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 5.3 Two Comparison Models This subsection introduces two alternative models featuring monetary Non neutrality. They will be used in the next section and throughout the paper as a point of comparison with solutions of the Calvo model. The first model is classical, in the sense that it features perfectly flexible prices. The second is Keynesian because prices are not able to adjust fully to inflation. Along with the menu cost model discussed in Section 8, it is an example of a state-dependent pricing model. Crucially, it does not feature staggered optimization. Prices will typically change in every period. I layout the equivalence between the existing solution of Calvo and the Rotemberg model, I extend this equivalence to the original Lucas model, allowing me to view it as part of the observational equivalence claim of Lucas Jr [1976]. Finally, I briefly review the rest of the critique. 5.3.1 Lucas Model Lucas Jr [1972] is the original rational expectations monetary model. Markets are perfectly competitive and anticipated monetary shocks are impotent. Monetary non-neutrality arises because of imperfect information rather than sticky prices. Nevertheless, by adding a money demand function, I am able to derive surprising similarity to New Keynesian models with price rigidity. There are a collection of price-taking households.[34] Each produces a single good, using just its own labor.[35] The choice problem simplifies to a static labor-leisure trade-of The first order condition balances marginal revenue and marginal cost so Solving and log-linearizing gives an expression for an individual firm’s supply curve For parsimony, a reduced form aggregate demand curve replaces the Euler equation (18). Idiosyncratic taste shocks, reflected by the disturbance zt(i) in the individual firms demand schedule, are essential to the Lucas model. I assume they are independent and identically distributed across producers and time, with tail restrictions such that they will cancel out in aggregate almost surely. Here is the log-linear form There is linear aggregation so whilst Monetary non-neutrality is entirely driven by unexpected shocks. There is no role for price dispersion here, since the firms are price-takers and the distribution of relative price is unaffected by inflation. Unlike its Keynesian counterparts, the Lucas model would be unaffected by trend inflation. Using the individual supply curve (62), demand curve (64), steady state deviation (68) and the Lucas supply curve, (71) and (72) implies the expression for relative price deviation and from (69) and (70) This allows me to derive the following implicit formulation final form for the slope of the Lucas surprise Phillips curve 5.3.2 Rotemberg Pricing This model Rotemberg [1982] is perhaps the simplest formulation of state dependent pricing and the one most commonly compared to Calvo. The idea is that firms face a convex cost of changing prices and therefore they do not adjust immediately to their optimal flexible price. The problem is that convex costs of price changing imply firms change their prices every period through small adjustments. This runs contrary to an overwhelming body of evidence of infrequent large adjustment discussed in Appendix H.1.3. Therefore the first order conditions simplify to Finally, the resource constraint reflecting a wedge between production and consumption caused by the cost of price changing replaces (20) Author:
(1) David Staines. This paper is available on arxiv under CC 4.0 license. [34] It is not common to have differentiated products and price-taking behavior. For motivation imagine there were actually multiple households producing the same good and therefore competition between them or the possibility to set-up firms with a consequent free entry condition. Recall the Bertrand paradox that two competing firms can implement a competitive equilibrium. [35] This formulation comes from Romer [2012]. The idea is to avoid the households using the economy-wide labor market to deduce the aggregate price level. In the original paper, Lucas used an overlapping generations structure where households produced in one period then sold output in the next. Recent literature has developed more sophisticated stories about dispersed private information and strategic interaction. I do not pursue this approach as I am looking for a benchmark model. 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 5.3 Two Comparison Models This subsection introduces two alternative models featuring monetary Non neutrality. They will be used in the next section and throughout the paper as a point of comparison with solutions of the Calvo model. The first model is classical, in the sense that it features perfectly flexible prices. The second is Keynesian because prices are not able to adjust fully to inflation. Along with the menu cost model discussed in Section 8, it is an example of a state-dependent pricing model. Crucially, it does not feature staggered optimization. Prices will typically change in every period. I layout the equivalence between the existing solution of Calvo and the Rotemberg model, I extend this equivalence to the original Lucas model, allowing me to view it as part of the observational equivalence claim of Lucas Jr [1976]. Finally, I briefly review the rest of the critique. 5.3.1 Lucas Model Lucas Jr [1972] is the original rational expectations monetary model. Markets are perfectly competitive and anticipated monetary shocks are impotent. Monetary non-neutrality arises because of imperfect information rather than sticky prices. Nevertheless, by adding a money demand function, I am able to derive surprising similarity to New Keynesian models with price rigidity. There are a collection of price-taking households.[34] Each produces a single good, using just its own labor.[35] The choice problem simplifies to a static labor-leisure trade-of The first order condition balances marginal revenue and marginal cost so Solving and log-linearizing gives an expression for an individual firm’s supply curve For parsimony, a reduced form aggregate demand curve replaces the Euler equation (18). Idiosyncratic taste shocks, reflected by the disturbance zt(i) in the individual firms demand schedule, are essential to the Lucas model. I assume they are independent and identically distributed across producers and time, with tail restrictions such that they will cancel out in aggregate almost surely. Here is the log-linear form There is linear aggregation so whilst Monetary non-neutrality is entirely driven by unexpected shocks. There is no role for price dispersion here, since the firms are price-takers and the distribution of relative price is unaffected by inflation. Unlike its Keynesian counterparts, the Lucas model would be unaffected by trend inflation. Using the individual supply curve (62), demand curve (64), steady state deviation (68) and the Lucas supply curve, (71) and (72) implies the expression for relative price deviation and from (69) and (70) This allows me to derive the following implicit formulation final form for the slope of the Lucas surprise Phillips curve 5.3.2 Rotemberg Pricing This model Rotemberg [1982] is perhaps the simplest formulation of state dependent pricing and the one most commonly compared to Calvo. The idea is that firms face a convex cost of changing prices and therefore they do not adjust immediately to their optimal flexible price. The problem is that convex costs of price changing imply firms change their prices every period through small adjustments. This runs contrary to an overwhelming body of evidence of infrequent large adjustment discussed in Appendix H.1.3. Therefore the first order conditions simplify to Finally, the resource constraint reflecting a wedge between production and consumption caused by the cost of price changing replaces (20) 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 [34] It is not common to have differentiated products and price-taking behavior. For motivation imagine there were actually multiple households producing the same good and therefore competition between them or the possibility to set-up firms with a consequent free entry condition. Recall the Bertrand paradox that two competing firms can implement a competitive equilibrium. [35] This formulation comes from Romer [2012]. The idea is to avoid the households using the economy-wide labor market to deduce the aggregate price level. In the original paper, Lucas used an overlapping generations structure where households produced in one period then sold output in the next. Recent literature has developed more sophisticated stories about dispersed private information and strategic interaction. I do not pursue this approach as I am looking for a benchmark model.

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