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 u x1 y y u x u 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 v x11, x 22 x 11 x 12 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.