Competitive Equilibrium in a Radial Network

In a competitive market over a network with finite capacity, we can focus without loss of generality on simple allocations, where the net transaction over a link is equal to the gross transaction over the same link. If a simple allocation can be sustained by an equilibrium, then the allocation is efficient. If an equilibrium allocation is equivalent to a simple allocation and if the simple allocation is no longer sustained by an equilibrium, then the original equilibrium allocation is not efficient. For any efficient allocation, an equivalent simple allocation exists that can be sustained by a competitive equilibrium.