# The Income Fluctuation Problem and the Evolution of Wealth

## TL;DR

This paper studies a comprehensive household savings model with state-dependent returns and income, establishing conditions for solution existence, uniqueness, and properties of wealth distribution, including Pareto tails.

## Contribution

It extends classic models by allowing multiple state-dependent, correlated processes and derives conditions for wealth distribution characteristics.

## Key findings

- Solutions exist, are unique, and globally computable.
- Wealth dynamics are stationary, ergodic, and geometrically mixing.
- Wealth distribution exhibits Pareto tails.

## Abstract

We analyze the household savings problem in a general setting where returns on assets, non-financial income and impatience are all state dependent and fluctuate over time. All three processes can be serially correlated and mutually dependent. Rewards can be bounded or unbounded and wealth can be arbitrarily large. Extending classic results from an earlier literature, we determine conditions under which (a) solutions exist, are unique and are globally computable, (b) the resulting wealth dynamics are stationary, ergodic and geometrically mixing, and (c) the wealth distribution has a Pareto tail. We show how these results can be used to extend recent studies of the wealth distribution. Our conditions have natural economic interpretations in terms of asymptotic growth rates for discounting and return on savings.