Rouche’s Theorem: Conditions for Preventing Recursive Equilibrium
by Keynesian TechnologyDecember 11th, 2024
Too Long; Didn't Read
This part extends Proposition 17, using Rouche’s theorem to establish conditions that prevent the existence of recursive equilibrium. The cases presented show how the number of roots outside the unit circle (zero, one, three, or four) determines equilibrium outcomes, with price dispersion playing a crucial role.
Author:
(1) David Staines.
Table of Links
4 Calvo Framework and 4.1 Household’s Problem
4.3 Household Equilibrium Conditions
4.5 Nominal Equilibrium Conditions
4.6 Real Equilibrium Conditions and 4.7 Shocks
5.2 Persistence and Policy Puzzles
6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems
7 General Linearized Phillips Curve
8 Existence Results and 8.1 Main Results
9.2 Algebraic Aspects (I) Singularities and Covers
9.3 Algebraic Aspects (II) Homology
9.4 Algebraic Aspects (III) Schemes
9.5 Wider Economic Interpretations
10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs
10.4 Microeconomic Interpretation
Appendices
A Proof of Theorem 2 and A.1 Proof of Part (i)
B Proofs from Section 4 and B.1 Individual Product Demand (4.2)
B.2 Flexible Price Equilibrium and ZINSS (4.4)
B.4 Cost Minimization (4.6) and (10.4)
C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence
D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium
D.2 Profits and Long-Run Growth
E Slopes and Eigenvalues and E.1 Slope Coefficients
E.4 Rouche’s Theorem Conditions
F Abstract Algebra and F.1 Homology Groups
F.4 Marginal Costs and Inflation
G Further Keynesian Models and G.1 Taylor Pricing
G.3 Unconventional Policy Settings
H Empirical Robustness and H.1 Parameter Selection
I Additional Evidence and I.1 Other Structural Parameters
I.3 Trend Inflation Volatility
E.4 Rouche’s Theorem Conditions
This part augments Proposition 17 with more cases derived from Rouche’s theorem.
Proposition 31. Any of the following conditions would prevent the existence of a recursive equilibrium.
Proof. Following Proposition 17, it is sufficient to note that the cases (I), (II), (III) and (IV) correspond respectively to the conditions guaranteeing zero, one, three and four roots outside the unit circle.
If price dispersion were jettisoned the system would be governed by (163), in which case the sufficient condition would be
whilst the following would imply non-existence
This paper is available on arxiv under CC 4.0 license.
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