Chapter 13 — Crisis: Coupled Exposure and Transformation Networks
Part IV · The Fire Bench · Week 12
K→𝒢→Q→Ξ
Two networks, one fire. When shortfalls force sales, sales load routes and mark neighbors, and the coupled system clears at a fixed point — sometimes orderly, sometimes a spiral.
Learning objectives
- LOS 13.2 — State the coupled model of exposure and transformation networks.
- LOS 13.3 — Prove the Coupled Clearing Theorem and compute its least and greatest fixed points.
- LOS 13.7 — Compute equilibrium-disjointness tolerated-failure counts.
The laboratory module
Module 13 — The Fire Bench. A coupled sandbox, a haircut anatomizer, a redundancy auditor, and a concentration census.
Three firms, two fates (Example 13.7). Firms \(A,B,C\) each owe 9, hold cash 2 and one asset; the sale route’s loaded price is \(P(s)=\max(12-5s,1)\) with \(s\) forced sellers; a firm is forced when \(2+P(s)-9<0\). Firm \(A\) suffers a cash-destroying shock.
- Iterating from \(\{A\}\): \(s=1\), \(P=7\), neighbors mark \(2+7-9=0\) — not negative. \(\;S^- = \{A\}\): the orderly equilibrium.
- Iterating from everybody: \(s=3\), \(P=1\), equity \(2+1-9=-6<0\) for all. \(\;S^+ = \{A,B,C\}\): the spiral.
The gap is \(\{B,C\}\) — two firms whose fate is an equilibrium selection, not a fundamental. The crisis route book (Appendix B.5) has a tolerated-failure count of 1: routes disjoint in counterparties but not in equilibria.
Guided experiments
- Reproduce both fixed points of Example 13.7, then find the smallest joint cash injection that empties the gap.
- Give a firm two routes sharing an arranger and show the auditor refusing them disjointness credit.
- Enter the route book into the auditor, reproduce the tolerated-failure count of one, and find the cheapest addition that raises it to two.
ch13 · 8/8 PASS