Two-resource path packing as minimum-cost circulation

Consider nonnegative route variables with constraints $u\le a$, $v\le b$, $v+w\le c$, $w+z\le d$, and a linear reward $p_{u}u+p_{v}v+p_{w}w+p_{z}z$. It is a minimum-cost flow problem: use arcs $s\to t$ of capacity $a$ and cost $-p_u$, $s\to L$ of capacity $c$ and cost zero, $L\to t$ of capacity $b$ and cost $-p_v$, $L\to R$ of unlimited capacity and cost $-p_w$, $s\to R$ of unlimited capacity and cost $-p_z$, and $R\to t$ of capacity $d$ and cost zero. Close the network by an unlimited zero-cost arc $t\to s$. The four forward path flows are exactly $u,v,w,z$, so conservation gives the resource constraints. This reduction does not assert that general multi-commodity flow is ordinary minimum-cost flow.