:~:text=An%20Euler%20equation%20is%20a,variables%20along%20an%20optimal%20path](https://mitsloan.mit.edu/shared/ods/documents?DocumentID=4171#:%7E:text=An%20Euler%20equation%20is%20a,variables%20along%20an%20optimal%20path).

With respect to the intertemporal utility function for consumption delineated in the paper above, u'(c\_(t+1))/u'(c\_(t))=beta^(-1)\*R^(-1), is there a more fundamental reasoning that goes into this equation? In other words, are we supposed to accept that lower interest rates at time t will increase consumption at time t+1, or is there a mathematical proof for this?

This question is in light of the fact that there was and is [skepticism about this stylized fact](https://cepr.org/voxeu/columns/did-low-interest-rates-boost-households-consumption). I'm not saying it is right or wrong, but that maybe the relationship should not be a foundational bedrock of macroeconomics without more motivation.

**Tags:** macroeconomics

> **Comments:**
> **username_1**: Quoting from page 2 of your linked document: "*The mathematics was developed by Bernoulli, Euler, Lagrange and others centuries ago*" The paper also lists references for where the equation originated...  

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## 2 answers

### Answer by username_2 on 2022-08-20. Score: 3

1. This is not a *stylized fact*. It is a *result of a stylized theoretical model* that, if anything, assumes uncertainty away.
2. With uncertainty included, the "Euler formula" can be written (discrete time), from the point of view of the household standing at time $t$

$$u'(c\_t) =\beta R\_{t}\,\mathbb{E} \big[ u'(c\_{t+1})\mid t\big]\implies \frac{\mathbb{E} \big[ u'(c\_{t+1})\mid t\big]}{u'(c\_t)} = \frac{1}{\beta R\_{t}}$$

Now, how can this be described in words?

"If returns on savings today are increased, $R\_t \uparrow$, I will form a consumption *plan* $\{c\_t, c\_{t+1}\}$ where the left hand ratio will be lower than before the increase in returns".

How can I actually make this ratio lower?

By decreasing my *current* consumption (leading to higher marginal utility today) and by *planning* to increase it in the future.

So what the formula really says, is that higher interest rates will increase savings today, with the *intent* to consume more tomorrow (which is presumably feasible due to these higher savings).

Meeting reality, we should check in the data

1. Do higher interest rates increase savings?
2. Suppose they do. Do we detect this positive relation between interest rates today and consumption tomorrow?

If we don't, we start thinking about what other forces left out of the core theoretical model may be important after all... and the second link the OP provides says

> Low interest rates have stimulated consumption of durable goods, but
> the expansionary effect is partially dampened by households’ desire to
> deleverage voluntarily.

In English, households had debts already, and they wanted to lower their debt, something that "dampened" the effect predicted by the Euler equation.

Now, note, that that study is concerned with *lowering* interest rates, to push consumption *today*. This is the reverse effect, but comes from the same logic. "If $R\_t \downarrow$ etc."


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### Answer by username_3 on 2022-08-14. Score: 3

Here is a mathematical proof of the Euler equation. Suppose the household is alive in periods $t$ and $t+1$. We assume that the househould wants to maximise lifetime utility, which is equal to
$$u(c\_{t})+\beta u(c\_{t+1}),$$
where the $\beta \in [0,1]$ term is the discount rate. The house hold has a budget constraint:
$$c\_{t}+s= y\_{t}, \quad c\_{t+1} = R s + y\_{t+1},$$
where $s$ denotes savings, and $y\_{t}$ denotes period $t$ income. We can substitute out $s$ to obtain the lifetime budget constraint:
$$c\_{t}+R^{-1}c\_{t+1} = y\_{t} + R^{-1}y\_{t+1}.$$
Thus, the household faces the following utility maximisation problem:
\begin{align}\max\_{c\_{t}, c\_{t+1}}u(c\_{t})+\beta u(c\_{t+1})\\s.t. c\_{t}+R^{-1}c\_{t+1} = y\_{t} + R^{-1}y\_{t+1}\end{align}
The Lagrangian for this problem is:
$$\mathscr{L}=u(c\_{t})+\beta u(c\_{t+1})+\lambda (c\_{t}+R^{-1}c\_{t+1} - y\_{t} -R^{-1}y\_{t+1}).$$
Assuming the utility function is `well-behaved' (increasing and concave), the solution to the maximisation problem satisfies the first-order conditions:
\begin{align}
u'(c\_{t})=\lambda,\\
\beta u'(c\_{t+1})=R^{-1}\lambda.
\end{align}
The Euler equation comes from these first-order conditions. We simply substitute $\beta R u'(c\_{t+1})= \lambda$ into the first to obtain:
$$u'(c\_{t})=\beta R u'(c\_{t+1}),$$
which is equivalent to
$$\frac{u'(c\_{t+1})}{u'(c\_{t})}=\beta^{-1} R^{-1}.$$

The intuition isthat when $R$ is higher, future consumption is `cheaper'; because there is a greater return on savings. Thus, the household is better off trading off some of its present consumption for future consumption.


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