We study the design of mechanisms that implement Lindahl or Walrasian allocations and whose Nash equilibria are dynamically stable for a wide class of adaptive dynamics. We argue that supermodularity is not a desirable stability criterion in this mechanism design context, focusing instead on contractive mechanisms. We provide necessary and sufficient conditions for a mechanism to Nash implement Lindahl or Walrasian allocations, show that these conditions are inconsistent with the contraction property when message spaces are one-dimensional, and then show how to use additional dimensions to achieve dynamic stability while gaining budget balance out of equilibrium.