2nd/10_Wiki/Topic_Programming/Backend/Solow Growth Model.md

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