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 5.4 Lucas Critique This part begins by explaining the Lucas critique and its various interpretations. The second subsection sets out observational equivalences and relates them to the Lucas critique framework. This should be viewed as a failure of existing New Keynesian economics. I will go onto demonstrate that the approximation I construct "passes" this crucial aspect of the Lucas critique by breaking this equivalence. In Appendix C.2.3, I confirm that there are (correctly solved) multiple models of recent vintage that are still ensnared. 5.4.1 Interpretations 1. Mapping Micro to Macro Lucas Jr [1976] heralded a new approach to macroeconomics that emphasised that models should be based on aggregation of the optimizing behaviour of agents. This methodology has become ubiquitous in modern research, providing an underlying motivation for all the New Keynesian models in this text. Researchers however have neglected to check solutions to DSGE models preserve the mapping from micro to macro. This paper analyses and exemplifies two threats to this connection. There are a priori reduced form approximations used in econometric work that do not represent neighboring stochastic systems because of failure to account for singularities. Principle 1 links this phenomenon back to the breakdown of constraints in individual optimization problems, akin to the common understanding of micro foundations. Secondly, there arise instances where the local approximation would be correct but ex post the dynamics of the reduced form are not consistent with the existence of nearby non-stochastic systems. In fact, these two forces combine to obscure any connection between the behavior of approximation solutions and the developments and mechanisms of an underlying non-linear New Keynesian economy. 2. Observational Equivalence The sole formal contribution of Lucas Jr [1976] was to demonstrate the equivalence between a new classical Phillips curve and a then standard Keynesian formulation of the relationship. It is a seminal result of macroeconometrics. I am able to show that it still applies to popular New Keynesian formulations but breaks down when Calvo is solved correctly. It demonstrates a fundamental advantage of Keynesian over Classical monetary models. 3. Breakdown of Statistical Relationships Lucas argued that statistical relationships like the Phillips curve[38] would break down when exploited for policy purposes.[39] This argument gained credence in the 1970s when there was stagflation (simultaneous increases in unemployment and inflation). The idea was central to the rise of New Classical economics, indeed it spurred the creation of DSGE. The result is only partially correct. He was right that the Phillips curve is not policy invariant, as the policy rule parameters affected the optimal supply decisions with drawbacks to monetary activism. He was also correct that there need not be an upward sloping relationship between inflation and output in equilibrium. On the other hand, this does not rule out systematic countercylical policy consistent with the Keynesian consensus (see Snowdon and Vane [2005]). Lastly, the comparative statics, part of Theorem 2, can be viewed as an analysis of policy regime changes of the kind considered by Lucas. 5.4.2 Observational Equivalence This part formalizes the traditional interpretation of the Lucas critique and proves that, subject to parametric conditions, it extends to the current benchmark solution of Calvo and the Rotemberg Phillips curves. The equivalence between these approximations is not novel, although the link back to Lucas is. Proposition 11. (Lucas Critique) There can arise an observational equivalence between the Singular Calvo, Rotemberg and Lucas Phillips curves provided that ω > η. Proof. Proposition 8 ensures the expectation terms vanish from the two forward-looking models. The rest of the proof is constructive. Equating (54) and (91) yields which sets Rotemberg equal to Calvo. Deploying (81) and equating shows equivalence to the Lucas model when the bound follows because these variances have to be strictly positive. The interpretation is that provided the Phillips curve is sufficiently steep, then the methods of macro econometrics cannot be used to distinguish between these three underlying designs for the Phillips curve. As an empirical matter, this is doubtful. Empirical specifications of the Phillips curve strongly favor a flat slope (ω < 1), whilst micro econometric evidence strongly supports an inelastic labor supply (η > 1).[40] Furthermore, there appears to be greater volatility of monetary shocks relative to technology shocks than the model would support. These issues are taken up in detail, in Appendices H and I. The broader interpretation is that these solutions are just barely "passing" the observational equivalence aspect of the Lucas critique. There are many dynamic variants of the Lucas model, featuring more sophisticated information frictions (see Angeletos et al. [2021] and Maćkowiak et al. [2023]). They typically generating flatter inflation-output trade-offs (Afrouzi and Yang [2021]). Thus with respect to these versions, forward-looking Phillips curves may still be ensnared by the critique. Conversely, by overturning the singular Calvo solution, I can rule out all these models too. We can do better than this. The Divine Coincidence and lack of dynamics have helped motivate a move away from DSGE back to ad hoc models. With new methods and better results, I anticipate a reversal. Moreover, armed with corrected approximations to Calvo around ZINSS, it should be possible to search for similar observational equivalences, in a wider class of DSGE. The link with Taylor pricing, established in Section 10, is a starting point. Author:
(1) David Staines. This paper is available on arxiv under CC 4.0 license. [38] He had in mind the original Phillips curve, a negative relationship between unemployment and inflation estimated by Phillips [1958]. [39] Similar ideas were advanced in the context of monetary targeting by Goodhart [1975] without formal mathematical underpinning. The principles are of wide application throughout economics. 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 5.4 Lucas Critique This part begins by explaining the Lucas critique and its various interpretations. The second subsection sets out observational equivalences and relates them to the Lucas critique framework. This should be viewed as a failure of existing New Keynesian economics. I will go onto demonstrate that the approximation I construct "passes" this crucial aspect of the Lucas critique by breaking this equivalence. In Appendix C.2.3, I confirm that there are (correctly solved) multiple models of recent vintage that are still ensnared. 5.4.1 Interpretations 1. Mapping Micro to Macro 1. Mapping Micro to Macro Lucas Jr [1976] heralded a new approach to macroeconomics that emphasised that models should be based on aggregation of the optimizing behaviour of agents. This methodology has become ubiquitous in modern research, providing an underlying motivation for all the New Keynesian models in this text. Researchers however have neglected to check solutions to DSGE models preserve the mapping from micro to macro. This paper analyses and exemplifies two threats to this connection. There are a priori reduced form approximations used in econometric work that do not represent neighboring stochastic systems because of failure to account for singularities. Principle 1 links this phenomenon back to the breakdown of constraints in individual optimization problems, akin to the common understanding of micro foundations. Secondly, there arise instances where the local approximation would be correct but ex post the dynamics of the reduced form are not consistent with the existence of nearby non-stochastic systems. In fact, these two forces combine to obscure any connection between the behavior of approximation solutions and the developments and mechanisms of an underlying non-linear New Keynesian economy. 2. Observational Equivalence 2. Observational Equivalence The sole formal contribution of Lucas Jr [1976] was to demonstrate the equivalence between a new classical Phillips curve and a then standard Keynesian formulation of the relationship. It is a seminal result of macroeconometrics. I am able to show that it still applies to popular New Keynesian formulations but breaks down when Calvo is solved correctly. It demonstrates a fundamental advantage of Keynesian over Classical monetary models. 3. Breakdown of Statistical Relationships 3. Breakdown of Statistical Relationships Lucas argued that statistical relationships like the Phillips curve[38] would break down when exploited for policy purposes.[39] This argument gained credence in the 1970s when there was stagflation (simultaneous increases in unemployment and inflation). The idea was central to the rise of New Classical economics, indeed it spurred the creation of DSGE. The result is only partially correct. He was right that the Phillips curve is not policy invariant, as the policy rule parameters affected the optimal supply decisions with drawbacks to monetary activism. He was also correct that there need not be an upward sloping relationship between inflation and output in equilibrium. On the other hand, this does not rule out systematic countercylical policy consistent with the Keynesian consensus (see Snowdon and Vane [2005]). Lastly, the comparative statics, part of Theorem 2, can be viewed as an analysis of policy regime changes of the kind considered by Lucas. 5.4.2 Observational Equivalence This part formalizes the traditional interpretation of the Lucas critique and proves that, subject to parametric conditions, it extends to the current benchmark solution of Calvo and the Rotemberg Phillips curves. The equivalence between these approximations is not novel, although the link back to Lucas is. Proposition 11. (Lucas Critique) There can arise an observational equivalence between the Singular Calvo, Rotemberg and Lucas Phillips curves provided that ω > η. Proposition 11. (Lucas Critique) There can arise an observational equivalence between the Singular Calvo, Rotemberg and Lucas Phillips curves provided that ω > η. Proof . Proposition 8 ensures the expectation terms vanish from the two forward-looking models. The rest of the proof is constructive. Equating (54) and (91) yields Proof which sets Rotemberg equal to Calvo. Deploying (81) and equating shows equivalence to the Lucas model when the bound follows because these variances have to be strictly positive. The interpretation is that provided the Phillips curve is sufficiently steep, then the methods of macro econometrics cannot be used to distinguish between these three underlying designs for the Phillips curve. As an empirical matter, this is doubtful. Empirical specifications of the Phillips curve strongly favor a flat slope (ω < 1), whilst micro econometric evidence strongly supports an inelastic labor supply (η > 1).[40] Furthermore, there appears to be greater volatility of monetary shocks relative to technology shocks than the model would support. These issues are taken up in detail, in Appendices H and I. The broader interpretation is that these solutions are just barely "passing" the observational equivalence aspect of the Lucas critique. There are many dynamic variants of the Lucas model, featuring more sophisticated information frictions (see Angeletos et al. [2021] and Maćkowiak et al. [2023]). They typically generating flatter inflation-output trade-offs (Afrouzi and Yang [2021]). Thus with respect to these versions, forward-looking Phillips curves may still be ensnared by the critique. Conversely, by overturning the singular Calvo solution, I can rule out all these models too. We can do better than this. The Divine Coincidence and lack of dynamics have helped motivate a move away from DSGE back to ad hoc models. With new methods and better results, I anticipate a reversal. Moreover, armed with corrected approximations to Calvo around ZINSS, it should be possible to search for similar observational equivalences, in a wider class of DSGE. The link with Taylor pricing, established in Section 10, is a starting point. 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 [38] He had in mind the original Phillips curve, a negative relationship between unemployment and inflation estimated by Phillips [1958]. [39] Similar ideas were advanced in the context of monetary targeting by Goodhart [1975] without formal mathematical underpinning. The principles are of wide application throughout economics.

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