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Econs 501-Microeconomic Theory I
WELCOME!!!
TA: PakSing Choi (Sunny)
MWG and Munoz-Garcia
Wednesdays (Assignments)
Recitations: Fridays 1-2pm (Hulbert 23)
Course website:
◦ https://anaespinolaarredondo.com/econs-501-microeconomic-theory-i/
Advanced Microeconomic Theory 1
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Preferences and Utility
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How can we formally describe an individual’s
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
individual’s preference can be mathematically
represented with a utility function.
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Preference and Choice
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Preference and Choice
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Advantages:
Advanced Microeconomic Theory 6
Preference-based approach:
◦ More tractable when the set of alternatives
𝑋has many elements.
Choice-based approach:
◦ It is based on observables (actual choices)
rather than on unobservables (I.P)
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Preference-Based Approach
Preferences: “attitudes” of the decision-
maker towards a set of possible alternatives
𝑋.
For any 𝑥, 𝑦 ∈ 𝑋, how do you compare 𝑥and 𝑦?
I prefer 𝑥 to 𝑦 (𝑥 ≻ 𝑦)
I prefer 𝑦 to 𝑥 (𝑦 ≻ 𝑥)
I am indifferent (𝑥 ∼ 𝑦)
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Preferences
Advanced Microeconomic Theory 8
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Preference-Based Approach
Completeness:
◦ For an pair of alternatives 𝑥, 𝑦 ∈ 𝑋, the
individual decision maker:
𝑥 ≻ 𝑦, or
𝑦 ≻ 𝑥, or
both, i.e., 𝑥 ∼ 𝑦
Advanced Microeconomic Theory 9
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Preference-Based Approach
Not all binary relations satisfy Completeness.
Example:
◦ “Is the brother of”: John ⊁ Bob and Bob ⊁John if they are not brothers.
◦ “Is the father of”: John ⊁ Bob and Bob ⊁ John if the two individuals are not related.
Not all pairs of alternatives are comparable according to these two relations.
Advanced Microeconomic Theory 10
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Preference-Based Approach
Advanced Microeconomic Theory 11
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Preference-Based Approach
Advanced Microeconomic Theory 12
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Preference-Based Approach
Advanced Microeconomic Theory 13
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Preference-Based Approach
Advanced Microeconomic Theory 14
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Preference-Based Approach
Sources of intransitivity:
a) Indistinguishable alternatives
a) Examples?
b) Framing effects
c) Aggregation of criteria
d) Change in preferences
a) Examples?
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Preference-Based Approach
• Example 1.1 (Indistinguishable alternatives):
◦ Take 𝑋 = ℝ as a piece of pie and 𝑥 ≻ 𝑦 if 𝑥 ≥ 𝑦 −1 (𝑥 + 1 ≥ 𝑦) but 𝑥~𝑦 if 𝑥 − 𝑦 < 1(indistinguishable).
◦ Then,
1.5~0.8 since 1.5 − 0.8 = 0.7 < 1
0.8~0.3 since 0.8 − 0.3 = 0.5 < 1
◦ By transitivity, we would have 1.5~0.3, but in fact
1.5 ≻ 0.3 (intransitive preference relation).
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Preference-Based Approach
Other examples:
◦ similar shades of gray paint
◦ milligrams of sugar in your coffee
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Utility Function
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Utility Function
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Utility Function
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Desirability
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Desirability
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Desirability
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Desirability
Advanced Microeconomic Theory 24
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Desirability
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Desirability
Advanced Microeconomic Theory 26
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Desirability
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Indifference sets
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Upper contour set (UCS){y +: y x}2
Indifference set{y +: y ~ x}2
Lower contour set (LCS){y +: y x}2
x1
x2
x
Indifference sets
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Indifference sets
Note:
◦ 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.
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Convexity of Preferences
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Convexity of Preferences
Convexity 1
Advanced Microeconomic Theory 32
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Convexity of Preferences
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Convexity of Preferences
Convexity 2
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Convexity of Preferences
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x1
x2
λx + (1 λ)y z
UCS
x
x
yy
z
Convexity of Preferences
Strictly convex preferences
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Convexity of Preferences
Convexity but not strict convexity
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– 𝜆𝑥 + 1 − 𝜆 𝑦~𝑧
– Such preference relation
is represented by utility
function such as
𝑢 𝑥1, 𝑥2 = 𝑎𝑥1 + 𝑏𝑥2
where 𝑥1 and 𝑥2 are
substitutes.
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Convexity of Preferences
Convexity but not strict convexity
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– 𝜆𝑥 + 1 − 𝜆 𝑦~𝑧
– Such preference relation
is represented by utility
function such as
𝑢 𝑥1, 𝑥2 = min{𝑎𝑥1, 𝑏𝑥2}
where 𝑎, 𝑏 > 0.
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Convexity of Preferences
Example 1.6
39
𝑢 𝑥1, 𝑥2 Satisfies
convexity
Satisfies strict
convexity
𝑎𝑥1 + 𝑏𝑥2 √ X
min{𝑎𝑥1, 𝑏𝑥2} √ X
𝑎𝑥1
12 + 𝑏𝑥2
12 √ √
𝑎𝑥12 + 𝑏𝑥2
2 X X
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Convexity of Preferences
1) Taste for diversification:
◦ An individual with
convex preferences
prefers the convex
combination of
bundles 𝑥 and 𝑦,
than either of those
bundles alone.
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• Interpretation of convexity
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Convexity of Preferences
Interpretation of convexity
2) Diminishing marginal rate of substitution:
𝑀𝑅𝑆1,2 ≡𝜕𝑢/𝜕𝑥1
𝜕𝑢/𝜕𝑥2
◦ MRS describes the additional amount of good 1 that the consumer needs to receive in order to keep her utility level unaffected.
◦ 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.
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x1
x2
A
B
C
D
1 unit = x2
1 unit = x2
x1 x1
Convexity of Preferences
Diminishing marginal rate of substitution
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Convexity of Preferences
Advanced Microeconomic Theory 43
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Convexity of Preferences
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Quasiconcavity
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Quasiconcavity
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Quasiconcavity
Quasiconcavity
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Quasiconcavity
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1x
2x
u x u y
x
y
1x y
1u x y
Quasiconcavity
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Quasiconcavity
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Quasiconcavity
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Quasiconcavity
Concavity implies quasiconcavity
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Quasiconcavity
Advanced Microeconomic Theory 53
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1x
2x
u
1 1
4 41 2 1 1,u x x x x
Quasiconcavity
Concave and quasiconcave utility function (3D)
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𝑢(𝑥1, 𝑥2) = 𝑥1
14𝑥2
14
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Quasiconcavity
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2x1x
v
6 6
4 41 2 1 1,v x x x x
Quasiconcavity
Convex but quasiconcave utility function (3D)
56
𝑣(𝑥1, 𝑥2) = 𝑥1
64𝑥2
64
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Quasiconcavity
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Quasiconcavity
•Advanced Microeconomic Theory 58
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Quasiconcavity
Example 1.7 (continued):
◦ Let us consider the case of only two goods,
𝐿 = 2.
◦ Then, an individual prefers a bundle 𝑥 =(𝑥1, 𝑥2) to another bundle 𝑦 = (𝑦1, 𝑦2) iff 𝑥contains more units of both goods than
bundle 𝑦, i.e., 𝑥1 ≥ 𝑦1 and 𝑥2 ≥ 𝑦2.
◦ For illustration purposes, let us take bundle
such as (2,1).
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Quasiconcavity
Example 1.7 (continued):
Advanced Microeconomic Theory 60
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Quasiconcavity
Example 1.7 (continued):
1) UCS:
◦ The upper contour set of bundle (2,1) contains
bundles (𝑥1, 𝑥2) with weakly more than 2 units
of good 1 and/or weakly more than 1 unit of
good 2:
𝑈𝐶𝑆 2,1 = {(𝑥1, 𝑥2) ≿ (2,1) ⟺ 𝑥1 ≥ 2, 𝑥2 ≥ 1}
◦ The frontiers of the UCS region also represent
bundles preferred to (2,1).
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Quasiconcavity
Example 1.7 (continued):
2) LCS:
◦ The bundles in the lower contour set of
bundle (2,1) contain fewer units of both
goods:
𝐿𝐶𝑆 2,1 = {(2,1) ≿ (𝑥1, 𝑥2) ⟺ 𝑥1 ≤ 2, 𝑥2 ≤ 1}
◦ The frontiers of the LCS region also
represent bundles with fewer unis of
either good 1 or good 2.
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Quasiconcavity
Advanced Microeconomic Theory 63
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Quasiconcavity
Example 1.7 (continued):
4) Regions A and B:
◦ Region 𝐴 contains bundles with more units of
good 2 but fewer units of good 1 (the opposite
argument applies to region 𝐵).
◦ The consumer cannot compare bundles in
either of these regions against bundle 2,1 .
◦ 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.
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Quasiconcavity
Example 1.7 (continued):
5) Preference relation is not complete:
◦ Completeness requires for every pair 𝑥and 𝑦, either 𝑥 ≿ 𝑦 or 𝑦 ≿ 𝑥 (or both).
◦ Consider two bundles 𝑥, 𝑦 ∈ ℝ+2 with
bundle 𝑥 containing more units of good 1
than bundle 𝑦 but fewer units of good 2,
i.e., 𝑥1 > 𝑦1 and 𝑥2 < 𝑦2 (as in Region B)
◦ Then, we have neither 𝑥 ≿ 𝑦 nor 𝑦 ≿ 𝑥.
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Quasiconcavity
Example 1.7 (continued):
6) Preference relation is transitive:
◦ Transitivity requires that, for any three
bundles 𝑥, 𝑦 and 𝑧, if 𝑥 ≿ 𝑦 and 𝑦 ≿ 𝑧then 𝑥 ≿ 𝑧.
◦ Now 𝑥 ≿ 𝑦 and 𝑦 ≿ 𝑧 means 𝑥𝑙 ≥ 𝑦𝑙 and
𝑦𝑙 ≥ 𝑧𝑙 for all 𝑙 goods.
◦ Then, 𝑥𝑙 ≥ 𝑧𝑙 implies 𝑥 ≿ 𝑧.
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Quasiconcavity
Example 1.7 (continued):
7) Preference relation is strongly monotone:
◦ 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.
◦ Now 𝑥 ≥ 𝑦 and 𝑥 ≠ 𝑦 implies that 𝑥𝑙 ≥ 𝑦𝑙 for all good 𝑙 and 𝑥𝑘 > 𝑦𝑘 for at least one good 𝑘.
◦ Thus, 𝑥 ≥ 𝑦 and 𝑥 ≠ 𝑦 implies 𝑥 ≿ 𝑦 and not 𝑦 ≿ 𝑥.
◦ Thus, we can conclude that 𝑥 ≻ 𝑦.
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Quasiconcavity
Example 1.7 (continued):
8) Preference relation is strictly convex:
◦ Strict convexity requires that if 𝑥 ≿ 𝑧 and 𝑦 ≿𝑧 and 𝑥 ≠ 𝑧, then 𝛼𝑥 + 1 − 𝛼 𝑦 ≻ 𝑧 for all
𝛼 ∈ 0,1 .
◦ Now 𝑥 ≿ 𝑧 and 𝑦 ≿ 𝑧 implies that 𝑥𝑙 ≥ 𝑦𝑙 and
𝑦𝑙 ≥ 𝑧𝑙 for all good 𝑙.
◦ 𝑥 ≠ 𝑧 implies, for some good 𝑘, we must have
𝑥𝑘 > 𝑧𝑘.
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Quasiconcavity
Example 1.7 (continued):
◦ Hence, for any 𝛼 ∈ 0,1 , we must have that
𝛼𝑥𝑙 + 1 − 𝛼 𝑦𝑙 ≥ 𝑧𝑙 for all good 𝑙
𝛼𝑥𝑘 + 1 − 𝛼 𝑦𝑘 > 𝑧𝑘 for some 𝑘
◦ Thus, we have that 𝛼𝑥 + 1 − 𝛼 𝑦 ≥ 𝑧 and 𝛼𝑥 + 1 − 𝛼 𝑦 ≠ 𝑧, and so
𝛼𝑥 + 1 − 𝛼 𝑦 ≿ 𝑧
and not 𝑧 ≿ 𝛼𝑥 + 1 − 𝛼 𝑦
◦ Therefore, 𝛼𝑥 + 1 − 𝛼 𝑦 ≻ 𝑧.
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Common Utility Functions
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Common Utility Functions
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Common Utility Functions
◦ Marginal utilities: 𝜕𝑢
𝜕𝑥1> 0 and
𝜕𝑢
𝜕𝑥2> 0
◦ A diminishing MRS
𝑀𝑅𝑆𝑥1,𝑥2=
𝛼𝐴𝑥1𝛼−1𝑥2
𝛽
𝛽𝐴𝑥1𝛼𝑥2
𝛽−1=
𝛼𝑥2
𝛽𝑥1
which is decreasing in 𝑥1.
Hence, indifference curves become flatter as 𝑥1
increases.
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2x
1x
IC
A
B
CD
1 unit 1 unit
2in x
2x
Common Utility Functions
Cobb-Douglas preference
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Common Utility Functions
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Common Utility Functions
Perfect substitutes
75
2A
A
2BB
Aslope
B
2x
1x
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Common Utility Functions
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Common Utility Functions
Advanced Microeconomic Theory 77
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Common Utility Functions
Perfect complements
78
2x
1x
2
1 2
1u A
2 2u A
2 1x x
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Common Utility Functions
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Common Utility Functions
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Common Utility Functions
CES preferences
Advanced Microeconomic Theory 81
2x
1x
8
1
0.2
0
Cobb-Douglas
Perfect complement
Perfect substitutes
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Common Utility Functions
◦ CES utility function is often presented as
𝑢 𝑥1, 𝑥
2= 𝑎𝑥1
𝜌+ 𝑏𝑥2
𝜌1𝜌
where 𝜌 ≡𝜎−1
𝜎.
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Common Utility Functions
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Common Utility Functions
MRS of quasilinear preferences
Advanced Microeconomic Theory 84
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Common Utility Functions
◦ For 𝑢 𝑥1, 𝑥2 = 𝑣 𝑥1 + 𝑏𝑥2, the marginal utilities are
𝜕𝑢
𝜕𝑥2= 𝑏 and
𝜕𝑢
𝜕𝑥1=
𝜕𝑣
𝜕𝑥1
which implies
𝑀𝑅𝑆𝑥1,𝑥2=
𝜕𝑣
𝜕𝑥1
𝑏
◦ Quasilinear preferences are often used to represent the consumption of goods that are relatively insensitive to income.
◦ Examples: garlic, toothpaste, etc.
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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 𝑋 is continuous if it is preserved under limits.
◦ That is, for any sequence of pairs
(𝑥𝑛, 𝑦𝑛) 𝑛=1∞ with 𝑥𝑛 ≿ 𝑦𝑛 for all 𝑛
and lim𝑛→∞
𝑥𝑛 = 𝑥 and lim𝑛→∞
𝑦𝑛 = 𝑦, the preference
relation is maintained in the limiting points, i.e., 𝑥 ≻ 𝑦.
Advanced Microeconomic Theory 86
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Continuous Preferences
◦ Intuitively, there can be no sudden jumps (i.e.,
preference reversals) in an individual preference
over a sequence of bundles.
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Continuous Preferences
Lexicographic preferences are not continuous
◦ Consider the sequence 𝑥𝑛 =1
𝑛, 0 and 𝑦𝑛 =
(0,1), where 𝑛 = {0,1,2,3, … }.
◦ The sequence 𝑦𝑛 = (0,1) is constant in 𝑛.
◦ The sequence 𝑥𝑛 =1
𝑛, 0 is not:
It starts at 𝑥1 = 1,0 , and moves leftwards to
𝑥2 =1
2, 0 , 𝑥3 =
1
3, 0 , etc.
Advanced Microeconomic Theory 88
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x1
x2
1
10 ⅓ ½¼
x 4 x 3 x 2 x 1
y n, n, y 1 = y 2 = = y n
lim x n = (0,0)n
Continuous Preferences
Thus, the individual prefers:𝑥1 = 1,0 ≻ 0,1 = 𝑦1
𝑥2 =1
2, 0 ≻ 0,1 = 𝑦2
𝑥3 =1
3, 0 ≻ 0,1 = 𝑦3
⋮
But, lim
𝑛→∞𝑥𝑛 = 0,0 ≺ 0,1
= lim𝑛→∞
𝑦𝑛
Preference reversal!
Advanced Microeconomic Theory 89
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Existence of Utility Function
If a preference relation satisfies monotonicity and
continuity, then there exists a utility function 𝑢(∙)representing such preference relation.
Proof:
◦ Take a bundle 𝑥 ≠ 0.
◦ By monotonicity, 𝑥 ≿ 0, where 0 = (0,0, … , 0).
That is, if bundle 𝑥 ≠ 0, it contains positive amounts
of at least one good and, it is preferred to bundle 0.
Advanced Microeconomic Theory 90
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Existence of Utility Function
◦ Define bundle 𝑀 as the bundle where all
components coincide with the highest
component of bundle 𝑥:
𝑀 = max𝑘
{𝑥𝑘} , … , max𝑘
{𝑥𝑘}
◦ Hence, by monotonicity, 𝑀 ≿ 𝑥.
◦ Bundles 0 and 𝑀 are both on the main
diagonal, since each of them contains the
same amount of good 𝑥1 and 𝑥2.
Advanced Microeconomic Theory 91
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x2
x1
Existence of Utility Function
Advanced Microeconomic Theory 92
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Existence of Utility Function
◦ By continuity and monotonicity, there exists a
bundle that is indifferent to 𝑥 and which lies
on the main diagonal.
◦ By monotonicity, this bundle is unique
Otherwise, modifying any of its components
would lead to higher/lower indifference curves.
◦ Denote such bundle as
𝑡 𝑥 , 𝑡 𝑥 , … , 𝑡(𝑥)
◦ Let 𝑢 𝑥 = 𝑡 𝑥 , which is a real number.
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Existence of Utility Function
◦ Applying the same steps for another bundle 𝑦 ≠ 𝑥, we
obtain
𝑡 𝑦 , 𝑡 𝑦 , … , 𝑡(𝑦)
and let 𝑢 𝑦 = 𝑡 𝑦 , which is also a real number.
◦ We know that
𝑥~ 𝑡 𝑥 , 𝑡 𝑥 , … , 𝑡(𝑥)𝑦~ 𝑡 𝑦 , 𝑡 𝑦 , … , 𝑡(𝑦)
𝑥 ≿ 𝑦
◦ Hence, by transitivity, 𝑥 ≿ 𝑦 iff
𝑥~ 𝑡 𝑥 , 𝑡 𝑥 , … , 𝑡(𝑥) ≿ 𝑡 𝑦 , 𝑡 𝑦 , … , 𝑡(𝑦) ~𝑦
Advanced Microeconomic Theory 94
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Existence of Utility Function
◦ And by monotonicity,
𝑥 ≿ 𝑦 ⟺ 𝑡 𝑥 ≥ 𝑡 𝑦 ⟺ 𝑢(𝑥) ≥ 𝑢(𝑦)
◦ Note: A utility function can satisfy continuity
but still be non-differentiable.
For instance, the Leontief utility function,
min{𝑎𝑥1,𝑏𝑥2}, is continuous but cannot be
differentiated at the kink.
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Choice Based Approach
We now focus on the actual choice behavior
rather than individual preferences.
◦ From the alternatives in set 𝐴, which one would
you choose?
A choice structure (ℬ, 𝑐(∙)) contains two
elements:
1) ℬ is a family of nonempty subsets of 𝑋, so that
every element of ℬ is a set 𝐵 ⊂ 𝑋.
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Choice Based Approach
◦ Example 1: In consumer theory, 𝐵 is a
particular set of all the affordable bundles for
a consumer, given his wealth and market
prices.
◦ Example 2: 𝐵 is a particular list of all the
universities where you were admitted, among
all universities in the scope of your
imagination 𝑋, i.e., 𝐵 ⊂ 𝑋.
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Choice Based Approach
2) 𝑐(∙) is a choice rule that selects, for each budget set 𝐵, a subset of elements in 𝐵, with the interpretation that 𝑐(𝐵) are the chosen elements from 𝐵.
◦ Example 1: In consumer theory, 𝑐(𝐵) would be the bundles that the individual chooses to buy, among all bundles he can afford in budget set 𝐵;
◦ Example 2: In the example of the universities, 𝑐(𝐵) would contain the university that you choose to attend.
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Choice Based Approach
◦ Note:
If 𝑐(𝐵) contains a single element, 𝑐(⋅) is a
function;
If 𝑐(𝐵) contains more than one element, 𝑐(⋅) is correspondence.
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Choice Based Approach
Example 1.10 (Choice structures):
◦ Define the set of alternatives as
𝑋 = {𝑥, 𝑦, 𝑧}
◦ Consider two different budget sets
𝐵1 = {𝑥, 𝑦} and 𝐵2 = {𝑥, 𝑦, 𝑧}
◦ Choice structure one (ℬ, 𝑐1(∙))𝑐1 𝐵1 = 𝑐1 𝑥, 𝑦 = {𝑥}
𝑐1 𝐵2 = 𝑐1 𝑥, 𝑦, 𝑧 = {𝑥}
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Choice Based Approach
• Example 1.10 (continued):
◦ Choice structure two (ℬ, 𝑐2(∙))𝑐2 𝐵1 = 𝑐2 𝑥, 𝑦 = {𝑥}
𝑐2 𝐵2 = 𝑐2 𝑥, 𝑦, 𝑧 = {𝑦}
◦ 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.
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Consistency on Choices: the
Weak Axiom of Revealed
Preference (WARP)
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WARP
Weak Axiom of Revealed Preference (WARP): The choice structure (ℬ, 𝑐(∙))satisfies the WARP if:
1) for some budget set 𝐵 ∈ ℬ with 𝑥, 𝑦 ∈ 𝐵, we have that element 𝑥 is chosen, 𝑥 ∈ 𝑐(𝐵),then
2) for any other budget set 𝐵′ ∈ ℬ where alternatives 𝑥 and 𝑦 are also available, 𝑥, 𝑦 ∈ 𝐵′, and where alternative 𝑦 is chosen, 𝑦 ∈ 𝑐(𝐵′), then we must have that alternative 𝑥 is chosen as well, 𝑥 ∈ 𝑐(𝐵′).
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WARP
Example 1.11 (Checking WARP in
choice structures):
◦ Take budget set 𝐵 = {𝑥, 𝑦} with the choice
rule of 𝑐 𝑥, 𝑦 = 𝑥.
◦ Then, for budget set 𝐵′ = {𝑥, 𝑦, 𝑧}, the “legal”
choice rules are either:
𝑐 𝑥, 𝑦, 𝑧 = {𝑥}, or
𝑐 𝑥, 𝑦, 𝑧 = {𝑧}, or
𝑐 𝑥, 𝑦, 𝑧 = {𝑥, 𝑧}
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WARP
Example 1.11 (continued):
◦ This implies, individual decision-maker cannot
select
𝑐 𝑥, 𝑦, 𝑧 ≠ {𝑦}𝑐 𝑥, 𝑦, 𝑧 ≠ {𝑦, 𝑧}𝑐 𝑥, 𝑦, 𝑧 ≠ {𝑥, 𝑦}
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WARP
Example 1.12 (More on choice
structures satisfying/violating WARP:
◦ Take budget set 𝐵 = {𝑥, 𝑦} with the choice
rule of 𝑐 𝑥, 𝑦 = {𝑥, 𝑦}.
◦ Then, for budget set 𝐵′ = {𝑥, 𝑦, 𝑧}, the “legal”
choices according to WARP are either:
𝑐 𝑥, 𝑦, 𝑧 = {𝑥, 𝑦}, or
𝑐 𝑥, 𝑦, 𝑧 = {𝑧}, or
𝑐 𝑥, 𝑦, 𝑧 = {𝑥, 𝑦, 𝑧}
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WARP
Example 1.12 (continued):
◦ Choice rule satisfying WARP
Advanced Microeconomic Theory 107
B
B
C(B )
C(B)
y
x
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WARP
Example 1.12 (continued):
◦ Choice rule violating WARP
Advanced Microeconomic Theory 108
B
B
C(B )
C(B)
y
x
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Consumption Sets
Consumption set: a subset of the
commodity space ℝ𝐿, denoted by 𝑥 ⊂ ℝ𝐿,
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 𝑥 as a
vector of 𝐿 components.
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Consumption Sets
Physical constraint on the labor market
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x
Beer inSeattleat noon
Beer inBarcelonaat noon
Consumption Sets
Consumption at two different locations
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Consumption Sets
Convex consumption sets:
◦ A consumption set 𝑋 is convex if, for two
consumption bundles 𝑥, 𝑥′ ∈ 𝑋, the bundle
𝑥′′ = 𝛼𝑥 + 1 − 𝛼 𝑥′
is also an element of 𝑋 for any 𝛼 ∈ (0,1).
◦ 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.
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Consumption Sets: Economic Constraints
Assumptions on the price vector in ℝ𝐿:
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 𝐿 goods, i.e.,
𝑝 ≫ 0 for every good 𝑘.
Some prices could be negative, such as pollution.
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Consumption Sets: Economic Constraints
3) Price taking assumption: a consumer’s
demand for all 𝐿 goods represents a small
fraction of the total demand for the good.
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Consumption Sets: Economic Constraints
Bundle 𝑥 ∈ ℝ+𝐿 is affordable if
𝑝1𝑥1 + 𝑝2𝑥2 + ⋯ + 𝑝𝐿𝑥𝐿 ≤ 𝑤
or, in vector notation, 𝑝 ∙ 𝑥 ≤ 𝑤.
Note that 𝑝 ∙ 𝑥 is the total cost of buying bundle 𝑥 =(𝑥1, 𝑥2, … , 𝑥𝐿) at market prices 𝑝 = (𝑝1, 𝑝2, … , 𝑝𝐿), and 𝑤 is the total wealth of the consumer.
When 𝑥 ∈ ℝ+𝐿 then the set of feasible consumption
bundles consists of the elements of the set:
𝐵𝑝,𝑤 = {𝑥 ∈ ℝ+𝐿 : 𝑝 ∙ 𝑥 ≤ 𝑤}
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x1
x2
wp2
wp1
p2
p1- (slope)
{x +:p x = w}2
Consumption Sets: Economic Constraints
𝑝1𝑥1 + 𝑝2𝑥2 = 𝑤 ⟹
𝑥2 =𝑤
𝑝2−
𝑝1
𝑝2𝑥1
Advanced Microeconomic Theory 116
• Example: 𝐵𝑝,𝑤 = {𝑥 ∈ ℝ+2 : 𝑝1𝑥1 + 𝑝2𝑥2 ≤ 𝑤}
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x1
x3
x2
Consumption Sets: Economic Constraints
• Example: 𝐵𝑝,𝑤 = {𝑥 ∈ ℝ+3 : 𝑝1𝑥1 + 𝑝2𝑥2 + 𝑝3𝑥3 ≤
𝑤}◦ Budget hyperplane
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Consumption Sets: Economic Constraints
Price vector 𝑝 is orthogonal to the budget line
𝐵𝑝,𝑤.
◦ Note that 𝑝 ∙ 𝑥 = 𝑤 holds for any bundle 𝑥 on the
budget line.
◦ Take any other bundle 𝑥′ which also lies on 𝐵𝑝,𝑤.
Hence, 𝑝 ∙ 𝑥′ = 𝑤.
◦ Then,
𝑝 ∙ 𝑥′ = 𝑝 ∙ 𝑥 = 𝑤
𝑝 ∙ 𝑥′ − 𝑥 = 0 or 𝑝 ∙ ∆𝑥 = 0
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Consumption Sets: Economic Constraints
◦ Since this is valid for any two bundles on the
budget line, then 𝑝 must be perpendicular to
∆𝑥 on 𝐵𝑝,𝑤.
◦ This implies that the price vector is
perpendicular (orthogonal) to 𝐵𝑝,𝑤.
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Consumption Sets: Economic Constraints
The budget set 𝐵𝑝,𝑤 is convex.
◦ We need that, for any two bundles 𝑥, 𝑥′ ∈𝐵𝑝,𝑤, their convex combination
𝑥′′ = 𝛼𝑥 + 1 − 𝛼 𝑥′
also belongs to the 𝐵𝑝,𝑤, where 𝛼 ∈ (0,1).
◦ Since 𝑝 ∙ 𝑥 ≤ 𝑤 and 𝑝 ∙ 𝑥′ ≤ 𝑤, then
𝑝 ∙ 𝑥′′ = 𝑝𝛼𝑥 + 𝑝 1 − 𝛼 𝑥′= 𝛼𝑝𝑥 + 1 − 𝛼 𝑝𝑥′ ≤ 𝑤
Advanced Microeconomic Theory 120