<p>They both relate in one way or the other to extensions of the same mathematical object, namely the <a href="https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation" rel="noreferrer">Cauchy-Euler differential equation</a>, that has the form</p>

<p>$$a_{n} x^n f^{(n)}(x) + a_{n-1} x^{n-1} f^{(n-1)}(x) + \cdots + a_0 f(x) = 0 \tag{1}$$</p>

<p>where $f^{(n)}$ denotes the $n$-th derivative of the function $f$. Its 1st-order version is</p>

<p>$$a_{1} x f^{(1)}(x) +  a_0 f(x) = 0\tag{2}$$</p>

<p><strong>A. "Eulers theorem for homogeneous functions".</strong></p>

<p>Consider the 1st-order Cauchy-Euler equation, in a <em>multivariate</em> extension:</p>

<p>$$ a_1\mathbf x'\cdot  \nabla f(\mathbf x) + a_0f(\mathbf x) = 0 \tag{3}$$</p>

<p><a href="https://en.wikipedia.org/wiki/Homogeneous_function#Positive_homogeneity" rel="noreferrer">Euler's theorem for homogeneous functions</a> says essentially that <em>if</em> a multivariate function is homogeneous of degree $r$, then it satisfies the multivariate first-order Cauchy-Euler equation, with $a_1 = -1, a_0 =r$.</p>

<p><strong>B. "Euler's equation in consumption."</strong><br>
Consider a univariate but <em>non-linear</em> extension for the 1st-order Cauchy-Euler differential equation:</p>

<p>$$a_{1}g(x) f^{(1)}(x) +  a_0 f(x) = 0 \tag{4}$$</p>

<p>Now set $x=t$ (i.e. equal to time), and $f(x) = C(t)$ (say, per capita consumption). Then $(4)$ is written</p>

<p>$$a_{1}g(t) \frac{dC(t)}{dt} +   a_0 C(t) = 0 $$</p>

<p>$$\implies \dot C = -\frac {a_0}{a_{1}g(t)}C(t) \tag{5}$$</p>

<p>What we realize is that the solution to the intertemporal utility maximization problem is described by an equation of the form of $(5)$, with suitable values and forms  for $a_0, a_1, g(t)$. For example, in the Ramsey model, we could set $a_0=-1, a_1 = 1, g(t) = (r(t)-\rho)^{-1}$</p>

<p>The discrete analogue (i.e. for discrete time) of the Cauchy-Euler non-linear extension is </p>

<p>$$a_{1}g(t) \Delta C_{t+1} +   a_0 C_t = 0 $$</p>

<p>$$\implies C_{t+1} = \left (1-\frac {a_0}{a_1 g(t)}\right) C_t$$</p>

<p>and again we see that the solution to the intertemporal utility problem in discrete time satisfies a Cauchy-Euler 1st order equation.  </p>

<p>Why the "Cauchy" name prefix has dropped out in the economics literature, I don't know.</p>

CC BY-SA 4.0
2018-08-20T12:08:40.037Z
2018-08-20T18:00:11.997Z
2018-08-20T18:00:11.997Z
Alecos Papadopoulos
<p>Can anyone give me connection and intuition behind each of the following euler's equation-</p>

<p>Euler's equation in production function represents that total factor payment equals degree of homogeneity times output, given factors are paid according to marginal productivity. 
Refer to this - <a href="http://www.applet-magic.com/euler.htm" rel="nofollow noreferrer">http://www.applet-magic.com/euler.htm</a></p>

<p>And Euler's equation in consumption which seeks to find optimal consumption path where marginal utility lost due to little less consumption today matches the expected utility gain from higher future consumption.
Refer to this - <a href="https://www.quora.com/What-is-the-Euler-condition" rel="nofollow noreferrer">https://www.quora.com/What-is-the-Euler-condition</a></p>

<p>1.Can someone please explain is there any connection between the two? </p>

<ol start="2">
<li>And what are the motive (intuition) behind their use ?</li>
</ol>

CC BY-SA 4.0
2018-08-20T12:30:49.643Z
Thank you for noting, I have edited the question. And the question 'can they be used together' was an irrelevant one.
Elina Gilbert
<macroeconomics><mathematical-economics><production-function><representative-agent>
Euler's Theorem
Elina Gilbert
# Euler's Theorem

(Asked by: username_0 on 2018-08-19)

Can anyone give me connection and intuition behind each of the following euler's equation-

Euler's equation in production function represents that total factor payment equals degree of homogeneity times output, given factors are paid according to marginal productivity.
Refer to this - <http://www.applet-magic.com/euler.htm>

And Euler's equation in consumption which seeks to find optimal consumption path where marginal utility lost due to little less consumption today matches the expected utility gain from higher future consumption.
Refer to this - <https://www.quora.com/What-is-the-Euler-condition>

1.Can someone please explain is there any connection between the two?

2. And what are the motive (intuition) behind their use ?

**Tags:** macroeconomics, mathematical-economics, production-function, representative-agent
> **Comments:**
> **username_0**: Thank you for noting, I have edited the question. And the question 'can they be used together' was an irrelevant one.  

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## 1 answers

### Answer by username_1 on 2018-08-20. Score: 7 | Accepted answer

They both relate in one way or the other to extensions of the same mathematical object, namely the [Cauchy-Euler differential equation](https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation), that has the form

$$a\_{n} x^n f^{(n)}(x) + a\_{n-1} x^{n-1} f^{(n-1)}(x) + \cdots + a\_0 f(x) = 0 \tag{1}$$

where $f^{(n)}$ denotes the $n$-th derivative of the function $f$. Its 1st-order version is

$$a\_{1} x f^{(1)}(x) + a\_0 f(x) = 0\tag{2}$$

**A. "Eulers theorem for homogeneous functions".**

Consider the 1st-order Cauchy-Euler equation, in a *multivariate* extension:

$$ a\_1\mathbf x'\cdot \nabla f(\mathbf x) + a\_0f(\mathbf x) = 0 \tag{3}$$

[Euler's theorem for homogeneous functions](https://en.wikipedia.org/wiki/Homogeneous_function#Positive_homogeneity) says essentially that *if* a multivariate function is homogeneous of degree $r$, then it satisfies the multivariate first-order Cauchy-Euler equation, with $a\_1 = -1, a\_0 =r$.

**B. "Euler's equation in consumption."**  
Consider a univariate but *non-linear* extension for the 1st-order Cauchy-Euler differential equation:

$$a\_{1}g(x) f^{(1)}(x) + a\_0 f(x) = 0 \tag{4}$$

Now set $x=t$ (i.e. equal to time), and $f(x) = C(t)$ (say, per capita consumption). Then $(4)$ is written

$$a\_{1}g(t) \frac{dC(t)}{dt} + a\_0 C(t) = 0 $$

$$\implies \dot C = -\frac {a\_0}{a\_{1}g(t)}C(t) \tag{5}$$

What we realize is that the solution to the intertemporal utility maximization problem is described by an equation of the form of $(5)$, with suitable values and forms for $a\_0, a\_1, g(t)$. For example, in the Ramsey model, we could set $a\_0=-1, a\_1 = 1, g(t) = (r(t)-\rho)^{-1}$

The discrete analogue (i.e. for discrete time) of the Cauchy-Euler non-linear extension is

$$a\_{1}g(t) \Delta C\_{t+1} + a\_0 C\_t = 0 $$

$$\implies C\_{t+1} = \left (1-\frac {a\_0}{a\_1 g(t)}\right) C\_t$$

and again we see that the solution to the intertemporal utility problem in discrete time satisfies a Cauchy-Euler 1st order equation.

Why the "Cauchy" name prefix has dropped out in the economics literature, I don't know.


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