Chapter 3: Preferences

Transcription

1 Econ 401 Price Theory Chapter 3: Preferences Instructor: Hiroki Watanabe Summer / 62 1 Introduction 2 Preference Relations 3 Assumptions Rational Preferences Well-Behaved Preferences 4 Indifference Curves Indifference Curves Preferred Sets Convexity and Indifference Curves 5 Example Preferences Perfect Complements 6 Trinity Marginal Willingness to Pay Comparison to Relative Prices 7 Summary 2 / 62

2 Chapter 2 covered $ part of consumer theory. Now we move on to part. How do we describe? 3 / 62 Fact 1 Numbers are easy to compare (Chapter 4). 2 Bundles are hard to compare (Chapter 3). 1 x C = 1 vs y C = (x C, x T ) = (10, 3) vs (y C, y T ) = (3, 12). 4 / 62

3 We assume that a decision-maker always chooses his most preferred alternative from his set of available alternatives. We need to some tools to describe decision-maker s preferences. 5 / 62 1 Introduction 2 Preference Relations 3 Assumptions Rational Preferences Well-Behaved Preferences 4 Indifference Curves Indifference Curves Preferred Sets Convexity and Indifference Curves 5 Example Preferences Perfect Complements 6 Trinity Marginal Willingness to Pay Comparison to Relative Prices 7 Summary 6 / 62

4 Comparing two different bundles x = (x 1, x 2 ) and y = (y 1, y 2 ). Preference (x y): You like x at least as much as y. Say x is preferred to y. Indifference (x y): Consuming x or y doesn t make any difference to you. If you prefer x = (x C, x T ) = (10, 3) to y = (y C, y T ) = (3, 6), we write (10, 3) (3, 6). If you prefer (y C, y T ) = (3, 6) to x = (x C, x T ), we write (3, 6) (10, 3). If you are indifferent between (x C, x T ) = (10, 3) and (y C, y T ) = (3, 6), we write (10, 3) (3, 6). 7 / 62 Note x y is equivalent to x y and y x, i.e., if you like x at least as much as y and y at least as much as x at the same time, then you re indifferent between x and y. 8 / 62

5 1 Introduction 2 Preference Relations 3 Assumptions Rational Preferences Well-Behaved Preferences 4 Indifference Curves Indifference Curves Preferred Sets Convexity and Indifference Curves 5 Example Preferences Perfect Complements 6 Trinity Marginal Willingness to Pay Comparison to Relative Prices 7 Summary 9 / 62 Rational Preferences Rational Preferences Your preference is rational if your preference relation is complete and transitive. 1 Complete: For any bundle x = (x 1, x 2 ), y = (y 1, y 2 ), you can say whether x y, y x or both. 2 Transitive: For x = (x 1, x 2 ), y = (y 1, y 2 ), and z = (z 1, z 2 ), if x y and y z, then x z. 10 / 62

6 Rational Preferences Example At Kayak s, the following combinations of cheesecakes and tea are available: x = (x C, x T ) = (1, 2). y = (y C, y T ) = (2, 1). z = (z C, z T ) = (3, 4). Greg s preferences are given by y x, x z, y z. Dharma s preferences are given by x y, z x. Which one has rational preferences? 11 / 62 Rational Preferences Greg s preferences are complete because he compares all the choices. Greg s preferences are transitive because y x, x z should imply y z and he does have that relation. Dharma s preferences are not complete because she doesn t compare y and z. Dharma s preferences are not transitive because z x, x y should imply z y but she doesn t have that. 12 / 62

7 Rational Preferences We exclude irrational agents like Dharma (she cannot make a choice between y and z). While Greg s preferences ARE rational, his preference y = (2, 1) z = (3, 4) doesn t sound convincing. 13 / 62 Well-Behaved Preferences Well-Behaved Preferences Your preferences are well-behaved if your preferences are 1 Monotonic 2 Convex. 14 / 62

8 Well-Behaved Preferences Monotonicity More of any commodity is always preferred. If x 1 y 1 and x 2 y 2, then x y. If is monotonic, for x = (x C, x T ) = (1, 2) z = (z C, z T ) = (3, 4) x because 3 1 and 4 2. The bundles lying to the northeast of x is preferred to x itself. 15 / 62 Well-Behaved Preferences T 4 z C 16 / 62

9 Well-Behaved Preferences Convex Preferences If you prefer mixtures of bundles are preferred to the bundles themselves, your preferences are convex. If your preferences are convex and 1 9 x = y =, then z = = x / 62 Well-Behaved Preferences 18 / 62

10 Well-Behaved Preferences The mixture doesn t have to be z = = x z = = x z = = x / 62 Well-Behaved Preferences 10 Tea (x 2 ) x=(1, 9) y=(9, 1) Cheesecakes (x 1 ) 1 20 / 62

11 1 Introduction 2 Preference Relations 3 Assumptions Rational Preferences Well-Behaved Preferences 4 Indifference Curves Indifference Curves Preferred Sets Convexity and Indifference Curves 5 Example Preferences Perfect Complements 6 Trinity Marginal Willingness to Pay Comparison to Relative Prices 7 Summary 21 / 62 Indifference Curves Now we have tool to describe Greg s preferences: &. For example, we can describe Greg s preferences as: (2, 2) (1, 1) (1, 3) (2, 2) (3, 2) (2, 2) (3, 2) (1, 1). 22 / 62

12 Indifference Curves While does represent what Greg prefers, it is hard to visualize it. We have a device called the indifference curve to picture preferences. Indifference Curves The indifference curve at a bundle x = (x 1, x 2 ) is a collection of bundles that is equally preferred to x. If a bundle x = (x 1, x 2 ) and y = (y 1, y 2 ) is on the same indifference curve, then Greg is indifferent between them: x y. 23 / 62 Indifference Curves 24 / 62

13 Indifference Curves 25 / 62 Figure: Indifference Curves 26 / 62

14 Indifference Curves Greg s indifference curve does not go across another indifference curve if Greg is rational. 27 / 62 Indifference Curves Figure: x and z is on the same indifference curve so that x z. likewise z y. By transitivity, x y. x and y are on the different curves so that x is not y. contradiction. 28 / 62

15 Preferred Sets We have another visual aid: Preferred Set A preferred set to a bundle x = (x 1, x ) is a collection of 2 all the bundles that is preferred to x. If y x, then y is in the preferred set of a bundle x. 29 / 62 Preferred Sets 30 / 62

16 Preferred Sets If Greg s rational, his indifference curve is downward sloping O / 62 Preferred Sets Exercise Greg s preferences for cheesecakes and tea (x C, x T ) are given by: (2, 2) (1, 1) (1, 4) (2, 2) (3, 1) (1, 4) (2, 3) (2, 2). 1 Sketch an indifference curve at (x C, x T ) = (2, 2). 2 Is a bundle (2, 3) contained in the preferred set to (2, 2)? 3 How about (1, 1)? 1 32 / 62

17 Preferred Sets 5 4 Tea (x T ) Cheesecakes (x C ) 1 33 / 62 Convexity and Indifference Curves What does Greg s preferred set look like when his preferences exhibit convexity? 34 / 62

18 Convexity and Indifference Curves 35 / 62 Figure: Convexity and Indifference Curves 36 / 62

19 Convexity and Indifference Curves 37 / 62 Figure: 1 Introduction 2 Preference Relations 3 Assumptions Rational Preferences Well-Behaved Preferences 4 Indifference Curves Indifference Curves Preferred Sets Convexity and Indifference Curves 5 Example Preferences Perfect Complements 6 Trinity Marginal Willingness to Pay Comparison to Relative Prices 7 Summary 38 / 62

20 Two extreme examples of preferences and one example of linear preferences. 39 / 62 If Greg always regards units of commodities 1 and 2 as equivalent, then the commodities are perfect substitutes. E.g., Coke & Pepsi. 40 / 62

21 The indifference curve at (x C, x P ) = (2, 2) includes ,,,, (What do the bundles above have in common?) The indifference curve at (x C, x P ) = (2, 0) includes 2 1 0,,, (They all sum up to 2) / six packs (x 6 ) bottles (x 1 ) 1 42 / 62

22 The commodities do not have to be interchangeable on the one-to-one basis. Consider a bundle of six-packs and bottles of Corona (x 6, x 1 ): (0, 12) (1, 6) (0, 12) (2, 0). They all satisfy 6x 6 + x 1 = 12, i.e., any bundle on the indifference curve at (12, 0) contains 12 bottles of Corona in total. 43 / 62 Likewise, They all satisfy It is likely that (4, 0) (3, 6) (4, 0) (2, 12) (4, 0) (1, 18) (4, 0) (0, 24). x 1 + 6x 6 = 24. (4, 0) (2, 0). 44 / 62

23 24 18 bottles (x 1 ) six packs (x 6 ) 1 45 / 62 Exercise Draw the indifference curve at the bundle (x 5, x 1 ) = (2, 10), where x 5 denotes # of nickels and x 1 denotes # pennies. Note (2, 10) amounts to 20 cents. What other combinations of pennies and nickels brings you 20 cents? 46 / 62

24 20 15 Pennies (x 1 ) Nickels (x 5 ) 1 47 / 62 Perfect Complements Perfect substitutes are completely interchangeable and one commodity is replaced by the other commodity. Perfect complements are the other extreme case: Perfect Complements If Greg always consumes commodities 1 and 2 in fixed proportion, then the commodities are perfect complements. Only the # of pairs of the two commodities (as opposed to # of each commodities) determines his preferences. E.g., left glove and right glove. 48 / 62

25 Perfect Complements 49 / 62 Figure: Perfect Complements Exercise Draw Greg s indifference curve at (cereal, milk)= (2, 5). Greg says he can t have cereals without milk and the only time he has milk is when he eats his cereals. Greg s preferred cereal-milk ratio is 2 to / 62

26 Perfect Complements Milk (x M ) Cereal (x C ) 1 51 / 62 1 Introduction 2 Preference Relations 3 Assumptions Rational Preferences Well-Behaved Preferences 4 Indifference Curves Indifference Curves Preferred Sets Convexity and Indifference Curves 5 Example Preferences Perfect Complements 6 Trinity Marginal Willingness to Pay Comparison to Relative Prices 7 Summary 52 / 62

27 Marginal Willingness to Pay Just like the slope of the budget line has the meaning (recall trinity), the slope of the indifference curve has an important meaning. Marginal Willingness to Pay If Greg is just willing to give up one slice of cheesecake for a cups of tea, then a is called his marginal willingness to pay. What does he mean by "just willing"? E.g., if (4, 5) (3, 7), then Greg is just willing to give up 1 slice of cheesecake for 2 cups of tea. He is indifferent between those two bundles. MWTP is 2. If (8, 2) (7, 6), then Greg is just willing to give up 1 slice of cheesecake for 4 cups of tea. His MWTP at (8, 2) is / 62 Marginal Willingness to Pay In general xc x T xc 1. x T + MRS "Marginal" = additional one unit. 54 / 62

28 Marginal Willingness to Pay bottles (x 1 ) six packs (x 6 ) 1 55 / 62 Marginal Willingness to Pay The slope of indifference curves represents the marginal willingness to pay. MWTP is also referred to as marginal rate of substitution. 56 / 62

29 Marginal Willingness to Pay T 1 O C 57 / 62 Marginal Willingness to Pay MRS usually varies depending on the bundle x. Consider two bundles: x = (x C, x T ) = (1, ) y = (y C, y T ) = ( , 1). MRS at x is pretty large: Greg is willing to give up lots of cups of tea for a slice of cheesecake. MRS at y is pretty small: Greg is willing to give up only few cups of tea for a slice of cheesecake (he already has many cheesecakes) / 62

30 Marginal Willingness to Pay 59 / 62 Figure: Comparison to Relative Prices Notice the difference: Relative price is the cups of tea Greg has to sell (give up) to get one slice of cheesecakes to keep to his budget. Marginal willingness to pay is the cups of tea Greg is willing to give up to get one slice of cheesecakes to remain as happy as before. 60 / 62

31 1 Introduction 2 Preference Relations 3 Assumptions Rational Preferences Well-Behaved Preferences 4 Indifference Curves Indifference Curves Preferred Sets Convexity and Indifference Curves 5 Example Preferences Perfect Complements 6 Trinity Marginal Willingness to Pay Comparison to Relative Prices 7 Summary 61 / 62 Describe preferences. Rational, well-behaved, perfect substitutes, perfect complements. Indifference curves, preferred sets. Trinity. 62 / 62