# The Income Fluctuation Problem with Capital Income Risk: Optimality and Stability

## TL;DR

This paper analyzes the income fluctuation problem considering capital income risk with serial correlation, establishing new results on solution existence, stability, and wealth distribution, and examining impacts on wealth inequality.

## Contribution

It introduces general conditions for solution existence, stability, and computation in models with correlated wealth returns and labor income, extending previous work.

## Key findings

- Stochastic volatility and persistence significantly affect wealth inequality.
- New methods for computing wealth distribution in complex income models.
- Model solutions are shown to be unique and stochastically stable.

## Abstract

This paper studies the income fluctuation problem with capital income risk (i.e., dispersion in the rate of return to wealth). Wealth returns and labor earnings are allowed to be serially correlated and mutually dependent. Rewards can be bounded or unbounded. Under rather general conditions, we develop a set of new results on the existence and uniqueness of solutions, stochastic stability of the model economy, as well as efficient computation of the ergodic wealth distribution. A variety of applications are discussed. Quantitative analysis shows that both stochastic volatility and mean persistence in wealth returns have nontrivial impact on wealth inequality.