  When a communication network's capacity increases, it is natural to want the
bandwidth allocated to increase to exploit this capacity. But, if the same
relative capacity increase occurs at each network resource, it is also natural
to want each user to see the same relative benefit, so the bandwidth allocated
to each route should remain proportional. We will be interested in bandwidth
allocations which scale in this \textit{iso-elastic} manner and, also, maximize
a utility function.
  Utility optimizing bandwidth allocations have been frequently studied, and a
popular choice of utility function are the weighted $\alpha$-fair utility
functions introduced by Mo and Walrand \cite{MoWa00}. Because weighted
$\alpha$-fair utility functions possess this iso-elastic property, they are
frequently used to form fluid models of bandwidth sharing networks. In this
paper, we present results that show, in many settings, the only utility
functions which are iso-elastic are weighted $\alpha$-fair utility functions.
  Thus, if bandwidth is allocated according to a network utility function which
scales with relative network changes then that utility function must be a
weighted $\alpha$-fair utility function, and hence, a control protocol that is
robust to the future relative changes in network capacity and usage ought to
allocate bandwidth inorder to maximize a weighted $\alpha$-fair utility
function.
