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 6.3 Literature Comparison This third subsection sets the new equilibrium concept in the context of the existing literature. This allows me to draw out salient features and delineate its applicability. Meyer-Gohde [2014] succinctly summarizes existing equilibrium notions in the following framework where k is a dummy variable: where k = 1 indicates the world is stochastic and k = 0 that it is non-stochastic. The three equilibrium concepts called deterministic steady state, stochastic steady state and ergodic mean are defined as follows Stochastic equilibrium has the property that Thus for the variables defining the recursive equilibrium (and affine combinations) stochastic equilibrium unites stochastic steady state and ergodic mean. Note that nonlinear functions of defining variables will not generally be at their mean value in the present period. For example, away from σ = 1, MC(Z) 6= EMC since in general MC(EZ) 6= EMC. [45] However, results need not carry over to continuous time. For example, Fernández-Villaverde et al. [2023] solve a heterogeneous agent model with nonlinear interactions between asset prices and precautionary saving. They find three stochastic steady states (corresponding to dZt = 0), of which two are locally stable. The reason is that in continuous time there is almost surely no incremental uncertainty[46] (as the future, present and past are arbitrarily close together). Therefore, there can be multiple stochastic steady states, in the way that there can be multiple non-stochastic steady states in discrete time. However, there is a unique ergodic invariant measure, with multiple steady states corresponding to a multi-modal distribution. The issue is that the model does not possess strong mixing; just because a fixed point is locally stable does not mean the economy is expected to stay there in the long-run.[47] It would be interesting to see how findings here carry over to models with multiple steady states and whether these techniques offer any traction in challenging continuous time environments. Beyond mathematical interest, it might help applied researchers decide between continuous and discrete time in specific applications. Finally, a concrete formulation of stochastic equilibrium may be of deeper theoretical interest. It has the desirable property that it is a fixed point concept associated with a single point in space, that is nevertheless determined entirely by out of equilibrium dynamics. This is clearly an improvement on previous disequilibrium modelling. Typically contributions have been static. Dynamic analysis has tended to use temporary equilibrium, where expectations are exogenous, as in Barro and Grossman [1971], Dreze [1975], Hahn [1978], Benassy [1990] and Grandmont [1982]. In fact, it can be seen as a synthesis of Keynesian adjustment and traditional general equilibrium theory.[48] Author:
(1) David Staines. This paper is available on arxiv under CC 4.0 license. [45] Z is unique up to Borel isomorphism (see Dudley [2018]). Therefore, you could redefine the cocycle with Z′ that included MC but left out one more of the variables in Z defined in Section 4. You would obtain an alternative map f ′ in which you could interpret that MC = EMC, although in return you would have to give up the possibility of giving this interpretation to another variable left out of Z ′ . With MC there would not be a problem but if higher moments were included you would have to verify the relevant expectations existed. [46] This could change if there were jump processes. [47] Formally, the incremental generating function from which the expectation is derived is weakly rather than strongly mixing, unlike f the corresponding object in the discrete time case constructed in Proposition 4. [48] The construction here speaks to Smale’s eighth problem about the inclusion of explicit price adjustment in general equilibrium. His inclusion of this economic problem in his list of major problems for twenty first century mathematics was esoteric (Smale [2000].) It harked back to Hilbert’s problems whose sixth referred to theoretical physics (Hilbert [1902].) I suspect it will receive extensive mathematical study. 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 6.3 Literature Comparison This third subsection sets the new equilibrium concept in the context of the existing literature. This allows me to draw out salient features and delineate its applicability. Meyer-Gohde [2014] succinctly summarizes existing equilibrium notions in the following framework where k is a dummy variable: where k = 1 indicates the world is stochastic and k = 0 that it is non-stochastic. The three equilibrium concepts called deterministic steady state, stochastic steady state and ergodic mean are defined as follows Stochastic equilibrium has the property that Thus for the variables defining the recursive equilibrium (and affine combinations) stochastic equilibrium unites stochastic steady state and ergodic mean. Note that nonlinear functions of defining variables will not generally be at their mean value in the present period. For example, away from σ = 1, MC(Z) 6= EMC since in general MC(EZ) 6= EMC. [45] However, results need not carry over to continuous time. For example, Fernández-Villaverde et al. [2023] solve a heterogeneous agent model with nonlinear interactions between asset prices and precautionary saving. They find three stochastic steady states (corresponding to dZt = 0), of which two are locally stable. The reason is that in continuous time there is almost surely no incremental uncertainty[46] (as the future, present and past are arbitrarily close together). Therefore, there can be multiple stochastic steady states, in the way that there can be multiple non-stochastic steady states in discrete time. However, there is a unique ergodic invariant measure, with multiple steady states corresponding to a multi-modal distribution. The issue is that the model does not possess strong mixing; just because a fixed point is locally stable does not mean the economy is expected to stay there in the long-run.[47] It would be interesting to see how findings here carry over to models with multiple steady states and whether these techniques offer any traction in challenging continuous time environments. Beyond mathematical interest, it might help applied researchers decide between continuous and discrete time in specific applications. Finally, a concrete formulation of stochastic equilibrium may be of deeper theoretical interest. It has the desirable property that it is a fixed point concept associated with a single point in space, that is nevertheless determined entirely by out of equilibrium dynamics. This is clearly an improvement on previous disequilibrium modelling. Typically contributions have been static. Dynamic analysis has tended to use temporary equilibrium, where expectations are exogenous, as in Barro and Grossman [1971], Dreze [1975], Hahn [1978], Benassy [1990] and Grandmont [1982]. In fact, it can be seen as a synthesis of Keynesian adjustment and traditional general equilibrium theory.[48] 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 [45] Z is unique up to Borel isomorphism (see Dudley [2018]). Therefore, you could redefine the cocycle with Z′ that included MC but left out one more of the variables in Z defined in Section 4. You would obtain an alternative map f ′ in which you could interpret that MC = EMC, although in return you would have to give up the possibility of giving this interpretation to another variable left out of Z ′ . With MC there would not be a problem but if higher moments were included you would have to verify the relevant expectations existed. [46] This could change if there were jump processes. [47] Formally, the incremental generating function from which the expectation is derived is weakly rather than strongly mixing, unlike f the corresponding object in the discrete time case constructed in Proposition 4. [48] The construction here speaks to Smale’s eighth problem about the inclusion of explicit price adjustment in general equilibrium. His inclusion of this economic problem in his list of major problems for twenty first century mathematics was esoteric (Smale [2000].) It harked back to Hilbert’s problems whose sixth referred to theoretical physics (Hilbert [1902].) I suspect it will receive extensive mathematical study.

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