ch04

2
A Numerical Illustration数例说明
• Assume that the individual’s MRS = 1 假定个人的MRS=1
– willing to trade one unit of x for one unit of y 愿意用一单位的x换取 一单位的y
• Suppose the price of x = $2 and the price of y = $1 假定x的 价格为$2，y的价格为$1 • The individual can be made better off trade 1 unit of x for 2 units of y in the marketplace如果在市场中,用1单位x换取2单位的 y，个人的境况可以变得更好
I = p1x1 + p2x2 +…+ pnxn
• Set up the Lagrangian建立拉格朗日函数：
L = U(x1,x2,…,xn) + (I - p1x1 - p2x2 -…- pnxn)
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The n-Good Case n种商品的情况
• First-order conditions for an interior maximum 内部最大值的 一阶条件为： L/x1 = U/x1 - p1 = 0 L/x2 = U/x2 - p2 = 0
• • • L/xn = U/xn - pn = 0 L/ = I - p1x1 - p2x2 - … - pnxn = 0
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Implications of First-Order Conditions 一阶条件的含义
• For any two goods对于任意两种商品
U / xi pi U / x j p j
• Add the individual’s utility map to show the utility-maximization process 加入个人的效用函数图，来说明效用最大化的过程
Quantity of y
A
The individual can do better than point A by reallocating his budget 通过重新分配他的预算，个人的情况可以变得比点A处更好 The individual cannot have point C because income is not large enough 个人不能达到点C，因为其收入不够
I py
If all income is spent on y, this is the amount of y that can be purchased 如果所有的收入用于y ，这是所能购 买的y的数量
The individual can afford to choose only combinations of x and y in the shaded triangle 个人只能够负担阴影三角形中的x和y的组合
L = U(x1,x2) + (I - p1xst-Order Conditions for a Maximum最大值的一阶条件
• First-order conditions for an interior maximum 内部最大值的 一阶条件为：
L/x1 = U/x1 - p1 = 0
– Similar reasoning applies whenever the MRS and the price ration differ同样推论适用于任何MRS与价格比不等的情况 – The condition for maximum utility must be the equality of these two 效用最大化的条件一定是这两者相等
Quantity of y
U1 U2 U3
At point A, the indifference curve is not tangent to the budget constraint 在点A，无差异曲线与预算约束线并不相切
Utility is maximized at point A 在点A处，效用达到最大化 A
Chapter 4 第四章
UTILITY MAXIMIZATON AND CHOICE 效用最大化和选择
1
Optimization Principle 最优化准则
• To maximize utility, given a fixed amount of income to spend, an individual will buy the goods and services为了最大化效 用，给定一定数量的收入，个人会购买商品和服务，直 到:
Quantity of x
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The n-Good Case n种商品的情况
• The individual’s objective is to maximize 个人的目标为最大化
utility = U(x1,x2,…,xn) subject to the budget constraint 受限于预算约束
• The individual’s objective is to maximize 个人的目标为最大化 utility = U(x1,x2) subject to the budget constraint 受限于预算约束 I = p1x1 + p2x2 • Set up the Lagrangian建立拉格朗日函数：
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Interpreting the Lagrangian Multiplier 拉格朗日乘子的含义
• At the margin, the price of a good represents the consumer’s evaluation of the utility of the last unit consumed在边际的概念上，商品的价格代表了消费者 对所消费的最后一个单位的效用的评价
C
B
U3 U2 U1 Point B is the point of utility maximization 点B是效用最大化的点
Quantity of x
5
First-Order Conditions for a Maximum 最大值的一阶条件
• Utility is maximized where the indifference curve is tangent to the budget constraint 效用在无差异曲线与预算约束线相切处，达到最大
B A
U2 U1
Quantity of x
10
Corner Solutions角解
• In some situations, individuals’ preferences may be such that they can maximize utility by choosing to consume only one of the goods 有些时候，个人的偏好可能使得他们可以在只消费一种商品时，达到 效用最大化
• The tangency rule is only a necessary condition 相切原则仅仅 是必要条件
– we need MRS to be diminishing 我们需要MRS是递减的
Quantity of y There is a tangency at point A, but the individual can reach a higher level of utility at point B 点A为一个切点，但个人可以在点B获得更高的效用
Quantity of y
slope of budget constraint
px py
dy dx U constant
B
slope of indifferen ce curve
U2
px dy MRS py dx U constant
Quantity of x
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First-Order Conditions for a Maximum 最大值的一阶条件
L/x2 = U/x2 - p2 = 0
U / x1 p1 U / x2 p2
• This implies that at the optimal allocation of income
这意味着在最优的收入分配处
p1 MRS (x1 for x2 ) p2
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Second-Order Conditions for a Maximum 最大值的二阶条件
– how much the consumer is willing to pay for the last unit 消费者愿意为最后一个单位 商品支付多少钱
pi
MUxi
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Corner Solutions角解
– that exhaust his or her total income 耗尽他（她）所有的收入 – for which the psychic rate of trade-off between any goods (the MRS) is equal to the rate at which goods can be traded for one another in the marketplace 任意商品间进行交换的比率（MRS）与商品间在 市场中交换的比率相同
• This implies that at the optimal allocation of income 这意味着在最优的收入分配处
pi MRS ( xi for x j ) pj
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Interpreting the Lagrangian Multiplier 拉格朗日乘子的含义