Achdou, Han, Lasry, Lions and Moll (2022) recast the Aiyagari-Bewley-Huggett model of income and wealth distribution in continuous time as a coupled Hamilton-Jacobi-Bellman / Kolmogorov-forward system and proved that its stationary equilibrium is unique when the intertemporal elasticity of substitution (IES) is weakly greater than one and borrowing is prohibited; they explicitly asked for uniqueness under weaker assumptions than the IES condition. The problem is to determine whether the stationary equilibrium of the standard continuous-time Huggett (bond) or Aiyagari (production) economy remains unique when the IES is below one (e.g., CRRA utility with risk aversion above one, the empirically standard calibration), or to construct an economy in this class with multiple stationary equilibria. Later work has sharply circumscribed but not resolved the question: no known sufficient condition covers the standard model class with risk aversion above one (the sole exceptions, CARA-utility Huggett economies without a borrowing constraint, do not cover CRRA calibrations); a 2025 preprint shows that aggregate saving need not be increasing in the interest rate once risk aversion exceeds one, closing the natural proof route, and proves multiplicity only in continuous-time variants with government deficits, while numerical studies exhibit multiplicity in the discrete-time model with disaster income states. The rigorous open core for the standard economy survives.

Background

The Aiyagari-Bewley-Huggett model is the workhorse theory of income and wealth distribution in macroeconomics. A continuum of infinitely-lived households holds a single riskless asset aa and receives exogenous labor income that follows a finite-state Markov chain; this idiosyncratic income risk is uninsurable, so households engage in precautionary saving subject to a borrowing constraint. Households interact only through prices: in the Huggett version the asset is a bond in fixed supply and the interest rate clears the bond market, while in the Aiyagari version a representative firm with constant-returns-to-scale production rents capital and labor, and the interest and wage rates clear the factor markets. A stationary equilibrium is a situation in which prices are constant and the joint distribution of income and wealth is invariant, so that individual households churn through the distribution while its shape stays fixed.

Achdou, Han, Lasry, Lions and Moll (2022) recast this model in continuous time. The household's optimal control problem becomes a Hamilton-Jacobi-Bellman (HJB) equation, the nonlinear partial differential equation (PDE) characterizing the value function of a continuous-time stochastic optimization; the cross-sectional distribution of households evolves according to a Kolmogorov forward (Fokker-Planck) equation, the formal adjoint of the HJB operator. The stationary equilibrium is thus a coupled backward-forward system of the kind studied in mean field game (MFG) theory, where each atomistic agent optimizes against the population distribution generated by everyone's optimizing behavior. Two structural features matter here: the borrowing constraint enters only as a boundary condition on the HJB equation, and the stationary wealth distribution typically contains a Dirac point mass (a point mass in the distribution) at the borrowing constraint.

Within this framework the paper proves, as its Proposition 5, that if the intertemporal elasticity of substitution (IES) - the elasticity of consumption growth with respect to the interest rate, given by IES(c)=−u′(c)/(u′′(c)c)IES(c) = -u'(c)/(u''(c)c) for utility function uu and equal to 1/γ1/\gamma for constant-relative-risk-aversion (CRRA) utility with coefficient γ\gamma - is weakly greater than one for all consumption levels, and the borrowing constraint is a strict no-borrowing limit a≥0a \ge 0, then individual consumption is strictly decreasing in the interest rate, the stationary distribution shifts right in the sense of first-order stochastic dominance (more probability mass at higher wealth levels) as the rate rises, aggregate saving S(r)S(r) is strictly increasing in rr, and the stationary equilibrium is unique. The proof exploits a decomposition of the consumption response to an interest-rate change into substitution and income effects: with IES ≥1\ge 1 the substitution effect dominates. The extension to the production (Aiyagari) version needs one additional condition: the elasticity of substitution of the production function must exceed the capital share of output. Uniqueness is economically important because, without it, the model could in principle exhibit poverty traps, history dependence, and ill-defined comparative statics and policy counterfactuals.

The hypotheses of Proposition 5 are, however, stronger than economic practice suggests they need to be, and the paper says so explicitly: 'Future work should try to prove uniqueness under weaker assumptions than the IES being greater than one.' The question is of first-order empirical relevance because the most common quantitative calibrations use CRRA utility with γ≥2\gamma \ge 2, i.e. an IES of one half or less, which lies outside the covered regime; moreover the proof also relies on the strict no-borrowing limit, whereas many applications allow strictly negative wealth. In discrete time the analogous boundary is equally rigid: Light (published online 2018) proved uniqueness for the Aiyagari model with CRRA risk aversion ≤1\le 1 and elasticity of substitution ≥1\ge 1, and a 2024 survey of uniqueness of competitive equilibrium by Toda and Walsh concludes that risk aversion ≤1\le 1 remains the only robust sufficient condition known across heterogeneous-agent macro models, while risk aversion above one is widely used in applications.

Problem Statement

In the continuous-time Aiyagari-Bewley-Huggett framework of Achdou, Han, Lasry, Lions and Moll (2022), Proposition 5 establishes that the stationary equilibrium is unique under two hypotheses: the intertemporal elasticity of substitution satisfies IES(c)=−u′(c)/(u′′(c)c)≥1IES(c) = -u'(c)/(u''(c)c) \ge 1 for all consumption levels cc, and the borrowing constraint is a strict no-borrowing limit a≥0a \ge 0. The open problem declared by the authors is to prove uniqueness under weaker assumptions than the IES being greater than one, i.e. to determine whether this conclusion survives weakening the IES hypothesis. Specifically, either:

(i) prove that the stationary equilibrium of the standard continuous-time Huggett economy (bond in fixed supply) or Aiyagari economy (representative firm with constant-returns production) is unique under conditions strictly weaker than IES ≥1\ge 1 everywhere. The target is the class of utility functions with IES below one; CRRA utility with coefficient of relative risk aversion γ>1\gamma > 1 is the empirically central case that any such condition must cover, and it suffices to establish the result for either the Huggett or the Aiyagari variant. A proof may additionally permit a nonzero borrowing limit a‾<0\underline{a} < 0, but relaxing only the no-borrowing limit while keeping IES ≥1\ge 1 does not answer the question; or

(ii) construct, within this same standard model class (idiosyncratic income risk only, a single asset, interaction only through prices, fixed bond supply or a production economy without fiscal deficits), an explicit economy possessing two or more distinct stationary equilibria, with IES below one for all consumption levels (as with CRRA utility with coefficient of relative risk aversion γ>1\gamma > 1, for which IES=1/γ<1IES = 1/\gamma < 1) - hence necessarily an economy outside the hypotheses of Proposition 5, since Proposition 5 itself rules out multiplicity under its hypotheses.

The question must be resolved for the standard economy itself; multiplicity results for neighboring variants - discrete-time models, CARA utility without a borrowing constraint, or continuous-time economies with constant government deficits - do not answer it. Note that the aggregate-saving monotonicity route of the original proof is unavailable: it is now known (Höfer, 2025) that for CRRA γ>1\gamma > 1 aggregate saving need not be increasing in the interest rate.

References

  1. Achdou, Y., Han, J., Lasry, J.-M., Lions, P.-L., Moll, B. (2022). 'Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach.' Review of Economic Studies 89(1), 45-86. DOI: 10.1093/restud/rdab002. https://doi.org/10.1093/restud/rdab002
  2. Achdou, Y., Han, J., Lasry, J.-M., Lions, P.-L., Moll, B. (2017). 'Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach.' NBER Working Paper 23732 (earlier draft of the Restud article). DOI: 10.3386/w23732. https://doi.org/10.3386/w23732
  3. Höfer, F. (2025, revised June 2026). 'Stationary Heterogeneous-Agent Models in Continuous Time.' arXiv:2510.26065 (v2, revised 5 June 2026). https://arxiv.org/abs/2510.26065
  4. Light, B., Weintraub, G. Y. (2022). 'Mean Field Equilibrium: Uniqueness, Existence, and Comparative Statics.' Operations Research 70(1), 585-605. DOI: 10.1287/opre.2020.2090 (arXiv:1903.02273v3). https://doi.org/10.1287/opre.2020.2090
  5. Light, B. (2018, published online; Economic Theory 69(2), 435-450, 2020). 'Uniqueness of equilibrium in a Bewley-Aiyagari model.' DOI: 10.1007/s00199-018-1167-z. https://doi.org/10.1007/s00199-018-1167-z
  6. Light, B. (2026). 'Invariant Distributions in Nonlinear Markov Chains with Aggregators: Theory, Computation, and Applications.' Operations Research. DOI: 10.1287/opre.2024.1547 (arXiv:2303.15898). https://doi.org/10.1287/opre.2024.1547
  7. Açıkgöz, Ö. T. (2018). 'On the existence and uniqueness of stationary equilibrium in Bewley economies with production.' Journal of Economic Theory 173, 18-55. DOI: 10.1016/j.jet.2017.10.006. https://doi.org/10.1016/j.jet.2017.10.006
  8. Toda, A. A. (2017). 'Huggett economies with multiple stationary equilibria.' Journal of Economic Dynamics and Control 84, 77-90. DOI: 10.1016/j.jedc.2017.09.002. https://doi.org/10.1016/j.jedc.2017.09.002
  9. Toda, A. A., Walsh, K. J. (2024). 'Recent advances on uniqueness of competitive equilibrium.' Journal of Mathematical Economics 113, 103008. DOI: 10.1016/j.jmateco.2024.103008 (arXiv:2402.00998). https://doi.org/10.1016/j.jmateco.2024.103008
  10. Walsh, K. J., Young, E. R. (2024). 'Equilibrium Multiplicity in Aiyagari and Krusell-Smith.' Federal Reserve Bank of Cleveland Working Paper 24-13. DOI: 10.26509/frbc-wp-202413. https://doi.org/10.26509/frbc-wp-202413
  11. Hu, T.-W., Shmaya, E. (2019). 'Unique monetary equilibrium with inflation in a stationary Bewley-Aiyagari model.' Journal of Economic Theory 180, 368-382. DOI: 10.1016/j.jet.2019.01.003. https://doi.org/10.1016/j.jet.2019.01.003
  12. Kirkby, R. (2018, published online; Macroeconomic Dynamics 23(6), 2469-2508, 2019). 'Bewley-Huggett-Aiyagari Models: Computation, Simulation, and Uniqueness of General Equilibrium.' DOI: 10.1017/S1365100517000761. https://doi.org/10.1017/S1365100517000761
  13. Bayer, C., Rendall, A. D., Wälde, K. (2019). 'The invariant distribution of wealth and employment status in a small open economy with precautionary savings.' Journal of Mathematical Economics 85, 17-37. DOI: 10.1016/j.jmateco.2019.08.003. https://doi.org/10.1016/j.jmateco.2019.08.003
  14. Kaplan, G., Nikolakoudis, G., Violante, G. (2023). 'Price Level and Inflation Dynamics in Heterogeneous Agent Economies.' NBER Working Paper 31433. DOI: 10.3386/w31433. https://doi.org/10.3386/w31433
  15. Shigeta, Y. (2023). 'Existence of Invariant Measure and Stationary Equilibrium in a Continuous-Time One-Asset Aiyagari Model: A Case of Regular Controls Under Markov Chain Uncertainty.' SSRN 4510345. DOI: 10.2139/ssrn.4510345. https://doi.org/10.2139/ssrn.4510345