🔢 mathematics

Finite difference methods for a continuous-time heterogeneous agent model with recursive utility

This paper proposes and analyzes convergent finite difference algorithms, specifically Howard-Newton and Howard-Tarski-Kantorovich methods, to solve the discretized Hamilton-Jacobi-Bellman equations arising in continuous-time heterogeneous agent models with Epstein-Zin recursive utility, while also establishing the existence of solutions and deriving error estimates for the late resolution preference case.

Original authors: Yves Achdou, Qing Tang

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yves Achdou, Qing Tang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast, bustling city where millions of people are trying to figure out the best way to spend their money over their entire lives. Some people have steady jobs, others have volatile incomes, and everyone faces the risk of losing their job or getting a raise. They all want to balance enjoying life today with saving for a rainy day tomorrow.

This paper is about building a super-precise mathematical map to understand how these people make those decisions. The authors, Yves Achdou and Qing Tang, have created a new set of computational tools (algorithms) to solve a very complex puzzle involving how people feel about risk versus how much they like to delay gratification.

Here is a breakdown of their work using simple analogies:

1. The Two Different "Personalities" of Savers

In economics, there's a tricky concept called Epstein-Zin utility. Think of it as a way to measure a person's "financial personality" based on two distinct traits:

Risk Aversion: How much they hate losing money.
Intertemporal Substitution (EIS): How willing they are to swap spending today for spending tomorrow.

Usually, math models treat these as the same thing. But in the real world, they are different. You might be very scared of losing money (high risk aversion) but also very willing to wait to buy a house (high EIS).

The authors split the problem into two scenarios based on how people feel about when they find out the future:

The "Late Resolution" Crowd (θ ≥ 1): These people prefer to keep the uncertainty hanging over them for a while. They are okay with not knowing the future immediately.
The "Early Resolution" Crowd (0 < θ < 1): These people are anxious. They want to know the outcome now so they can stop worrying. They prefer to resolve uncertainty early.

2. The Map-Making Challenge (The HJB Equation)

To predict how these people behave, the authors use a giant equation called the Hamilton-Jacobi-Bellman (HJB) equation.

The Analogy: Imagine trying to draw a topographical map of a mountain range where the terrain changes depending on how many people are walking on it.
The Problem: The math is incredibly jagged. If you try to use standard map-drawing tools (standard algorithms), they break because the "terrain" (the math) has sharp cliffs and infinite slopes in certain directions.
The Solution: The authors built a new kind of "grid" (a Finite Difference Method) that fits the jagged terrain perfectly. They discretized the continuous world into small, manageable steps, like turning a smooth curve into a staircase that approximates the curve.

3. The Two Different Climbing Strategies

Because the "Late" and "Early" resolution groups have different mathematical properties, the authors had to invent two different ways to climb the mountain to find the solution.

Strategy A: The "Howard-Newton" Hike (For the Late Resolution Crowd)

The Metaphor: Imagine you are hiking up a hill that is perfectly smooth and bowl-shaped (convex). You can use a GPS that tells you exactly which direction is steepest and take a giant, confident step toward the top.
How it works: The authors use a method called Howard-Newton. It's a two-step dance:
1. Guess a path: Assume a spending plan.
2. Newton's Step: Use a powerful mathematical shortcut (Newton's method) to instantly correct that guess to get closer to the perfect plan.
The Result: They proved this method always converges to the one and only correct answer. They also showed that their digital map is incredibly accurate, with errors shrinking linearly as they make the grid finer.

Strategy B: The "Howard-Tarski-Kantorovich" Ladder (For the Early Resolution Crowd)

The Metaphor: Now imagine the hill is bumpy, jagged, and has no single "top." It's more like a maze with many possible paths. You can't just take a giant step; you have to be careful.
The Problem: The standard "GPS" (Newton's method) might get lost or bounce around forever because the math isn't smooth here.
The Solution: The authors used a strategy inspired by a famous math theorem (Tarski-Kantorovich).
- The Analogy: Imagine you are trying to find a hidden treasure in a room. You don't know exactly where it is, but you know it's somewhere between a "floor" (a guaranteed low value) and a "ceiling" (a guaranteed high value).
- The Method: They start at the floor and slowly, step-by-step, climb up. They check if their current guess is a "sub-solution" (a safe, low estimate). If it is, they move up. Because of the specific math properties of this group, they proved that if you keep climbing up from the bottom, you will eventually hit the "ceiling" of the solution.
- The Twist: They also built a "downward" version that starts at the ceiling and climbs down. In their computer tests, both the "up" and "down" ladders met at the exact same spot, suggesting there is likely only one true answer, even though the math is too messy to prove it yet.

4. Why This Matters

The authors didn't just invent these algorithms; they proved they work.

They showed that for the "Late Resolution" group, their method is guaranteed to find the unique solution and gave a precise formula for how fast it gets accurate.
For the "Early Resolution" group, they proved their method finds a valid solution and that it is the "minimal" (lowest possible) valid solution, which is a crucial mathematical guarantee.

Summary

In short, this paper is a guidebook for navigating a very difficult economic landscape. The authors realized that different types of people (those who like to wait for news vs. those who want news now) require different mathematical tools to model. They built two specialized "climbing gear" sets (algorithms) to solve the equations for each group, proving that these tools are reliable, accurate, and will always lead to a solution. This allows economists to better simulate how fiscal and monetary policies affect the wealth distribution of a society.

Technical Summary: Finite Difference Methods for a Continuous-Time Heterogeneous Agent Model with Recursive Utility

Problem Statement
This paper addresses the computational challenge of solving continuous-time heterogeneous agent (HA) models, specifically Aiyagari-Bewley-Huggett type models, where agents possess Epstein-Zin recursive utility. Unlike standard Constant Relative Risk Aversion (CRRA) utility, Epstein-Zin utility allows for the separation of risk aversion ( $\gamma$ ) and the elasticity of intertemporal substitution (EIS, $\psi$ ). The model is formulated as a system of coupled Partial Differential Equations (PDEs): Hamilton-Jacobi-Bellman (HJB) equations for the value functions and Fokker-Planck-Kolmogorov (FPK) equations for the distribution of wealth.

The primary difficulties in solving the HJB equations in this context are twofold:

The Hamiltonian takes infinite values for negative momentum (corresponding to borrowing constraints).
The utility aggregator depends non-linearly on the value function $v$ itself, a feature specific to recursive utility. This dependence complicates standard iterative solution methods, such as the Howard algorithm, which are typically used for CRRA utility.

The behavior of the system changes fundamentally based on the parameter $\theta = \frac{1-\psi^{-1}}{1-\gamma}$ . The case $\theta \ge 1$ corresponds to "late resolution" of uncertainty, while $0 < \theta < 1$ corresponds to "early resolution." The paper notes that while the $\theta \ge 1$ case has been studied (e.g., in [4]), the $0 < \theta < 1$ case presents significant theoretical and numerical hurdles, particularly regarding the existence and uniqueness of solutions.

Methodology
The authors propose a finite difference method using monotone (upwind) schemes to discretize the state-constrained HJB equations on a bounded grid. The core of the methodology involves designing specific iterative algorithms tailored to the two distinct preference regimes:

Late Resolution Case ( $\theta \ge 1$ ):
- In this regime, the system satisfies a comparison principle, ensuring the uniqueness of the viscosity solution.
- The authors propose a Howard-Newton algorithm. This method involves an outer loop that updates the policy and an inner loop that solves the resulting linearized system using Newton's method.
- The algorithm relies on the strict concavity of the Hamiltonian with respect to the value function in this regime.
Early Resolution Case ( $0 < \theta < 1$ ):
- Here, the standard comparison principle may not hold, and the Hamiltonian is convex with respect to the value function.
- The authors define a map $\Gamma$ such that a solution to the HJB equation is a fixed point of $\Gamma$ . They prove that this map is monotone and invariant within a specific interval defined by discrete sub- and supersolutions (barriers).
- Leveraging the Tarski-Kantorovich fixed point theorem, they propose a Howard-Tarski-Kantorovich algorithm. This method uses an outer loop to update the value function based on the monotonicity of $\Gamma$ and an inner Howard loop to solve the policy evaluation step.
- Two variants are presented: an "upward" variant (Algorithm 5.1) starting from a subsolution to converge to the minimal solution, and a "downward" variant (Algorithm 5.2) starting from a supersolution to converge to the maximal solution.

Key Contributions

Algorithmic Design: The paper introduces and rigorously analyzes convergent iterative algorithms for discretized HJB equations with recursive utility. It distinguishes between the Howard-Newton approach for $\theta \ge 1$ and the Howard-Tarski-Kantorovich approach for $0 < \theta < 1$ .
Convergence Proofs: The authors prove the convergence of these iterative algorithms. For the $\theta \ge 1$ case, this implies the existence and uniqueness of the solution to the discretized HJB equations. For the $0 < \theta < 1$ case, the convergence guarantees the existence of at least one solution (specifically, minimal and maximal solutions).
Error Estimates: In the late resolution case ( $\theta \ge 1$ ), the paper derives a priori error estimates between the unique solution of the continuous HJB equation and its discrete counterpart. The authors establish a convergence rate of $O(\Delta x)$ , utilizing the strong regularity of the continuous solutions to avoid the "doubling of variables" technique often required in viscosity solution theory.
Existence Results: The analysis provides existence proofs for solutions to the discretized equations in both preference regimes, a non-trivial result given the non-standard dependence of the utility function on the value function.

Results

Numerical Validation: The authors present numerical experiments for three test cases with varying parameters ( $\gamma, \psi$ ) covering both $\theta > 1$ and $0 < \theta < 1$ .
Algorithm Performance:
- For $\theta \ge 1$ , the Howard-Newton algorithm produces a non-decreasing sequence of grid functions, confirming the theoretical monotonicity.
- For $0 < \theta < 1$ , the Howard-Tarski-Kantorovich algorithms (both upward and downward variants) produce monotonic sequences converging to the minimal and maximal solutions, respectively.
Convergence Rate: Numerical tests on a fixed interest rate scenario (Test 1 HJB) confirm the theoretical $O(\Delta x)$ convergence rate. The error curves on a log-log scale run parallel to a reference line of slope 1.
Policy Behavior: The simulations illustrate saving policies and asset distributions. Notably, in a specific test case (Test 4 HJB) with low transition risk, the model captures a scenario where high-income agents do not save (savings are zero at the borrowing limit), a behavior consistent with theoretical predictions regarding precautionary motives.

Significance and Claims
The paper claims to provide a rigorous computational framework for HA models with Epstein-Zin utility, a class of models that has become indispensable for quantifying the distributional effects of fiscal and monetary policy. The authors emphasize that their work fills a gap in the literature by offering a mathematically sound method for the $0 < \theta < 1$ case, where standard methods fail due to the lack of a comparison principle.

While the paper does not prove the uniqueness of the solution for the $0 < \theta < 1$ case (noting it remains an open question), the authors observe in their experiments that both the upward and downward algorithms converge to the same solution. They modestly suggest this implies uniqueness in practice, even if it is not yet theoretically established. The work extends the applicability of finite difference methods, previously popular for CRRA utility, to the more complex recursive utility setting, providing a foundation for future quantitative macroeconomic analysis.