 User-network decomposition of concave utility maximization

Maximize a sum of increasing strictly concave utilities $U_r(x_r)$ subject to $Ax\le C$, $x\ge 0$. Suppose capacities are positive, every route is nonempty, and $U_r^{\prime }(0)=\infty$. At its unique positive optimizer $x^{\ast }$, the Karush-Kuhn-Tucker conditions give nonnegative resource congestion prices $p^{\ast }$ and route congestion prices $y_r^{\ast }=U_r^{\prime }(x_r^{\ast })$. Choosing ${\nu }_r^{\ast }=x_r^{\ast }{e}^{y_r^{\ast }}$ makes each user maximize $U_r({\nu }_r{e}^{-y_r^{\ast }})-y_r^{\ast }{\nu }_r{e}^{-y_r^{\ast }}$ at ${\nu }_r^{\ast }$. It also makes $x^{\ast }$ maximize the entropy maximization for loss-network occupancy objective, since its gradient is $\log ({\nu }_r^{\ast }/x_r^{\ast })=(A^T{p}^{\ast }{)}_r$. Thus user and network optima implement the same system optimum.