A Beginner's Guide to Homology

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

F Abstract Algebra

This section supplies an introductory primer on homological algebra, De Rham cohomology and category theory, complimentary to the informal discussion in the text. Each occupies a subsection. At the end, a short part is devoted to additional details of a calculation from Proposition 20, which applies the methods here to the ultimate goal of the paper.

F.1 Homology Groups

This part develops the theory of singular homology to support the claims and analysis in the main text. Other popular homology constructions are discussed in Jacobson [1980]. Informally, the singular homology is built by a succession of mappings between n dimensional blocks and the paths around their boundaries.

F.1.1 Singular Simplex

its vertices are the k standard unit vectors and the origin. [112] Familiar examples include:

F.1.2 Singular Chain Complex

F.1.3 Homotopy Invariance

for all n ≥ 0. This means that homology groups are topologically invariants. In particular, if X is a connected contractible space, then all its homology groups are 0 except

I finish with a sketch of the proof of homotopy invariance of singular homology. A continuous map f : X → Y induces a homomorphism

It is easy to see that

where f♯ is a chain map which descends to homomorphisms on homology

It is left to show that, if f and g are homotopically equivalent, then f∗ = g∗. This implies that if f is a homotopy equivalence then f∗ is an isomorphism. Let F : X × [0, 1] → Y be a homotopy, that takes f to g on the level of chains. It defines a homomorphism

Therefore, they are homologous, which settles the claim.