  In the context of the multi-dimensional infinite horizon optimal
consumption-investment problem with proportional transaction costs, we provide
the first order expansion in small transact costs. Similar to the
one-dimensional derivation in our accompanying paper [42], the asymptotic
expansion is expressed in terms of a singular ergodic control problem, and our
arguments are based on the theory of viscosity solutions, and the techniques of
homogenization which leads to a system of corrector equations. In contrast with
the one-dimensional case, no explicit solution of the first corrector equation
is available anymore. Finally, we provide some numerical results which
illustrate the structure of the first order optimal controls.
