Chapter Three
Preferences
偏好
Structure
?Describe preferences
?Indifference curves (无差异曲线 )
?Well-behaved preferences
?Marginal rate of substitution (边际替
代率 )
Rationality in Economics
? Behavioral Postulate:
A decisionmaker always chooses its
most preferred alternative from its
set of available alternatives.
?So to model choice we must model
decision-makers’ preferences.
Preference Relations
?Comparing two different consumption
bundles,x and y,
–strict preference (严格偏好 ),x is
more preferred than is y (x y).
–Indifference (无差异 ),x is exactly as
preferred as is y (x~y),
–weak preference (弱偏好 ),x is as at
least as preferred as is y (x y).
p
~f
Preference Relations
?Preference relations are ordinal
relations; i.e,they state only the
order in which bundles are
preferred.
Preference Relations
?x y and y x imply x ~ y.~f ~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 (完备性 ),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 (反身性 ),Any bundle x is
always at least as preferred as itself;
i.e.
x x.~f
Assumptions about Preference
Relations
?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.
x y and y z x z.~f ~f ~f
Indifference Curves
?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
x2
x1
I(x’)
x
I(x)
WP(x),the set of
bundles weakly
preferred to x.
Weakly Preferred Set (弱偏好集 )
Weakly Preferred Set (弱偏好集 )
x2
x1
WP(x),the set of
bundles weakly
preferred to x.
WP(x)
includes
I(x).
x
I(x)
Strictly Preferred Set (严格偏好集 )
x2
x1
SP(x),the set of
bundles strictly
preferred to x,
does not
include
I(x).
x
I(x)
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.
Examples
?Perfect substitutes (完全替代 )
?Perfect complements (完全互补 )
?Satiation (餍足 )
?Discrete goods (离散商品 )
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 total
amount 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
I1
45o
5
9
5 9
Each of (5,5),(5,9)
and (9,5) contains
5 pairs so each is
equally preferred.
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
Satiation
(bliss)
point
Indifference Curves Exhibiting
Satiation
x2
x1
Bet
ter
Satiation
(bliss)
point
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
?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
Gas-
oline
Car0 1 2 3 4
Indifference,curves”
are collections of
discrete points.
Well-Behaved Preferences
?A preference relation is,well-
behaved” if it is
–Monotonic (单调 ) and convex (凸状 ).
?Monotonicity,More of any commodity
is always preferred (i.e,no satiation
and every commodity is a good).
Well-Behaved Preferences
?Convexity,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
x2+y2
2
x1 y1x1+y
1 2
x
y
z = x+y2 is strictly preferred to both x and 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.
Slopes of Indifference Curves
?The slope of an indifference curve is
its marginal rate-of-substitution
(MRS).
?How can a MRS be calculated?
?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.
Marginal willingness to pay.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.
Diminishing MRS
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.
Preferences
偏好
Structure
?Describe preferences
?Indifference curves (无差异曲线 )
?Well-behaved preferences
?Marginal rate of substitution (边际替
代率 )
Rationality in Economics
? Behavioral Postulate:
A decisionmaker always chooses its
most preferred alternative from its
set of available alternatives.
?So to model choice we must model
decision-makers’ preferences.
Preference Relations
?Comparing two different consumption
bundles,x and y,
–strict preference (严格偏好 ),x is
more preferred than is y (x y).
–Indifference (无差异 ),x is exactly as
preferred as is y (x~y),
–weak preference (弱偏好 ),x is as at
least as preferred as is y (x y).
p
~f
Preference Relations
?Preference relations are ordinal
relations; i.e,they state only the
order in which bundles are
preferred.
Preference Relations
?x y and y x imply x ~ y.~f ~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 (完备性 ),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 (反身性 ),Any bundle x is
always at least as preferred as itself;
i.e.
x x.~f
Assumptions about Preference
Relations
?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.
x y and y z x z.~f ~f ~f
Indifference Curves
?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
x2
x1
I(x’)
x
I(x)
WP(x),the set of
bundles weakly
preferred to x.
Weakly Preferred Set (弱偏好集 )
Weakly Preferred Set (弱偏好集 )
x2
x1
WP(x),the set of
bundles weakly
preferred to x.
WP(x)
includes
I(x).
x
I(x)
Strictly Preferred Set (严格偏好集 )
x2
x1
SP(x),the set of
bundles strictly
preferred to x,
does not
include
I(x).
x
I(x)
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.
Examples
?Perfect substitutes (完全替代 )
?Perfect complements (完全互补 )
?Satiation (餍足 )
?Discrete goods (离散商品 )
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 total
amount 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
I1
45o
5
9
5 9
Each of (5,5),(5,9)
and (9,5) contains
5 pairs so each is
equally preferred.
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
Satiation
(bliss)
point
Indifference Curves Exhibiting
Satiation
x2
x1
Bet
ter
Satiation
(bliss)
point
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
?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
Gas-
oline
Car0 1 2 3 4
Indifference,curves”
are collections of
discrete points.
Well-Behaved Preferences
?A preference relation is,well-
behaved” if it is
–Monotonic (单调 ) and convex (凸状 ).
?Monotonicity,More of any commodity
is always preferred (i.e,no satiation
and every commodity is a good).
Well-Behaved Preferences
?Convexity,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
x2+y2
2
x1 y1x1+y
1 2
x
y
z = x+y2 is strictly preferred to both x and 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.
Slopes of Indifference Curves
?The slope of an indifference curve is
its marginal rate-of-substitution
(MRS).
?How can a MRS be calculated?
?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.
Marginal willingness to pay.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.
Diminishing MRS
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.
