The Impact of Calvo Pricing and the Taylor Rule on Economic Modeling and Household Behavior

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

4 Calvo Framework

This section lays out the principal model used throughout the paper featuring Calvo pricing and a Taylor rule. The assumptions concerning error terms present throughout the paper are discussed. The section winds up with a crucial finite dimensional characterization of the system.

4.1 Household’s Problem

This part lays out the households optimization problem that underpins the demand side of the economy. There is a single representative household[20] that chooses consumption C and labor supply L, so as to maximize the following objective function

subject to the budget constraint

There is a unit continuum of firms. Π(i) is profit from an individual firm i given by

Note in a stochastic environment firms need not make the same profits even with a symmetric equilibrium. Since when price rigidity is introduced, firms with the same demand curve will charge different prices, depending on when they last re-optimized and can therefore make different levels of profit.[21] The budget constraint states that the uses for nominal income (consumption and saving) must be equal to the sources of income (wealth, labor and dividend income).

Finally, there are two constraints

The first is a bond market clearing condition. It is simplification designed to allow me to ignore the effect of government spending or debt management. It is not important at this stage because the analysis focuses on limiting cases where, without nominal frictions, Ricardian equivalence ensures that debt levels do not matter (Barro [1974]). The second is a transversality condition. It is necessary for a well-defined optimization problem. It forces the household to honor their debts. Otherwise they would seek to borrow an unbounded amount and never repay. Giglio et al. [2016] provides model-independent support.

[20] You may be surprised with a model where the world is organized into one happy family. It simplifies the dynamic exposition. Werning [2016] provides useful aggregation results, however none survive a full stochastic setting. The bifurcation analysis, the backbone of the Phillips curve, goes through without reference to a specific Euler.

[21] In fact non-resetters need not have positive profit expectations or even a non-negative stock price, although, I will show this does not apply in our quantitative setting. Alternatively, one could swerve around this problem by imagining there are a continuum of households who each own one firm and an efficient long-term insurance market with contracts set arbitrarily far back in the past (so that there is no conditioning on pricing history). In this case the market will absorb all idiosyncratic risk associated with non-optimal price setting behavior.