Preferences and Utility How can we formally describe an individual’s preference for different amounts of a good?
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.
2 Preference and Choice
3 Preference and Choice
4 Advantages:
Preference-based approach: ◦ More tractable when the set of alternatives 푋has many elements.
Choice-based approach: ◦ It is based on observables (actual choices) rather than on unobservables (I.P)
Advanced Microeconomic Theory 5 Preference-Based Approach
Preferences: “attitudes” of the decision- maker towards a set of possible alternatives 푋.
For any 푥, 푦 ∈ 푋, how do you compare 푥 and 푦? I prefer 푥 to 푦 (푥 ≻ 푦) I prefer 푦 to 푥 (푦 ≻ 푥) I am indifferent (푥 ∼ 푦)
6 Preference-Based Approach
By asking: We impose the assumption: Tick one box Completeness: individuals must (i.e., not refrain from compare any two alternatives, answering) even the ones they don’t know. Tick only one box The individual is capable of comparing any pair of alternatives. Don’t add any new box in We don’t allow the individual to which the individual says, “I specify the intensity of his love 푥 and hate 푦” preferences.
7 Preference-Based Approach
Completeness: ◦ For an pair of alternatives 푥, 푦 ∈ 푋, the individual decision maker: . 푥 ≻ 푦, or . 푦 ≻ 푥, or . both, i.e., 푥 ∼ 푦
Advanced Microeconomic Theory 8 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 9 Preference-Based Approach
Advanced Microeconomic Theory 10 Preference-Based Approach
Advanced Microeconomic Theory 11 Preference-Based Approach
Advanced Microeconomic Theory 12 Preference-Based Approach
Advanced Microeconomic Theory 13 Preference-Based Approach
Sources of intransitivity: a) Indistinguishable alternatives a) Examples? b) Framing effects c) Aggregation of criteria d) Change in preferences a) Examples?
14 Preference-Based Approach
• Example 1.1 (Indistinguishable alternatives): ◦ Take 푋 = ℝ as a piece of pie and 푥 ≻ 푦 if 푥 ≥ 푦 − 1 (푥 + 1 ≥ 푦) but 푥~푦 if 푥 − 푦 < 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).
15 Preference-Based Approach
Other examples: ◦ similar shades of gray paint
◦ milligrams of sugar in your coffee
16 Preference-Based Approach
17 Preference-Based Approach
18 Preference-Based Approach
19 Preference-Based Approach
Example 1.3 (continued): ◦ By majority of these considerations: 푀퐼푇 ≿ณ 푊푆푈 ≿ณ 퐻표푚푒 푈푛푖푣 ≿ณ 푀퐼푇 criteria 1 & 3 criteria 1 & 2 criteria 2 & 3 ◦ Transitivity is violated due to a cycle.
◦ A similar argument can be used for the aggregation of individual preferences in group decision-making: . Every person in the group has a different (transitive) preference relation but the group preferences are not necessarily transitive (“Condorcet paradox”).
20 Preference-Based Approach
21 Utility Function
22 Utility Function
23 Utility Function
24 Desirability
25 Desirability
26 Desirability
27 Desirability
Advanced Microeconomic Theory 28 Desirability
29 Desirability
Advanced Microeconomic Theory 30 Desirability
31 Indifference sets
32 Indifference sets
x2 Upper contour set (UCS) 2 {y +: y x}
x
Indifference set 2 {y +: y ~ x}
Lower contour set (LCS) 2 {y +: y x}
x1
33 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.
34 Convexity of Preferences
35 Convexity of Preferences
Convexity 1
Advanced Microeconomic Theory 36 Convexity of Preferences
37 Convexity of Preferences
Convexity 2
38 Convexity of Preferences
39 Convexity of Preferences
Strictly convex preferences
x2 x x λx + (1 λ)y z
UCS z y y
x1
40 Convexity of Preferences
Convexity but not strict convexity
– 휆푥 + 1 − 휆 푦~푧 – Such preference relation is represented by utility function such as
푢 푥1, 푥2 = 푎푥1 + 푏푥2
where 푥1 and 푥2 are substitutes.
41 Convexity of Preferences
Convexity but not strict convexity
– 휆푥 + 1 − 휆 푦~푧 – Such preference relation is represented by utility function such as
푢 푥1, 푥2 = min{푎푥1, 푏푥2} where 푎, 푏 > 0.
42 Convexity of Preferences
Example 1.6
푢 푥1, 푥2 Satisfies Satisfies strict convexity convexity
푎푥1 + 푏푥2 √ X
min{푎푥1, 푏푥2} √ X 1 1 2 2 √ √ 푎푥1 + 푏푥2 2 2 푎푥1 + 푏푥2 X X
43 Convexity of Preferences
• Interpretation of convexity 1) Taste for diversification: ◦ An individual with convex preferences prefers the convex combination of bundles 푥 and 푦, than either of those bundles alone.
44 Convexity of Preferences
Interpretation of convexity 2) Diminishing marginal rate of substitution: 휕푢/휕푥1 푀푅푆1,2 ≡ 휕푢/휕푥2 ◦ 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.
45 Convexity of Preferences
Diminishing marginal rate of substitution
x2 A
1 unit = x2 B
C
1 unit = x2 D
x1 x1 x1
46 Convexity of Preferences
Advanced Microeconomic Theory 47 Convexity of Preferences
48 Quasiconcavity
49 Quasiconcavity
50 Quasiconcavity
Quasiconcavity
51 Quasiconcavity
52 Quasiconcavity
x2
x xy1
uxy1
y u xu y
x1
53 Quasiconcavity
54 Quasiconcavity
55 Quasiconcavity
Concavity implies quasiconcavity
56 Quasiconcavity
Advanced Microeconomic Theory 57 Quasiconcavity
Concave and quasiconcave utility function (3D)
111 1 푢(푥 , 푥 ) = 푥444푥4 u x11, x 22 x 11 2 x 1
u
x1
x2
58 Quasiconcavity
59 Quasiconcavity
Convex but quasiconcave utility function (3D)
666 6 푣(푥 , 푥 ) = 푥444푥4 vxxx 12111, x 2 1 2
v
x1 x2
60 Quasiconcavity
61 Quasiconcavity
•Advanced Microeconomic Theory 62 Quasiconcavity
Example 1.7 (continued): ◦ Let us consider the case of only two goods, 퐿 = 2.
◦ Then, an individual prefers a bundle 푥 = (푥1, 푥2) to another bundle 푦 = (푦1, 푦2) iff 푥 contains more units of both goods than bundle 푦, i.e., 푥1 ≥ 푦1 and 푥2 ≥ 푦2. ◦ For illustration purposes, let us take bundle such as (2,1).
63 Quasiconcavity
Example 1.7 (continued):
Advanced Microeconomic Theory 64 Quasiconcavity
Example 1.7 (continued): 1) UCS: ◦ The upper contour set of bundle (2,1) contains bundles (푥1, 푥2) with weakly more than 2 units of good 1 and/or weakly more than 1 unit of good 2:
푈퐶푆 2,1 = {(푥1, 푥2) ≿ (2,1) ⟺ 푥1 ≥ 2, 푥2 ≥ 1} ◦ The frontiers of the UCS region also represent bundles preferred to (2,1).
65 Quasiconcavity
Example 1.7 (continued): 2) LCS: ◦ The bundles in the lower contour set of bundle (2,1) contain fewer units of both goods:
퐿퐶푆 2,1 = {(2,1) ≿ (푥1, 푥2) ⟺ 푥1 ≤ 2, 푥2 ≤ 1} ◦ The frontiers of the LCS region also represent bundles with fewer unis of either good 1 or good 2.
66 Quasiconcavity
Advanced Microeconomic Theory 67 Quasiconcavity
Example 1.7 (continued): 4) Regions A and B: ◦ Region 퐴 contains bundles with more units of good 2 but fewer units of good 1 (the opposite argument applies to region 퐵). ◦ 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.
68 Quasiconcavity
Example 1.7 (continued): 5) Preference relation is not complete: ◦ Completeness requires for every pair 푥 and 푦, either 푥 ≿ 푦 or 푦 ≿ 푥 (or both). 2 ◦ Consider two bundles 푥, 푦 ∈ ℝ+ with bundle 푥 containing more units of good 1 than bundle 푦 but fewer units of good 2, i.e., 푥1 > 푦1 and 푥2 < 푦2 (as in Region B) ◦ Then, we have neither 푥 ≿ 푦 nor 푦 ≿ 푥.
69 Quasiconcavity
Example 1.7 (continued): 6) Preference relation is transitive: ◦ Transitivity requires that, for any three bundles 푥, 푦 and 푧, if 푥 ≿ 푦 and 푦 ≿ 푧 then 푥 ≿ 푧.
◦ Now 푥 ≿ 푦 and 푦 ≿ 푧 means 푥푙 ≥ 푦푙 and 푦푙 ≥ 푧푙 for all 푙 goods.
◦ Then, 푥푙 ≥ 푧푙 implies 푥 ≿ 푧.
70 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 푥 ≥ 푦 and 푥 ≠ 푦 implies that 푥푙 ≥ 푦푙 for all good 푙 and 푥푘 > 푦푘 for at least one good 푘. ◦ Thus, 푥 ≥ 푦 and 푥 ≠ 푦 implies 푥 ≿ 푦 and not 푦 ≿ 푥. ◦ Thus, we can conclude that 푥 ≻ 푦.
71 Quasiconcavity
Example 1.7 (continued): 8) Preference relation is strictly convex: ◦ Strict convexity requires that if 푥 ≿ 푧 and 푦 ≿ 푧 and 푥 ≠ 푧, then 훼푥 + 1 − 훼 푦 ≻ 푧 for all 훼 ∈ 0,1 .
◦ Now 푥 ≿ 푧 and 푦 ≿ 푧 implies that 푥푙 ≥ 푦푙 and 푦푙 ≥ 푧푙 for all good 푙. ◦ 푥 ≠ 푧 implies, for some good 푘, we must have 푥푘 > 푧푘.
72 Quasiconcavity
Example 1.7 (continued): ◦ Hence, for any 훼 ∈ 0,1 , we must have that 훼푥푙 + 1 − 훼 푦푙 ≥ 푧푙 for all good 푙 훼푥푘 + 1 − 훼 푦푘 > 푧푘 for some 푘 ◦ Thus, we have that 훼푥 + 1 − 훼 푦 ≥ 푧 and 훼푥 + 1 − 훼 푦 ≠ 푧, and so 훼푥 + 1 − 훼 푦 ≿ 푧 and not 푧 ≿ 훼푥 + 1 − 훼 푦 ◦ Therefore, 훼푥 + 1 − 훼 푦 ≻ 푧.
73 Common Utility Functions
74 Common Utility Functions
75 Common Utility Functions
◦ Marginal utilities: 휕푢 휕푢 > 0 and > 0 휕푥1 휕푥2 ◦ A diminishing MRS 훼퐴푥훼−1푥훽 훼푥 푀푅푆 = 1 2 = 2 푥1,푥2 훽−1 훼 훽푥1 훽퐴푥1 푥2 which is decreasing in 푥1. . Hence, indifference curves become flatter as 푥1 increases.
76 Common Utility Functions
Cobb-Douglas preference
x 2
A in x 2 B
C x D 2 IC x 1 unit 1 unit 1
77 Common Utility Functions
78 Common Utility Functions
Perfect substitutes
x 2 A 2A slope B
A
B 2B x1
79 Common Utility Functions
80 Common Utility Functions
Advanced Microeconomic Theory 81 Common Utility Functions
Perfect complements
x2 xx 21 2 uA2 2
uA1 1 2 x1
82 Common Utility Functions
83 Common Utility Functions
84 Common Utility Functions
CES preferences
x2
0Perfect complement
0.2 1 Cobb-Douglas 8 Perfect substitutes x1
Advanced Microeconomic Theory 85 Common Utility Functions
◦ CES utility function is often presented as 1 푢 푥 , 푥 = 푎푥𝜌 + 푏푥𝜌 𝜌 1 2 1 2 𝜎−1 where 휌 ≡ . 𝜎
86 Common Utility Functions
87 Common Utility Functions
MRS of quasilinear preferences
Advanced Microeconomic Theory 88 Common Utility Functions
◦ For 푢 푥1, 푥2 = 푣 푥1 + 푏푥2, the marginal utilities are 휕푢 휕푢 휕푣 = 푏 and = 휕푥2 휕푥1 휕푥1 which implies 휕푣 푀푅푆 = 휕푥1 푥1,푥2 푏 ◦ Quasilinear preferences are often used to represent the consumption of goods that are relatively insensitive to income. ◦ Examples: garlic, toothpaste, etc.
89 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 푋 is continuous if it is preserved under limits. ◦ That is, for any sequence of pairs 푛 푛 ∞ 푛 푛 (푥 , 푦 ) 푛=1 with 푥 ≿ 푦 for all 푛 and lim 푥푛 = 푥 and lim 푦푛 = 푦, the preference 푛→∞ 푛→∞ relation is maintained in the limiting points, i.e., 푥 ≻ 푦.
Advanced Microeconomic Theory 90 Continuous Preferences
◦ Intuitively, there can be no sudden jumps (i.e., preference reversals) in an individual preference over a sequence of bundles.
Advanced Microeconomic Theory 91 Continuous Preferences
Lexicographic preferences are not continuous 1 ◦ Consider the sequence 푥푛 = , 0 and 푦푛 = 푛 (0,1), where 푛 = {0,1,2,3, … }. ◦ The sequence 푦푛 = (0,1) is constant in 푛. 1 ◦ The sequence 푥푛 = , 0 is not: 푛 . It starts at 푥1 = 1,0 , and moves leftwards to 1 1 푥2 = , 0 , 푥3 = , 0 , etc. 2 3
Advanced Microeconomic Theory 92 Continuous Preferences
Thus, the individual prefers: x2 y n, n, y 1 = y 2 = = y n 푥1 = 1,0 ≻ 0,1 = 푦1 1 1 푥2 = , 0 ≻ 0,1 = 푦2 2 1 푥3 = , 0 ≻ 0,1 = 푦3 3 ⋮ lim x n = (0,0) But, n lim 푥푛 = 0,0 ≺ 0,1 푛→∞ = lim 푦푛 푛→∞ x 4 x 3 x 2 x 1 Preference reversal! 0 ¼ ⅓ ½ 1 x1
Advanced Microeconomic Theory 93 Existence of Utility Function
If a preference relation satisfies monotonicity and continuity, then there exists a utility function 푢(∙) representing such preference relation.
Proof: ◦ Take a bundle 푥 ≠ 0. ◦ By monotonicity, 푥 ≿ 0, where 0 = (0,0, … , 0). . That is, if bundle 푥 ≠ 0, it contains positive amounts of at least one good and, it is preferred to bundle 0.
Advanced Microeconomic Theory 94 Existence of Utility Function
◦ Define bundle 푀 as the bundle where all components coincide with the highest component of bundle 푥:
푀 = max{푥푘} , … , max{푥푘} 푘 푘 ◦ Hence, by monotonicity, 푀 ≿ 푥.
◦ Bundles 0 and 푀 are both on the main diagonal, since each of them contains the same amount of good 푥1 and 푥2.
Advanced Microeconomic Theory 95 Existence of Utility Function
x2
x1
Advanced Microeconomic Theory 96 Existence of Utility Function
◦ By continuity and monotonicity, there exists a bundle that is indifferent to 푥 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 푡 푥 , 푡 푥 , … , 푡(푥)
◦ Let 푢 푥 = 푡 푥 , which is a real number.
Advanced Microeconomic Theory 97 Existence of Utility Function
◦ Applying the same steps for another bundle 푦 ≠ 푥, we obtain 푡 푦 , 푡 푦 , … , 푡(푦) and let 푢 푦 = 푡 푦 , which is also a real number. ◦ We know that 푥~ 푡 푥 , 푡 푥 , … , 푡(푥) 푦~ 푡 푦 , 푡 푦 , … , 푡(푦) 푥 ≿ 푦
◦ Hence, by transitivity, 푥 ≿ 푦 iff 푥~ 푡 푥 , 푡 푥 , … , 푡(푥) ≿ 푡 푦 , 푡 푦 , … , 푡(푦) ~푦
Advanced Microeconomic Theory 98 Existence of Utility Function
◦ And by monotonicity, 푥 ≿ 푦 ⟺ 푡 푥 ≥ 푡 푦 ⟺ 푢(푥) ≥ 푢(푦)
◦ Note: A utility function can satisfy continuity but still be non-differentiable. . For instance, the Leontief utility function, min{푎푥1,푏푥2}, is continuous but cannot be differentiated at the kink.
Advanced Microeconomic Theory 99 Social and Reference-Dependent Preferences We now examine social, as opposed to individual, preferences. Consider additively separable utility functions of the form 푢푖(푥푖, 푥) = 푓(푥푖) + 푔푖(푥) where ◦ 푓(푥푖) captures individual 푖’s utility from the monetary amount that he receives, 푥푖; ◦ 푔푖(푥) measures the utility/disutility he derives from the distribution of payoffs 푥 = (푥1, 푥2, . . . , 푥푁) among all 푁 individuals.
Advanced Microeconomic Theory 100 Social and Reference-Dependent Preferences Fehr and Schmidt (1999): ◦ For the case of two players,
푢푖(푥푖, 푥푗) = 푥푖 − 훼푖 max 푥푗 − 푥푖, 0 − 훽푖 max 푥푖 − 푥푗, 0
where 푥푖 is player 푖's payoff and 푗 ≠ 푖.
◦ Parameter 훼푖 represents player 푖’s disutility from envy
. When 푥푖 < 푥푗, max 푥푗 − 푥푖, 0 = 푥푗 − 푥푖 > 0 but max 푥푖 − 푥푗, 0 = 0. . Hence, 푢푖(푥푖, 푥푗) = 푥푖 − 훼푖(푥푗 − 푥푖).
Advanced Microeconomic Theory 101 Social and Reference-Dependent Preferences
◦ Parameter 훽푖 ≥ 0 captures player 푖's disutility from guilt
. When 푥푖 > 푥푗, max 푥푖 − 푥푗, 0 = 푥푖 − 푥푗 > 0 but max 푥푗 − 푥푖, 0 = 0.
. Hence, 푢푖 푥푖, 푥푗 = 푥푖 − 훽푖(푥푖 − 푥푗).
◦ Players’ envy is stronger than their guilt, i.e., 훼푖 ≥ 훽푖 for 0 ≤ 훽푖 < 1. . Intuitively, players (weakly) suffer more from inequality directed at them than inequality directed at others.
Advanced Microeconomic Theory 102 Social and Reference-Dependent Preferences ◦ Thus players exhibit “concerns for fairness” (or “social preferences”) in the distribution of payoffs.
◦ If 훼푖 = 훽푖 = 0 for every player 푖, individuals only care about their material payoff 푢푖(푥푖, 푥푗) = 푥푖. . Preferences coincide with the individual preferences.
Advanced Microeconomic Theory 103 Social and Reference-Dependent Preferences Fehr and Schmidt’s (1999) preferences
xi 45o-line
IC2
IC1
xj
Advanced Microeconomic Theory 104 Social and Reference-Dependent Preferences Bolton and Ockenfels (2000): ◦ Similar to Fehr and Schmidt (1999), but allow for non-linearities
푥푖 푢푖 푥푖, 푥푖+푥푗
where 푢푖(∙) increases in 푥푖 (i.e., selfish component) decreases in the share of total payoffs that individual 푥 푖 enjoys, 푖 (i.e., social preferences) 푥푖+푥푗
Advanced Microeconomic Theory 105 Social and Reference-Dependent Preferences ◦ For instance, 1 푥푖 푥푖 2 푢푖 푥푖, = 푥푖 − 훼 푥푖+푥푗 푥푖+푥푗
◦ Letting 푢 = 푢 and solving for 푥푗 yields 2 2 푥푖 훼 − 푢 − 푥푖 푥푗 = 2 푢 − 푥푖 which produces non-linear indifference curves.
Advanced Microeconomic Theory 106 Social and Reference-Dependent Preferences Andreoni and Miller (2002): ◦ A CES utility function 1 𝜌 𝜌 𝜌 푢푖(푥푖, 푥푗) = 훼푥푖 + 1 − 훼 푥푗
where 푥푖 and 푥푗 are the monetary payoff of individual 푖 rather than the amounts of goods.
◦ If individual 푖 is completely selfish, i.e., 훼 = 1, 푢(푥푖) = 푥푖
Advanced Microeconomic Theory 107 Social and Reference-Dependent Preferences ◦ If 훼 ∈ (0,1), parameter 휌 captures the elasticity of substitution between individual 푖's and 푗's payoffs.
. That is, if 푥푗 decreases by one percent, 푥푖 needs to be increased by 휌 percent for individual 푖 to maintain his utility level unaffected.
Advanced Microeconomic Theory 108 Choice Based Approach
We now focus on the actual choice behavior rather than individual preferences. ◦ From the alternatives in set 퐴, which one would you choose?
A choice structure (ℬ, 푐(∙)) contains two elements: 1) ℬ is a family of nonempty subsets of 푋, so that every element of ℬ is a set 퐵 ⊂ 푋.
Advanced Microeconomic Theory 109 Choice Based Approach
◦ Example 1: In consumer theory, 퐵 is a particular set of all the affordable bundles for a consumer, given his wealth and market prices.
◦ Example 2: 퐵 is a particular list of all the universities where you were admitted, among all universities in the scope of your imagination 푋, i.e., 퐵 ⊂ 푋.
Advanced Microeconomic Theory 110 Choice Based Approach
2) 푐(∙) is a choice rule that selects, for each budget set 퐵, a subset of elements in 퐵, with the interpretation that 푐(퐵) are the chosen elements from 퐵. ◦ Example 1: In consumer theory, 푐(퐵) would be the bundles that the individual chooses to buy, among all bundles he can afford in budget set 퐵; ◦ Example 2: In the example of the universities, 푐(퐵) would contain the university that you choose to attend.
Advanced Microeconomic Theory 111 Choice Based Approach
◦ Note: . If 푐(퐵) contains a single element, 푐(⋅) is a function; . If 푐(퐵) contains more than one element, 푐(⋅ ) is correspondence.
Advanced Microeconomic Theory 112 Choice Based Approach
Example 1.10 (Choice structures): ◦ Define the set of alternatives as 푋 = {푥, 푦, 푧}
◦ Consider two different budget sets
퐵1 = {푥, 푦} and 퐵2 = {푥, 푦, 푧}
◦ Choice structure one (ℬ, 푐1(∙)) 푐1 퐵1 = 푐1 푥, 푦 = {푥} 푐1 퐵2 = 푐1 푥, 푦, 푧 = {푥}
Advanced Microeconomic Theory 113 Choice Based Approach • Example 1.10 (continued):
◦ Choice structure two (ℬ, 푐2(∙)) 푐2 퐵1 = 푐2 푥, 푦 = {푥} 푐2 퐵2 = 푐2 푥, 푦, 푧 = {푦} ◦ 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 114 Consistency on Choices: the Weak Axiom of Revealed Preference (WARP)
Advanced Microeconomic Theory 115 WARP
Weak Axiom of Revealed Preference (WARP): The choice structure (ℬ, 푐(∙)) satisfies the WARP if: 1) for some budget set 퐵 ∈ ℬ with 푥, 푦 ∈ 퐵, we have that element 푥 is chosen, 푥 ∈ 푐(퐵), then 2) for any other budget set 퐵′ ∈ ℬ where alternatives 푥 and 푦 are also available, 푥, 푦 ∈ 퐵′, and where alternative 푦 is chosen, 푦 ∈ 푐(퐵′), then we must have that alternative 푥 is chosen as well, 푥 ∈ 푐(퐵′).
Advanced Microeconomic Theory 116 WARP
Example 1.11 (Checking WARP in choice structures): ◦ Take budget set 퐵 = {푥, 푦} with the choice rule of 푐 푥, 푦 = 푥. ◦ Then, for budget set 퐵′ = {푥, 푦, 푧}, the “legal” choice rules are either: 푐 푥, 푦, 푧 = {푥}, or 푐 푥, 푦, 푧 = {푧}, or 푐 푥, 푦, 푧 = {푥, 푧}
Advanced Microeconomic Theory 117 WARP
Example 1.11 (continued): ◦ This implies, individual decision-maker cannot select 푐 푥, 푦, 푧 ≠ {푦} 푐 푥, 푦, 푧 ≠ {푦, 푧} 푐 푥, 푦, 푧 ≠ {푥, 푦}
Advanced Microeconomic Theory 118 WARP
Example 1.12 (More on choice structures satisfying/violating WARP: ◦ Take budget set 퐵 = {푥, 푦} with the choice rule of 푐 푥, 푦 = {푥, 푦}. ◦ Then, for budget set 퐵′ = {푥, 푦, 푧}, the “legal” choices according to WARP are either: 푐 푥, 푦, 푧 = {푥, 푦}, or 푐 푥, 푦, 푧 = {푧}, or 푐 푥, 푦, 푧 = {푥, 푦, 푧}
Advanced Microeconomic Theory 119 WARP
Example 1.12 (continued): ◦ Choice rule satisfying WARP
B
C(B) x y
C(B )
B
Advanced Microeconomic Theory 120 WARP
Example 1.12 (continued): ◦ Choice rule violating WARP
B C(B)
x
y C(B )
B
Advanced Microeconomic Theory 121 Consumption Sets
Consumption set: a subset of the commodity space ℝ퐿, denoted by 푥 ⊂ ℝ퐿, 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 푥 as a vector of 퐿 components.
Advanced Microeconomic Theory 122 Consumption Sets
Physical constraint on the labor market
Advanced Microeconomic Theory 123 Consumption Sets
Consumption at two different locations
Beer in Seattle at noon
x
Beer in Barcelona at noon
Advanced Microeconomic Theory 124 Consumption Sets
Convex consumption sets: ◦ A consumption set 푋 is convex if, for two consumption bundles 푥, 푥′ ∈ 푋, the bundle 푥′′ = 훼푥 + 1 − 훼 푥′ is also an element of 푋 for any 훼 ∈ (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.
Advanced Microeconomic Theory 125 Consumption Sets: Economic Constraints
Assumptions on the price vector in ℝ퐿:
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 퐿 goods, i.e., 푝 ≫ 0 for every good 푘. Some prices could be negative, such as pollution.
Advanced Microeconomic Theory 126 Consumption Sets: Economic Constraints
3) Price taking assumption: a consumer’s demand for all 퐿 goods represents a small fraction of the total demand for the good.
Advanced Microeconomic Theory 127 Consumption Sets: Economic Constraints
퐿 Bundle 푥 ∈ ℝ+ is affordable if 푝1푥1 + 푝2푥2 + ⋯ + 푝퐿푥퐿 ≤ 푤 or, in vector notation, 푝 ∙ 푥 ≤ 푤.
Note that 푝 ∙ 푥 is the total cost of buying bundle 푥 = (푥1, 푥2, … , 푥퐿) at market prices 푝 = (푝1, 푝2, … , 푝퐿), and 푤 is the total wealth of the consumer.
퐿 When 푥 ∈ ℝ+ then the set of feasible consumption bundles consists of the elements of the set: 퐿 퐵푝,푤 = {푥 ∈ ℝ+: 푝 ∙ 푥 ≤ 푤}
Advanced Microeconomic Theory 128 Consumption Sets: Economic Constraints
2 • Example: 퐵푝,푤 = {푥 ∈ ℝ+: 푝1푥1 + 푝2푥2 ≤ 푤}
x2
푝1푥1 + 푝2푥2 = 푤 ⟹
w p1 푤 푝 p - (slope) 1 2 p2 푥2 = − 푥1 푝2 푝2
2 {x +:p x = w}
w x1 p1
Advanced Microeconomic Theory 129 Consumption Sets: Economic Constraints
3 • Example: 퐵푝,푤 = {푥 ∈ ℝ+: 푝1푥1 + 푝2푥2 + 푝3푥3 ≤ 푤} ◦ Budget hyperplane x3
x1
x2 Advanced Microeconomic Theory 130 Consumption Sets: Economic Constraints
Price vector 푝 is orthogonal to the budget line 퐵푝,푤. ◦ Note that 푝 ∙ 푥 = 푤 holds for any bundle 푥 on the budget line.
◦ Take any other bundle 푥′ which also lies on 퐵푝,푤. Hence, 푝 ∙ 푥′ = 푤. ◦ Then, 푝 ∙ 푥′ = 푝 ∙ 푥 = 푤 푝 ∙ 푥′ − 푥 = 0 or 푝 ∙ ∆푥 = 0
Advanced Microeconomic Theory 131 Consumption Sets: Economic Constraints
◦ Since this is valid for any two bundles on the budget line, then 푝 must be perpendicular to ∆푥 on 퐵푝,푤.
◦ This implies that the price vector is perpendicular (orthogonal) to 퐵푝,푤.
Advanced Microeconomic Theory 132 Consumption Sets: Economic Constraints
The budget set 퐵푝,푤 is convex. ◦ We need that, for any two bundles 푥, 푥′ ∈ 퐵푝,푤, their convex combination 푥′′ = 훼푥 + 1 − 훼 푥′
also belongs to the 퐵푝,푤, where 훼 ∈ (0,1).
◦ Since 푝 ∙ 푥 ≤ 푤 and 푝 ∙ 푥′ ≤ 푤, then 푝 ∙ 푥′′ = 푝훼푥 + 푝 1 − 훼 푥′ = 훼푝푥 + 1 − 훼 푝푥′ ≤ 푤
Advanced Microeconomic Theory 133