# 经济代写|微观经济学代写Microeconomics代考|ECON106

## 经济代写|微观经济学代写Microeconomics代考|GRAPHICAL CHARACTERIZATION OF THE SOLUTION

Figure $7.2$ shows two isocost lines and the isoquant corresponding to $Q_0$ units of output. The solution to the firm’s cost-minimization problem occurs at point $A$, where the isoquant is just tangent to an isocost line. That is, of all the input combinations along the isoquant, point $A$ provides the firm with the lowest level of cost.
To verify this, consider other points in Figure $7.2$, such as $E, F$, and $G$ :

• Point $G$ is off the $Q_0$ isoquant altogether. Although this input combination could produce $Q_0$ units of output, in using it the firm would be wasting inputs (i.e., point $G$ is technically inefficient). This point cannot be optimal because input combination $A$ also produces $Q_0$ units of output but uses fewer units of labor and capital.
• Points $E$ and $F$ are technically efficient, but they are not cost-minimizing because they are on an isocost line that corresponds to a higher level of cost than the isocost line passing through the cost-minimizing point $A$. By moving from point $E$ to $A$ or from $F$ to $A$, the firm can produce the same amount of output, but at a lower total cost.
Note that the slope of the isoquant at the cost-minimizing point $A$ is equal to the slope of the isocost line. In Chapter 6, we saw that the negative of the slope of the isoquant is equal to the marginal rate of technical substitution of labor for capital,
• $\operatorname{MRTS}{L, K}$, and that $M R T S{L, K}=M P_L / M P_{K^*}$. As we just illustrated, the slope of an isocost line is $-w / r$. Thus, the cost-minimizing condition occurs when:
• slope of isoquant $=$ slope of isocost line
• • \begin{aligned} • -M R T S_{L, K} &=-\frac{w}{r} \ • \frac{M P_L}{M P_K} &=\frac{w}{r} • \end{aligned} •
• ratio of marginal products $=$ ratio of input prices
• In Figure 7.2, the optimal input combination $A$ is an interior optimum. An interior optimum involves positive amounts of both inputs $(L>0$ and $K>0)$, and the optimum occurs at a tangency between the isoquant and an isocost line. Equation (7.1) tells us that at an interior optimum, the ratio of the marginal products of labor and capital equals the ratio of the price of labor to the price of capital. We could also rewrite equation (7.1) to state the optimality condition in this form:
• $$• \frac{M P_L}{w}=\frac{M P_K}{r} •$$
• Expressed this way, this condition tells us that at a cost-minimizing input combination, the additional output per dollar spent on labor services equals the additional output per dollar spent on capital services. Thus, if we are minimizing costs, we get equal “bang for the buck” from each input. (Recall that we obtained a similar condition at the solution to a consumer’s utility-maximization problem in Chapter 4.)
• To see why equation (7.2) must hold, consider a non-cost-minimizing point in Figure 7.2, such as $E$. At point $E$, the slope of the isoquant is more negative than the slope of the isocost line. Therefore, $-\left(M P_L / M P_K\right)<-(w / r)$, or $M P_L / M P_K>w / r$, or $M P_L / w>M P_{\Lambda} / r$

## 经济代写|微观经济学代写Microeconomics代考|CORNER POINT SOLUTIONS

In discussing the theory of consumer behavior in Chapter 4 , we studied corner point solutions: optimal solutions at which we did not have a tangency between a budget line and an indifference curve. We can also have corner point solutions to the cost-minimization problem. Figure $7.3$ illustrates this case. The cost-minimizing input combination for producing $Q_0$ units of output occurs at point $A$, where the firm uses no capital.

At this corner point, the isocost line is flatter than the isoquant. Mathematically, this says $-\left(M P_L / M P_K\right)<-(w / r)$, or equivalently, $M P_L / M P_K>w / r$. Another way to write this would be
$$\frac{M P_L}{w}>\frac{M P_K}{r}$$
Thus, at the corner solution at point $A$, the marginal product per dollar spent on labor exceeds the marginal product per dollar spent on capital services. If you look closely at other points along the $Q_0$ unit isoquant, you see that isocost lines are always flatter than the isoquant. Hence, $M P_L / w>M P_{\mathrm{L}} / r$ holds for all input combinations along the $Q_0$ isoquant. A corner solution at which no capital is used can be thought of as a response to a situation in which every additional dollar spent on labor yields more output than every additional dollar spent on capital. In this situation, the firm should substitute labor for capital until it uses no capital at all, as illustrated in Learning-By-Doing Exercise 7.3.

# 微观经济学代考

## 经济代写|微观经济学代写Microeconomics代考|GRAPHICAL CHARACTERIZATION OF THE SOLUTION

• 观点G关闭了问0完全等量。虽然这种输入组合可以产生问0产出单位，在使用它时，公司会浪费投入（即点G技术上效率低下）。这一点不是最优的，因为输入组合一个也生产问0产出单位，但使用较少单位的劳动力和资本。
• 积分和和F在技​​术上是有效的，但它们不是成本最小化的，因为它们位于等成本线上，该等成本线对应于比通过成本最小化点的等成本线更高的成本水平一个. 通过从点移动和至一个或来自F至一个，企业可以生产相同数量的产出，但总成本较低。
请注意，等产量线在成本最小化点的斜率一个等于等成本线的斜率。在第六章，我们看到等产量线斜率的负值等于劳动力对资本的边际技术替代率，
• $\operatorname{MRTS} {L, K},一个nd吨H一个吨MRTS {L, K}=M P_L / M P_{K^*}.一个秒在和j在秒吨一世升升在秒吨r一个吨和d,吨H和秒升欧p和欧F一个n一世秒欧C欧秒吨升一世n和一世秒-w/r$。因此，成本最小化条件发生在：
• 等产量斜率=等成本线的斜率
• $$• \begin{对齐} • -MRT S_{L, K} &=-\frac{w}{r} \ • \frac{M P_L}{M P_K} &=\frac{w}{r} • \结束{对齐} •$$
• 边际产品比率=投入价格比率
• 在图 7.2 中，最优输入组合一个是内部最优。内部最优涉及两个输入的正数(大号>0和钾>0)，并且最优出现在等产量线和等成本线之间的相切处。等式（7.1）告诉我们，在内部最优条件下，劳动力和资本的边际产品之比等于劳动力价格与资本价格之比。我们还可以重写等式 (7.1) 以这种形式陈述最优条件：
• $$• \frac{M P_L}{w}=\frac{M P_K}{r} •$$
• 以这种方式表达，这个条件告诉我们，在成本最小化的投入组合中，花在劳务上的每一美元的额外产出等于花在资本服务上的每一美元的额外产出。因此，如果我们正在最小化成本，我们会从每个输入中获得相等的“物超所值”。（回想一下，我们在第 4 章中解决消费者效用最大化问题时得到了类似的条件。）
• 要了解为什么等式 (7.2) 必须成立，请考虑图 7.2 中的非成本最小化点，例如和. 在点和，等产量线的斜率比等成本线的斜率更负。所以，−(米P大号/米P钾)<−(在/r)， 或者米P大号/米P钾>在/r， 或者米P大号/在>米P大号/r

## 经济代写|微观经济学代写Microeconomics代考|CORNER POINT SOLUTIONS

$$\frac{M P_L}{w}>\frac{M P_K}{r}$$

$M P_L / w>M P_{\mathrm{L}} / r$ 对所有输入组合都成立 $Q_0$ 等量。一个不使用资本的角落解决方案可以被认为是对 这样一种情况的回应，在这种情况下，每多花一美元在劳动力上比每多花一美元在资本上产生更多的产 出。在这种情况下，公司应该用劳动力代替资本，直到完全不使用资本，如边做边学练习 $7.3$ 所示。

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