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Price Levels in Heterogeneous-Agent Models

Felix Höfer

TL;DR

The paper develops a rigorous continuous-time framework for the Fiscal Theory of the Price Level with heterogeneous agents, cast as a mean-field game. It proves the existence of stationary Huggett and Aiyagari equilibria, including multiplicity for small deficits, and establishes regularity of value functions, an invariant distribution, and continuity properties with respect to the interest rate. A key technical contribution is the constrained viscosity-solution analysis of the household problem, together with an invariant-measure theory that shows aggregate savings explode as the discount rate is approached. The results reveal how price-level multiplicity arises from fiscal deficits in heterogeneous-agent settings and connect the Aiyagari model with capital to the simpler Huggett case via a CRRA scaling limit, with convergence as the capital elasticity tends to zero. These findings advance the mathematical understanding of FTPL in realistic economies and offer a robust foundation for exploring policy-induced price dynamics in the presence of idiosyncratic income risk.

Abstract

We study a model of the Fiscal Theory of the Price Level (FTPL) in a Bewley-Huggett-Aiyagari framework with heterogeneous agents. The model is set in continuous time, and ex post heterogeneity arises due to idiosyncratic, uninsurable income shocks. Such models have a natural interpretation as mean-field games, introduced by Huang, Caines, and Malhamé and by Lasry and Lions. We highlight this connection and discuss the existence and multiplicity of stationary equilibria in models with and without capital. Our focus is on the mathematical analysis, and we prove the existence of two equilibria in which the government runs constant primary deficits, which in turn implies the existence of multiple price levels.

Price Levels in Heterogeneous-Agent Models

TL;DR

The paper develops a rigorous continuous-time framework for the Fiscal Theory of the Price Level with heterogeneous agents, cast as a mean-field game. It proves the existence of stationary Huggett and Aiyagari equilibria, including multiplicity for small deficits, and establishes regularity of value functions, an invariant distribution, and continuity properties with respect to the interest rate. A key technical contribution is the constrained viscosity-solution analysis of the household problem, together with an invariant-measure theory that shows aggregate savings explode as the discount rate is approached. The results reveal how price-level multiplicity arises from fiscal deficits in heterogeneous-agent settings and connect the Aiyagari model with capital to the simpler Huggett case via a CRRA scaling limit, with convergence as the capital elasticity tends to zero. These findings advance the mathematical understanding of FTPL in realistic economies and offer a robust foundation for exploring policy-induced price dynamics in the presence of idiosyncratic income risk.

Abstract

We study a model of the Fiscal Theory of the Price Level (FTPL) in a Bewley-Huggett-Aiyagari framework with heterogeneous agents. The model is set in continuous time, and ex post heterogeneity arises due to idiosyncratic, uninsurable income shocks. Such models have a natural interpretation as mean-field games, introduced by Huang, Caines, and Malhamé and by Lasry and Lions. We highlight this connection and discuss the existence and multiplicity of stationary equilibria in models with and without capital. Our focus is on the mathematical analysis, and we prove the existence of two equilibria in which the government runs constant primary deficits, which in turn implies the existence of multiple price levels.

Paper Structure

This paper contains 17 sections, 34 theorems, 219 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Huggett and Aiyagari models
  3. Connection to mean-field games
  4. Main results
  5. Discussion and related literature
  6. Household problem
  7. Viscosity characterization
  8. Invariant distribution
  9. Proof of Theorem \ref{['thm:invariant-distribution']}
  10. Proof of Proposition \ref{['prop:lower-interest-bound']}
  11. Proof of Proposition \ref{['prop:linearity-of-A-in-tau']}
  12. Proofs of Proposition \ref{['prop:continuity-G*']} and Theorem \ref{['thm:explosion-at-rho']}
  13. General equilibrium
  14. Huggett model
  15. Aiyagari model
  16. ...and 2 more sections

Key Result

Proposition 1.3

Given a tax-and-transfer function $\tau(\cdot)$, let $\mu^*\in\mathscr{P}(\mathscr{X})$ satisfy $0<\int a \mu^*(\mathrm{d} a,\mathrm{d} z)<\infty$. Then, $\mu^*$ is a stationary equilibrium of the above mean-field game if and only if there exists a stationary monetary Huggett equilibrium $\Xi=(\tau(

Figures (3)

  • Figure 1: (Huggett) Plot of $r\mapsto A(r,\tau)$ and $r\mapsto \tau/r$ for $\tau\in(0,1)$.
  • Figure 2: (Huggett) Plots of $r\mapsto A(r,\tau)$ and $r\mapsto \tau/r$ for two values of $\tau<0$.
  • Figure 3: (Aiyagari) Plots of $r\mapsto A(r,1,\tau)$ and $r\mapsto S(r)$ for $\tau<0$ and two values of $\alpha\in(0,1)$

Theorems & Definitions (67)

  • Remark 1.1
  • Remark 1.2
  • Proposition 1.3
  • Remark 1.4: Typical vs representative
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Proposition 1.11
  • Proposition 1.12
  • Proposition 1.13
  • ...and 57 more