From eec4594d8bc0a49f2b82aad528c6e59cd630c7f0 Mon Sep 17 00:00:00 2001 From: nisha617 Date: Thu, 9 Oct 2025 18:00:17 +1100 Subject: [PATCH 1/7] Add files via upload --- lectures/lqcontrol.md | 382 ++++++++++++++++++++---------------------- 1 file changed, 182 insertions(+), 200 deletions(-) diff --git a/lectures/lqcontrol.md b/lectures/lqcontrol.md index 5b23b48e..005a0b65 100644 --- a/lectures/lqcontrol.md +++ b/lectures/lqcontrol.md @@ -38,27 +38,27 @@ tags: [hide-output] ## 概述 -线性二次(LQ)控制指的是一类在几乎所有科学领域都有应用的动态优化问题。 +线性二次(linear–quadratic,LQ)控制指的是一类在几乎所有科学领域都有应用的动态优化问题。 -本讲座介绍LQ控制及其经济应用。 +正如我们将看到的,LQ 系统具有简洁的结构,使其成为解决各类经济问题的主力工具。 -我们将会看到,LQ系统具有简单的结构,使其成为处理各种经济问题的优秀工具。 +此外,尽管线性二次结构看似具有约束性,它的实际灵活性远超最初印象。 此外,虽然线性二次结构具有局限性,但实际上它比最初看起来要灵活得多。 -这些主题在下文中会反复出现。 +这些主题将在下文中会反复出现。 从数学角度来看,LQ控制问题与{doc}`卡尔曼滤波`密切相关 * 线性二次控制问题和卡尔曼滤波问题的递归表述都涉及矩阵**黎卡提方程**。 -* 线性控制和线性滤波问题的经典表述使用类似的矩阵分解(参见[这节课](https://python-advanced.quantecon.org/lu_tricks.html)和[这节课](https://python-advanced.quantecon.org/classical_filtering.html))。 +* 线性控制和线性滤波问题的经典表述使用类似的矩阵分解(参见[这个讲座](https://python-advanced.quantecon.org/lu_tricks.html)和[这个讲座](https://python-advanced.quantecon.org/classical_filtering.html))。 -在阅读以下内容时,了解这些内容会很有帮助: +阅读下文时,熟悉以下内容会更有帮助: * 矩阵运算 * 随机变量向量 -* 动态规划和贝尔曼方程(参见{doc}`这节课 `和{doc}`这节课 `) +* 动态规划和贝尔曼方程(参见{doc}`这个讲座 `和{doc}`这个讲座 `) 关于LQ控制的更多阅读材料,请参见: @@ -72,27 +72,21 @@ tags: [hide-output] ```{code-cell} ipython import matplotlib.pyplot as plt -import matplotlib as mpl -FONTPATH = "fonts/SourceHanSerifSC-SemiBold.otf" -mpl.font_manager.fontManager.addfont(FONTPATH) -plt.rcParams['font.family'] = ['Source Han Serif SC'] - -plt.rcParams["figure.figsize"] = (11, 5) #设置默认图形大小 import numpy as np from quantecon import LQ ``` ## 引言 -LQ中的"线性"部分指的是状态的线性运动规律,而"二次"部分则指的是偏好。 +LQ 中的“线性(linear)”部分指的是状态变量的线性运动规律,而“二次(quadratic)”部分指的是偏好。 -让我们先从前者开始,然后讨论后者,最后将它们组合成一个优化问题。 +我们将从前者开始,然后讨论后者,最后把它们结合起来构成一个优化问题。 ### 运动规律 -设$x_t$是描述某个经济系统状态的向量。 +设 $x_t$ 是描述某个经济系统状态的向量。 -假设$x_t$遵循以下线性运动规律 +假设 $x_t$ 遵循以下线性运动规律 ```{math} :label: lq_lom @@ -103,27 +97,26 @@ x_{t+1} = A x_t + B u_t + C w_{t+1}, 这里 -* $u_t$是一个"控制"向量,包含了决策者面对当前状态$x_t$时可用的选择 -* $\{w_t\}$是一个不相关的零均值冲击过程,满足$\mathbb E w_t w_t' = I$,其中右边是单位矩阵 +* $u_t$ 是一个“控制”向量,包含了决策者面对当前状态 $x_t$ 时可用的选择 +* $\{w_t\}$ 是一个均值为零、互不相关的冲击过程,满足 $\mathbb E w_t w_t' = I$,其中右边是单位矩阵 -关于维度 +关于维度的规定如下: -* $x_t$是$n \times 1$维,$A$是$n \times n$维 -* $u_t$是$k \times 1$维,$B$是$n \times k$维 -* $w_t$是$j \times 1$维,$C$是$n \times j$维 +* $x_t$ 是 $n \times 1$ 维,$A$ 是 $n \times n$ 维 +* $u_t$ 是 $k \times 1$ 维,$B$ 是 $n \times k$ 维 +* $w_t$ 是 $j \times 1$ 维,$C$ 是 $n \times j$ 维 #### 示例1 考虑一个家庭预算约束,表示为 $$ - a_{t+1} + c_t = (1 + r) a_t + y_t $$ -这里$a_t$是资产,$r$是固定利率,$c_t$是当前消费,$y_t$是当前非金融收入。 +这里 $a_t$ 是资产,$r$ 是固定利率,$c_t$ 是当期消费,$y_t$ 是当期非金融收入。 -如果我们假设$\{ y_t \}$是序列不相关的且服从$N(0, \sigma^2)$分布,那么,令$\{ w_t \}$为标准正态分布,我们可以将系统写作 +假设 $\{ y_t \}$ 是序列不相关的,且服从 $N(0, \sigma^2)$ 分布。令$\{ w_t \}$为标准正态分布,我们可以将系统写作 $$ a_{t+1} = (1 + r) a_t - c_t + \sigma w_{t+1} @@ -131,18 +124,17 @@ $$ 这显然是{eq}`lq_lom`的一个特例,其中资产是状态变量,消费是控制变量。 -(lq_hhp)= #### 示例2 -前一个模型的一个不现实的特征是非金融收入的均值为零且经常为负。 +前一个模型中有一个不太现实的特征:非金融收入的均值为零,且经常为负值。 这个问题可以通过添加一个足够大的均值来轻松解决。 -因此在这个例子中,我们令$y_t = \sigma w_{t+1} + \mu$,其中$\mu$是某个正实数。 +因此在这个例子中,我们令 $y_t = \sigma w_{t+1} + \mu$,其中 $\mu$ 是某个正实数。 -另一个值得引入的改变(我们很快就会明白原因)是将控制变量从消费改为消费与某个"理想"数量$\bar c$的偏差。 +另一个值得引入的改变(稍后我们将看到其用途)是将控制变量从消费改为消费与某个“理想”数量 $\bar c$ 的偏离量。 -(大多数参数化情况下,$\bar c$ 相对于每个时期可获得的消费量来说都很大,因此家庭希望增加消费。) +(在多数参数设定下,$\bar c$ 相对可行消费水平较大,因此家庭倾向于增加消费。) 因此,我们现在将控制变量设为 $u_t := c_t - \bar c$。 @@ -156,10 +148,10 @@ a_{t+1} = (1 + r) a_t - u_t - \bar c + \sigma w_{t+1} + \mu 我们如何将这个新系统写成方程 {eq}`lq_lom` 的形式? -如果像前面的例子一样,我们把 $a_t$ 作为状态变量,那么我们会遇到一个问题: +如果我们像前面的例子一样,把 $a_t$ 作为状态变量,那么会遇到一个问题: 运动方程右侧包含一些常数项。 -这意味着我们处理的是一个*仿射*函数,而不是线性函数 +这意味着我们处理的是一个*仿射*(affine)函数,而不是线性函数 (回顾 {ref}`这个讨论 `)。 幸运的是,我们可以通过添加一个额外的状态变量来轻松解决这个问题。 @@ -203,8 +195,7 @@ u_t + \right) w_{t+1} ``` - -第一行等价于{eq}`lq_lomwc`。 +那么上式第一行等价于{eq}`lq_lomwc`。 此外,该模型现在是线性的,可以通过设定以下参数写成{eq}`lq_lom`的形式 @@ -248,14 +239,14 @@ C := ### 偏好 -在LQ模型中,目标是最小化损失流,其中时间t的损失由以下二次表达式给出 +在LQ模型中,目标是最小化损失流,其中第 $t$ 期的损失由以下二次表达式给出 ```{math} :label: lq_pref_flow x_t' R x_t + u_t' Q u_t ``` - +其中, * 假设 $R$ 是 $n \times n$ 的对称非负定矩阵。 * 假设 $Q$ 是 $k \times k$ 的对称正定矩阵。 @@ -263,36 +254,37 @@ x_t' R x_t + u_t' Q u_t 实际上,对于许多经济问题,可以放宽对 $R$ 和 $Q$ 的正定性条件。只需要 $R$ 和 $Q$ 的某些子矩阵是非负定的即可。详见 {cite}`HansenSargent2008`。 ``` +(lq_hhp)= #### 示例1 -一个满足这些假设的非常简单的例子是将 $R$ 和 $Q$ 取为单位矩阵,这样当前损失为 +一个满足上述假设的简单例子是令 $R$ 和 $Q$ 都为单位矩阵,此时当期损失为: $$ x_t' I x_t + u_t' I u_t = \| x_t \|^2 + \| u_t \|^2 $$ -因此,对于状态和控制而言,损失都被度量为与原点的平方距离。 +因此,对于状态变量和控制变量,损失都被度量为与原点的平方距离。 -(实际上,一般情况 {eq}`lq_pref_flow` 也可以用这种方式理解,但 $R$ 和 $Q$ 定义了其他 - 非欧几里得 - 与零向量的"距离"概念。) +(实际上,一般情形 {eq}`lq_pref_flow` 也可以用这种方式理解,只是 $R$ 与 $Q$ 刻画的是非欧几里得意义下与零向量的“距离”。) -直观上,我们通常可以将状态 $x_t$ 理解为偏离目标的程度,例如 +直观上,我们可以将状态变量 $x_t$ 理解为对某个目标的偏离,例如: -* 通货膨胀偏离某个目标水平的程度 +* 通胀率偏离某个目标水平的程度 * 企业资本存量偏离某个理想数量的程度 -目标是使状态接近目标值,同时节约地使用控制变量。 +目标是在节约使用控制变量的同时,使状态尽量靠近目标值。 #### 示例2 -在{ref}`之前研究的家庭问题 `中,设定$R=0$和$Q=1$得到的偏好为 +在{ref}`之前研究的家庭问题 `中,设定 $R=0$ 和 $Q=1$ 得到的偏好为 $$ x_t' R x_t + u_t' Q u_t = u_t^2 = (c_t - \bar c)^2 $$ -在这种设定下,家庭当前的损失是消费与理想水平$\bar c$的平方差。 +在这种设定下,家庭当前的损失是消费与理想水平$\bar c$的差的平方。 -## 最优性 -- 有限时域 +## 最优性 —— 有限期问题 ```{index} single: LQ Control; Optimality (Finite Horizon) ``` @@ -301,9 +293,9 @@ $$ ### 目标函数 -我们将从有限时域情况开始,终止时间为$T \in \mathbb N$。 +我们将从有限期情形开始,终止时间为$T \in \mathbb N$。 -在这种情况下,目标是选择一系列控制变量$\{u_0, \ldots, u_{T-1}\}$来最小化目标函数 +在这种情形下,目标是选择一系列控制变量 $\{u_0, \ldots, u_{T-1}\}$ 来最小化目标函数 ```{math} :label: lq_object @@ -315,40 +307,38 @@ $$ \right\} ``` -受制于运动方程{eq}`lq_lom`和初始状态$x_0$。 +其约束为运动规律{eq}`lq_lom`和初始状态 $x_0$。 -这里引入的新对象是$\beta$和矩阵$R_f$。 +这里引入的新对象是 $\beta$ 和矩阵 $R_f$。 -标量$\beta$是折现因子,而$x' R_f x$给出与状态$x$相关的终端损失。 +标量 $\beta$ 是折现因子,而 $x' R_f x$ 给出与状态 $x$ 相关的终值损失。 注释: -* 我们假设$R_f$是$n \times n$的、对称且非负定的矩阵。 -* 我们允许$\beta = 1$,因此包含了未折现的情况。 -* $x_0$本身可能是随机的,在这种情况下,我们要求它与冲击序列$w_1, \ldots, w_T$相互独立。 +* 我们假设 $R_f$ 是 $n \times n$ 的、对称且非负定的矩阵。 +* 我们允许 $\beta = 1$,因此包含了未折现的情形。 +* $x_0$ 本身可能是随机的,在这种情形下,我们要求它与冲击序列 $w_1, \ldots, w_T$ 相互独立。 (lq_cp)= ### 信息 到目前为止,我们还忽略了一个约束,即解决这个LQ问题的决策者只知道现在和过去,而不知道未来。 -为了说明这一点,考虑控制序列$\{u_0, \ldots, u_{T-1}\}$。 +为了说明这一点,考虑控制序列 $\{u_0, \ldots, u_{T-1}\}$。 -在选择这些控制时,决策者可以考虑冲击的影响 +当决策者在选择这些控制变量时,他可以考虑冲击序列 ${w_1, \ldots, w_T}$ 对系统的影响。 -系统中的 $\{w_1, \ldots, w_T\}$。 +然而,我们通常假定(此处亦然),第 $t$ 期的控制 $u_t$ 只能基于当前与过去的冲击信息。 -然而,通常假设(在此也将假设)时刻 $t$ 的控制 $u_t$ 只能基于过去和当前的冲击来做出。 - -用高深的[测度论](https://en.wikipedia.org/wiki/Measure_%28mathematics%29)术语来说,就是 $u_t$ 必须对由 $x_0, w_1, w_2, \ldots, w_t$ 生成的 $\sigma$-代数可测。 +用[测度论](https://baike.baidu.com/item/%E6%B5%8B%E5%BA%A6%E8%AE%BA/918407)术语来说,就是 $u_t$ 必须对由 $x_0, w_1, w_2, \ldots, w_t$ 生成的 $\sigma$-代数可测。 这实际上等价于说 $u_t$ 可以写成形式 $u_t = g_t(x_0, w_1, w_2, \ldots, w_t)$,其中 $g_t$ 是某个 Borel 可测函数。 -(几乎所有在应用中有用的函数都是 Borel 可测的,所以从直观上理解,你可以把最后那句话理解为"对某个函数 $g_t$") +(几乎所有在应用中有意义的函数都是 Borel 可测的,所以从直观上理解,你可以把最后那句话理解为"对某个函数 $g_t$") -现在注意到 $x_t$ 最终将取决于 $x_0, w_1, w_2, \ldots, w_t$ 的实现值。 +注意,$x_t$ 最终取决于 $x_0, w_1, w_2, \ldots, w_t$ 的实现值。 -事实上,$x_t$ 总结了决策者为了最优设置控制所需的所有历史冲击信息。 +事实上,$x_t$ 总结了所有历史冲击中与最优控制相关的全部信息。 更准确地说,可以证明任何最优控制 $u_t$ 都可以表示为仅与当前状态相关的函数。 @@ -356,17 +346,17 @@ $$ 实际上,前面的讨论适用于所有标准动态规划问题。 -LQ情况的特殊之处在于——正如我们即将看到的——最优的 $u_t$ 原来是 $x_t$ 的线性函数。 +LQ情形的特殊之处在于——正如我们即将看到的——最优控制 $u_t$ 实际上是 $x_t$ 的线性函数。 -### 解决方案 +### 解 -为了解决有限时域LQ问题,我们可以使用基于向后归纳的动态规划策略,这在概念上类似于{doc}`本讲座 `中采用的方法。 +为了解决有限期LQ问题,我们可以使用基于逆向归纳的动态规划策略,这在概念上类似于{doc}`本讲座 `中采用的方法。 -由于很快就会明白的原因,我们首先引入符号 $J_T(x) = x' R_f x$。 +为了后续推导方便,我们首先引入符号 $J_T(x) = x' R_f x$。 现在考虑决策者在倒数第二个时期的问题。 -具体来说,假设时间为$T-1$,且状态为$x_{T-1}$。 +具体来说,假设时间为 $T-1$,且状态为 $x_{T-1}$。 决策者必须权衡当前损失和(折现的)最终损失,因此需要求解 @@ -377,7 +367,7 @@ x_{T-1}' R x_{T-1} + u' Q u + \beta \, \} $$ -在这个阶段,定义以下函数会比较方便 +此时我们定义函数: ```{math} :label: lq_lsm @@ -389,20 +379,20 @@ x' R x + u' Q u + \beta \, \} ``` -函数$J_{T-1}$被称为$T-1$时期的价值函数,而$J_{T-1}(x)$可以被理解为从$T-1$时期状态$x$开始,当决策者采取最优行为时的总"未来损失"。 +函数 $J_{T-1}$ 被称为 $T-1$ 时期的价值函数,而 $J_{T-1}(x)$ 可以被理解为从 $T-1 $时期状态 $x$ 开始,当决策者采取最优行为时的总"未来损失"。 现在让我们回到$T-2$时期。 -对于$T-2$时期的决策者来说,$J_{T-1}(x)$的作用类似于终端损失$J_T(x) = x' R_f x$对$T-1$时期决策者的作用。 +对于$T-2$时期的决策者来说,$J_{T-1}(x)$ 的作用类似于终端损失 $J_T(x) = x' R_f x$ 对 $T-1$ 时期决策者的作用。 也就是说,$J_{T-1}(x)$ 概括了转移到状态 $x$ 所关联的未来损失。 -决策者选择她的控制 $u$ 来权衡当前损失和未来损失,其中 +决策者需选择控制 $u$,在当前损失与未来损失之间作权衡,其中: * 下一期的状态是 $x_{T-1} = Ax_{T-2} + B u + C w_{T-1}$,因此取决于当前控制的选择。 -* 进入状态 $x_{T-1}$ 的"成本"是 $J_{T-1}(x_{T-1})$。 +* 进入状态 $x_{T-1}$ 的“成本”是 $J_{T-1}(x_{T-1})$。 -因此她的问题是 +因此他的问题是 $$ \min_u @@ -423,7 +413,7 @@ x' R x + u' Q u + \beta \, \} $$ -现在向后归纳的模式已经很清楚了。 +现在逆向的模式已经很清楚了。 具体来说,我们通过以下方式定义一系列价值函数 $\{J_0, \ldots, J_T\}$ @@ -438,15 +428,15 @@ x' R x + u' Q u + \beta \, J_T(x) = x' R_f x $$ -第一个等式是动态规划理论中的贝尔曼方程,专门用于有限时域LQ问题。 +第一个等式是动态规划理论中的贝尔曼方程,专门用于有限期LQ问题。 现在我们有了$\{J_0, \ldots, J_T\}$,我们可以获得最优控制。 作为第一步,让我们看看价值函数是什么样的。 -事实证明,每个$J_t$都具有形式$J_t(x) = x' P_t x + d_t$,其中$P_t$是一个$n \times n$矩阵,而$d_t$是一个常数。 +事实证明,每个 $J_t$ 都具有形式 $J_t(x) = x' P_t x + d_t$,其中 $P_t$ 是一个 $n \times n$ 矩阵,而 $d_t$ 是一个常数。 -我们可以通过归纳法证明这一点,从$P_T := R_f$和$d_T = 0$开始。 +我们可以通过归纳法证明这一点,从 $P_T := R_f$ 和 $d_T = 0$ 开始。 使用这个符号,{eq}`lq_lsm`变为 @@ -460,7 +450,7 @@ x' R x + u' Q u + \beta \, \} ``` -为了得到最小值,我们可以对右边项关于$u$求导,并令其等于零。 +为了得到最小值,我们可以对上式的右侧关于 $u$ 求导,并令其等于零。 应用{ref}`矩阵微积分 `的相关规则,得到 @@ -470,7 +460,7 @@ x' R x + u' Q u + \beta \, u = - (Q + \beta B' P_T B)^{-1} \beta B' P_T A x ``` -将此代回{eq}`lq_fswb`并重新整理得到 +将此代回{eq}`lq_fswb`并整理,得到 $$ @@ -494,9 +484,9 @@ P_{T-1} = R - \beta^2 A' P_T B (Q + \beta B' P_T B)^{-1} B' P_T A + d_{T-1} := \beta \mathop{\mathrm{trace}}(C' P_T C) ``` -(这个代数运算是一个很好的练习 --- 我们把它留给你。) +(这个代数运算是一个很好的练习——我们把它留给读者。) -如果我们继续按这种方式向后推导,很快就会清楚 $J_t (x) = x' P_t x + d_t$ 如我们所说,其中 $\{P_t\}$ 和 $\{d_t\}$ 满足递归式 +如果我们继续按这种方式向后推导,很快就会清楚 $J_t (x) = x' P_t x + d_t$,其中 $\{P_t\}$ 和 $\{d_t\}$ 满足递归式 ```{math} :label: lq_pr @@ -529,9 +519,9 @@ F_t := (Q + \beta B' P_{t+1} B)^{-1} \beta B' P_{t+1} A 这些是我们{ref}`上面讨论过的`线性最优控制策略。 -特别是,由{eq}`lq_oc`和{eq}`lq_lom`给出的控制序列解决了我们的有限时域LQ问题。 +特别是,由{eq}`lq_oc`和{eq}`lq_lom`给出的控制序列解决了我们的有限期LQ问题。 -更准确地说,序列$u_0, \ldots, u_{T-1}$由以下式子给出 +更准确地说,由以下式子给出的序列 $u_0, \ldots, u_{T-1}$, 在我们的约束条件下实现了{eq}`lq_object`的最小值:对于$t = 0, \ldots, T-1$, ```{math} :label: lq_xud @@ -541,39 +531,37 @@ u_t = - F_t x_t x_{t+1} = (A - BF_t) x_t + C w_{t+1} ``` -对于$t = 0, \ldots, T-1$,在我们的约束条件下实现了{eq}`lq_object`的最小值。 ## 实现 -我们将使用[QuantEcon.py](http://quantecon.org/quantecon-py)中的[lqcontrol.py](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lqcontrol.py)代码来求解有限和无限时域线性二次控制问题。 +我们将使用[QuantEcon.py](http://quantecon.org/quantecon-py)中的[lqcontrol.py](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lqcontrol.py)模块来求解有限和无限期线性二次控制问题。 在该模块中,各种更新、模拟和不动点方法被封装在一个名为`LQ`的类中,包括 * 实例数据: - * 所需参数$Q, R, A, B$和可选参数$C, \beta, T, R_f, N$用于指定给定的LQ模型 - -* 在无限视界情况下,将 $T$ 和 $R_f$ 设为 `None` -* 在确定性情况下,将 `C = None`(或零) -* 值函数和策略数据 - * 有限视界情况下的 $d_t, P_t, F_t$ - * 无限视界情况下的 $d, P, F$ + * 所需参数 $Q, R, A, B$ 和可选参数 $C, \beta, T, R_f, N$ 用于指定一个LQ模型 + * 在无限期情形下,将 $T$ 和 $R_f$ 设为 `None` + * 在确定性情形下,设 `C = None`(或零) + * 值函数和策略数据 + * 有限期情形下的 $d_t, P_t, F_t$ + * 无限期情形下的 $d, P, F$ * 方法: - * `update_values` --- 通过 {eq}`lq_pr`、{eq}`lq_dd` 和 {eq}`lq_oc` 将 $d_t, P_t, F_t$ 更新为它们在 $t-1$ 时的值 - * `stationary_values` --- 计算无限视界情况下的 $P, d, F$ - * `compute_sequence` ---- 给定 $x_0$ 并假设标准正态冲击,模拟 $x_t, u_t, w_t$ 的动态 + * `update_values` —— 通过 {eq}`lq_pr`、{eq}`lq_dd` 和 {eq}`lq_oc` 将 $d_t, P_t, F_t$ 更新为它们在 $t-1$ 时的值 + * `stationary_values` —— 计算无限期情形下的 $P, d, F$ + * `compute_sequence` —— 给定 $x_0$ 并假设标准正态冲击,模拟 $x_t, u_t, w_t$ 的动态 (lq_mfpa)= ### 一个应用 -早期凯恩斯模型假设家庭从当前收入中的边际消费倾向是恒定的。 +早期的凯恩斯模型假设家庭对当前收入的边际消费倾向是常数。 -数据与边际消费倾向的恒定性相矛盾。 +然而,数据与这一假设相矛盾。 -作为回应,米尔顿·弗里德曼、弗兰科·莫迪利亚尼等人基于消费者对跨期平滑消费流的偏好构建了模型。 +作为回应,Milton Friedman、Franco Modigliani 等学者建立了基于消费者跨期平滑消费偏好的模型。 (参见{cite}`Friedman1956`或{cite}`ModiglianiBrumberg1954`。) -这些模型的一个特点是家庭通过购买和出售金融资产使消费流比收入流更加平滑。 +这些模型的一个特点是,家庭通过购买和出售金融资产使消费流比收入流更加平滑。 {ref}`上述 `家庭储蓄问题体现了这些观点。 @@ -588,17 +576,17 @@ x_{t+1} = (A - BF_t) x_t + C w_{t+1} \right\} ``` -受预算约束序列$a_{t+1} = (1 + r) a_t - c_t + y_t, \ t \geq 0$的限制。 +预算约束序列为 $a_{t+1} = (1 + r) a_t - c_t + y_t, \ t \geq 0$。 -这里$q$是一个较大的正常数,其作用是促使消费者在生命末期将债务目标控制为零。 +这里 $q$ 是一个大于零的、较大的常数,其作用是促使消费者在生命末期使债务归零。 -(没有这样的约束,最优选择就是在每个时期选择$c_t = \bar c$,相应地让资产进行调整。) +(若没有此约束,最优选择将是令 $c_t = \bar c$,并让资产调整以维持该水平。) 和之前一样,我们设定 $y_t = \sigma w_{t+1} + \mu$ 和 $u_t := c_t - \bar c$,之后约束条件可以写成 {eq}`lq_lomwc` 的形式。 我们看到这个约束条件可以通过设定 $x_t = (a_t \; 1)'$ 并使用 {eq}`lq_lowmc2` 中的定义,转化为 LQ 形式 $x_{t+1} = Ax_t + Bu_t + Cw_{t+1}$。 -为了与这个状态和控制相匹配,目标函数 {eq}`lq_pio` 可以通过选择以下参数写成 {eq}`lq_object` 的形式: +为了与这个状态和控制变量相匹配,目标函数 {eq}`lq_pio` 可以通过选择以下参数写成 {eq}`lq_object` 的形式: $$ Q := 1, @@ -689,9 +677,9 @@ for ax in axes: plt.show() ``` -顶部面板显示了模拟中消费 $c_t$ 和收入 $y_t$ 的时间路径。 +上图的第一面板显示了模拟中消费 $c_t$ 和收入 $y_t$ 的时间路径。 -正如消费平滑讨论中所预期的那样,消费的时间路径比收入的时间路径要平滑得多。 +正如关于消费平滑的讨论中所预期的那样,消费的时间路径比收入的时间路径要平滑得多。 (但请注意,在生命周期末期,消费变得更加不规则,这是因为零期末资产要求对消费选择的影响更大。) @@ -707,9 +695,9 @@ $$ 这些结果对参数变化相对稳健。 -例如,让我们把 $\beta$ 从 $1 / (1 + r) \approx 0.952$ 增加到 $0.96$,同时保持其他参数不变。 +例如,我们把 $\beta$ 从 $1 / (1 + r) \approx 0.952$ 增加到 $0.96$,同时保持其他参数不变。 -这个消费者比之前的更有耐心,因此对后期消费值赋予相对更大的权重。 +这个消费者比之前的更有耐心,因此对后期消费赋予相对更大的权重。 ```{code-cell} ipython3 --- @@ -752,7 +740,7 @@ for ax in axes: plt.show() ``` -现在我们有一个缓慢上升的消费流,以及在中期出现驼峰形状的资产积累来为不断增长的消费提供资金。 +现在我们有一个缓慢上升的消费路径,以及在中期出现驼峰形状的资产积累来为不断增长的消费提供资金。 然而,基本特征保持不变:相对于收入而言,消费是平滑的,且资产与累积的未预期收入呈强烈的正相关。 @@ -762,15 +750,15 @@ plt.show() ### 时变参数 -在某些情况下,允许$A, B, C, R$和$Q$依赖于$t$可能是可取的。 +在某些情形下,我们可能希望允许参数 $A, B, C, R, Q$ 随时间 $t$ 变化。 -为了简单起见,我们在下面的实现中选择不处理这个扩展。 +为简化分析,我们在后文实现中未考虑这一扩展。 -然而,这种一般性的损失并不像你最初想象的那么大。 +不过,这样的简化并不会造成显著的普遍性损失。 -事实上,我们可以通过适当选择状态变量来处理许多具有时变参数的模型。 +事实上,通过适当扩展状态变量,我们仍可处理许多具有时变参数的模型。 -一个说明在{ref}`下面 `给出。 +{ref}`下面 `给出了一个简单说明。 要了解更多示例和更系统的处理方法,请参见{cite}`HansenSargent2013`第2.4节。 @@ -789,9 +777,9 @@ plt.show() \right\} ``` -我们的结果可以直接扩展到这种情况。 +我们的结果可以直接扩展到这种情形。 -{eq}`lq_pr`中的序列$\{P_t\}$变为 +{eq}`lq_pr`中的序列 $\{P_t\}$ 变为 ```{math} :label: lq_pr_cp @@ -813,7 +801,7 @@ u_t = - F_t x_t F_t := (Q + \beta B' P_{t+1} B)^{-1} (\beta B' P_{t+1} A + N) ``` -序列$\{d_t\}$与{eq}`lq_dd`中保持不变。 + 式{eq}`lq_dd`中的序列 $\{d_t\}$保持不变。 我们让感兴趣的读者自行验证这些结果(计算过程虽然冗长但并不特别困难)。 @@ -823,7 +811,7 @@ F_t := (Q + \beta B' P_{t+1} B)^{-1} (\beta B' P_{t+1} A + N) ```{index} single: LQ Control; Infinite Horizon ``` -最后,我们考虑无限视界情况,带有{ref}`交叉项 `,动态方程不变,目标函数为 +最后,我们考虑无限期情形,带有{ref}`交叉项 `,而系统动态与目标函数的形式保持不变: ```{math} :label: lq_object_ih @@ -834,15 +822,15 @@ F_t := (Q + \beta B' P_{t+1} B)^{-1} (\beta B' P_{t+1} A + N) \right\} ``` -在无限视界情况下,最优策略只有在时间本身是状态向量$x_t$的一个分量时才会依赖于时间。 +在无限期情形下,最优策略只有在时间 $t$ 本身是状态向量 $x_t$ 的一个分量时才会依赖于时间。 -换句话说,存在一个固定矩阵$F$,使得对所有$t$都有$u_t = -F x_t$。 +换句话说,存在一个固定矩阵$F$,使得对所有 $t$ 都有 $u_t = -F x_t$。 -决策规则随时间保持不变是直观的——毕竟,决策者在每个阶段都面临相同的无限视界,只是当前状态在变化。 +决策规则随时间保持不变是直观的——毕竟,决策者在每个阶段都面临相同的无限期,只是当前状态在变化。 -不出所料,$P$和$d$也是常数。 +不出所料,$P$ 和 $d$也是常数。 -稳态矩阵$P$是[离散时间代数黎卡提方程](https://en.wikipedia.org/wiki/Algebraic_Riccati_equation)的解。 +稳态矩阵 $P$ 是[离散时间代数黎卡提方程](https://blog.csdn.net/weixin_36815313/article/details/111773535#:~:text=%E4%BB%A3%E6%95%B0%20Riccati%20%E6%96%B9%E7%A8%8B%EF%BC%88algebr)的解。 (riccati_equation)= ```{math} @@ -855,7 +843,7 @@ P = R - (\beta B' P A + N)' 方程 {eq}`lq_pr_ih` 也被称为 *LQ 贝尔曼方程*,将给定的 $P$ 映射到 {eq}`lq_pr_ih` 右侧的映射被称为 *LQ 贝尔曼算子*。 -这个模型的平稳最优策略是 +这个模型的时间不变的最优策略是 ```{math} :label: lq_oc_ih @@ -894,15 +882,15 @@ d (lq_nsi)= ### 应用1:与年龄相关的收入过程 -{ref}`此前 ` 我们研究了一个产生消费平滑的永久收入模型。 +{ref}`此前 ` 我们研究了一个永久收入模型,该模型产生了消费平滑的结果。 -该模型一个不切实际的特点是假设随机收入过程的均值不依赖于消费者的年龄。 +然而,该模型有一个不太现实的假设:随机收入过程的均值不随消费者年龄而变化。 -一个更现实的收入曲线是在职业生涯早期上升,在中期达到顶峰,可能在后期下降,并在退休期间下降更多。 +一个更现实的收入轨迹是:在工作早期阶段上升,在中期达到峰值,随后逐渐下降,并在退休阶段进一步下滑。 -在本节中,我们将使用年龄的多项式函数,将这种上升和下降模拟为一个对称的倒"U"形。 +在本节中,我们将使用关于年龄的多项式函数,对这种对称的倒 “U” 形收入模式进行建模。 -和之前一样,消费者试图最小化 +和之前一样,消费者最小化以下目标函数: ```{math} :label: lq_pip @@ -913,17 +901,17 @@ d \right\} ``` -满足约束条件 $a_{t+1} = (1 + r) a_t - c_t + y_t, \ t \geq 0$。 +约束条件为 $a_{t+1} = (1 + r) a_t - c_t + y_t, \ t \geq 0$。 -对于收入,我们现在取 $y_t = p(t) + \sigma w_{t+1}$,其中 $p(t) := m_0 + m_1 t + m_2 t^2$。 +对于收入,我们现在假设 $y_t = p(t) + \sigma w_{t+1}$,其中 $p(t) := m_0 + m_1 t + m_2 t^2$。 (在{ref}`下一节 `中,我们将使用一些技巧来实现一个更复杂的模型。) -系数 $m_0, m_1, m_2$ 的选择使得 $p(0)=0, p(T/2) = \mu,$ 且 $p(T)=0$。 +系数 $m_0, m_1, m_2$ 的选择使得 $p(0)=0, p(T/2) = \mu$,且 $p(T)=0$。 -你可以确认规格 $m_0 = 0, m_1 = T \mu / (T/2)^2, m_2 = - \mu / (T/2)^2$ 满足这些约束。 +你可以确认 $m_0 = 0, m_1 = T \mu / (T/2)^2, m_2 = - \mu / (T/2)^2$ 满足上述条件。 -要将其转化为LQ设定,考虑预算约束,它变为 +为了将该问题写入 LQ 框架,预算约束可写为: ```{math} :label: lq_hib @@ -931,11 +919,11 @@ d a_{t+1} = (1 + r) a_t - u_t - \bar c + m_1 t + m_2 t^2 + \sigma w_{t+1} ``` -由于 $a_{t+1}$ 是 $(a_t, 1, t, t^2)$ 的线性函数,这提示我们将这四个变量作为状态向量 $x_t$。 +注意,$a_{t+1}$ 是 $(a_t, 1, t, t^2)$ 的线性函数,因此我们将这四个变量作为状态向量 $x_t$。 -一旦选定了合适的状态和控制(回想 $u_t = c_t - \bar c$) -完成这些设置后,其余规格就相对容易确定了。 +一旦对状态变量与控制变量作出合适的选择(回忆 $u_t = c_t - \bar c$ ), +其余部分的设定就相对容易确定。 因此,对于动态系统,我们设定 @@ -983,9 +971,9 @@ C := \right) ``` -如果你使用这个规格展开表达式 $x_{t+1} = A x_t + B u_t + C w_{t+1}$,你会发现资产按照期望的方式遵循{eq}`lq_hib`,且其他状态变量也会适当更新。 +如果你使用这个设定展开表达式 $x_{t+1} = A x_t + B u_t + C w_{t+1}$,你会发现资产按照期望的方式遵循{eq}`lq_hib`,且其他状态变量也会适当更新。 -为了实现偏好规格{eq}`lq_pip`,我们取 +为了实现偏好函数{eq}`lq_pip`,我们取 ```{math} :label: lq_4sp @@ -1030,16 +1018,16 @@ q & 0 & 0 & 0 \\ (lq_nsi2)= ### 应用2:包含退休的永久收入模型 -在{ref}`前一个应用`中,我们使用多项式生成了一个倒U形的收入动态,并将其置于LQ框架中。 +在{ref}`前一个应用`中,我们使用多项式生成了一个倒“U”形的收入动态,并将其置于LQ框架中。 可以说,这个收入过程仍然包含一些不切实际的特征。 更常见的收入模式是 -1. 收入在工作生涯中增长,围绕着上升趋势波动,在晚年增长趋势趋于平缓 +1. 在整个工作期内,收入随时间上升,并围绕上升趋势波动;到中后期增长趋缓; 1. 退休后,收入较低但相对稳定(非金融收入) -令$K$为退休日期,我们可以用以下方式表达这些收入动态 +令 $K$ 为退休日期,我们可以用以下方式表达这些收入动态 ```{math} :label: lq_cases @@ -1051,33 +1039,32 @@ s & \quad \text{otherwise } \end{cases} ``` -这里 +其中 -* $p(t) := m_1 t + m_2 t^2$,其中系数$m_1, m_2$的选择使得$p(K) = \mu$且$p(0) = p(2K)=0$ -* $s$是退休收入 +* $p(t) := m_1 t + m_2 t^2$,其中系数 $m_1, m_2$ 的选择使得 $p(K) = \mu$ 且 $p(0) = p(2K)=0$ +* $s$ 是退休收入 我们假设偏好保持不变,由{eq}`lq_pio`给出。 -预算约束也保持不变,由$a_{t+1} = (1 + r) a_t - c_t + y_t$给出。 +预算约束也保持不变,由 $a_{t+1} = (1 + r) a_t - c_t + y_t$ 给出。 -我们的目标是解决这个问题,并使用本讲中描述的LQ技术模拟路径。 +我们的目标是利用本讲所介绍的 LQ 技术来求解该问题,并模拟相应的路径。 事实上,这是一个不简单的问题,因为在 $K$ 处动态方程{eq}`lq_cases`的拐点使得很难将运动规律表示为固定系数的线性系统。 -然而,我们仍然可以通过适当地连接两个组成部分的LQ问题来使用我们的LQ方法。 +然而,我们仍可通过连接两个 LQ 子问题的方式来求解。 这两个LQ问题分别描述了消费者在工作期间(`lq_working`)和退休期间(`lq_retired`)的行为。 -(这是可行的,因为在生命的这两个不同阶段,各自的收入过程[多项式趋势和常数]都符合LQ框架。) +(这是可行的,因为在生命的这两个不同阶段,各自的收入过程(多项式趋势和常数)都符合LQ框架。) 基本思路是,尽管整个问题不是单一的时不变LQ问题,但它仍然是一个动态规划问题,因此我们可以在每个阶段使用适当的贝尔曼方程。 基于这个逻辑,我们可以: -1. 通过常规的向后归纳程序求解`lq_retired`,从退休结束时向退休开始时迭代。 - +1. 通过常规的逆向归纳程序求解`lq_retired`,从退休终点向前迭代。 1. 将通过此过程生成的退休初始值函数作为终端条件 $R_f$,输入到 `lq_working` 规范中。 -1. 从这个选定的 $R_f$ 开始,通过向后归纳法求解 `lq_working`,一直迭代回工作生涯开始时。 +1. 从这个选定的 $R_f$ 开始,通过逆向归纳法求解 `lq_working`,一直迭代回工作生涯开始时。 这个过程给出了整个生命周期的值函数序列和最优策略。 @@ -1088,32 +1075,32 @@ s & \quad \text{otherwise } ``` -模拟中使用的完整参数集在{ref}`lqc_ex2`中讨论,那里要求你重现这个图形。 +模拟中使用的完整参数集在{ref}`lqc_ex2`中讨论,并要求读者复现该图。 再次强调,在模拟中可以观察到的主要特征是消费平滑。 -资产路径与标准生命周期理论相符,表现为生命早期的负储蓄,随后转为储蓄。 +资产路径与标准生命周期理论相符,年轻时期存在负储蓄,随后进入正储蓄阶段。 资产在退休时达到峰值,之后逐渐下降。 (lqc_mwac)= ### 应用3:具有调整成本的垄断 -考虑一个面临随机逆需求函数的垄断者 +考虑一个垄断者面临以下逆需求函数: $$ p_t = a_0 - a_1 q_t + d_t $$ -这里$q_t$是产量,需求冲击$d_t$遵循 +这里 $q_t$ 是产量,需求冲击 $d_t$ 遵循 $$ d_{t+1} = \rho d_t + \sigma w_{t+1} $$ -其中$\{w_t\}$是独立同分布的标准正态分布。 +其中 $\{w_t\}$ 是独立同分布的标准正态分布。 -垄断者最大化当前和未来利润的期望贴现和 +垄断者最大化未来折现利润流的期望值: ```{math} :label: lq_object_mp @@ -1123,22 +1110,22 @@ $$ \sum_{t=0}^{\infty} \beta^t \pi_t \right\} -\quad \text{where} \quad +\quad \text{其中} \quad \pi_t := p_t q_t - c q_t - \gamma (q_{t+1} - q_t)^2 ``` 这里 -* $\gamma (q_{t+1} - q_t)^2$表示调整成本 -* $c$是平均生产成本 +* $\gamma (q_{t+1} - q_t)^2$ 表示调整成本 +* $c$ 是平均生产成本 + +这一问题可以被表述为 LQ 控制问题并进行求解与模拟。 -这可以被表述为一个LQ问题,然后求解和模拟, -但首先让我们研究这个问题并试图获得一些直观认识。 +在正式求解前,我们先研究这个问题以得到一些直观理解。 -思考这个问题的一种方式是考虑如果$\gamma = 0$会发生什么。 +思考这个问题的一种方式是考虑如果 $\gamma = 0$ 会发生什么。 -没有调整成本就没有跨期权衡,所以 -垄断者将在每个时期选择产量以最大化当期利润。 +没有调整成本就没有跨期权衡,所以垄断者将在每个时期选择产量以最大化当期利润。 不难证明利润最大化的产出为 @@ -1146,10 +1133,10 @@ $$ \bar q_t := \frac{a_0 - c + d_t}{2 a_1} $$ -根据这个讨论,对于一般的$\gamma$值,我们可以预期: +根据这个讨论,对于一般的 $\gamma$ 值,我们可以预期: -* 如果$\gamma$接近零,那么$q_t$会相对紧密地跟踪$\bar q_t$的时间路径。 -* 如果$\gamma$较大,那么$q_t$会比$\bar q_t$更平滑,因为垄断者试图避免调整成本。 +* 如果 $\gamma$ 接近零,那么 $q_t$ 会相对紧密地跟踪 $\bar q_t$ 的时间路径。 +* 如果 $\gamma$ 较大,那么 $q_t$ 会比 $\bar q_t$ 更平滑,因为垄断者试图避免调整成本。 这种直觉是正确的。 @@ -1169,23 +1156,23 @@ $$ ``` -为了生成这些图表,我们将垄断者问题转换为LQ问题。 +为了生成这些图,我们将垄断者问题转换为一个LQ问题。 -这种转换的关键在于选择正确的状态 --- 这有点像一门艺术。 +这种转换的关键在于选择正确的状态变量——这有点像一门艺术。 -这里我们取$x_t = (\bar q_t \;\, q_t \;\, 1)'$,而控制变量选择为$u_t = q_{t+1} - q_t$。 +这里我们取 $x_t = (\bar q_t \;\, q_t \;\, 1)'$,而控制变量选择为 $u_t = q_{t+1} - q_t$。 我们还对利润函数做了轻微调整。 在{eq}`lq_object_mp`中,当期利润为$\pi_t := p_t q_t - c q_t - \gamma (q_{t+1} - q_t)^2$。 -现在让我们在{eq}`lq_object_mp`中用$\hat \pi_t := \pi_t - a_1 \bar q_t^2$替换$\pi_t$。 +现在让我们在{eq}`lq_object_mp`中用 $\hat \pi_t := \pi_t - a_1 \bar q_t^2$ 替换 $\pi_t$。 -这对解决方案没有影响,因为$a_1 \bar q_t^2$不依赖于控制变量。 +这对最优解没有影响,因为 $a_1 \bar q_t^2$ 不依赖于控制变量。 -(实际上,我们只是在{eq}`lq_object_mp`中添加了一个常数项,而优化器不受常数项影响。) +(实际上,我们只是在{eq}`lq_object_mp`中添加了一个常数项,而优化结果不受常数项影响。) -进行这种替换的原因是,正如你将能够验证的那样,$\hat \pi_t$可以简化为简单的二次形式 +进行这种替换的原因是,正如你将能够验证的那样,$\hat \pi_t$ 可以简化为简单的二次形式 $$ \hat \pi_t = -a_1 (q_t - \bar q_t)^2 - \gamma u_t^2 @@ -1205,11 +1192,11 @@ $$ \right\} ``` -现在找到 $R$ 和 $Q$ 使得{eq}`lq_object_mp2`可以写成{eq}`lq_object_ih`的形式相对简单。 +现在找到 $R$ 和 $Q$, 使得{eq}`lq_object_mp2`可以写成{eq}`lq_object_ih`的形式。 此外,通过写出状态每个元素的动态方程,可以找到{eq}`lq_lom`中的矩阵 $A, B$ 和 $C$。 -{ref}`lqc_ex3`要求你完成这个过程,并重现前面的图表。 +{ref}`lqc_ex3`要求你完成这个过程,并复现前面的图表。 ## 练习 @@ -1217,7 +1204,7 @@ $$ ```{exercise} :label: lqc_ex1 -复现上面{ref}`所示`的多项式收入图。 +复现{ref}`上面所示`的多项式收入曲线图。 参数为 $r = 0.05, \beta = 1 / (1 + r), \bar c = 1.5, \mu = 2, \sigma = 0.15, T = 50$ 和 $q = 10^4$。 ``` @@ -1226,9 +1213,9 @@ $$ :class: dropdown ``` -这是一个解决方案。 +以下是一种可行的解法。 -我们使用一些高级绘图命令来获得特定的样式 — 你可以使用更简单的命令。 +我们使用了一些绘图样式指令以获得更美观的输出 —— 当然你也可以使用更简洁的绘图代码。 这个模型是一个具有驼峰形收入的LQ永久收入/生命周期模型 @@ -1314,11 +1301,11 @@ plt.show() ````{exercise} :label: lqc_ex2 -复现上面{ref}`所示的工作和退休图表 `。 +复现{ref}`上面所示`关于工作和退休的图。 参数为 $r = 0.05, \beta = 1 / (1 + r), \bar c = 4, \mu = 4, \sigma = 0.35, K = 40, T = 60, s = 1$ 和 $q = 10^4$。 -要理解整体过程,请仔细阅读包含该图表的章节。 +为理解整体流程,请仔细阅读该图所在的章节。 ```{hint} :class: dropdown @@ -1329,26 +1316,24 @@ plt.show() 对于`lq_working`, $x_t, A, B, C$ 可以按照{eq}`lq_lowmc3`中的方式选择。 -* 请记住,选择 $m_1, m_2$ 使得 $p(K) = \mu$ 且 $p(2K)=0$。 - -对于`lq_retired`,使用相同的 $x_t$ 和 $u_t$ 定义,但修改 $A, B, C$ 以对应固定收入 $y_t = s$。 +* 请记住,选择 $m_1, m_2$ 使得 $p(K) = \mu$ 且 $p(2K)=0$。 -对于`lq_retired`,按照{eq}`lq_4sp`设置偏好。 +对于`lq_retired`,使用相同的 $x_t$ 和 $u_t$ 定义,但修改 $A, B, C$ 以对应固定收入 $y_t = s$。 -对于`lq_working`,偏好相同,除了 $R_f$ 应该 +对于`lq_retired`,按照{eq}`lq_4sp`设置偏好。 -被退休时期通过迭代`lq_retired`得到的最终值函数所替代。 +对于`lq_working`,偏好相同,但其终值条件 $R_f$ 应替换为由`lq_retired`从退休期末逆向迭代至退休起点所得到的最终值函数。 -通过仔细处理,可以将这两个独立模型的模拟结果拼接在一起生成完整的模拟。 +通过仔细处理,可以将这两个独立模型的模拟结果拼接在一起生成完整的模拟。 ```` ```{solution-start} lqc_ex2 :class: dropdown ``` -这是一个永久收入/生命周期模型,工作期间收入呈多项式增长,退休后收入固定。 +这是一个永久收入/生命周期模型,工作期间收入呈多项式增长,退休后收入固定。 -该模型通过组合两个LQ规划问题来求解,正如讲座中所述。 +该模型通过组合两个LQ规划问题来求解,正如讲座中所述。 ```{code-cell} ipython3 # 模型参数 @@ -1456,7 +1441,7 @@ plt.show() ```{exercise} :label: lqc_ex3 -复现上述垄断者应用中的图形 {ref}`given above `。 +复现{ref}`上述 `垄断者应用中的图形。 参数设置为 $a_0 = 5, a_1 = 0.5, \sigma = 0.15, \rho = 0.9, \beta = 0.95$ 和 $c = 2$,而 $\gamma$ 在1到50之间变化 @@ -1469,8 +1454,7 @@ plt.show() 第一个任务是找到定义LQ问题的矩阵 $A, B, C, Q, R$。 -回顾一下 $x_t = (\bar q_t \;\, q_t \;\, 1)'$,而 -$u_t = q_{t+1} - q_t$。 +回顾一下 $x_t = (\bar q_t \;\, q_t \;\, 1)'$,而 $u_t = q_{t+1} - q_t$。 令 $m_0 := (a_0 - c) / 2a_1$ 且 $m_1 := 1 / 2 a_1$,我们 可以写成 $\bar q_t = m_0 + m_1 d_t$,然后经过一些 @@ -1480,11 +1464,9 @@ $$ \bar q_{t+1} = m_0 (1 - \rho) + \rho \bar q_t + m_1 \sigma w_{t+1} $$ -根据我们对 $u_t$ 的定义,$q_t$ 的动态方程为 -$q_{t+1} = q_t + u_t$。 +根据我们对 $u_t$ 的定义,$q_t$ 的动态方程为 $q_{t+1} = q_t + u_t$。 -使用这些事实,你应该能够构建正确的 -$A, B, C$ 矩阵(然后与下面解决方案代码中的矩阵进行对照)。 +使用这些事实,你应该能够构建正确的$A, B, C$ 矩阵(然后与下面代码中的矩阵进行对照)。 通过检查目标函数可以找到合适的 $R, Q$ 矩阵,为方便起见,我们在此重复该目标函数: @@ -1497,7 +1479,7 @@ $$ \right\} $$ -我们的解决方案代码如下: +我们的代码如下: ```{code-cell} ipython3 # 模型参数 From d080c0d6d21d73c09ea722df44ef98d2edb99a0f Mon Sep 17 00:00:00 2001 From: nisha617 Date: Mon, 13 Oct 2025 11:13:48 +1100 Subject: [PATCH 2/7] Update lqcontrol.md --- lectures/lqcontrol.md | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/lectures/lqcontrol.md b/lectures/lqcontrol.md index 005a0b65..ddc78945 100644 --- a/lectures/lqcontrol.md +++ b/lectures/lqcontrol.md @@ -72,6 +72,11 @@ tags: [hide-output] ```{code-cell} ipython import matplotlib.pyplot as plt +import matplotlib as mpl +FONTPATH = "fonts/SourceHanSerifSC-SemiBold.otf" +mpl.font_manager.fontManager.addfont(FONTPATH) +plt.rcParams['font.family'] = ['Source Han Serif SC'] + import numpy as np from quantecon import LQ ``` @@ -195,6 +200,7 @@ u_t + \right) w_{t+1} ``` + 那么上式第一行等价于{eq}`lq_lomwc`。 此外,该模型现在是线性的,可以通过设定以下参数写成{eq}`lq_lom`的形式 From 827fd89bf041acf601931f9bd003687c14113db0 Mon Sep 17 00:00:00 2001 From: nisha617 Date: Mon, 13 Oct 2025 23:19:44 +1100 Subject: [PATCH 3/7] Update lqcontrol.md --- lectures/lqcontrol.md | 1 - 1 file changed, 1 deletion(-) diff --git a/lectures/lqcontrol.md b/lectures/lqcontrol.md index ddc78945..340f9647 100644 --- a/lectures/lqcontrol.md +++ b/lectures/lqcontrol.md @@ -171,7 +171,6 @@ a_{t+1} = (1 + r) a_t - u_t - \bar c + \sigma w_{t+1} + \mu a_{t+1} \\ 1 \end{array} - \right) = \left( \begin{array}{cc} From 088665ac5da86264a664237275889ada99af9864 Mon Sep 17 00:00:00 2001 From: Humphrey Yang Date: Wed, 15 Oct 2025 14:58:19 +1100 Subject: [PATCH 4/7] update figures --- .../lqcontrol/solution_lqc_ex1.png | Bin 38484 -> 61802 bytes .../lqcontrol/solution_lqc_ex2.png | Bin 40248 -> 56383 bytes .../lqcontrol/solution_lqc_ex3_g1.png | Bin 43109 -> 69052 bytes .../lqcontrol/solution_lqc_ex3_g10.png | Bin 37169 -> 67531 bytes .../lqcontrol/solution_lqc_ex3_g50.png | Bin 39643 -> 63551 bytes lectures/lqcontrol.md | 29 ++++++++---------- 6 files changed, 13 insertions(+), 16 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Python 3 (ipykernel) + language: python --- (lqc)= @@ -29,10 +31,9 @@ kernelspec: 除了Anaconda中已有的库外,本讲座还需要以下库: -```{code-cell} ipython ---- -tags: [hide-output] ---- +```{code-cell} ipython3 +:tags: [hide-output] + !pip install quantecon ``` @@ -70,7 +71,7 @@ tags: [hide-output] 让我们从一些导入开始: -```{code-cell} ipython +```{code-cell} ipython3 import matplotlib.pyplot as plt import matplotlib as mpl FONTPATH = "fonts/SourceHanSerifSC-SemiBold.otf" @@ -705,9 +706,8 @@ $$ 这个消费者比之前的更有耐心,因此对后期消费赋予相对更大的权重。 ```{code-cell} ipython3 ---- -tags: [output_scroll] ---- +:tags: [output_scroll] + # 计算解并模拟 lq = LQ(Q, R, A, B, C, beta=0.96, T=T, Rf=Rf) x0 = (0, 1) @@ -1296,7 +1296,6 @@ for ax in axes: ax.grid() ax.set_xlabel('时间') ax.legend(ncol=2, **legend_args) - plt.show() ``` @@ -1436,7 +1435,6 @@ for ax in axes: ax.grid() ax.set_xlabel('时间') ax.legend(ncol=2, **legend_args) - plt.show() ``` @@ -1528,7 +1526,7 @@ q = xp[1, :] # 绘制模拟结果 fig, ax = plt.subplots(figsize=(10, 6.5)) -# 一些复杂的绘图设置 -- 如果你愿意可以简化 +# 一些复杂的绘图设置 bbox = (0., 1.01, 1., .101) legend_args = {'bbox_to_anchor': bbox, 'loc': 3, 'mode': 'expand'} p_args = {'lw': 2, 'alpha': 0.6} @@ -1538,11 +1536,10 @@ ax.set(xlabel='时间', xlim=(0, max(time))) ax.plot(time, q_bar, 'k-', lw=2, alpha=0.6, label=r'$\bar q_t$') ax.plot(time, q, 'b-', lw=2, alpha=0.6, label='$q_t$') ax.legend(ncol=2, **legend_args) -s = f'动态过程,其中 $\gamma = {γ}$' -ax.text(max(time) * 0.6, 1 * q_bar.max(), s, fontsize=14) +s = fr'给定$\gamma = {γ}$下的动态过程' +ax.text(min(time) + 1, 1 * q_bar.max(), s, fontsize=14) plt.show() ``` ```{solution-end} ``` - From 2b84bc5c65993a47cf2581a29c60a77d63298cfd Mon Sep 17 00:00:00 2001 From: nisha617 Date: Wed, 15 Oct 2025 15:18:18 +1100 Subject: [PATCH 5/7] Update lqcontrol.md --- lectures/lqcontrol.md | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/lectures/lqcontrol.md b/lectures/lqcontrol.md index b2ecc8de..faa1d18a 100644 --- a/lectures/lqcontrol.md +++ b/lectures/lqcontrol.md @@ -802,7 +802,7 @@ P_T = R_f :label: lq_oc_cp u_t = - F_t x_t -\quad \text{where} \quad +\quad \text{其中} \quad F_t := (Q + \beta B' P_{t+1} B)^{-1} (\beta B' P_{t+1} A + N) ``` @@ -994,7 +994,6 @@ R := 0 & 0 & 0 & 0 \end{array} \right) - \quad \text{和} \quad R_f := \left( @@ -1190,7 +1189,6 @@ $$ \min \mathbb E \, - \sum_{t=0}^{\infty} \beta^t \left\{ a_1 ( q_t - \bar q_t)^2 + \gamma u_t^2 From 5e444e302fa086f0a150b3199b3ada8c8deb4833 Mon Sep 17 00:00:00 2001 From: nisha617 Date: Wed, 15 Oct 2025 19:12:02 +1100 Subject: [PATCH 6/7] Update lqcontrol.md --- lectures/lqcontrol.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/lectures/lqcontrol.md b/lectures/lqcontrol.md index faa1d18a..542b8a78 100644 --- a/lectures/lqcontrol.md +++ b/lectures/lqcontrol.md @@ -533,7 +533,7 @@ F_t := (Q + \beta B' P_{t+1} B)^{-1} \beta B' P_{t+1} A :label: lq_xud u_t = - F_t x_t -\quad \text{with} \quad +\quad \text{且} \quad x_{t+1} = (A - BF_t) x_t + C w_{t+1} ``` @@ -1038,8 +1038,8 @@ q & 0 & 0 & 0 \\ y_t = \begin{cases} -p(t) + \sigma w_{t+1} & \quad \text{if } t \leq K \\ -s & \quad \text{otherwise } +p(t) + \sigma w_{t+1} & \quad \text{如果 } t \leq K \\ +s & \quad \text{其他情况} \end{cases} ``` From 72d6a1f6036ceed23c6f19e5e05d6e8b51397981 Mon Sep 17 00:00:00 2001 From: nisha617 Date: Wed, 15 Oct 2025 22:11:42 +1100 Subject: [PATCH 7/7] Update lqcontrol.md --- lectures/lqcontrol.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/lectures/lqcontrol.md b/lectures/lqcontrol.md index 542b8a78..7f0e5f5b 100644 --- a/lectures/lqcontrol.md +++ b/lectures/lqcontrol.md @@ -792,7 +792,7 @@ plt.show() P_{t-1} = R - (\beta B' P_t A + N)' (Q + \beta B' P_t B)^{-1} (\beta B' P_t A + N) + \beta A' P_t A -\quad \text{with } \quad +\quad \text{且} \quad P_T = R_f ```