09

Growth Dynamics — Expected Value and Limits

Parametric derivation of the theoretical 1.8% steady-state growth rate and the 30% single-year maximum

Learning Objectives

  • Derive the theoretical endogenous growth rate (~1.8%) from CDR regression coefficients
  • Explain the connection between 1.8% and Euler's constant: ≈ ¼e²
  • Interpret the parametric integral derivation of maximum endogenous growth (~30%) and its economic meaning (overheating threshold)
  • Read the CDR growth field as a vector field — interpret its curl (institutional "twisting" cost) and divergence (outward expansionary pull), and derive the geometric return on investment
1.8%
Long-run expected endogenous growth rate (= ¼e²)
30%
Theoretical maximum single-year growth (overheating threshold)
−12.9%
US contraction 1932 (Great Depression worst year)
25%
Libya 2017 — highest empirical growth found
9.6%
Geometric ROI = divergence / |curl| at C=D=R=1
Expected Endogenous Growth in g (Parametric Derivation)
Expected growth = (1/2)[β0 + (β̂C−β̂ê) + βD + βR + β̂CDR + βN]
= (1/2)(−0.00051 + (1.534346−1.295617) + 0.116963 + 0.275395 − 0.98133 + 0.388146)
= 0.018698 per unit ≈ 1.8%
95% CI: 1.8 ± 1.96 × 2.3% = {−2.7%, 6.3%}. The mature country convergence rate (USA, UK, Germany) falls within this interval. This value is a fixed constant of the model — it does not vary with ê (see simulator below for the value that does).

Growth Simulator: Maximum Endogenous Growth vs. Entrepreneurship Saturation (ê) real formula, live

—
Theoretical max single-year growth
1.8%
Long-run steady state (constant)
—
Overheating risk
Uses the real published formula: Max growth = β0 + (βC−βê)/2 + βD/2 + βR/2 + β̂CDR·ê/4 + βN/2. At ê=0.85 this evaluates to ≈30%, matching P1 §4.3. Sustained growth well above the 1.8% steady state risks economic overheating — excess demand over supply capacity.
Key Result
"This theoretical 1.8% is numerically equal to what economists have observed empirically as the steady-state rate to which countries converge as they develop. This derivation explains the previously observed but unexplained 1.8% and brings that mystery to an end." — P1, §4.1
The CDR Field as a Vector Quantity — Curl, Divergence & Geometric Return on Investment (P1 §4.3–4.4)
F(C,D,R) = FC i + FD j + FR k
∇×F = (∂FR/∂D − ∂FD/∂R)i − (∂FR/∂C − ∂FC/∂R)j + (∂FD/∂C − ∂FC/∂D)k
Evaluated at C = D = R = 1, ê = 0.85, with βD = 0.116963, βR = 0.275395, βCDR = −0.98133:
∇×F ≈ −0.16 i − 0.80 j + 0.96 k · ∇·F ≈ 0.18511
ROIgeometric = divergence / |curl| = 0.18511 / (|−0.16| + |−0.80| + |0.96|) = 0.18511 / 1.92 ≈ 9.6%
Treating F as the marginal-return vector of the CDR growth field, curl measures how the marginal return to one pillar twists against the others — the geometric expression of the decision-making friction discussed in §6 — while divergence measures the field's net outward, expansionary pull. Analogous to defining investment return as profit over investment, Ridley & Llaugel (2022) define a geometric return on investment as divergence over total curl magnitude: even at full institutional saturation (C=D=R=1), only about a tenth of the CDR field's raw expansionary force survives as realizable growth — the rest is absorbed by the coordination "twist" among Capitalism, Democracy, and Rule of Law. See P1 §4.3–4.4 for the full derivation.

Historical Comparisons

  • US WWII spurt: 17.7% growth in 1941 — approaching but not exceeding the theoretical 30% ceiling.
  • Great Depression: −12.9% contraction in 1932, the worst year on record.
  • Libya, 2017: 25% growth — highest empirical figure found in the dataset, still under the 30% theoretical maximum.
Sources
P1 · §4.1 Expected endogenous growth derivation (1.8%), §4.2 Parametric integral average, §4.3 Maximum growth (30%) & the CDR vector field (curl/divergence, geometric ROI), §4.4 Discussion (overheating, decision-making cost, geography extension to 90% R²), §5 Conclusions, pp. 6–11 P2 · §4 Parametric derivation optimal growth rate (3.7% for full CDR model), §5 Conclusions, pp. 11–13