1. Homework:

6. Indifference Curves Definition: For any bundle a and a preference relation over bundles, the indifference curve through a is the set of all bundles that are indifferent to a in that preference relation. Basically, it’s the curve the traces out which bundles are equally good. An indifference map is the set of all indifference curves. Note that an indifference map will include all possible bundles, as long as preferences are complete.

7. Indifference Curves Remember: we assume preferences are monotonic, complete, and transitive. Are Upward sloping curves possible? No! UNC Basketball National Championships UNC Football National Championships

8. Indifference Curves Remember: we assume preferences are monotonic, complete, and transitive. Are Upward sloping curves possible? No! UNC Basketball National Championships UNC Football National Championships A B

9. Indifference Curves Remember: we assume preferences are monotonic, complete, and transitive. Are thick indifference curves possible? No. UNC Basketball National Championships UNC Football National Championships

10. Indifference Curves Remember: we assume preferences are monotonic, complete, and transitive. Are thick indifference curves possible? No. UNC Basketball National Championships UNC Football National Championships A B

11. Indifference Curves Remember: we assume preferences are monotonic, complete, and transitive. Can indifference curves cross? No.

12. Indifference Curves Remember: we assume preferences are monotonic, complete, and transitive. Can indifference curves cross? No. A B C

13. Indifference Curves Good 2 Good 1 I1I1 I2I2 I3I3 I4I4 I5I5 I6I6 I7I7

14. UNC Basketball National Championships UNC Football National Championships Preferred A B C Indifference Curves

15. Review: Indifference curves must be: 1.Everywhere 2.Thin 3.Non-increasing 4.Non-Crossing

16. Indifference Curves Pizza Coke I1I1 I2I2 x y

17. Indifference Curves Pizza Coke I1I1 I2I2 x y

18. Indifference Curves Review: Indifference curves representing convex preferences must be: 1.Everywhere 2.Thin 3.Non-increasing 4.Non-Crossing 5.Convex (bowed inward)

19. Indifference Curves Indifference curves tell us how people make tradeoffs Pizza Coke

20. Indifference Curves What does an extremely bowed indifference curve mean? Good 2 Good 1 I1I1

21. Indifference Curves This curve represents perfect complements, where goods are consumed only in fixed proportions Good 2 Good 1 I1I1

22. Indifference Curves What does a straight indifference curve mean? Good 2 I1I1 Good 1

23. Indifference Curves This curve represents perfect substitutes, where goods are interchangeable at some fixed ratio. Good 2 I1I1 Good 1

24. Indifference Curves Intermediately bowed curves are imperfect substitutes or complements. More curved means that the goods are more complementary—the individual wants to consume them together. Less curved means one can be used in place of the other—as a substitute. Pizza Coke

25. Indifference Curves We’d like to be able to quantify what trades individuals are willing to make between various goods Good 2 Good 1 I1I1

26. Indifference Curves Marginal Rate of Substitution The MRS between good 1 and good 2 ( MRS 12 ) is: 1.The maximum amount of Good 2 the individual would be willing to give up to get one more unit of Good 1 2.The minimum amount of Good 2 the individual would need to receive to give up one unit of Good 1. Good 2 Good 1 I1I1

27. Indifference Curves Good 2 Good 1 I1I1

28. Indifference Curves

29. Homework: Finish Petranka Ch. 3 Have a great weekend!

30. Utility

31. Preferences are rational and monotonic iff an increasing utility representation exists.

32. Utility Ordinal interpretation of utility functions: In this class, we will not be interested in cardinal utility—that is, the exact value of the utility function. Instead, we will be interested in the ordinal properties of utility—that is, which value of the utility function is higher.

33. Utility Ordinal comparisons: Kate is taller than Jenny. 5 is more than 3. Pete likes making $100,000/year more than making $20,000/year. Cardinal comparisons Kate is 3 inches taller than Jenny. 5 is 2 more than 3. Pete likes making $100,000/year twice as much as $20,000/year.

34. Utility

35. Monotonic Transformations Definition: f(x) is monotonic if f(x)>f(y) iff x>y. If f’(x)>0 for all x, then f is monotonic.

36. Utility Uses for cardinal utility In order to decide what allocation of resources is better, we usually need cardinal utility. Why? Because interpersonal comparisons of utility are impossible with ordinal preferences.

37. Utility Uses for cardinal utility Examples: Is a monopoly bad? CS and PS require that everyone gets the same cardinal utility from money. When Bill Gates gives money to impoverished people in Africa, do they gain more welfare than he loses? Using the concept of decreasing marginal returns to wealth, we’d guess that they gain much more than he loses. But without cardinal utility we have no way to analyze that in an economic model.

38. Utility Why ordinal utility? There are infinitely many utility functions that can represent a set of preferences, so preferences alone can’t determine cardinal utility. Specifically, any monotonic transformation of a utility function represents the same preferences.

39. Utility

41. Preferences Transitive +Complete Monotonic Continuous Utility function: Exists Increasing Continuous

42. Indifference Curves We need to get an idea of what utility functions correspond to what type of preferences.

43. Utility Good 2 Good 1

44. Utility Good 2 Good 1

45. Utility Pizza Coke

46. Utility Exercise: come up with utility functions U(x,y) to satisfy the following preferences: 1.Your preferences are continuous 2.You like x five times more than y, and would always trade 5x for a unit of y. 3.You always want to consume twice as many x as y. 4.You like having more y than x and you like having some of both more than a lot of one or the other. You always strictly prefer having another unit of either good. You have 10 minutes.

47. Indifference Curves Marginal Rate of Substitution The MRS between good 1 and good 2 ( MRS 12 ) is: 1.The maximum amount of Good 2 the individual would be willing to give up to get one more unit of Good 1 2.The minimum amount of Good 2 the individual would need to receive to give up one unit of Good 1. Good 2 Good 1 I1I1

48. Indifference Curves Good 2 Good 1 I1I1

49. Utility

50. Utility also gives us an easy way to find the marginal rate of substitution (MRS). That is, the rate you’d trade A for B at is the marginal utility of A (additional utility of a little more A) divided by the marginal utility of B.

51. Utility

52. Good 1 Good 2

53. Utility Good 1 Good 2

54. Constrained Optimization We’ve already learned how to optimize a function without constraints: 1.Take partial derivatives and set them equal to zero. 2.Solve the system of equations. However, there are almost always constraints on our choices. Examples: Choosing the time you spend studying vs leisure each day—you only have 24 hours in a day. Choosing what to spend your income one—you have a finite budget.

55. Constrained Optimization Good 2 I1I1 I2I2 I3I3 I4I4 I5I5

56. Constrained Optimization The budget set is the set of all possible bundles an individual can choose from. Usually, it’s the area under a downward sloping line that has all non-negative values for every good in the bundle. Meals Budget Set Books

57. Constrained Optimization Meals Budget Set Books

58. Constrained Optimization Meals Budget Set Books

59. Constrained Optimization Meals Books

60. Constrained Optimization

61. Break time!

62. Meals Books I1I1 I2I2 I3I3 I4I4 I5I5 Preferred Budget Line

63. Constrained Optimization Can a point below the budget line be optimal if preferences are complete, transitive, convex, and monotonic? No! Any optimal bundle MUST be on the budget line. A B

64. Constrained Optimization Given rational, convex, monotonic preferences and smooth indifference curves, is it possible for an optimal bundle to include both goods and not be where the indifference curve is tangent to the budget line? No! Meals I1I1 I2I2 I3I3 I4I4 I5I5 Books

65. Constrained Optimization Given rational, convex, monotonic preferences, a smooth indifference curve, and an optimal bundle with nonzero quantities of all goods, the indifference curve must be tangent to the budget line. Meals I1I1 I2I2 I3I3 I4I4 I5I5 Books

66. Constrained Optimization

67. Special cases: Corner Solutions: If the slope of the indifference curve never equals the slope of the budget constraint, clearly MRS=MRT is not possible. In this case, the individual doesn’t consume one or more goods. Kinked indifference curves If the indifference curve is kinked, then the derivative may not be defined at the optimum.

68. Constrained Optimization Meals

69. Constrained Optimization Special cases: Corner cases: If the slope of the indifference curve never equals the slope of the budget constraint, clearly MRS=MRT is not possible. In this case, the individual consumes only one good. Generally, you can check corner solutions against interior solutions and other corner solutions directly. Whichever solution has the highest utility is the optimum. Meals

70. Constrained Optimization Special cases: Kinked indifference curves If the indifference curve is kinked, then the derivative may not be defined at the optimum. Drawing the indifference curves and budget constraint may be helpful. Check for solutions on the non kinked portion of the curve. Meals Books

71. Constrained Optimization Exercise: For each combination of budget constraint and indifference curve, draw the optimum and give the approximate optimal bundle of x and y. Note that the utility functions are quasilinear in x, so every indifference curve looks the same, just transposed to the left or right. Budget Constraints:Indifference curves:

72. Constrained Optimization Homework: Homework due Friday Read Petranka through 5.3

73. Constrained Optimization Given rational, convex, monotonic preferences, a smooth indifference curve, and an optimal bundle with nonzero quantities of all goods, the indifference curve must be tangent to the budget line. Meals I1I1 I2I2 I3I3 I4I4 I5I5 Books

74. Constrained Optimization

75. Special cases: Corner cases: If the slope of the indifference curve never equals the slope of the budget constraint, clearly MRS=MRT is not possible. In this case, the individual consumes only one good. Generally, you can check corner solutions against interior solutions and other corner solutions directly. Whichever solution has the highest utility is the optimum. Meals

76. Constrained Optimization Special cases: Kinked indifference curves If the indifference curve is kinked, then the derivative may not be defined at the optimum. Drawing the indifference curves and budget constraint may be helpful. Check for solutions on the non kinked portion of the curve. Meals Books

77. Constrained Optimization Strictly convex preferences display a diminishing marginal rate of substitution. In other words, as an individual gets more of Good A, she needs to receive less of Good B in order to be willing to give up some of Good A Pizza Coke

78. Constrained Optimization

80. Restricted Conditions to use Lagrangian techniques to find a constrained maximum 1.The function we want to maximize is continuous. 2.The function we want to maximize has a derivative that is continuous. 3.The function we want to maximize is quasi-concave. 4.Our constraint set is defined by an equality. 5.Our constraint set is defined by a weakly concave function.

81. Constrained Optimization Restricted Conditions to use Lagrangian techniques to find a constrained maximum 1.The function we want to maximize is continuous. If not, we could have very ugly functions

82. Constrained Optimization Restricted Conditions to use Lagrangian techniques to find a constrained maximum 2.The function we want to maximize has a derivative that is continuous. If not, we have a kinked function, and if the maximum is at the kink we can’t take derivatives to find the optimality condition.

83. Constrained Optimization

85. Restricted Conditions to use Lagrangian techniques to find a constrained maximum 5.Our constraint set is defined by a weakly concave function. Linear functions are weakly concave, so the budget line satisfies this condition.

86. Constrained Optimization

90. Homework: Homework due Friday Finish Chapter 5 in Petranka

91. Constrained Optimization