I have the following system of difference equations. Fix any $(\alpha,K,p)\in\mathbb R_+\times\mathbb R_+\times (0,1)$. Let $u_0=1, \nu_0=0$. The sequence $(a_t,w_t,\nu_t,u_t)_{t\geq 1}$ is defined as follows: \begin{align*} &a_t=\frac{e^{\frac{K}{\alpha}(\nu_{t-1}-1)}-1}{\frac{K}{\alpha}(\nu_{t-1}-1)}; \qquad w_t=\frac{u_{t-1}}{1-\nu_{t-1}}(1-a_t);\\ &\nu_t=\frac{1-e^{-K(a_tp+w_t)}}{K(a_tp+w_t)}(1-p)+\frac{a_tp\left(1-\frac{1-e^{-K(a_tp+w_t)}}{K(a_tp+w_t)}\right)}{a_tp+w_t}; \qquad u_t=\frac{w_t\left(1-\frac{1-e^{-K(a_tp+w_t)}}{K(a_tp+w_t)}\right)}{a_tp+w_t}. \end{align*}
I numerically observe that for any $t$, $a_t, \nu_t$ are increasing in $\alpha$, and $w_t, u_t$ are decreasing in $\alpha$. However, I'm not able to prove it. What I'm able to show is that $\frac{\partial a_t}{\partial \alpha}\geq 0$, $\frac{\partial a_t}{\partial \nu_{t-1}}\geq 0$, $\frac{\partial\nu_{t}}{\partial a_t}\geq 0$, $\frac{\partial\nu_t}{\partial w_t}\leq 0$, $\frac{\partial u_t}{\partial a_t}\leq 0$, and $\frac{\partial u_t}{\partial w_t}\geq 0$. I'm not able to complete the missing piece with respect to $w_t$. Can anyone help me with this? Thank you so much!!
