6 商品间的需求关系

x x py py x y I
income effect (-) if x is normal
U constant
substitution effect (+)
combined effect (ambiguous)
8
Cont’d
• To prove that cross-price substitution effect must be positive under Hicksian demand curve: • Because MRS= y/ x=(y/ py)(py / x)<0, and y/ py<0, py / x>0, i.e., x / py>0.
– two goods are gross substitutes if xi /pj > 0 – two goods are gross complements if xi /pj < 0
总替代品与总互补品的概念强调价格变化后的总效应。
5
Gross Substitutes
Quantity of y
• 希克斯第二需求定律（了解）
– 大多数商品是替代关系
19
总定义的非对称性源于收入效应的存在。
11
Asymmetry of the Gross Definitions
• Suppose a quasi-linear utility function（拟 线性效用） for two goods is given by
U(x,y) = ln x + y Figure?
总定义的非对称性 • One undesirable characteristic of the gross definitions of substitutes and complements is that they are not symmetric • It is possible for x1 to be a substitute for x2 and at the same time for x2 to be a complement of x1 • Any example: a leader and a driver
xi p j
U constant
x j pi
U constant
15
Substitutability with Many Goods
• Once the utility-maximizing model is extended to many goods, a wide variety of demand patterns become possible • According to Hicks’ second law of demand, “most” goods must be substitutes.
3
Substitutes and Complements
• Two goods are substitutes if one good may replace the other in use
– examples: tea & coffee, butter & margarine
• Two goods are complements if they are used together
x i p j 0
U constant
– two goods are net complements if
x i p j 0
U constant
14
Net Substitutes and Complements
• This definition looks only at the shape of the indifference curve • This definition is perfectly symmetric, according to Shephard’s lemma and Young’s theorem（杨格定理）
Chapter 6
DEMAND RELATIONSHIPS AMONG GOODS 商品间的需求关系
Slides prepared by Linda Ghent (Eastern Illinois University) Modified by FANG Mingyue (Capital University of Economics and Business)
– this implies that the change in the price of any good induces income and substitution effects that may change the quantity of every good demanded
10
Asymmetry of the Gross Definitions
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Net Substitutes and Complements
• The concepts of net substitutes and complements focuses solely on substitution effects (利用希克斯需求函数来定义)
– two goods are net substitutes if
U1 U0
y1 y0
x/py < 0
Quantity of x
7Baidu Nhomakorabea
x0 x1
py 下降
A Mathematical Treatment
• The change in x caused by changes in py can be shown by a Slutsky-type equation
U1
y1
y0
x/py > 0
Quantity of x
6
U0 x1 x0
py 下降
Gross Complements
Quantity of y
For nearly L-shaped curve, there is little of substitution effect and some income effect, so x increases In this case, we call x and y gross complements
– examples: coffee & sugar, fish & chips
Is there any relationships that are both substituting and complementary?
4
Gross Substitutes and Complements
• The concepts of gross substitutes and complements include both substitution and income effects
9
Gross Substitutes and Complements
• For the case of many goods, we can generalize the Slutsky analysis
x i x i p j p j x i xj I
U constant
or ei,j=eci,j-sjei,I
1
主要内容
• 两种商品的情形 • 总替代品与总互补品 • 净替代品与净互补品 • 希克斯第二需求定律
2
The Two-Good Case
• From Px->x to Py->x • Consider two goods case: if Py, what happens to x? • It depends on the relationship.
• Since the negativity of the substitution effect implies that eiic 0, it must be the case that c
e
j i
ij
0
一种商品对于其他所有商品的补偿性交叉价格弹性 非负，即大多数商品是（净）替代品。 18
x ic x ic x ic p1 p2 ... pn 0 p1 p2 pn
17
Substitutability with Many Goods
• In elasticity terms, we get
e e ... e 0
c i1 c i2 c in
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Substitutability with Many Goods
• To prove this, we can start with the compensated demand function
xc(p1,…pn,V)
• Applying Euler’s theorem, function yields
Important Points to Note
x x c x y • 交叉价格效应的斯拉茨基方程 p y p y I • 总替代品与总互补品的数学定义
• 总替代品与总互补品定义的非对称性
– 源于收入效应的模糊性和相对大小的不确定性
• 净替代品与净互补品的数学定义
c xi c x j • 净替代品与净互补品的定义对称（ p p ） j i
• Setting up the Lagrangian
L = ln x + y + (I – pxx – pyy)
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Cont’d
• Applying FOCs and budget constraint, we can find the Marshallian demand for x and y • x=py /px y = (I – py)/py => pyy/I=1-py /I => (1) x/py >0 and (2) y/px=0 • (1) x and y are gross substitutes • (2) but spending on y is independent of px , so they are neither gross substitutes nor gross complements
For nearly linear curve, substitution effect is so large but income effect is little, so there is less x and more y In this case, we call x and y gross substitutes