Optimal Consumption and Investment Problem with Consumption Ratcheting in Luxury Goods

Geonwoo Kim and Junkee Jeon ()
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Geonwoo Kim: School of Natural Sciences, Seoul National University of Science and Technology, Seoul 01811, Republic of Korea
Junkee Jeon: Department of Applied Mathematics, Kyung Hee University, Yongin 17104, Republic of Korea

Mathematics, 2025, vol. 13, issue 22, 1-19

Abstract: This paper investigates an infinite-horizon optimal consumption and investment problem for an agent who consumes two types of goods: necessities and luxuries. The agent derives utility from both goods but faces a ratcheting constraint on luxury consumption, which prohibits any decline in its level over time. This constraint captures the irreversible nature of high living standards or luxury habits often observed in real economies. We formulate the problem in a complete financial market with a risk-free asset and a risky stock and solve it analytically using the dual–martingale method. The dual problem is shown to reduce to a family of optimal stopping problems, from which we derive explicit closed-form solutions for the value function and optimal policies. Our results reveal that the ratcheting constraint generates asymmetric consumption dynamics: necessities adjust freely, whereas luxuries exhibit downward rigidity. As a consequence, the marginal propensity to consume necessities declines with wealth, while luxury consumption and portfolio risk exposure increase more sharply compared to the benchmark case without ratcheting. The model provides a continuous-time microfoundation for persistent high consumption levels and greater risk-taking among wealthy individuals.

Keywords: consumption ratcheting; luxury goods; duality; optimal stopping; portfolio choice; marginal propensity to consume; risk-taking (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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