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 E Slopes and Eigenvalues This section contains details involving eigenvalues and slope coefficients, omitted from the main text. The first subsection gives the general expressions for the slope coefficients. The second finishes off the derivation of the general linearized solution of a DSGE. The third constructs the characteristic equations. The final part covers the general existence conditions implied by Rouche’s theorem. E.1 Slope Coefficients This subsection begins by gathering together the expression for the non-linear Phillips curve coefficients. The expressions are grouped into three subsubsections dealing in turn with the Phillips curve, the Euler and the price dispersion relation. The fourth covers the Phillips curve in the general small noise limit case. The final part solves for the singular surfaces around ZINSS. E.1.1 Phillips Curve (116) E.1.2 Euler Coefficients (117) E.1.3 Price Dispersion E.1.4 Trend Inflation Phillips Curve This final part lays out the general Phillips curve in the small noise limit with arbitrary time preference and trend inflation. It compliments discussions of trend inflation and allows me to complete the proofs of Theorem 8 and Proposition 23. The justification of Footnote 92 is immediate. Remark 39. Symmetry arises in the error terms under trend inflation as under ZINSS. This reflects the inefficiency of aggregate fluctuations discovered in Theorem 6, which holds at any rate of trend inflation. However, it cannot be solved for without finding an exact form for the entire model which is beyond the perimeter of this study. Remark 40. Symmetry breaks down in non-degenerate stochastic steady state, where non-vanishing uncertainty creates additional inter-temporal distortions, hence, the more complicated expressions in Section 6.2. E.1.5 Singular Surfaces This part completes the solution of the boundary surface. It justifies (5) and extends it to the |ε| limit, whilst generalizing it to the whole parameter space. Parametize the surface as follows The bifurcated Phillips curve takes the form where κ was defined back in (44). Substitute into the main Phillips curve (116) and suppressing the errors reveals Substituting in the wall-crossing constraint from Proposition 20 Equating the coefficients between (302) with (304) substituting in the primitives and splitting numerators from denominators, as before, yields Finally, down the limit where β → 1 and σ = 1. Note that these coefficients have clear predicted signs. This paper is available on arxiv under CC 4.0 license. Author: (1) David Staines. Author: Author: (1) David Staines. Table of Links Abstract Abstract 1 Introduction 1 Introduction 2 Mathematical Arguments 2 Mathematical Arguments 3 Outline and Preview 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.2 Preferences 4.3 Household Equilibrium Conditions 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 4.8 Recursive Equilibrium 5 Existing Solutions 5 Existing Solutions 5.1 Singular Phillips Curve 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.3 Two Comparison Models 5.4 Lucas Critique 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.2 Equilibrium Construction 6.3 Literature Comparison 6.3 Literature Comparison 6.4 Equilibrium Analysis 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.1 Slope Coefficients 7.2 Error Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.2 Key Proofs 8.3 Discussion 8.3 Discussion 9 Bifurcation Analysis 9 Bifurcation Analysis 9.1 Analytic Aspects 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.2 Econometric Duality 10.3 Coefficient Properties 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 10.4 Microeconomic Interpretation 11 Policy Rule 11 Policy Rule 12 Conclusions and References 12 Conclusions and References Appendices Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.2 Behaviour of ∆ A.3 Proof Part (iii) A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.2 Basic Categories F.3 De Rham Cohomology F.3 De Rham Cohomology F.4 Marginal Costs and Inflation F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.2 Lucas Critique I.3 Trend Inflation Volatility I.3 Trend Inflation Volatility E Slopes and Eigenvalues This section contains details involving eigenvalues and slope coefficients, omitted from the main text. The first subsection gives the general expressions for the slope coefficients. The second finishes off the derivation of the general linearized solution of a DSGE. The third constructs the characteristic equations. The final part covers the general existence conditions implied by Rouche’s theorem. E.1 Slope Coefficients This subsection begins by gathering together the expression for the non-linear Phillips curve coefficients. The expressions are grouped into three subsubsections dealing in turn with the Phillips curve, the Euler and the price dispersion relation. The fourth covers the Phillips curve in the general small noise limit case. The final part solves for the singular surfaces around ZINSS. E.1.1 Phillips Curve (116) E.1.1 Phillips Curve (116) E.1.2 Euler Coefficients (117) E.1.2 Euler Coefficients (117) E.1.3 Price Dispersion E.1.3 Price Dispersion E.1.4 Trend Inflation Phillips Curve E.1.4 Trend Inflation Phillips Curve This final part lays out the general Phillips curve in the small noise limit with arbitrary time preference and trend inflation. It compliments discussions of trend inflation and allows me to complete the proofs of Theorem 8 and Proposition 23. The justification of Footnote 92 is immediate. Remark 39. Symmetry arises in the error terms under trend inflation as under ZINSS. This reflects the inefficiency of aggregate fluctuations discovered in Theorem 6, which holds at any rate of trend inflation. However, it cannot be solved for without finding an exact form for the entire model which is beyond the perimeter of this study. Remark 39. Remark 40. Symmetry breaks down in non-degenerate stochastic steady state, where non-vanishing uncertainty creates additional inter-temporal distortions, hence, the more complicated expressions in Section 6.2. Remark 40. E.1.5 Singular Surfaces E.1.5 Singular Surfaces This part completes the solution of the boundary surface. It justifies (5) and extends it to the |ε| limit, whilst generalizing it to the whole parameter space. Parametize the surface as follows The bifurcated Phillips curve takes the form where κ was defined back in (44). Substitute into the main Phillips curve (116) and suppressing the errors reveals Substituting in the wall-crossing constraint from Proposition 20 Equating the coefficients between (302) with (304) substituting in the primitives and splitting numerators from denominators, as before, yields Finally, down the limit where β → 1 and σ = 1. Note that these coefficients have clear predicted signs. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

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