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.2 Flexible Price Equilibrium and ZINSS (4.4) This subsection clarifies remarks concerning the relationship between the optimal reset price and the flexible price equilibrium, including their coincidence when there is zero inflation. To achieve this I introduce a little mathematical machinery, that I understand to be original. Under monopolistic competition, I find that the optimal price is a mark-up m = 1/(θ − 1) over marginal costs, so in real terms It is worth briefly motivating this definition. Continuity and increasing requirements support the idea of the mean as an average and rule out alternative measures of location, such as median and mode.[106] The function space assumption allows us to include expectations of future variables, as part of our average ensemble. It is this averaging property that underpins ergodicity and mixing in the model. Proof. Weak measurability comes from the structure of the optimization problem. Monotonicity and continuity follow from the following implicit function argument, where the left-hand side is (27), combined with (199) and the gross inflation definition (23) on the right-hand side Hence Corollary 3. ZINSS is equivalent to a flexible price equilibrium. Therefore uniquely at ZINSS the reset price constraint is not binding. This confirms the breakdown of nominal rigidity as the root cause of the bifurcation. The other weighted measures feature expectations of non-linear terms and are therefore not amenable to our definition. Remark 36. These results carry over easily to alternative settings with finite price lives. This includes Taylor pricing and models where the price spell is truncated, such as Wolman [1999] and Dixon and Le Bihan [2012]. The final result justifies a comment in Footnote 25 in Section 4.4. It explains the distinction between the reset price and the flexible price. Moreover, it presages the idea of a limiting fringe of flexible price firms, used in Section 9.5 to make distinctions between real and nominal rigidity. Proposition 28. The magnitude of the change in the reset price weakly exceeds the change in the price of a firm that always had flexible prices, with equality only when neither price changes. Proof. Since price re-setters are randomly assigned here, by a standard law of large numbers argument, the reset price inflation is given by A flexible price firm will always set its price equal to the general price level (202) corresponds to Proposition 1, the economic idea that the relative price of the re-setter is increasing in inflation. Intuitively, with rigid prices to generate inflation or deflation, re-setters have to adjust more aggressively than in the flexible price world, in order to make up for the portion of the price level that does not change. This is why we cannot naively compare observed price changes with a classical benchmark to obtain the spillover from rigid prices to flexible prices. This paper is available on arxiv under CC 4.0 license. [105] A reflexive space has the property that the set of possible ensemble averages and the set of possible realizations coincide. [106] Consult Bullen [2003] for a review of mean concepts and their inequalities. 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.2 Flexible Price Equilibrium and ZINSS (4.4) This subsection clarifies remarks concerning the relationship between the optimal reset price and the flexible price equilibrium, including their coincidence when there is zero inflation. To achieve this I introduce a little mathematical machinery, that I understand to be original. Under monopolistic competition, I find that the optimal price is a mark-up m = 1/(θ − 1) over marginal costs, so in real terms It is worth briefly motivating this definition. Continuity and increasing requirements support the idea of the mean as an average and rule out alternative measures of location, such as median and mode.[106] The function space assumption allows us to include expectations of future variables, as part of our average ensemble. It is this averaging property that underpins ergodicity and mixing in the model. Proof . Weak measurability comes from the structure of the optimization problem. Monotonicity and continuity follow from the following implicit function argument, where the left-hand side is (27), combined with (199) and the gross inflation definition (23) on the right-hand side Proof Hence Corollary 3. ZINSS is equivalent to a flexible price equilibrium. Corollary 3. ZINSS is equivalent to a flexible price equilibrium. Therefore uniquely at ZINSS the reset price constraint is not binding. This confirms the breakdown of nominal rigidity as the root cause of the bifurcation. The other weighted measures feature expectations of non-linear terms and are therefore not amenable to our definition. Remark 36. These results carry over easily to alternative settings with finite price lives. This includes Taylor pricing and models where the price spell is truncated, such as Wolman [1999] and Dixon and Le Bihan [2012]. Remark 36. These results carry over easily to alternative settings with finite price lives. This includes Taylor pricing and models where the price spell is truncated, such as Wolman [1999] and Dixon and Le Bihan [2012]. The final result justifies a comment in Footnote 25 in Section 4.4. It explains the distinction between the reset price and the flexible price. Moreover, it presages the idea of a limiting fringe of flexible price firms, used in Section 9.5 to make distinctions between real and nominal rigidity. Proposition 28. The magnitude of the change in the reset price weakly exceeds the change in the price of a firm that always had flexible prices, with equality only when neither price changes. Proposition 28. The magnitude of the change in the reset price weakly exceeds the change in the price of a firm that always had flexible prices, with equality only when neither price changes. Proof . Since price re-setters are randomly assigned here, by a standard law of large numbers argument, the reset price inflation is given by Proof A flexible price firm will always set its price equal to the general price level (202) corresponds to Proposition 1, the economic idea that the relative price of the re-setter is increasing in inflation. Intuitively, with rigid prices to generate inflation or deflation, re-setters have to adjust more aggressively than in the flexible price world, in order to make up for the portion of the price level that does not change. This is why we cannot naively compare observed price changes with a classical benchmark to obtain the spillover from rigid prices to flexible prices. 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 [105] A reflexive space has the property that the set of possible ensemble averages and the set of possible realizations coincide. [106] Consult Bullen [2003] for a review of mean concepts and their inequalities.

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