  This paper considers exponential utility indifference pricing for a
multidimensional non-traded assets model, and provides two linear
approximations for the utility indifference price. The key tool is a
probabilistic representation for the utility indifference price by the solution
of a functional differential equation, which is termed \emph{pseudo linear
pricing rule}. We also provide an alternative derivation of the quadratic BSDE
representation for the utility indifference price.
