diff --git a/lectures/scalar_dynam.md b/lectures/scalar_dynam.md index 5926a3f..4d238ef 100644 --- a/lectures/scalar_dynam.md +++ b/lectures/scalar_dynam.md @@ -18,314 +18,273 @@ kernelspec: ``` (scalar_dynam)= -# Dynamics in One Dimension +# 一维动力学 +## 概述 -## Overview +在经济学中，许多变量依赖于它们的过去值。 -In economics many variables depend on their past values +例如，我们有理由相信去年的通货膨胀会影响今年的通货膨胀。 +（也许去年的高通胀会导致人们要求更高的工资作为补偿，这将导致今年的价格进一步上涨。） -For example, it seems reasonable to believe that inflation last year with affects inflation this year. - -(Perhaps high inflation last year will lead people to demand higher wages to -compensate, which will feed into higher prices this year.) - -Letting $\pi_t$ be inflation this year and $\pi_{t-1}$ be inflation last year, we -can write this relationship in a general form as +让$\pi_t$表示今年的通货膨胀率，$\pi_{t-1}$表示去年的通货膨胀率，我们可以用一般形式将这种关系写成： $$ \pi_t = f(\pi_{t-1}) $$ -where $f$ is some function describing the relationship between the variables. +其中$f$是描述变量之间关系的某个函数。 -This equation is an example of one-dimensional discrete time dynamic system. +这个方程是一维离散时间动态系统的一个例子。 -In this lecture we cover the foundations of one-dimensional discrete time -dynamics. +在本讲座中，我们将介绍一维离散时间动力学的基础知识。 -(While most quantitative models have two or more state variables, the -one-dimensional setting is a good place to learn foundations -and understand key concepts.) +（虽然大多数定量模型有两个或更多的状态变量，但一维设置是学习基础知识和理解关键概念的好地方。） -Let's start with some standard imports: +让我们从一些标准导入开始： ```{code-cell} ipython import matplotlib.pyplot as plt import numpy as np ``` +## 一些定义 +本节阐述了我们关注的对象和研究的属性类型。 -## Some definitions - -This section sets out the objects of interest and the kinds of properties we study. +### 函数组合 -### Composition of functions +对于本讲座，您应该了解以下内容： -For this lecture you should know the following. +如果 -If +* $g$ 是从 $A$ 到 $B$ 的函数，且 -* $g$ is a function from $A$ to $B$ and -* $f$ is a function from $B$ to $C$, +* $f$ 是从 $B$ 到 $C$ 的函数， -then the **composition** $f \circ g$ of $f$ and $g$ is defined by +那么 $f$ 和 $g$ 的**组合** $f \circ g$ 定义为 $$ (f \circ g)(x) = f(g(x)) $$ -For example, if +例如，如果 -* $A=B=C=\mathbb R$, the set of real numbers, -* $g(x)=x^2$ and $f(x)=\sqrt{x}$, then $(f \circ g)(x) = \sqrt{x^2} = |x|$. +* $A=B=C=\mathbb R$，即实数集， -If $f$ is a function from $A$ to itself, then $f^2$ is the composition of $f$ -with itself. +* $g(x)=x^2$ 且 $f(x)=\sqrt{x}$，那么 $(f \circ g)(x) = \sqrt{x^2} = |x|$。 -For example, if $A = (0, \infty)$, the set of positive numbers, and $f(x) = -\sqrt{x}$, then +如果 $f$ 是从 $A$ 到其自身的函数，那么 $f^2$ 是 $f$ 与自身的组合。 + +例如，如果 $A = (0, \infty)$，即正数集，且 $f(x) = \sqrt{x}$，那么 $$ f^2(x) = \sqrt{\sqrt{x}} = x^{1/4} $$ -Similarly, if $n$ is a positive integer, then $f^n$ is $n$ compositions of $f$ with -itself. - -In the example above, $f^n(x) = x^{1/(2^n)}$. - +同样，如果 $n$ 是正整数，那么 $f^n$ 是 $f$ 与自身的 $n$ 次组合。 +在上面的例子中，$f^n(x) = x^{1/(2^n)}$。 -### Dynamic systems +### 动态系统 +**（离散时间）动态系统**是一个集合 $S$ 和一个将集合 $S$ 映射回自身的函数 $g$。 -A **(discrete time) dynamic system** is a set $S$ and a function $g$ that sends -set $S$ back into to itself. +动态系统的例子包括： +* $S = (0, 1)$ 且 $g(x) = \sqrt{x}$ -Examples of dynamic systems include +* $S = (0, 1)$ 且 $g(x) = x^2$ -* $S = (0, 1)$ and $g(x) = \sqrt{x}$ -* $S = (0, 1)$ and $g(x) = x^2$ -* $S = \mathbb Z$ (the integers) and $g(x) = 2 x$ +* $S = \mathbb Z$（整数集）且 $g(x) = 2 x$ +另一方面，如果 $S = (-1, 1)$ 且 $g(x) = x+1$，那么 $S$ 和 $g$ 不构成动态系统，因为 $g(1) = 2$。 -On the other hand, if $S = (-1, 1)$ and $g(x) = x+1$, then $S$ and $g$ do not -form a dynamic system, since $g(1) = 2$. +* $g$ 并不总是将 $S$ 中的点映射回 $S$。 -* $g$ does not always send points in $S$ back into $S$. +我们关心动态系统是因为我们可以用它们来研究动态！ -We care about dynamic systems because we can use them to study dynamics! - -Given a dynamic system consisting of set $S$ and function $g$, we can create -a sequence $\{x_t\}$ of points in $S$ by setting +给定由集合 $S$ 和函数 $g$ 组成的动态系统，我们可以通过设定 ```{math} :label: sdsod + x_{t+1} = g(x_t) - \quad \text{ with } - x_0 \text{ given}. + \quad \text{其中} + x_0 \text{给定} ``` -This means that we choose some number $x_0$ in $S$ and then take +来创建 $S$ 中点的序列 $\{x_t\}$。 + +这意味着我们选择 $S$ 中的某个数 $x_0$，然后取 ```{math} :label: sdstraj + x_0, \quad x_1 = g(x_0), \quad - x_2 = g(x_1) = g(g(x_0)), \quad \text{etc.} + x_2 = g(x_1) = g(g(x_0)), \quad \text{等等} ``` -This sequence $\{x_t\}$ is called the **trajectory** of $x_0$ under $g$. - -In this setting, $S$ is called the **state space** and $x_t$ is called the -**state variable**. +这个序列 $\{x_t\}$ 被称为 $x_0$ 在 $g$ 下的**轨迹**。 -Recalling that $g^n$ is the $n$ compositions of $g$ with itself, -we can write the trajectory more simply as +在这种情况下，$S$ 被称为**状态空间**，$x_t$ 被称为**状态变量**。 +回想一下 $g^n$ 是 $g$ 与自身的 $n$ 次组合， +我们可以更简单地将轨迹写为 $$ - x_t = g^t(x_0) \quad \text{ for } t = 0, 1, 2, \ldots + x_t = g^t(x_0) \quad \text{对于} t = 0, 1, 2, \ldots $$ -In all of what follows, we are going to assume that $S$ is a subset of -$\mathbb R$, the real numbers. +在接下来的所有内容中，我们假设 $S$ 是 $\mathbb R$（实数集）的子集。 -Equation {eq}`sdsod` is sometimes called a **first order difference equation** +方程 {eq}`sdsod` 有时被称为**一阶差分方程** -* first order means dependence on only one lag (i.e., earlier states such as $x_{t-1}$ do not enter into {eq}`sdsod`). +* 一阶意味着只依赖于一个滞后（即，像 $x_{t-1}$ 这样的早期状态不会出现在 {eq}`sdsod` 中）。 +### 示例：线性模型 +动态系统的一个简单例子是当 $S=\mathbb R$ 且 $g(x)=ax + b$ 时，其中 $a, b$ 是常数（有时称为"参数"）。 -### Example: a linear model - -One simple example of a dynamic system is when $S=\mathbb R$ and $g(x)=ax + -b$, where $a, b$ are constants (sometimes called ``parameters''). - -This leads to the **linear difference equation** - +这导致了**线性差分方程** $$ x_{t+1} = a x_t + b - \quad \text{ with } - x_0 \text{ given}. + \quad \text{其中} + x_0 \text{已给定}。 $$ - - -The trajectory of $x_0$ is +$x_0$ 的轨迹是 ```{math} :label: sdslinmodpath x_0, \quad a x_0 + b, \quad -a^2 x_0 + a b + b, \quad \text{etc.} +a^2 x_0 + a b + b, \quad \text{等等} ``` -Continuing in this way, and using our knowledge of {doc}`geometric series -`, we find that, for any $t = 0, 1, 2, \ldots$, +继续这样下去，并利用我们对几何级数的知识，我们发现，对于任何 $t = 0, 1, 2, \ldots$， ```{math} :label: sdslinmod + x_t = a^t x_0 + b \frac{1 - a^t}{1 - a} + ``` -We have an exact expression for $x_t$ for all non-negative integer $t$ and hence a full -understanding of the dynamics. +我们得到了所有非负整数 $t$ 的 $x_t$ 的精确表达式，从而完全理解了其动态。 -Notice in particular that $|a| < 1$, then, by {eq}`sdslinmod`, we have +特别注意，如果 $|a| < 1$，那么根据上式，我们有 ```{math} :label: sdslinmodc -x_t \to \frac{b}{1 - a} \text{ as } t \to \infty +x_t \to \frac{b}{1 - a} \text{ 当 } t \to \infty ``` -regardless of $x_0$. +无论 $x_0$ 为何值。 -This is an example of what is called global stability, a topic we return to -below. +这是所谓全局稳定性的一个例子，我们将在后面再次讨论这个话题。 +### 示例：非线性模型 +在上面的线性例子中，我们得到了 $x_t$ 关于任意非负整数 $t$ 和 $x_0$ 的精确解析表达式。 +这使得动态分析变得非常容易。 -### Example: a nonlinear model +然而，当模型是非线性时，情况可能会大不相同。 -In the linear example above, we obtained an exact analytical expression for -$x_t$ in terms of arbitrary non-negative integer $t$ and $x_0$. - -This made analysis of dynamics very easy. - -When models are nonlinear, however, the situation can be quite different. - -For example, in a later lecture {doc}`solow`, we will study the Solow-Swan growth model, which has dynamics +例如，在后面的索洛讲座中，我们将研究索洛-斯旺增长模型，其动态为 ```{math} :label: solow_lom2 k_{t+1} = s A k_t^{\alpha} + (1 - \delta) k_t -``` - -Here $k=K/L$ is the per capita capital stock, $s$ is the saving rate, $A$ is the total factor productivity, $\alpha$ is the capital share, and $\delta$ is the depreciation rate. - -All these parameter are positive and $0 < \alpha, \delta < 1$. - -If you try to iterate like we did in {eq}`sdslinmodpath`, you will find that -the algebra gets messy quickly. - -Analyzing the dynamics of this model requires a different method (see below). +``` +这里 $k=K/L$ 是人均资本存量，$s$ 是储蓄率，$A$ 是全要素生产率，$\alpha$ 是资本份额，$\delta$ 是折旧率。 +所有这些参数都是正数，且 $0 < \alpha, \delta < 1$。 +如果你尝试像我们在线性模型中那样迭代，你会发现代数运算很快就变得复杂。 +分析这个模型的动态需要一种不同的方法（见下文）。 -## Stability +## 稳定性 -Consider a dynamic system consisting of set $S \subset \mathbb R$ and -$g$ mapping $S$ to $S$. +考虑一个动态系统，由集合 $S \subset \mathbb R$ 和将 $S$ 映射到 $S$ 的函数 $g$ 组成。 (scalar-dynam:steady-state)= -### Steady states +### 稳态 -A **steady state** of this system is a -point $x^*$ in $S$ such that $x^* = g(x^*)$. +该系统的**稳态**是 $S$ 中的一个点 $x^*$，满足 $x^* = g(x^*)$。 -In other words, $x^*$ is a **fixed point** of the function $g$ in -$S$. +换句话说，$x^*$ 是函数 $g$ 在 $S$ 中的一个**不动点**。 -For example, for the linear model $x_{t+1} = a x_t + b$, you can use the -definition to check that +例如，对于线性模型 $x_{t+1} = a x_t + b$，你可以使用定义来验证： -* $x^* := b/(1-a)$ is a steady state whenever $a \not= 1$, -* if $a = 1$ and $b=0$, then every $x \in \mathbb R$ is a - steady state, -* if $a = 1$ and $b \not= 0$, then the linear model has no steady - state in $\mathbb R$. +* 当 $a \not= 1$ 时，$x^* := b/(1-a)$ 是一个稳态， +* 如果 $a = 1$ 且 $b=0$，那么每个 $x \in \mathbb R$ 都是稳态， +* 如果 $a = 1$ 且 $b \not= 0$，那么这个线性模型在 $\mathbb R$ 中没有稳态。 (scalar-dynam:global-stability)= -### Global stability +### 全局稳定性 -A steady state $x^*$ of the dynamic system is called -**globally stable** if, for all $x_0 \in S$, +如果对于所有的 $x_0 \in S$，都有 $$ -x_t = g^t(x_0) \to x^* \text{ as } t \to \infty +x_t = g^t(x_0) \to x^* \text{ 当 } t \to \infty $$ -For example, in the linear model $x_{t+1} = a x_t + b$ with $a -\not= 1$, the steady state $x^*$ +那么动态系统的稳态 $x^*$ 被称为**全局稳定**的。 + +例如，在线性模型 $x_{t+1} = a x_t + b$ 中，当 $a \not= 1$ 时，稳态 $x^*$ -* is globally stable if $|a| < 1$ and -* fails to be globally stable otherwise. +* 如果 $|a| < 1$，则是全局稳定的， -This follows directly from {eq}`sdslinmod`. +* 否则不是全局稳定的。 +这直接从方程 {eq}`sdslinmod` 得出。 -### Local stability +### 局部稳定性 -A steady state $x^*$ of the dynamic system is called -**locally stable** if there exists an $\epsilon > 0$ such that +如果存在一个 $\epsilon > 0$，使得 $$ | x_0 - x^* | < \epsilon \; \implies \; -x_t = g^t(x_0) \to x^* \text{ as } t \to \infty +x_t = g^t(x_0) \to x^* \text{ 当 } t \to \infty $$ -Obviously every globally stable steady state is also locally stable. +那么动态系统的稳态 $x^*$ 被称为**局部稳定**的。 + +显然，每个全局稳定的稳态也是局部稳定的。 -Here is an example where the converse is not true. +下面是一个反之不成立的例子。 ```{prf:example} -Consider the self-map $g$ on $\mathbb{R}$ defined by $g(x)=x^2$. The fixed point $1$ is not stable. +考虑 $\mathbb{R}$ 上的自映射 $g$，定义为 $g(x)=x^2$。不动点 $1$ 不是稳定的。 -For example, $g^t (x)\to\infty$ for any $x>1$. +例如，对于任何 $x>1$，$g^t (x)\to\infty$。 -However, $0$ is locally stable, because $-1 k_t$。 -Clearly, +* 如果在这一点上 $g$ 位于45度线之下，那么我们有 $k_{t+1} < k_t$。 -* If $g$ lies above the 45-degree line at this point, then we have $k_{t+1} > k_t$. -* If $g$ lies below the 45-degree line at this point, then we have $k_{t+1} < k_t$. -* If $g$ hits the 45-degree line at this point, then we have $k_{t+1} = k_t$, so $k_t$ is a steady state. +* 如果在这一点上 $g$ 与45度线相交，那么我们有 $k_{t+1} = k_t$，所以 $k_t$ 是一个稳态。 -For the Solow-Swan model, there are two steady states when $S = \mathbb R_+ = -[0, \infty)$. +对于索洛-斯旺模型，当 $S = \mathbb R_+ = [0, \infty)$ 时，有两个稳态。 -* the origin $k=0$ -* the unique positive number such that $k = s z k^{\alpha} + (1 - \delta) k$. +* 原点 $k=0$ -By using some algebra, we can show that in the second case, the steady state is +* 唯一的正数，使得 $k = s z k^{\alpha} + (1 - \delta) k$。 + +通过一些代数运算，我们可以证明在第二种情况下，稳态是 $$ k^* = \left( \frac{sz}{\delta} \right)^{1/(1-\alpha)} $$ -### Trajectories +### 轨迹 -By the preceding discussion, in regions where $g$ lies above the 45-degree line, we know that the trajectory is increasing. +根据前面的讨论，在 $g$ 位于45度线之上的区域，我们知道轨迹是递增的。 -The next figure traces out a trajectory in such a region so we can see this more clearly. +下图追踪了这样一个区域内的轨迹，使我们能更清楚地看到这一点。 -The initial condition is $k_0 = 0.25$. +初始条件是 $k_0 = 0.25$。 ```{code-cell} ipython k0 = 0.25 @@ -473,21 +432,19 @@ k0 = 0.25 plot45(g, xmin, xmax, k0, num_arrows=5, var='k') ``` -We can plot the time series of per capita capital corresponding to the figure above as -follows: +我们可以按照上图所示，绘制人均资本随时间变化的时间序列图，具体如下： ```{code-cell} ipython ts_plot(g, xmin, xmax, k0, var='k') ``` -Here's a somewhat longer view: +这里是一个稍长一些的视角： ```{code-cell} ipython ts_plot(g, xmin, xmax, k0, ts_length=20, var='k') ``` -When per capita capital stock is higher than the unique positive steady state, we see that -it declines: +当人均资本存量高于唯一的正稳态值时，我们可以看到它呈下降趋势： ```{code-cell} ipython k0 = 2.95 @@ -495,24 +452,24 @@ k0 = 2.95 plot45(g, xmin, xmax, k0, num_arrows=5, var='k') ``` -Here is the time series: +这里是一个时间序列： ```{code-cell} ipython ts_plot(g, xmin, xmax, k0, var='k') ``` -### Complex dynamics - -The Solow-Swan model is nonlinear but still generates very regular dynamics. +### 复杂动态 -One model that generates irregular dynamics is the **quadratic map** +索洛-斯旺模型是非线性的，但仍然产生非常规律的动态。 +一个能产生不规则动态的模型是**二次映射** $$ g(x) = 4 x (1 - x), \qquad x \in [0, 1] $$ -Let's have a look at the 45-degree diagram. +让我们来看看45度图。 + ```{code-cell} ipython xmin, xmax = 0, 1 @@ -521,56 +478,51 @@ g = lambda x: 4 * x * (1 - x) x0 = 0.3 plot45(g, xmin, xmax, x0, num_arrows=0) ``` - -Now let's look at a typical trajectory. +现在让我们来看一个典型的轨迹。 ```{code-cell} ipython plot45(g, xmin, xmax, x0, num_arrows=6) ``` +注意它是多么不规则。 -Notice how irregular it is. - -Here is the corresponding time series plot. +这是相应的时间序列图。 ```{code-cell} ipython ts_plot(g, xmin, xmax, x0, ts_length=6) ``` - -The irregularity is even clearer over a longer time horizon: +在更长的时间范围内，这种不规则性甚至更加明显： ```{code-cell} ipython ts_plot(g, xmin, xmax, x0, ts_length=20) ``` -## Exercises +## 练习 ```{exercise} :label: sd_ex1 -Consider again the linear model $x_{t+1} = a x_t + b$ with $a -\not=1$. +再次考虑线性模型 $x_{t+1} = a x_t + b$，其中 $a \not=1$。 + +唯一的稳态为 $b / (1 - a)$。 -The unique steady state is $b / (1 - a)$. +当 $|a| < 1$ 时，该稳态是全局稳定的。 -The steady state is globally stable if $|a| < 1$. +尝试通过观察一系列初始条件来图形化地说明这一点。 -Try to illustrate this graphically by looking at a range of initial conditions. +在 $a \in (-1, 0)$ 和 $a \in (0, 1)$ 的情况下，你注意到了什么区别？ -What differences do you notice in the cases $a \in (-1, 0)$ and $a -\in (0, 1)$? +使用 $a=0.5$ 然后使用 $a=-0.5$ 并研究轨迹。 -Use $a=0.5$ and then $a=-0.5$ and study the trajectories. +在整个过程中设置 $b=1$。 -Set $b=1$ throughout. ``` ```{solution-start} sd_ex1 :class: dropdown ``` -We will start with the case $a=0.5$. - -Let's set up the model and plotting region: +我们将从 $a=0.5$ 的情况开始。 +让我们设置模型和绘图区域： ```{code-cell} ipython a, b = 0.5, 1 @@ -578,23 +530,22 @@ xmin, xmax = -1, 3 g = lambda x: a * x + b ``` -Now let's plot a trajectory: +现在让我们来绘制轨迹： ```{code-cell} ipython x0 = -0.5 plot45(g, xmin, xmax, x0, num_arrows=5) ``` -Here is the corresponding time series, which converges towards the steady -state. +这是相应的时间序列，它收敛于稳态。 ```{code-cell} ipython ts_plot(g, xmin, xmax, x0, ts_length=10) ``` -Now let's try $a=-0.5$ and see what differences we observe. +现在让我们尝试 $a=-0.5$ 并观察有什么不同。 -Let's set up the model and plotting region: +让我们设置模型和绘图区域： ```{code-cell} ipython a, b = -0.5, 1 @@ -602,27 +553,23 @@ xmin, xmax = -1, 3 g = lambda x: a * x + b ``` -Now let's plot a trajectory: +现在让我们来绘制轨迹： ```{code-cell} ipython x0 = -0.5 plot45(g, xmin, xmax, x0, num_arrows=5) ``` - -Here is the corresponding time series, which converges towards the steady -state. +这是相应的时间序列，它收敛于稳态。 ```{code-cell} ipython ts_plot(g, xmin, xmax, x0, ts_length=10) ``` -Once again, we have convergence to the steady state but the nature of -convergence differs. +我们再次观察到收敛于稳态的情况，但收敛的性质有所不同。 -In particular, the time series jumps from above the steady state to below it -and back again. +特别是，时间序列在稳态上方和下方来回跳跃。 -In the current context, the series is said to exhibit **damped oscillations**. +在当前的语境中，这个序列被称为呈现**阻尼振荡**。 ```{solution-end} -``` +``` \ No newline at end of file diff --git a/lectures/solow.md b/lectures/solow.md index ede2bfa..cdb1a4e 100644 --- a/lectures/solow.md +++ b/lectures/solow.md @@ -380,10 +380,12 @@ $$ :label: solow_ex1 绘制稳态下人均消费 $c$ 作为储蓄率 $s$ 的函数图，其中 $0 \leq s \leq 1$。 + 使用柯布-道格拉斯生产函数 $f(k) = A k^\alpha$。 + 设定 $A=2.0, \alpha=0.3,$ 且 $\delta=0.5$。 -此外，找出大约能使 $c^*(s)$ 最大化的 $s$ 值，并在图中显示。 +此外，找出大约能使 $c^*(s)$ 最大化的 $s$ 值，并在图中显示。 ``` ```{solution-start} solow_ex1 @@ -409,6 +411,7 @@ c_star = (1 - s_grid) * A * k_star ** alpha ``` 让我们使用 [scipy.optimize.minimize_scalar](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize_scalar.html#scipy.optimize.minimize_scalar) 来找出使 $c^*$ 最大化的 $s$ 值。 + 我们将使用 $-c^*(s)$，因为 `minimize_scalar` 函数是用来寻找最小值的。