Skip to content

[action-translation] resync: cass_koopmans_1.md #108

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
mmcky merged 2 commits into from
Jul 19, 2026
Merged

[action-translation] resync: cass_koopmans_1.md #108

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
2 commits
Select commit Hold shift + click to select a range
410bc4c
🔄 resync cass_koopmans_1.md
mmcky Jul 18, 2026
02e1493
Restore trailing newline (engine issue tracked in QuantEcon/action-tr…
mmcky Jul 19, 2026
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
6 changes: 6 additions & 0 deletions .translate/state/cass_koopmans_1.md.yml
View file
Open in desktop
Original file line number Diff line number Diff line change
@@ -0,0 +1,6 @@
source-sha: bce028b3121056e0bbb7b57e48b356aa8da050d2
synced-at: "2026-07-18"
model: claude-sonnet-5
mode: RESYNC
section-count: 9
tool-version: 0.17.0
29 changes: 23 additions & 6 deletions lectures/cass_koopmans_1.md
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -9,6 +9,23 @@ kernelspec:
display_name: Python 3 (ipykernel)
language: python
name: python3
translation:
title: Cass-Koopmans模型
headings:
Overview: 概述
The Model: 模型
'The Model::Digression: Aggregation Theory': 插话：聚合理论
'The Model::Digression: Aggregation Theory::An Economy': 一个经济体
Planning Problem: 规划问题
Planning Problem::Useful Properties of Linearly Homogeneous Production Function: 线性齐次生产函数的有用性质
Planning Problem::First-order necessary conditions: 一阶必要条件
Shooting Algorithm: 打靶算法
Setting Initial Capital to Steady State Capital: 将初始资本设定为稳态资本
A Turnpike Property: 收费公路性质（Turnpike property）
A Limiting Infinite Horizon Economy: 极限无限期经济
Stable Manifold and Phase Diagram: 稳定流形和相图
Concluding Remarks: 结论
Concluding Remarks::Exercise: 练习
---

(cass_koopmans_1)=
Expand Down Expand Up @@ -570,8 +587,8 @@ plt.show()
@jit
def bisection(pp, c0, k0, T=10, tol=1e-4, max_iter=500, k_ter=0, verbose=True):

# 设置初始边界
c0_upper = pp.f(k0)
# 猜测c0的初始边界
c0_upper = pp.f(k0) + (1 - pp.δ) * k0
Comment on lines +590 to +591
c0_lower = 0

i = 0
Expand Down Expand Up @@ -659,7 +676,7 @@ $$
1=\beta \frac{u'(\bar{C})}{u'(\bar{C})}[f'(\bar{K})+(1-\delta)]
$$

定义 $\beta = \frac{1}{1+\rho}$，得到
定义 $\beta = \frac{1}{1+\rho}$，并化简得到

$$
1+\rho = 1[f'(\bar{K}) + (1-\delta)]
Expand Down Expand Up @@ -842,7 +859,7 @@ plot_saving_rate(pp, 0.3, k_ss/3, [250, 150, 75, 50], k_ss=k_ss)
合适的做法是将终端条件{eq}`constraint4`替换为

$$
\lim_{T \rightarrow +\infty} \beta^T u'(C_T) K_{T+1} = 0
\lim_{T \rightarrow +\infty} \beta^T u'(C_T) K_{T+1} = 0 ,
$$
Comment on lines 861 to 863

收敛到最优稳态的路径将满足以上条件。
Expand Down Expand Up @@ -971,11 +988,11 @@ c_vec2, k_vec2 = bisection(pp, 1e-3, 1e-3, T=200, k_ter=Ks)

* 蓝线表示方程 {eq}`eq:tildeC` 所描述的不动点 $C = \tilde C (K)$ 的图像。
* 红线表示方程 {eq}`eq:tildeK` 所描述的不动点 $K = \tilde K(C)$ 的图像。
* 绿线表示从时间0时任意 $K_0$ 开始收敛到稳态的稳定流形。
* 绿线表示从时间0时任意 $K_0$ 开始收敛到稳态所描绘出的稳定路径。
* 对于给定的 $K_0$，射击算法将 $C_0$ 设置为绿线上的坐标，以启动一条收敛到最优稳态的路径。
Comment on lines 989 to 992
* 绿线上的箭头显示了动态方程{eq}`eq:systemdynamics`推动连续对 $(K_{t+1}, C_t)$ 的方向。

除了显示三条曲线外，图{numref}`stable_manifold`还绘制了箭头来指示当给定 $K_0$ 时，$C_0$不在绿线所示稳定流形上时，动态方程{eq}`eq:systemdynamics`驱动系统的方向。
除了这三条曲线外，图{numref}`stable_manifold`还绘制了箭头，指示当给定 $K_0$ 时，若 $C_0$ 不在绿线所示的稳定流形上，动态方程{eq}`eq:systemdynamics`将把系统驱动向何处。

* 如果对给定的$K_0$，$C_0$设置在绿线以下，则积累了过多的资本

Expand Down
Loading