  We consider an optimal consumption/investment problem to maximize expected
utility from consumption. In this market model, the investor is allowed to
choose a portfolio which consists of one bond, one liquid risky asset (no
transaction costs) and one illiquid risky asset (proportional transaction
costs). We fully characterize the optimal consumption and trading strategies in
terms of the solution of the free boundary ODE with an integral constraint. We
find an explicit characterization of model parameters for the well-posedness of
the problem, and show that the problem is well-posed if and only if there
exists a shadow price process. Finally, we describe how the investor's optimal
strategy is affected by the additional opportunity of trading the liquid risky
asset, compared to the simpler model with one bond and one illiquid risky
asset.
