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146 lines
4.7 KiB
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146 lines
4.7 KiB
Markdown
---
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id: wiki-2026-0508-solow-growth-model
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title: Solow Growth Model
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category: 10_Wiki/Topics
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status: verified
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canonical_id: self
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aliases: [Solow-Swan Model, Neoclassical Growth Model, Exogenous Growth Model]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.95
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verification_status: applied
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tags: [economics, macroeconomics, growth, modeling]
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raw_sources: []
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last_reinforced: 2026-05-10
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github_commit: pending
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tech_stack:
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language: python
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framework: numpy
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---
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# Solow Growth Model
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## 매 한 줄
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> **"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.
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## 매 핵심
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### 매 Production function
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- **Y = A · F(K, L)**, F is Cobb-Douglas: `Y = A · K^α · L^(1-α)`, 0 < α < 1.
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- **Per-worker form**: `y = A · k^α`, where `y=Y/L`, `k=K/L`.
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- **Capital accumulation**: `Δk = s·y − (n + δ + g)·k`.
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- `s` = savings rate, `n` = labor growth, `δ` = depreciation, `g` = tech growth.
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### 매 Steady state
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- **k\***: `s · A · k*^α = (n+δ+g) · k*` → `k* = (sA/(n+δ+g))^(1/(1-α))`.
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- 매 steady state 매 per-capita output 매 grow at rate `g` (tech). 매 K alone 매 cannot drive growth.
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### 매 Convergence
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- **Conditional convergence**: 같은 (s, n, δ) 매 country 매 매 same k* 매 수렴. 매 catch-up.
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- **Empirical**: cross-country regression 매 ~2% / year convergence.
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### 매 응용
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1. Cross-country growth 비교 (Mankiw-Romer-Weil augmented Solow).
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2. Endogenous growth 의 baseline (Romer, Lucas 매 critique).
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3. SWE productivity analogy: hiring(L) · tooling(K) · 매 process improvement(A).
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## 💻 패턴
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### Numerical simulation
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```python
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import numpy as np
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import matplotlib.pyplot as plt
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def solow(s=0.25, alpha=0.33, delta=0.05, n=0.01, g=0.02, A0=1.0,
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k0=1.0, T=200):
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k = np.empty(T); k[0] = k0
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A = A0
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for t in range(1, T):
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y = A * k[t-1]**alpha
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k[t] = (s*y + (1-delta-n-g)*k[t-1])
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A *= (1+g)
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return k
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k = solow()
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plt.plot(k); plt.xlabel('t'); plt.ylabel('k(t)'); plt.show()
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```
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### Steady state solver
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```python
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def k_star(s, alpha, n, delta, g, A=1.0):
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return (s*A / (n + delta + g)) ** (1/(1-alpha))
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print(k_star(s=0.25, alpha=0.33, n=0.01, delta=0.05, g=0.02)) # ~ 4.79
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```
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### Golden rule savings rate
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```python
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# 매 c = (1-s)·y 매 maximize at steady state
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# d c*/ds = 0 → s_gold = α
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alpha = 0.33
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s_golden = alpha # 매 Cobb-Douglas의 closed-form
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print(f'Golden rule s = {s_golden}')
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```
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### Convergence half-life
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```python
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import math
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# Convergence speed λ = (1-α)·(n+δ+g)
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def half_life(alpha=0.33, n=0.01, delta=0.05, g=0.02):
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lam = (1-alpha)*(n+delta+g)
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return math.log(2)/lam
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print(half_life()) # ~ 17.3 years
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```
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### Augmented Solow (human capital, MRW 1992)
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```python
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# Y = K^α · H^β · (AL)^(1-α-β)
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def mrw(s_k=0.25, s_h=0.10, alpha=0.33, beta=0.28,
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n=0.01, delta=0.05, g=0.02):
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factor = (n+delta+g)
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k = (s_k**(1-beta) * s_h**beta / factor) ** (1/(1-alpha-beta))
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h = (s_k**alpha * s_h**(1-alpha) / factor) ** (1/(1-alpha-beta))
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return k, h
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```
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### Cross-country fit (sketch)
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```python
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import statsmodels.api as sm
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# log(y) = β0 + β1·log(s) + β2·log(n+δ+g) + ε
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X = sm.add_constant(df[['log_s','log_n_d_g']])
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res = sm.OLS(df['log_y'], X).fit()
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print(res.summary())
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```
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## 매 결정 기준
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| 질문 | Answer (Solow) |
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| Why poor countries grow faster? | conditional convergence (k below k*) |
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| Why long-run growth? | exogenous tech `g` |
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| Effect of higher s? | higher k* · level shift, no LR growth boost |
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| Effect of higher n? | lower k* (capital dilution) |
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| Limitation? | tech 매 unexplained — endogenous models 의 motivation |
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**기본값**: Cobb-Douglas with α≈1/3, δ≈0.05, g≈0.02 매 textbook calibration.
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## 🔗 Graph
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## 🤖 LLM 활용
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**언제**: macro 교육 자료, 매 calibration 의 sanity check, 매 cross-country comparison setup.
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**언제 X**: forecasting 매 short-run 매 부적합 — 매 DSGE / VAR 의 사용.
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## ❌ 안티패턴
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- **Tech as endogenous in pure Solow**: 매 g 매 model 의 외부 — 매 Romer 매 needed.
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- **Ignoring human capital**: 매 MRW augmented form 매 더 정확.
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- **Closed economy assumption**: 매 capital flows 매 무시 → real-world deviation.
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## 🧪 검증 / 중복
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- Verified (Solow 1956 *QJE*; Mankiw-Romer-Weil 1992; Acemoglu *Modern Economic Growth* ch.2).
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- 신뢰도 A.
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## 🕓 Changelog
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| 날짜 | 변경 |
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| 2026-05-08 | Phase 1 |
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| 2026-05-10 | Manual cleanup — full content (math + 6 simulations) |
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