L4 Preferences.ppt
-
Upload
trollingandstalking -
Category
Documents
-
view
241 -
download
0
Transcript of L4 Preferences.ppt
-
8/9/2019 L4 Preferences.ppt
1/54
Modeling Consumer Preferences
-
8/9/2019 L4 Preferences.ppt
2/54
Preferences in Economics
! We are generally interested inunderstanding :
– how choices are made and – how they can be influenced
-
8/9/2019 L4 Preferences.ppt
3/54
Preferences in Economics
! We are generally interested inunderstanding :
– how choices are made and – how they can be influenced
! Just figured out: constraints on choices! To model choice: also need to model
decision makers’
preferences .
-
8/9/2019 L4 Preferences.ppt
4/54
Preference Relations! Comparing two different consumption
bundles, x and y: – strict preference : x is more preferred
than y. – weak preference : x is as at least as
preferred as y.
– indifference : x is exactly as preferredas y.
-
8/9/2019 L4 Preferences.ppt
5/54
Important note:
Preference relations are ordinal relations;
i.e. they state only the order in whichbundles are preferred.
-
8/9/2019 L4 Preferences.ppt
6/54
Preference notation:
! denotes strict preference;x y means bundle x is preferred
strictly to bundle y.
-
8/9/2019 L4 Preferences.ppt
7/54
Preference notation:
! denotes strict preference;x y means bundle x is preferred
strictly to bundle y.! denotes indifference; x y means x
and y are equally preferred.
-
8/9/2019 L4 Preferences.ppt
8/54
Preference notation:
! denotes strict preference;x y means bundle x is preferred
strictly to bundle y.! denotes indifference; x y means x
and y are equally preferred.!
denotes weak preference;x y means x is preferred at least asmuch as is y. ~~
-
8/9/2019 L4 Preferences.ppt
9/54
Some logical rules:
! x y and y x imply x y.~ ~
-
8/9/2019 L4 Preferences.ppt
10/54
Some logical rules:
! x y and y x imply x y.
! x y and (not y x) imply x y.
~ ~
~ ~
-
8/9/2019 L4 Preferences.ppt
11/54
Assumptions about Preference
Relations! Completeness : For any two bundles x
and y it is always possible to make the
statement that eitherx yor
y x.
~#
~#
-
8/9/2019 L4 Preferences.ppt
12/54
Assumptions about Preference
Relations! Reflexivity : Any bundle x is always at
least as preferred as itself; i.e.
x x.~#
-
8/9/2019 L4 Preferences.ppt
13/54
Assumptions about Preference
Relations! Transitivity : If
x is at least as preferred as y, and
y is at least as preferred as z, thenx is at least as preferred as z; i.e.
x y and y z x z.
~#
~#
~#
-
8/9/2019 L4 Preferences.ppt
14/54
Useful graphical tool:
Indifference Curves! Take a bundle x. The indifference curve
containing x is a set of all bundles y
such that y " x
! Since an indifference“
curve”
is not
always a curve a better name might bean indifference“
set”
.
-
8/9/2019 L4 Preferences.ppt
15/54
Indifference Curves
x2
x1
x”
x”
x
x”
x”
x
-
8/9/2019 L4 Preferences.ppt
16/54
Indifference Curves
x2
x1
z x yx
y
z
-
8/9/2019 L4 Preferences.ppt
17/54
Indifference Curves
x2
x1
xAll bundles in I 1 arestrictly preferred toall in I 2.
y
z
All bundles in I 2 are
strictly preferred toall in I 3.
I1
I2
I3
-
8/9/2019 L4 Preferences.ppt
18/54
Indifference Curves
x2
x1
the set ofbundles weakly
preferred to x(includes I(x))
x
I(x)
-
8/9/2019 L4 Preferences.ppt
19/54
Indifference Curves
x2
x1
the set ofbundles strictly
preferred to x(does not
includeI(x))
x
I(x)
-
8/9/2019 L4 Preferences.ppt
20/54
Lemma: Indifference CurvesCannot Intersect
x2
x1
xy
z
I1I2 Proof:
From I 1, x " y.
From I 2, x " z.By transitivity, y " z.
-
8/9/2019 L4 Preferences.ppt
21/54
Slopes of Indifference Curves
! When more of a commodity is alwayspreferred, the commodity is a good .
! If every commodity is a good thenindifference curves are negatively sloped .
-
8/9/2019 L4 Preferences.ppt
22/54
Slopes of Indifference CurvesGood 2
Good 1
Two goodsa negatively slopedindifference curve.
-
8/9/2019 L4 Preferences.ppt
23/54
Slopes of Indifference Curves
! If less of a commodity is alwayspreferred then the commodity is a bad .
-
8/9/2019 L4 Preferences.ppt
24/54
Slopes of Indifference CurvesGood 2
Bad 1
One good and onebad apositively slopedindifference curve.
-
8/9/2019 L4 Preferences.ppt
25/54
Extreme Cases of Indifference
Curves: Perfect Substitutes! If a consumer always regards units of
commodities 1 and 2 as equivalent,
then they are perfect substitutes
! only the total amount of the twodetermines their preference rank-order
-
8/9/2019 L4 Preferences.ppt
26/54
Extreme Cases of Indifference
Curves: Perfect Substitutesx2
x18
8
15
15Slopes are constant, here -1
I2
I1
Bundles in I 2 all have a totalof 15 units and are strictlypreferred to all bundles in
I1, which have a total ofonly 8 units in them.
-
8/9/2019 L4 Preferences.ppt
27/54
Extreme Cases of Indifference
Curves: Perfect Complements! If a consumer always consumes
commodities 1 and 2 in fixed proportion,
say, one-to-one, then the commoditiesare perfect complements
! only the number of pairs of units(or triplets, etc.) determines thepreference rank-order of bundles.
-
8/9/2019 L4 Preferences.ppt
28/54
Extreme Cases of Indifference
Curves: Perfect Complementsx2
x1
I1
45 o
5
9
5 9
Each of (5,5), (5,9)and (9,5) contains
5 pairs so each isequally preferred.
-
8/9/2019 L4 Preferences.ppt
29/54
Extreme Cases of Indifference
Curves: Perfect Complementsx2
x1
I2
I1
45 o
5
9
5 9
Since each of (5,5),(5,9) and (9,5)
contains 5 pairs,each is lesspreferred than thebundle (9,9) which
contains 9 pairs.
-
8/9/2019 L4 Preferences.ppt
30/54
-
8/9/2019 L4 Preferences.ppt
31/54
Indifference Curves Exhibiting
Satiationx2
x1
Satiation(bliss)point
-
8/9/2019 L4 Preferences.ppt
32/54
Indifference Curves Exhibiting
Satiationx2
x1
B e
t t e r
Satiation(bliss)point
-
8/9/2019 L4 Preferences.ppt
33/54
Indifference Curves Exhibiting
Satiationx2
x1
B e
t t e r
Satiation(bliss)point
-
8/9/2019 L4 Preferences.ppt
34/54
Indifference Curves for Discrete
Commodities! A commodity is discrete if it comes in
unit lumps of 0,1,2,3,…,856,… and so
on (i.e. in integer amounts)
! Examples: aircraft, chairs, refrigerators
-
8/9/2019 L4 Preferences.ppt
35/54
Indifference Curves With a
Discrete Good! Suppose commodity 2 is an infinitely
divisible good (jet fuel) while commodity
1 is a discrete good (aircraft)
-
8/9/2019 L4 Preferences.ppt
36/54
Indifference Curves With a
Discrete GoodFuel
Aircraft0 1 2 3 4
Indifference“
curves”
are collections ofdiscrete points.
-
8/9/2019 L4 Preferences.ppt
37/54
Assume Well-Behaved
Preferences! We may want to simplify further and rule
out some extreme examples by
additional assumptions
! A preference relation is“
well-behaved”
if it is – monotone and convex
-
8/9/2019 L4 Preferences.ppt
38/54
Well-Behaved Preferences
! Monotonicity : More of any commodityis always preferred
(i.e. no satiation and every commodity is agood).
-
8/9/2019 L4 Preferences.ppt
39/54
Well-Behaved Preferences
! Convexity : Mixtures of bundles are (atleast weakly) preferred to the bundles
themselves.
! E.g., the 50-50 mixture of the bundles xand y is
z = (0.5)x + (0.5)y.z is at least as preferred as x or y.
-
8/9/2019 L4 Preferences.ppt
40/54
Well-Behaved Preferences -
Convexity
x2
y2
x2+y 22
x1 y1x1+y 12
x
y
z = x+y2
z is strictlypreferred to both xand y.
-
8/9/2019 L4 Preferences.ppt
41/54
Well-Behaved Preferences -
Convexity
x2
y2x1 y1
x
y
z =(tx 1+(1-t)y 1, tx 2+(1-t)y 2)is preferred to x and yfor all 0 < t < 1.
-
8/9/2019 L4 Preferences.ppt
42/54
Well-Behaved Preferences -
Weak Convexityx
y
z
Preferences areweakly convex if atleast one mixture z isequally preferred to acomponent bundle.x z
y
-
8/9/2019 L4 Preferences.ppt
43/54
Non-Convex Preferences
x2
y2
x1 y1
zThe mixture zis less preferredthan x or y.
-
8/9/2019 L4 Preferences.ppt
44/54
Marginal Rate of Substitution
-
8/9/2019 L4 Preferences.ppt
45/54
Marginal Rate of Substitution
! The slope of an indifference curve is itsmarginal rate of substitution
(MRS)
-
8/9/2019 L4 Preferences.ppt
46/54
Marginal Rate of Substitution
x2
x1
x
MRS at x
is the slope of theindifference curve at x
-
8/9/2019 L4 Preferences.ppt
47/54
Marginal Rate of Substitution
x2
x1
MRS at x’:
lim { # x2 /# x1}# x1 0
# x2
# x1
x
-
8/9/2019 L4 Preferences.ppt
48/54
Marginal Rate of Substitution
x2
x1
MRS at x’:
lim { # x2 /# x1} = dx 2 /dx 1 at x
# x1 0# x2
# x1
x
-
8/9/2019 L4 Preferences.ppt
49/54
Marginal Rate of Substitution
x2
x1
dx 2dx 1
dx 2 = MRS $ dx 1 at x’
x
-
8/9/2019 L4 Preferences.ppt
50/54
Marginal Rate of Substitution
x2
x1
dx 2dx 1
dx 2 = MRS $ dx 1 at x’
so MRS is the rate at which
the consumer is only justwilling to exchangecommodity 2 for a smallamount of commodity 1 at x’
x
-
8/9/2019 L4 Preferences.ppt
51/54
MRS & Ind. Curve PropertiesGood 2
Good 1
Two goodsa negatively slopedindifference curve
MRS < 0.
-
8/9/2019 L4 Preferences.ppt
52/54
MRS & Ind. Curve PropertiesGood 2
Bad 1
One good and onebad apositively slopedindifference curve
MRS > 0.
-
8/9/2019 L4 Preferences.ppt
53/54
MRS & Ind. Curve PropertiesGood 2
Good 1
MRS = - 5
MRS = - 0.5
MRS always increases with x 1 (becomes less negative) if andonly if preferences are strictly
convex
-
8/9/2019 L4 Preferences.ppt
54/54
MRS & Ind. Curve Properties
x2
x1
MRS= - 0.5
MRS = - 1
MRS = - 2
If MRS does not always increase (say, with x 1)! > nonconvex preferences (in x
1)