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 D.2 Profits and Long-Run Growth The first two subsubsections house the proofs of the two main propositions concerning long-run profits. The final subsubsection contains a final proof and fortifies empirical arguments concerning labour supply made in the profit section. D.2.1 Proof of Proposition 13 Proof. The proof starts with the lower bound where there are three cases and then finishes with the upper bound. For the lower bound the order of argument is (ii), (iii) then (i). Aggregate profit can be written as where the subscript indicates the time since the last reset. Cross-multiplying, taking expectations and using the non-stochastic limit, to cancel out terms in the preference shock ψ from both sides, reveals that in the patient limit; with the weight coming from the terms in (25) and the context of the resetting firms problem because the demand curve slopes down and marginal costs are constant, we can see that For Assumption 2 (iii), the proof comes from an analysis of covariance and (strict) stochastic monotonicity properties. By application of Theorem 1.A.3 and 1.A.8 of Shaked and Shanthikumar [2007] to the cases of above and below average profit, we know that the covariance is strictly positive. Applying the definition of covariance. Therefore there will be three channels. A substitution effect from (lower) wages onto labor supply and two income effects, one from the change in wages and the other from the change in profits. Theorem 2 tells us that labor supply rises and wages fall. The strategy is to show that the first two effects net out to reduce labor supply. This means that profits must decrease, to generate a wealth effect that increases labor supply. To demonstrate this point, it is sufficient to show that the labor supply curve is positively sloped. Totally differentiating with respect to L Simplifying yields Since the right hand side is always positive, the sign of the derivative is determined by the bracket. Now substitute the production function in to create the determinant D.2.2 Proof of Proposition 14 Proof. The result comes from a chain of inequalities. From the principle of optimality, it is clear that the firm is weakly better off when it can change prices. Furthermore, since we know that profit function is strictly concave, the optimization problem at any time will have a unique solution. Therefore, we can upgrade to strict inequalities where I am using (25) (the profit function). Since the demand curve slopes down and there are no increasing returns to scale, we know that D.2.3 Additional Arguments The evidence on long-run labor supply elasticity is mixed. The consensus estimate appears to be zero associated with Doran [2014] and Berger et al. [2018], in which case the main result goes through. Ashenfelter et al. [2010] argues that the likely range is between -0.2 and 0.2 with their study at the lower range. The concern with unemployment and underemployment, in Mas and Pallais [2019], favours a positive figure. There is more discussion in the parametization Appendix I. Proof. With balanced growth it has been established that σ = 1, whilst the labor supply and profit conditions take the form and the proof will be complete if we can argue that the left hand bracket is strictly positive with probability one. To do so suppose the converse, then it follows from the definition of balanced growth in stochastic equilibrium that 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 D.2 Profits and Long-Run Growth The first two subsubsections house the proofs of the two main propositions concerning long-run profits. The final subsubsection contains a final proof and fortifies empirical arguments concerning labour supply made in the profit section. D.2.1 Proof of Proposition 13 D.2.1 Proof of Proposition 13 Proof. The proof starts with the lower bound where there are three cases and then finishes with the upper bound. For the lower bound the order of argument is (ii), (iii) then (i). Aggregate profit can be written as where the subscript indicates the time since the last reset. Cross-multiplying, taking expectations and using the non-stochastic limit, to cancel out terms in the preference shock ψ from both sides, reveals that in the patient limit; with the weight coming from the terms in (25) and the context of the resetting firms problem because the demand curve slopes down and marginal costs are constant, we can see that For Assumption 2 (iii), the proof comes from an analysis of covariance and (strict) stochastic monotonicity properties. By application of Theorem 1.A.3 and 1.A.8 of Shaked and Shanthikumar [2007] to the cases of above and below average profit, we know that the covariance is strictly positive. Applying the definition of covariance. Therefore there will be three channels. A substitution effect from (lower) wages onto labor supply and two income effects, one from the change in wages and the other from the change in profits. Theorem 2 tells us that labor supply rises and wages fall. The strategy is to show that the first two effects net out to reduce labor supply. This means that profits must decrease, to generate a wealth effect that increases labor supply. To demonstrate this point, it is sufficient to show that the labor supply curve is positively sloped. Totally differentiating with respect to L Simplifying yields Since the right hand side is always positive, the sign of the derivative is determined by the bracket. Now substitute the production function in to create the determinant D.2.2 Proof of Proposition 14 D.2.2 Proof of Proposition 14 Proof. The result comes from a chain of inequalities. From the principle of optimality, it is clear that the firm is weakly better off when it can change prices. Furthermore, since we know that profit function is strictly concave, the optimization problem at any time will have a unique solution. Therefore, we can upgrade to strict inequalities where I am using (25) (the profit function). Since the demand curve slopes down and there are no increasing returns to scale, we know that D.2.3 Additional Arguments D.2.3 Additional Arguments The evidence on long-run labor supply elasticity is mixed. The consensus estimate appears to be zero associated with Doran [2014] and Berger et al. [2018], in which case the main result goes through. Ashenfelter et al. [2010] argues that the likely range is between -0.2 and 0.2 with their study at the lower range. The concern with unemployment and underemployment, in Mas and Pallais [2019], favours a positive figure. There is more discussion in the parametization Appendix I. Proof. With balanced growth it has been established that σ = 1, whilst the labor supply and profit conditions take the form and the proof will be complete if we can argue that the left hand bracket is strictly positive with probability one. To do so suppose the converse, then it follows from the definition of balanced growth in stochastic equilibrium 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

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