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 10 Econometric and Theoretical Implications This section focuses on the most basic economic theory embedded in the coefficients of the Phillips curve. It sketches out the fundamental policy and econometric implications of this theory. I chart a new course for macroeconomic policy analysis and suggest new avenues for econometric research. 10.1 Identification and Trade-offs The first task of this section is to prove that price rigidity implies first order trade-offs between stabilization objectives. The second is to show that this naturally resolves econometric problems with identification. These conditions constitute natural well-posedness conditions for dynamic stochastic models. 10.1.1 No Divine Coincidence Here, I establish the existence of trade-offs in monetary policy setting, previously thought not to exist in such benchmark setting. In particular, it overturns Proposition 6 for the Calvo model. I begin with an interesting preliminary result of interest throughout this subsection. The simple proof occupies Appendix G.3.1. It is instructive to discuss further this special case. The first viewpoint is topological, it complements the bifurcation analysis of Section 9. The second focus on theoretical interpretations. Remark 28. This special case amounts to the Central Bank engineering financial frictions, which move the economy back to an ad hoc demand curve. This was traditionally how Keynesian models were constructed. Microfounded demand curves were seen as a restrictive afterthought. I have overturned this view. They are a necessary stabilizing force. This symbiosis of Keynesian and Classical mechanisms is a subtle yet significant feature of modern Keynesian economics. Nevertheless, the practical significance is limited. The policy would be inconsistent with central banks mandate to develop and stabilize financial markets.[91] Theorem 8. (No Divine Coincidence) Under Calvo pricing, even if the government is able to correct static market failures, there will always arise a first order welfare loss from business cycle fluctuations. Remark 29. This trade-off confirms the constraint interpretation of the boundary surface, prevalent in the last section. Remark 30. This morphs into a general stabilization trade-off away from ZINSS. This is because successfully stabilizing the economy to first order, around any stochastic equilibrium, would move the economy back to the neighborhood of the non-stochastic equilibrium, since the mixing property would ensure second order terms vanished. The problem is there are two cost-push shocks in the Phillips curve and only one policy instrument. This is the simplest possible stabilization trade-off. Although a priority, it lies beyond the scope of this paper to analyze the policy prescriptions. It also some mathematical significance. Remark 31. The demise of Divine Coincidence demonstrates that the cost of economic fluctuations has a non-degenerate Lipschitz bound as a function of the shock size. This is a basic regularity property of the optimal policy problem. 10.1.2 Persistence and Identification Here I prove identification and persistence, stemming from structural shocks, without real rigidity, in opposition to Propositions 8 and 9. Remark 32. This no persistence result breaks down in the large noise limit because of price dispersion but might arise if price dispersion were ignored. The no persistence policy rule is counter-intuitive. Central banks seek to raise the profile of interest rates in response to increases in inflation. However, further analysis of the policy rule in Appendix G.3 is unable to rule it out. This means there is no direct counter to Proposition 8. Elsewhere in the text, I persist with non-negative reaction coefficients. Remark 33. Persistence is necessary and sufficient for identification of the structural model (without ad hoc shocks). This can be seen from the solution in Appendix E.2, which theoretically implies an infinite number of over-identifying conditions through construction of a suitable auto-regression. The (over)identification of structural parameters follows swiftly apart from θ, which will be dealt with in a subsequent paper. It is surely a step forward to no longer rely on unexplained and poorly motivated shocks for identification, particularly when they are not compatible with rational expectations (Proposition 7). This pathology with persistence and the surprising non-existence results point towards optimal monetary policy. Optimal monetary policy should yield stronger regularity of solution, including the prospect of unconditional existence in many cases. It is likely to deliver clearer empirical predictions and may even simplify the solution or improve its fit. It should be a natural next step for central bank economists to include their own optimizing efforts in the regular process of monetary policy modelling. This paper is available on arxiv under CC 4.0 license. [91] Moreover, this is not how one would model credit constraints; even if some people were shut out of the financial system. There would be capitalists who owned the firms’ and if they were credit constrained, they would choose to close down their business. Hence we should always expect there to be some unconstrained agents. A genuine heterogeneous agent model is considerably outside the scope of the paper. 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 10 Econometric and Theoretical Implications This section focuses on the most basic economic theory embedded in the coefficients of the Phillips curve. It sketches out the fundamental policy and econometric implications of this theory. I chart a new course for macroeconomic policy analysis and suggest new avenues for econometric research. 10.1 Identification and Trade-offs The first task of this section is to prove that price rigidity implies first order trade-offs between stabilization objectives. The second is to show that this naturally resolves econometric problems with identification. These conditions constitute natural well-posedness conditions for dynamic stochastic models. 10.1.1 No Divine Coincidence Here, I establish the existence of trade-offs in monetary policy setting, previously thought not to exist in such benchmark setting. In particular, it overturns Proposition 6 for the Calvo model. I begin with an interesting preliminary result of interest throughout this subsection. The simple proof occupies Appendix G.3.1. It is instructive to discuss further this special case. The first viewpoint is topological, it complements the bifurcation analysis of Section 9. The second focus on theoretical interpretations. Remark 28. This special case amounts to the Central Bank engineering financial frictions, which move the economy back to an ad hoc demand curve. Remark 28. This special case amounts to the Central Bank engineering financial frictions, which move the economy back to an ad hoc demand curve. This was traditionally how Keynesian models were constructed. Microfounded demand curves were seen as a restrictive afterthought. I have overturned this view. They are a necessary stabilizing force. This symbiosis of Keynesian and Classical mechanisms is a subtle yet significant feature of modern Keynesian economics. This was traditionally how Keynesian models were constructed. Microfounded demand curves were seen as a restrictive afterthought. I have overturned this view. They are a necessary stabilizing force. This symbiosis of Keynesian and Classical mechanisms is a subtle yet significant feature of modern Keynesian economics. Nevertheless, the practical significance is limited. The policy would be inconsistent with central banks mandate to develop and stabilize financial markets.[91] Theorem 8. (No Divine Coincidence) Under Calvo pricing, even if the government is able to correct static market failures, there will always arise a first order welfare loss from business cycle fluctuations. Remark 29. This trade-off confirms the constraint interpretation of the boundary surface, prevalent in the last section. Remark 29. This trade-off confirms the constraint interpretation of the boundary surface, prevalent in the last section. Remark 30. This morphs into a general stabilization trade-off away from ZINSS. This is because successfully stabilizing the economy to first order, around any stochastic equilibrium, would move the economy back to the neighborhood of the non-stochastic equilibrium, since the mixing property would ensure second order terms vanished. Remark 30. This morphs into a general stabilization trade-off away from ZINSS. This is because successfully stabilizing the economy to first order, around any stochastic equilibrium, would move the economy back to the neighborhood of the non-stochastic equilibrium, since the mixing property would ensure second order terms vanished. The problem is there are two cost-push shocks in the Phillips curve and only one policy instrument. This is the simplest possible stabilization trade-off. Although a priority, it lies beyond the scope of this paper to analyze the policy prescriptions. It also some mathematical significance. Remark 31. The demise of Divine Coincidence demonstrates that the cost of economic fluctuations has a non-degenerate Lipschitz bound as a function of the shock size. This is a basic regularity property of the optimal policy problem. Remark 31. The demise of Divine Coincidence demonstrates that the cost of economic fluctuations has a non-degenerate Lipschitz bound as a function of the shock size. This is a basic regularity property of the optimal policy problem. 10.1.2 Persistence and Identification Here I prove identification and persistence, stemming from structural shocks, without real rigidity, in opposition to Propositions 8 and 9. Remark 32. This no persistence result breaks down in the large noise limit because of price dispersion but might arise if price dispersion were ignored. Remark 32. This no persistence result breaks down in the large noise limit because of price dispersion but might arise if price dispersion were ignored. The no persistence policy rule is counter-intuitive. Central banks seek to raise the profile of interest rates in response to increases in inflation. However, further analysis of the policy rule in Appendix G.3 is unable to rule it out. This means there is no direct counter to Proposition 8. Elsewhere in the text, I persist with non-negative reaction coefficients. Remark 33. Persistence is necessary and sufficient for identification of the structural model (without ad hoc shocks). This can be seen from the solution in Appendix E.2, which theoretically implies an infinite number of over-identifying conditions through construction of a suitable auto-regression. Remark 33. Persistence is necessary and sufficient for identification of the structural model (without ad hoc shocks). This can be seen from the solution in Appendix E.2, which theoretically implies an infinite number of over-identifying conditions through construction of a suitable auto-regression. The (over)identification of structural parameters follows swiftly apart from θ, which will be dealt with in a subsequent paper. It is surely a step forward to no longer rely on unexplained and poorly motivated shocks for identification, particularly when they are not compatible with rational expectations (Proposition 7). This pathology with persistence and the surprising non-existence results point towards optimal monetary policy. Optimal monetary policy should yield stronger regularity of solution, including the prospect of unconditional existence in many cases. It is likely to deliver clearer empirical predictions and may even simplify the solution or improve its fit. It should be a natural next step for central bank economists to include their own optimizing efforts in the regular process of monetary policy modelling. 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 [91] Moreover, this is not how one would model credit constraints; even if some people were shut out of the financial system. There would be capitalists who owned the firms’ and if they were credit constrained, they would choose to close down their business. Hence we should always expect there to be some unconstrained agents. A genuine heterogeneous agent model is considerably outside the scope of the paper.

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