Author:
(1) David Staines.
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
G.3 Unconventional Policy Settings
This final subsection looks at monetary policy at the unconventional settings discussed in Theorem 8, Propositions 23 and 24. The main result housed in the first subsection here is a proof of Proposition 22. Implicitly both rest on Theorem 3 and Proposition 16. There is some further discussion. The special case of no persistence occupies the second part.
G.3.1 Proof of Proposition 23
Proof. Focus on the case where the central bank seeks to implement Divine Coincidence, with a policy that destroys the present demand shock alongside the present output deviation, in order to stabilize the demand side of the economy.
This is without loss of generality, as other arguments would merely generate additional persistence. Appendix E.1.4 implies that it would have to set
This would imply
substituting into (176) would create a Phillips curve, where inflation evolves autonomously facing two non-degenerate shock terms. It is clear from It is clear from (298), (299) and (302) that
whilst ˜b0 = 1 is unchanged. This implies the characteristic equation takes the form
It is clear this equation factorizes with two roots inside the unit circle ensuring persistence
Remark 42. The error structure would have surprising properties at this policy setting. When β → 1, it would take the form
This would reverse the direction of the initial impulse response. The fact that this case is ruled out ex post is an example of the model preserving equilibriating mechanisms. In this case, it is likely to be inflation rising in response to demand shocks. This asymmetry is possible in opposition to Theorem 6 because monetary policy has created a new source of market failure, by destroying households’ inter-temporal substitution possibilities.
G.3.2 No Persistence Case
Proposition 33. At standard output smoothing settings, the policy in Proposition 24 that eliminates persistence is feasible.
Consider the standard case where β → 1, σ = 1. The policy reaction to inflation can be expressed as
Thus the economy is governed by
Thus the eigenvalue polynomial reads
Expanding out and removing the substitution (54) yields
In fact, the exact roots could be calculated from the expression in Footnote 102. The next section should make it easy to appreciate the prevalence of parameter uncertainty.
