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 Abstract In this paper, a mathematically rigorous solution overturns existing wisdom regarding New Keynesian Dynamic Stochastic General Equilibrium. I develop a formal concept of stochastic equilibrium. I prove uniqueness and necessity, when agents are patient, across a wide class of dynamic stochastic models. Existence depends on appropriately specified eigenvalue conditions. Otherwise, no solution of any kind exists. I construct the equilibrium for the benchmark Calvo New Keynesian. I provide novel comparative statics with the non-stochastic model of independent mathematical interest. I uncover a bifurcation between neighbouring stochastic systems and approximations taken from the Zero Inflation Non-Stochastic Steady State (ZINSS). The correct Phillips curve agrees with the zero limit from the trend inflation framework. It contains a large lagged inflation coefficient and a small response to expected inflation. The response to the output gap is always muted and is zero at standard parameters. A neutrality result is presented to explain why and to align Calvo with Taylor pricing. Present and lagged demand shocks enter the Phillips curve so there is no Divine Coincidence and the system is identified from structural shocks alone. The lagged inflation slope is increasing in the inflation response, embodying substantive policy trade-offs. The Taylor principle is reversed, inactive settings are necessary for existence, pointing towards inertial policy. The observational equivalence idea of the Lucas critique is disproven. The bifurcation results from the breakdown of the constraints implied by lagged nominal rigidity, associated with cross-equation cancellation possible only at ZINSS. This creates a singular surface inside which the inter-temporal transmission mechanism breaks down and the error terms cancel out. There is a dual relationship between restrictions on the econometrician and constraints on repricing firms. Thus if the model is correct, goodness of fit will jump. Key Words: Stochastic Equilibrium, Mean Field Game, Bifurcation Analysis, Phillips Curve, Econometric Duality, Lucas Critique, Macroeconometrics, Monetary Policy Rules, Divine Coincidence. JEL Classifications: B22, C02, C50, C52, C61, C62, C65, D50, E12, E17, E31, E52. AMS Classifications: (Primary) 91B51, (Secondary) 41A99, 49A80, 54H25, 57R55, 60B99, 62M99, 91A15, 91A16, 91A50, 91B02, 91B50, 91B51, 91B52. 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 Abstract In this paper, a mathematically rigorous solution overturns existing wisdom regarding New Keynesian Dynamic Stochastic General Equilibrium. I develop a formal concept of stochastic equilibrium. I prove uniqueness and necessity, when agents are patient, across a wide class of dynamic stochastic models. Existence depends on appropriately specified eigenvalue conditions. Otherwise, no solution of any kind exists. I construct the equilibrium for the benchmark Calvo New Keynesian. I provide novel comparative statics with the non-stochastic model of independent mathematical interest. I uncover a bifurcation between neighbouring stochastic systems and approximations taken from the Zero Inflation Non-Stochastic Steady State (ZINSS). The correct Phillips curve agrees with the zero limit from the trend inflation framework. It contains a large lagged inflation coefficient and a small response to expected inflation. The response to the output gap is always muted and is zero at standard parameters. A neutrality result is presented to explain why and to align Calvo with Taylor pricing. Present and lagged demand shocks enter the Phillips curve so there is no Divine Coincidence and the system is identified from structural shocks alone. The lagged inflation slope is increasing in the inflation response, embodying substantive policy trade-offs. The Taylor principle is reversed, inactive settings are necessary for existence, pointing towards inertial policy. The observational equivalence idea of the Lucas critique is disproven. The bifurcation results from the breakdown of the constraints implied by lagged nominal rigidity, associated with cross-equation cancellation possible only at ZINSS. This creates a singular surface inside which the inter-temporal transmission mechanism breaks down and the error terms cancel out. There is a dual relationship between restrictions on the econometrician and constraints on repricing firms. Thus if the model is correct, goodness of fit will jump. Key Words: Stochastic Equilibrium, Mean Field Game, Bifurcation Analysis, Phillips Curve, Econometric Duality, Lucas Critique, Macroeconometrics, Monetary Policy Rules, Divine Coincidence. JEL Classifications: B22, C02, C50, C52, C61, C62, C65, D50, E12, E17, E31, E52. AMS Classifications: (Primary) 91B51, (Secondary) 41A99, 49A80, 54H25, 57R55, 60B99, 62M99, 91A15, 91A16, 91A50, 91B02, 91B50, 91B51, 91B52. 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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