Yonsei University
Chae-Yeon Xon
Yonsei University, School of Economics
Home/Resources/Geometry, Dynamics, and Operators for Economic Models

Geometry, Dynamics, and Operators for Economic Models

This book develops the mathematical tools needed to differentiate equilibrium systems, trace policy paths, analyze dynamic responses, work with inverse problems, and certify optimization claims. Each tool is tied to an economic question and accompanied by a diagnostic that separates a valid calculation from a stronger conclusion.

18 chapters90 focused topics

Local Geometry and Policy Paths

III-01

Differentiating Implicit Models

Many economic outcomes are defined by equilibrium equations rather than explicit formulas. The implicit-function calculation converts derivatives of those equations into derivatives of the equilibrium, provided the relevant Jacobian is square and nonsingular. Inverse derivatives and low-rank updates then make comparative statics reusable across states, policies, and information changes.

  1. Multivariable differentiation
  2. Implicit equilibrium maps
  3. Second-order responses
  4. Low-rank updates
  5. Regularity and branch failure
→
III-02

State Spaces, Fibers, Tangent Directions, and Metrics

An economic state may live on a constrained space such as a simplex, a moment manifold, or a family of distributions. Observable reports partition that space into fibers, and tangent directions describe locally feasible changes. A metric records the units or loss used to measure those changes, so spectra and minimum-distance statements must be interpreted with the metric attached.

  1. Smooth state spaces
  2. Restricted derivatives
  3. Distributions and frames
  4. The role of a metric
  5. Coordinates for distribution states
→
III-03

Constrained Policy Motion and Lifts

A desired path in reported aggregates generally leaves many compatible paths in the underlying state. A lift chooses one state velocity that implements the report velocity. The choice depends on feasibility, an implementation metric, and any drift already present in the state, so a lift is an economic specification rather than a neutral inverse.

  1. Report and state motion
  2. Implementation metrics
  3. Pseudoinverse lifts
  4. Drift and feasibility
  5. Minimal connection language
→
III-04

Curvature, Transport, and Holonomy

When two policy directions are lifted to the state space, applying them in different orders can produce different latent outcomes. Their commutator is the leading local measure of this order effect. Transport also changes target covectors, so a closed path in reported variables may alter both the hidden state and the response assigned to the next policy.

  1. Vector fields and order
  2. Curvature of a lift
  3. Target changes along paths
  4. Covector transport
  5. Levels of closure
→
III-05

Small-Loop Error Rates and Experimental Design

Finite policy loops contain curvature, higher-order terms, randomization error, and numerical error. Reversing or mirroring loops isolates components with different symmetry, while two loop scales help estimate the order of the remainder. These devices turn a geometric identity into an experimental design with an explicit resolution limit.

  1. Rectangle expansions
  2. Reversal and mirroring
  3. Two-scale calculations
  4. Designable policy directions
  5. Local power
→
III-06

Dynamic Quotients and Additive-State Realization

A static report may fail because future targets depend on states reached after transitions or policy lifts. Dynamic sufficiency therefore closes the target span under transport through the model. Finite-state systems permit an explicit closure algorithm, while an additive state representation requires further structure on how individual states combine.

  1. Accessible states
  2. Transported target spans
  3. Finite closure algorithms
  4. Additive frames
  5. The Gorman connection
→

Linear Dynamics and Modes

III-07

Linear Dynamics: Stability and Response Spaces

For x_{t+1}=Ax_t+Bu_t and y_t=Cx_t, the transition matrix governs persistence, while B and C govern which shocks enter and which consequences are seen. Reachability and observability organize the resulting response space. Stability alone says little about the span or value of feasible policy responses.

  1. Linear state equations
  2. Stability
  3. Impulse-response maps
  4. Reachability and observability
  5. Compression criteria
→
III-08

Exact Root Counts and Equilibrium Determinacy

Dynamic equilibrium arguments often reduce to the number of characteristic roots inside or outside the unit circle. Signed economic interactions can produce cancellations hidden by absolute-value bounds. Exact polynomial calculations and root-count certificates therefore matter whenever determinacy rests on a boundary count.

  1. Signed and absolute structure
  2. Fixed-point systems
  3. Characteristic polynomials
  4. Unit-circle root counts
  5. Economic determinacy
→
III-09

Poles, Cancellation, and Generalized Modes

Repeated characteristic roots can generate polynomial-times-exponential responses, and input-output maps can cancel internal modes. A determinant records algebraic roots; an observable transfer function records poles after cancellation. Generalized eigenvectors or Jordan chains are needed when a repeated observable pole remains.

  1. Transfer functions
  2. Cancellations and multiplicity
  3. Generalized modes
  4. Jordan and realization views
  5. Nearly repeated roots
→

Function Spaces, Inverse Problems, and Risk Certificates

III-10

Function Spaces and Legal Operator Algebra

An inverse formula is meaningful only after the domain, codomain, norm, and range are fixed. An operator can be injective while its inverse is unbounded or defined only on a strict range. Compositions that look algebraically valid may therefore be undefined on candidate functions used in an identification proof.

  1. Spaces and norms
  2. Operator domains
  3. Meaning of an inverse
  4. Boundedness and theorem tools
  5. Composition and graph cores
→
III-11

Integral, Multiplication, Adjoint, and Smoothing Operators

Conditional expectation and measurement systems naturally produce integral and multiplication operators. Their adjoints identify which queries can be represented, while smoothing attenuates high-frequency components and makes inversion unstable. The legal norm depends on the probability space and cannot be transferred casually between L1 and L2 arguments.

  1. Probability integral operators
  2. Multiplication operators
  3. Adjoints
  4. Fourier smoothing
  5. Inverse domains
→
III-12

Spectral Projections and Cross-Candidate Identification

Candidate latent models may each admit stable inversion on their own ranges while comparisons between candidates remain undefined. Spectral band projections create common finite-resolution objects, but uniform stability across expanding bands imposes stronger conditions. Those conditions can conflict with the smoothing that makes a measurement model empirically plausible.

  1. Band projections
  2. Uniform stability
  3. Costs of stronger conditions
  4. Shared candidate structure
  5. Probability reconstruction
→
III-13

Hilbert Target Operators and Distribution States

A finite-dimensional perturbation can affect an entire response curve. Mapping the perturbation into a Hilbert space of targets yields an integrated Gram operator whose spectrum ranks policy-relevant directions. When the state itself is a distribution, tangent restrictions and sieve approximations determine which perturbations the calculation represents.

  1. Finite-to-integrated targets
  2. Integrability
  3. Spectra of target operators
  4. Distribution tangents
  5. Sieve approximation
→
III-14

Conditional Generators, Poincare Inequalities, and Graph Certificates

Sufficiency can be tested through variation within an observation fiber. A conditional energy penalizes derivatives or differences along hidden directions. Under a spectral gap, small energy controls distance from a fiberwise constant function; on a finite graph the same logic becomes an exact Laplacian certificate.

  1. Conditional energy
  2. Generator assumptions
  3. Spectral gaps
  4. Residual bounds
  5. Finite-graph analogues
→

Sets, Optimization, and Sequential Information

III-15

Set Functions, Mobius Inversion, and Modularity

When outcomes depend on bundles of inputs or options, a set function records the value of every feasible subset. Mobius inversion isolates interaction terms and minimal enabling combinations. Modularity is a strong restriction: it makes reference-set increments portable across bundles, while monotonicity alone permits complements and substitutes.

  1. Boolean lattices
  2. Minimal enabling sets
  3. Mobius inversion
  4. Representability
  5. Modularity
→
III-16

KKT Conditions, Linear Duality, and Sharpness

Constrained economic calculations require feasibility, complementary slackness, and a treatment of corners. Linear-program duality turns a bound into a certificate and Farkas alternatives turn infeasibility into a witness. A sharp result additionally requires that the bound be attained by an admissible economic construction.

  1. Constrained optimization
  2. Corners
  3. Linear-program duality
  4. Infeasibility certificates
  5. Minimum enabling cost
→
III-17

Matrix Optimization and Information Allocation

Information design frequently chooses a positive-semidefinite precision or retention matrix under budget and technological restrictions. Schur complements translate quadratic risk bounds into block-matrix constraints. Matrix KKT conditions and dual gaps provide certificates, while fixed spectra restrict which coordinate-wise information allocations are attainable.

  1. Positive-semidefinite order
  2. Schur complements
  3. Semidefinite programs
  4. Directional optimization
  5. Isospectral information design
→
III-18

Sequential Gaussian Learning and Filtering

A Gaussian signal updates precision by a rank-one term and shifts the posterior mean by a gain times the forecast error. Repeated measurements learn only in persistently excited directions. When the latent state changes, the prediction step prevents the static accumulation of precision and produces a Riccati recursion.

  1. Gaussian conditioning
  2. Rank-one precision updates
  3. Cumulative excitation
  4. Changing latent states
  5. Numerical stability
→