Review Class Two
Outlines
 A typical consumer will satisfy himself as
much as possible with limited resources.
 How to describe “limited resources”?
 Budget constraint or feasible set
 How to describe “satisfaction”, or say,
“desire”?
 Preference
Outlines
 The central definition of the consumer
theory is demand.
 Three factors of demand are price,
income and preference.
 Demand is the feasible, conditional and
optimal choice.
 The task facing to us is to attain the
demand, so we should grasp the analysis
of preference.
Outlines
 Two key definitions of preference:
 Consumption set
 Preference relation
 From Preference to Utility function
Consumption set
 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.
 Four properties a consumption set should
satisfy.
 Consumption set is the basis of the whole
theory of preference.
Preference relation
 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.
Axiom 1 Completeness:
or y is w.p. to x
x and y.
x is w.p. to y
for any pair of
Preference relation
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.
Rationality is defined as Axiom 1,2 and
3,that is to say, a rational consumer is
able to make a choice and the choices
he makes are consistent.
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 ,which is called the
then we say
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.
Examples
 Perfect substitutes and
perfect complements.
Goods, bads, and neutrals.
Satiation.
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.
Utility function
 If we want to use a continuous U.F. to
describe the given preference ,the case
must satisfy Axiom1,2,3 and the
assumption of Continuity.
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.
Relation between U.F. and IC:
1 Draw a diagonal line and label
each indifference curve with
how far it is from the origin.
Relation between U.F. and IC:
 2 The indifference curves
are
the projections of contours
of
u = u ( x1, x2 ).
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
Relation among Preference, IC and
U.F.
 对于一种特定的偏好，可以用至少一个效用函数
进行刻画，把偏好关系转化为函数关系来讨论，
反过来，对于一个特定的效用函数形式，只能够
描述一种特定的偏好；
 无差异曲线既可以从效用函数得到，也可以从定
义得到，但根本依据是定义。给定特定的效用函
数形式，可以划出无数组无差异曲线束，然而无
差异曲线束的形状却是唯一的；
 一种特定的偏好与一种特定的无差异曲线的形状
一一对应。
 Quasilinear preferences:
All indifference curves are vertically
(or horizontally) shifted copies of a
single one, for example u (x1, x2) = v
(x1) + x2 .
How to find the parameter?
 u (x1, x2) = ax1 + bx2
(perfect substitutes);
 u (x1, x2) = min{ax1, bx2}
(perfect complements).
Thank you!