Network Expectations in HANK: Variational Message Passing as Approximate Bayesian Learning

Rational expectations (RE) in Heterogeneous Agent New Keynesian (HANK) models require each household to forecast the evolving cross-sectional distribution of wealth—a demand that, as Moll (2025) argues, leads to the Master equation, an intractable infinite-dimensional fixed-point problem. Standard RE-HANK solution methods such as SSJ sidestep this problem through linearization, but in doing so they silently eliminate the very distributional complexity that RE would require agents to forecast, potentially imposing implicit cognitive constraints while claiming rational expectations. Responding to this critique, we replace RE with approximate Bayesian learning on a household-level factor graph, embedded in a temporary equilibrium (TE) framework where current-period prices are the outcome of market clearing and are therefore unavailable when agents form beliefs. Each agent maintains a perceived law of motion (PLM) for aggregate prices and updates its parameters via Variational Message Passing (VMP), exchanging precision-weighted messages with neighbors on a social network while minimizing variational free energy, which decomposes into interpretable accuracy and complexity terms. We compare VMP-NET—in which neighbor interactions enrich the information structure—against a Moll-TE benchmark that updates PLM coefficients via constant-gain SGD alone (Krusell and Smith, 1998; Moll and Ryzhik, 2025) and SSJ-RE in a standard one-asset HANK with endogenous labor supply. Both bounded-rationality specifications produce cumulative output multipliers that approximately bracket the RE benchmark (10.8 and 9.5 versus 9.3), while the welfare gap of approximately 8.5% in consumption equivalents quantifies the structural distance between local variational inference and model-consistent expectations. A precision sweep shows that tightening the cognitive constraint in VMP produces output dynamics increasingly similar to SSJ-RE, suggesting that linearization may act as an implicit capacity limitation, although the underlying generating mechanisms differ.