INTERMIC_LPPT_Ch4_Student.pp Businessstx

Business

1. CHAPTER 4 Utility Copyright © 2019 Hal R. Varian

2. Preferences: A Reminder x ≻ y : x is to y . x ~ y : x and y are . x ≿ y : x is y . Copyright © 2019 Hal R. Varian

3. Utility Functions – 1 A preference relation that is complete, reflexive, transitive, and continuous can be represented by a . means that small changes to a consumption bundle cause only small changes to the preference level. Copyright © 2019 Hal R. Varian

4. Utility Functions – 2 A utility function u ( x ) represents a preference relation if and only if: x ʹ ≿ x ʺ ⇔ u ( x ʹ) ?? u ( x ʺ) x ʹ ≻ x ʺ ⇔ u ( x ʹ) ?? u ( x ʺ) x ʹ ~ x ʺ ⇔ u ( x ʹ) ?? u ( x ʺ) Copyright © 2019 Hal R. Varian

5. Utility Functions – 3 Utility is an concept. E.g., if u ( x ) = 6 and u ( y ) = 2 then bundle x is strictly preferred to bundle y . But x is not preferred as is y . Copyright © 2019 Hal R. Varian

6. Utility Functions & Indifference Curves – 1 Consider the bundles (4, 1), (2, 3), and (2, 2). Suppose (2, 3) ≻ (4, 1) ~ (2, 2). Assign to these bundles any numbers that preserve the preference ordering; e.g., Call these numbers . Copyright © 2019 Hal R. Varian

7. Utility Functions & Indifference Curves – 2 An indifference curve contains bundles. Equal preference  level. Therefore, all bundles on an indifference curve have level. Copyright © 2019 Hal R. Varian

8. Utility Functions & Indifference Curves – 3 So the bundles (4, 1) and (2, 2) are on the indifference curve with utility level u º 4. But the bundle (2, 3) is on the indifference curve with utility level u º 6. In an indifference curve diagram, this preference information looks as follows: Copyright © 2019 Hal R. Varian

9. Utility Functions & Indifference Curves – 4 Copyright © 2019 Hal R. Varian

10. Utility Functions & Indifference Curves – 5 Another way to visualize this same information is to plot the utility level on a vertical axis. Copyright © 2019 Hal R. Varian

11. Utility Functions & Indifference Curves – 6 Copyright © 2019 Hal R. Varian

12. Utility Functions & Indifference Curves – 7 This 3D visualization of preferences can be made more informative by adding into it the two indifference curves. Copyright © 2019 Hal R. Varian

13. Utility Functions & Indifference Curves – 8 Copyright © 2019 Hal R. Varian

14. Utility Functions & Indifference Curves – 9 Comparing more bundles will create a larger collection of all indifference curves and a better description of the consumer’ s preferences. Copyright © 2019 Hal R. Varian

15. Utility Functions & Indifference Curves – 10 Copyright © 2019 Hal R. Varian

16. Utility Functions & Indifference Curves – 11 As before, this can be visualized in 3D by plotting each indifference curve at the height of its utility index. Copyright © 2019 Hal R. Varian

17. Utility Functions & Indifference Curves – 12 Copyright © 2019 Hal R. Varian

18. Utility Functions & Indifference Curves – 13 Comparing all possible consumption bundles gives the complete collection of the consumer’ s indifference curves, each with its assigned utility level. This complete collection of indifference curves completely represents the consumer’ s . Copyright © 2019 Hal R. Varian

19. Utility Functions & Indifference Curves – 14 The following eight diagrams illustrate the connection between the two-dimensional look at utility functions and indifference curves to the three-dimensional look. Copyright © 2019 Hal R. Varian

20. Utility Functions & Indifference Curves – 15 Copyright © 2019 Hal R. Varian

21. Utility Functions & Indifference Curves – 16 Copyright © 2019 Hal R. Varian

22. Utility Functions & Indifference Curves – 17 Copyright © 2019 Hal R. Varian

23. Utility Functions & Indifference Curves – 18 Copyright © 2019 Hal R. Varian

24. Utility Functions & Indifference Curves – 19 Copyright © 2019 Hal R. Varian

25. Utility Functions & Indifference Curves – 20 Copyright © 2019 Hal R. Varian

26. Utility Functions & Indifference Curves – 21 Copyright © 2019 Hal R. Varian

27. Utility Functions & Indifference Curves – 22 Copyright © 2019 Hal R. Varian

28. Utility Functions & Indifference Curves – 23 The collection of all indifference curves for a given preference relation is an “ .” An indifference map is equivalent to a . Copyright © 2019 Hal R. Varian

29. Utility Functions – 4 There is no utility function representation of a preference relation. Suppose represents a preference relation. Again consider the bundles (4, 1), (2, 3), and (2, 2). Copyright © 2019 Hal R. Varian

30. Utility Functions – 5 so, = ; = ; = . Since ?? ?? Copyright © 2019 Hal R. Varian

31. Utility Functions – 6 Define Then, . . So again, V preserves the same order as u and therefore represents . Copyright © 2019 Hal R. Varian

32. Utility Functions – 7 Define Then, . So again, W preserves the same order as u and therefore represents . Copyright © 2019 Hal R. Varian

33. Utility Functions – 8 If u is a utility function that represents a preference relation and f is a strictly increasing function, then V = f ( u ) is also a utility function representing . This is known as a “ .” Copyright © 2019 Hal R. Varian

34. Goods, Bads, and Neutrals – 1 A good is a commodity which utility with additional consumption. A bad is a commodity which utility with additional consumption. A neutral is a commodity which utility with additional consumption. Copyright © 2019 Hal R. Varian

35. Goods, Bads, and Neutrals – 2 Copyright © 2019 Hal R. Varian

36. Some Other Utility Functions and Their Indifference Curves – 1 Instead of consider This utility function is an example of two goods that are . What do the indifference curves for this “ perfect substitution” utility function look like? Copyright © 2019 Hal R. Varian

37. Perfect Substitution Indifference Curves All are linear and parallel. Copyright © 2019 Hal R. Varian

38. Some Other Utility Functions and Their Indifference Curves – 2 Instead of consider This utility function is an example of two goods that are . What do the indifference curves for this “ perfect complementarity” utility function look like? Copyright © 2019 Hal R. Varian

39. Perfect-Complementarity Indifference Curves Copyright © 2019 Hal R. Varian

40. Some Other Utility Functions and Their Indifference Curves – 3 A utility function of the form is linear in just x 2 and is called “ .” E.g., Copyright © 2019 Hal R. Varian

41. Quasi-linear Indifference Curves Copyright © 2019 Hal R. Varian

42. Some Other Utility Functions and Their Indifference Curves – 4 Any utility function of the form with a > 0 and b > 0 is called a “Cobb-Douglas” utility function. E.g., Copyright © 2019 Hal R. Varian

43. Cobb-Douglas Indifference Curves Copyright © 2019 Hal R. Varian

44. Marginal Utilities – 1 Marginal means “ .” The marginal utility of commodity i is the of total utility as the quantity of commodity i consumed changes. I.e., Copyright © 2019 Hal R. Varian

45. Marginal Utilities – 2 Copyright © 2019 Hal R. Varian E.g., if then

46. Marginal Utilities and Marginal Rates of Substitution – 1 The general equation for an indifference curve is k is a . Totally differentiating this identity gives . Copyright © 2019 Hal R. Varian

47. Marginal Utilities and Marginal Rates of Substitution – 2 can be rearranged algebraically to be: . This is the . Copyright © 2019 Hal R. Varian

48. MU and MRS: An Example – 1 Suppose Then and . So, . Copyright © 2019 Hal R. Varian

49. MU and MRS: An Example – 2 Copyright © 2019 Hal R. Varian

50. MRS for Quasi-linear Utility Functions – 1 A quasi-linear utility function is of the form and . So, . Copyright © 2019 Hal R. Varian

51. MRS for Quasi-linear Utility Functions – 2 MRS = − f ’ (x 1 ) does not depend upon x 2 so the of indifference curves for a quasi-linear utility function is a long any line for which x 1 is constant. What does that make the indifference map for a quasi-linear utility function look like? Copyright © 2019 Hal R. Varian

52. MRS for Quasi-linear Utility Functions – 3 x 1 ″ Copyright © 2019 Hal R. Varian

53. Monotonic Transformations & Marginal Rates of Substitution – 1 As previously discussed, applying a monotonic transformation to a utility function representing a preference relation simply creates another utility function representing . What happens to marginal rates of substitution when a monotonic transformation is applied? Copyright © 2019 Hal R. Varian

54. Monotonic Transformations & Marginal Rates of Substitution – 2 For the . Create What is the MRS for V ? Which is the as the MRS for u . Copyright © 2019 Hal R. Varian

55. Monotonic Transformations & Marginal Rates of Substitution – 3 More generally, if V = f ( u ) where f is a strictly increasing function, then the MRS is by this positive monotonic transformation. Copyright © 2019 Hal R. Varian

56. Credits This concludes the Lecture PowerPoint presentation for Chapter 4 of Intermediate Microeconomics, 9e For more resources, please visit http://digital.wwnorton.com/intermicro9media . Copyright © 2019 Hal R. Varian