Lecture # 05 Consumer Preferences and the Concept of Utility (cont.) Lecturer: Martin Paredes

1. Lecture # 05 Consumer Preferences and the Concept of Utility (cont.) Lecturer: Martin Paredes

2. Outline • Indifference Curves (end) • The Marginal Rate of Substitution • The Utility Function • Marginal Utility • Some Special Functional Forms

3. Indifference Curves Definition: An Indifference Curve is the set of all baskets for which the consumer is indifferent Definition: An Indifference Map illustrates the set of indifference curves for a particular consumer

4. Properties of Indifference Maps: • Completeness • Each basket lies on only one indifference curve • Monotonicity • Indifference curves have negative slope • Indifference curves are not “thick”

5. y Monotonicity • A x

6. y Monotonicity Preferred to A • A x

7. y Monotonicity Preferred to A • A Less preferred x

8. y Monotonicity Preferred to A • A Less preferred IC1 x

9. y Indifference Curves are NOT Thick B • • A IC1 x

10. Properties of Indifference Maps: 3. Transitivity • Indifference curves do not cross 4. Averages preferred to extremes • Indifference curves are bowed toward the origin (convex to the origin).

11. y Indifference Curves Cannot Cross • Suppose a consumer is indifferent between A and C • Suppose that B preferred to A. IC1 B • • A C • x

12. y Indifference Curves Cannot Cross • It cannot be the case that an IC contains both B and C • Why? because, by definition of IC the consumer is: • Indifferent between A & C • Indifferent between B & C • Hence he should be indifferent between A & B (by transitivity). • => Contradiction. IC2 IC1 B • • A C • x

13. y Averages Preferred to Extremes A • • IC1 B x

14. y Averages Preferred to Extremes A • (.5A, .5B) • • IC1 B x

15. y Averages Preferred to Extremes A • (.5A, .5B) • IC2 • IC1 B x

16. Marginal Rate Of Substitution There are several ways to define the Marginal Rate of Substitution Definition 1: It is the maximum rate at which the consumer would be willing to substitute a little more of good x for a little less of good y in order to leave the consumer just indifferent between consuming the old basket or the new basket

17. Marginal Rate Of Substitution Definition 2: It is the negative of the slope of the indifference curve: MRSx,y = — dy (for a constant level of dx preference)

18. Diminishing Marginal Rate Of Substitution An indifference curve exhibits a diminishing marginal rate of substitution: • The more of good x you have, the more you are willing to give up to get a little of good y. • The indifference curves • Get flatter as we move out along the horizontal axis • Get steeper as we move up along the vertical axis.

19. Example: The Diminishing Marginal Rate of Substitution

20. The Utility Function Definition: The utility function measures the level of satisfaction that a consumer receives from any basket of goods.

21. The Utility Function • The utility function assigns a number to each basket • More preferred baskets get a higher number than less preferred baskets. • Utility is an ordinal concept • The precise magnitude of the number that the function assigns has no significance.

22. Ordinal and Cardinal Ranking • Ordinalranking gives information about the order in which a consumer ranks baskets • E.g. a consumer may prefer A to B, but we cannot know how much more she likes A to B • Cardinal ranking gives information about the intensity of a consumer’s preferences. • We can measure the strength of a consumer’s preference for A over B.

23. Example: Consider the result of an exam • An ordinal ranking lists the students in order of their performance • E.g., Harry did best, Sean did second best, Betty did third best, and so on. • A cardinal ranking gives the marks of the exam, based on an absolute marking standard • E.g. Harry got 90, Sean got 85, Betty got 80, and so on.

24. The Utility Function Implications of an ordinal utility function: • Difference in magnitudes of utility have no interpretation per se • Utility is not comparable across individuals • Any transformation of a utility function that preserves the original ranking of bundles is an equally good representation of preferences. eg. U = xy U = xy + 2 U = 2xy all represent the same preferences.

25. y Example: Utility and a single indifference curve 5 2 10 = xy 0 x 2 5

26. y Example: Utility and a single indifference curve Preference direction 5 20 = xy 2 10 = xy 0 x 2 5

27. Marginal Utility Definition: The marginal utility of good x is the additional utility that the consumer gets from consuming a little more of x MUx = dU dx • It is is the slope of the utility function with respect to x. • It assumes that the consumption of all other goods in consumer’s basket remain constant.

28. Diminishing Marginal Utility Definition: The principle of diminishing marginal utility states that the marginal utility of a good falls as consumption of that good increases. Note: A positive marginal utility implies monotonicity.

29. Example: Relative Income and Life Satisfaction (within nations) Relative IncomePercent > “Satisfied” Lowest quartile 70 Second quartile 78 Third quartile 82 Highest quartile 85 Source: Hirshleifer, Jack and D. Hirshleifer, Price Theory and Applications. Sixth Edition. Prentice Hall: Upper Saddle River, New Jersey. 1998.

30. Marginal Utility and the Marginal Rate of Substitution We can express the MRS for any basket as a ratio of the marginal utilities of the goods in that basket • Suppose the consumer changes the level of consumption of x and y. Using differentials: dU = MUx . dx + MUy . dy • Along a particular indifference curve, dU = 0, so: 0 = MUx . dx + MUy . dy

31. Marginal Utility and the Marginal Rate of Substitution • Solving for dy/dx: dy = _ MUx dx MUy • By definition, MRSx,y is the negative of the slope of the indifference curve: MRSx,y = MUx MUy

32. Marginal Utility and the Marginal Rate of Substitution • Diminishing marginal utility implies the indifference curves are convex to the origin (implies averages preferred to extremes)

33. Example: • U= (xy)0.5 • MUx=y0.5/2x0.5 • MUy=x0.5/2y0.5 • Marginal utility is positive for both goods: • => Monotonicity satisfied • Diminishing marginal utility for both goods • => Averages preferred to extremes • Marginal rate of substitution: • MRSx,y = MUx = y • MUy x • Indifference curves do not intersect the axes

34. y Example: Graphing Indifference Curves IC1 x

35. y Example: Graphing Indifference Curves Preference direction IC2 IC1 x

36. Special Functional Forms • Cobb-Douglas (“Standard case”) U = Axy where:  +  = 1; A, , positive constants Properties: MUx = Ax-1y MUy = Axy-1 MRSx,y = y x

37. y Example: Cobb-Douglas IC1 x

38. y Example: Cobb-Douglas Preference direction IC2 IC1 x

39. Special Functional Forms • 2. Perfect Substitutes: • U = Ax + By where: A,B are positive constants Properties: MUx = A MUy = B MRSx,y = A (constant MRS) B

40. Example: Perfect Substitutes (butter and margarine) y IC1 0 x

41. Example: Perfect Substitutes (butter and margarine) y IC2 IC1 0 x

42. Example: Perfect Substitutes (butter and margarine) y Slope = -A/B IC2 IC3 IC1 0 x

43. Special Functional Forms • 3. Perfect Complements: • U = min {Ax,By} where: A,B are positive constants Properties: MUx = A or 0 MUy = B or 0 MRSx,y = 0 or  or undefined

44. Example: Perfect Complements (nuts and bolts) y IC1 0 x

45. Example: Perfect Complements (nuts and bolts) y IC2 IC1 0 x

46. Special Functional Forms • 4. Quasi-Linear Utility Functions: • U = v(x) + Ay where: A is a positive constant, and v(0) = 0 Properties: MUx = v’(x) MUy = A MRSx,y = v’(x) (constant for any x) A

47. y Example: Quasi-linear Preferences (consumption of beverages) IC1 • 0 x

48. y Example: Quasi-linear Preferences (consumption of beverages) IC2 IC’s have same slopes on any vertical line IC1 • • 0 x

49. Summary • Characterization of consumer preferences without any restrictions imposed by budget • Minimal assumptions on preferences to get interesting conclusions on demand…seem to be satisfied for most people. (ordinal utility function)