Ergodic Theory and Markov Chains Reveal Unique Stable Solutions in DSGE Models

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 2 Mathematical Arguments This paper has two mathematical objectives, one analytic the other algebraic. The first is to analyze the most basic qualitative and quantitative properties governing existence of equilibrium. The second to describe the algebraic structures around ZINSS that give rise to the erroneous conclusions holding back modern macroeconomics. The subsequent discussion is heuristic rather than technically precise.[15] The central analytic contribution is a powerful fixed-point result. In fact, Theorem 3 provides necessary and sufficient conditions for any solution to exist, in a wide class of DSGE models, for a popular limiting case. Away from this limit, the condition is a requirement for standard statistical inference. The proof has two parts. The first step uses an argument from the Markov chains literature to show that the model has to mix. This follows easily from two standard features of these models: firstly, the existence of a sufficiently smooth cocycle in the mathematical expectation (coming from rational expectations) and secondly, the blow up on the boundary of the relevant objective functions of the optimizing agents, along a popular limit. The second step is motivated by control theory of linear rational expectations systems. It employs the local regularity properties from a stochastic Grobman-Hartman theorem of Coayla-Teran et al. [2007] to prove that only a unique stable solution is consistent with equilibrium existence. This arises where the number of eigenvalues inside the unit circle matches the number of jump variables and the number outside the total predetermined variables. Standard calculations yield functional conditions containing non-linear functions evaluated at the ergodic invariant measure. Applications follow swiftly. Theorem 4 uses basic ergodic theory to demonstrate that the probabilistic trajectory of the mean field system is unique, overturning previous non-rigorous work. The main focus of the paper is on limits where the noise is small. This allows me to derive quantitative estimates of the structural parameters supporting equilibrium existence using elementary methods. Polydromy arises because one variable, which is second order around the non-stochastic steady state, is first order around the stochastic steady state, although existence conditions are unaffected. For future functional analysis, I provide a mathematical rigidity result stemming from ergodicity. The comparative statics in Theorem 2 can also be interpreted in this light as (sharp) a priori quantitative moment estimates, reflecting model-specific equilibrium restrictions. They could surely be profitably combined in future to yield quantitative descriptions of the function space supporting this and other DSGE. Moreover, in Theorem 8, I prove a basic regularity result pertaining to the optimal policy problem that should be more amenable to existing functional analysis goals and techniques. The De Rham cohomology underpins the bifurcation analysis. In Decomposition 1, I calculate the singular surface around ZINSS. Theorem 6 and extensions connect the structure of the singular surface to the efficiency properties of ZINSS and the absence of distortions from large shocks. The disproof of Divine Coincidence, in Theorem 8, confirms the constraints interpretation. More abstract techniques including schemes and categories help to generalize and interpret these bifurcations. In Theorem 7, scheme theory is used to show that cross-equation cancellations are the ultimate source of all bifurcations in DSGE systems (with well-defined solutions) and admits speedy generalization. Furthermore, the idea of "near solutions", constructed via scheme theory, allow me to formalize Stochastic Bifurcation and to explain how the effect attributed to trend inflation conceals an error in the Phillips curve approximated at ZINSS. Econometric duality results from the confluence of constrained optimization, scheme and cohomology theory. Finally, categories allow me to axiomize an aspect of the Lucas critique. The paper contributes to two strands of mathematical and statistical literature. The first concerns mean field games. These are a joint endeavour between mathematics, economics, engineering and other disciplines. Some prominent papers include Jovanovic and Rosenthal [1988], Caines et al. [2006], Lasry and Lions [2006], Lasry and Lions [2007a], Guéant et al. [2011] and Cardaliaguet et al. [2019]. The closest to macroeconomics include Lasry and Lions [2007b], Nourian and Caines [2013], Bensoussan et al. [2016], Lasry and Lions [2018] and Cardaliaguet et al. [2020] that consider games with small and large players. Macroeconomic applications, so far, include Achdou et al. [2022] who construct a mean field solution to an incomplete markets model and Porretta and Rossi [2022] who study economic growth in the context of knowledge diffusion. Lastly, Alvarez et al. [2023] focus on a restricted variant of the Calvo model studied here.[16] [17] Results in adjacent areas include Carmona et al. [2016], Lacker [2016] and Carmona and Delarue [2018], who study common noise (aggregate shocks) but in finite time or with restrictive monotonicity conditions, as in Bertucci et al. [2021].[18] Gomes et al. [2010], Chau et al. [2017], Saldi et al. [2018a], Saldi et al. [2018b] and Saldi [2019] are formulated in discrete time. Finally, Cardaliaguet and Souganidis [2022] attack a (representative agent) model with no idiosyncratic noise. The closest contributions to my own concern ergodicity and non-existence. In ergodic mean field games, players maximize a long-term average objective function. Cao et al. [2022] show that discounted mean field games approach their ergodic counterpart as time preference recedes. This parallels my focus on stochastic equilibrium here. Secondly, Cirant and Ghilli [2022] prove a nonexistence result. The culprit is a strong complementarity (externality), in their case, it is effectively in the economy’s growth process. Decomposition 2 has this flavour here, albeit with a different source. With direct macroeconomic application, Alvarez et al. [2023] uncovers non-existence, when strategic complementarities in pricing cross a critical threshold. Both results focus on the limit of finite time, with a boundary condition designed to represent long-run equilibrium, rather than a true infinite horizon with an endogenous destination. A unique feature of my approach is an emphasis on game-theoretic constructive methods. The limiting construct is inspired by the folk theorem and the eigenvalue condition functions like a dynamic incentive compatibility constraint for the entire economy. This paper constitutes the first progress on constructing rigorous solutions to a general class of mean field games with common noise over infinite time- which is surely the major open problem of the whole literature (Achdou et al. [2014]). This paper is formulated in discrete time, partly to reduce the technical burden on readers. Nevertheless, I am confident that techniques and concepts will carry over into continuous time, under some circumstances. An example, discussed in Section 6, shows that there may be differences in qualitative behavior between discrete and continuous time. It should be possible to analyze this time-scaling behavior, both to help economists decide which solutions to use and for its mathematical beauty. Both the a priori duality result and the ex post blow ups are connected to a small literature on phase transition of test statistics, which studies conditions where statistical estimators cease to converge. Examples include Cover [1965], Silvapulle [1981], Albert and Anderson [1984], Santner and Duffy [1986], Sur et al. [2019], Candès and Sur [2020] and McCracken [2020]. I am confident statistical theory will expand along many frontiers to accommodate the deep and multi-faceted predictions of stochastic equilibrium theory. In my epsilon management I will deploy the asymptotic notation x << y to mean "x is negligible relative to y".[19] Unless otherwise stated, all background results are available in Aliprantis and Border [2007] or Royden and Fitzpatrick [2010]. Jacobson [2009] and Jacobson [1980] might be helpful references for the abstract algebra with Eisenbud and Harris [2006] useful for schemes. The main text will emphasize intuition with more formality and some preliminaries in the Appendices. Author: (1) David Staines. This paper is available on arxiv under CC 4.0 license. [15] It is helpful to define a few terms from mathematical analysis for the benefit of economists. A quantitative estimate is where an unknown function or constant is bounded using only the model’s primitives. Mathematical rigidity is where a collection of objects can be described from a smaller set of their properties. Universality is where models with different formulations display common behavior or structures. They will appear repeatedly in this section. [16] They use the price gap formulation of the period profit function (Woodford [2009]), which avoids the singularities in the existing approximation, at the expense of simplifying dynamics. [17] Bilal [2023] develops perturbation techniques using the master equation but does not prove existence results. His focus on reflecting boundaries suits his main application to job ladders and unemployment but contrasts with the blow-up condition here. [18] Monotonicity conditions are popular in the mean field game literature but are restrictive because macroeconomic time series typically display non-monotone responses particularly when there are shocks to monetary policy. [19] This corresponds to x = o(y) in the notation of Tao [April 2, 2012], where further background can be found. 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 2 Mathematical Arguments This paper has two mathematical objectives, one analytic the other algebraic. The first is to analyze the most basic qualitative and quantitative properties governing existence of equilibrium. The second to describe the algebraic structures around ZINSS that give rise to the erroneous conclusions holding back modern macroeconomics. The subsequent discussion is heuristic rather than technically precise.[15] The central analytic contribution is a powerful fixed-point result. In fact, Theorem 3 provides necessary and sufficient conditions for any solution to exist, in a wide class of DSGE models, for a popular limiting case. Away from this limit, the condition is a requirement for standard statistical inference. The proof has two parts. The first step uses an argument from the Markov chains literature to show that the model has to mix. This follows easily from two standard features of these models: firstly, the existence of a sufficiently smooth cocycle in the mathematical expectation (coming from rational expectations) and secondly, the blow up on the boundary of the relevant objective functions of the optimizing agents, along a popular limit. The second step is motivated by control theory of linear rational expectations systems. It employs the local regularity properties from a stochastic Grobman-Hartman theorem of Coayla-Teran et al. [2007] to prove that only a unique stable solution is consistent with equilibrium existence. This arises where the number of eigenvalues inside the unit circle matches the number of jump variables and the number outside the total predetermined variables. Standard calculations yield functional conditions containing non-linear functions evaluated at the ergodic invariant measure. Applications follow swiftly. Theorem 4 uses basic ergodic theory to demonstrate that the probabilistic trajectory of the mean field system is unique, overturning previous non-rigorous work. The main focus of the paper is on limits where the noise is small. This allows me to derive quantitative estimates of the structural parameters supporting equilibrium existence using elementary methods. Polydromy arises because one variable, which is second order around the non-stochastic steady state, is first order around the stochastic steady state, although existence conditions are unaffected. For future functional analysis, I provide a mathematical rigidity result stemming from ergodicity. The comparative statics in Theorem 2 can also be interpreted in this light as (sharp) a priori quantitative moment estimates, reflecting model-specific equilibrium restrictions. They could surely be profitably combined in future to yield quantitative descriptions of the function space supporting this and other DSGE. Moreover, in Theorem 8, I prove a basic regularity result pertaining to the optimal policy problem that should be more amenable to existing functional analysis goals and techniques. The De Rham cohomology underpins the bifurcation analysis. In Decomposition 1, I calculate the singular surface around ZINSS. Theorem 6 and extensions connect the structure of the singular surface to the efficiency properties of ZINSS and the absence of distortions from large shocks. The disproof of Divine Coincidence, in Theorem 8, confirms the constraints interpretation. More abstract techniques including schemes and categories help to generalize and interpret these bifurcations. In Theorem 7, scheme theory is used to show that cross-equation cancellations are the ultimate source of all bifurcations in DSGE systems (with well-defined solutions) and admits speedy generalization. Furthermore, the idea of "near solutions", constructed via scheme theory, allow me to formalize Stochastic Bifurcation and to explain how the effect attributed to trend inflation conceals an error in the Phillips curve approximated at ZINSS. Econometric duality results from the confluence of constrained optimization, scheme and cohomology theory. Finally, categories allow me to axiomize an aspect of the Lucas critique. The paper contributes to two strands of mathematical and statistical literature. The first concerns mean field games. These are a joint endeavour between mathematics, economics, engineering and other disciplines. Some prominent papers include Jovanovic and Rosenthal [1988], Caines et al. [2006], Lasry and Lions [2006], Lasry and Lions [2007a], Guéant et al. [2011] and Cardaliaguet et al. [2019]. The closest to macroeconomics include Lasry and Lions [2007b], Nourian and Caines [2013], Bensoussan et al. [2016], Lasry and Lions [2018] and Cardaliaguet et al. [2020] that consider games with small and large players. Macroeconomic applications, so far, include Achdou et al. [2022] who construct a mean field solution to an incomplete markets model and Porretta and Rossi [2022] who study economic growth in the context of knowledge diffusion. Lastly, Alvarez et al. [2023] focus on a restricted variant of the Calvo model studied here.[16] [17] Results in adjacent areas include Carmona et al. [2016], Lacker [2016] and Carmona and Delarue [2018], who study common noise (aggregate shocks) but in finite time or with restrictive monotonicity conditions, as in Bertucci et al. [2021].[18] Gomes et al. [2010], Chau et al. [2017], Saldi et al. [2018a], Saldi et al. [2018b] and Saldi [2019] are formulated in discrete time. Finally, Cardaliaguet and Souganidis [2022] attack a (representative agent) model with no idiosyncratic noise. The closest contributions to my own concern ergodicity and non-existence. In ergodic mean field games, players maximize a long-term average objective function. Cao et al. [2022] show that discounted mean field games approach their ergodic counterpart as time preference recedes. This parallels my focus on stochastic equilibrium here. Secondly, Cirant and Ghilli [2022] prove a nonexistence result. The culprit is a strong complementarity (externality), in their case, it is effectively in the economy’s growth process. Decomposition 2 has this flavour here, albeit with a different source. With direct macroeconomic application, Alvarez et al. [2023] uncovers non-existence, when strategic complementarities in pricing cross a critical threshold. Both results focus on the limit of finite time, with a boundary condition designed to represent long-run equilibrium, rather than a true infinite horizon with an endogenous destination. A unique feature of my approach is an emphasis on game-theoretic constructive methods. The limiting construct is inspired by the folk theorem and the eigenvalue condition functions like a dynamic incentive compatibility constraint for the entire economy. This paper constitutes the first progress on constructing rigorous solutions to a general class of mean field games with common noise over infinite time- which is surely the major open problem of the whole literature (Achdou et al. [2014]). This paper is formulated in discrete time, partly to reduce the technical burden on readers. Nevertheless, I am confident that techniques and concepts will carry over into continuous time, under some circumstances. An example, discussed in Section 6, shows that there may be differences in qualitative behavior between discrete and continuous time. It should be possible to analyze this time-scaling behavior, both to help economists decide which solutions to use and for its mathematical beauty. Both the a priori duality result and the ex post blow ups are connected to a small literature on phase transition of test statistics, which studies conditions where statistical estimators cease to converge. Examples include Cover [1965], Silvapulle [1981], Albert and Anderson [1984], Santner and Duffy [1986], Sur et al. [2019], Candès and Sur [2020] and McCracken [2020]. I am confident statistical theory will expand along many frontiers to accommodate the deep and multi-faceted predictions of stochastic equilibrium theory. In my epsilon management I will deploy the asymptotic notation x << y to mean "x is negligible relative to y".[19] Unless otherwise stated, all background results are available in Aliprantis and Border [2007] or Royden and Fitzpatrick [2010]. Jacobson [2009] and Jacobson [1980] might be helpful references for the abstract algebra with Eisenbud and Harris [2006] useful for schemes. The main text will emphasize intuition with more formality and some preliminaries in the Appendices. 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 [15] It is helpful to define a few terms from mathematical analysis for the benefit of economists. A quantitative estimate is where an unknown function or constant is bounded using only the model’s primitives. Mathematical rigidity is where a collection of objects can be described from a smaller set of their properties. Universality is where models with different formulations display common behavior or structures. They will appear repeatedly in this section. [16] They use the price gap formulation of the period profit function (Woodford [2009]), which avoids the singularities in the existing approximation, at the expense of simplifying dynamics. [17] Bilal [2023] develops perturbation techniques using the master equation but does not prove existence results. His focus on reflecting boundaries suits his main application to job ladders and unemployment but contrasts with the blow-up condition here. [18] Monotonicity conditions are popular in the mean field game literature but are restrictive because macroeconomic time series typically display non-monotone responses particularly when there are shocks to monetary policy. [19] This corresponds to x = o(y) in the notation of Tao [April 2, 2012], where further background can be found.