## 经济代写|微观经济学代写Microeconomics代考|ECON1001

2023年1月5日

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## 经济代写|微观经济学代写Microeconomics代考|Production Space and Production Function

If we consider production as a process of transforming a bundle of inputs (of production factors) into a bundle of outputs (of products) then we can describe any production process by means of production function as a mapping $f: \mathbb{R}{+}^n \rightarrow \mathbb{R}{+}^m$ that assigns at most one vector $\mathbf{y}=\left(y_1, \ldots, y_m\right) \in \mathbb{R}{+}^m$ of production outputs (of final products) to any vector $\mathbf{x}=\left(x_1, \ldots, x_n\right) \in \mathbb{R}{+}^n$ of inputs of production factors.

For the sake of simplicity, we assume that $n=2, m=1$. This way we limit ourselves to production processes in which one product is produced by using two production factors.

Definition 4.1 A production process is a vector $\mathbf{z}=(\mathbf{x}, y) \in \mathbb{R}{+}^3$, consisting of a vector $\mathbf{x}=\left(x_1, x_2\right) \in \mathbb{R}{+}^2$ of inputs of production factors and of a variable $y \in \mathbb{R}_{+}$ which describes a quantity of a product that can be produced by a given vector of inputs of production factors.

Note 4.1 When by a given technology it is possible to obtain any (not necessarily maximum) quantity of a product using a given vector $\mathbf{x}=\left(x_1, x_2\right) \in \mathbb{R}_{+}^2$ of inputs of production factors then we relate such situations to technologically feasible productions processes.

Definition 4.2 A production space is a set $Z=\left{\mathrm{z}=(\mathbf{x}, y) \in \mathbb{R}{+}^3\right.$ $\left.\mid \mathbf{x} \in \mathbb{R}{+}^2, y \in \mathbb{R}{+}\right} \subseteq \mathbb{R}{+}^3$ of all technologically feasible production processes with a norm ${ }^2|\mathbf{z}|=\max \left{|y|,\left|x_1\right|,\left|x_2\right|\right}=\max \left{y, x_1, x_2\right}$ defined on this set.

When by a given technology it is possible to obtain maximum quantity $y \in \mathbb{R}{+}$ of a product using a given vector $\mathbf{x}=\left(x_1, x_2\right) \in \mathbb{R}{+}^2$ of inputs of production factors then we say that it is a technologically feasible production process.

Definition 4.3 A production function ${ }^3$ is a mapping $f: \mathbb{R}{+}^2 \rightarrow \mathbb{R}{+}$which assigns maximum quantity $y \in \mathbb{R}{+}$of a product that can be produced when using a vector $\mathbf{x}=\left(x_1, x_2\right) \in \mathbb{R}{+}^2$ of inputs of production factors.

## 经济代写|微观经济学代写Microeconomics代考|Substitutability and Complementarity of Production Factors

Let us use our knowledge of the topic of substitutability of consumer goods to discuss the substitutability of production factors. ${ }^5$
Definition 4.8 Production isoquant is a set:
$$G=\left{\mathbf{x} \in \mathbb{R}{+}^2 \mid f(\mathbf{x})=y=\text { const. } \geq 0\right} \subset \mathbb{R}{+}^2,$$
consisting of all vectors of production factors’ inputs by which the output can be produced at a fixed level $y=$ const. $\geq 0$.

Definition 4.9 A marginal rate of substitution of the first production factor by the second production factor in a vector $\mathbf{x}=\left(x_1, x_2\right) \in \mathbb{R}{+}^2$ of inputs by which the output can be produced at level $y_0=$ const. $>0$ is an expression: $$\sigma{12}\left(x_1, x_2\right)=\lim {\substack{\Delta x_1 \rightarrow 0 \ \Delta x_1<0}} \frac{\Delta x_2}{\Delta x_1} \cong-\frac{\mathrm{d} x_2}{\mathrm{~d} x_1},$$ which describes by approximately how many units one should raise the quantity of the second production factor in a vector $\mathbf{x}=\left(x_1, x_2\right) \in \mathbb{R}{+}^2$ of inputs when input of the first production factor has been reduced by one (notional) unit, in order to keep the output level $y_0=$ const. $>0$ unchanged.

Note $4.5$ The minus sign in Definition $4.9$ results from the fact that the input of the first production factor has been reduced.

# 微观经济学代考

## 经济代写|微观经济学代写Microeconomics代考|Substitutability and Complementarity of Production Factors

$\mathrm{G}=\backslash$ left $\left{\backslash \operatorname{mathbf}{x} \backslash\right.$ in $\backslash \operatorname{mathbb}{\mathrm{R}}{+}^{\wedge} 2 \backslash \operatorname{mid} f(\backslash \operatorname{mathbf}{x})=y=$

$$\sigma 12\left(x_1, x_2\right)=\lim \Delta x_1 \rightarrow 0 \Delta x_1<0 \frac{\Delta x_2}{\Delta x_1} \cong-\frac{\mathrm{d} x_2}{\mathrm{~d} x_1}$$ 中第二个生产要素的数量应该增加多少个单位，当第一 个生产要素的投入减少了一个 (名义上的) 单位，以保 持产出水平 const。不变。 $\mathbf{x}=\left(x_1, x_2\right) \in \mathbb{R}+{ }^2$ $y_0=>0$

$4.9$

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