Vol. I, Ch. 10seed 26110
DCT Laboratory — Volume I, Chapter 10
Enterprise Capital Architecture
Seed 26110 · Companion to the chapter and AXIOM Module AXIOM-10
Capital beyond finance, in motion: three stocks (Financial, Human, Technological)
under the accumulation law , two allocation
policies through a multiplicative productivity function, and the
accumulation/depletion asymmetry — destroyed in one quarter, rebuilt in ten.
Mirrored in DCT_V1_Ch10_Lab.xlsx.
import numpy as np
import matplotlib.pyplot as plt
plt.rcParams['figure.dpi']=110
import numpy as np
SEED = 26110
K0 = np.array([40.0, 30.0, 20.0]) # Financial, Human, Technological
DELTA = np.array([0.02, 0.06, 0.10])
ALPHA = np.array([0.30, 0.40, 0.30])
A_TFP, N = 10.0, 20
I_A = np.array([2.0, 2.0, 2.0]) # balanced allocation
I_B = np.array([4.0, 1.0, 1.0]) # finance-heavy allocation
def path(I, n=N, k0=K0, shock=None):
"""shock = (quarter, index, multiplier) applied AFTER accumulation that quarter."""
ks = np.empty((n+1, 3)); ks[0] = k0
for k in range(n):
ks[k+1] = (1-DELTA)*ks[k] + I
if shock and k+1 == shock[0]:
ks[k+1, shock[1]] *= shock[2]
return ks
def output(ks):
return A_TFP*np.prod(ks**ALPHA, axis=1)
def recovery_quarters():
base = path(I_A)
pre = base[10, 1] # Human capital before the shock
sh = path(I_A, shock=(10, 1, 0.7))
later = sh[11:, 1]
below = int((later < pre).sum()) # quarters spent below pre-shock level
return pre, below
def reference_values():
ka, kb = path(I_A), path(I_B)
ya, yb = output(ka), output(kb)
pre, rec = recovery_quarters()
return {
"F20_A": round(float(ka[20,0]),4), "H20_A": round(float(ka[20,1]),4),
"T20_A": round(float(ka[20,2]),4),
"Y20_A": round(float(ya[20]),4), "Y20_B": round(float(yb[20]),4),
"cumY_gap_A_minus_B": round(float(ya[1:].sum()-yb[1:].sum()),4),
"H10_preshock": round(float(pre),4),
"recovery_quarters": rec,
}
if __name__ == "__main__":
[print(f"{k:22s} {v}") for k,v in reference_values().items()]F20_A 59.9435 H20_A 32.3663 T20_A 20.0 Y20_A 337.0303 Y20_B 269.6156 cumY_gap_A_minus_B 800.4275 H10_preshock 31.5379 recovery_quarters 10
Panel 1 — Two allocations of the same budget
Six units of investment per quarter, split balanced (2,2,2) vs finance-heavy (4,1,1). Financial capital depreciates slowest — and the finance-heavy policy still loses: output is multiplicative in the stocks (Capital Productivity Theorem), so starving the fast-depreciating stocks costs more than stacking the durable one earns. Cumulative gap over five years: 800 output units.
ka, kb = path(I_A), path(I_B)
ya, yb = output(ka), output(kb)
t = np.arange(N+1)/4
fig, axes = plt.subplots(1, 2, figsize=(10,4.1))
for i,(nm,c) in enumerate(zip(["Financial","Human","Technological"],["#0B3D2E","#C8A24B","#1B6B52"])):
axes[0].plot(t, ka[:,i], c=c, lw=2.2, label=nm+" (balanced)")
axes[0].plot(t, kb[:,i], c=c, lw=1.6, ls="--")
axes[0].set(xlabel="years", ylabel="stock", title="Stocks: solid = balanced, dashed = finance-heavy")
axes[0].legend(frameon=False, fontsize=9); axes[0].grid(alpha=.25)
axes[1].plot(t, ya, c="#C8A24B", lw=2.4, label=f"balanced → Y₂₀ = {ya[20]:.1f}")
axes[1].plot(t, yb, c="#8A8F8B", lw=2.2, ls="--", label=f"finance-heavy → Y₂₀ = {yb[20]:.1f}")
axes[1].set(xlabel="years", ylabel="output Y", title="Productivity: Y = A·F^0.3 H^0.4 T^0.3")
axes[1].legend(frameon=False, fontsize=9); axes[1].grid(alpha=.25)
plt.tight_layout(); plt.show()
print(f"cumulative output gap (balanced − finance-heavy): {ya[1:].sum()-yb[1:].sum():.4f}")
cumulative output gap (balanced − finance-heavy): 800.4275
Panel 2 — The asymmetry: one quarter down, ten quarters back
A 30% human-capital destruction at quarter 10 (a botched reorganization, a talent exodus). Under unchanged investment, the stock needs 10 quarters to regain its pre-shock level — Depletion Reduces Adaptability, Asymmetrically (Prop.), measured rather than asserted.
base = path(I_A); sh = path(I_A, shock=(10,1,0.7))
pre, rec = recovery_quarters()
fig, ax = plt.subplots(figsize=(8.2,4.2))
ax.plot(t, base[:,1], c="#8A8F8B", lw=2, ls="--", label="no shock")
ax.plot(t, sh[:,1], c="#C8A24B", lw=2.4, label="30% destruction at q10")
ax.axhline(pre, c="#0B3D2E", lw=1, ls=":", label=f"pre-shock level {pre:.2f}")
ax.axvspan(2.5, 2.5+rec/4, color="#B0532F", alpha=.08)
ax.set(xlabel="years", ylabel="Human capital H", title=f"Recovery takes {rec} quarters (seed 26110)")
ax.legend(frameon=False); ax.grid(alpha=.25); plt.tight_layout(); plt.show()
print(f"pre-shock H at q10: {pre:.4f} quarters below pre-shock after destruction: {rec}")
pre-shock H at q10: 31.5379 quarters below pre-shock after destruction: 10
Panel 3 — Capital dependency, one edge
Human-capital investment is funded from financial capital: raise and falls by the same amount. The Capital Dependency Theorem in a one-line sweep — each capital's best level is not independent of the others'.
print(f"{'I_H':>5s} {'I_F':>5s} {'Y at q20':>10s}")
for ih in [1.0, 2.0, 3.0, 4.0]:
I = np.array([6.0-ih-1.0, ih, 1.0])
y = output(path(I))[20]
print(f"{ih:5.1f} {6.0-ih-1.0:5.1f} {y:10.4f}")I_H I_F Y at q20 1.0 4.0 269.6156 2.0 3.0 304.9242 3.0 2.0 320.9460 4.0 1.0 320.1353
Validation — agrees with DCT_V1_Ch10_Lab.xlsx
ref = reference_values()
expected = {"F20_A":59.9435,"H20_A":32.3663,"T20_A":20.0,"Y20_A":337.0303,"Y20_B":269.6156,
"cumY_gap_A_minus_B":800.4275,"H10_preshock":31.5379,"recovery_quarters":10}
for k,v in expected.items():
assert abs(ref[k]-v)<5e-4, f"MISMATCH {k}"
print(f"PASS {k:22s} {ref[k]}")
print("\nAll checkpoints agree — seed 26110.")PASS F20_A 59.9435 PASS H20_A 32.3663 PASS T20_A 20.0 PASS Y20_A 337.0303 PASS Y20_B 269.6156 PASS cumY_gap_A_minus_B 800.4275 PASS H10_preshock 31.5379 PASS recovery_quarters 10 All checkpoints agree — seed 26110.
Next: Exercises 10.9–10.12 (Part C) sweep δ, α, and the shock size; AXIOM-10's allocation bench makes the budget split draggable. Solutions: IM Ch. 10.