# Optimal Consumption under a Habit-Formation Constraint: the Deterministic Case

arXiv: https://arxiv.org/abs/2012.02277 · HTML: https://mlsanity.com/paper/2012.02277/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2020.

## Built on by (most cited first, 2 of 2)

- [On the Convergence of Jacobian-Free Backpropagation for Optimal Control Problems with Implicit Hamiltonians](https://mlsanity.com/paper/2602.00921/) (2026)
- [Consumer Autonomy or Illusion? Rethinking Consumer Agency in the Age of Algorithms](https://mlsanity.com/paper/2508.13440/) (2025)

## Abstract (arXiv)

We formulate and solve a deterministic optimal consumption problem to maximize the discounted CRRA utility of an individual's consumption-to-habit process assuming she only invests in a riskless market and that she is unwilling to consume at a rate below a certain proportion α∈(0,1] of her consumption habit. Increasing α, increases the degree of addictiveness of habit formation, with α=0 (respectively, α=1) corresponding to non-addictive (respectively, completely addictive) model. We derive the optimal consumption policies explicitly in terms of the solution of a nonlinear free-boundary problem, which we analyze in detail. Impatient individuals (or, equivalently, those with more addictive habits) always consume above the minimum rate; thus, they eventually attain the minimum wealth-to-habit ratio. Patient individuals (or, equivalently, those with less addictive habits) consume at the minimum rate if their wealth-to-habit ratio is below a threshold, and above it otherwise. By consuming patiently, these individuals maintain a wealth-to-habit ratio that is greater than the minimum acceptable level. Additionally, we prove that the optimal consumption path is hump-shaped if the initial wealth-to-habit ratio is either: (1) larger than a high threshold; or (2) below a low threshold and the agent is more risk seeking (that is, less risk averse). Thus, we provide a simple explanation for the consumption hump observed by various empirical studies.