  We study the fair division problem of allocating multiple resources among a
set of agents with Leontief preferences that are each required to complete a
finite amount of work, which we term "limited demands". We examine the behavior
of the classic Dominant Resource Fairness (DRF) mechanism in this setting and
show it is fair but only weakly Pareto optimal and inefficient in many natural
examples. We propose as an alternative the Least Cost Product (LCP) mechanism,
a natural adaptation of Maximum Nash Welfare to this setting. We characterize
the structure of allocations of the LCP mechanism in this setting, show that it
is Pareto efficient, and that it satisfies the relatively weak fairness
property of sharing incentives. While we prove it satisfies the stronger
fairness property of (expected) envy freeness in some special cases, we provide
a counterexample showing it does not do so in general, a striking contrast to
the "unreasonable fairness" of Maximum Nash Welfare in other settings.
Simulations suggest, however, that these violations of envy freeness are rare
in randomly generated examples.
