Author:
(1) David Staines.
Table of Links
4 Calvo Framework and 4.1 Household’s Problem
4.3 Household Equilibrium Conditions
4.5 Nominal Equilibrium Conditions
4.6 Real Equilibrium Conditions and 4.7 Shocks
5.2 Persistence and Policy Puzzles
6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems
7 General Linearized Phillips Curve
8 Existence Results and 8.1 Main Results
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
10.4 Microeconomic Interpretation
Appendices
A Proof of Theorem 2 and A.1 Proof of Part (i)
B Proofs from Section 4 and B.1 Individual Product Demand (4.2)
B.2 Flexible Price Equilibrium and ZINSS (4.4)
B.4 Cost Minimization (4.6) and (10.4)
C Proofs from Section 5, and C.1 Puzzles, Policy and 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.4 Rouche’s Theorem Conditions
F Abstract Algebra and F.1 Homology Groups
F.4 Marginal Costs and Inflation
G Further Keynesian Models and G.1 Taylor Pricing
G.3 Unconventional Policy Settings
H Empirical Robustness and H.1 Parameter Selection
I Additional Evidence and I.1 Other Structural Parameters
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].
This paper is available on arxiv under CC 4.0 license.