Econs 501-Microeconomic Theory I
Transcript of Econs 501-Microeconomic Theory I
![Page 1: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/1.jpg)
Econs 501-Microeconomic Theory I
WELCOME!!! TA: Will Ottenheimer (Office hours Tuesdays 1
โ 2 pm) MWG and Munoz-Garcia Mondays (Assignments) Recitations: Fridays 12-1pm (Hulbert 23) Course website:โฆ https://anaespinolaarredondo.com/econs-501-
microeconomic-theory-i/
Advanced Microeconomic Theory 1
![Page 2: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/2.jpg)
Preferences and Utility
![Page 3: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/3.jpg)
How can we formally describe an individualโs preference for different amounts of a good?
3
How can we represent his preference for a particular list of goods (a bundle) over another?
We will examine under which conditions an individualโs preference can be mathematically represented with a utility function.
![Page 4: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/4.jpg)
Preference and Choice
4
![Page 5: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/5.jpg)
Preference and Choice
5
![Page 6: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/6.jpg)
Advantages:
Advanced Microeconomic Theory 6
Preference-based approach: โฆ More tractable when the set of alternatives
๐๐has many elements.
Choice-based approach: โฆ It is based on observables (actual choices)
rather than on unobservables (I.P)
![Page 7: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/7.jpg)
Preference-Based Approach
Preferences: โattitudesโ of the decision-maker towards a set of possible alternatives ๐๐.
For any ๐ฅ๐ฅ, ๐ฆ๐ฆ โ ๐๐, how do you compare ๐ฅ๐ฅand ๐ฆ๐ฆ? I prefer ๐ฅ๐ฅ to ๐ฆ๐ฆ (๐ฅ๐ฅ โป ๐ฆ๐ฆ) I prefer ๐ฆ๐ฆ to ๐ฅ๐ฅ (๐ฆ๐ฆ โป ๐ฅ๐ฅ) I am indifferent (๐ฅ๐ฅ โผ ๐ฆ๐ฆ)
7
![Page 8: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/8.jpg)
Preferences
Advanced Microeconomic Theory 8
![Page 9: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/9.jpg)
Preference-Based Approach
Completeness:โฆ For an pair of alternatives ๐ฅ๐ฅ, ๐ฆ๐ฆ โ ๐๐, the
individual decision maker: ๐ฅ๐ฅ โป ๐ฆ๐ฆ, or ๐ฆ๐ฆ โป ๐ฅ๐ฅ, or both, i.e., ๐ฅ๐ฅ โผ ๐ฆ๐ฆ
Advanced Microeconomic Theory 9
![Page 10: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/10.jpg)
Preference-Based Approach
Not all binary relations satisfy Completeness.
Example:โฆ โIs the brother ofโ: John โ Bob and Bob โ
John if they are not brothers.โฆ โIs the father ofโ: John โ Bob and Bob โ John
if the two individuals are not related.
Not all pairs of alternatives are comparable according to these two relations.
Advanced Microeconomic Theory 10
![Page 11: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/11.jpg)
Preference-Based Approach
Advanced Microeconomic Theory 11
![Page 12: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/12.jpg)
Preference-Based Approach
Advanced Microeconomic Theory 12
![Page 13: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/13.jpg)
Preference-Based Approach
Advanced Microeconomic Theory 13
![Page 14: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/14.jpg)
Preference-Based Approach
Advanced Microeconomic Theory 14
![Page 15: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/15.jpg)
Preference-Based Approach
Sources of intransitivity:a) Indistinguishable alternatives
a) Examples?
b) Framing effectsc) Aggregation of criteriad) Change in preferences
a) Examples?
15
![Page 16: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/16.jpg)
Preference-Based Approach
โข Example 1.1 (Indistinguishable alternatives):โฆ Take ๐๐ = โ as a piece of pie and ๐ฅ๐ฅ โป ๐ฆ๐ฆ if ๐ฅ๐ฅ โฅ ๐ฆ๐ฆ โ
1 (๐ฅ๐ฅ + 1 โฅ ๐ฆ๐ฆ) but ๐ฅ๐ฅ~๐ฆ๐ฆ if ๐ฅ๐ฅ โ ๐ฆ๐ฆ < 1(indistinguishable).โฆ Then,
1.5~0.8 since 1.5 โ 0.8 = 0.7 < 10.8~0.3 since 0.8 โ 0.3 = 0.5 < 1
โฆ By transitivity, we would have 1.5~0.3, but in fact 1.5 โป 0.3 (intransitive preference relation).
16
![Page 17: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/17.jpg)
Preference-Based Approach
Other examples: โฆ similar shades of gray paint
โฆ milligrams of sugar in your coffee
17
![Page 18: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/18.jpg)
Utility Function
18
![Page 19: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/19.jpg)
Utility Function
19
![Page 20: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/20.jpg)
Utility Function
20
![Page 21: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/21.jpg)
Desirability
21
![Page 22: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/22.jpg)
Desirability
22
![Page 23: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/23.jpg)
Desirability
23
![Page 24: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/24.jpg)
Desirability
Advanced Microeconomic Theory 24
![Page 25: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/25.jpg)
Desirability
25
![Page 26: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/26.jpg)
Desirability
Advanced Microeconomic Theory 26
![Page 27: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/27.jpg)
Desirability
27
![Page 28: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/28.jpg)
Indifference sets
28
![Page 29: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/29.jpg)
Upper contour set (UCS){y โ โ+: y โฟ x}2
Indifference set{y โ โ+: y ~ x}2
Lower contour set (LCS){y โ โ+: y โพ x}2
x1
x2
x
Indifference sets
29
![Page 30: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/30.jpg)
Indifference sets
Note: โฆ Strong monotonicity (and monotonicity)
implies that indifference curves must be negatively sloped. Hence, to maintain utility level unaffected along all
the points on a given indifference curve, an increase in the amount of one good must be accompanied by a reduction in the amounts of other goods.
30
![Page 31: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/31.jpg)
Convexity of Preferences
31
![Page 32: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/32.jpg)
Convexity of Preferences
Convexity 1
Advanced Microeconomic Theory 32
![Page 33: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/33.jpg)
Convexity of Preferences
33
![Page 34: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/34.jpg)
Convexity of Preferences
Convexity 2
34
![Page 35: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/35.jpg)
Convexity of Preferences
35
![Page 36: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/36.jpg)
x1
x2
ฮปx + (1โฮป)y โปz
UCS
xx
yy
z
Convexity of Preferences Strictly convex preferences
36
![Page 37: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/37.jpg)
Convexity of Preferences Convexity but not strict convexity
37
โ ๐๐๐ฅ๐ฅ + 1 โ ๐๐ ๐ฆ๐ฆ~๐ง๐งโ Such preference relation
is represented by utility function such as
๐ข๐ข ๐ฅ๐ฅ1, ๐ฅ๐ฅ2 = ๐๐๐ฅ๐ฅ1 + ๐๐๐ฅ๐ฅ2
where ๐ฅ๐ฅ1 and ๐ฅ๐ฅ2 are substitutes.
![Page 38: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/38.jpg)
Convexity of Preferences Convexity but not strict convexity
38
โ ๐๐๐ฅ๐ฅ + 1 โ ๐๐ ๐ฆ๐ฆ~๐ง๐งโ Such preference relation
is represented by utility function such as
๐ข๐ข ๐ฅ๐ฅ1, ๐ฅ๐ฅ2 = min{๐๐๐ฅ๐ฅ1, ๐๐๐ฅ๐ฅ2}
where ๐๐, ๐๐ > 0.
![Page 39: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/39.jpg)
Convexity of Preferences
Example 1.6
39
๐ข๐ข ๐ฅ๐ฅ1, ๐ฅ๐ฅ2 Satisfiesconvexity
Satisfies strict convexity
๐๐๐ฅ๐ฅ1 + ๐๐๐ฅ๐ฅ2 โ X
min{๐๐๐ฅ๐ฅ1, ๐๐๐ฅ๐ฅ2} โ X
๐๐๐ฅ๐ฅ1
12 + ๐๐๐ฅ๐ฅ2
12 โ โ
๐๐๐ฅ๐ฅ12 + ๐๐๐ฅ๐ฅ2
2 X X
![Page 40: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/40.jpg)
Convexity of Preferences
1) Taste for diversification: โฆ An individual with
convex preferences prefers the convex combination of bundles ๐ฅ๐ฅ and ๐ฆ๐ฆ, than either of those bundles alone.
40
โข Interpretation of convexity
![Page 41: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/41.jpg)
Convexity of Preferences
Interpretation of convexity2) Diminishing marginal rate of substitution:
๐๐๐๐๐๐1,2 โก ๐๐๐๐/๐๐๐ฅ๐ฅ1๐๐๐๐/๐๐๐ฅ๐ฅ2
โฆ MRS describes the additional amount of good 1 that the consumer needs to receive in order to keep her utility level unaffected.
โฆ A diminishing MRS implies that the consumer needs to receive increasingly larger amounts of good 1 in order to accept further reductions of good 2.
41
![Page 42: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/42.jpg)
x1
x2
A
B
C
D
1 unit = โx2
1 unit = โx2
โx1 โโx1
Convexity of Preferences
Diminishing marginal rate of substitution
42
![Page 43: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/43.jpg)
Convexity of Preferences
Advanced Microeconomic Theory 43
![Page 44: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/44.jpg)
Convexity of Preferences
44
![Page 45: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/45.jpg)
Quasiconcavity
45
![Page 46: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/46.jpg)
Quasiconcavity
46
![Page 47: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/47.jpg)
Quasiconcavity
Quasiconcavity
47
![Page 48: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/48.jpg)
Quasiconcavity
48
![Page 49: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/49.jpg)
1x
2x
( ) ( )u x u y=
x
y
( )1x yฮฑ ฮฑ+ โ
( )( )1u x yฮฑ ฮฑ+ โ
Quasiconcavity
49
![Page 50: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/50.jpg)
Quasiconcavity
50
![Page 51: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/51.jpg)
Quasiconcavity
51
![Page 52: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/52.jpg)
Quasiconcavity
Concavity implies quasiconcavity
52
![Page 53: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/53.jpg)
Quasiconcavity
Advanced Microeconomic Theory 53
![Page 54: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/54.jpg)
1x2x
u
( )1 14 4
1 2 1 1,u x x x x= +
Quasiconcavity
Concave and quasiconcave utility function (3D)
54
๐ข๐ข(๐ฅ๐ฅ1, ๐ฅ๐ฅ2) = ๐ฅ๐ฅ1
14๐ฅ๐ฅ2
14
![Page 55: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/55.jpg)
Quasiconcavity
55
![Page 56: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/56.jpg)
2x 1x
v
( )6 64 4
1 2 1 1,v x x x x=
Quasiconcavity
Convex but quasiconcave utility function (3D)
56
๐ฃ๐ฃ(๐ฅ๐ฅ1, ๐ฅ๐ฅ2) = ๐ฅ๐ฅ1
64๐ฅ๐ฅ2
64
![Page 57: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/57.jpg)
Quasiconcavity
57
![Page 58: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/58.jpg)
Quasiconcavity
โขAdvanced Microeconomic Theory 58
![Page 59: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/59.jpg)
Quasiconcavity
Example 1.7 (continued):โฆ Let us consider the case of only two goods,
๐ฟ๐ฟ = 2.
โฆ Then, an individual prefers a bundle ๐ฅ๐ฅ =(๐ฅ๐ฅ1, ๐ฅ๐ฅ2) to another bundle ๐ฆ๐ฆ = (๐ฆ๐ฆ1, ๐ฆ๐ฆ2) iff ๐ฅ๐ฅcontains more units of both goods than bundle ๐ฆ๐ฆ, i.e., ๐ฅ๐ฅ1 โฅ ๐ฆ๐ฆ1 and ๐ฅ๐ฅ2 โฅ ๐ฆ๐ฆ2.
โฆ For illustration purposes, let us take bundle such as (2,1).
59
![Page 60: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/60.jpg)
Quasiconcavity
Example 1.7 (continued):
Advanced Microeconomic Theory 60
![Page 61: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/61.jpg)
Quasiconcavity
Example 1.7 (continued):1) UCS:โฆ The upper contour set of bundle (2,1) contains
bundles (๐ฅ๐ฅ1, ๐ฅ๐ฅ2) with weakly more than 2 units of good 1 and/or weakly more than 1 unit of good 2:
๐๐๐๐๐๐ 2,1 = {(๐ฅ๐ฅ1, ๐ฅ๐ฅ2) โฟ (2,1) โบ ๐ฅ๐ฅ1 โฅ 2, ๐ฅ๐ฅ2 โฅ 1}
โฆ The frontiers of the UCS region also represent bundles preferred to (2,1).
61
![Page 62: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/62.jpg)
Quasiconcavity
Example 1.7 (continued):2) LCS: โฆ The bundles in the lower contour set of
bundle (2,1) contain fewer units of both goods:
๐ฟ๐ฟ๐๐๐๐ 2,1 = {(2,1) โฟ (๐ฅ๐ฅ1, ๐ฅ๐ฅ2) โบ ๐ฅ๐ฅ1 โค 2, ๐ฅ๐ฅ2 โค 1}
โฆ The frontiers of the LCS region also represent bundles with fewer unis of either good 1 or good 2.
62
![Page 63: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/63.jpg)
Quasiconcavity
Advanced Microeconomic Theory 63
![Page 64: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/64.jpg)
Quasiconcavity
Example 1.7 (continued):4) Regions A and B: โฆ Region ๐ด๐ด contains bundles with more units of
good 2 but fewer units of good 1 (the opposite argument applies to region ๐ต๐ต).
โฆ The consumer cannot compare bundles in either of these regions against bundle 2,1 .
โฆ For him to be able to rank one bundle against another, one of the bundles must contain the same or more units of all goods.
64
![Page 65: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/65.jpg)
Quasiconcavity
Example 1.7 (continued):5) Preference relation is not complete:โฆ Completeness requires for every pair ๐ฅ๐ฅ
and ๐ฆ๐ฆ, either ๐ฅ๐ฅ โฟ ๐ฆ๐ฆ or ๐ฆ๐ฆ โฟ ๐ฅ๐ฅ (or both).โฆ Consider two bundles ๐ฅ๐ฅ, ๐ฆ๐ฆ โ โ+
2 with bundle ๐ฅ๐ฅ containing more units of good 1 than bundle ๐ฆ๐ฆ but fewer units of good 2, i.e., ๐ฅ๐ฅ1 > ๐ฆ๐ฆ1 and ๐ฅ๐ฅ2 < ๐ฆ๐ฆ2 (as in Region B)
โฆ Then, we have neither ๐ฅ๐ฅ โฟ ๐ฆ๐ฆ nor ๐ฆ๐ฆ โฟ ๐ฅ๐ฅ.
65
![Page 66: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/66.jpg)
Quasiconcavity
Example 1.7 (continued):6) Preference relation is transitive:โฆ Transitivity requires that, for any three
bundles ๐ฅ๐ฅ, ๐ฆ๐ฆ and ๐ง๐ง, if ๐ฅ๐ฅ โฟ ๐ฆ๐ฆ and ๐ฆ๐ฆ โฟ ๐ง๐งthen ๐ฅ๐ฅ โฟ ๐ง๐ง.
โฆ Now ๐ฅ๐ฅ โฟ ๐ฆ๐ฆ and ๐ฆ๐ฆ โฟ ๐ง๐ง means ๐ฅ๐ฅ๐๐ โฅ ๐ฆ๐ฆ๐๐ and๐ฆ๐ฆ๐๐ โฅ ๐ง๐ง๐๐ for all ๐๐ goods.
โฆ Then, ๐ฅ๐ฅ๐๐ โฅ ๐ง๐ง๐๐ implies ๐ฅ๐ฅ โฟ ๐ง๐ง.
66
![Page 67: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/67.jpg)
Quasiconcavity
Example 1.7 (continued):7) Preference relation is strongly monotone:โฆ Strong monotonicity requires that if we
increase one of the goods in a given bundle, then the newly created bundle must be strictly preferred to the original bundle.
โฆ Now ๐ฅ๐ฅ โฅ ๐ฆ๐ฆ and ๐ฅ๐ฅ โ ๐ฆ๐ฆ implies that ๐ฅ๐ฅ๐๐ โฅ ๐ฆ๐ฆ๐๐ for all good ๐๐ and ๐ฅ๐ฅ๐๐ > ๐ฆ๐ฆ๐๐ for at least one good ๐๐.
โฆ Thus, ๐ฅ๐ฅ โฅ ๐ฆ๐ฆ and ๐ฅ๐ฅ โ ๐ฆ๐ฆ implies ๐ฅ๐ฅ โฟ ๐ฆ๐ฆ and not ๐ฆ๐ฆ โฟ ๐ฅ๐ฅ.
โฆ Thus, we can conclude that ๐ฅ๐ฅ โป ๐ฆ๐ฆ.67
![Page 68: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/68.jpg)
Quasiconcavity
Example 1.7 (continued):8) Preference relation is strictly convex:โฆ Strict convexity requires that if ๐ฅ๐ฅ โฟ ๐ง๐ง and ๐ฆ๐ฆ โฟ
๐ง๐ง and ๐ฅ๐ฅ โ ๐ง๐ง, then ๐ผ๐ผ๐ฅ๐ฅ + 1 โ ๐ผ๐ผ ๐ฆ๐ฆ โป ๐ง๐ง for all ๐ผ๐ผ โ 0,1 .
โฆ Now ๐ฅ๐ฅ โฟ ๐ง๐ง and ๐ฆ๐ฆ โฟ ๐ง๐ง implies that ๐ฅ๐ฅ๐๐ โฅ ๐ฆ๐ฆ๐๐ and ๐ฆ๐ฆ๐๐ โฅ ๐ง๐ง๐๐ for all good ๐๐.
โฆ ๐ฅ๐ฅ โ ๐ง๐ง implies, for some good ๐๐, we must have ๐ฅ๐ฅ๐๐ > ๐ง๐ง๐๐.
68
![Page 69: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/69.jpg)
Quasiconcavity
Example 1.7 (continued):โฆ Hence, for any ๐ผ๐ผ โ 0,1 , we must have that
๐ผ๐ผ๐ฅ๐ฅ๐๐ + 1 โ ๐ผ๐ผ ๐ฆ๐ฆ๐๐ โฅ ๐ง๐ง๐๐ for all good ๐๐๐ผ๐ผ๐ฅ๐ฅ๐๐ + 1 โ ๐ผ๐ผ ๐ฆ๐ฆ๐๐ > ๐ง๐ง๐๐ for some ๐๐
โฆ Thus, we have that ๐ผ๐ผ๐ฅ๐ฅ + 1 โ ๐ผ๐ผ ๐ฆ๐ฆ โฅ ๐ง๐ง and ๐ผ๐ผ๐ฅ๐ฅ + 1 โ ๐ผ๐ผ ๐ฆ๐ฆ โ ๐ง๐ง, and so
๐ผ๐ผ๐ฅ๐ฅ + 1 โ ๐ผ๐ผ ๐ฆ๐ฆ โฟ ๐ง๐งand not ๐ง๐ง โฟ ๐ผ๐ผ๐ฅ๐ฅ + 1 โ ๐ผ๐ผ ๐ฆ๐ฆ
โฆ Therefore, ๐ผ๐ผ๐ฅ๐ฅ + 1 โ ๐ผ๐ผ ๐ฆ๐ฆ โป ๐ง๐ง.
69
![Page 70: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/70.jpg)
Common Utility Functions
70
![Page 71: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/71.jpg)
Common Utility Functions
71
![Page 72: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/72.jpg)
Common Utility Functionsโฆ Marginal utilities:
๐๐๐๐๐๐๐ฅ๐ฅ1
> 0 and ๐๐๐๐๐๐๐ฅ๐ฅ2
> 0
โฆ A diminishing MRS
๐๐๐๐๐๐๐ฅ๐ฅ1,๐ฅ๐ฅ2 =๐ผ๐ผ๐ด๐ด๐ฅ๐ฅ1
๐ผ๐ผโ1๐ฅ๐ฅ2๐ฝ๐ฝ
๐ฝ๐ฝ๐ด๐ด๐ฅ๐ฅ1๐ผ๐ผ๐ฅ๐ฅ2
๐ฝ๐ฝโ1 =๐ผ๐ผ๐ฅ๐ฅ2
๐ฝ๐ฝ๐ฅ๐ฅ1
which is decreasing in ๐ฅ๐ฅ1. Hence, indifference curves become flatter as ๐ฅ๐ฅ1
increases.
72
![Page 73: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/73.jpg)
2x
1xIC
A
B
CD
1 unit 1 unit
2in xโโ
2xโ
Common Utility Functions
Cobb-Douglas preference
73
![Page 74: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/74.jpg)
Common Utility Functions
74
![Page 75: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/75.jpg)
Common Utility Functions
Perfect substitutes
75
2A
A
2BB
AslopeB
= โ
2x
1x
![Page 76: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/76.jpg)
Common Utility Functions
76
![Page 77: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/77.jpg)
Common Utility Functions
Advanced Microeconomic Theory 77
![Page 78: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/78.jpg)
Common Utility Functions
Perfect complements
78
2x
1x
2 ฮฒฮฑฮฒฮฑ
1 2
ฮฒฮฑ
1u Aฮฑ=2 2u Aฮฑ=
2 1x xฮฒฮฑ
=
![Page 79: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/79.jpg)
Common Utility Functions
79
![Page 80: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/80.jpg)
Common Utility Functions
80
![Page 81: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/81.jpg)
Common Utility Functions
CES preferences
Advanced Microeconomic Theory 81
2x
1xฯ = โ 8ฯ =
1ฯ =0.2ฯ =
0ฯ =
} Cobb-Douglas
Perfect complement
Perfect substitutes
![Page 82: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/82.jpg)
Common Utility Functionsโฆ CES utility function is often presented as
๐ข๐ข ๐ฅ๐ฅ1, ๐ฅ๐ฅ
2= ๐๐๐ฅ๐ฅ1
๐๐ + ๐๐๐ฅ๐ฅ2๐๐
1๐๐
where ๐๐ โก ๐๐โ1๐๐
.
82
![Page 83: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/83.jpg)
Common Utility Functions
83
![Page 84: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/84.jpg)
Common Utility Functions
MRS of quasilinear preferences
Advanced Microeconomic Theory 84
![Page 85: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/85.jpg)
Common Utility Functions
โฆ For ๐ข๐ข ๐ฅ๐ฅ1, ๐ฅ๐ฅ2 = ๐ฃ๐ฃ ๐ฅ๐ฅ1 + ๐๐๐ฅ๐ฅ2, the marginal utilities are
๐๐๐๐๐๐๐ฅ๐ฅ2
= ๐๐ and ๐๐๐๐๐๐๐ฅ๐ฅ1
= ๐๐๐ฃ๐ฃ๐๐๐ฅ๐ฅ1
which implies
๐๐๐๐๐๐๐ฅ๐ฅ1,๐ฅ๐ฅ2 =๐๐๐๐
๐๐๐ฅ๐ฅ1๐๐
โฆ Quasilinear preferences are often used to represent the consumption of goods that are relatively insensitive to income.โฆ Examples: garlic, toothpaste, etc.
85
![Page 86: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/86.jpg)
Continuous Preferences
In order to guarantee that preference relations can be represented by a utility function we need continuity.
Continuity: A preference relation defined on ๐๐ is continuous if it is preserved under limits. โฆ That is, for any sequence of pairs
(๐ฅ๐ฅ๐๐, ๐ฆ๐ฆ๐๐) ๐๐=1โ with ๐ฅ๐ฅ๐๐ โฟ ๐ฆ๐ฆ๐๐ for all ๐๐
and lim๐๐โโ
๐ฅ๐ฅ๐๐ = ๐ฅ๐ฅ and lim๐๐โโ
๐ฆ๐ฆ๐๐ = ๐ฆ๐ฆ, the preference relation is maintained in the limiting points, i.e., ๐ฅ๐ฅ โป ๐ฆ๐ฆ.
Advanced Microeconomic Theory 86
![Page 87: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/87.jpg)
Continuous Preferences
โฆ Intuitively, there can be no sudden jumps (i.e., preference reversals) in an individual preference over a sequence of bundles.
Advanced Microeconomic Theory 87
![Page 88: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/88.jpg)
Continuous Preferences
Lexicographic preferences are not continuous
โฆ Consider the sequence ๐ฅ๐ฅ๐๐ = 1๐๐
, 0 and ๐ฆ๐ฆ๐๐ =(0,1), where ๐๐ = {0,1,2,3, โฆ }.โฆ The sequence ๐ฆ๐ฆ๐๐ = (0,1) is constant in ๐๐.
โฆ The sequence ๐ฅ๐ฅ๐๐ = 1๐๐
, 0 is not:
It starts at ๐ฅ๐ฅ1 = 1,0 , and moves leftwards to ๐ฅ๐ฅ2 = 1
2, 0 , ๐ฅ๐ฅ3 = 1
3, 0 , etc.
Advanced Microeconomic Theory 88
![Page 89: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/89.jpg)
x1
x2
1
10 โ ยฝยผx 4 x 3 x 2 x 1
y n, โn, y 1 = y 2 = โฆ = y n
lim x n = (0,0)n โโ
Continuous Preferences
Thus, the individual prefers:๐ฅ๐ฅ1 = 1,0 โป 0,1 = ๐ฆ๐ฆ1
๐ฅ๐ฅ2 = 12
, 0 โป 0,1 = ๐ฆ๐ฆ2
๐ฅ๐ฅ3 = 13
, 0 โป 0,1 = ๐ฆ๐ฆ3
โฎ But,
lim๐๐โโ
๐ฅ๐ฅ๐๐ = 0,0 โบ 0,1= lim
๐๐โโ๐ฆ๐ฆ๐๐
Preference reversal!
Advanced Microeconomic Theory 89
![Page 90: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/90.jpg)
Existence of Utility Function
If a preference relation satisfies monotonicity and continuity, then there exists a utility function ๐ข๐ข(๏ฟฝ)representing such preference relation.
Proof:โฆ Take a bundle ๐ฅ๐ฅ โ 0. โฆ By monotonicity, ๐ฅ๐ฅ โฟ 0, where 0 = (0,0, โฆ , 0). That is, if bundle ๐ฅ๐ฅ โ 0, it contains positive amounts
of at least one good and, it is preferred to bundle 0.
Advanced Microeconomic Theory 90
![Page 91: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/91.jpg)
Existence of Utility Functionโฆ Define bundle ๐๐ as the bundle where all
components coincide with the highest component of bundle ๐ฅ๐ฅ:
๐๐ = max๐๐
{๐ฅ๐ฅ๐๐} , โฆ , max๐๐
{๐ฅ๐ฅ๐๐}
โฆ Hence, by monotonicity, ๐๐ โฟ ๐ฅ๐ฅ.
โฆ Bundles 0 and ๐๐ are both on the main diagonal, since each of them contains the same amount of good ๐ฅ๐ฅ1 and ๐ฅ๐ฅ2.
Advanced Microeconomic Theory 91
![Page 92: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/92.jpg)
x2
x1
Existence of Utility Function
Advanced Microeconomic Theory 92
![Page 93: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/93.jpg)
Existence of Utility Functionโฆ By continuity and monotonicity, there exists a
bundle that is indifferent to ๐ฅ๐ฅ and which lies on the main diagonal.โฆ By monotonicity, this bundle is uniqueOtherwise, modifying any of its components
would lead to higher/lower indifference curves.
โฆ Denote such bundle as ๐ก๐ก ๐ฅ๐ฅ , ๐ก๐ก ๐ฅ๐ฅ , โฆ , ๐ก๐ก(๐ฅ๐ฅ)
โฆ Let ๐ข๐ข ๐ฅ๐ฅ = ๐ก๐ก ๐ฅ๐ฅ , which is a real number.
Advanced Microeconomic Theory 93
![Page 94: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/94.jpg)
Existence of Utility Function
โฆ Applying the same steps for another bundle ๐ฆ๐ฆ โ ๐ฅ๐ฅ, we obtain
๐ก๐ก ๐ฆ๐ฆ , ๐ก๐ก ๐ฆ๐ฆ , โฆ , ๐ก๐ก(๐ฆ๐ฆ)and let ๐ข๐ข ๐ฆ๐ฆ = ๐ก๐ก ๐ฆ๐ฆ , which is also a real number.
โฆ We know that ๐ฅ๐ฅ~ ๐ก๐ก ๐ฅ๐ฅ , ๐ก๐ก ๐ฅ๐ฅ , โฆ , ๐ก๐ก(๐ฅ๐ฅ)๐ฆ๐ฆ~ ๐ก๐ก ๐ฆ๐ฆ , ๐ก๐ก ๐ฆ๐ฆ , โฆ , ๐ก๐ก(๐ฆ๐ฆ)
๐ฅ๐ฅ โฟ ๐ฆ๐ฆ
โฆ Hence, by transitivity, ๐ฅ๐ฅ โฟ ๐ฆ๐ฆ iff๐ฅ๐ฅ~ ๐ก๐ก ๐ฅ๐ฅ , ๐ก๐ก ๐ฅ๐ฅ , โฆ , ๐ก๐ก(๐ฅ๐ฅ) โฟ ๐ก๐ก ๐ฆ๐ฆ , ๐ก๐ก ๐ฆ๐ฆ , โฆ , ๐ก๐ก(๐ฆ๐ฆ) ~๐ฆ๐ฆ
Advanced Microeconomic Theory 94
![Page 95: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/95.jpg)
Existence of Utility Functionโฆ And by monotonicity,
๐ฅ๐ฅ โฟ ๐ฆ๐ฆ โบ ๐ก๐ก ๐ฅ๐ฅ โฅ ๐ก๐ก ๐ฆ๐ฆ โบ ๐ข๐ข(๐ฅ๐ฅ) โฅ ๐ข๐ข(๐ฆ๐ฆ)
โฆ Note: A utility function can satisfy continuity but still be non-differentiable. For instance, the Leontief utility function,
min{๐๐๐ฅ๐ฅ1,๐๐๐ฅ๐ฅ2}, is continuous but cannot be differentiated at the kink.
Advanced Microeconomic Theory 95
![Page 96: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/96.jpg)
Choice Based Approach
We now focus on the actual choice behavior rather than individual preferences.โฆ From the alternatives in set ๐ด๐ด, which one would
you choose?
A choice structure (โฌ, ๐๐(๏ฟฝ)) contains two elements:1) โฌ is a family of nonempty subsets of ๐๐, so that
every element of โฌ is a set ๐ต๐ต โ ๐๐.
Advanced Microeconomic Theory 96
![Page 97: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/97.jpg)
Choice Based Approachโฆ Example 1: In consumer theory, ๐ต๐ต is a
particular set of all the affordable bundles for a consumer, given his wealth and market prices.
โฆ Example 2: ๐ต๐ต is a particular list of all the universities where you were admitted, among all universities in the scope of your imagination ๐๐, i.e., ๐ต๐ต โ ๐๐.
Advanced Microeconomic Theory 97
![Page 98: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/98.jpg)
Choice Based Approach
2) ๐๐(๏ฟฝ) is a choice rule that selects, for each budget set ๐ต๐ต, a subset of elements in ๐ต๐ต, with the interpretation that ๐๐(๐ต๐ต) are the chosen elements from ๐ต๐ต.
โฆ Example 1: In consumer theory, ๐๐(๐ต๐ต) would be the bundles that the individual chooses to buy, among all bundles he can afford in budget set ๐ต๐ต;
โฆ Example 2: In the example of the universities, ๐๐(๐ต๐ต) would contain the university that you choose to attend.
Advanced Microeconomic Theory 98
![Page 99: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/99.jpg)
Choice Based Approachโฆ Note: If ๐๐(๐ต๐ต) contains a single element, ๐๐(โ ) is a
function; If ๐๐(๐ต๐ต) contains more than one element, ๐๐(โ
) is correspondence.
Advanced Microeconomic Theory 99
![Page 100: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/100.jpg)
Choice Based Approach
Example 1.10 (Choice structures):โฆ Define the set of alternatives as
๐๐ = {๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง}
โฆ Consider two different budget sets๐ต๐ต1 = {๐ฅ๐ฅ, ๐ฆ๐ฆ} and ๐ต๐ต2 = {๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง}
โฆ Choice structure one (โฌ, ๐๐1(๏ฟฝ))๐๐1 ๐ต๐ต1 = ๐๐1 ๐ฅ๐ฅ, ๐ฆ๐ฆ = {๐ฅ๐ฅ}
๐๐1 ๐ต๐ต2 = ๐๐1 ๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง = {๐ฅ๐ฅ}
Advanced Microeconomic Theory 100
![Page 101: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/101.jpg)
Choice Based Approach
โข Example 1.10 (continued):โฆ Choice structure two (โฌ, ๐๐2(๏ฟฝ))
๐๐2 ๐ต๐ต1 = ๐๐2 ๐ฅ๐ฅ, ๐ฆ๐ฆ = {๐ฅ๐ฅ}๐๐2 ๐ต๐ต2 = ๐๐2 ๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง = {๐ฆ๐ฆ}
โฆ Is such a choice rule consistent?We need to impose a consistency requirement
on the choice-based approach, similar to rationality assumption on the preference-based approach.
Advanced Microeconomic Theory 101
![Page 102: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/102.jpg)
Consistency on Choices: the Weak Axiom of Revealed Preference (WARP)
Advanced Microeconomic Theory 102
![Page 103: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/103.jpg)
WARP Weak Axiom of Revealed Preference
(WARP): The choice structure (โฌ, ๐๐(๏ฟฝ))satisfies the WARP if:1) for some budget set ๐ต๐ต โ โฌ with ๐ฅ๐ฅ, ๐ฆ๐ฆ โ ๐ต๐ต,
we have that element ๐ฅ๐ฅ is chosen, ๐ฅ๐ฅ โ ๐๐(๐ต๐ต),then
2) for any other budget set ๐ต๐ตโฒ โ โฌ where alternatives ๐ฅ๐ฅ and ๐ฆ๐ฆ are also available, ๐ฅ๐ฅ, ๐ฆ๐ฆ โ ๐ต๐ตโฒ, and where alternative ๐ฆ๐ฆ is chosen, ๐ฆ๐ฆ โ ๐๐(๐ต๐ตโฒ), then we must have that alternative ๐ฅ๐ฅ is chosen as well, ๐ฅ๐ฅ โ ๐๐(๐ต๐ตโฒ).
Advanced Microeconomic Theory 103
![Page 104: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/104.jpg)
WARP
Example 1.11 (Checking WARP in choice structures):โฆ Take budget set ๐ต๐ต = {๐ฅ๐ฅ, ๐ฆ๐ฆ} with the choice
rule of ๐๐ ๐ฅ๐ฅ, ๐ฆ๐ฆ = ๐ฅ๐ฅ. โฆ Then, for budget set ๐ต๐ตโฒ = {๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง}, the โlegalโ
choice rules are either:๐๐ ๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง = {๐ฅ๐ฅ}, or ๐๐ ๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง = {๐ง๐ง}, or
๐๐ ๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง = {๐ฅ๐ฅ, ๐ง๐ง}
Advanced Microeconomic Theory 104
![Page 105: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/105.jpg)
WARP
Example 1.11 (continued):โฆ This implies, individual decision-maker cannot
select ๐๐ ๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง โ {๐ฆ๐ฆ}
๐๐ ๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง โ {๐ฆ๐ฆ, ๐ง๐ง}๐๐ ๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง โ {๐ฅ๐ฅ, ๐ฆ๐ฆ}
Advanced Microeconomic Theory 105
![Page 106: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/106.jpg)
WARP
Example 1.12 (More on choice structures satisfying/violating WARP:โฆ Take budget set ๐ต๐ต = {๐ฅ๐ฅ, ๐ฆ๐ฆ} with the choice
rule of ๐๐ ๐ฅ๐ฅ, ๐ฆ๐ฆ = {๐ฅ๐ฅ, ๐ฆ๐ฆ}.โฆ Then, for budget set ๐ต๐ตโฒ = {๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง}, the โlegalโ
choices according to WARP are either:๐๐ ๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง = {๐ฅ๐ฅ, ๐ฆ๐ฆ}, or
๐๐ ๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง = {๐ง๐ง}, or ๐๐ ๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง = {๐ฅ๐ฅ, ๐ฆ๐ฆ, ๐ง๐ง}
Advanced Microeconomic Theory 106
![Page 107: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/107.jpg)
WARP
Example 1.12 (continued):โฆ Choice rule satisfying WARP
Advanced Microeconomic Theory 107
B
Bโ
C(Bโ)
C(B)
yx
![Page 108: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/108.jpg)
WARP
Example 1.12 (continued):โฆ Choice rule violating WARP
Advanced Microeconomic Theory 108
B
Bโ
C(Bโ)
C(B)
y
x
![Page 109: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/109.jpg)
Consumption Sets Consumption set: a subset of the
commodity space โ๐ฟ๐ฟ, denoted by ๐ฅ๐ฅ โ โ๐ฟ๐ฟ,whose elements are the consumption bundles that the individual can conceivably consume, given the physical constrains imposed by his environment.
Let us denote a commodity bundle ๐ฅ๐ฅ as a vector of ๐ฟ๐ฟ components.
Advanced Microeconomic Theory 109
![Page 110: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/110.jpg)
Consumption Sets
Physical constraint on the labor market
Advanced Microeconomic Theory 110
![Page 111: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/111.jpg)
x
Beer inSeattleat noon
Beer inBarcelona
at noon
Consumption Sets
Consumption at two different locations
Advanced Microeconomic Theory 111
![Page 112: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/112.jpg)
Consumption Sets
Convex consumption sets:โฆ A consumption set ๐๐ is convex if, for two
consumption bundles ๐ฅ๐ฅ, ๐ฅ๐ฅโฒ โ ๐๐, the bundle ๐ฅ๐ฅโฒโฒ = ๐ผ๐ผ๐ฅ๐ฅ + 1 โ ๐ผ๐ผ ๐ฅ๐ฅโฒ
is also an element of ๐๐ for any ๐ผ๐ผ โ (0,1).
โฆ Intuitively, a consumption set is convex if, for any two bundles that belong to the set, we can construct a straight line connecting them that lies completely within the set.
Advanced Microeconomic Theory 112
![Page 113: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/113.jpg)
Consumption Sets: Economic Constraints
Assumptions on the price vector in โ๐ฟ๐ฟ:
1) All commodities can be traded in a market, at prices that are publicly observable. This is the principle of completeness of markets It discards the possibility that some goods
cannot be traded, such as pollution.
2) Prices are strictly positive for all ๐ฟ๐ฟ goods, i.e., ๐๐ โซ 0 for every good ๐๐. Some prices could be negative, such as pollution.
Advanced Microeconomic Theory 113
![Page 114: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/114.jpg)
Consumption Sets: Economic Constraints
3) Price taking assumption: a consumerโs demand for all ๐ฟ๐ฟ goods represents a small fraction of the total demand for the good.
Advanced Microeconomic Theory 114
![Page 115: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/115.jpg)
Consumption Sets: Economic Constraints
Bundle ๐ฅ๐ฅ โ โ+๐ฟ๐ฟ is affordable if
๐๐1๐ฅ๐ฅ1 + ๐๐2๐ฅ๐ฅ2 + โฏ + ๐๐๐ฟ๐ฟ๐ฅ๐ฅ๐ฟ๐ฟ โค ๐ค๐คor, in vector notation, ๐๐ ๏ฟฝ ๐ฅ๐ฅ โค ๐ค๐ค.
Note that ๐๐ ๏ฟฝ ๐ฅ๐ฅ is the total cost of buying bundle ๐ฅ๐ฅ =(๐ฅ๐ฅ1, ๐ฅ๐ฅ2, โฆ , ๐ฅ๐ฅ๐ฟ๐ฟ) at market prices ๐๐ = (๐๐1, ๐๐2, โฆ , ๐๐๐ฟ๐ฟ), and ๐ค๐ค is the total wealth of the consumer.
When ๐ฅ๐ฅ โ โ+๐ฟ๐ฟ then the set of feasible consumption
bundles consists of the elements of the set:๐ต๐ต๐๐,๐ค๐ค = {๐ฅ๐ฅ โ โ+
๐ฟ๐ฟ : ๐๐ ๏ฟฝ ๐ฅ๐ฅ โค ๐ค๐ค}
Advanced Microeconomic Theory 115
![Page 116: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/116.jpg)
x1
x2
wp2
wp1
p2
p1- (slope)
{x โโ+:p โx = w}2
Consumption Sets: Economic Constraints
๐๐1๐ฅ๐ฅ1 + ๐๐2๐ฅ๐ฅ2 = ๐ค๐ค โน
๐ฅ๐ฅ2 =๐ค๐ค๐๐2
โ๐๐1
๐๐2๐ฅ๐ฅ1
Advanced Microeconomic Theory 116
โข Example: ๐ต๐ต๐๐,๐ค๐ค = {๐ฅ๐ฅ โ โ+2 : ๐๐1๐ฅ๐ฅ1 + ๐๐2๐ฅ๐ฅ2 โค ๐ค๐ค}
![Page 117: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/117.jpg)
x1
x3
x2
Consumption Sets: Economic Constraints
โข Example: ๐ต๐ต๐๐,๐ค๐ค = {๐ฅ๐ฅ โ โ+3 : ๐๐1๐ฅ๐ฅ1 + ๐๐2๐ฅ๐ฅ2 + ๐๐3๐ฅ๐ฅ3 โค
๐ค๐ค}โฆ Budget hyperplane
Advanced Microeconomic Theory 117
![Page 118: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/118.jpg)
Consumption Sets: Economic Constraints
Price vector ๐๐ is orthogonal to the budget line ๐ต๐ต๐๐,๐ค๐ค.โฆ Note that ๐๐ ๏ฟฝ ๐ฅ๐ฅ = ๐ค๐ค holds for any bundle ๐ฅ๐ฅ on the
budget line.โฆ Take any other bundle ๐ฅ๐ฅโฒ which also lies on ๐ต๐ต๐๐,๐ค๐ค.
Hence, ๐๐ ๏ฟฝ ๐ฅ๐ฅโฒ = ๐ค๐ค.โฆ Then,
๐๐ ๏ฟฝ ๐ฅ๐ฅโฒ = ๐๐ ๏ฟฝ ๐ฅ๐ฅ = ๐ค๐ค๐๐ ๏ฟฝ ๐ฅ๐ฅโฒ โ ๐ฅ๐ฅ = 0 or ๐๐ ๏ฟฝ โ๐ฅ๐ฅ = 0
Advanced Microeconomic Theory 118
![Page 119: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/119.jpg)
Consumption Sets: Economic Constraints
โฆ Since this is valid for any two bundles on the budget line, then ๐๐ must be perpendicular to โ๐ฅ๐ฅ on ๐ต๐ต๐๐,๐ค๐ค.
โฆ This implies that the price vector is perpendicular (orthogonal) to ๐ต๐ต๐๐,๐ค๐ค.
Advanced Microeconomic Theory 119
![Page 120: Econs 501-Microeconomic Theory I](https://reader030.fdocuments.in/reader030/viewer/2022020621/61e6ec3ae0e59027783dc457/html5/thumbnails/120.jpg)
Consumption Sets: Economic Constraints
The budget set ๐ต๐ต๐๐,๐ค๐ค is convex.โฆ We need that, for any two bundles ๐ฅ๐ฅ, ๐ฅ๐ฅโฒ โ
๐ต๐ต๐๐,๐ค๐ค, their convex combination๐ฅ๐ฅโฒโฒ = ๐ผ๐ผ๐ฅ๐ฅ + 1 โ ๐ผ๐ผ ๐ฅ๐ฅโฒ
also belongs to the ๐ต๐ต๐๐,๐ค๐ค, where ๐ผ๐ผ โ (0,1).
โฆ Since ๐๐ ๏ฟฝ ๐ฅ๐ฅ โค ๐ค๐ค and ๐๐ ๏ฟฝ ๐ฅ๐ฅโฒ โค ๐ค๐ค, then ๐๐ ๏ฟฝ ๐ฅ๐ฅโฒโฒ = ๐๐๐ผ๐ผ๐ฅ๐ฅ + ๐๐ 1 โ ๐ผ๐ผ ๐ฅ๐ฅโฒ
= ๐ผ๐ผ๐๐๐ฅ๐ฅ + 1 โ ๐ผ๐ผ ๐๐๐ฅ๐ฅโฒ โค ๐ค๐ค
Advanced Microeconomic Theory 120