Econ 4601 Urban & Regional Economics. Lecture 4: Utility. Instructor: Hiroki Watanabe. Summer 2010

Transcription

1 Econ 601 Urban & Regional Economics Lecture : Utility Instructor: Hiroki Watanabe Summer 0 1 / 1 1 Introduction Utility Function Utility Function Ordinal Property Indifference Curves 3 Examples Perfect Compliments MRS & Utility Function MRS 5 Summary / 1

2 Fact 1 Numbers are easy to compare (Lecture. Bundles are hard to compare (Lecture 3. (x 1, x is hard to deal with: (3, (,. On the other hand, 3 is apparently larger than 1 for example. It might be a good idea to represent (3, by 3 and (, by 1. Then we can easily see (3, (, as 3 1. Today s focus: Quantification of preference relations. 3 / 1 1 Introduction Utility Function Utility Function Ordinal Property Indifference Curves 3 Examples Perfect Compliments MRS & Utility Function MRS 5 Summary / 1

3 Utility Function A bundle of six-packs and bottles of Corona: x = (x 6, x 1. One way to represent Greg s preferences by numbers is to assign the total # of bottles 6x 6 + x 1 = T(x 6, x 1 to (x 6, x 1. Is this assignment reasonable? (, 0 (1, 6 corresponds to T(, 0 = 1 = T(1, 6. (, 0 (1, 3 corresponds to T(, 0 = 1 9 = T(1, 3. 5 / 1 Utility Function A utility function u(x 1, x assigns a number (called utility level to a bundle (x 1, x. 1 ( 6 / 1

4 Utility Function Preference Relations & Utility Function A utility function u(x represents preference relations if x y exactly when u(x u(y. Note that the left-hand side is comparison between bundles (hard while right-hand side is comparison between numbers (easy. Also note x y exactly when u(x = u(y. 7 / 1 Ordinal Property Utility is an ordinal concept. In the Corona example, we could have assigned 3 for each bottle: u(x 6, x 1 = 18x 6 + 3x 1. Assignment is still reasonable: (, 0 (1, 6 corresponds to T(, 0 = 36 = T(1, 6. (, 0 (1, 3 corresponds to T(, 0 = 36 7 = T(1, 3. 8 / 1

5 Ordinal Property A number has two functions: 1 to mark the absolute value: $3, 81 F etc. to arrange elements in sequence: 1st runner, 3rd day, MTWTF, abc... Utility functions focus on ➋ (ordinal concept. The ordering matters but the value itself does not. 1 + = 3 but Mon+Tue is not Wed. To represent with x y, u(x = 1 and u(y =.9. v(x = 380 and v(y = / 1 Ordinal Property Exercise Greg s preferences for cheesecakes and tea (x C, x T are given by (1, (1, 1, (1, (,, (, (3, 1, (3, (1,. Assign utility levels for each bundle to represent his preferences. / 1

6 Ordinal Property 5 Tea (x T Cheesecakes (x C 11 / 1 Indifference Curves The indifference curve to x = (x 1, x is a collection of bundles that are indifferent to x. Equivalently, we can find the indifference curve to x = (x 1, x by collecting the bundles that has the same utility level as (x 1, x. 1 / 1

7 Indifference Curves 13 / 1 Figure: 1 Introduction Utility Function Utility Function Ordinal Property Indifference Curves 3 Examples Perfect Compliments MRS & Utility Function MRS 5 Summary 1 / 1

8 Consider Greg s preferences for left gloves and right gloves x = (x 1, x. u(x 1, x = x 1 + x. 15 / 1 16 / 1

9 Perfect Compliments Consider Greg s preferences for left gloves and right gloves x = (x 1, x. u(x 1, x = min {x 1, x } = x1 if x 1 < x x if x x / 1 Perfect Compliments Figure: 18 / 1

10 One way to represent convex preferences is to use Cobb-Douglas utility function where a > 0 and b > 0. u(x 1, x = x a 1 xb, The Cobb-Douglas utility functions is commonly used in academics. E.g. u(x 1, x = x 1 x. 19 / Tea (x T Cheesecakes (x C 0 / 1

11 1 / 1 Figure: Another commonly used utility function is a quasilinear utility function of the form E.g. u(x 1, x = x 1 + x. x 1 is money for example. u(x 1, x = x 1 + v(x. / 1

12 Tea (x T Money (x 1 3 / 1 1 Introduction Utility Function Utility Function Ordinal Property Indifference Curves 3 Examples Perfect Compliments MRS & Utility Function MRS 5 Summary / 1

13 MRS Recall trinity: the slope of the indifference curve marks the marginal rate of substitution. With utility functions, we can compute the slope. 5 / 1 A bundle of six-packs and bottles of Corona: x = (x 6, x 1. u(x 6, x 1 = 6x 6 + x 1. Greg s marginal rate of substitution at (x 6, x 1 = (1, 1 is the slope of the indifference curve at (1, 1. u(1, 1 = 18. The indifference curve at (1, 1 satisfies MRS at (1, 1 is 6. 6x 6 + x 1 = 18 x 1 = 6x / 1

14 18 1 Bottles (x Six Packs (x 6 7 / 1 Consider instead and find MRS at (1, 8. u(x C, x T = x C x T, The indifference curve at (1, 8 is a collection of (x 1, x such that x 1 x = 8 x = 8 x 1. The slope of x = 8 x 1 at x 1 = 1 is Greg s MRS. 8 / 1

15 9 / 1 Figure: u(x 1, x = x 1 + x. MRS is constant at each x regardless of the amount of x / 1

16 Tea (x T Money (x 1 31 / 1 1 Good 1 exhibits diminishing marginal utility if MRS grows smaller as good 1 increases against a fixed amount of good. Good exhibits diminishing marginal utility if MRS grows larger as good increases against a fixed amount of good 1. 1 Fix x T = 3. First slice of cheesecake is worth cups of tea (MRS= at (1, th slice of cheesecake is worth.007 cups of tea (MRS=.007 at (300, 3. Fix x C = 7. First cup of tea is worth 3 slices of cheesecake (MRS= 1 at (7, th cup of tea is worth.00 slices of cheesecake (MRS= 1.00 = 500 at (7, / 1

17 Cobb-Douglas utility is one example of utility functions that exhibits diminishing marginal utility in both goods. Fix x T = 6 and look at the development of the slopes along the line (diminishing marginal utility in cheesecakes. Fix x C = and look at the development of the slopes along the line (diminishing marginal utility in tea. 33 / 1 8 Tea (x T Cheesecakes (x C 3 / 1

18 Is there any commodity that does not exhibit diminishing marginal utility? There are preferences that do not exhibit diminishing marginal utility in one of or both commodities. Quasilinear preferences: diminishing marginal utility only in commodity. Perfect substitutes: marginal utility is constant in both commodities. 35 / Tea (x T Money (x 1 36 / 1

19 In a sense, you can t have too much money. One unit of x 1 is always worth cups of tea regardless of the current amount of x 1. On the other hand, tea can be too much. An additional cup of tea worth less as total cups of tea increases. 37 / Bottles (x Six Packs (x 6 38 / 1

20 Note this does not mean Greg never gets bored with Corona. Corona will probably exhibit diminishing marginal utility against chicken wings. It all depends on what pair of commodities you put under examination. 39 / 1 1 Introduction Utility Function Utility Function Ordinal Property Indifference Curves 3 Examples Perfect Compliments MRS & Utility Function MRS 5 Summary 0 / 1

21 Quantify preferences. What does it mean to represent preferences? Utility functions for perfect substitutes, perfect compliments, Cobb-Douglass and quasilinear preferences. Marginal rate of substitution on a graph. Diminishing marginal utility. 1 / 1