MU x2 p2
...
MU xn pn
• is the marginal utility of income (“收入”的边 际效用).
13
库恩-塔克条件/松弛互补条件
• When corner solutions are involved, FOCs are complementary-slackness（松弛互补）: L/xi = U/xi - pi = 0 for xi > 0 • and L/xi = U/xi - pi < 0, then xi = 0 • Because it means that
– if MRS is diminishing, then indifference curves are strictly convex (无差异曲线是严格凸的)
• If MRS is not diminishing, then we must check second-order conditions to ensure that we are at a maximum
pi MRS ( x i for x j ) pj
消费者心理上的替代率必须等于市场上的替代率。
12
Interpreting the Lagrangian Multiplier
U / x1 U / x 2 U / x n ... p1 p2 pn

MU x1 p1
1
主要内容
效用最大化 最优化原则
一阶条件 vs. 二阶条件 内点解 vs. 角点解 N 种商品的情形
间接效用函数
一次总付原则
支出函数及其性质
2
Optimization Principle
最优化原则 • To maximize utility, given a fixed amount of income to spend, an individual will buy the goods and services:
px py
dpe of indifferen ce curve
px dy py dx
Quantity of x
U constant
U2
MRS
U constant
6
Second-Order Conditions for a Maximum
• The tangency rule is only necessary but not sufficient unless we assume that MRS is diminishing
• In some situations, individuals’ preferences may be such that they can maximize utility by choosing to consume only one of the goods
Quantity of y
U1 U2 U3
U / x i MU xi pi
Corner Solutions
– any good whose price exceeds its marginal value to the consumer will not be purchased 14
补充：库恩-塔克条件
含有三个变量，两个约束方程的最大化问题
y/x = px/py
• Since + = 1:
pyy = (/)pxx = [(1- )/]pxx
• Substituting into the budget constraint:
I = pxx + [(1- )/]pxx = (1/)pxx
17
C-D Demand Functions
• The higher is the relative price of x (or y), the smaller the share of income spent on x (or y) (比较静态分析)
I py
If all income is spent on y, this is the amount of y that can be purchased
The slope of the constraint is –px/py, i.e. MRS
If all income is spent on x, this is the amount of x that can be purchased
19
CES Demand
• Assume that = 0.5 (original form?)
U(x,y) = x0.5 + y0.5 (单调变换)
• Setting up the Lagrangian:
L = x0.5 + y0.5 + (I - pxx - pyy)
• By FOCs, we get:
构造拉格朗日函数：
库恩-塔克条件为：
15
C-D Demand Functions
• Cobb-Douglas utility function (+ =1):
U(x,y) = xy
• Setting up the Lagrangian:
L = xy + (I - pxx - pyy)
There is a tangency at point A, but U(A)<U(C)<U(B)
B A
1.在凸偏好的情况下，相切是效用最大化 的充分必要条件（最优解可能不唯一）。
C
U2 U1
2.若无差异曲线是严格凸的（MRS递减） ，则效用最大化的最优解唯一。
Quantity of x
8
Corner Solutions 角点解
• Solving for x yields
I x* px
• Solving for y yields
I y* py
py y py I I I py
消费者在商品y上的 支出占收入的比重 18
px x px I , I I px
消费者在商品x上的 支出占收入的比重
5
U1
First-Order Conditions for a Maximum
• By tangency, we get the FOC of maximizing utility, dy/dx at budget line and indifference curve
Quantity of y
slope of budget constraint
I px
Quantity of x
4
First-Order Conditions for a Maximum
• We can add the individual’s utility map to show the utility-maximization process
Quantity of y
C-D Demand Functions
• The Cobb-Douglas utility function is limited in its ability to explain actual consumption behavior. Why?
– =1（回忆CES效用函数的替代弹性）
• A more general functional form might be more useful in explaining consumption decisions
x* I px px [1 ] py
y* I py py [1 ] px
20
CES Demand
• In these demand functions, the share of income spent on either x or y is not a constant
– depends on the ratio of the two prices（px/py）
11
Implications of First-Order Conditions
• For any two goods,
U / xi U / x j U / xi pi pi pj U / x j p j
• Recall that MRSxy=Ux/Uy, so we have an important rule
• The individual’s objective is to maximize
utility = U(x1,x2,…,xn)
s.t.
I = p1x1 + p2x2 +…+ pnxn
• Set up the Lagrangian:
L = U(x1,x2,…,xn) + (I - p1x1 - p2x2 -…- pnxn)
10
The n-Good Case
• First-order conditions for an interior maximum:
L/x1 = U/x1 - p1 = 0 L/x2 = U/x2 - p2 = 0 • • • L/xn = U/xn - pn = 0 L/ = I - p1x1 - p2x2 - … - pnxn = 0
3
The Budget Constraint
预算约束
• Assume that an individual has I dollars to allocate between good x and good y pxx + pyy I→y= I/py-(px/py)x
Quantity of y
A C B