Why Output and Steady States Must Change to Restore Economic Balance

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

A.3 Proof Part (iii)

This part proves the claim concerning output.

Note that the condition is sharp in θ, by considering the limits respectively as α → 0 and π → −100%. Now the Jensen-Chebyshev inequality implies that the left-hand side strictly exceeds the right-hand side, evaluated at the non-stochastic steady state. This implies that ∆ or Y must differ from their non-stochastic steady state values to restore equilibrium. The right-hand side is clearly increasing in ∆. Therefore to correct the imbalance ∆ would have to fall. However, this induces a contradiction since ∆ is convex in π for the relevant parametization. Hence Y must change to equilibriate the system. To see that it must fall take the derivative of the right-hand side

since when σ ≥ 1 the second term is non-negative and therefore unable to cancel the strictly positive first term.

Sharpness depends on parametric assumptions. Suppose 0 < σ < 1 and consider the joint limit, where the economy approaches any non-stochastic steady state and A → 0. There are two competing terms. The first represented by the top two lines above is always positive and the second is negative by assumption.

The idea is to show that sufficiently close to these limiting cases the second term will dominate. To achieve this consider the ratio of the negative term to the positive term. The proof will be complete if its limit is greater than unity.

The expression is the following product