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 D Stochastic Equilibrium This section aligns with Section 6 of the main text. It begins by setting out the non-stochastic equilibrium conditions, before moving onto to the three omitted proofs and addressing the claims made in Remark 8. D.1 Non-Stochastic Equilibrium This subsection divided in two the first part details the closed form of the solution in terms of primitive parameters. The second explains why we must restrict the model to allow for economic growth and de-trending. D.1.1 Exact Form With functional forms in, the non-stochastic steady state is as follows where I have used the price-setting relation. Therefore, thanks to constant returns the wage rate is Price dispersion is Inverting the marginal cost function and substituting in the price dispersion relationship yields The resource constraint yields the solution for labor supply The aggregate profit rate is substituting into the utility function reveals that Finally, present discounted welfare is In line with the discussion in the text about profitability with trend inflation, firms will make different profits depending on when they last reset their price. Denoting the time since the last reset by the age of its price a D.1.2 Economic Growth and De-trending It is useful to examine the consequences of allowing for economic growth. To do so suppose there is exogenous technological progress at rate g so To justify the practice of linear de-trending standard in empirical work, a balanced growth path is required. However, the following result demonstrates that this necessitates a specific parametric restriction. Proposition 29. There exists a balanced growth path if and only if σ = 1. Proof. There can be a balanced growth path if and only if we can solve for detrended output Y/A that does not depend on the growing variable A. Repeating the steps used to derive the steady state with no growth reveals that where k is a function of only the behavioural parameters that could be calculated directly from (261). It is clear independence only arises when the exponent is zero which corresponds to σ = 1. Therefore, I select σ = 1 for quantitative work here, since I am working with small noise limits and desire consistency with standard econometric practice. Although, this value is not ubiquitous in the literature, in the calibration section I demonstrate that it is consistent with a body of empirical evidence and prior work. In fact, Theorem 3 and all the subsequent analysis would go through in this case. Expected utility on the balanced growth path (BGP) would be augmented by a term reflecting the value of future growth. [111] Adding capital would avoid this knife-edge condition, at the expense of complicating the rest of the analysis. 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 D Stochastic Equilibrium This section aligns with Section 6 of the main text. It begins by setting out the non-stochastic equilibrium conditions, before moving onto to the three omitted proofs and addressing the claims made in Remark 8. D.1 Non-Stochastic Equilibrium This subsection divided in two the first part details the closed form of the solution in terms of primitive parameters. The second explains why we must restrict the model to allow for economic growth and de-trending. D.1.1 Exact Form D.1.1 Exact Form With functional forms in, the non-stochastic steady state is as follows where I have used the price-setting relation. Therefore, thanks to constant returns the wage rate is Price dispersion is Inverting the marginal cost function and substituting in the price dispersion relationship yields The resource constraint yields the solution for labor supply The aggregate profit rate is substituting into the utility function reveals that Finally, present discounted welfare is In line with the discussion in the text about profitability with trend inflation, firms will make different profits depending on when they last reset their price. Denoting the time since the last reset by the age of its price a D.1.2 Economic Growth and De-trending D.1.2 Economic Growth and De-trending It is useful to examine the consequences of allowing for economic growth. To do so suppose there is exogenous technological progress at rate g so To justify the practice of linear de-trending standard in empirical work, a balanced growth path is required. However, the following result demonstrates that this necessitates a specific parametric restriction. Proposition 29. There exists a balanced growth path if and only if σ = 1. Proposition 29. Proof. There can be a balanced growth path if and only if we can solve for detrended output Y/A that does not depend on the growing variable A. Repeating the steps used to derive the steady state with no growth reveals that where k is a function of only the behavioural parameters that could be calculated directly from (261). It is clear independence only arises when the exponent is zero which corresponds to σ = 1. Therefore, I select σ = 1 for quantitative work here, since I am working with small noise limits and desire consistency with standard econometric practice. Although, this value is not ubiquitous in the literature, in the calibration section I demonstrate that it is consistent with a body of empirical evidence and prior work. In fact, Theorem 3 and all the subsequent analysis would go through in this case. Expected utility on the balanced growth path (BGP) would be augmented by a term reflecting the value of future growth. [111] Adding capital would avoid this knife-edge condition, at the expense of complicating the rest of the analysis. 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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