Comparative dynamics in forward-looking equation I have set up and solved an optimization problem with time $t$ endogenous state variables, $\alpha_t$ and $\beta_t$ and choice variable $s_t$. After some manipulation, the first-order condition for $s_t$ is of the form:

$f(\alpha_t,\beta_t,s_t,s_{t+1})=0$

where $f(\cdot)$ is a non-linear and contains expectations over future realizations of shocks. In some specifications there is no explicit solution for $s_t$.

I want to derive testable implications from the underlying theory. In particular, I am interested in the signs of:

$\dfrac{\partial s_t}{\partial \alpha_t}$ and $\dfrac{\partial s_t}{\partial \beta_t}$.

How should I treat $s_{t+1}$ when I take the total derivative? Should I treat it as a constant, or include it? What's the rationale? If more elaboration is needed, do let me know and I'll be happy to explain in more detail the problem.

Thanks!
Try writing your equation explicitly with a conditional expectation operator:

$$ \mathbb{E}_t f(\alpha_t, \beta_t, s_t, s_{t+1}) = 0$$

Hopefully this clarifies things.  You can take the total derivative as you normally would with any function, however $s_{t+1}$ is a random variable since it is not predetermined.  Ultimately you are considering all cases of $s_{t+1}$ weighted by their probability.

Since $\mathbb{E}_t$ integrates over shocks, you can differentiate under the integral sign by Leibniz's integration rule.