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We study the allocation of divisible goods to competing agents via a marketmechanism, focusing on agents with Leontief utilities. The majority of theeconomics and mechanism design literature has focused on \emph{linear} prices,meaning that the cost of a good is proportional to the quantity purchased.Equilibria for linear prices are known to be exactly the maximum Nash welfareallocations.  \emph{Price curves} allow the cost of a good to be any (increasing) functionof the quantity purchased. We show that price curve equilibria are not limitedto maximum Nash welfare allocations with two main results. First, we show thatan allocation can be supported by strictly increasing price curves if and onlyif it is \emph{group-domination-free}. A similarly characterization holds forweakly increasing price curves. We use this to show that given any allocation,we can compute strictly (or weakly) increasing price curves that support it (orshow that none exist) in polynomial time. These results involve a connection tothe \emph{agent-order matrix} of an allocation, which may have otherapplications. Second, we use duality to show that in the bandwidth allocationsetting, any allocation maximizing a CES welfare function can be supported byprice curves.

Markets Beyond Nash Welfare for Leontief Utilities