Optimal investment and consumption with labor income in incomplete markets

TL;DR

This paper develops a comprehensive framework for optimal consumption and investment strategies in incomplete markets with labor income, accounting for borrowing constraints and providing dual characterizations under broad conditions.

Contribution

It introduces a novel dual approach for optimal plans in incomplete markets with labor income, emphasizing mathematical generality and the impact of future income streams.

Findings

Existence and uniqueness of optimal plans under general conditions

Dual characterization using martingale deflators and decreasing parts

First-order dependence of strategies on future income/liabilities

Abstract

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.