Optimal Savings with Preference for Wealth

2025-09-15

AI Summary

This paper investigates optimal savings and consumption decisions for agents with wealth-dependent preferences—i.e., utility functions that explicitly depend on aggregate wealth level—under uncertainty. Methodologically, it formulates an infinite-horizon stochastic dynamic programming problem, imposes a power utility specification, and employs stochastic process theory and convex optimization to establish rigorous existence and uniqueness of the optimal consumption policy. The key contribution is the first formal demonstration of the asymptotic linearity of the consumption function at high wealth levels: specifically, when relative wealth risk aversion is lower than relative consumption risk aversion, the marginal propensity to consume converges to zero asymptotically. This result provides an endogenous theoretical explanation for the empirically observed high savings rates among affluent households, aligning closely with both macroeconomic and microeconomic evidence. Moreover, it extends the canonical life-cycle framework by enriching the modeling of wealth motives beyond standard time-separable utility.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Behavioral Game Theory

Application Category

• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience

Abstract

The consumption function maps current wealth and the exogenous state to current consumption. We prove the existence and uniqueness of a consumption function when the agent has a preference for wealth. When the period utility functions are restricted to power functions, we prove that the consumption function is asymptotically linear as wealth tends to infinity and provide a complete characterization of the asymptotic slopes. When the risk aversion with respect to wealth is less than that for consumption, the asymptotic slope is zero regardless of other model parameters, implying wealthy households save a large fraction of their income, consistent with empirical evidence.

Problem

• Existence and uniqueness of consumption function with wealth preference
• Asymptotic linearity of consumption function for power utility
• Wealthy households' high savings rate under specific risk aversion

Innovation

• Proves existence of consumption function with wealth preference
• Characterizes asymptotic linearity in power utility cases
• Links low wealth risk aversion to high savings rates