Stochastic Equilibrium the Lucas Critique and Keynesian Economics: Equilibrium Construction

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

6.2 Equilibrium Construction

Here I characterize the stochastic equilibrium of the Calvo New Keynesian economy. The main tool is Birkhoff’s ergodic theorem. The stochastic steady state resembles its non-stochastic counterpart, except there are expectation terms attached to future realizations of non-linear functions of the variables (π, y, ∆, ψ, A) underpinning the recursive equilibrium, described back in Section 4.8 and Proposition 4. Current values of this set can be interpreted as their long run mathematical expectation. Existence results will be established in Section 8.

Starting with the Phillips curve, in particular the numerator of the reset price relationship from Proposition 1 given by combining infinite horizon expression (28) and recursive form (30)

combining gives the basis for the Stochastic Equilibrium Phillips curve.

Price dispersion evolves as follows

It is clear that the asymptote in all these equations corresponds to the case where ∆ grows unbounded. The Euler equation takes the form

Hence the equilibrium real interest rate, known as the natural rate of interest rate, is

Similar expressions could be written for the quantities U and Π but are suppressed in the interest of space.

[44] This integral is technically a Bochner or strong integral, the natural extension of Lebesgue integral to infinite dimensional space, whereas that of (28) is a more general Pettis integral.