\begin{equation}
    \begin{split}
        &\sup_{\pi_t\in\mathcal{A}^r_t}\biggl\{\int_{-\infty}^\infty \biggl[-K_t \,\biggl(\nu - \frac{\gamma_t\,\partial_xH^{(\lambda)} + D_t\,x}{2\,K_t}\biggr)^2 - \frac{\lambda}{q-1}\,(\pi_t^*(\nu))^{q-1}\biggr]\,\pi^*_t(\nu)\,d\nu \biggr\}\\
        =& -\biggl(\frac{1-q}{\lambda\,q}\biggr)^{\frac{1}{q-1}}\psi_t^{\frac{1}{q-1}+\frac{3}{2}}\frac{\sqrt{\pi}\,\Gamma(\frac{1}{1-q}-\frac{3}{2})}{\sqrt{K_t}}\biggl(\frac{1}{2\,\Gamma(\frac{1}{1-q})} - \frac{1}{(1-q)\,\Gamma(\frac{1}{1-q})}\biggr)\\
        =& \frac{q+1}{3\,q-1}\,\psi_t\,.
    \end{split}    
\end{equation}
