Yonsei University
Chae-Yeon Xon
Yonsei University, School of Economics
Home/Resources/Geometry, Dynamics, and Operators for Economic Models/III-08
Linear Dynamics and Modes · Chapter 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.

Conceptual map

  1. III-08.01Signed and absolute structure
  2. III-08.02Fixed-point systems
  3. III-08.03Characteristic polynomials
  4. III-08.04Unit-circle root counts
  5. III-08.05Economic determinacy

1. Determinacy is a boundary root count

A dynamic equilibrium condition typically fixes how many roots must lie on each side of the unit circle. The certificate must preserve signs, count multiplicity, and exclude roots on the boundary.

Let the linearized system imply a real monic characteristic polynomial \(p(z)=\operatorname{det}(zI-A)\). Define \(N_{in}\), \(N_{out}\), and \(N_{unit}\) as its root counts inside, outside, and on \(\lvert z\rvert=1\), each counted with algebraic multiplicity. An economic determinacy statement adds a model-specific mapping between these counts and the number of predetermined or forward variables.

Definition 1 · Certified root count

A root count is certified when exact coefficient signs or interval enclosures establish the count and establish \(N_{unit}=0\). A floating list of approximated roots is a diagnostic because rounding can move a near-boundary root across the circle.

2. A complete quadratic Schur test

Proposition 1 · Jury conditions for a real quadratic

Both roots of \(p(z)=z^{2}+a_{1}z+a_{2}\) lie strictly inside the unit circle exactly when

\[1+a_{1}+a_{2}>0, \quad 1-a_{1}+a_{2}>0, \quad 1-a_{2}>0.\](1)

Derivation. Apply the bilinear map \(z=(1+s)/(1-s)\), which sends the open left half-plane to the open unit disk. After multiplying by \((1-s)^{2}\), the transformed polynomial has coefficients 1+\(a_{1}+a_{2}\), 2(\(1-a_{2}\)), and \(1-a_{1}+a_{2}\). The second-order Hurwitz criterion requires these coefficients to have the same positive sign, which gives (1). ∎

3. Signed cancellation defeats an absolute bound

Consider the rational matrix

\[A=\begin{bmatrix}\frac{3}{5} & \frac{3}{5} \\ \frac{-3}{5} & \frac{3}{5}\end{bmatrix}.\](2)

Its infinity norm is 6/5, so the sufficient contraction test \(\lVert A\rVert_{\infty}<1\) fails. The same is true for the row-sum test applied to \(\lvert A\rvert\). The signed characteristic polynomial is

\[p(z)=z^{2}-(\frac{6}{5})z+\frac{18}{25}.\](3)

The three values in (1) are 13/25, 73/25, and 7/25. All are positive, so both roots are inside the unit circle. Direct calculation gives \(3/5\pm(3/5)i\), each with modulus \(\sqrt{18}/5\approx0.84853\). The exact certificate records the stabilizing effect of the negative cross-interaction that the absolute matrix discards.

4. Higher-degree workflow and boundary cases

For a higher-degree rational polynomial, construct a Schur–Cohn table, a Jury table, or map the unit circle to the imaginary axis and use a Routh–Hurwitz count. Exact rational arithmetic is available when the linearized coefficients are rational. With estimated coefficients, use coefficient intervals or verified complex-root enclosures and report every boundary margin.

Failure case · An unreported unit root

If \(p(1)=0\) or \(p(-1)=0\), strict root counts fail at the boundary. A repeated root also changes the number of independent modal solutions. A determinacy claim must specify how the economic boundary conditions treat those cases.

An absolute small-gain bound remains useful: when it passes, it gives a convenient sufficient certificate. Its failure leaves the exact signed system unresolved until a root-count calculation is performed.

5. Implementation, exercises, and sources

Generate the characteristic polynomial from the signed equations before numerical rounding. Evaluate exact boundary tests first, compute high-precision roots as a cross-check, and compare the product and sum of the computed roots with the polynomial coefficients. Record the distance of every root from the unit circle.

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Exercises

  1. Verify (3) directly from \(\operatorname{det}(zI-A)\) and reproduce all three Jury margins.
  2. Replace 3/5 by a scalar \(c\). Find the interval in which the exact system is stable while the infinity-norm contraction test fails.
  3. Construct a cubic with one root outside and two inside the unit circle and certify the count after the bilinear transformation.
Partial solutions

1. The determinant is \((z-3/5)^{2}+9/25\), which expands to (3). Substitution into (1) gives 13/25, 73/25, and 7/25. 2. With 3/5 replaced by \(c>0\), the roots are \(c(1\pm i)\) and have modulus \(\sqrt{2}c\). Exact stability requires \(c<1/\sqrt{2}\), while the infinity-norm test requires \(2c<1\). The gap interval is \(1/2\le c<1/\sqrt{2}\).

  1. Eliahu I. Jury, Theory and Application of the z-Transform Method.Discrete-time stability tables.
  2. Felix R. Gantmacher, The Theory of Matrices, Volume II.Root-location criteria and matrix pencils.
  3. Olivier Blanchard and Charles Kahn (1980), “The Solution of Linear Difference Models under Rational Expectations,” Econometrica 48, 1305–1311.Economic mapping from roots to equilibrium restrictions.

6. Audit checkpoint

Preserve signed coefficients, state the unit-circle convention, count multiplicities, test boundary roots, report exact or interval margins, and show how the economic model maps the certified count into a determinacy conclusion.

7. Scope boundary

The chapter supplies finite-polynomial root certificates used in linearized economic systems. General complex analysis, nonlinear global dynamics, and a full solution theory for rational-expectations models lie beyond this treatment.

Prerequisites