Chapter Three
Preferences
偏好
Structure



3.1 Preference relations
3.2 Assumptions about preferences
3.3 Graphical representation of
preferences—indifference curve (无差异曲线)



3.3.1 definitions and its properties
3.3.2 Slope and marginal rate of substitution (边
际替代率)
3.3.3 Shapes and different preferences
Rationality in Economics


Behavioral Postulate:
A decision-maker always chooses its most
preferred alternative from its set of available
alternatives.
So to model choice we must model decisionmakers’ preferences.
3.1 Preference Relations
Comparing two different consumption
bundles, x and y:



strict preference (严格偏好): more preferred (x
y).
Indifference (无差异): exactly as preferred as
(x~y).
weak preference (弱偏好): at least as preferred
as (x f y).
~
p

3.2 Assumptions about Preference
Relations





1) Completeness
2) Reflexivity
3) Transitivity
4) Monotone
5) Convexity
1) Completeness (完备性)

For any two bundles x and y it is always
possible to make the statement that either
x f
y
~
or
yf
~ x
or both.
2) Reflexivity (反身性)

Any bundle x is always at least as preferred
as itself; i.e.
x
f x.
~
3) Transitivity (传递性)


If
x is at least as preferred as y, and
y is at least as preferred as z, then
x is at least as preferred as z; i.e.
f z
xf
y
and
y
x
z.
f
~
~
~
Rational preferences: 1) +3).
4) Monotone (单调性)


Monotone: More of any commodity is always
preferred.
The commodities that consumers prefer less
of are called bads.
5) Convexity (凸性)

Convexity: Mixtures of bundles are (at least
weakly) preferred to the bundles themselves.
Strict Convexity.
x
x2
x+y is strictly preferred
z=
2 to both x and y.
x2+y2
2
y
y2
x1
x1+y
1
2
y1
Strict Convexity.
x
x2
z =(tx1+(1-t)y1, tx2+(1-t)y2)
is preferred to x and y
for all 0 < t < 1.
y
y2
x1
y1
Strict Convexity.
x
x2
y2
x1
Preferences are strictly convex
when all mixtures z
are strictly
z
preferred to their
component
bundles x and y.
y
y1
Weak Convexity.
Preferences are
weakly convex if at
least one mixture z
is equally preferred
to a component
bundle.
x’
z’
x
z
y
y’
Non-Convex Preferences
x2
The mixture z
is less preferred
than x or y.
z
y2
x1
y1
More Non-Convex Preferences
x2
The mixture z
is less preferred
than x or y.
z
y2
x1
y1
Well-Behaved Preferences

A preference relation is “well-behaved” if it is

Monotonic and convex.
3.3 Indifference Curve
3.3.1 Definitions and Properties

Indifference curve: a curve representing all
combinations of goods that provide a
consumer with the same level of utility.
Indifference Curves
x2
x’ ~ x” ~ x”’
x’
x”
x”’
x1
Indifference Curves
p
x
z
x
z
y
x1
p
x2
y
Indifference Curves
I1
x2
All bundles in I1 are
strictly preferred to
all in I2.
x
z
I2
y
I3
All bundles in I2 are
strictly preferred to
all in I3.
x1
Indifference Map

There are many indifference curves that represent
different utility levels.

Your satisfaction increases as you move to higher
(i.e., to the northeast) indifference curves.

Why?
Weakly Preferred Set (弱偏好集)
x2
WP(x), the set of
x bundles weakly
preferred to x.
I(x)
I(x’)
x1
Strictly Preferred Set (严格偏好集)
x2
SP(x), the set of
x bundles strictly
preferred to x,
does not
include
I(x)
I(x).
x1
Indifference curve cannot intersect
Proof by contradiction
Movie

B
A
U2
C
U1
CD
Well-behaved Indifference Curve

If preferences are monotonic and (strict)
convex, then the indifference curve that
represents the preferences is well-behaved.
Well-behaved Indifference Curve
x2
x1
3.3.2 Slope of indifference curve





The slope of indifference curve measures the
marginal rate of substitution (MRS).
MRS: The amount of good x2 that a
consumer would give up to obtain one more
unit of good x1 while holding utility constant.
MRS=- D x2 /D x1---is positive
The slope is D x2 /D x1---is negative.
Different from the book.
Marginal Rate of Substitution
x2
D x2
x’
MRS at x’ is
lim {Dx2/Dx1}
Dx1
0
= dx2/dx1 at x’
D x1
x1
Marginal Rate of Substitution
x2
dx2 x’
dx1
dx2 = MRS*dx1 so, at x’, MRS is
the rate at which the consumer
is only just willing to exchange
commodity 2 for a small
amount of commodity 1.
Marginal willingness to pay.
x1
Slope and goods
Good 2
Slope < 0.
Good 1
Good and bad
Good 2
Slope > 0.
Bad 1
Diminishing MRS

Diminishing MRS: There
is a decline in the
amount of good x2 that
the consumer will give up
for an additional unit of
good x1.

Why diminishing?
Movies
A
B
C D
CDs
3.3.3 The shapes of indifference curves

The shape of an indifference curve
describes a consumer‘s willingness to
subtitute one good for another.

What would a movie lover’s indifference
curves look like?
Movie Fanatic

Movies
U3
U2
U1
CD
CD Fanatic

Movies
U3
U1
U2
CD
Extreme Cases of Indifference Curves




Perfect substitutes (完全替代)
Perfect complements (完全互补)
Satiation (饱和)
Discrete goods (离散商品)
Perfect substitutes and perfect
complements





Perfect substitutes: a consumer is willing to
substitute one good for the other at a constant rate.
MRS is (slope =).
Total bundles matter.
Perfect complements: Two goods that are always
bought in the same ratio.
MRS is
. Indifference curves are shaped as
right angles.
Perfect Substitutes
x2
15 I2
8
I1
Bundles in I2 all have a total
of 15 units and are strictly
preferred to all bundles in
I1, which have a total of
only 8 units in them.
x1
8
15
Perfect Complements


If a consumer always consumes commodities
1 and 2 in fixed proportion (e.g. one-to-one),
then the commodities are perfect
complements .
The number of pairs determines the
preference rank-order of bundles.
Extreme Cases of Indifference Curves;
Perfect Complements
x2
45o
9
5
Each of (5,5), (5,9)
and (9,5) contains
5 pairs so each is
equally preferred.
I1
5
9
x1
Extreme Cases of Indifference Curves;
Perfect Complements
x2
Since each of (5,5),
(5,9) and (9,5)
contains 5 pairs,
each is less
I2 preferred than the
bundle (9,9) which
I1 contains 9 pairs.
45o
9
5
5
9
x1
Satiation


A bundle strictly preferred to any other is a
satiation point or a bliss point.
What do indifference curves look like for
preferences exhibiting satiation?
Indifference Curves Exhibiting
Satiation
x2
Satiation
(bliss)
point
x1
Indifference Curves Exhibiting
Satiation
x2
Better
Satiation
(bliss)
point
x1
Indifference Curves Exhibiting
Satiation
x2
Better
Satiation
(bliss)
point
x1
Monotone and Satiation

What is the relationship between monotone
and Satiation?
Discrete Commodities


A commodity is infinitely divisible if it can be
acquired in any quantity; e.g. water or sugar.
A commodity is discrete if it comes in unit
lumps of 1, 2, 3, … and so on; e.g. cars,
ships and refrigerators.
Indifference Curves for Discrete
Commodities

Suppose commodity 2 is an infinitely divisible
good (sugar) while commodity 1 is a discrete
good (car). What do indifference “curves”
look like?
Indifference Curves With a Discrete
Good
Sugar
Indifference “curves”
are collections of
discrete points.
0
1
2
3
4 Car