Bifurcation and Its Impact on DSGE: Mathematical Insights from the Calvo Model

Table of Links

Abstract

1 Introduction

2 Mathematical Arguments

3 Outline and Preview

4 Calvo Framework and 4.1 Household’s Problem

4.2 Preferences

4.3 Household Equilibrium Conditions

4.4 Price-Setting Problem

4.5 Nominal Equilibrium Conditions

4.6 Real Equilibrium Conditions and 4.7 Shocks

4.8 Recursive Equilibrium

5 Existing Solutions

5.1 Singular Phillips Curve

5.2 Persistence and Policy Puzzles

5.3 Two Comparison Models

5.4 Lucas Critique

6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems

6.2 Equilibrium Construction

6.3 Literature Comparison

6.4 Equilibrium Analysis

7 General Linearized Phillips Curve

7.1 Slope Coefficients

7.2 Error Coefficients

8 Existence Results and 8.1 Main Results

8.2 Key Proofs

8.3 Discussion

9 Bifurcation Analysis

9.1 Analytic Aspects

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

10.2 Econometric Duality

10.3 Coefficient Properties

10.4 Microeconomic Interpretation

11 Policy Rule

12 Conclusions and References

Appendices

A Proof of Theorem 2 and A.1 Proof of Part (i)

A.2 Behaviour of ∆

A.3 Proof Part (iii)

B Proofs from Section 4 and B.1 Individual Product Demand (4.2)

B.2 Flexible Price Equilibrium and ZINSS (4.4)

B.3 Price Dispersion (4.5)

B.4 Cost Minimization (4.6) and (10.4)

B.5 Consolidation (4.8)

C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence

C.2 Extending No 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.2 Linearized DSGE Solution

E.3 Eigenvalue Conditions

E.4 Rouche’s Theorem Conditions

F Abstract Algebra and F.1 Homology Groups

F.2 Basic Categories

F.3 De Rham Cohomology

F.4 Marginal Costs and Inflation

G Further Keynesian Models and G.1 Taylor Pricing

G.2 Calvo Wage Phillips Curve

G.3 Unconventional Policy Settings

H Empirical Robustness and H.1 Parameter Selection

H.2 Phillips Curve

I Additional Evidence and I.1 Other Structural Parameters

I.2 Lucas Critique

I.3 Trend Inflation Volatility

9 Bifurcation Analysis

This section sets out to give precise mathematical explanation of what is taking place around the ZINSS of the Calvo model. A number of concepts from algebra, topology and analysis are introduced. Every mathematical construction has intuitive economic appeal. There is discussion of the significance and salience of bifurcation in macroeconomics.

The first part focuses on analysis. The goal is to review how the results here square up with basic real analysis and topology. There follow three subsections introducing ideas from algebra. The first two are topological, the last is geometric. I begin by introducing singularities and covers, with a view to aiding our understanding of market failure. Next a subsection on homology allows me to rigorously discuss how singularities give rise to "holes" in the state space and how they destroy the dynamics of the underlying model. Finally, I introduce the algebraic machinery of schemes to rigorously study limiting approximations and dig into the root cause of the mathematical pathology.

There are two main theorems. The first proves the singularity construction, foretold back in Section 3 and a little more besides. It buttresses the principal decomposition of the paper. The closing subsection ensures all abstract objects relate back to primitive economic phenomenon. The second provides a tight link between bifurcation, the lag polynomial and the infinite horizon solution. Supporting appendices on the dual cohomology theory, topological groups and categories can be found in Section F of the Appendix. There is also space to consider robustness and extensions.