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Learning Paths to Multi-Sector Equilibrium: Belief Dynamics Under Uncertain Returns to Scale

Stefano Nasini, Rabia Nessah, Bertrand Wigniolle

TL;DR

The paper develops a dynamic Bayesian framework for learning unknown returns-to-scale in a multi-sector general equilibrium with private information. Firms alternate between input optimization and Bayesian belief updating (MAP) based on noisy production observations, leading to path-dependent and path-independent learning schemes. It derives conditions under which firms can learn the true returns-to-scale and shows that short-memory learning typically outperforms full-history updating due to Bayesian filtering effects. The analysis highlights how endogenously generated input decisions carry all necessary information for learning, and it extends to higher-dimensional settings, offering a foundation for studying belief convergence in complex economies.

Abstract

This paper explores the dynamics of learning in a multi-sector general equilibrium model where firms operate under incomplete information about their production returns to scale. Firms iteratively update their beliefs using maximum a-posteriori estimation, derived from observed production outcomes, to refine their knowledge of their returns to scale. The implications of these learning dynamics for market equilibrium and the conditions under which firms can effectively learn their true returns to scale are the key objects of this study. Our results shed light on how idiosyncratic shocks influence the learning process and demonstrate that input decisions encode all pertinent information for belief updates. Additionally, we show that a long-memory (path-dependent) learning which keeps track of all past estimations ends up having a worse performance than a short-memory (path-independent) approach.

Learning Paths to Multi-Sector Equilibrium: Belief Dynamics Under Uncertain Returns to Scale

TL;DR

The paper develops a dynamic Bayesian framework for learning unknown returns-to-scale in a multi-sector general equilibrium with private information. Firms alternate between input optimization and Bayesian belief updating (MAP) based on noisy production observations, leading to path-dependent and path-independent learning schemes. It derives conditions under which firms can learn the true returns-to-scale and shows that short-memory learning typically outperforms full-history updating due to Bayesian filtering effects. The analysis highlights how endogenously generated input decisions carry all necessary information for learning, and it extends to higher-dimensional settings, offering a foundation for studying belief convergence in complex economies.

Abstract

This paper explores the dynamics of learning in a multi-sector general equilibrium model where firms operate under incomplete information about their production returns to scale. Firms iteratively update their beliefs using maximum a-posteriori estimation, derived from observed production outcomes, to refine their knowledge of their returns to scale. The implications of these learning dynamics for market equilibrium and the conditions under which firms can effectively learn their true returns to scale are the key objects of this study. Our results shed light on how idiosyncratic shocks influence the learning process and demonstrate that input decisions encode all pertinent information for belief updates. Additionally, we show that a long-memory (path-dependent) learning which keeps track of all past estimations ends up having a worse performance than a short-memory (path-independent) approach.

Paper Structure

This paper contains 32 sections, 22 theorems, 213 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The model
  3. The consumer.
  4. The producers.
  5. Market clearing.
  6. Decision-estimation dynamics.
  7. The equilibrium
  8. The learning paths
  9. Path-dependent (non-Markovian) learning
  10. Limit properties and convergence under the path-dependent learning
  11. Path-independent (Markovian) learning
  12. Limit properties and convergence under path-independent learning
  13. A Bayesian (deterministic belief) filter with endogenous feedback
  14. Extension to high-dimensional learning
  15. Conclusions
  16. ...and 17 more sections

Key Result

Proposition 1

We have the following equilibrium labor and prices, for each sector $i \in \mathcal{N}$:

Figures (6)

  • Figure 1: Learning path $\{\zeta_i(t)\}_t$ with $\tau=0.1$.
  • Figure 2: Learning path $\{\zeta_i(t)\}_t$ with $\tau=0.01$.
  • Figure 3: Histograms of $w(t) \delta(t)/GDP$ for three different economies.
  • Figure 4: Learning path $\{\zeta_i(t)\}_t$ with $\sigma_i = 0.1$ and $\tau=0.1$
  • Figure 5: Learning path $\{\zeta_i(t)\}_t$ with $\sigma_i = 0.1$ and $\tau=0.01$.
  • ...and 1 more figures

Theorems & Definitions (46)

  • Proposition 1
  • Lemma 1
  • Proposition 2: MAP dynamics (path-dependent)
  • Proposition 3
  • Proposition 4: Expectation and variance of the MAP estimator (path-dependent)
  • Proposition 5: Mode of the MAP estimator (path-dependent)
  • Proposition 6: Convergence of the expectation (path-dependent)
  • Proposition 7: Demographic expansion (path-dependent)
  • Proposition 8: Convergence of the mode (path-dependent)
  • Example 1
  • ...and 36 more