TWO-SECTOR WORKED EXAMPLE
Paper Section 5
A two-dimensional economic manifold with explicit metric, curvature, and geodesics. Demonstrates that cross-coordinate coupling produces constant negative curvature.
> Metric: ds² = (dr¹)² + eαr¹(dr²)²
> When sector 1 has positive returns (r¹ > 0), adjustments in sector 2 become more "costly"
> This asymmetric coupling creates inherent instability (negative curvature)
PARAMETERS
GEOMETRIC QUANTITIES
Gaussian Curvature K:
-0.1250
Scalar Curvature R:
-0.2500
Geometry Type:
Hyperbolic
GEODESIC DIVERGENCE
Geodesic Interpretation
Geodesics are paths of least resistance. In negative curvature, nearby geodesics diverge: small perturbations amplify. This is the geometric signature of systemic fragility.
> Each line is a geodesic from the origin in a different direction
> Negative curvature causes paths to spread apart (diverge) faster than in flat space
> Increase α to see stronger divergence
METRIC FIELD g22 = eαr¹
Metric Coefficient
The metric coefficient g22 determines the "cost" of moving in the r2 direction. It depends on r1, creating the coupling that produces curvature.
ECONOMIC INTERPRETATION
| Mathematical Object | Economic Meaning |
|---|---|
| r¹, r² | Log-returns of sector 1 and sector 2 |
| g22 = eαr¹ | Cost of adjusting sector 2 depends on sector 1's state |
| K = -α²/8 < 0 | Asymmetric coupling creates inherent instability |
| Geodesics | Friction-minimizing economic trajectories |
| Geodesic divergence | Small perturbations amplify (systemic fragility) |
NAVIGATION