\begin{equation}
    \frac{d \rho^*}{dU}=\frac{\alpha\gamma_{P,\max}(1-V)}{\gamma_{R,\max}V} \frac{c_U\beta_r }{(c_R(\beta_r+U)+\frac{c_P\gamma_{P,\max}(1-V)(\beta_R+U)}{\gamma_{R,\max}V}-c_U U)^2}>0, 
\end{equation}
