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Let $x^{\ast }$ maximize SYSTEM. Under the usual positive-capacity, nonempty-route assumptions, the feasible set is compact, contains a positive point, and has a Slater condition point. Strict concavity makes the optimum unique. The condition $U_r^{\prime }(0)=\infty$ forces $x_r^{\ast }>0$: moving from any optimum with a zero component slightly toward a positive feasible point gives an infinite positive directional derivative in that component and only finite losses in the positive components.

By the Karush-Kuhn-Tucker conditions, there are link Lagrange multipliers $p_j^{\ast }\ge 0$ withSet the route congestion price $y_r^{\ast }=(A^T{p}^{\ast }{)}_r=U_r^{\prime }(x_r^{\ast })$ and chooseThen $x_r^{\ast }={\nu }_r^{\ast }{e}^{-y_r^{\ast }}$, as required. For USER with fixed $y_r^{\ast }$, the substitution $x={\nu }_r{e}^{-y_r^{\ast }}$ is a bijection of the nonnegative half-line, and its objective becomes $U_r(x)-y_r^{\ast }x$. This is a strictly concave function whose derivative vanishes at $x_r^{\ast }$, so its unique maximizer is $x_r^{\ast }$ and the corresponding USER optimizer is ${\nu }_r^{\ast }$.

For NETWORK at ${\nu }^{\ast }$, its objective derivative at $x^{\ast }$ isThus $x^{\ast },p^{\ast }$ obey its Karush-Kuhn-Tucker conditions, with the same feasibility and complementary slackness. The NETWORK objective is a strictly concave function, so $x^{\ast }$ is also its unique maximizer. The triple $(x^{\ast },y^{\ast },{\nu }^{\ast })$ solves all three problems simultaneously. This is a user-network decomposition of concave utility maximization.