Euler equation in Continuous time VS Discrete time I have seen the euler equation in discrete time for the baseline neoclassical growth model written as:
$$\frac{U'(c_{t+1})}{U'(c_{t})}=\frac{1}{\beta(1+r)}$$
however I have also seen the euler equation for the continous time equivalent written as:
$$\frac{U''(c(t))}{U'(c(t))}\dot{c}(t)=r-f'(k)$$
Ignoring the right hand sides of these equations (which vary based on model set up) and exclusively looking at the left hand side I see two very different formulations of the same equation.
My question is: &quot;How does the Euler equation in Continous time represent the same thing conceptually when compared to its discrete time counter part?&quot;
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 &quot;derivative&quot; of the discrete setting (when allowing the time interval to go to zero).