Finite-horizon Optimal Consumption and Investment with Costly Income Adjustment

Abstract

We study optimal consumption and investment for an investor who receives labor income until a fixed retirement date, can raise the income rate irreversibly at a constant price per unit of income flow, and cannot borrow against future income. A pathwise argument shows that no income is purchased during a final period before retirement whose length depends only on the interest rate and the price. Before this period the dual problem has two free boundaries: a lower one generated by the borrowing constraint and an upper one, at which income is purchased, that escapes to infinity as the final period begins. The analysis of this problem is our main mathematical contribution. During the final period it is a single obstacle problem, which we solve, together with the regularity of its free boundary, by established methods. Before that period a partial hodograph transformation straightens both free boundaries and turns the problem into a quasilinear parabolic equation with oblique boundary conditions on a fixed interval. Comparison principles, explicit barriers and regularity theory for quasilinear equations then give existence, uniqueness and smoothness. Transforming back and gluing the two constructions yields global regularity: both free boundaries are infinitely differentiable and the value function is a classical solution at every positive wealth level. A dual budget process verifies optimality. We also derive coupled Volterra integral equations for the free boundaries, prove exponential convergence to the perpetual problem under a scalar condition, and find numerically a purchase threshold that is U-shaped in the time to retirement.