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Mid-exam review class
Consumer theory
 A typical consumer will satisfy himself as
much as possible with limited resources.
 First I should use proper tools to describe
the “limited resources” and “satisfaction”
Budget Set
 Two important definitions:
 The budget set consists of all
bundles that are affordable at the
given prices and income.
 The budget line consists of all
bundles that just exhaust the
consumer’s income.
 You should describe the above two
definitions in arithmetical and
geometrical ways!
Preference
 Def.1 Consumption set is the set which
consists of all consumption bundles, or
say, consumption plans which the typical
consumer desires, regardless whether
they are feasible or not.
 Def.2 If the binary relation defined on the
consumption set satisfies Axiom 1,2 and 3,
then this binary relation is said to be the
Preference relation.
Preference
Axiom 1 Completeness:
or y is w.p. to x
x and y.
x is w.p. to y
for any pair of
Axiom 2 Reflexivity: x is w.p. to x for any
bundle x.
Axiom 3 Transitivity: If x is w.p. to y and
y is w.p. to z, then x is w.p. to z.
Strict preference relation and
indifference relation
 Def.3 If the binary relation defined on the consumption
set satisfies
2
1
1
2 and
x
n
.
w
.
px
x w.px
1
2
then we say x  x
,which is called the strict
preference relation.
Def.4 If the binary relation defined on the consumption
set satisfies
2
1
1
2 and
x
w
.px
x w.px
x1 ~ x 2
then we say
,which is called the
indifference relation,
Indifference curves
 Def. 5 An indifference curve is a set of
consumption bundles with the same
desire level of a representative consumer.
 Now we can conclude that for a
specific preference, there’s a unique
shape of indifference curves
corresponding to it.
 We can use a cluster of IC to describe
a specific preference.
Well-behaved preference
 Def.6 A given preference is called well-behaved
preference if it satisfies Axiom 1,2,3,4 and 5.
 Axiom 4monotonic (meaning
more is better)
 Axiom
5convex
(meaning
average
are
preferred
to
extremes).
Utility function
 Utility function is a way to describe
preference.
 Def. 7 A mapping u: R 2  R is called the

utility function which stands for the preference
relation if the mapping satisfies u
(x)≥u
( y ) if and only if
bundle x is w.p. to bundle y.
Positive Monotonic transformation
 Lemma: Ordinal utility holds that the size
of the utility difference between any two
consumption bundles doesn’t matter.
 So what we care is only the ordinal
represented by the amount of utility
function.
 Positive Monotonic transformation of the
utility function represents the same
preference as the original utility function.
Positive Monotonic transformation
 For a given preference, there’s at least
one utility function to describe it.
u
2 A utility function
 Take a slice at given utility level
 Project down to get contours
U(x1,x2)
The indifference
curve
0
x2
Demand
 Three factors of demand:
 price, income and preference
 Now we can obtain the demand function,
and carry out the comparative statics.
Demand function




Key point:
Lagrangian method
Two properties:
1)
MU
p
i
MU j
 2)

i
pj
MU i MU j


pi
pj
Demand function
 Warning: the two arithmetical properties only
hold for the tangency case.
 In order to obtain the demand, we can also use
the geometrical tools.
 Find the highest IC in the budget set!
 Interior solution: Tangency case and kinky case
 Boundary solution:
 Some examples
Comparative statics(1)
 How does the demand of one good
change corresponding to the change in
income with the prices constant?
 Def. income offer curve and Engle curve
 For different types of goods
 For different types of preferences (H.M)
Comparative statics(1)
 Def. normal goods(p97) and inferior
goods(p98)
 Def. luxury goods and necessary
goods(p101)
 To make the def. more clearly, we can
impose the def. of income elasticity of
demand.
Comparative statics(1)
 Arithmetical form:
dQ
I
Q
Q


dI
dI
I
dQ
 Geometrical meaning:
 The ratio of the slope of the ray and the slope of
the tangent line of some point on the Engle
curve.
Comparative statics(1)
 Strict def.
 Normal goods:
 Inferior goods:
 Luxury goods:
 Necessary goods:
 0
 0
 1
  0,1
Comparative statics(2)
 How does the demand of one good
change corresponding to the change in its
price with the income and price of the
other good constant?
 Def. price offer curve and personal
demand curve
 For different types of goods
 For different types of preferences (H.M)
Comparative statics(2)




Ordinary goods
Giffen goods
Substitutes
Complements
Q1
0
p1
Q1
0
p1
Q2
0
p1
Q2
0
p1
 Distinguish the ordinary and Giffen goods in the
figures of price offer curve and demand curve
画四种曲线的思路
 总原则：直观切入，代数辅助。
 1、为了有好的引导，应先求出两种商品的
需求函数；
 2、根据定义和函数形式开始直观作图；
 3、对比函数形式，检验图像的正确性；
 4、最后要检查曲线（尤其是价格提供曲线
和需求曲线）的起点、终点和渐近线。
注意：
 原点一般来说都会在收入提供曲线和恩格
尔曲线上面
 在分析价格提供曲线和个人需求曲线时，
我们要尤其注意起点、终点和渐近线。而
起点和终点分别是价格趋向于无穷大和零
的情形。对此李杰老师认为这两个极限点
可取可不取；而我认为对于两个极限情形
我们都取空心点，毕竟极限情形是无法达
到的。
Deeper sight
 Two problems about price and income we
should think about seriously:
 We have known How the demand of one
good changes corresponding to the
change in its price with the income and
price of the other good constant, but do
you know the mechanism that forms the
ordinary good or Giffen good? (Ch 8)
 Do you know the source of income? (Ch 9)
Chapter 8
 Slutsky and Hicks decomposition:
 The common ground: the process of the
change in demand corresponding to the
change in the relative price can be
decomposed into substitution effect and
income effect.
Chapter 8
 Def. of substitution effect : the difference
between the new optimal choice and the
original optimal choice with the real income
unchanged and the relative price adjusted.
 Def. of income effect : the difference
between the final optimal choice and the
optimal choice with the real income unchanged
and the relative price adjusted.
Chapter 8
 Slutsky and Hicks decomposition:
 The differentia: the def. of “unchanged real income”
 Slutsky:
 After the change in the relative price, the typical
consumer is still able to buy the original optimal
consumption bundle. (Keep the purchasing
power constant before and after the change in
the relative price.)
Chapter 8
 Hicks:
 After the change in the relative price, the
typical consumer is still able to reach the
same level of satisfaction as the original
optimal consumption bundle. (Keep the
utility constant before and after the
change in the relative price.)
 Figures.
Slutsky decomposition
Hicks decomposition
希克斯分解
Y
U1
U2
E
E'
G
X
O
替代
效应
收入
效应
Chapter 8
 Warning :
 Subs. Effect is non-positive, while the sign of
the income effect isn’t unique, depending on the
type of the good.
 Can we compare the subs. Effects?
 Giffen goods must be the inferior goods, but not
vice versa.
 The law of demand.
 Some examples.
Chapter 8
Chapter 8
Chapter 8
Chapter 8
Chapter 9
 To explain the source of “income”, we impose a
def. of “endowment”, which means the initial
holding of different goods.
 Budget constraint:
 We can see the endowment is always on the
budget line, independent of the relative price.
Chapter 9
Chapter 9
 Figure: Gross demand, net demand or net
supply
 Now in the equilibrium, we should find the
optimal choice (gross demand) and make
it clear that the decision the typical
consumer makes (net demand or net
supply).
Figure 1
Y
A'
U"
U'
U
A
W
A"
E'
E
E"
W'
W"
O
B"
B
用
B'
X
Figure 2
初始禀赋改变Ⅱ
Y
U
A
XW
W
E
Y
W'
X'W
O
X
YW
X
Y'W
B
Chapter 9
 Comparative
statics:
 Income(1
and 2) figure
 Price :figure
Chapter 9
 How to obtain the price offer curve?
 Endowment point must be on the price
offer curve, and the line through w and a,
which is another point on the price offer
curve is the budget line corresponding to
a.
 The area which the price offer curve must
be in.
提供曲线
Y
l2
效用增加
A
此处效用最小
l1
W
效用增加
U
O
X
B
Chapter 9
 From personal gross demand curve to net
demand or supply curve (Def.) (figure)
 Two examples: (figures)
 Impose a quantity tax on the consumption
of good 1
 Analysis of the labor supply curve (from
the price offer curve to the labor supply
curve)
个人净需求曲线
个人总需求曲线
PX
个人净供给曲线
PX
PX
A
P
O
X
O
X
WX
B
O
X
劳动供给曲线的推导
I
W2
E3
W1
E2
W0
W
R
O
W
W2
A3
A2
W1
A1
W0
O
L
L2
L1
Applications of two-good model
Summary：
 在分析具体背景之后，要找准应该把那些因素抽
象为两种商品。这个是分析问题的突破口。
 确定主旨句：这个经济主体要在一定的约束条件
下追求自身利益的最大化。
 接下来就做翻译工作，刻画好“约束”和“自身
利益”，正如前面所述，这就是你们分析的出发
点。
 展开相应的几何分析或代数推导，建议两者联合
运用。
 最后得到我们想要的“最优选择”，有需要还要
进行比较静态分析。
Outlines
 Applications of two-good model with
endowment point
 Intertemporal choices (Fisher)
 Insurance market (Stiglitz)
Intertemporal choices
 The key sentence:
 A typical and forward-looking consumer
will make a consumption decision to
maximize his lifetime utility with the
intertemporal budget constraint.
 How to describe the intertemporal budget
constraint and the preference?
Stiglitz: how does the Insurance
market operates?
 You’re the risk averser and have a risky asset:
(W1 ,W2 , P)
 You can go into the insurance market and spread
your risk:
(W1  K ,W2  (1   ) K )
 Questions:1 how to fix the premium?

2 what level of insured amount should
you choose?
Chapter 14
 Here we have discussed the two-good
model, and we have learned how to use
this model to solve the optimal problem
facing a typical agent.
 But why does a typical consumer come
into a market, or say, take part in the
transaction?
 Because a consumer will earn the
transaction benefits from it.
Chapter 14
 The transaction benefits a typical
consumer will earn in the market is
considered as consumer’s surplus.
 Def. of (net) consumer’s surplus: a
buyer’s willingness to pay minus the
amount the buyer actually pays.
 Figures
 Arithmetical way（这是定积分的运用，一
定会考核的内容）
Chapter 14
Chapter 14
Change in the price
Chapter 14
Producer’s surplus
 Def. of producer’s surplus: the amount a
seller is paid for a good minus the seller’s
variable cost.
 Figures
 Arithmetical way
Chapter 14
Producer’s surplus
Chapter 14
Market surplus
 Market (total) surplus=Consumer’s
surplus + Producer’s surplus
 Market surplus can reflect the transaction
benefits earned by the consumer and the
producer.
 Why will the transaction happen?
 The answer is transaction benefits.
Chapter 14
Market surplus
Chapter 15
Outlines
 Horizontal summation------- Competitive
equilibrium
 Two important elasticity:
 Income elasticity of demand
 Price elasticity of demand
 Taxes
Chapter 15
Horizontal summation
 The market is made up of many agents, so in
order to obtain the market demand or supply we
can use the method of Horizontal summation.
 Method: You should draw the rough figure and
find the kinky point, and then you can calculate
the functional form with some different domains.
At last you should make “price” the independent
variable.
 (这是一定要考核的内容）
Chapter 15
Horizontal summation
Chapter 15
Competitive equilibrium
 In the perfectly competitive market, we
can reach the competitive equilibrium,
which is the intersection of market supply
and demand curve.
Key points of Chapter 1
 Welfare economics:
 the study of how the allocation of
resources affects economic well-being
 Two very important definitions about the
welfare economics:
Key points of Chapter 1
A Pareto improvement is a change to make some people
better off without hurting anybody else.
An economic situation is Pareto efficient if there is no way to
make any person better off without hurting anybody else.
 Pareto efficiency or
Pareto optimal :
 if there is already no way to make any more
Pareto improvement.
Key points of Chapter 1
 Competitive market
 Monopoly
 Shortage
Chapter 15
Price elasticity of demand
 Def. of Elasticity: a measure of the
responsiveness of quantity demanded or
quantity supplied to one of its determinants
 Def. of Price elasticity of demand: a measure
of how much the quantity demanded of a good
responds to a change in the price of that good,
computed as the percentage change in quantity
demanded divided by the percentage change in
price.
Chapter 15
Price elasticity of demand





dQ
Arithmetical form:
P Q
Q
P 

dP
dP dQ
P
Geometrical meaning: the slope of the ray over the
slope of the tangent line of one point on the
demand curve.
Warning:
Price elastic is usually negative.
Price or income elasticity is the point elasticity.
Chapter 15
Application of price elastic:
Analysis of revenue
 Suppose the producer is a monopoly, and
then he can control the price or quantity in
order to affect the revenue. (Can he
control the price and quantity at the same
time?)
 1 He controls the price
 2 He controls the quantity
 1
 2
Chapter 15
Application of price elastic:
dR Analysis of revenue
 Q( P)(1   P )
dP
dR
1
 MR(Q)  P(Q)(1 
)
dQ
P
Taxes
 Here we only consider the quantity tax
 Two important conclusions:
 In the case of quantity tax the values of
Pp and Pr, and the relation Pp=Pr+T are
independent of tax objects.
 The steeper the curve is, the more tax
burden, and vice versa.
Deadweight loss
考试注意
 七道大题，里面会有小题，所以要抓紧时间。
 中英文答题皆可。不会在语言上扣分。
 请大家答题时千万不要长篇大论，在开头就明确
表明自己观点。图形要清晰（很关键），代数推
导会帮你省墨水的。一句话：“少说废话。”
 你们千万不要用套模型的观念去做题，一定要根
据具体的问题展开。如果你碰到难以攻克的地方，
请你思考定义。定势思维会很吃亏的！