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Intro Utility Function Indifference Curves Examples Trinity Econ 4935 Urban Economics Lecture CT2: Utility Function Instructor: Hiroki Watanabe Fall 2012 Watanabe Intro Econ 4935 Utility Function CT2 Utility Function Indifference Curves 1 Introduction 2 Utility Function 3 Indifference Curves 4 Examples 5 Trinity 6 Now We Know Watanabe Intro Econ 4935 Utility Function Indifference Curves Introduction Consumer Theory Overview Task for Today 2 Utility Function 3 Indifference Curves 4 Examples 5 Trinity 6 Now We Know Econ 4935 Trinity CT2 Utility Function 1 Watanabe Examples 1 / 44 Examples CT2 Utility Function 2 / 44 Trinity 3 / 44 Intro Utility Function Indifference Curves Examples Trinity Consumer Theory Overview 1 What do consumers face? 2 What do consumers want? 3 How do consumers resolve conflict above? 4 How do consumers resolve conflict above in conjunction with location choice? CT1 CT2 CT3 from Lecture 1A onwards Watanabe Intro Econ 4935 Utility Function CT2 Utility Function Indifference Curves Examples 4 / 44 Trinity Task for Today 1 C = 1 vs yC = 10. 2 (C , T ) = (10, 3) vs (yC , yT ) = (3, 12). Fact 1.1 (Comparing Bundles) 1 Numbers are easy to compare. 2 Bundles are hard to compare. Watanabe Intro Econ 4935 Utility Function CT2 Utility Function Indifference Curves Examples 5 / 44 Trinity Task for Today Today’s focus: quantification of our preferences. Watanabe Econ 4935 CT2 Utility Function 6 / 44 Intro Utility Function Indifference Curves 1 Introduction 2 Utility Function Utility Function Ordinal Property 3 Indifference Curves 4 Examples 5 Trinity 6 Now We Know Watanabe Intro Econ 4935 Utility Function Examples Trinity CT2 Utility Function Indifference Curves Examples 7 / 44 Trinity Utility Function Example 2.1 (Corona) A bundle of six-packs and bottles of Corona: = (6 , 1 ). One way to represent Liz’s preferences by a number is to assign the total # of bottles 66 + 1 = T(6 , 1 ) to a bundle (6 , 1 ). Is this assignment reasonable? T(2, 0) = T(2, 0) = Watanabe Intro Econ 4935 Utility Function T(1, 6) = T(1, 3) = CT2 Utility Function Indifference Curves Examples 8 / 44 Trinity Utility Function Definition 2.2 (Utility Function) A utility function (C , T ) assigns a number (called utility level) to a bundle (C , T ). Watanabe Econ 4935 CT2 Utility Function 9 / 44 Intro Utility Function Indifference Curves Examples Trinity Ordinal Property Utility is an ordinal concept. In Corona example Example 2.1 , we could have assigned 3 ☺’s instead of 1 ☺ for each bottle: (6 , 1 ) = 186 + 31 . Assignment is still reasonable: (2, 0) = (2, 0) = Watanabe Intro (1, 6) = (1, 3) = Econ 4935 Utility Function CT2 Utility Function Indifference Curves Examples 10 / 44 Trinity Ordinal Property Assigned values of utility level does not matter so long as they represent what we prefer to consume. Watanabe Intro Econ 4935 Utility Function CT2 Utility Function Indifference Curves Examples 1 Introduction 2 Utility Function 3 Indifference Curves Art of Drawing What We Cannot See Example 4 Examples 5 Trinity 6 Now We Know Watanabe Econ 4935 CT2 Utility Function 11 / 44 Trinity 12 / 44 Intro Utility Function Indifference Curves Examples Trinity Art of Drawing What We Cannot See While (·) does represent what Liz prefers, it is hard to visualize. We have a handy little device to show her liking on a graph. Definition 3.1 (Indifference Curves) The indifference curve at a bundle = (C , T ) is a collection of bundles that is equally preferred to . If a bundle y = (yC , yT ) and z = (zC , zT ) are on the same indifference curve, then Liz is indifferent between them. Watanabe Intro Econ 4935 CT2 Utility Function Utility Function Indifference Curves Examples 13 / 44 Trinity Art of Drawing What We Cannot See We can trace the indifference curve at = (C , T ) by collecting the bundles that yield the same utility level as (C , T ). Watanabe Intro Econ 4935 CT2 Utility Function Utility Function Indifference Curves Examples 14 / 44 Trinity Example Take Example 2.1 . Question 3.2 (Indifference Curve) Liz’s utility function for (6 , 1 ) is (6 , 1 ) = 66 + 1 . Trace indifference curves at (6 , 1 ) = (3, 0) and (3, 12). Watanabe Econ 4935 CT2 Utility Function 15 / 44 Intro Utility Function Indifference Curves Examples Trinity Example 30 Indifference Curves 54 30 24 48 Bottles x1 24 18 42 18 12 36 12 6 6 0 0 Watanabe Intro 1 2 3 Six−Packs x6 Econ 4935 Utility Function Indifference Curves Introduction 2 Utility Function 3 Indifference Curves 4 Examples Perfect Substitutes Perfect Complements Cobb-Douglas Utility Quasilinear Utility 5 Trinity 6 Now We Know Intro Econ 4935 Utility Function 5 CT2 Utility Function 1 Watanabe 4 Examples 16 / 44 Trinity CT2 Utility Function Indifference Curves Examples 17 / 44 Trinity Perfect Substitutes Consider Liz’s preferences for Coke and Pepsi = (C , P ). (C , P ) = C + P . In this case, Coke and Pepsi are perfect substitutes. Watanabe Econ 4935 CT2 Utility Function 18 / 44 Intro Utility Function Indifference Curves Examples Trinity Perfect Substitutes 5 Indifference Curves 9 5 4 8 Pepsi x (oz) 4 P 3 7 3 2 6 2 1 1 0 0 Watanabe Intro 1 2 3 Coke xC (oz) Econ 4935 4 5 CT2 Utility Function Utility Function Indifference Curves 19 / 44 Examples Trinity Utility Level u(x) Perfect Substitutes 20 18 16 14 12 10 8 6 4 2 0 10 9 8 7 6 5 4 Pepsi x (oz) P Watanabe Intro Econ 4935 Utility Function 3 2 1 0 0 1 2 3 4 5 6 7 8 9 Coke xC (oz) CT2 Utility Function Indifference Curves 10 Examples 20 / 44 Trinity Perfect Complements Consider Liz’s preferences for cereal and milk = (C , M ). C if C < M (C , M ) = min {C , M } = M if M ≤ C . In this case, cereal and milk are perfect complements. Watanabe Econ 4935 CT2 Utility Function 21 / 44 Intro Utility Function Indifference Curves Examples Trinity Perfect Complements 5 Indifference Curves Milk x (oz) 4 4 3 M 3 2 2 1 1 0 0 Watanabe Intro 1 2 3 Cereal xC (oz) Econ 4935 4 5 CT2 Utility Function Utility Function Indifference Curves 22 / 44 Examples Trinity Utility Level u(x) Perfect Complements 10 9 8 7 6 5 4 3 2 1 0 10 9 8 7 6 5 4 Milk x (oz) M Watanabe Intro 3 2 1 0 0 Econ 4935 Utility Function 1 2 3 4 5 6 7 8 9 Cereal xC (oz) CT2 Utility Function Indifference Curves 10 Examples 23 / 44 Trinity Cobb-Douglas Utility The most commonly used utility function is Cobb-Douglas utility function: (C , T ) = C bT , where > 0 and b > 0.1 E.g. (C , T ) = C T . 1 Thanks to ordinality of utility function, you can use () := log[()] = log(C ) + b log(T ) to represent the same preferences. Watanabe Econ 4935 CT2 Utility Function 24 / 44 Intro Utility Function Indifference Curves Examples Trinity Cobb-Douglas Utility 10 90 Indifference Curves 9 80 8 70 Tea xT (cups) 7 60 6 50 5 40 4 30 3 20 2 10 1 0 0 Watanabe Intro 1 2 3 4 5 6 7 Cheese xC (slices) Econ 4935 8 9 10 CT2 Utility Function Utility Function Indifference Curves 25 / 44 Examples Trinity Utility Level u(x) Cobb-Douglas Utility 100 90 80 70 60 50 40 30 20 10 0 10 9 8 7 6 5 4 Tea x (cups) 3 2 1 T Watanabe Intro Econ 4935 Utility Function 0 0 1 2 3 4 5 6 7 9 8 10 Cheese xC (slices) CT2 Utility Function Indifference Curves Examples 26 / 44 Trinity Quasilinear Utility Another commonly used utility function is a quasilinear utility function of the form (1 , 2 ) = 1 + (2 ). E.g. (1 , 2 ) = 1 + p 2 . 1 is the number of baskets for example. Watanabe Econ 4935 CT2 Utility Function 27 / 44 Intro Utility Function Indifference Curves Examples Trinity Quasilinear Utility 10 Indifference Curves Tea xT (oz) 8 6 12 4 2 2 Intro 10 Watanabe 2 8 6 4 0 0 4 6 8 Composite Goods xC (baskets) Econ 4935 10 CT2 Utility Function Utility Function Indifference Curves 28 / 44 Examples Trinity Quasilinear Utility Utility Level u(x) 12 8 4 0 10 9 8 7 6 5 4 3 2 Tea x (oz) 1 T Watanabe Intro 0 0 Econ 4935 Utility Function 1 2 3 4 7 8 9 10 Composite Goods xC (baskets) Indifference Curves Introduction 2 Utility Function 3 Indifference Curves 4 Examples 5 Trinity Marginal Willingness to Pay Comparison to Relative Prices 6 Now We Know Econ 4935 6 CT2 Utility Function 1 Watanabe 5 Examples CT2 Utility Function 29 / 44 Trinity 30 / 44 Intro Utility Function Indifference Curves Examples Trinity Marginal Willingness to Pay Just like the slope of the budget line has the meaning (recall trinity), the slope of the indifference curve carries two important meanings. Fact 5.1 (Trinity on the Preference Side) The following are the same: 1 The slope of indifference curve. 2 Marginal willingness to pay. 3 Marginal rate of substitution. Watanabe Intro Econ 4935 Utility Function CT2 Utility Function Indifference Curves Examples 31 / 44 Trinity Marginal Willingness to Pay Definition 5.2 (Marginal Willingness to Pay) If Liz is just willing to give up cups of tea for a slice of cheesecake, then is called her marginal willingness to pay. Watanabe Intro Econ 4935 Utility Function CT2 Utility Function Indifference Curves Examples 32 / 44 Trinity Marginal Willingness to Pay E.g., if (4, 10) = (5, 6), then Liz is just willing to give up 4 cups of tea for an additional slice of cheesecake. She is indifferent between those two bundles. Her MWTP at (4, 10) is 4. If (8, 4) = (9, 3), Her MWTP at (8, 4) is 1. Watanabe Econ 4935 CT2 Utility Function 33 / 44 Intro Utility Function Indifference Curves Examples Trinity Marginal Willingness to Pay MWTP is also referred to as marginal rate of substitution (MRS). The slope of indifference curves represents the marginal willingness to pay. Take Watanabe Intro Example 2.1 Econ 4935 . CT2 Utility Function Utility Function Indifference Curves 34 / 44 Examples Trinity Marginal Willingness to Pay 30 Indifference Curves 54 30 24 48 Bottles x1 24 18 42 18 12 36 12 6 6 0 0 Watanabe Intro 1 Econ 4935 2 3 Six−Packs x6 4 5 CT2 Utility Function Utility Function Indifference Curves Examples 35 / 44 Trinity Marginal Willingness to Pay Example 2.1 is a rather unusual case. MRS usually varies depending on the bundle . Consider two bundles: = (C , T ) = (1, 29384720396) y = (yC , yT ) = (3240894603, 1). MRS at is pretty large (why?) MRS at y is pretty small (why?) Watanabe Econ 4935 CT2 Utility Function 36 / 44 Intro Utility Function Indifference Curves Examples Trinity Marginal Willingness to Pay I.e., people prefer a balanced combination of two goods rather than something extreme. This tendency is called convex preferences. Watanabe Intro Econ 4935 CT2 Utility Function Utility Function Indifference Curves 37 / 44 Examples Trinity Marginal Willingness to Pay 10 90 Indifference Curves 9 80 8 70 Tea xT (cups) 7 60 6 50 5 40 4 30 3 20 2 10 1 0 0 Watanabe Intro 1 Econ 4935 Utility Function 2 3 4 5 6 7 Cheese xC (slices) 8 9 10 CT2 Utility Function Indifference Curves Examples 38 / 44 Trinity Comparison to Relative Prices Notice the difference: Relative price is the cups of tea Liz has to sell (give up) to get one slice of cheesecakes to keep to her budget. Marginal willingness to pay is the cups of tea Liz is willing to give up to get one slice of cheesecakes to remain as happy as before. Watanabe Econ 4935 CT2 Utility Function 39 / 44 Intro Utility Function Indifference Curves 1 Introduction 2 Utility Function 3 Indifference Curves 4 Examples 5 Trinity 6 Now We Know Watanabe Intro Econ 4935 Utility Function Examples Trinity CT2 Utility Function Indifference Curves Examples 40 / 44 Trinity Quantification of preferences. Utility functions for perfect substitutes, perfect complements, Cobb-Douglass and quasilinear preferences. Trinity. Watanabe Intro Econ 4935 Utility Function CT2 Utility Function Indifference Curves Examples 41 / 44 Trinity Map du Jour Source http://bigthink.com/strange-maps/ 312-the-population-of-chinas-provinces-compared Watanabe Econ 4935 CT2 Utility Function 42 / 44 Intro Utility Function Indifference Curves Examples Trinity Airline du Jour Today’s color theme is provided by courtesy of Watanabe Intro Econ 4935 Utility Function Northwest Airlines CT2 Utility Function Indifference Curves Examples 43 / 44 Trinity Index Cobb-Douglas utility function, 24 convex preferences, 37 indifference curve, 13 marginal rate of substitution, 31, 34 marginal willingness to pay, 31, 32 Watanabe Econ 4935 perfect complements, 21 perfect substitutes, 18 quasilinear utility function, 27 trinity on the preference side, 31 utility function, 9 CT2 Utility Function 44 / 44