  We prove that the displacement problem of inhomogeneous elastostatics in a
two--dimensional exterior Lipschitz domain has a unique solution with finite
Dirichlet integral $\u$, vanishing uniformly at infinity if and only if the
boundary datum satisfies a suitable compatibility condition (Stokes' paradox).
Moreover, we prove that it is unique under the sharp condition $\u=o(\log r)$
and decays uniformly at infinity with a rate depending on the elasticities. In
particular, if these last ones tend to a homogeneous state at large distance,
then $\u=O(r^{-\alpha})$, for every $\alpha<1$.
