### 数学代写|数学生态学作业代写Mathematical Ecology代考| Major Types of Production Functions

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• Statistical Inference 统计推断
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• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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|Cobb–Douglas Production Function

$$y=A x_{1}^{a_{1}} \times \ldots \times x_{n}^{a_{n}}$$
has the following characteristics:
$$\begin{array}{cl} \partial f / \partial x_{i}=\alpha_{i}\left(y / x_{i}\right), & h_{i j}=\alpha_{j} x_{i} / \alpha_{i} x_{j}, \quad \varepsilon_{i}=\alpha_{i}, \quad \varepsilon=\alpha_{1}+\ldots+\alpha_{n}, \ \sigma_{i j}=1, \quad i, j=1, \ldots, n, \end{array}$$
i.e., the elasticity of substitution for any pair $(i, j)$ of inputs is equal to one. The returns to scale are increasing in the case $\alpha_{1}+\ldots+\alpha_{n}>1$, decreasing in the case $\alpha_{1}+\ldots+\alpha_{n}<1$, and constant at $\alpha_{1}+\ldots+\alpha_{n}=1$.
By taking the logarithm of both sides of $(2.9)$, we obtain a linear expression
$$\ln y=\ln A+\sum_{i=1}^{n} \alpha_{i} \ln x_{i}$$
that after differentiation becomes
$$y^{\prime} / y=\alpha_{1}\left(x_{1}^{\prime} / x_{1}\right)+\ldots+\alpha_{n}\left(x_{n}^{\prime} / x_{n}\right)$$
i.e., the growth rate of the output in the Cobb-Douglas production function is equal to the weighted sum of the growth rates of the inputs.

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

Production functions with two inputs, called two-factor production functions, are the most common in economics and are usually written as
$$Q=F(K, L)$$
where
$Q$ is the output,
$K$ is the amount of capital used,
$L$ is the amount of labor used.
The capital $K$ reflects the total cost of the equipment, machines, buildings, etc., used in production process. Such production functions are characterized by single values of the marginal rate of substitution $h$ and the elasticity of substitution $\sigma$ between capital and labor.

A two-factor production function is called the neoclassical production function, if it satisfies the following properties:

1. Essentiality of inputs:
$$F(K, 0)=F(0, L)=0$$
2. Positive and diminishing returns:
$$\partial F / \partial K>0, \quad \partial F / \partial L>0, \quad \partial^{2} F / \partial K^{2}<0, \quad \partial^{2} F / \partial L^{2}<0 .$$
3. Constant returns to scale: $F(K, L)$ is a linearly homogeneous function,
$$F(l K, l L)=l F(K, L) \quad \text { for } l>0 .$$
4. The Inada conditions: the marginal products of capital and labor satisfy
$$\lim {K \rightarrow 0} \frac{\partial F}{\partial K}=\infty, \quad \lim {L \rightarrow 0} \frac{\partial F}{\partial L}=\infty, \quad \lim {K \rightarrow \infty} \frac{\partial F}{\partial K}=0, \quad \lim {L \rightarrow \infty} \frac{\partial F}{\partial L}=0$$
The Inada conditions mean that the production increases very fast if the production input (capital or labor) is low and increases slowly, whereas the production increase is very slow if the production input has been already abundant and more is added. Property 1 holds if the other three properties hold.

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

$$Q=A K^{a} L^{\beta}, \quad \alpha>0, \quad \beta>0$$
where the total factor productivity A reflects the level of technology. In the general case when $\alpha+\beta \neq 1$, the Cobb-Douglas production is not neoclassical because it does not satisfy Property 3 of constant returns. The Cobb-Douglas production at $\alpha+\beta=1$ is neoclassical and can be presented in the standard and intensive forms as
$$Q=A K^{\alpha} L^{1-a} \text { or } q=A k^{a}, \quad 0<\alpha<1$$
Then, the marginal products of capital and labor of $(2.22)$ are
$$\partial Q / \partial K=\alpha A k^{a-1}, \quad \partial Q / \partial L=(1-\alpha) A k^{a},$$
the marginal rate of substitution is $h=k(1-\alpha) / \alpha$, the output elasticity of capital is $\varepsilon_{K}=\alpha$, the total output elasticity is $\varepsilon=1$, and the elasticity of substitution is $\sigma=1$. The graph of the Cobb-Douglas production function (2.23) in the intensive form is shown Fig. 2.1 with a black curve and is typical for the neoclassical production functions. The output $f(k)$ increases indefinitely when the capital per capita $k \rightarrow \infty$, which reflects the Inada condition (2.17). Some economists consider such an increase to be unrealistic.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Cobb–Douglas Production Function

∂F/∂X一世=一种一世(是/X一世),H一世j=一种jX一世/一种一世Xj,e一世=一种一世,e=一种1+…+一种n, σ一世j=1,一世,j=1,…,n,

ln⁡是=ln⁡一种+∑一世=1n一种一世ln⁡X一世

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

ķ是使用的资本金额，

1. 输入的本质：
F(ķ,0)=F(0,大号)=0
2. 正收益和递减收益：
∂F/∂ķ>0,∂F/∂大号>0,∂2F/∂ķ2<0,∂2F/∂大号2<0.
3. 规模报酬不变：F(ķ,大号)是一个线性齐次函数，
F(lķ,l大号)=lF(ķ,大号) 为了 l>0.
4. 稻田条件：资本和劳动的边际产品满足
林ķ→0∂F∂ķ=∞,林大号→0∂F∂大号=∞,林ķ→∞∂F∂ķ=0,林大号→∞∂F∂大号=0
稻田条件意味着如果生产投入（资本或劳动力）低并且缓慢增加，则生产增加非常快，而如果生产投入已经充足并且增加更多，则生产增加非常缓慢。如果其他三个属性成立，则属性 1 成立。

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

∂问/∂ķ=一种一种ķ一种−1,∂问/∂大号=(1−一种)一种ķ一种,

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