Chapter Three
Preferences
消费者偏好
Where Are We in the Course?
We are studying the 1st of the three
blocks of microeconomics,Consumer
behavior,production theory,and market
equilibrium
Within the 1st block,we are working on
the 2nd of the three components,choice
set,preference,and consumer demand
What Do We Mean by
Preference? (偏好 )
It refers to the ordered relationship
among alternative choices given by an
economic agent.
In most economic literature,consumer
preference is treated as the ultimate
exogenous element.
Preference Relations
Comparing two different consumption
bundles,x and y,
–strict preference,x is more preferred
than is y.
–weak preference,x is as at least as
preferred as is y.
–Indifference,x is exactly as preferred
as is y.
Notations
denotes strict preference;
~ denotes indifference;
denotes weak preference;
p
~f
Preference Relations
x y and y x imply x ~ y.
x y and (not y x) imply x y.
~f ~f
~f ~f
p
Assumptions about Preference
Relations
Completeness,For any two bundles
x and y it is always possible to make
the statement that either
x y
or
y x.
~f
~f
Assumptions about Preference
Relations
Reflexivity,Any bundle x is always
at least as preferred as itself; i.e.
x x.~f
Assumptions about Preference
Relations
Transitivity,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.
x y and y z x z.~f ~f ~f
Indifference Curves
无差异曲线 (或,无差异集 )
Take a reference bundle x’,The set
of all bundles equally preferred to x’
is the indifference curve containing
x’; the set of all bundles y ~ x’.
Since an indifference,curve” is not
always a curve a better name might
be an indifference,set”.
Indifference Curves
x2
x1
x”
x”’
x’ ~ x” ~ x”’x’
Indifference Curves
x2
x1
z x yp px
y
z
Indifference Curves
x2
x1
x
All bundles in I1 are
strictly preferred to
all in I2.
y
z
All bundles in I2 are
strictly preferred to
all in I3.
I1
I2
I3
Indifference Curves
x2
x1
WP(x),the set of
bundles weakly
preferred to x.
WP(x)
includes
I(x).
x
I(x)
Indifference Curves
x2
x1
SP(x),the set of
bundles strictly
preferred to x,
does not
include
I(x).
x
I(x)
Indifference Curves Cannot
Intersect ! (不相交 !)
x2
x1
x
y
z
I1 I2
From I1,x ~ y,From I2,x ~ z.
Therefore y ~ z.
Indifference Curves Cannot
Intersect
x2
x1
x
y
z
I1 I2
From I1,x ~ y,From I2,x ~ z.
Therefore y ~ z,But from I1
and I2 we see y z,a
contradiction.
p
Slopes of Indifference Curves
When more of a commodity is always
preferred,the commodity is a good.
If every commodity is a good then
indifference curves are negatively
sloped.
Slopes of Indifference Curves
Good 2
Good 1
Two goods
a negatively sloped
indifference curve.
Slopes of Indifference Curves
If less of a commodity is always
preferred then the commodity is a
bad.
Slopes of Indifference Curves
Good 2
Bad 1
One good and one
bad a
positively sloped
indifference curve.
Extreme Cases of Indifference
Curves; Perfect Substitutes
If a consumer always regards units
of commodities 1 and 2 as
equivalent,then the commodities are
perfect substitutes and only the sum
of the two commodities in bundles
determines their preference rank-
order,
Extreme Cases of Indifference
Curves; Perfect Substitutes
x2
x18
8
15
15 Slopes are constant at - 1.I2
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.
Extreme Cases of Indifference
Curves; 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 and only the number of
pairs of units of the two commodities
determines the preference rank-order
of bundles,
Extreme Cases of Indifference
Curves; Perfect Complements
x2
x1
I2
I1
45o
5
9
5 9
Since each of (5,5),
(5,9) and (9,5)
contains 5 pairs,
each is less
preferred than the
bundle (9,9) which
contains 9 pairs.
Preferences Exhibiting 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
x1
Bet
ter
Satiation
(bliss)
point
Indifference Curves for Discrete
Commodities
A commodity is infinitely divisible if
it can be acquired in any quantity;
e.g,water or cheese.
A commodity is discrete if it comes
in unit lumps of 1,2,3,… and so on;
e.g,aircraft,ships and refrigerators.
Indifference Curves for Discrete
Commodities
(study this yourself)
Suppose commodity 2 is an infinitely
divisible good (gasoline) while
commodity 1 is a discrete good
(aircraft),What do indifference
“curves” look like?
Indifference Curves With a
Discrete Good
(study this yourself)
Gas-
oline
Aircraft0 1 2 3 4
Indifference,curves”
are collections of
discrete points.
Well-Behaved Preferences
A preference relation is,well-
behaved” if it is
–monotonic and convex.
Monotonicity,More of any
commodity is always preferred (i.e.
no satiation and every commodity is
a good).
Well-Behaved Preferences
Convexity (凸性 ),Mixtures of bundles
are (at least weakly) preferred to the
bundles themselves,E.g.,the 50-50
mixture of the bundles x and y is
z = (0.5)x + (0.5)y.
z is at least as preferred as x or y,
Well-Behaved Preferences --
Convexity.
x2
y2
x1 y1
x
y
z =(tx1+(1-t)y1,tx2+(1-t)y2)
is preferred to x and y
for all 0 < t < 1.
Well-Behaved Preferences --
Convexity.
x2
y2
x1 y1
x
y
Preferences are strictly convex
when all mixtures z
are strictly
preferred to their
component
bundles x and y.
z
Well-Behaved Preferences --
Weak Convexity.
x’
y’
z’
Preferences are
weakly convex if at
least one mixture z
is equally preferred
to a component
bundle.
x z
y
Non-Convex Preferences
x2
y2
x1 y1
z
The mixture z
is less preferred
than x or y.
More Non-Convex Preferences
x2
y2
x1 y1
z
The mixture z
is less preferred
than x or y.
Marginal Rate-of-Substitution
(边际替代率 )
The slope of an indifference curve is its
marginal rate-of-substitution (MRS) (边际替代率 )
How can a MRS be calculated?
Marginal Rate of Substitution
x2
x1
x’
MRS at x’ is the slope of the
indifference curve at x’
Marginal Rate of Substitution
x2
x1
MRS at x’ is
lim {Dx2/Dx1}
Dx1 0
= dx2/dx1 at x’Dx2
Dx1
x’
Marginal Rate of Substitution
x2
x1
dx2
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.x’
MRS & Ind,Curve Properties
Good 2
Good 1
Two goods
a negatively sloped
indifference curve
MRS < 0.
MRS & Ind,Curve Properties
Good 2
Bad 1
One good and one
bad a
positively sloped
indifference curve
MRS > 0.
MRS & Ind,Curve Properties
Good 2
Good 1
MRS = - 5
MRS = - 0.5
MRS always increases with x1
(becomes less negative) if and
only if preferences are strictly
convex.
MRS & Ind,Curve Properties
x1
x2 MRS = - 0.5
MRS = - 5
MRS decreases
(becomes more negative)
as x1 increases
nonconvex preferences
MRS & Ind,Curve Properties
x2
x1
MRS
= - 0.5
MRS = - 1
MRS = - 2
MRS is not always increasing as
x1 increases nonconvex
preferences.
Summary,
The Key Concept of this Chapter
It is the indifference curve (IC)!
It consists of all the consumption bundles
that are indifferent to the consumer;
The shape of an IC tells a lot about the
consumer’s preference:
--- Its slope is called the marginal rate-of-
substitution;
--- We can define convexity of a preference which
corresponds to the shape of the IC.
Preferences
消费者偏好
Where Are We in the Course?
We are studying the 1st of the three
blocks of microeconomics,Consumer
behavior,production theory,and market
equilibrium
Within the 1st block,we are working on
the 2nd of the three components,choice
set,preference,and consumer demand
What Do We Mean by
Preference? (偏好 )
It refers to the ordered relationship
among alternative choices given by an
economic agent.
In most economic literature,consumer
preference is treated as the ultimate
exogenous element.
Preference Relations
Comparing two different consumption
bundles,x and y,
–strict preference,x is more preferred
than is y.
–weak preference,x is as at least as
preferred as is y.
–Indifference,x is exactly as preferred
as is y.
Notations
denotes strict preference;
~ denotes indifference;
denotes weak preference;
p
~f
Preference Relations
x y and y x imply x ~ y.
x y and (not y x) imply x y.
~f ~f
~f ~f
p
Assumptions about Preference
Relations
Completeness,For any two bundles
x and y it is always possible to make
the statement that either
x y
or
y x.
~f
~f
Assumptions about Preference
Relations
Reflexivity,Any bundle x is always
at least as preferred as itself; i.e.
x x.~f
Assumptions about Preference
Relations
Transitivity,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.
x y and y z x z.~f ~f ~f
Indifference Curves
无差异曲线 (或,无差异集 )
Take a reference bundle x’,The set
of all bundles equally preferred to x’
is the indifference curve containing
x’; the set of all bundles y ~ x’.
Since an indifference,curve” is not
always a curve a better name might
be an indifference,set”.
Indifference Curves
x2
x1
x”
x”’
x’ ~ x” ~ x”’x’
Indifference Curves
x2
x1
z x yp px
y
z
Indifference Curves
x2
x1
x
All bundles in I1 are
strictly preferred to
all in I2.
y
z
All bundles in I2 are
strictly preferred to
all in I3.
I1
I2
I3
Indifference Curves
x2
x1
WP(x),the set of
bundles weakly
preferred to x.
WP(x)
includes
I(x).
x
I(x)
Indifference Curves
x2
x1
SP(x),the set of
bundles strictly
preferred to x,
does not
include
I(x).
x
I(x)
Indifference Curves Cannot
Intersect ! (不相交 !)
x2
x1
x
y
z
I1 I2
From I1,x ~ y,From I2,x ~ z.
Therefore y ~ z.
Indifference Curves Cannot
Intersect
x2
x1
x
y
z
I1 I2
From I1,x ~ y,From I2,x ~ z.
Therefore y ~ z,But from I1
and I2 we see y z,a
contradiction.
p
Slopes of Indifference Curves
When more of a commodity is always
preferred,the commodity is a good.
If every commodity is a good then
indifference curves are negatively
sloped.
Slopes of Indifference Curves
Good 2
Good 1
Two goods
a negatively sloped
indifference curve.
Slopes of Indifference Curves
If less of a commodity is always
preferred then the commodity is a
bad.
Slopes of Indifference Curves
Good 2
Bad 1
One good and one
bad a
positively sloped
indifference curve.
Extreme Cases of Indifference
Curves; Perfect Substitutes
If a consumer always regards units
of commodities 1 and 2 as
equivalent,then the commodities are
perfect substitutes and only the sum
of the two commodities in bundles
determines their preference rank-
order,
Extreme Cases of Indifference
Curves; Perfect Substitutes
x2
x18
8
15
15 Slopes are constant at - 1.I2
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.
Extreme Cases of Indifference
Curves; 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 and only the number of
pairs of units of the two commodities
determines the preference rank-order
of bundles,
Extreme Cases of Indifference
Curves; Perfect Complements
x2
x1
I2
I1
45o
5
9
5 9
Since each of (5,5),
(5,9) and (9,5)
contains 5 pairs,
each is less
preferred than the
bundle (9,9) which
contains 9 pairs.
Preferences Exhibiting 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
x1
Bet
ter
Satiation
(bliss)
point
Indifference Curves for Discrete
Commodities
A commodity is infinitely divisible if
it can be acquired in any quantity;
e.g,water or cheese.
A commodity is discrete if it comes
in unit lumps of 1,2,3,… and so on;
e.g,aircraft,ships and refrigerators.
Indifference Curves for Discrete
Commodities
(study this yourself)
Suppose commodity 2 is an infinitely
divisible good (gasoline) while
commodity 1 is a discrete good
(aircraft),What do indifference
“curves” look like?
Indifference Curves With a
Discrete Good
(study this yourself)
Gas-
oline
Aircraft0 1 2 3 4
Indifference,curves”
are collections of
discrete points.
Well-Behaved Preferences
A preference relation is,well-
behaved” if it is
–monotonic and convex.
Monotonicity,More of any
commodity is always preferred (i.e.
no satiation and every commodity is
a good).
Well-Behaved Preferences
Convexity (凸性 ),Mixtures of bundles
are (at least weakly) preferred to the
bundles themselves,E.g.,the 50-50
mixture of the bundles x and y is
z = (0.5)x + (0.5)y.
z is at least as preferred as x or y,
Well-Behaved Preferences --
Convexity.
x2
y2
x1 y1
x
y
z =(tx1+(1-t)y1,tx2+(1-t)y2)
is preferred to x and y
for all 0 < t < 1.
Well-Behaved Preferences --
Convexity.
x2
y2
x1 y1
x
y
Preferences are strictly convex
when all mixtures z
are strictly
preferred to their
component
bundles x and y.
z
Well-Behaved Preferences --
Weak Convexity.
x’
y’
z’
Preferences are
weakly convex if at
least one mixture z
is equally preferred
to a component
bundle.
x z
y
Non-Convex Preferences
x2
y2
x1 y1
z
The mixture z
is less preferred
than x or y.
More Non-Convex Preferences
x2
y2
x1 y1
z
The mixture z
is less preferred
than x or y.
Marginal Rate-of-Substitution
(边际替代率 )
The slope of an indifference curve is its
marginal rate-of-substitution (MRS) (边际替代率 )
How can a MRS be calculated?
Marginal Rate of Substitution
x2
x1
x’
MRS at x’ is the slope of the
indifference curve at x’
Marginal Rate of Substitution
x2
x1
MRS at x’ is
lim {Dx2/Dx1}
Dx1 0
= dx2/dx1 at x’Dx2
Dx1
x’
Marginal Rate of Substitution
x2
x1
dx2
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.x’
MRS & Ind,Curve Properties
Good 2
Good 1
Two goods
a negatively sloped
indifference curve
MRS < 0.
MRS & Ind,Curve Properties
Good 2
Bad 1
One good and one
bad a
positively sloped
indifference curve
MRS > 0.
MRS & Ind,Curve Properties
Good 2
Good 1
MRS = - 5
MRS = - 0.5
MRS always increases with x1
(becomes less negative) if and
only if preferences are strictly
convex.
MRS & Ind,Curve Properties
x1
x2 MRS = - 0.5
MRS = - 5
MRS decreases
(becomes more negative)
as x1 increases
nonconvex preferences
MRS & Ind,Curve Properties
x2
x1
MRS
= - 0.5
MRS = - 1
MRS = - 2
MRS is not always increasing as
x1 increases nonconvex
preferences.
Summary,
The Key Concept of this Chapter
It is the indifference curve (IC)!
It consists of all the consumption bundles
that are indifferent to the consumer;
The shape of an IC tells a lot about the
consumer’s preference:
--- Its slope is called the marginal rate-of-
substitution;
--- We can define convexity of a preference which
corresponds to the shape of the IC.
