Finite-horizon ratcheting with perpetual commitment: optimal consumption, investment, and lock-in effects
Abstract
We study optimal consumption and portfolio policies for an agent with a finite planning horizon and an irreversible consumption ratcheting constraint. During the planning horizon, the agent may increase consumption but cannot reduce it. After the terminal date, the consumption level reached by that time is permanently locked in, and the agent continues to consume at that level for the rest of life, where death occurs randomly. The problem is well defined whenever initial wealth is large enough to support the current consumption floor indefinitely. Using duality theory in complete markets, we decompose the problem into a continuum of optimal stopping problems through a layer-cake representation. The dual problem can then be interpreted in terms of American put option pricing on the shadow price process, with a strike determined by the relation between the risk-free rate and the effective discount rate. We characterize the optimal consumption policy through a free boundary driven by the shadow price, and show that the wealth-to-consumption ratio is reflected at an endogenous boundary. A central finding is that the lifetime lock-in effect makes the agent more cautious than in the standard finite-horizon ratcheting model: upward consumption adjustments occur less frequently, and the optimal risky share is uniformly lower for any given wealth-to-consumption ratio. This stronger precautionary behavior arises because any increase in consumption before the terminal date also raises the permanently committed consumption level afterward, thereby creating an additional lock-in cost.