  In this article we study a multi-asset version of the Merton investment and
consumption problem with proportional transaction costs. In general it is
difficult to make analytical progress towards a solution in such problems, but
we specialise to a case where transaction costs are zero except for sales and
purchases of a single asset which we call the illiquid asset.
  Assuming agents have CRRA utilities and asset prices follow exponential
Brownian motions we show that the underlying HJB equation can be transformed
into a boundary value problem for a first order differential equation. The
optimal strategy is to trade the illiquid asset only when the fraction of the
total portfolio value invested in this asset falls outside a fixed interval.
Important properties of the multi-asset problem (including when the problem is
well-posed, ill-posed, or well-posed only for large transaction costs) can be
inferred from the behaviours of a quadratic function of a single variable and
another algebraic function.
