Heterogeneous Agents New Keynesian Models

• HANK models are macroeconomic frameworks that incorporate agent-specific heterogeneity and financial constraints into the standard New Keynesian paradigm.
• They employ analytic solution techniques like Jordan canonical decomposition to trace micro-level shocks and aggregate dynamics efficiently.
• Advanced numerical and econometric methods, including deep learning approaches, enable robust estimation of policy transmission and distributional impacts.

Heterogeneous Agents New Keynesian (HANK) models constitute a class of macroeconomic frameworks that integrate agent-specific heterogeneity into the standard New Keynesian paradigm. Rooted in the recognition that real-world agents (households, firms) face idiosyncratic risks and financial constraints, HANK models depart from representative agent structures, embedding rich cross-sectional state spaces. Contemporary HANK research encompasses analytic solution construction, econometric implications, numerical global solution techniques, nonstationarity, and policy transmission mechanisms, as detailed in theoretical and computational advances across recent literature.

1. Structural Foundations and Model Specification

The HANK framework systematically augments traditional New Keynesian models, incorporating heterogeneity in agent states such as income, wealth, investment constraints, and behavioral expectations. Fundamentally, each agent is modeled as solving dynamic optimization problems subject to various frictions—e.g., incomplete markets, liquidity constraints, monopolistic competition, imperfect information, or adjustment costs. A canonical HANK specification consists of an intertemporal optimality condition for each agent $i$, combined with general equilibrium constraints for aggregate prices and quantities.

For instance, in a linear autoregressive macro setup, the evolution of output growth rates $x_{(i)}(t)$ and sentiment $y_{(i)}(t)$ for agent $i$ at time $t$ is given by

$\begin{split} x_{(i)}(t+1) &= (1 - \alpha)x_{(i)}(t) + \alpha \left[ \bar{y}(t) + \epsilon_{(i)}(t) \right], \ y_{(i)}(t+1) &= (1 - \beta) y_{(i)}(t) - \beta \left[ \bar{x}(t) + \eta_{(i)}(t) \right], \end{split}$

where $\bar{x}(t), \bar{y}(t)$ denote weighted cross-sectional averages (using convex weights $a$ and $b$) (Chen, 2015). The system's joint evolution may be succinctly encoded as

$z_{t+1} = M z_t + \gamma_t,$

with $z_t = (x_t, y_t)^\top$ and $M$ a block matrix depending on heterogeneity and aggregation details.

2. Analytic Solution Techniques and Dynamic Behavior

A distinguishing methodological advance in HANK modeling is the use of explicit solution techniques, notably Jordan canonical decomposition, to obtain closed-form representations of agent dynamics and induced aggregate cycles (Chen, 2015). By decomposing the transition matrix $M$,

$M = Q J Q^{-1},$

one attains explicit solutions for the state vector evolution: $$z_{t+1} = Q J^{t+1} Q^{-1} z_0 + \sum_{i=0}^t Q J^i Q^{-1} \gamma_{t-i}.$$ This approach obviates iterative aggregation in simulations, enabling direct tracing of joint fluctuations, oscillatory modes, and nonstationarity via the eigenstructure of $M$. Both stationarity and explosive dynamics are governed by the modulus and sign of eigenvalues; for example, if $|\lambda_i| < 1$ for all $i$, variances converge, while $|\lambda_i| \geq 1$ can induce persistent or unbounded behavior.

Such explicit solutions are rarely available in standard (nonlinear or numerically solved) HANK models and underlie deeper insights into how micro-level shocks propagate to aggregate variables.

3. Aggregation, Business Cycles, and Distributional Dynamics

Aggregation mechanisms in HANK frameworks translate micro-level variations to aggregate macroeconomic dynamics. Weighted averages of agent-level outcomes (e.g., growth rates) induce an explicit second-order difference equation for the aggregate cycle: $$\Delta^2 \bar{x}(t) + (\alpha + \beta) \Delta \bar{x}(t) + 2\alpha\beta \bar{x}(t) = h(t),$$ where $h(t)$ encodes aggregate shocks from micro-level disturbances (Chen, 2015). The solution exhibits damped or periodic trajectories, with parameters governing amplitude and frequency: $\bar{x}(t) = c_1 |\rho_1|^t \cos(c_2 + \omega t),$ where $$\omega = -\arctan (\kappa_1^{-1} \sqrt{|\Delta_1|})$$ and $\Delta_1 = \kappa_1^2 - 4\kappa_2$. Complex roots dictate cyclical fluctuations; parameter regimes justify damped-pendulum dynamics, common in macroeconomic business cycle models.

Crucially, the induced aggregate dynamics may mask persistent distributional shifts within heterogeneous agent populations, necessitating tracking of the entire cross-sectional distribution rather than a few summary statistics.

4. Computational Approaches: Global Solution and Neural Networks

Numerical HANK models involve high-dimensional state spaces, rendering classical solution techniques (e.g., perturbation, simulation-based moment matching) burdensome. Modern computational strategies approximate the agent distribution via finite agents, discrete grids, or projection onto basis functions (Gu et al., 19 Jun 2024). The equilibrium is then characterized by solving a nonlinear master equation: $0 = \mathcal{L} V(x, z, g),$ where $V$ is the value function of a representative agent (or its derivative), $z$ is an aggregate shock, and $g$ parameterizes the agent distribution.

Deep learning architectures, specifically Economic Model Informed Neural Networks (EMINN), are trained to satisfy master PDE residuals using sampling across state-space points. Loss functions incorporate economic shape constraints, and methods inspired by DGM or PINNs break the curse of dimensionality, yielding global solutions for complex heterogeneous agent models.

Applications span variants of Krusell–Smith, Aiyagari, Khan–Thomas, and spatial migration models, achieving residual levels around $10^{-5}$ and close quantitative agreement with traditional solution techniques. Projection methods employing eigenfunction bases isolate persistent distributional features, efficiently capturing aggregate price determinants.

5. Econometric Properties, Estimation, and Micro Data Integration

HANK models have precipitated new approaches for empirical identification and estimation. A prominent indirect inference methodology constructs a reduced-rank dynamic factor model from simulated micro data, exploiting low-rank structure as cross-sectional dimension grows (Iao et al., 17 Feb 2024). The infinite-order VAR representation collapses to a first-order VAR: $y_t = B_1^\infty y_{t-1} + a_t,$ where $B_1^\infty = G K$ incorporates filtering effects. The likelihood approximation, derived via Dynamic Mode Decomposition (DMD), enables fast and robust classical or Bayesian estimation compatible with sequence-space solution methods.

Simulation evidence demonstrates that exploiting the full cross-sectional distribution, rather than moments or aggregates, substantially reduces parameter bias and standard error. This motivates the use of richer micro data sources for model calibration, with implications for policy analysis in settings with pronounced heterogeneity.

6. Policy Transmission, Welfare Implications, and Comparative Statics

HANK models revise the New Keynesian transmission mechanism by embedding liquidity and financial constraints, incomplete markets, and distributional channels. Impact multipliers for monetary policy (e.g., $\Omega^p$, $\Omega^x$ for inflation and output) depend on heterogeneity parameters—such as the share of hand-to-mouth households $\lambda$, nominal rigidity parameters ($\eta_p$, $\eta_w$, $\psi_p$, $\psi_w$)—and Taylor rule responsiveness $\phi$ (Miyazaki, 16 Aug 2025). Conditions such as

$\phi > 1 + \frac{1}{\kappa_2}$

guarantee amplification of transmission.

The modified aggregate welfare loss function,

$$\mathbb{W}_t = -\frac{\gamma+\varphi}{2} x_t^2 - \frac{\eta_w}{2}(\pi_t^w)^2 - \frac{\tilde{\eta}_p}{2}(\pi_t^p)^2 - \frac{1}{2} \lambda^2 \left( \gamma\delta_c^2 + \varphi\delta_n^2 + \eta_w\delta_w^2 \right)d_t^2 + \text{t.i.p.},$$

features a heterogeneity–adjusted inflation penalty $\tilde{\eta}_p$, which rises with heterogeneity and liquidity constraints, increasing the social cost of inflation and modifying the policy trade-off.

Comparative statics delineate how amplification of monetary policy effects intensifies with greater heterogeneity and nominal rigidity, whereas policy impacts are dampened by more aggressive Taylor rule settings.

7. Open Problems, Extensions, and Methodological Innovations

Recent advancements address persistent challenges in HANK modeling:

• Distributional Dynamics: Time-interlaced, self-consistent master systems formalize the endogenous evolution of the agent cross-sectional distribution (Lyasoff, 2023). The development of transport operators (e.g., $\mathcal{T}_{t,x}^y(F)$) and backward–forward coupling frameworks clarify the limits of approximate aggregation and the importance of retaining full distributional information.
• Bounded Rationality: N-bounded foresight equilibrium models relax the rational expectations assumption, allowing agents to forecast only $N$ steps ahead and treat variables as constant thereafter (Islah et al., 23 Feb 2025). This lowers computational burden and incorporates behavioral realism (time inconsistency, limited attention), while endogenous forecast errors inject additional variability into equilibrium outcomes.
• Mean-Field Games: Explicit mean-field game approaches model strategic interdependence between heterogeneous groups (experts and households), exploring boom–bust cycles and financial frictions with analytically tractable Nash equilibrium representations (Vu et al., 15 Feb 2025).
• Reinforcement Learning Synthesis: MARL-BC integrates deep multi-agent reinforcement learning with RBC and HANK paradigms, recovering benchmark results in limit cases, and efficiently simulating rich heterogeneity without exhaustive specification (Gabriele et al., 14 Oct 2025).

Contemporary HANK research is thus characterized by convergence of analytic, numerical, and empirical methodologies—enabling rigorous paper of heterogeneity, transmission mechanisms, and policy design in high-dimensional, frictional economies. Ongoing investigations target the quantification of aggregation errors, further generalization of computational techniques, and robust micro–macro data integration.