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

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• Statistical Inference 统计推断
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• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
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## 数学代写|数学生态学作业代写Mathematical Ecology代考|Production Functions and Their Types

A production function describes a relationship
$$y=f\left(x_{1}, \ldots, x_{n}\right)$$
between the aggregate product output $y$ and the productive inputs $x_{1}, \ldots, x_{n}$ that can include labor, capital, knowledge (human capital), energy consumption, raw materials, natural resources (land, water, minerals), and others. The output $y$ and inputs $x_{i}$ are assumed to be identical. For example, the labor is the quantity of workers indistinguishable in a productive sense.

Henceforth, we will often use the following definition. The function $r(t)=f^{\prime}(t) /$ $f(t)$ is the relative rate of the function $f(t)$ and is often referred to as the growth rate of $f(t)$. If $r \equiv$ const, then $f(t)=C \exp (r t)$.

Economists often use the notation $\dot{f}$ for the derivative of a function $f$ in time. We will keep the standard notation $f^{\prime}$.

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

Commonly accepted properties of production functions are:

1. Essentiality of inputs: If at least one $x_{i}=0$, then $y=0$, i.e., production is not possible without any of the inputs.
2. Positive returns: $\partial f / \partial x_{i}>0, i=1, \ldots, n$, i.e., the output increases if an input increases.
3. Diminishing returns: The Hessian matrix
$$H=\left[\begin{array}{ccc} \partial^{2} f / \partial x_{1}^{2} & \ldots & \partial^{2} f / \partial x_{1} \partial x_{n} \ \ldots & \ldots & \ldots \ \partial^{2} f / \partial x_{n} \partial x_{1} & \ldots & \partial^{2} f / \partial x_{n}^{2} \end{array}\right]$$
is negatively definite. It means that if only one input $x_{i}$ increases and the other inputs $x_{j}, j \neq i$, remain constant, then the efficiency of using the input $x_{i}$ decreases.
4. Proportional returns to scale: $f(\mathbf{x})$ is a homogeneous function of degree $\gamma>0$, i.e.,
$$f(l \mathbf{x})=l^{\eta} f(\mathbf{x}), \quad l \in R^{1}, \quad l>0, \quad \mathbf{x}=\left(x_{1}, \ldots, x_{n}\right)$$
The production function $f(\mathbf{x})$ exhibits increasing returns to scale at $\gamma>1$, decreasing returns to scale at $\gamma<1$, and constant returns to scale at $\gamma=1$. The increasing returns mean that a $1 \%$ increase in the levels of all inputs leads to a greater than the $1 \%$ increase of the output $y$.

In the case of constant returns to scale, the function $f(\mathbf{x})$ is linearly homogeneous: $f(l \mathbf{x})=l f(\mathbf{x})$, and the output increases linearly with respect to a proportional increase of all inputs: a $1 \%$ increase of all inputs produces exactly the $1 \%$ increase of the output. Then, the condition $(2.2)$ is reduced to
$$\partial^{2} f / \partial x_{i}^{2}<0, \quad i=1, \ldots, n$$

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

The major characteristics of production functions are

• The average product $f\left(x_{1}, \ldots, x_{n}\right) / x_{i}$ of the $i$-th input is the output per one unit of the input $x_{i}$ spent, $i=1, \ldots, n$.
• The marginal product $\partial f / \partial x_{i}$ of the $i$-th input describes the additional output obtained due to the increase of the $i$-th input quantity by one unit.
• The isoquant is the set of all possible combinations of inputs $\mathbf{x}=\left(x_{1}, \ldots, x_{n}\right)$ that yield the same level of the output $y=f(x)$. Along an isoquant, the differential of the function $f(\mathbf{x})$ is zero: $\sum_{i=1}^{n}\left(\partial f / \partial x_{i}\right) \mathrm{d} x_{i}=0$.The marginal rate of substitution between the inputs $i$ and $j$
• $$• h_{i j}=\left(\partial f / \partial x_{i}\right) /\left(\partial f / \partial x_{j}\right) •$$
• shows how many units of the $j$-th input are required to substitute one unit of the $i$-th input in order to produce the same level of the output $y$.
• The partial elasticity of output with respect to the input $i$
• $$• \varepsilon_{i}(\mathbf{x})=\left(\partial f(\mathbf{x}) / \partial x_{i}\right) /\left(f(\mathbf{x}) / x_{i}\right)=\partial \ln f(\mathbf{x}) / \partial \ln x_{i} •$$
• is the ratio between the marginal product and the average product of the $i$-th input. It describes the increase of the output $y$ when the $i$-th input increases by $1 \%$.
• The total output elasticity $\varepsilon(\mathbf{x})=\sum_{i=1}^{n} \varepsilon_{i}(\mathbf{x})$ describes the output increase under a proportional production scale extension. For a homogeneous production function $(2.3), \varepsilon(\mathbf{x})=\gamma$.
• The elasticity of substitution is a quantitative measure of a possibility of changes in the input combination to produce the same output. It is equal to the relative change in the ratio of the $i$-th and $j$-th inputs divided by the relative change in their marginal rate of substitution $h_{i j}$ :
• $$• \sigma_{i j}=\frac{\mathrm{d}\left(x_{i} / x_{j}\right)}{\left(x_{i} / x_{j}\right)} \times \frac{h_{i j}}{\mathrm{~d} h_{i j}}=\frac{\mathrm{d} \ln \left(x_{i} / x_{j}\right)}{\mathrm{d} \ln h_{i j}} •$$
• This characteristic shows the percentage change of the ratio $x_{i} / x_{j}$ of these inputs along an isoquant in order to change their marginal substitution rate by one percent. The larger the $\sigma_{i j}$, the greater the substitutability between the two inputs. The inputs $i$ and $j$ are perfect substitutes at $\sigma_{i j}=\infty$ and they are not substitutable at all at $\sigma_{i j}=0$. The elasticity of substitution is used for classification of various production functions.

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

1. 输入的必要性：如果至少有一个X一世=0， 然后是=0，即没有任何投入，生产是不可能的。
2. 正回报：∂F/∂X一世>0,一世=1,…,n，即，如果输入增加，则输出增加。
3. 收益递减：Hessian 矩阵
H=[∂2F/∂X12…∂2F/∂X1∂Xn ……… ∂2F/∂Xn∂X1…∂2F/∂Xn2]
是负定的。这意味着如果只有一个输入X一世增加和其他投入Xj,j≠一世，保持不变，则使用输入的效率X一世减少。
4. 比例回报：F(X)是度的齐次函数C>0， IE，
F(lX)=l这F(X),l∈R1,l>0,X=(X1,…,Xn)
生产函数F(X)表现出规模报酬递增C>1, 规模报酬递减C<1, 规模报酬不变C=1. 收益递增意味着1%所有投入水平的增加导致大于1%增加产量是.

∂2F/∂X一世2<0,一世=1,…,n

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

• 平均产品F(X1,…,Xn)/X一世的一世-th 输入是每单位输入的输出X一世花费，一世=1,…,n.
• 边际产品∂F/∂X一世的一世-th 输入描述了由于一世-第一个单位的输入数量。
• 等量线是所有可能的输入组合的集合X=(X1,…,Xn)产生相同水平的输出是=F(X). 沿等量线，函数的微分F(X)为零：∑一世=1n(∂F/∂X一世)dX一世=0. 投入品之间的边际替代率一世和j
• $$• h_{ij}=\left(\partial f / \partial x_{i}\right) /\left(\partial f / \partial x_{j}\right) •$$
• 显示有多少个单位j-th 输入需要替换一个单位一世-th 输入以产生相同级别的输出是.
• 产出相对于投入的部分弹性一世
• $$• \varepsilon_{i}(\mathbf{x})=\left(\partial f(\mathbf{x}) / \partial x_{i}\right) /\left(f(\mathbf{x}) / x_ {i}\right)=\partial \ln f(\mathbf{x}) / \partial \ln x_{i} •$$
• 是边际产量与平均产量的比值一世-th 输入。它描述了输出的增加是当。。。的时候一世-th 输入增加1%.
• 总产出弹性e(X)=∑一世=1ne一世(X)描述了按比例生产规模扩展下的产量增加。对于同质生产函数(2.3),e(X)=C.
• 替代弹性是对投入组合发生变化以产生相同产出的可能性的定量测量。它等于比率的相对变化一世-th 和j-th 投入除以其边际替代率的相对变化H一世j :
• $$• \sigma_{ij}=\frac{\mathrm{d}\left(x_{i} / x_{j}\right)}{\left(x_{i} / x_{j}\right)} \times \ frac{h_{ij}}{\mathrm{~d} h_{ij}}=\frac{\mathrm{d} \ln \left(x_{i} / x_{j}\right)}{\mathrm{ d} \ln h_{ij}} •$$
• 此特性显示比率的百分比变化X一世/Xj沿等量线将这些投入品的边际替代率改变 1%。越大的σ一世j，两个输入之间的可替代性越大。输入一世和j是完美的替代品σ一世j=∞而且它们根本不可替代σ一世j=0. 替代弹性用于对各种生产函数进行分类。

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