Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Unified Framework for Equilibrium Selection in DSGE Models

Mitsuhiro Okano

TL;DR

This paper reframes DSGE models as fixed-point selection systems characterized by a triplet (S, T, Π), where S encodes constraints, T embeds forward-looking self-reference, and Π selects a unique equilibrium from the generally multiplicity-laden Fix(T). It shows that Blanchard–Kahn conditions function as a specific selection operator Π_BK and that common solvers (QZ, OccBin) implement the same semantic operation, explaining their identical results across implementations. The work further demonstrates that alternative selectors (e.g., minimal-variance, fiscal anchoring) are feasible under indeterminacy, thereby treating equilibrium selection as a policy choice rather than a mathematical necessity. By applying the framework to a New Keynesian model, the paper clarifies how policy parameters interact with spectral structure to determine determinacy and demonstrates how regime-switching (OccBin) can be captured as an iterated application of selection operators. Overall, the (S, T, Π) framework provides a unified, verifiable language for comparing solution methods, diagnosing solver outputs, and exploring policy-driven selector choices in DSGE analysis.

Abstract

This paper characterizes DSGE models as fixed-point selection devices for self-referential economic specifications. We formalize this structure as $(S, T, Π)$: specification, self-referential operator, and equilibrium selector. The framework applies to any DSGE model through compositional pipelines where specifications are transformed, fixed points computed, and equilibria selected. We provide formal results and computational implementation for linear rational-expectations systems, reinterpreting Blanchard-Kahn conditions as a specific selection operator and verifying that standard solution methods (such as QZ decomposition and OccBin) realize this operation. We show that alternative selectors (minimal-variance, fiscal anchoring) become available under indeterminacy, revealing selection as a policy choice rather than a mathematical necessity. Our framework reveals the formal structure underlying DSGE solution methods, enabling programmatic verification and systematic comparison of selection rules.

A Unified Framework for Equilibrium Selection in DSGE Models

TL;DR

This paper reframes DSGE models as fixed-point selection systems characterized by a triplet (S, T, Π), where S encodes constraints, T embeds forward-looking self-reference, and Π selects a unique equilibrium from the generally multiplicity-laden Fix(T). It shows that Blanchard–Kahn conditions function as a specific selection operator Π_BK and that common solvers (QZ, OccBin) implement the same semantic operation, explaining their identical results across implementations. The work further demonstrates that alternative selectors (e.g., minimal-variance, fiscal anchoring) are feasible under indeterminacy, thereby treating equilibrium selection as a policy choice rather than a mathematical necessity. By applying the framework to a New Keynesian model, the paper clarifies how policy parameters interact with spectral structure to determine determinacy and demonstrates how regime-switching (OccBin) can be captured as an iterated application of selection operators. Overall, the (S, T, Π) framework provides a unified, verifiable language for comparing solution methods, diagnosing solver outputs, and exploring policy-driven selector choices in DSGE analysis.

Abstract

This paper characterizes DSGE models as fixed-point selection devices for self-referential economic specifications. We formalize this structure as : specification, self-referential operator, and equilibrium selector. The framework applies to any DSGE model through compositional pipelines where specifications are transformed, fixed points computed, and equilibria selected. We provide formal results and computational implementation for linear rational-expectations systems, reinterpreting Blanchard-Kahn conditions as a specific selection operator and verifying that standard solution methods (such as QZ decomposition and OccBin) realize this operation. We show that alternative selectors (minimal-variance, fiscal anchoring) become available under indeterminacy, revealing selection as a policy choice rather than a mathematical necessity. Our framework reveals the formal structure underlying DSGE solution methods, enabling programmatic verification and systematic comparison of selection rules.
Paper Structure (42 sections, 18 equations, 1 figure)

This paper contains 42 sections, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Axiomatic Framework: The (S, T, Π) Triad
  4. Basic Setup
  5. The Four Axioms
  6. Definition of DSGE System
  7. Minimality of the Axioms
  8. The Logic of Self-Reference: Disentangling Specification, Dynamics, and Selection
  9. Rational Expectations Equilibrium (REE) as Property, Not Primitive
  10. Blanchard--Kahn as Selection Operator
  11. Linear Rational Expectations Setting
  12. The Fixed-Point Set Fix(T)
  13. Selection via Stability: The Π_{BK} Operator
  14. QZ Decomposition and Spectral Filtering
  15. BK-as-Selection Theorem
  16. ...and 27 more sections

Figures (1)

  • Figure 1: Determinacy regions in $(\varphi_\pi,\varphi_y)$ space. Panel (a) shows where $\Pi_{BK}$ is defined; panel (b) shows how $\Pi_{MV}$ extends selection into indeterminate regions.