[lecture-backward-test.md] The source contains a bug: the steady-state formula omits population growth rate…

Summary

The source contains a bug: the steady-state formula omits population growth rate n in the denominator (s/δ should be s/(n+δ)). The code also has the same bug. The translation corrects both, and adds a comparative statics visualization and a clarifying note about the effects of n and s.

Details

Field	Value
File	lecture-backward-test.md
Section	## Steady State
Category	BUG FIX
Confidence	99% (high)
Source repo	test-translation-sync
Target repo	test-translation-sync.zh-cn

Analysis

The omission of n in the steady-state formula is a clear mathematical error — in the standard Solow model with population growth, the steady state satisfies s*k^α = (n+δ)k, giving k = (s/(n+δ))^(1/(1-α)). The source defines n=0.02 in the code but then fails to use it in the denominator, producing an incorrect numerical result. This is a high-confidence bug fix. The comparative statics visualization and clarifying notes are genuine content improvements that should also be considered for backport.

Suggested Changes

1. Bug fix in steady-state formula

Before:

$$
y^* = \left(\frac{s}{\delta}\right)^{\frac{\alpha}{1-\alpha}}
$$

After:

$$
y^* = \left(\frac{s}{n + \delta}\right)^{\frac{\alpha}{1-\alpha}}
$$

2. Bug fix in code: k_star computation

Before:

k_star = (s / delta) ** (1 / (1 - alpha))

After:

k_star = (s / (n + delta)) ** (1 / (1 - alpha))

3. Clarifying note on comparative statics

Before:

This means higher savings rates lead to higher steady-state output per worker.

After:

Note that the denominator includes the population growth rate $n$. This means higher savings rates lead to higher steady-state output per worker, while higher population growth rates reduce steady-state output per worker.

4. Code improvement: comparative statics plot

Before:

(no comparative statics visualization in source)

After:

Add a comparative statics plot showing steady-state output per worker as a function of the savings rate:

```python
# Comparative statics: effect of savings rate
s_values = np.linspace(0.05, 0.5, 50)
y_star_values = (s_values / (n + delta)) ** (alpha / (1 - alpha))

fig, ax = plt.subplots(figsize=(8, 5))
ax.plot(s_values, y_star_values, 'b-', linewidth=2)
ax.set_xlabel('Savings rate (s)', fontsize=12)
ax.set_ylabel('Steady-state output per worker (y*)', fontsize=12)
ax.set_title('Effect of Savings Rate on Steady-State Output', fontsize=14)
ax.axvline(x=s, color='r', linestyle='--', label=f's = {s}')
ax.legend()
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()
```

5. Code improvement: print statement clarity

Before:

print(f"Steady state capital: {k_star:.2f}")
print(f"Steady state output: {y_star:.2f}")

After:

print(f"Steady state capital per worker: {k_star:.2f}")
print(f"Steady state output per worker: {y_star:.2f}")

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Generated by resync backward on 2026-03-05.