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经济代写|Economics: the marginal rate of substitution微观经济学代写

经济代写

6.2. Economics: the marginal rate of substitution. Consider two goods 1 and 2 (if other goods are present, hold them constant). A bundle $y=\left(y_{1}, y_{2}\right)$ defines an indifference curve
$$I_{y}=\left{\left(x_{1}, x_{2}\right) \in \mathbb{R}{+}^{2}:\left(x{1}, x_{2}\right) \sim\left(y_{1}, y_{2}\right)\right} .$$
We now consider an amount $x_{1}$ of good 1 and look for the amount $x_{2}$ of good 2 such that $\left(x_{1}, x_{2}\right)$ is contained in $I_{y}$ (see figure IV.21 for an illustration). In this manner, we can sometimes define a function
$$I_{y}: x_{1} \mapsto x_{2} \text {. }$$
Note that we have used the symbol $I_{y}$ in two different ways, as a subset of the goods space and as a function. We do this to economize on symbols and also to make clear that the function $I_{y}$ is closely related to the indifference curve.
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TV. ORDINAL PREFERENCE THEORY
FIGURE IV.21. An implicit function
DEFINITION IV.30 (MARGINAL RATE OF SUBSTITUTION). If the function $I_{y}$ is differentiable and if preferences are monotonic, we call
$$M R S=\left|\frac{d I_{y}\left(x_{1}\right)}{d x_{1}}\right|$$
the marginal rate of substitution between good 1 and good 2 (or of good 2 for good 1).

We can readily interpret the marginal rate of substitution: if one additional unit of good 1 is consumed while good 2’s consumption is reduced by $M R S$ units, the consumer stays indifferent. We could also say: $M R S$ measures the willingness to pay for one additional unit of good 1 in terms of good 2 .
REMARK IV.2. Note that the above definition “does not work” if one of the goods is a bad. In that case, consuming more of good 1 (nice music) leaves the consumer indifferent if he endures more of good 2 (filthy smoky air). However, since we deal with goods (in the sense of monotonic preferences) interested in the slope of the indifference curve, we can simply calculate $\frac{d I_{y}\left(x_{1}\right)}{d x_{1}}$. In order to avoid tedious repetitions, we will not always point to the fact that we have monotonic preferences.

REMARK IV.3. Marginal this and marginal that is standard staple for economists. It is a somewhat peculiar way of saying that we consider the derivative of a function. Apart from the marginal rate of substitution, we will encounter “marginal utility”, “manginal cost”, “marginal revenue” etc.

1. MARGINAL RATE OF SUBSTITUTION
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As an example, consider the utility function given by $U\left(x_{1}, x_{2}\right)=$ $a x_{1}+b x_{2}, a>0$ and $b>0$, i.e., the case of perfect substitutes. Along an indifference curve, the utility is constant at some level $k$ so that we focus on all goods bundles $\left(x_{1}, x_{2}\right)$ fulfilling $a x_{1}+b x_{2}=k$. We find the slope of that indifference curve by

6.2.经济学：边际替代率。考虑两种商品 1 和 2（如果存在其他商品，则保持它们不变）。束 $y=\left(y_{1}, y_{2}\right)$ 定义了一条无差异曲线
$$I_{y}=\left{\left(x_{1}, x_{2}\right) \in \mathbb{R}{+}^{2}:\left(x{1}, x_{ 2}\right) \sim\left(y_{1}, y_{2}\right)\right} 。$$

$$I_{y}: x_{1} \mapsto x_{2} \text {。 }$$

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$$M R S=\left|\frac{d I_{y}\left(x_{1}\right)}{d x_{1}}\right|$$

1. 边际替代率
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例如，考虑由 $U\left(x_{1}, x_{2}\right)=$ $a x_{1}+b x_{2}, a>0$ 和 $b> 给出的效用函数0$，即完全替代的情况。沿着一条无差异曲线，效用在某个水平 $k$ 上是恒定的，因此我们关注所有满足 $a x_{1}+b x_ 的商品束$\left(x_{1}, x_{2}\right)${2}=k$。我们通过以下方式找到该无差异曲线的斜率

经济代考

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复分析代考

(1) 提到复变函数 ，首先需要了解复数的基本性左和四则运算规则。怎么样计算复数的平方根， 极坐标与 $x y$ 坐标的转换，复数的模之类的。这些在高中的时候囸本上都会学过。
(2) 复变函数自然是在复平面上来研究问题，此时数学分析里面的求导数之尖的运算就会很自然的 引入到复平面里面，从而引出解析函数的定义。那/研究解析函数的性贡就是关楗所在。最关键的 地方就是所谓的Cauchy一Riemann公式，这个是判断一个函数是否是解析函数的关键所在。
(3) 明白解析函数的定义以及性质之后，就会把数学分析里面的曲线积分 $a$ 的概念引入复分析中， 定义几乎是一致的。在引入了闭曲线和曲线积分之后，就会有出现复分析中的重要的定理: Cauchy 积分公式。 这个是易分析的第一个重要定理。