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 4 Calvo Framework This section lays out the principal model used throughout the paper featuring Calvo pricing and a Taylor rule. The assumptions concerning error terms present throughout the paper are discussed. The section winds up with a crucial finite dimensional characterization of the system. 4.1 Household’s Problem This part lays out the households optimization problem that underpins the demand side of the economy. There is a single representative household[20] that chooses consumption C and labor supply L, so as to maximize the following objective function subject to the budget constraint There is a unit continuum of firms. Π(i) is profit from an individual firm i given by Note in a stochastic environment firms need not make the same profits even with a symmetric equilibrium. Since when price rigidity is introduced, firms with the same demand curve will charge different prices, depending on when they last re-optimized and can therefore make different levels of profit.[21] The budget constraint states that the uses for nominal income (consumption and saving) must be equal to the sources of income (wealth, labor and dividend income). Finally, there are two constraints The first is a bond market clearing condition. It is simplification designed to allow me to ignore the effect of government spending or debt management. It is not important at this stage because the analysis focuses on limiting cases where, without nominal frictions, Ricardian equivalence ensures that debt levels do not matter (Barro [1974]). The second is a transversality condition. It is necessary for a well-defined optimization problem. It forces the household to honor their debts. Otherwise they would seek to borrow an unbounded amount and never repay. Giglio et al. [2016] provides model-independent support. Author:
(1) David Staines. This paper is available on arxiv under CC 4.0 license. [20] You may be surprised with a model where the world is organized into one happy family. It simplifies the dynamic exposition. Werning [2016] provides useful aggregation results, however none survive a full stochastic setting. The bifurcation analysis, the backbone of the Phillips curve, goes through without reference to a specific Euler. [21] In fact non-resetters need not have positive profit expectations or even a non-negative stock price, although, I will show this does not apply in our quantitative setting. Alternatively, one could swerve around this problem by imagining there are a continuum of households who each own one firm and an efficient long-term insurance market with contracts set arbitrarily far back in the past (so that there is no conditioning on pricing history). In this case the market will absorb all idiosyncratic risk associated with non-optimal price setting behavior. 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 4 Calvo Framework This section lays out the principal model used throughout the paper featuring Calvo pricing and a Taylor rule. The assumptions concerning error terms present throughout the paper are discussed. The section winds up with a crucial finite dimensional characterization of the system. 4.1 Household’s Problem This part lays out the households optimization problem that underpins the demand side of the economy. There is a single representative household[20] that chooses consumption C and labor supply L , so as to maximize the following objective function C L subject to the budget constraint There is a unit continuum of firms. Π( i ) is profit from an individual firm i given by i i Note in a stochastic environment firms need not make the same profits even with a symmetric equilibrium. Since when price rigidity is introduced, firms with the same demand curve will charge different prices, depending on when they last re-optimized and can therefore make different levels of profit.[21] The budget constraint states that the uses for nominal income (consumption and saving) must be equal to the sources of income (wealth, labor and dividend income). Finally, there are two constraints The first is a bond market clearing condition. It is simplification designed to allow me to ignore the effect of government spending or debt management. It is not important at this stage because the analysis focuses on limiting cases where, without nominal frictions, Ricardian equivalence ensures that debt levels do not matter (Barro [1974]). The second is a transversality condition. It is necessary for a well-defined optimization problem. It forces the household to honor their debts. Otherwise they would seek to borrow an unbounded amount and never repay. Giglio et al. [2016] provides model-independent support. 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 [20] You may be surprised with a model where the world is organized into one happy family. It simplifies the dynamic exposition. Werning [2016] provides useful aggregation results, however none survive a full stochastic setting. The bifurcation analysis, the backbone of the Phillips curve, goes through without reference to a specific Euler. [21] In fact non-resetters need not have positive profit expectations or even a non-negative stock price, although, I will show this does not apply in our quantitative setting. Alternatively, one could swerve around this problem by imagining there are a continuum of households who each own one firm and an efficient long-term insurance market with contracts set arbitrarily far back in the past (so that there is no conditioning on pricing history). In this case the market will absorb all idiosyncratic risk associated with non-optimal price setting behavior.

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The Impact of Calvo Pricing and the Taylor Rule on Economic Modeling and Household Behavior

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