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 B.5 Consolidation (4.8) Here, I derive the state space form of the non-linear model. The proof uses only techniques from basic analysis. The main complexity arises from the non-linear Phillips curve. B.5.1 Proof of Proposition 4 Proof. Start with the non-linear Phillips curve, formed from substituting the relationship between the optimal relative reset price and inflation derived in Proposition 1 into the left-hand side of (27). Now put (30) and (31) into the right-hand side. Finally, lag the relationship one period to reveal that where W. t are error terms reflecting the difference between expected and true values. They are white noise thanks to the rational expectation hypothesis. Subsequently, dependence on the current shocks will be suppressed. [108] where the formula is valid whenever the denominator is non-zero. Now rearranging the future price equation then substituting into the right-hand side and simplifying yields Moving (207) forward one period and substituting in reveals To prove this holds with probability one consider which appears in the denominator of (207) and (210) through (212), setting the expression equal to zero gives the equation of the singularity We can see that this function is surjective, by evaluating its derivative, which shows that it is strictly increasing. This means that the singularity can only occur at a point in time t+1, if inflation is equal to one particular value, determined by inflation in the previous period t. We know this must occur with probability zero as inflation is continuously distributed, since it is determined by the reset price that depends non-trivially on the continuously distributed shock processes. This can be converted into a relationship in expected inflation, by noting that equation (27) and the underlying optimization problems bound inflation above and below as follows To complete this section of the derivation all that is required is to substitute out terms in marginal costs and lagged income. Present marginal costs can be removed using (19) and (37) as follows leaving only lagged income which can be substituted out via the Euler equation. To this end let us define the following function Calling in the policy rule (35), including the output smoothing restriction gives which allows us to form a function substituting into (217) gives the appropriate form for the Phillips curve Finally, we can form the canonical Euler by inverting the Euler using then substituting in the Phillips curve gives the final section Lastly, by inverting (34), moving one period into the future and substituting in the Phillips curve and noting the expectation must exist to prevent transversality condition blowing again, which means that This paper is available on arxiv under CC 4.0 license. [108] The transversality condition prevents additional non-fundamental terms entering. [109] ZINSS corresponds to a particular case where both singularities arise simultaneously and "cancel out" from the right hand side of (205). The nature of the cocycle at the singularity is treated in the next section. 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 B.5 Consolidation (4.8) Here, I derive the state space form of the non-linear model. The proof uses only techniques from basic analysis. The main complexity arises from the non-linear Phillips curve. B.5.1 Proof of Proposition 4 Proof. Start with the non-linear Phillips curve, formed from substituting the relationship between the optimal relative reset price and inflation derived in Proposition 1 into the left-hand side of (27). Now put (30) and (31) into the right-hand side. Finally, lag the relationship one period to reveal that where W. t are error terms reflecting the difference between expected and true values. They are white noise thanks to the rational expectation hypothesis. Subsequently, dependence on the current shocks will be suppressed. [108] where the formula is valid whenever the denominator is non-zero. Now rearranging the future price equation then substituting into the right-hand side and simplifying yields Moving (207) forward one period and substituting in reveals To prove this holds with probability one consider which appears in the denominator of (207) and (210) through (212), setting the expression equal to zero gives the equation of the singularity We can see that this function is surjective, by evaluating its derivative, which shows that it is strictly increasing. This means that the singularity can only occur at a point in time t+1, if inflation is equal to one particular value, determined by inflation in the previous period t. We know this must occur with probability zero as inflation is continuously distributed, since it is determined by the reset price that depends non-trivially on the continuously distributed shock processes. This can be converted into a relationship in expected inflation, by noting that equation (27) and the underlying optimization problems bound inflation above and below as follows To complete this section of the derivation all that is required is to substitute out terms in marginal costs and lagged income. Present marginal costs can be removed using (19) and (37) as follows leaving only lagged income which can be substituted out via the Euler equation. To this end let us define the following function Calling in the policy rule (35), including the output smoothing restriction gives which allows us to form a function substituting into (217) gives the appropriate form for the Phillips curve Finally, we can form the canonical Euler by inverting the Euler using then substituting in the Phillips curve gives the final section Lastly, by inverting (34), moving one period into the future and substituting in the Phillips curve and noting the expectation must exist to prevent transversality condition blowing again, which means that 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 [108] The transversality condition prevents additional non-fundamental terms entering. [109] ZINSS corresponds to a particular case where both singularities arise simultaneously and "cancel out" from the right hand side of (205). The nature of the cocycle at the singularity is treated in the next section.

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