[Economics] Euler equation in Continuous time VS Discrete time

Solution 1:

One way to see (intuitively) the connection between the left hand sides is to write the discrete case as: $$ \frac{u'(c(t + \tau))}{u'(c(t))}, $$ for $\tau = 1$. Now if we generalise this to a setting where $\tau$ is now a variable in $\mathbb{R}$, this becomes a function of $\tau$. Taking the derivative with respect to $\tau$ and evaluating at $\tau = 0$, gives: $$ \frac{u''(c(t))}{u'(c(t))} \dot c(t). $$ which is the left hand side for the continuous case. As such the left hand side for the continuous case can be seen as the "derivative" of the discrete setting (when allowing the time interval to go to zero).

Solution 2:

You cannot completely ignore the RHS. Starting with $$\frac{U'(c_{t+1})}{U'(c_{t})}=RHS,$$ replace $t+1$ by $t+\Delta t$ to get $$\frac{U'(c(t+\Delta t))}{U'(c(t))}=RHS_{\Delta t},$$ where $RHS_{\Delta t}$ is the modified version of $RHS$ which contains terms depending on $\Delta t$, e.g. the modified discount factor. Expanding around $c(t)$ (and neglecting higher order terms for brevity) then gives you $$\frac{U''(c(t))}{U'(c(t))}\dot{c}(t)=\frac{RHS_{\Delta t}-1}{\Delta t},$$ and letting $\Delta t\rightarrow 0$, everything should fall into place.