2.  This will explain how consumers allocate their income over many goods.  This looks at individual’s decision making when faced with limited income and market- determined price.  This allows us to derive the market demand curve that we used in our supply-and- demand model and to make a variety of predictions about consumers’ responses to changes in prices and income. 2

3.  This is based on the following premises: ◦ Individual tastes or preferences determine the amount of pleasure people derive from the goods and services they consume. ◦ Consumers face constraints or limits on their choices. ◦ Consumers maximize their well-being or pleasure from consumption, subject to the constraints they face. 3

4.  Economists assume that consumers have a set of taste or preferences that they use to guide them in choosing between goods.  Properties of consumer preferences: ◦ completeness ◦ transitivity ◦ more is better ◦ continuity ◦ strict convexity 4

5.  This property holds that when facing a choice between any two bundles of goods, a consumer can rank them so that one and only one of the following relationships is true:  The consumer, ◦ prefers the first bundle to the second bundle ◦ prefers the second bundle to the first bundle, or ◦ is indifferent between the two bundles. 5

6.  Also known as rationality, this property is that a consumer’s preferences over bundle is consistent in the sense that, if the consumer weakly prefers Bundle z to Bundle y - that is, likes z at least as much as y - and weakly prefers Bundle y to Bundle x, the consumer also weakly prefers Bundle z to Bundle x. 6

7.  Also known as nonsatiation, holds that, all else the same, more of a commodity is better than less of it.  A good is defined as a commodity for which more is preferred to less, at least at some levels of consumption.  A bad is something for which less is preferred to more, such as pollution.  If consumers can freely dispose of excess goods, consumers can be no worse off with extra goods. 7

8.  This property holds that if a consumers prefers Bundle a to Bundle b, then the consumer prefers Bundle c to b if c is very close to a. 8

9.  This means that consumers prefer averages to extremes.  If Bundle a and Bundle b are distinct bundles and the consumer prefers both of these bundles to Bundle c, then the consumer prefers a weighted average of a and b, βa + (1-β)b (where 0< β<1), to Bundle c. 9

10.  One of the simplest ways to summarize information about a consumer’s preferences is to create a graphical interpretation – a map – of them.  For simplicity, we concentrate on choices between only two goods, but the model can be generalized to handle any number of goods. 10

11. Illustration: Lisa, who lives for fastfood, decides how many pizzas and burgers to eat. The various bundles of pizzas and burgers she might consume are shown in the table below. BundlePizzaBurger a3212 b3010 c1525 d1510 e2515 f3020 11

12. 0 152530 10 25 20 15 Pizza (q 1 ) Burger (q 1 ) f d c b a e A B 32 12

13.  The set of all bundles of goods that a consumer views as being equally desirable.  Suppose that consuming Bundle a, or Bundle e, or Bundle c will give Lisa the same amount of pleasure, then she is indifferent about consuming these bundles.  If we draw a curve connecting those bundles that Lisa is indifferent about, then this curve is called an indifference curve. 13

14. 0 152530 10 25 20 15 Pizza (q 1 ) Burger (q 1 ) f d c b a e I1I1 32 12 14

15.  A complete set of indifference curves.  We could draw an entire set of indifference curves through every possible bundles.  This is a summary of preferences.  Also known as preference maps. 15

16. 0 2530 10 25 20 15 Pizza (q 1 ) Burger (q 1 ) f d c a e I1I1 I0I0 I2I2 32 12 16

17.  Bundles on higher indifference curves are preferred to those on lower indifference curves.  There is an indifference curve through every possible bundle.  Indifference curves cannot cross.  Indifference curves slope downward.  Indifference curves cannot be thick. 17

18. 0 152530 10 25 20 15 Pizza (q 1 ) Burger (q 1 ) b a e I1I1 I2I2 32 12 18

19. 0 152530 10 25 20 15 Pizza (q 1 ) Burger (q 1 ) f d e I 32 12 19

20. 0 152530 10 25 20 15 Pizza (q 1 ) Burger (q 1 ) b a I 32 12 20

21.  Underlying our model of consumer behavior is the belief that consumers can compare various bundles of goods and decide which bundle gives them the greatest pleasure.  We can summarize a consumer’s preferences by assigning a numerical value to each possible bundle to reflect the consumer’s relative ranking of these bundles. 21

22.  Economists apply the term utility to this set of numerical values that reflect the relative rankings of various bundles of goods.  The statement that “Lisa prefers Bundle x to Bundle y” is equivalent to the statement that “Consuming Bundle x gives Lisa more utility than consuming Bundle y.  Lisa prefers x to y if Bundle x gives Lisa 10 utils – units of utility – and Bundle y gives her 8 utils. 22

23.  This is the relationship between utility measures and every possible bundle of goods.  If we know the utility function, we can summarize the information in indifference maps. 23

24.  Suppose that the utility, U, that Lisa gets from pizzas, q 1, and burgers, q 2 is  From this function, we know that the more Lisa consumes of either good, the greater her utility.  Using this function, we can determine whether she would be happier if she had Bundle x with 16 pizzas and 9 burgers or Bundle y with 13 of each. 24

25.  The utility she gets from Bundle x is  The utility she gets from Bundle y is  Therefore she prefers Bundle y to Bundle x. 25

26.  An indifference curve consists of all those bundles that correspond to a particular utility measure.  If Lisa’s utility function is U(q 1,q 2 ), then the expression for one of her indifference curves is  This expression determines all those bundles of q 1 and q 2 that give her U 0 utils of pleasure. 26

27. For example, if the utility function is then the indifference curve includes any (q 1,q 2 ) such that q 1 q 2 = 16, including the bundles (4,4), (2,8), (8,2), (1,16), and (16,1). 27

28. 0 1 16 8 4 2 1248 q1q1 q2q2 (1,16) (2,8) (4,4) (8,2) (16,1) I:U = 4 28

29.  It is the slope at a point in an indifference curve.  It is the maximum amount of one good that a consumer will sacrifice to obtain one more unit of another good.  It is a negative number because indifference curve is downward sloping. 29

30. q1q1 q2q2 I e 30

31.  We can use calculus to determine the MRS at a point on Lisa’s indifference curve.  The MRS depends on how much extra utility Lisa gets from a little more of each good.  This extra utility that a consumer gets from consuming the last unit of a good is called the marginal utility.  Given that Lisa’s utility function is U(q 1,q 2 ), the marginal utility that she gets from a little pizza, holding the quantity of burgers fixed is 31

32.  Similarly, the marginal utility from more burgers is  The slope of Lisa’s indifference curve or MRS can be determined using the changes in q 1 and q 2 that leave her utility unchanged, keeping her on her original indifference curve. 32

33.  Let q 2 (q 1 ) be the implicit function that shows how much q 2 is needed to keep Lisa’s utility constant given that she consumes q 1.  We want to know how much q 2 must change if we increase q 1 such that her utility remains constant.  This is expressed as 33

34.  Thus, we differentiate U 0 = U(q 1,q 2 (q 1 )) with respect to q 1  Since U 0 is constant then,  and 34

35.  Recall that  Thus, 35

36.  The marginal rate of substitution is  Thus the slope of the indifference curve is the negative of the ratio of the marginal utilities. 36

37. Illustration: Find the MRS of a Cobb-Douglas utility function given as where a is a positive constant. To get the MRS, find first U 1 and U 2. 37

38. 38

39.  Because the indifference curve is convex to the origin, as we move to the right along the indifference curve, the MRS becomes smaller in absolute value.  This reflect a diminishing marginal rate of substitution. 39

40.  When people have a lot of one good, they are willing to give up a relatively large amount of it to get a good of which they have relatively little.  However, after that first trade, they are willing to give up less of the first good to get the same amount of the second good. 40

41. 0 1 16 8 4 2 1248 q1q1 q2q2 (1,16) (2,8) (4,4) (8,2) (16,1) I 41

42.  Consumers maximize their well-being subject to constraints.  The most important constraint most of us face in deciding what to consume is our personal budget constraint.  For graphical simplicity, we assume that consumers spend their money on only two goods.  If Lisa spends all her budget, Y, on pizza and burgers, then 42

43.  where p 1 is the price of pizza, and p 2 is the price of burger  thus p 1 q 1 is the amount she spends on pizza and p 2 q 2 is the amount she spends on burgers.  This is her budget line or budget constraint.  The bundles of goods that can be bought if the entire budget is spent on those goods at given prices. 43

44. Given her budget constraint We can rearranged the terms as follows Finally, we can write her budget constraint as She can buy more burgers with a higher income, 44

45. the purchase of fewer pizzas, a lower price of burgers, 45

46. If p 1 = 1, p 2 = 2, Y = 50, then or a lower price of pizzas. 46

47. 0 q2q2 q1q1 L: p 1 = 1, p 2 = 2, Y = 50 Opportunity Set Opportunity set is the set of all bundles of q 1 and q 2 such that p 1 q 1 + p 2 q 2 ≤ Y 10 15 (15,10) (15,17.5) 17.5 30 (30,10) 47

48.  The slope of the budget line.  The trade-off the market imposes on the consumer in terms of the amount of one good the consumer must give up to obtain more of the other good.  The rate at which Lisa can trade burgers for pizzas in the market where the prices she pays and her income are fixed. 48

49. Because the price of a pizza is half of a burger (p 1 = 1 and p 2 = 2), the marginal rate of transformation that Lisa faces is An extra pizza costs her half an extra burger or, equivalently, an extra burger costs her two pizzas. 49

50.  Were it not for the budget constraint, consumers who prefer more to less would consume unlimited amounts of a least some good.  Well, they can’t have it all.  Instead, consumers maximize their well- being subject to their budget constraints.  To complete our analysis of consumer behavior, we have to determine the bundle of goods that maximizes well-being subject to the budget constraint. 50

51.  We want to determine which bundle within the opportunity set gives the consumer the highest level of utility.  To determine which bundle in the opportunity set gives Lisa the highest level of utility, we use her indifference curves.  We will show that her optimal bundle lies on the indifference curve that is tangent to the budget line. 51

52. 0 q2q2 q1q1 10 20 25 103050 I1I1 I3I3 I2I2 L f e d c a b 52

53. Lisa’s objective is to maximize her utility, U(q 1,q 2 ), subject to (s.t.) her budget constraint: Two alternative solutions: 1. Substitution Method 2. Lagrangian Method 53

54. We can substitute the budget constraint into the utility function. Given the budget constraint If we substitute this expression for q 1 in the utility function, U(q 1,q 2 ), we can rewrite Lisa’s problem as We can rewrite the budget constraint as 54

55. The first–order condition is obtained by setting the derivative of the utility function with respect to q 2 equal to zero. 55

56. Rearranging the terms, we have 56

57. We can write the equivalent Lagrangian problem as where λ is the Lagrange multiplier. 57

58. First-order conditions: 58

59. Illustration: Find the optimal value of q 1 andq 2 in terms of income, prices, and the positive constant a of a Cobb-Douglas utility function given as The Lagrangian function is 59

60. First-order conditions: 60

61. Substituting 61

62. Similarly, substituting into We have 62

63. Illustration: Find the budget share of q 1 and q 2 in terms of income, prices, and the positive constant a of a Cobb-Douglas utility function given as Let s 1 ands 2 be the budget shares of q 1 andq 2 respectively, then 63

64. Since thus Similarly, implies 64

65. 0 q2q2 q1q1 10 30 I2I2 E2E2 e E3E3 E1E1 65

66.  We can use calculus to solve the expenditure- minimizing problem.  Lisa’s objective is to minimize her expenditure, E, subject to the constraint that she hold her utility constant at  This can be written as 66

67.  The solution of this problem is an expression of the minimum expenditure as a function of the prices and the specified utility level:  We call this expression the expenditure function: the relationship showing the minimal expenditures necessary to achieve a specific utility level for a given set of prices. 67

68. Illustration: Find the expenditure function for a Cobb- Douglas utility function given as The Lagrangian function is 68

69. First-order conditions: 69

70. We can rewrite the last line as: thus 70

71. Similarly, substituting into we have 71

72. Substituting q 1 andq 2 into the indifference curve expression we have we can write the expenditure function as 72

73. For example, if a = ½, then 1–a = ½ and 73