### 数学代写|数学生态学作业代写Mathematical Ecology代考| Two-Factor CES Production Function

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## 数学代写|数学生态学作业代写Mathematical Ecology代考|Two-Factor CES Production Function

$$\begin{gathered} Q=A\left[\alpha(b K)^{\rho}+(1-\alpha)((1-b) L)^{\rho}\right]^{1 / \rho}, \quad \rho<1 \\ \text { or } q=A\left[\alpha(b k)^{\rho}+(1-\alpha)(1-b)^{\rho}\right]^{1 / \rho} . \end{gathered}$$ Here, the marginal product of capital (2.19) is $$\begin{gathered} \partial Q / \partial K=A \alpha b^{\rho}\left[\alpha b^{\rho}+(1-\alpha)(1-b)^{\rho} k^{-\rho}\right]^{(1-\rho) / \rho}, \\ h=(1-\alpha)(1-b)^{\rho} k^{1-\rho} /\left(a b^{\rho}\right), \quad \sigma=1 /(1-\rho) \end{gathered}$$ The CES production function is not neoclassical because the Inada conditions are violated. It is visible in Fig. 2.1. At a low degree of substitution $\sigma<1(\rho<0)$, its graph has a horizontal asymptote (see the brown curve in Fig. 2.1). When $\rho \rightarrow 0$, the CES production function approaches the Cobb-Douglas production function. At a high degree of substitution $\sigma>1(0<\rho<1)$, this function increases faster than the Cobb-Douglas one (see the red curve in Fig. 2.1). At $\sigma=\infty(\rho=1)$, the CES function becomes linear: $Q=A \alpha b K+A(1-\alpha)(1-b) L$. When $\rho \rightarrow-\infty$ $(\sigma \rightarrow 0)$, this production function approaches the Leontief production function $Q=\min [b K,(1-b) L]$ discussed next. There is essential economic evidence that the CES production function better fits many economic processes than the Cobb-Douglas production function. For this reason, the CES production function currently dominates in applied economic research.

We shall notice that some textbooks introduce the CES production function in a slightly different form as $Q=A\left[\alpha K^{\rho}+(1-\alpha) L^{\rho}\right]^{1 / \rho}$ and/or with the parameter $\rho$ replaced by $-\rho$ (then the new $\rho>-1$ ).

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Model Description

Let us consider an economy described by the following dynamic characteristics in the continuous time $t$ :
$Q(t)$-the total output produced at time $t$,
$C(t)$-the amount of consumption,
$I(t)$ the amount of gross investment,
$L(t)$ – the amount of labor,
$K(t)$-the amount of capital.
The Solow-Swan model is described by the following equations:
$$Q(t)=F(K(t), L(t))$$
i.e., the output $Q$ is determined by a neoclassical production function $F(K, L)$,
$$Q(t)=C(t)+I(t)$$
i.e., the output $Q$ is distributed between the consumption $C$ and the investment $I$,
$$K^{\prime}(t)=I(t)-\mu K(t), \quad \mu=\mathrm{const}>0,$$
i.e., the capital $K$ depreciates at a constant rate $\mu>0$ (a constant fraction of the capital leaves a production process at each point of time),
$$L^{\prime}(t)=\eta L(t), \quad \eta=\text { const } \geq 0$$
i.e., the labor $L(t)=L_{0} \exp (\eta t)$ grows at a constant exogenous rate $\eta$.
The structure of the Solow-Swan model is shown in Fig. 2.2. The part of the investment in the total product is known as the saving rate:
$$s(t)=I(t) / Q(t)$$
The saving rate is assumed to be constant in the classic Solow-Swan model:
$$I(t)=s Q(t), \quad 0<s<1, \quad s=\text { const. }$$
This assumption simplifies the investigation of the model and leads to a number of essential economic results. More advanced economic models (see next sections) consider the saving rate $s(t)$ as an endogenous control function.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Fundamental Equation of Model

Because the production function $F(K, L)$ is neoclassical and, therefore, linearly homogeneous, then $F(K, L)=L f(k)$ and the equation (2.33) leads to
$$K^{\prime}(t) / L(t)=f(k(t))-\mu k(t)$$
where the capital-labor ratio $k=K / L$ is defined as in (2.18). On the other side,
$$k^{\prime}(t)=K^{\prime}(t) / L(t)-\eta k(t)$$
by (2.34). Combining the last two equalities, we obtain the fundamental equation of the Solow-Swan model
$$k^{\prime}(t)=s f(k)-(\mu+\eta) k(t)$$
Thus, the dynamics of the model $(2.31)-(2.35)$ is reduced to one autonomous (not dependent on $t$ explicitly) differential equation (2.36) with respect to $k$.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Model Description

C(吨)- 消费量，

ķ(吨)- 资本数额。
Solow-Swan 模型由以下等式描述：

ķ′(吨)=一世(吨)−μķ(吨),μ=C这ns吨>0,

Solow-Swan 模型的结构如图 2.2 所示。投资在总产品中的部分称为储蓄率：
s(吨)=一世(吨)/问(吨)

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Fundamental Equation of Model

ķ′(吨)/大号(吨)=F(ķ(吨))−μķ(吨)

ķ′(吨)=ķ′(吨)/大号(吨)−这ķ(吨)

ķ′(吨)=sF(ķ)−(μ+这)ķ(吨)

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