  We provide an asymptotic expansion of the value function of a
multidimensional utility maximization problem from consumption with small
non-linear price impact. In our model cross-impacts between assets are allowed.
In the limit for small price impact, we determine the asymptotic expansion of
the value function around its frictionless version. The leading order
correction is characterized by a nonlinear second order PDE related to an
ergodic control problem and a linear parabolic PDE. We illustrate our result on
a multivariate geometric Brownian motion price model.
