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 H.2 Phillips Curve This subsection is split in two. The first part covers two alternative benchmarks. The second studies robustness at the standard settings and draws a comparison with the existing solution and empirics. Most emphasis is placed on the output slope coefficient in relation to macroeconometric estimates. H.2.1 Alternative Cases. Note that in the later case only the lagged inflation coefficient changes substantially from the text. Movements in the other coefficients are much smaller. It is interesting to look at how the cut-off value of the lag varies as the various parameters are varied. Information on the other coefficients in these cases comes in the next part. Note that the cut-off is always less than unity. [135] This reflects the fact that inflation has to be stationary. The following triptych of tables alter parameters one at a time, keeping the others at the standard setting. H.2.2 Benchmark Parametization In this subsection, I study changes in the Phillips curve slope coefficients when structural parameters are varied, over the ranges discussed in the previous section. I then perform a similar exercise for the existing solution. The difference remains stark. This should be a spur for future rigorous econometric research. The lagged inflation coefficient seems to be almost invariant to changes in the supply side, which is encouraging. The lead is also relatively unresponsive. This is encouraging from a Keynesian perspective. I anticipate the model will soon be augmented with additional frictions to better model the propagation of monetary shocks to labor and capital markets. I anticipate the model will soon be augmented with additional frictions to better model the propagation of monetary shocks to labor and capital markets. This will rest upon the robust foundation of our new understanding of inflation determination embodied in these coefficients. Overall, this part has provided a strong body of suggestive evidence that the new approximation provides a better fit than the existing coefficient by coefficient. Moreover, the stability of the critical slope coefficients, in response to plausible structural parameter uncertainty, is a showcase for robustness in macroeconomics. This exercise has surely been a useful prelude to future rigorous econometric investigation. This paper is available on arxiv under CC 4.0 license. [135] To see why this is true note that I can express the cut off as [139] This occurs when I adjust α to 0.8, when η = 1 it is higher at 0.333. [140] This conclusion is robust to any of the three deviation measures. 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 H.2 Phillips Curve This subsection is split in two. The first part covers two alternative benchmarks. The second studies robustness at the standard settings and draws a comparison with the existing solution and empirics. Most emphasis is placed on the output slope coefficient in relation to macroeconometric estimates. H.2.1 Alternative Cases. H.2.1 Alternative Cases. Note that in the later case only the lagged inflation coefficient changes substantially from the text. Movements in the other coefficients are much smaller. It is interesting to look at how the cut-off value of the lag varies as the various parameters are varied. Information on the other coefficients in these cases comes in the next part. Note that the cut-off is always less than unity. [135] This reflects the fact that inflation has to be stationary. The following triptych of tables alter parameters one at a time, keeping the others at the standard setting. H.2.2 Benchmark Parametization H.2.2 Benchmark Parametization In this subsection, I study changes in the Phillips curve slope coefficients when structural parameters are varied, over the ranges discussed in the previous section. I then perform a similar exercise for the existing solution. The difference remains stark. This should be a spur for future rigorous econometric research. The lagged inflation coefficient seems to be almost invariant to changes in the supply side, which is encouraging. The lead is also relatively unresponsive. This is encouraging from a Keynesian perspective. I anticipate the model will soon be augmented with additional frictions to better model the propagation of monetary shocks to labor and capital markets. I anticipate the model will soon be augmented with additional frictions to better model the propagation of monetary shocks to labor and capital markets. This will rest upon the robust foundation of our new understanding of inflation determination embodied in these coefficients. Overall, this part has provided a strong body of suggestive evidence that the new approximation provides a better fit than the existing coefficient by coefficient. Moreover, the stability of the critical slope coefficients, in response to plausible structural parameter uncertainty, is a showcase for robustness in macroeconomics. This exercise has surely been a useful prelude to future rigorous econometric investigation. 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 [135] To see why this is true note that I can express the cut off as [139] This occurs when I adjust α to 0.8, when η = 1 it is higher at 0.333. [140] This conclusion is robust to any of the three deviation measures.

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