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A Theory of Saving under Risk Preference Dynamics

Qingyin Ma, Xinxi Song, Alexis Akira Toda

TL;DR

The paper develops a general theory of saving under dynamically evolving risk preferences, allowing risk aversion to vary across states and over time. By embedding preference shocks in a Markov-switching framework and solving with a fixed-point approach, it derives that zero asymptotic MPCs are a robust feature whenever there is any probability of downward transitions in risk aversion, independent of return or discount-rate risk. The results yield a powerful mechanism for explaining persistently high saving rates among wealthy households, as precautionary saving remains active even at high wealth levels, and are complemented by quantitative IRFs showing substantial, state-dependent responses to risk-shock shocks. Overall, the theory provides a theoretically robust and empirically plausible account of wealth accumulation dynamics and offers new tools for analyzing saving behavior under heterogeneous, evolving risk attitudes.

Abstract

Empirical evidence shows that wealthy households have substantially higher saving rates and markedly lower marginal propensity to consume (MPC) than other groups. Existing theory cannot account for this pattern unless under restrictive assumptions on returns, discounting, and preferences. This paper develops a general theory of optimal savings with preference shocks, allowing risk aversion to vary across states and over time, and shows that incorporating such heterogeneity in risk attitudes fundamentally reshapes the asymptotic dynamics of consumption and saving. In particular, zero asymptotic MPCs (100% asymptotic saving rates) arise under markedly weaker conditions than in existing theory. Strikingly, such outcomes occur whenever there is a positive probability that agents become less risk averse in the future. Therefore, the vanishing MPC emerges as a generic feature rather than a knife-edge result of the optimal savings model, offering a more theoretically robust and empirically consistent account of the saving behavior of wealthy households.

A Theory of Saving under Risk Preference Dynamics

TL;DR

The paper develops a general theory of saving under dynamically evolving risk preferences, allowing risk aversion to vary across states and over time. By embedding preference shocks in a Markov-switching framework and solving with a fixed-point approach, it derives that zero asymptotic MPCs are a robust feature whenever there is any probability of downward transitions in risk aversion, independent of return or discount-rate risk. The results yield a powerful mechanism for explaining persistently high saving rates among wealthy households, as precautionary saving remains active even at high wealth levels, and are complemented by quantitative IRFs showing substantial, state-dependent responses to risk-shock shocks. Overall, the theory provides a theoretically robust and empirically plausible account of wealth accumulation dynamics and offers new tools for analyzing saving behavior under heterogeneous, evolving risk attitudes.

Abstract

Empirical evidence shows that wealthy households have substantially higher saving rates and markedly lower marginal propensity to consume (MPC) than other groups. Existing theory cannot account for this pattern unless under restrictive assumptions on returns, discounting, and preferences. This paper develops a general theory of optimal savings with preference shocks, allowing risk aversion to vary across states and over time, and shows that incorporating such heterogeneity in risk attitudes fundamentally reshapes the asymptotic dynamics of consumption and saving. In particular, zero asymptotic MPCs (100% asymptotic saving rates) arise under markedly weaker conditions than in existing theory. Strikingly, such outcomes occur whenever there is a positive probability that agents become less risk averse in the future. Therefore, the vanishing MPC emerges as a generic feature rather than a knife-edge result of the optimal savings model, offering a more theoretically robust and empirically consistent account of the saving behavior of wealthy households.

Paper Structure

This paper contains 21 sections, 26 theorems, 130 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Optimality results
  3. Problem Statement
  4. Optimality: Definitions and Fundamental Properties
  5. Existence and Computability of Optimal Consumption
  6. Fundamental Properties of Consumption and Saving
  7. Asymptotic Properties of Consumption and Saving
  8. Setups
  9. The Asymptotic MPCs
  10. The Asymptotic Saving Rate
  11. Quantitative Analysis
  12. Model Specification
  13. The Asymptotic Saving Rates
  14. Impulse Response Analysis
  15. Preliminaries
  16. ...and 6 more sections

Key Result

Proposition 2.1

If Assumption a:utility holds, then every feasible policy satisfying the first order and transversality conditions is an optimal policy.

Figures (3)

  • Figure 1: The optimal consumption function
  • Figure 2: The consumption and saving rates
  • Figure 3: Impulse responses to a positive one-unit risk aversion shock

Theorems & Definitions (56)

  • Proposition 2.1: Sufficiency of first order and transversality conditions
  • Theorem 2.1: Existence, uniqueness, and computability of optimal policies
  • Proposition 2.2: Monotonicity with respect to wealth
  • Proposition 2.3: Monotonicity with respect to income
  • Proposition 2.4: Threshold for saving decision
  • Theorem 3.1: Zero asymptotic MPCs
  • Theorem 3.2: Zero asymptotic MPCs under positive transition probability
  • Proposition 3.1: Impossibility of positive asymptotic MPCs without downward transitions
  • Theorem 3.3: Positive asymptotic MPCs under strictly persistent risk aversion
  • Remark 3.1
  • ...and 46 more