Euler's Theorem

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?

1. And what are the motive (intuition) behind their use ?
• Thank you for noting, I have edited the question. And the question 'can they be used together' was an irrelevant one. Aug 20, 2018 at 12:30

They both relate in one way or the other to extensions of the same mathematical object, namely the Cauchy-Euler differential 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 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.