(Bloomberg) -- Elliott Investment Management is seeking seats on the board of oil refiner Phillips 66, the latest effort in a multi-year campaign pushing the company to sell assets, improve operational performance and bolster board oversight. Most Read from BloombergTrump to Halt NY Congestion ...

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Oil company Phillips 66 faces federal charges related to alleged Clean Water Act violations

Oil company Phillips 66 has been federally indicted in connection with alleged violations of the Clean Water Act in California.

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Gas prices unlikely to be affected by closure of Phillips 66 refinery in Wilmington area: Expert

Phillips 66

phillips66

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.

Reassessing the Phillips Curve: Benchmarking, Robustness, and Structural Variations

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 G.2 Calvo Wage Phillips Curve This subsection shows a wage Phillips curve problem and how results of Theorem 8 apply. First, I introduce imperfect competition into the labor market before showing that the associated resetting problem is analogous to that of the price Phillips curve (26)-(31). G.2.1 Labor Unions In the benchmark model of Section 4 the labor market is perfectly competitive. To generate wage rigidity, we need to introduce imperfect substitutability between wage setters. The easiest way to achieve this is to introduce unions, which combine different types of labor into a composite labor service, that is leased to firms at a wage rate W. The aggregation is as follows This yields the labor demand system and the wage index G.2.2 Optimization Problem The union sets wages to maximize the welfare of its members. It gets to reset wages with probability αw each period. The problem can be written as follows where I have used the functional form for the disutility of labor. The first order condition is with Optimal reset wage can be viewed as a generalized weighted mean of flexible wages, according to Proposition 27 since which are set as a mark up over the marginal rate of substitution between labor and leisure, which is decreasing in the substitutability between labor varieties. Aggregate wages evolve according to The difference with the price level construction equation is that we are using real wages as opposed to nominal prices. The final and crucial step is to expand out the recursions (366) and (367) Linearizing these equations at ZINSS, it is clear that a common root in the lag polynomials will arise with This confirms we can apply Theorem 7 and conclude a bifurcation invalidates the linear approximations at ZINSS, without having to go through a detailed solution. Moreover, this analysis extends to larger models typically used in central banks, which feature both price and wage rigidity, like Christiano et al. [2005] and Smets and Wouters [2007]. 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 G.2 Calvo Wage Phillips Curve This subsection shows a wage Phillips curve problem and how results of Theorem 8 apply. First, I introduce imperfect competition into the labor market before showing that the associated resetting problem is analogous to that of the price Phillips curve (26)-(31). G.2.1 Labor Unions G.2.1 Labor Unions In the benchmark model of Section 4 the labor market is perfectly competitive. To generate wage rigidity, we need to introduce imperfect substitutability between wage setters. The easiest way to achieve this is to introduce unions, which combine different types of labor into a composite labor service, that is leased to firms at a wage rate W. The aggregation is as follows This yields the labor demand system and the wage index G.2.2 Optimization Problem G.2.2 Optimization Problem The union sets wages to maximize the welfare of its members. It gets to reset wages with probability αw each period. The problem can be written as follows where I have used the functional form for the disutility of labor. The first order condition is with Optimal reset wage can be viewed as a generalized weighted mean of flexible wages, according to Proposition 27 since which are set as a mark up over the marginal rate of substitution between labor and leisure, which is decreasing in the substitutability between labor varieties. Aggregate wages evolve according to The difference with the price level construction equation is that we are using real wages as opposed to nominal prices. The final and crucial step is to expand out the recursions (366) and (367) Linearizing these equations at ZINSS, it is clear that a common root in the lag polynomials will arise with This confirms we can apply Theorem 7 and conclude a bifurcation invalidates the linear approximations at ZINSS, without having to go through a detailed solution. Moreover, this analysis extends to larger models typically used in central banks, which feature both price and wage rigidity, like Christiano et al. [2005] and Smets and Wouters [2007]. 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

The Calvo Wage Phillips Curve and Labor Market Imperfections

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.

What the Phillips Curve Tells Us About Inflation

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.

Why the Phillips Curve Could Redefine Macroeconomic Policies

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 7.1 Slope Coefficients The first subsection has the derivation and the second houses the main results. 7.1.1 Derivation It is most convenient to start with (30) and (31) then apply Theorem 4 from Coayla-Teran et al. [2007], appropriate for (convergent) infinite horizon trajectories where I have used the following substitution derived from (19) and (31) The effect of changes in output on marginal revenues depends on the propensity for inter-temporal substitution reflected by σ (the inverse of the elasticity of substitution.) If σ > 1 the propensity to smooth consumption dominates the incentive to substitution over time, such that higher output today implies higher expected future marginal revenues- conversely, if σ < 1 the substitution exceeds the smoothing incentive so higher current output is associated with lower future marginal revenues. When σ = 1 the two forces balance out. Appendix D presents an intuitive argument for σ = 1 valid for this benchmark formulation. The error terms above reflect the difference between the actual and expected values of π, ℵ and i in period t + 1 that comes about because the future value of the structural jump errors are unknown and the model is linear in percentage deviation form. Now using the reset price equation to remove ℵ and i, the price level construction equation to express the reset price in terms of inflation and then manipulating terms in the lag operator yields Expanding the lag operator, collecting terms and passing expectations from time t yields Lagging the relationship and including the later period error term yields Finally, the log-linearized price dispersion relationship and its lagged form are respectively where I have used the expression for the stochastic steady of ∆ Combining (112), (114) and (116) yields the expression for the Phillips curve equivalent to (2) By substituting the Phillips curve into the Euler, we have For the price dispersion recursion we have These are sufficient to describe the canonical form. 7.1.2 Econometrics and Errors This part briefly discusses useful forms for econometric investigation, their basic properties and challenges. It is possible to remove price dispersion as follows[56] After taking lags and expanding some expectations, this forms the following Vector Autoregressive Moving Average (VARMA) (p, q) The first problem is a long term opportunity for econometrics and mathematical statistics. The second is a more pressing economic concern. It would arise in the benchmark model if one allowed for persistent Euler disturbances. Truncation strategies have been popular (see Ascari and Sbordone [2014]) but there may be scope for refinement. 7.1.3 Slope Coefficients Some of these expressions will be discussed and interpreted further in Section 10. 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 7.1 Slope Coefficients The first subsection has the derivation and the second houses the main results. 7.1.1 Derivation It is most convenient to start with (30) and (31) then apply Theorem 4 from Coayla-Teran et al. [2007], appropriate for (convergent) infinite horizon trajectories where I have used the following substitution derived from (19) and (31) The effect of changes in output on marginal revenues depends on the propensity for inter-temporal substitution reflected by σ (the inverse of the elasticity of substitution.) If σ > 1 the propensity to smooth consumption dominates the incentive to substitution over time, such that higher output today implies higher expected future marginal revenues- conversely, if σ < 1 the substitution exceeds the smoothing incentive so higher current output is associated with lower future marginal revenues. When σ = 1 the two forces balance out. Appendix D presents an intuitive argument for σ = 1 valid for this benchmark formulation. The error terms above reflect the difference between the actual and expected values of π, ℵ and i in period t + 1 that comes about because the future value of the structural jump errors are unknown and the model is linear in percentage deviation form. Now using the reset price equation to remove ℵ and i, the price level construction equation to express the reset price in terms of inflation and then manipulating terms in the lag operator yields Expanding the lag operator, collecting terms and passing expectations from time t yields t Lagging the relationship and including the later period error term yields Finally, the log-linearized price dispersion relationship and its lagged form are respectively where I have used the expression for the stochastic steady of ∆ Combining (112), (114) and (116) yields the expression for the Phillips curve equivalent to (2) By substituting the Phillips curve into the Euler, we have For the price dispersion recursion we have These are sufficient to describe the canonical form. 7.1.2 Econometrics and Errors This part briefly discusses useful forms for econometric investigation, their basic properties and challenges. It is possible to remove price dispersion as follows[56] After taking lags and expanding some expectations, this forms the following Vector Autoregressive Moving Average (VARMA) ( p, q ) p, q The first problem is a long term opportunity for econometrics and mathematical statistics. The second is a more pressing economic concern. It would arise in the benchmark model if one allowed for persistent Euler disturbances. Truncation strategies have been popular (see Ascari and Sbordone [2014]) but there may be scope for refinement. 7.1.3 Slope Coefficients Some of these expressions will be discussed and interpreted further in Section 10. 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

Understanding the Phillips Curve: How Economic Trends Shape Inflation and Growth

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 5.1 Singular Phillips Curve This subsection flags up the subtle error in the standard approximation that brings about a purely forward-looking solution. Linearizing the two components of the Phillips curve at ZINSS reveals a special structure, in particular, there is a common root L = 1/αβ in both lag polynomials. This means the system can be solved by elimination. Therefore, we do not have to lag one of the recursions to perform a substitution introducing lagged terms. The next result confirms that this is unrepresentative of the dynamics of any stochastic system because it only applies when inflation is at steady state where the approximation is irrelevant. Let Lℵ equal the zero of the lag polynomial in ℵ. Proof. By elementary manipulation it is clear that this amounts to proving that This follows immediately from Chebyshev’s correlation inequality (Lemma 1) which states that EAB ≥ EA EB for two strictly increasing functions with equality if and only the measure is degenerate. ZINSS is an example of a (measure zero) singularity not covered by Proposition 3. In fact it is the wall of the crossing which gives rise to (1). Bifurcation and common roots will be formally connected in Section 9. Thus we arrive at the incorrect step. Error 1. Cross Equation Cancellation in the Phillips curve This yields the forward-looking relationship. where the slope parameter is to derive an output gap expression first linearize the marginal cost function (38) Next linearize the production function (36) now using labor supply constraint and market clearing (19) and (20) The behaviour of price dispersion is governed by the linearization of (35) substituting in (π, ∆) = (0, 1) negates the first term leaving I can use the expression for ZINSS, given in Appendix D, to express the flexible price output in terms of the technology shock as Substituting into (47) reveals a proportional relationship between changes in marginal costs and the efficient output gap. Substituting into (43) yields the final form of the Phillips curve where the form of the composite parameter is Finally, the approximate Euler equation takes the form Author:
(1) David Staines. 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 5.1 Singular Phillips Curve This subsection flags up the subtle error in the standard approximation that brings about a purely forward-looking solution. Linearizing the two components of the Phillips curve at ZINSS reveals a special structure, in particular, there is a common root L = 1/αβ in both lag polynomials. This means the system can be solved by elimination. Therefore, we do not have to lag one of the recursions to perform a substitution introducing lagged terms. The next result confirms that this is unrepresentative of the dynamics of any stochastic system because it only applies when inflation is at steady state where the approximation is irrelevant. Let Lℵ equal the zero of the lag polynomial in ℵ. Proof . By elementary manipulation it is clear that this amounts to proving that Proof This follows immediately from Chebyshev’s correlation inequality (Lemma 1) which states that EAB ≥ EA EB for two strictly increasing functions with equality if and only the measure is degenerate. ZINSS is an example of a (measure zero) singularity not covered by Proposition 3. In fact it is the wall of the crossing which gives rise to (1). Bifurcation and common roots will be formally connected in Section 9. Thus we arrive at the incorrect step. Error 1. Cross Equation Cancellation in the Phillips curve Error 1. Cross Equation Cancellation in the Phillips curve This yields the forward-looking relationship. where the slope parameter is to derive an output gap expression first linearize the marginal cost function (38) Next linearize the production function (36) now using labor supply constraint and market clearing (19) and (20) The behaviour of price dispersion is governed by the linearization of (35) substituting in (π, ∆) = (0, 1) negates the first term leaving I can use the expression for ZINSS, given in Appendix D, to express the flexible price output in terms of the technology shock as Substituting into (47) reveals a proportional relationship between changes in marginal costs and the efficient output gap. Substituting into (43) yields the final form of the Phillips curve where the form of the composite parameter is Finally, the approximate Euler equation takes the form Author: (1) David Staines. Author: Author: (1) David Staines. 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

Why Economists Debate the Phillips Curve and Inflation Dynamics

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 3 Outline and Preview This final introductory section has two parts: the first summarizes the Phillips curve argument, whilst the second sets out the structure of the rest of the paper. 3.1 Phillips Curve A typical textbook New Keynesian Phillips curve currently looks like this The reason why is that as we approach ZINSS the Phillips curve is punctured by the following system of three singularities. The first is the wall of the crossing, which is necessary to the bifurcation. The second is a statistical restriction that destroys the error term. The third reflects the inter-temporal substitution incentive of the resetting firms. The crucial point is that substituting the singularities into the correct Phillips curve (2) simplifies to yield (1) the incorrect counterpart. The nature of the wall-crossing is that it is not possible to move in the other direction. This explains where macroeconomists went wrong for so many years. The precise parametric form of all coefficients are detailed later in the paper. The output gap term vanishing is specific to the parametization choices, which is discussed in the Appendix H alongside robustness exercises. The only nonstandard selection was a lower value for the inflation response, in the policy rule, in line with the anti-Taylor principle. 3.2 Roadmap The remainder of the paper is organized as follows. Section 4 sets out the New Keynesian framework. Section 5 focuses on existing solutions and details the Lucas critique. Section 6 lays out the stochastic equilibrium and a priori results. Section 7 derives the general Phillips curve. Section 8 proves the theoretical conditions governing existence and non-existence of equilibrium. Section 9 rigorously develops the bifurcation theory. Section 10 covers theoretical and econometric implications. Section 11 is devoted to the policy rule. Section 12 is for discussion. Author:
(1) David Staines. This paper is available on arxiv under CC 4.0 license. 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 3 Outline and Preview This final introductory section has two parts: the first summarizes the Phillips curve argument, whilst the second sets out the structure of the rest of the paper. 3.1 Phillips Curve A typical textbook New Keynesian Phillips curve currently looks like this The reason why is that as we approach ZINSS the Phillips curve is punctured by the following system of three singularities. The first is the wall of the crossing, which is necessary to the bifurcation. The second is a statistical restriction that destroys the error term. The third reflects the inter-temporal substitution incentive of the resetting firms. The crucial point is that substituting the singularities into the correct Phillips curve (2) simplifies to yield (1) the incorrect counterpart. The nature of the wall-crossing is that it is not possible to move in the other direction. This explains where macroeconomists went wrong for so many years. The precise parametric form of all coefficients are detailed later in the paper. The output gap term vanishing is specific to the parametization choices, which is discussed in the Appendix H alongside robustness exercises. The only nonstandard selection was a lower value for the inflation response, in the policy rule, in line with the anti-Taylor principle. 3.2 Roadmap The remainder of the paper is organized as follows. Section 4 sets out the New Keynesian framework. Section 5 focuses on existing solutions and details the Lucas critique. Section 6 lays out the stochastic equilibrium and a priori results. Section 7 derives the general Phillips curve. Section 8 proves the theoretical conditions governing existence and non-existence of equilibrium. Section 9 rigorously develops the bifurcation theory. Section 10 covers theoretical and econometric implications. Section 11 is devoted to the policy rule. Section 12 is for discussion. Author: (1) David Staines. Author: Author: (1) David Staines. 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

How Singularities Affect the Phillips Curve and Economic Policy

Matthew J Phillips is an entrepreneur from South Wales, U.K, and the creator of the Organic Client's program. Matthew has achieved feats such as closing multiple 4-figure clients weekly for his coaching business. He's also helped other coaches, consultants and mentors do the same. Some weeks ago, I sat down with Matthew and spoke on certain issues regarding his company and how he plans to move forward during these times. You'll also find this insightful. Let's have a look at what he had to say; So Matthew, why is now the time for your company to exist? More and more people are looking for a way to produce income in the chaos of COVID-19. We work with coaches, consultants, and mentors or those interested in becoming one. It seems to be a popular industry as there’s a low barrier to entry and many who are already coaches are in person trying to transition online. We supply the demand. What do you love about your team, and why are you the ones to solve this problem? We’re all results-oriented. We don’t care about flash, about the talk and all that jazz. We all care about the results. Every coach, consultant, or mentor we work worth our sole priority is ensuring they see success and changing their life. We show them how to create mouth-watering offers, generate streams of organic leads, and close their leads over the phone as high-paying clients. The three most common struggles you see and the only three things you need to master to have a thriving coaching, consulting, or mentorship business. If you weren’t building your startup, what would you be doing? I would be fitness coaching and continue to build my brand in the fitness industry like I was at 16 years old. I grew a following of 50K on Instagram and was sponsored by supplement companies etc. At the moment, how do you measure success? What are your metrics? Income. We care about helping coaches increase their clients to produce more income. For example, one of my clients from Montreal, Canada made $11,000 in under a month with his coaching business. Another, zero to $7,000 made in 5 days and a consultant from India closed 6 deals at $1,500 each totalling out to a total of $9,000 immediately upon implementation of my 1-1 program. In a few sentences, what do you offer to whom? We show coaches, consultants, and mentors how to create mouth-watering offers, generate streams of organic leads/sales appointments  - using inbound and outbound strategies for us then to show them how to close their leads as high-paying clients over the phone. We provide this service 1-1 through our main program The Organic Clients Program or we can breakdown that program  to fit precisely what they need. Help with content creation that drives leads, handling objections, etc. What’s most exciting about your traction to date? Our success stories and growth. My audience on Facebook is growing rapidly along with my FB group and we have new success stories pouring in daily along with interview and feature offers. Where do you think your growth will be next year? In the multiple 6 figure range along with branching out into other industries, for example, going back to my roots of fitness coaching where it all started for me before helping other coaches and starting a gym franchise/supplement company of my own. What’s your biggest threat? Hunger. As a young entrepreneur, 19 years old from the UK, South Wales, Cardiff to be specific. I’m hungry and I have options, now this helped me get to where I am although sometimes you can bite off too much than you can chew so I need to stay patient and not get off track with too many ventures and goals.

An Interview With Mathew J Phillips

Building Regulated AI Systems at Scale: Frameworks for Effective Governance

Miami, Florida, May 28th, 2025/Chainwire/--Today, Weed unveils its holistic wellness platform, a move aimed at reshaping the global conversation around well-being. Weed Weed Over the past five years, the team has directed more than $20 million toward intellectual-property acquisition and product development. The result is a convergence of advanced AI, decentralized Web3 design, and a brand with true cross-cultural resonance. Together, these elements open the door to a healthier, more participatory lifestyle economy. Mainstream AI tools rarely meaningfully address everyday health, a gap that Weed aims to close. By weaving technology with culture, the company recasts the word “weed” as shorthand for happiness, contentment, and personal fulfillment. While wellness is a process, well-being is a state of mind. Weed is the first global well-being brand. A New Chapter for Cannabis, Wellness, and Web3 Weed is more than cannabis; it represents a commitment to complete well-being. Built as a decentralized, community-first ecosystem, the platform spreads a message of practical self-care. It appeals to crypto natives and newcomers through token incentives, community governance, an upcoming AI assistant, and interactive digital content. The ambition is a circular well-being economy. With Weed, users do more than purchase; they help design, refine, and distribute what comes next. Extensive IP, Cultural Relevance, and Market Positioning Weed gained its edge by securing trademark protection for the WEED name across health, cosmetics, food, beverages, and many other industries. This rare portfolio creates a runway for Weed-branded products that tie directly into the Web3 stack. As legalization widens worldwide, early holders of the $WEED token obtain exposure to a brand positioned to shape the decade’s wellness landscape. Participation transforms consumers into stakeholders, aligning cultural impact with economic upside. What’s Next for Weed Chief executive Darin Phillips leads a multidisciplinary team preparing a global rollout of live events, celebrity collaborations, and product reveals. The launch of an AI agent will anchor this campaign and highlight how data-driven personalization can elevate everyday well-being. "At Weed, we believe in the love of others and the love of self. So no matter what the world throws at you, our motto is "Take the High Road." – Darin Philips "At Weed, we believe in the love of others and the love of self. So no matter what the world throws at you, our motto is "Take the High Road." – Darin Philips For further information, users can visit Weed's Discord Discord Discord About Weed Weed is the world’s first global well-being brand powered by AI and Web3. Bridging advanced technology with cultural consciousness, Weed redefines wellness as a participatory and decentralized lifestyle movement. Weed Weed Through its $WEED token, trademark-backed IP portfolio, and commitment to community-first innovation, Weed empowers individuals to design their journey toward health, happiness, and self-fulfillment. With a cross-industry presence spanning health, food, cosmetics, and beyond, Weed is cultivating a new era of holistic living, one that puts people, purpose, and peace of mind at the center. Contact Senior Growth Analyst Luki Radarblock Weed luki@radarblock.xyz This story was published as a press release by Chainwire under HackerNoon’s Business Blogging Program. This story was published as a press release by Chainwire under HackerNoon’s Business Blogging Program. Program Program

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