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• Statistical Inference 统计推断
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Dynamic Analysis

A dynamic analysis of such problems is more complex and requires sophisticated mathematical tools. The results provided below are obtained employing the max= imum principle from Sect. 2.4.

Necessary Condition for an Extremum: If the function $s(t), t \in[0, T]$, is a solution of the optimization problem $(2.45)-(2.47)$, then:
(a) There exists a continuous function $\hat{\lambda}(t), t \in[0, T]$, called the dual or adjoint variable, that satisfies the dual equation
$$\hat{\lambda}^{\prime}(t)=(\mu+r) \hat{\lambda}(t)-[1-s(t)+\hat{\lambda}(t) s(t)] f^{\prime}(k(t)),$$
with the terminal transversality condition
$$\left[k(T)-k_{T}\right] \mathrm{e}^{-r T} \hat{\lambda}(T)=0,$$
where the corresponding state variable $k(t), t \in[0, T]$, is found from (2.46).
(b) $s(t)$ maximizes $[1-s(t)+\hat{\lambda}(t) s(t)]$ at each point $t \in[0, T]$.
The proof of this result follows from Corollary $2.1$ of Sect. 2.4. Namely, the current-value Hamiltonian (2.69) for the optimal control problem (2.45)-(2.47) is constructed as
$$\hat{H}(s, k, \hat{\lambda})=f(k)(1-s)+\hat{\lambda}[s f(k)-\mu k],$$
and, then, the dual equation (2.48) is obtained from $(2.70)$ as $\hat{\lambda}^{\prime}=r \hat{\lambda}-\partial H / \partial k$, the state equation (2.46) fits $k^{\prime}=\partial H / \partial \hat{\lambda}$, and $s(t)$ maximizes $H(s, k, \hat{\lambda})$.

Extremum Condition for an Interior Solution. The maximum principle is constructed specifically to handle the case of boundary solutions: $s(t)=0$ or $s(t)=1$ in the domain $0 \leq s(t) \leq 1$ at some instants $t$. The possibility of boundary (or corner) solutions essentially complicates the optimal control dynamics. If a solution is known to be interior in the domain, then the optimality conditions

become simpler. Namely, by Corollary $2.2$ from Sect. $2.4$, if $0<s(t)<1$, then the optimal $s(t)$ satisfies $\partial H / \partial s=0$.

Let us utilize this optimality condition for the optimization problem (2.45)-(2.47). Taking the derivative of $(2.50)$ in $s$, we obtain $\partial \hat{H} / \partial s=f(k)(\hat{\lambda}-1)$. If a priori $0<s(t)<1$ for $t \in[0, T]$, then $\partial H / \partial s=0$ and, therefore, $\hat{\lambda}(t)=1$. Substituting $\hat{\lambda}$ to $(2.48)$, we obtain
$$0=\mu+r-f^{\prime}(k(t))$$
which is the same golden rule of capital accumulation $(2.40)$ as obtained during static optimization in the Solow-Swan model of Sect. 2.2.

Structure of Solution: Using the extremum condition (2.48) and (2.49) and rewriting $(2.50)$ as $\hat{H}(s, k, \hat{\lambda})=s(\hat{\lambda}-1) f(k)-\hat{\lambda} \mu k+f(k)$, we can show that $s(t)=0$ maximizes $\hat{H}(s, k, \hat{\lambda})$ at $\hat{\lambda}(t)<1$ and $s(t)=0$ maximizes $\hat{H}(s, k, \hat{\lambda})$ at $\hat{\lambda}(t)>1$. If $\hat{\lambda}(t)=1$, then $\hat{H}(s, k, \hat{\lambda})$ does not depend on $s$ and the optimal $k^{}$ is found from (2.48), which is the same as the golden rule of capital accumulation (2.40). Thus, the solution $s(t), t \in[0, T]$, of the optimization problem $(2.45)-(2.47)$ is $$s(t)=\left{\begin{array}{ccl} 0 & \text { when } & \hat{\lambda}(t)<1 \\ s^{*} & \text { when } & \hat{\lambda}(t)=1, \\ 1 & \text { when } & \hat{\lambda}(t)>1 \end{array}\right.$$ where $0}<1$ is the optimal (golden-rule) saving rate $(2.41)$ in the Solow-Swan model. When $s(t)=s^{}$, the corresponding trajectory is $k(t)=k^{}$, where the unique $k^{*}$ is found from (2.40).

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Long-Term and Transition Dynamics

Because of the specifics of economic optimization problems, their dynamic analysis is usually split into two steps: the investigation of a long-term dynamics and the investigation of the transition dynamics. In many problems, the long-term dynamics is independent of initial conditions of the problem and coincides with the steady state solution of the model. Then, the transition dynamics describes how the optimal trajectory approaches the steady state.

The solution $s(t), k(t), t \in[0, T]$, of the optimization problem $(2.45)-(2.47)$ in the case $k_{0}<k^{*}<k_{\mathrm{T}}$ is illustrated in Fig. 2.3.

The transition (short-term) dynamics of the problem (2.45)-(2.47) is common for well-formulated economic problems. The optimal trajectory $s(t), k(t)$ approaches the best steady state solution $\left(s^{}, k^{}\right)$ on the initial interval $\left[0, \theta_{1}\right]$ and the transition dynamics ends at the instant $\theta_{1}$ such that $k\left(\theta_{1}\right)=k^{*}$.

The optimal trajectory $s(t), k(t)$ leaves the steady state solution $\left(s^{}, k^{}\right)$ at some instant $\theta_{2}<T$ near the right end of the planning horizon $[0, T]$. This behavior illustrates the so-called end-of-horizon effect and is also common in economic problems. Even if the terminal condition is absent, such effects still take place and even become more substantial. In particular, if $k_{T}=0$, then there is no investments at the end $\left[\theta_{2}, T\right]$ of planning horizon.

Mathematically, this end-of-horizon effect appears because the optimal trajectory $k(t)$ must satisfy the transversality condition $(2.49)$. This condition becomes less restrictive at $T=\infty$. Non-importance of the transversality condition for the infinite-horizon problem $(2.31)-(2.35)$ was pointed out by $K$. Shell in [8]. It will be shown in the next section that the end-of-horizon effect is absent in the infinitehorizon problem.

The trajectory $s(t) \equiv s^{}, k(t) \equiv k^{}$ over $\left[\theta_{1}, \theta_{2}\right]$ represents the long-term dynamics of the optimization problem. The optimal saving rate $s(t)$ coincides with the constant golden-rule saving rate $s$ in the Solow-Swan model on a certain interior part $\left[\theta_{1}, \theta_{2}\right]$ of the planning period $[0, T]$. The length of $\left[\theta_{1}, \theta_{2}\right]$ becomes larger when $T$ increases. It means that a turnpike property holds for the optimization problem $(2.45)-(2.47)$, where the turnpike trajectory is $s_{T} \equiv s^{*}$.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Dynamic Analysis

The dynamic analysis of the Solow-Ramsey model includes new mathematical challenges such as the convergence of the improper integral (2.53) along the optimal trajectory $c$. The condition for this convergence in our model $(2.31)-(2.34),(2.53)$ is simply
$$r>\eta,$$
where $\eta$ is the given growth rate of labor in (2.35). However, finding such conditions becomes more complicated in more advanced models (see for example Chap. 3).

Under (2.54), the extremum conditions remain the same, (2.48)-(2.52), as in the Solow-Shell model. Using (2.48) and (2.52), we can show that the solution $s(t)$, $t \in[0, \infty)$, of the problem (2.53) in the model (2.31)-(2.34) is
$s(t)=\left{\begin{array}{ccc}1 & \text { at } & 0 \leq t<\theta_{1} \ s^{} & \text { at } & \theta_{1} \leq t<\infty\end{array},\right.$, with the golden-rule saving rate $s^{\circ}$ and capital per capita $k^{}$ in the Solow-Swan model. Also, it can be shown that the transversality condition (2.49) is reduced to the inequality (2.54). So the transversality condition is less important in the infinitehorizon problem in the sense that it does not directly affect the solution dynamics.
On the qualitative side, the behavior of the optimal trajectories appears to be simpler than in the finite-horizon Solow-Shell model (2.45)-(2.47). The solution $(s(t), k(t)), t \in[0, \infty)$, of the optimization problem (2.53) in the case $k_{0}<k^{*}$ is illustrated in Fig. $2.3$ by blue curves.

The transition dynamics of the problem (2.53) over the interval $\left[0, \theta_{1}\right]$ is the same as for the Solow-Shell model. The optimal trajectory $(s(t), k(t))$ approaches the steady state $\left(s^{}, k^{}\right)$ and the transition dynamics ends at the instant $\theta_{1}$ such that $k\left(\theta_{1}\right)=k^{*}$.

The long-term dynamics is $s(t) \equiv s^{}, k(t) \equiv k^{}$ over $\left[\theta_{1}, \infty\right)$, i.e., the optimal saving rate $s(t)$ coincides with the constant golden-rule saving rate $s^{*}$ in the Solow-Swan model starting with the time $\theta_{1}$. As shown in Fig. 2.3, the optimal

trajectory $s(t), k(t)$ does not leave the steady state $\left(s^{}, k^{}\right)$ because the end-of-horizon effects are absent in infinite-horizon problems.

The considered optimization versions of the Solow-Swan, Solow-Shell, and Solow-Ramsey models, are classified by the economic theory as the models of exogenous growth because they cannot generate an endogenous growth when the labor $L(t)$ is constant. However, even small modifications of these models can lead to the endogenous growth. For instance, if we replace the neoclassic production function in the model equation (2.31) with the CES or $A K$ production function, then the corresponding models are able to generate an endogenous growth. Models with endogenous growth are discussed in Sect. 3.4.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Dynamic Analysis

(a) 存在一个连续函数λ^(吨),吨∈[0,吨]，称为对偶或伴随变量，满足对偶方程
λ^′(吨)=(μ+r)λ^(吨)−[1−s(吨)+λ^(吨)s(吨)]F′(ķ(吨)),

[ķ(吨)−ķ吨]和−r吨λ^(吨)=0,

(二)s(吨)最大化[1−s(吨)+λ^(吨)s(吨)]在每个点吨∈[0,吨].

H^(s,ķ,λ^)=F(ķ)(1−s)+λ^[sF(ķ)−μķ],

0=μ+r−F′(ķ(吨))

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Dynamic Analysis

Solow-Ramsey 模型的动态分析包括新的数学挑战，例如沿最优轨迹的不正确积分 (2.53) 的收敛C. 我们模型中这种收敛的条件(2.31)−(2.34),(2.53)简直就是
r>这,

$s(t)=\left{1 在 0≤吨<θ1 s 在 θ1≤吨<∞，\对。,在一世吨H吨H和G这ld和n−r在l和s一种在一世nGr一种吨和s^{\circ}一种ndC一种p一世吨一种lp和rC一种p一世吨一种k^{}一世n吨H和小号这l这在−小号在一种n米这d和l.一种ls这,一世吨C一种nb和sH这在n吨H一种吨吨H和吨r一种ns在和rs一种l一世吨是C这nd一世吨一世这n(2.49)一世sr和d在C和d吨这吨H和一世n和q在一种l一世吨是(2.54).小号这吨H和吨r一种ns在和rs一种l一世吨是C这nd一世吨一世这n一世sl和ss一世米p这r吨一种n吨一世n吨H和一世nF一世n一世吨和H这r一世和这npr这bl和米一世n吨H和s和ns和吨H一种吨一世吨d这和sn这吨d一世r和C吨l是一种FF和C吨吨H和s这l在吨一世这nd是n一种米一世Cs.这n吨H和q在一种l一世吨一种吨一世在和s一世d和,吨H和b和H一种在一世这r这F吨H和这p吨一世米一种l吨r一种j和C吨这r一世和s一种pp和一种rs吨这b和s一世米pl和r吨H一种n一世n吨H和F一世n一世吨和−H这r一世和这n小号这l这在−小号H和ll米这d和l(2.45)−(2.47).吨H和s这l在吨一世这n(s(t), k(t)), t \in[0, \infty),这F吨H和这p吨一世米一世和一种吨一世这npr这bl和米(2.53)一世n吨H和C一种s和k_{0}<k^{*}一世s一世ll在s吨r一种吨和d一世nF一世G.蓝色曲线为 2.3 美元。 问题（2.53）在区间内的转换动态[0,θ1]与 Solow-Shell 模型相同。最优轨迹(s(吨),ķ(吨))接近稳态$\left(s^{ }, k^{ }\right)一种nd吨H和吨r一种ns一世吨一世这nd是n一种米一世Cs和nds一种吨吨H和一世ns吨一种n吨\theta_{1}s在CH吨H一种吨k\left(\theta_{1}\right)=k^{*}$。 长期动态是s(吨)≡s,ķ(吨)≡ķ超过[θ1,∞)，即最优储蓄率s(吨)与恒定的黄金法则储蓄率相吻合s∗在从时间开始的 Solow-Swan 模型中θ1. 如图 2.3 所示，最优 弹道s(吨),ķ(吨)不会离开稳定状态$\left(s^{ }, k^{ }\right)\$，因为在无限视野问题中不存在视野结束效应。

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