Understanding the Phillips Curve: How Economic Trends Shape Inflation and Growth

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

7.1 Slope Coefficients

The first subsection has the derivation and the second houses the main results.

7.1.1 Derivation

It is most convenient to start with (30) and (31) then apply Theorem 4 from Coayla-Teran et al. [2007], appropriate for (convergent) infinite horizon trajectories

where I have used the following substitution derived from (19) and (31)

The effect of changes in output on marginal revenues depends on the propensity for inter-temporal substitution reflected by σ (the inverse of the elasticity of substitution.) If σ > 1 the propensity to smooth consumption dominates the incentive to substitution over time, such that higher output today implies higher expected future marginal revenues- conversely, if σ < 1 the substitution exceeds the smoothing incentive so higher current output is associated with lower future marginal revenues. When σ = 1 the two forces balance out. Appendix D presents an intuitive argument for σ = 1 valid for this benchmark formulation.

The error terms above reflect the difference between the actual and expected values of π, ℵ and i in period t + 1 that comes about because the future value of the structural jump errors are unknown and the model is linear in percentage deviation form. Now using the reset price equation to remove ℵ and i, the price level construction equation to express the reset price in terms of inflation and then manipulating terms in the lag operator yields

Expanding the lag operator, collecting terms and passing expectations from time t yields

Lagging the relationship and including the later period error term yields

Finally, the log-linearized price dispersion relationship and its lagged form are respectively where I have used the expression for the stochastic steady of ∆

Combining (112), (114) and (116) yields the expression for the Phillips curve equivalent to (2)

By substituting the Phillips curve into the Euler, we have

For the price dispersion recursion we have

These are sufficient to describe the canonical form.

7.1.2 Econometrics and Errors

This part briefly discusses useful forms for econometric investigation, their basic properties and challenges.

It is possible to remove price dispersion as follows[56]

After taking lags and expanding some expectations, this forms the following Vector Autoregressive Moving Average (VARMA) (p, q)

The first problem is a long term opportunity for econometrics and mathematical statistics. The second is a more pressing economic concern. It would arise in the benchmark model if one allowed for persistent Euler disturbances. Truncation strategies have been popular (see Ascari and Sbordone [2014]) but there may be scope for refinement.

7.1.3 Slope Coefficients

Some of these expressions will be discussed and interpreted further in Section 10.