Microeconomic Stability Through Staggered Optimization

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

10.4 Microeconomic Interpretation

The stability property of the model can be repurposed to give a new window on the traditional microeconomic theory of general equilibrium. One of the central claims of this paper is that when there is no trend inflation and small noise, the economy does not need systematic central bank intervention to maintain economic stability. This may be surprising to modern Keynesian macroeconomists but not to those specialising in general equilibrium theory, where there has been a great deal of work on equilibrium stability.

The model has to be re-imagined with idiosyncratic rather than aggregate risk (common noise). The aggregate economy will be in non-stochastic equilibrium thanks to any suitable law of large numbers. Therefore interest rates will be independent of individual shocks, which is isomorphic to having an inactive policy setting (ay = aπ = 0). In the small noise √ ε limit, the aggregate dynamics will be repeated at the idiosyncratic level so long as there is firm-specific labor.[99]

The result assumes market incompleteness; with complete markets firms would fully insure against idiosyncratic risk. The focus on small noise may seem unusual given that idiosyncratic shocks are typically large relative to aggregate shocks. Nevertheless, it seems plausible in the case of nominal rigidity where there a substantial body of evidence favouring flexible adjustment in response to large shocks, consistent with the predictions of state-dependent pricing models (see for example Boivin et al. [2009]). Finally, an intriguing alternative explanation is that producers facing large shocks insure on a near competitive market, selecting allocations that are √ ε away from full insurance.[100]

Thus, I have demonstrated that a natural benchmark free market equilibrium is stable under staggered optimization, in response to small noise. I feel this is a great improvement upon previous attempts to incorporate dynamics into general equilibrium microeconomics. These focused on tatonnement processes, based on unrealistically flexible out of equilibrium adjustment.[101] Staggered optimization should feature more prominently in microeconomic theory in future.

[99] For simplicity throughout the main text, I have assumed there is perfect competition in the labor market. This is probably unrealistic but alternative flexible wage arrangements, like bargaining solutions or additional mark-downs, could be accommodated into θ and would therefore not appear.

[100] It would be natural to think of √ ε as the approximate size of the excess. It is possible to have |ε| equilibria with ∆ adjusted to reflect demand heterogeniety. I will leave this to others. However, it would not impact first order dynamics, under the assumption here of firm-specific labor (recalling Footnote 9).

[101] The idea originated with Walras [2014]. The framework struggles to generate clear or intuitive empirical predictions (see Sonnenschein [1972], Sonnenschein [1973], Mantel [1974] and Debreu [1974]). For further perspective consult Weintraub [1993].