chapter three preferences 偏好. structure 3.1 preference relations 3.2 assumptions about...
TRANSCRIPT
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 decision-makers’ 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 y).
~
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 y or y x
or both.
~
~
2) Reflexivity (反身性 ) Any bundle x is always at least as preferred
as itself; i.e.
x x.~
3) Transitivity (传递性 ) If
x is at least as preferred as y, andy is at least as preferred as z, thenx is at least as preferred as z; i.e.
x y and y z x z. 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.
xx22
yy22
xx22+y+y22
22
xx11 yy11xx11+y+y11
22
x
y
z = x+y
2is strictly preferred is strictly preferred to both x and y.to both x and y.
Strict Convexity.
xx22
yy22
xx11 yy11
x
y
z =(tx1+(1-t)y1, tx2+(1-t)y2)is preferred to x and y for all 0 < t < 1.
Strict Convexity.
xx22
yy22
xx11 yy11
x
y
Preferences are strictly convex when all mixtures z
are strictly preferred to their component bundles x and y.
z
Weak Convexity.
x’
y’
z’
Preferences are weakly convex if at least one mixture z is equally preferred to a component bundle.
xz
y
Non-Convex Preferences
xx22
yy22
xx11 yy11
zz
Better The mixture zThe mixture zis less preferredis less preferredthan x or y.than x or y.
More Non-Convex Preferences
xx22
yy22
xx11 yy11
zz
BetterThe mixture zThe mixture zis less preferredis less preferredthan x or y.than x or y.
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
xx22
xx11
x”x”
x”’x”’
x’ x’ x” x” x”’ x”’x’
Indifference Curves
xx22
xx11
zz xx yy
x
y
z
Indifference Curves
x2
x1
x
All bundles in I1 arestrictly preferred to all in I2.
y
z
All bundles in I2 are strictly preferred to all in I3.
I1
I2
I3
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?
x2
x1
I(x’)
x
I(x)
WP(x), the set of bundles weakly preferred to x.
Weakly Preferred Set (弱偏好集 )
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)
Proof by contradiction
Movie
CDUU11
UU22AA
CC
BB
Indifference curve cannot intersect
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
xx22
xx11
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=- x2/x1---is positive
The slope is x2/x1---is negative. Different from the book.
3.3.2 Slope of indifference curve
Marginal Rate of Substitution
xx22
xx11
MRS at x’ isMRS at x’ is lim { lim {xx22//xx11}}
xx11 0 0
= dx= dx22/dx/dx11 at x’ at x’xx22
xx11
x’x’
Marginal Rate of Substitution
xx22
x1
dxdx22
dxdx11
dxdx22 = MRS*dx = MRS*dx11 so, at x’, MRS is so, at x’, MRS is
the rate at which the consumer the rate at which the consumer is only is only just willingjust willing to exchange to exchange commodity 2 for a small commodity 2 for a small amount of commodity 1.amount of commodity 1.Marginal willingness to pay.Marginal willingness to pay.
x’x’
Slope and goods
Better
Better
Worse
Worse
Good 2Good 2
Good 1Good 1
Slope < 0.Slope < 0.
Good and bad
Better
Better
Wors
e
Wors
e
Good 2Good 2
Bad 1Bad 1
Slope > 0.Slope > 0.
Movies
CDs
A
D
B
C
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?
Diminishing MRS
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?
Movies
CD
U1
U2
U3
Movie Fanatic
Movies
CDU1
U2
U3
CD Fanatic
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
xx22
xx1188
88
1515
1515 I2
I1
Bundles in IBundles in I22 all have a total all have a total
of 15 units and are strictlyof 15 units and are strictlypreferred to all bundles inpreferred to all bundles in I I11, which have a total of, which have a total of
only 8 units in them. only 8 units in them.
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
xx22
xx11
I1
4545oo
55
99
55 99
Each of (5,5), (5,9) and (9,5) contains5 pairs so each is equally preferred.
Extreme Cases of Indifference Curves; Perfect Complements
xx22
xx11
I2
I1
4545oo
55
99
55 99
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.
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
xx22
xx11
SatiationSatiation(bliss)(bliss)pointpoint
Indifference Curves Exhibiting Satiation
xx22
xx11
BetteBette
rr
BetteBetterr
Bet
teB
ette
rr
SatiationSatiation(bliss)(bliss)pointpoint
Indifference Curves Exhibiting Satiation
xx22
xx11
BetteBette
rr
BetteBetterr
Bet
teB
ette
rr
SatiationSatiation(bliss)(bliss)pointpoint
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
SugarSugar
CarCar00 11 22 3 44
Indifference “curves”Indifference “curves”are collections ofare collections ofdiscrete points.discrete points.