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My starting equations are the following:

$V^{e}=w+\beta.((1-s).V^{e}+s.V^{u}(t))\\V^{u}(t)=\begin{matrix} \\max \\a(t) \end{matrix}b(t)-\psi.a(t)+\beta.[\pi(a(t)).V^{e}+(1-\pi(a(t))).V^{u}(t+1)]\\\pi(a(t))=1-exp(-\phi.a(t))\\b(t)=b.t^{-\mu}$

All the parameters are fixed, except from $a$ and $\phi$. When I use the first order condition on $V^{u}(t+1)$ with respect to $a(t)$, I obtain the following condition: $$\beta.\pi'.[V^{e}-V^{u}(t+1)]=\psi$$ My goal is to find, after calibration of the parameters, for which $\phi$ we have $V^{u}(t)>V^{e}$ after 13 periods. I have tried to put $V^{u}(t)$ on the right side of the first expression and $V^{u}(t+1)$ on the right side of the second one. After several substitutions, I get the following expression:$$V^{u}(t)=\frac{1}{\frac{1-\pi-\pi.\beta}{1-\pi}-\frac{s}{1-\beta(1-s)}}.[b(t)-\psi.a(t)+\frac{1-\pi}{2-\pi}.(\frac{\psi.\pi}{\pi'}+\frac{\pi.\beta.w}{(1-\pi).(1-\beta(1-s))})]$$ But this gets me nowhere near my problem. Does anyone have any idea on how I can have an optimization of $a(t)$, and therefore an a value for $\phi$ ? Thanks in advance.

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