Kuhn Tucker Maximization
After substituting for $c_{1t}$ and $c_{2t}$, the problem can be rewritten as :
\begin{eqnarray*} \max_{q_t^e} \ &amp; \frac{(1- \pi) (n\omega_t)^{-\gamma} }{-\gamma} \left((1-q_t^e)  + z_tq_t^e\right)^{-\gamma} + \frac{\pi (AR\theta\omega_t)^{-\gamma } }{-\gamma z_t^{ -\gamma }} \left((1-q_t^e)  + z_tq_t^e\right)^{-\gamma} \\ \text{s.t. } &amp; q_t^e \in [0, 1]\end{eqnarray*} 

which is same as
\begin{eqnarray*} \max_{q_t^e} \ &amp; \alpha \left((1-q_t^e)  + z_tq_t^e\right)^{-\gamma}  \\ \text{s.t. } &amp; q_t^e \in [0, 1]\end{eqnarray*}
where $\alpha  = \dfrac{(1- \pi) (n\omega_t)^{-\gamma} }{-\gamma}  + \dfrac{\pi (AR\theta\omega_t)^{-\gamma } }{-\gamma z_t^{ -\gamma }} $

Observe that this is equivalent to solving :
\begin{eqnarray*} \max_{q_t^e} \ &amp;  (1-q_t^e)  + z_tq_t^e  \\ \text{s.t. } &amp; q_t^e \in [0, 1]\end{eqnarray*}

This is an objective that is linear in choice variable $q_t^e$, which is increasing in $q_t^e$ when $z_t &gt; 1$, decreasing in $q_t^e$ when $z_t &lt; 1$, and is a constant for $z_t = 1$. Consequently, the solution is
\begin{eqnarray*} q_t^e \in \begin{cases} \{1\} &amp; \text{if } z_t &gt; 1 \\ \{0\} &amp; \text{if } z_t &lt; 1 \\ [0, 1] &amp; \text{if } z_t = 1 \end{cases} \end{eqnarray*}