### 经济代写|微观经济学代写Microeconomics代考|PACC6007

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• Statistical Inference 统计推断
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• (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

## 经济代写|微观经济学代写Microeconomics代考|Substitute, Independent and Complementary Goods

The substitutability concerns only these consumption bundles $\mathbf{x}=\left(x_1, x_2\right) \in \mathbb{R}_{+}^2$ whose utility is the same.

Any two goods are called substitute goods (substitutes) if in order to keep some given level of utility of a consumption bundle $\mathbf{x} \in \mathbb{R}_{+}^2$ when quantity of one of the goods is reduced (raised) one needs to compensate this change by appropriate increase (decrease) in quantity of the other good in a consumption bundle $\mathbf{x}$.

If in order to keep some given level of utility of a consumption bundle $\mathbf{x} \in \mathbb{R}_{+}^2$ when quantity of one of the goods is reduced (raised) one does need to compensate for this change by any increase (decrease) in quantity of the other good in a consumption bundle $\mathbf{x}$, then such two consumer goods are called independent goods.

Any two goods are called complementary goods (complements) if in order to change the utility of a given consumption bundle $\mathbf{x} \in \mathbb{R}_{+}^2$ one needs to simultaneously raise or reduce quantities of both goods in the bundle.

Note 2.11 When classifying consumption bundles in substitutes or independent goods one takes into account only these goods which are considered in bundles with the same utility level (bundles indifferent to each other).

Note 2.12 To state if any two goods in a consumption bundle $\mathbf{x}=\left(x_1, x_2\right) \in \mathbb{R}{+}^2$ are complements to each other one needs to determine whether an increase (decrease) in the utility of this bundle requires a simultaneous raise (reduction) in quantities of both goods. If there is no such need then the goods are called not complementary. Let us define measures of the substitutability of consumer goods. For this purpose let us assume that we are given: (1) a differentiable utility function $u: \mathbb{R}{+}^2 \rightarrow \mathbb{R}$,

(2) an indifference curve – a set
$$G(u)=\left{\mathbf{x} \in \mathbb{R}_{+}^2 \mid u(\mathbf{x})=u=\text { const. }>0\right} .$$
of all bundles with the same reference utility $u=$ const. $>0$.

## 经济代写|微观经济学代写Microeconomics代考|Marshallian Demand Function

Let us consider a market for two consumer goods where:
$i=1,2$ – consumer goods (products and services),
$X=\mathbb{R}{+}^2-$ a goods space, $\mathbf{p}=\left(p_1, p_2\right) \in \mathbb{R}{+}^2$-a vector of prices of consumer goods,
$\mathbf{x}=\left(x_1, x_2\right) \in B \subset \mathbb{R}{+}^2-$ a bundle of goods that the consumer wants to purchase (a consumption bundle), $B=\left{\mathbf{x}=\left(x_1, x_2\right) \in \mathbb{R}{+}^2 \mid x_1 \leq b_1, x_2 \leq b_2\right}$-a supply set,
$b_i, i=1,2$-supply of $i$-th consumer good, ${ }^{22}$
$I \in$ int $\mathbb{R}{+}-$a consumer’s income, ${ }^{23}$ $u: \mathbb{R}{+}^2 \rightarrow \mathbb{R}-$ a utility function describing the preferences of a consumer (describing a relation of consumer preference).
$D(\mathbf{p}, I)=\left{\mathbf{x} \in \mathbb{R}{+}^2 \mid p_1 x_1+p_2 x_2 \leq I\right} \subset X=\mathbb{R}{+}^2-$ a set of all consumption bundles whose value is not greater than the consumer’s income (a budget set),
Definition 2.26 A bundle $\mathbf{x}$ is called a limit of a sequence $\left{\mathbf{x}^i\right}_{i=1}^{+\infty}$ if a limit of sequence of metric values $\lim {i \rightarrow+\infty} d\left(\mathbf{x}^i, \mathbf{x}\right)=0$, which can be written as $$\lim {i \rightarrow+\infty} \mathbf{x}^i=\mathbf{x} \text { or } \quad \mathbf{x}^i \rightarrow_{i \rightarrow+\infty} \mathbf{x} .$$
Definition 2.27 The budget set $D(\mathbf{p}, I) \subset X=\mathbb{R}{+}^2$ is a closed set because: (2.38) $\forall \mathbf{x}^i \in D(\mathbf{p}, I) \lim {i \rightarrow+\infty} \mathbf{x}^i=\mathbf{x} \Rightarrow \mathbf{x} \in D(\mathbf{p}, I)$.
Definition $2.28$ The budget set $D(\mathbf{p}, I) \subset X=\mathbb{R}_{+}^2$ is a bounded set because:
$$\forall \mathbf{x}^1, \mathbf{x}^2 \in D(\mathbf{p}, I) \quad \exists \mathrm{N}>0 \quad d\left(\mathbf{x}^1, \mathbf{x}^2\right)<N$$

# 微观经济学代考

## 经济代写|微观经济学代写Microeconomics代考|Substitute, Independent and Complementary Goods

(2) 一条无差异曲线一一一个集合
$G(u)=\backslash$ left $\left{\backslash m a t h b f{x} \backslash\right.$ in $\backslash m a t h b b{R}_{-}{+}^{\wedge} 2 \backslash$ mid $u(\backslash \operatorname{mathbf}{x})=u=\backslash \operatorname{text}{$ 常数。 $}>0 \backslash$ 右 $}$

## 经济代写|微观经济学代写Microeconomics代考|Marshallian Demand Function

$i=1,2$ – 消费品（产品和服务)，
$X=\mathbb{R}+{ }^2$ 一货品空间， $\mathbf{p}=\left(p_1, p_2\right) \in \mathbb{R}+{ }^2$-消费品价格的向量，
$\mathbf{x}=\left(x_1, x_2\right) \in B \subset \mathbb{R}+{ }^2$ 一消费者想要购买的一束商品（消费束），
$B=\backslash$ left $\left{\backslash\right.$ mathbf ${x}=\backslash \operatorname{left}\left(x_{-} 1, x_{-} \backslash \backslash\right.$ ight $) \backslash$ in $\backslash m a t h b b{R}{+}^{\wedge} 2 \backslash m i d x_{-} 1 \backslash$ leq b_1, $x_{-} 2 \backslash$ leq b_2\right } } \text { – 一套补给 }

$b_i, i=1,2$-供应 $i$-i肖费品， 22
$I \in$ 整数 $\mathbb{R}+$ 一消费者的收入， ${ }^{23} u: \mathbb{R}+{ }^2 \rightarrow \mathbb{R}$ 一描述消费者偏好的效用函数（描述消费者偏好的关

$\lim i \rightarrow+\infty d\left(\mathbf{x}^i, \mathbf{x}\right)=0$, 可以写成
$$\lim i \rightarrow+\infty \mathbf{x}^i=\mathbf{x} \text { or } \quad \mathbf{x}^i \rightarrow_{i \rightarrow+\infty} \mathbf{x}$$

$$\forall \mathbf{x}^1, \mathbf{x}^2 \in D(\mathbf{p}, I) \quad \exists \mathrm{N}>0 \quad d\left(\mathbf{x}^1, \mathbf{x}^2\right)<N$$

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