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

C.2 Extending No Persistence

This subsection generalizes lack of persistence in two directions. The first is to show that in the absence of price dispersion, it extends to all approximations from ZINSS. Similar results are well-known for Rotemberg. The second shows that even when the output gap is adjusted to allow technology shocks, inflation still responds to demand shocks only on impact.

C.2.1 Higher Order Dynamics

C.2.2 Dynamics with Unobserved Natural Rate

Although no persistence breaks down, propagation problems still arise, in particular, with respect to inflation and aggregate demand shocks. The destruction of endogenous persistence when shock persistence takes limiting values is a theme common to the whole paper. This analysis has appeared before. The task at hand is to diagonalize (240) and substitute into the forward solution of (239) and then compute the persistence. First solve (241)-(243), this yields

The relevant inverse matrix is

using the forward solution and several rounds of substitution yields the following expression for inflation

Deploying the persistent properties of the errors, communicated in Subsection 4.7, this yields

It is intuitive that a positive supply shock drives down inflation. Away from the two limiting cases where it vanishes, as with Divine Coincidence, there is positive persistence, measured by the product moment correlation coefficient

It is still pathological to Keynesian intuition that inflation does not respond to demand shocks. For output the expression is

Again, the desired limiting cases arise. There can be overshooting or undershooting with respect to the efficient benchmark.110 However, the underlying message is that this system cannot generate persistent responses from inflation to demand shocks.