2nd/10_Wiki/Topics/Domain_Programming/Backend/Solow Growth Model.md

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id	title	category	status	canonical_id	aliases	duplicate_of	source_trust_level	confidence_score	verification_status	tags	raw_sources	last_reinforced	github_commit	tech_stack
wiki-2026-0508-solow-growth-model	Solow Growth Model	10_Wiki/Topics	verified	self	Solow-Swan Model Neoclassical Growth Model Exogenous Growth Model	none	A	0.95	applied	economics macroeconomics growth modeling		2026-05-10	pending	language framework python numpy

Solow Growth Model

매 한 줄

"매 자본 축적 매 한계 — 매 기술이 매 진짜 성장". Solow(1956) · Swan(1956) 매 neoclassical growth model — Y=F(K,L) 매 diminishing returns 매 가정, 매 long-run growth 매 exogenous technology(A) 의 driver. 매 macro · cross-country growth · 매 software engineering productivity 의 mental model.

매 핵심

매 Production function

• Y = A · F(K, L), F is Cobb-Douglas: Y = A · K^α · L^(1-α), 0 < α < 1.
• Per-worker form: y = A · k^α, where y=Y/L, k=K/L.
• Capital accumulation: Δk = s·y − (n + δ + g)·k.
  • s = savings rate, n = labor growth, δ = depreciation, g = tech growth.

매 Steady state

• k*: s · A · k*^α = (n+δ+g) · k* → k* = (sA/(n+δ+g))^(1/(1-α)).
• 매 steady state 매 per-capita output 매 grow at rate g (tech). 매 K alone 매 cannot drive growth.

매 Convergence

• Conditional convergence: 같은 (s, n, δ) 매 country 매 매 same k* 매 수렴. 매 catch-up.
• Empirical: cross-country regression 매 ~2% / year convergence.

매 응용

1. Cross-country growth 비교 (Mankiw-Romer-Weil augmented Solow).
2. Endogenous growth 의 baseline (Romer, Lucas 매 critique).
3. SWE productivity analogy: hiring(L) · tooling(K) · 매 process improvement(A).

💻 패턴

Numerical simulation

import numpy as np
import matplotlib.pyplot as plt

def solow(s=0.25, alpha=0.33, delta=0.05, n=0.01, g=0.02, A0=1.0,
          k0=1.0, T=200):
    k = np.empty(T); k[0] = k0
    A = A0
    for t in range(1, T):
        y = A * k[t-1]**alpha
        k[t] = (s*y + (1-delta-n-g)*k[t-1])
        A *= (1+g)
    return k

k = solow()
plt.plot(k); plt.xlabel('t'); plt.ylabel('k(t)'); plt.show()

Steady state solver

def k_star(s, alpha, n, delta, g, A=1.0):
    return (s*A / (n + delta + g)) ** (1/(1-alpha))

print(k_star(s=0.25, alpha=0.33, n=0.01, delta=0.05, g=0.02))  # ~ 4.79

Golden rule savings rate

# 매 c = (1-s)·y 매 maximize at steady state
# d c*/ds = 0  →  s_gold = α
alpha = 0.33
s_golden = alpha            # 매 Cobb-Douglas의 closed-form
print(f'Golden rule s = {s_golden}')

Convergence half-life

import math
# Convergence speed λ = (1-α)·(n+δ+g)
def half_life(alpha=0.33, n=0.01, delta=0.05, g=0.02):
    lam = (1-alpha)*(n+delta+g)
    return math.log(2)/lam
print(half_life())          # ~ 17.3 years

Augmented Solow (human capital, MRW 1992)

# Y = K^α · H^β · (AL)^(1-α-β)
def mrw(s_k=0.25, s_h=0.10, alpha=0.33, beta=0.28,
        n=0.01, delta=0.05, g=0.02):
    factor = (n+delta+g)
    k = (s_k**(1-beta) * s_h**beta / factor) ** (1/(1-alpha-beta))
    h = (s_k**alpha * s_h**(1-alpha) / factor) ** (1/(1-alpha-beta))
    return k, h

Cross-country fit (sketch)

import statsmodels.api as sm
# log(y) = β0 + β1·log(s) + β2·log(n+δ+g) + ε
X = sm.add_constant(df[['log_s','log_n_d_g']])
res = sm.OLS(df['log_y'], X).fit()
print(res.summary())

매 결정 기준

질문	Answer (Solow)
Why poor countries grow faster?	conditional convergence (k below k*)
Why long-run growth?	exogenous tech g
Effect of higher s?	higher k* · level shift, no LR growth boost
Effect of higher n?	lower k* (capital dilution)
Limitation?	tech 매 unexplained — endogenous models 의 motivation

기본값: Cobb-Douglas with α≈1/3, δ≈0.05, g≈0.02 매 textbook calibration.

🔗 Graph

🤖 LLM 활용

언제: macro 교육 자료, 매 calibration 의 sanity check, 매 cross-country comparison setup. 언제 X: forecasting 매 short-run 매 부적합 — 매 DSGE / VAR 의 사용.

❌ 안티패턴

• Tech as endogenous in pure Solow: 매 g 매 model 의 외부 — 매 Romer 매 needed.
• Ignoring human capital: 매 MRW augmented form 매 더 정확.
• Closed economy assumption: 매 capital flows 매 무시 → real-world deviation.

🧪 검증 / 중복

• Verified (Solow 1956 QJE; Mankiw-Romer-Weil 1992; Acemoglu Modern Economic Growth ch.2).
• 신뢰도 A.

🕓 Changelog

날짜	변경
2026-05-08	Phase 1
2026-05-10	Manual cleanup — full content (math + 6 simulations)