Dual Formulation of the Optimal Consumption problem with Multiplicative Habit Formation

TL;DR

This paper develops a dual formulation for the optimal consumption problem with multiplicative habit formation, overcoming non-concavity and path-dependency issues to establish strong duality and derive relations between primal and dual controls.

Contribution

It introduces a novel dual formulation for a complex habit formation model using Fenchel's Duality, enabling analytical evaluation and better understanding of optimal controls.

Findings

Established strong duality for the problem.

Derived duality relations linking primal and dual controls.

Developed an analytical mechanism to evaluate approximation accuracy.

Abstract

This paper provides a dual formulation of the optimal consumption problem with internal multiplicative habit formation. In this problem, the agent derives utility from the ratio of consumption to the internal habit component. Due to this multiplicative specification of the habit model, the optimal consumption problem is not strictly concave and incorporates irremovable path-dependency. As a consequence, standard Lagrangian techniques fail to supply a candidate for the corresponding dual formulation. Using Fenchel's Duality Theorem, we manage to identify a candidate formulation and prove that it satisfies strong duality. On the basis of this strong duality result, we are able to derive duality relations that stipulate how the optimal primal controls depend on the optimal dual controls and vice versa. {Moreover, using the dual formulation, we develop an analytical evaluation mechanism to bound the accuracy of approximations to the optimal solutions.