A worker's optimization problem is:
\begin{aligned} \max_{e, s} \quad & v=\frac{(\lambda + i)U_{emp}+(1-e)U_{unemp}}{i(1-e+\lambda+i)}\\ \textrm{s.t.} \quad & \lambda = \frac{n(1-e)\left(\frac{w}{\beta(ae)^{\beta}} \right)^{\frac{1}{\beta -1}}}{1-n\left(\frac{w}{\beta(ae)^{\beta}} \right)^{\frac{1}{\beta -1}}}\\ & U_{emp}=w-s-\frac{1}{(w-s)(1-e)} \\ & U_{unemp}=\frac{s\lambda}{1-e}-\frac{1-e}{s\lambda}\\ \end{aligned} where $0\leq i \leq 1$, $0\leq e \leq 1$, $0\leq a \leq 1$, $0\leq \beta \leq 1$, $0\leq \lambda \leq 1$, $w > s > 0$, $n> 0$.
A firm's optimization problem is:
\begin{aligned} \max_{w, L} \quad & \pi=y - wL\\ \textrm{s.t.} \quad & y = (a e L)^{\beta}\\ & e= e(w) \\ \end{aligned} where the second condition, $e=e(w)$, is the solution of the first optimization problem.
Now, from the solutions of the above optimization problems, I would like to find conduct comparative static for $\frac{\partial e^*}{\partial w^*}$, $\frac{\partial w^*}{\partial a}$, $\frac{\partial e^*}{\partial a}$, and $\frac{\partial s^*}{\partial a}$.
Here is my attempt: The first-order conditions of the first optimization problem are $\frac{\partial v}{\partial e}=0$, $\frac{\partial v}{\partial s}=0$, which are obtained as implicit functions. And the two first-order conditions of the second optimization problem are simplied to $\frac{\partial e}{\partial w}=\frac{e}{w}$. I know I have to use the implicit function theorem, but confused how to proceed from here.