  We consider the problem of optimal consumption from labor income and
investment in a general incomplete semimartingale market. The economic agent
cannot borrow against future income, so the total wealth is required to be
positive at (all or some) previous times. Under very general conditions, we
show that an optimal consumption and investment plan exists and is unique, and
provide a dual characterization in terms of martingale deflators and decreasing
parts, which allow for a limit that charges only the times when the
no-borrowing constraint is binding. The analysis relies on the
infinite-dimensional parametrization of the income/liability streams and,
therefore, provides the first-order dependence of the optimal investment and
consumption plans on future income/liabilities (as well as a pricing rule). An
emphasis is placed on mathematical generality.
