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Detecting Instability in Economic Models Using Algebraic Geometry
by Keynesian TechnologyDecember 8th, 2024
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This section applies algebraic geometry concepts, specifically schemes, rings, and the Zariski topology, to analyze bifurcations in Dynamic Stochastic General Equilibrium (DSGE) models. The first part lays out the foundational mathematics, introducing affine varieties, presheaves, and sheaves. The second part demonstrates how these tools can be used to detect bifurcations by identifying changes in the structure of solutions to systems of equations. The Grobman-Hartman theorem is used to develop a fail-safe test for bifurcations in DSGE models, allowing for the detection of breakdowns in standard approximations.
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
9.4 Algebraic Aspects (III) Schemes
This subsection is split in two. The first part is mathematical, where I introduce the central technical concepts of schemes, rings, ideals and the Zariski topology. The second part applies these techniques to prove necessary and sufficient conditions for bifurcation between the underlying non-linear DSGE and the dynamics of the linear approximation about an equilibrium point.
9.4.1 Preliminaries Algebraic Geometry
To appreciate the final argument it is necessary to develop some competence with the methods of algebraic geometry. The underlying principle is that one can associate a system of simultaneous equations with their roots.[74] This is formalized by a local ring system in the case of a linear system and algebraic variety for polynomials.[75]
The particular concept we need is an affine variety. A preliminary definition is helpful.
Intuitively, an affine space is the minimal structure required to determine changes in the multiplicity of solutions for a system of simultaneous linear equations. It starts from the familiar notion of a vector space and then dispenses with angles and distances; as such there is no need for an origin. Points represent translations or displacements rather than positions. One can think of the affine subspace as resulting from translating the linear subspace (away from the origin) by addition of the translation vector. In finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace and therefore must contain the origin of the vector space.[76]
It is now possible to lay out the crucial sequence of topological objects.
The idea is that each of the surfaces (1)-(5) represent prime ideals corresponding to the distinct components of the solution.[79] The Zariski topology will come into its own when we start associating systems of equations with their lag polynomial. The next part is devoted to describing the machinery of these local constructs. Appendix F.2 offers help with category theory.
Definition 13. Let X be a topological space. A presheaf of sets F on X consists of the following data:
• For each open set U of X a set F(U). This set is sometimes also denoted Γ(U, F). The elements in this set are called the sections of F over U.
• For each inclusion of open sets V ⊆ U, a function resV, U : F(U) → F(V ) called a restriction morphism (maps from one sheaf to another). The restriction morphisms are required to satisfy two additional categorical properties:
Definition 14. A sheaf is a presheaf with two additional requirements
Note that there is an order relation induced by reverse inclusion U ≤ V iff U ⊆ V . [82] The intuition is that sheaves permit us to rigorously speak about sequences of approximations with stalks, allowing us to specify a point around which to consider these limits. I will always choose ZINSS.
Before I finish by looking at schemes, it is necessary to introduce two relatively straightforward terms. The Spectrum R of a ring is the collection of all prime ideals. You can think of it as the set of all surfaces forming solutions of polynomial equations. A Ringed Space (X, OX ) is a topological space X together with a sheaf of rings OX on X. The sheaf OX is called the Structure Sheaf of X. A Locally Ringed Space is a ringed space such that all stalks are all local rings.[83]
The theory develops by parametizing systems of solutions. As before, the foundation are sets of linear equations.
One can think of a scheme as being covered by "coordinate charts", which are affine schemes. The definition means that schemes are obtained by gluing together affine schemes, using the Zariski topology. In particular, X comes with a sheaf OX , which assigns to every open subset U a commutative ring OX (U), called the ring of regular functions on U. Sheaves will be used to map between the parameter set γ (more properly an appropriate subset) and the family of perturbation expansions of the DSGE model and associated statistical objects like goodness of fit metrics.
Definition 17. The localization at x of an affine scheme is the affine local ring attached to that point by the sheaf F
This can be thought of as an approximation taken from the point x, whereas the stalk would be the sequence of approximations in the (deleted) neighborhood of x. It is now possible to provide a scheme theoretic formalization of the bifurcation seen in the Calvo model. First a generic bifurcation is represented by the breakdown of the following commuting diagram.[84]
The idea is that movements through the system of rings reflect movements through the underlying space in a way that preserves the topological structure in both spaces, i.e. the fundamental group G in the base space and the Zariski in the rings. Where this mapping breaks down, a bifurcation arises. The focus will be on spaces where ι is well-behaved, reflecting smoothness of the model primitives and the bifurcation arises because the ring becomes reducible. This is where the Krull dimension of the solution set (prime ideal at x) declines because of some cancellation in the equation system.
9.4.2 Detecting Bifurcations
It is common for macroeconomists, at central banks in particular, to build large models to study the interplay of multiple frictions and sectors. Conducting a full bifurcation analysis would likely be infeasible or technically challenging. Fortunately, I am able to use the machinery of the previous subsection to construct a readily implementable fail-safe test for fixed points about which standard approximations will fail.
Proof. The "if" part is a consequence of the De Rham cohomology and our stochastic Grobman-Hartman theorem. De Rham’s theorem ensures continuous differentiability away from singularities (where the dimension changes) whilst Theorem 3 rules out non-hyperbolic dynamics, allowing us to apply GrobmanHartman. The "only if" part follows from the definition of the Zariski topology, as the dimension of a linear system and the fact that homeomorphisms have a common dimension.
Remark 26. The result is a Grobman-Hartman style theorem for DSGEs without kinks. Borrowing constraints, an effective lower bound on nominal interest rates, or tax thresholds are economic phenomena that would generate kinks that would make G non-differentiable and prevent us from applying the theorem. Nevertheless, these instances could be analyzed piece-wise. Thus the approach is general.
Informally, you can check for bifurcations by ensuring there are no common roots in the lag polynomial of the first order conditions. This is because assuming a solution exists you can only have bifurcations where unrepresentative cross-equation cancellations arise. This should become a standard feature of the output of DSGE solution packages.
This paper is available on arxiv under CC 4.0 license.
[74] This is an extension of the fundamental theorem of algebra, which states that an nth order polynomial is determined by the n complex roots it factorizes into (allowing multiplicity).
[75] Recall that a ring R is a set equipped with two binary operators + and × satisfying the following three ring axioms
1. R is an Abelian group under addition meaning
- (a + b) + c = a + (b + c) for all a, b, c in R (that is, + is associative).
- a + b = b + a for all a, b in R (that is, + is commutative).
- There is a zero element in R, such that a + 0 = a for all a in R (that is, 0 is the additive identity).
- For each a in R there exists −a in R, such that a + (−a) = 0 (that is, −a is the additive inverse of a).
2. R is a monoid under multiplication
• (a × b) ◦ c = a × (b × c) for all a, b, c in R (that is, ◦ is associative).
• There is a unit element in R, such that a × 1 = a and 1 × a = a for all a in R (that is, 1 is the multiplicative identity).
3. Multiplication is distributive with respect to addition implying
• a × (b + c) = (a × b) + (a × c) for all a, b, c in R (left distributivity)
• (b + c) × a = (b × a) + (c × a) for all a, b, c in R (right distributivity).
The family of square matrices with the usual operations of addition and matrix multiplication is a ring, as is the set of all continuous functions on the real line.
[79] An ideal would be any subset of these.
[80] It is not possible to extend a holomorphic to the entire complex plane, since Liouville’s theorem states that the only bounded constant function can be entire (holomorphic throughout the complex plane, as explained in Stein and Shakarchi [2010]). It is not possible to glue two distinct constant functions together to make another constant function.
[81] "Direct limit" is a term from category theory. Appendix Section E.2 explains how the direct limit coincides with the limit from common analysis when we consider classes of approximations.
[82] An element of a stalk is an equivalence class of elements xU ∈ F(U), such that where two such sections xU and xV are considered equivalent if the restrictions of the two sections coincide on some neighborhood of x.
[83] A local ring is defined by a unique maximal ideal I. A maximal ideal contains all proper ideals and is smaller than the ring itself. All our ideals are maximal by the weak Nullstellensatz theorem, which informally carries over the idea from the Fundamental theorem of algebra, that polynomials can be uniquely determined by their roots to systems of equations.
[84] It can be viewed as a functor between categories of pointed topological spaces local to a certain point, for example ZINSS in Calvo.
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