Skip to content
Dynamic Corporate Transformation

Vol. I, Ch. 10 · Part 3. Enterprise Architectures · Week 5

Enterprise Capital Architecture

Learning outcomes

After completing this chapter, the reader should be able to:

  1. Define enterprise capital mathematically: durable, accumulable, depreciable, productive stocks carried on the state architecture.
  2. Distinguish among the forms of enterprise capital by their formal characteristics—depreciation, build time, tradability, separability—not by label alone.
  3. Construct enterprise capital vectors as declared measurements on the ESA, with carriers, units, and indicators.
  4. Analyze capital interactions: conversion channels with efficiencies, and complementarities with their bottleneck and supermodularity consequences.
  5. Evaluate capital dependencies through the capital dependency graph, inheriting the propagation discipline of Chapter 9.
  6. Measure the capital architecture: stocks, flows, productivity, and the conservation audit that detects undeclared channels.
  7. Explain capital accumulation exactly: geometric dynamics, the sustaining-investment identity, and time-to-build floors.
  8. Explain capital depletion and its asymmetry with accumulation—fast to burn, slow to rebuild—and its effect on the executable operator set.
  9. Interpret enterprise capital dynamics as Part II dynamics on a distinguished sub-state, with allocation as the control.
  10. Prepare capital representations for performance attribution (Chapter 11), risk (Chapter 12), and optimization (Chapter 15 and Volume II).

Reading guide

Work through the chapter in section order; the full development, proofs, and worked examples are in the book — this page indexes them and does not replace them.

  1. Motivation for Enterprise Capital Architecture

    Motivation for Enterprise Capital Architecture

  2. Capital Beyond Finance

    Capital Beyond Finance

  3. The Enterprise Capital Architecture

    The Enterprise Capital Architecture

  4. The Forms of Enterprise Capital

    The Forms of Enterprise Capital

  5. Capital Dependencies

    Capital Dependencies

  6. Capital Accumulation and Depletion

    Capital Accumulation and Depletion

  7. Capital Allocation and Productivity

    Capital Allocation and Productivity

  8. Architecture Consistency and Capital Dynamics

    Architecture Consistency and Capital Dynamics

  9. Worked Examples

    Worked Examples

  10. Preparation for Enterprise Performance Architecture

    Preparation for Enterprise Performance Architecture

  11. Chapter Summary

    Chapter Summary

  12. Exercises

    Exercises

  13. Notes and Sources

    Notes and Sources

On the map

AXIOM

This chapter is instrumented by:

Launch the module, load the chapter model, modify inputs, run the optimization, and compare against the worked examples in the book.

Exercises

19 exercises, grouped A concept checks · B mathematical · C computational · D enterprise applications. Starred (★) exercises are on the advanced track. Full solutions appear in the Instructor's Manual, Chapter 10.

A. Concept checks

  1. 10.1
    State the four stock-semantics properties of Definition (see book) and test each against three candidate "capitals": a brand, a quarterly revenue figure, and an exclusive license; classify each as capital, flow, or neither, with one sentence per test.
  2. 10.2
    Using Table (see book), explain why organizational capital's profile (untradable, non-separable, low δ\delta, long build) makes it simultaneously the most defensible and the most dangerous form to deplete.
  3. 10.3
    Give one enterprise situation each for the bottleneck regime and the complementarity regime of Theorem (see book), and name the observable that would tell you which regime you are in.
  4. 10.4
    Explain "capital is a rental" to a budget committee in four sentences, including the identity I=δC∗I = \delta C^{\ast} and one concrete consequence of a one-year rent holiday.
  5. 10.5
    What is the plug of Theorem (see book)(iii), why is a persistent plug informative rather than embarrassing, and what two failures does a nonzero plug disambiguate between?

B. Mathematical exercises

  1. 10.6
    Prove Theorem (see book)(i) by direct induction, then re-derive it as a corollary of Theorem 7.
  2. 10.7
    Derive the time-to-build floor in full: from the gap recursion Cˉ−Ck+1≥(1−δ)(Cˉ−Ck)\bar C - C_{k+1} \ge (1-\delta)(\bar C - C_k), prove no admissible policy exceeds the full-throttle path at any date, and compute k0.1k_{0.1} for δ∈{0.05,0.15,0.30}\delta \in \{0.05, 0.15, 0.30\}.
  3. 10.8
    Extend Theorem (see book)(ii) to chains: for a conversion path 1→2→⋯→m1 \to 2 \to \cdots \to m executed sequentially, show the end-to-end efficiency is ∏ηi,i+1\prod \eta_{i,i+1} and the total leak of converting amount TT is T(1−∏η)T(1 - \prod \eta); relate the product form to Theorem 6.
  4. 10.9 ★
    State and prove the equalization conditions of Theorem (see book)(i) as the KKT conditions of the concave program max⁡F(C+a)\max F(\Cvec + \mathbf{a}) s.
  5. 10.10
    For F=min⁡(C1/1, C2/2)F = \min(C_1/1,\ C_2/2) with C0=(10,30)\Cvec_0 = (10, 30) and budget B=20B = 20: compute the optimal allocation, the capacity path as a function of budget spent, and the kink point where the bottleneck rotates; sketch capacity versus budget.
  6. 10.11 ★
    For F(C1,C2)=C1αC2βF(C_1, C_2) = C_1^{\alpha} C_2^{\beta} with α,β>0\alpha, \beta > 0, α+β<1\alpha + \beta < 1: verify strict complementarity (compute the cross-partial), solve the equalization conditions explicitly for the optimal split of budget BB, and show the funded ratio a1/a2a_1/a_2 depends on the existing stocks—one-sided histories tilt optimal funding toward the neglected complement.
  7. 10.12
    Formalize the asymmetry of Proposition (see book): for a stock at floor cc suffering a write-off of size ww, show the mandatory repurchase period is k(w)≥ln⁡((Cˉ−c+w)/(Cˉ−c))/ln⁡(1/(1−δ))k(w) \ge \ln\bigl((\bar C - c + w)/(\bar C - c)\bigr) / \ln(1/(1-\delta)) and expand for small ww to obtain the marginal repurchase time w/(δ(Cˉ−c))w / (\delta(\bar C - c)) periods per unit written off.
  8. 10.13
    Prove the localization claim of Theorem (see book)(iii): if exactly one channel is undeclared, show πi,k\pi_{i,k} equals that channel's flow (with sign determined by its direction), and explain how the correlation of π\pi with candidate drivers identifies the channel.

C. Computational exercises

  1. 10.14
    (With AXIOM-10 or the Chapter 10 notebook.
  2. 10.15
    (With AXIOM-10 or the Chapter 10 notebook.
  3. 10.16
    (With AXIOM-10 or the Chapter 10 notebook.

D. Enterprise applications

  1. 10.17
    Construct the capital vector for an enterprise you know: seven or more forms with carriers, units, two indicators each (Table (see book) style), estimated δ\delta with one sentence of evidence, and error bars—then flag the form whose measurement you trust least and why it matters most.
  2. 10.18
    Write the sustaining-investment memo: for three stocks from Exercise (see book), compute the rent δC∗\delta C^{\ast}, compare against actual maintenance flows, and state which stocks are being silently liquidated—one page, ending with the single reallocation you would propose.
  3. 10.19
    Audit one past or planned transformation as capital conversion: declare its channels (II, OO, TT with η\eta estimates), compute the conversion bill, run the plug logic on the slow stocks (COC_O, CRC_R), and price the repurchase liability of Proposition (see book)—the pro-forma line the original business case omitted.

Downloads

All three companions consume the same seeded engine (26110), so their numbers agree by construction — the MFMF convention, carried forward.