The Calvo Wage Phillips Curve and Labor Market Imperfections

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

G.2 Calvo Wage Phillips Curve

This subsection shows a wage Phillips curve problem and how results of Theorem 8 apply. First, I introduce imperfect competition into the labor market before showing that the associated resetting problem is analogous to that of the price Phillips curve (26)-(31).

G.2.1 Labor Unions

In the benchmark model of Section 4 the labor market is perfectly competitive. To generate wage rigidity, we need to introduce imperfect substitutability between wage setters. The easiest way to achieve this is to introduce unions, which combine different types of labor into a composite labor service, that is leased to firms at a wage rate W. The aggregation is as follows

This yields the labor demand system

and the wage index

G.2.2 Optimization Problem

The union sets wages to maximize the welfare of its members. It gets to reset wages with probability αw each period. The problem can be written as follows

where I have used the functional form for the disutility of labor. The first order condition is

with

Optimal reset wage can be viewed as a generalized weighted mean of flexible wages, according to Proposition 27 since

which are set as a mark up over the marginal rate of substitution between labor and leisure, which is decreasing in the substitutability between labor varieties. Aggregate wages evolve according to

The difference with the price level construction equation is that we are using real wages as opposed to nominal prices. The final and crucial step is to expand out the recursions (366) and (367)

Linearizing these equations at ZINSS, it is clear that a common root in the lag polynomials will arise with

This confirms we can apply Theorem 7 and conclude a bifurcation invalidates the linear approximations at ZINSS, without having to go through a detailed solution. Moreover, this analysis extends to larger models typically used in central banks, which feature both price and wage rigidity, like Christiano et al. [2005] and Smets and Wouters [2007].