Chapter 1 — The Problem of Capital
Part I · The Miniature Capital System · Week 1
K→𝒢→Q→Ξ
Chapter 1 builds the smallest model in which the book’s central phenomenon can occur: two capital systems that agree in every respect a scalar can read — state, cash flows, information, even the transformations each can currently execute — and that nevertheless differ in value. The difference is the fourth coordinate: the capacity to alter the menu itself.
Learning objectives
By the end of this chapter you should be able to:
- LOS 1.1 — State the four transactions that embarrass scalar measures of capital, and name for each what the scalar reads versus what actually moved.
- LOS 1.2 — Distinguish resource, claim, value, wealth, and capital as working definitions, and explain why the theory cannot treat them as interchangeable.
- LOS 1.3 — Compute the three-layer backward-induction decomposition of the miniature system: state value, path value, and architecture premium.
- LOS 1.4 — Explain, via the miniature, what a scalar reading of a capital system cannot see, and why System A and System B tie on every scalar yet differ in value.
- LOS 1.5 — Derive the central proposition informally: value separates into what the state is worth, what existing pathways are worth, and what the capacity to make pathways is worth.
The laboratory module
Module 1 — The Miniature Capital System. The module implements Example 1.7 with sliders for the operating payoffs, the listing cost, the securitization cost, the minting cost \(\kappa\), and the probability \(q\), and displays the decomposition of Proposition 1.9 as a waterfall: \(V_{\text{state}}\), the path increment, the architecture premium.
The book’s numbers, which the module and every companion artifact reproduce exactly:
| Quantity | Value | Meaning |
|---|---|---|
| \(V_{\text{state}}\) | 90 | the current state, operated as it stands |
| path increment | 10 | the right to traverse the existing listing pathway on good news |
| architecture premium | 5.5 | the value of the capacity to mint the securitization pathway |
| \(V_B\) | 105.5 | System B’s full value (System A = 100) |
| \(\kappa^{*}\) | 7.5 | minting cost at which the premium vanishes |
| joint premium | 6.5 | two pathways add: \(6.5 = 5.5 + 1\) |
Guided experiments
- Reproduce the waterfall — recover \(90 + 10 + 5.5 = 105.5\) from the default parameters.
- Raise \(\kappa\) until the premium dies — verify \(\kappa^{*} = 7.5\) against Proposition 1.9 and explain the number in one sentence.
- Both-states securitization — lower the securitization spread until the minted edge pays in both states; observe what happens to the state-contingency claim of §1.3.3.
- Premium without current-option value — find a parameterization in which listing is never exercised yet the premium survives, demonstrating that architecture value does not require current-option value.
- The variants bench — mint the second pathway of Example 1.10(iii), verify the premia add to 6.5, then switch on the shared-budget toggle and watch additivity fail (the seed of Exercise 1.13).
ch01 · 8/8 PASS every artifact reproduces the book numbers by construction.
Exercises
Grouped as throughout the book: A concept checks, B computations, C proofs and extensions, D counterexamples, E modeling, F laboratory, G research. Full solutions are in the Instructor’s Manual, Chapter 1.
A · Concept checks
- 1.1 For a leveraged buyout, a dividend recap, a first-time securitization, a tokenized offering, a covenant amendment, and an appraisal markup, identify which components of the capital system change and which scalar readings move. Hint: only one of the four coordinates need move to change value; find it in each case.
- 1.3 Explain in at most five sentences, without formulas, why Systems A and B tie on every reading yet differ in value — using the phrase “currently executable” at least once. Hint: the distinguishing right operates on the menu, and is recorded in no coordinate a scalar reads.
B · Computations
- 1.4 Recompute the miniature with \(q = 0.4\) and listing cost 12; give the full decomposition and reconcile the new premium with Proposition 1.9’s threshold logic. Hint: the premium is the mint’s value over the stripped system — not the listing option’s value. State the exercised set per state before computing.
- 1.5 Find (a) the minting cost \(\kappa^{*}\) at which the premium vanishes and (b) the securitization exercise cost \(c^{*}\) at which it vanishes with \(\kappa=2\). Hint: \(\kappa^{*}=7.5\) prices the capacity (\(\Xi\)); \(c^{*}\) prices the pathway itself (\(\mathcal{G}\)) — two different quantities, which is the point.
- 1.6 Price System B’s meta-capacity as a Geske compound option and verify the value is 5.5. Hint: the arithmetic is the easy part — the graded paragraph is the concession about what the option analogy does not establish (the unaware / non-enumerable case).
C · Proofs and extensions
- 1.8 Prove \(V_{\text{path}} \geq V_{\text{state}}\), with equality iff listing is exercised in no state; characterize the equality region.
- 1.10 Prove for \(n\) mintable pathways that if the pathways are exercise-independent, the architecture premium of the set equals the sum of stand-alone premia; identify exactly where each independence clause is used. Hint: this is, ten chapters early, the counting argument behind the Separation Theorem’s \(2^n\) meta-states.
Statements and hints are surfaced here; complete solutions live in the Instructor’s Manual.
Notes
This chapter is the toy every later chapter’s machinery must reproduce. Table 1.5 in the text tracks what each chapter does to it; the same miniature returns in Chapter 8 (priced under stochastic discounting), Chapter 9 (the Separation Theorem in general), and Chapter 10 (where the premium exceeds any enumeration).