2. 1 Chapter 4 - Consumer Choice People can’t have everything; their choices are always constrained by factors such as time and money. In this chapter we will cover: 4.1 The budget constraint 4.2 Shifts in the budget constraint 4.3 Maximizing Utility 4.4 Minimizing Expenditure 4.5 Revealed Preferences

3. 2 4.1 The Budget Constraint If an individual only consumes 2 goods or services (x and y), their consumption is affected by 3 exogenous variables: –The price of x –The price of y –Income

4. 3 Price of x = P x Price of y = P y Income = I Total expenditure on basket (x,y): P x x + P y y Assume only two goods available: x and y The Basket is Affordable if total expenditure does not exceed total Income: P x x + P y y ≤ I

5. 4 The set of baskets that are affordable is the consumer’s BUDGET SET The BUDGET CONSTRAINT defines the set of baskets that the consumer may purchase given the income available: P x x+P y y=I The graphable BUDGET LINE is the set of baskets that are just affordable: y=I/P y -(P x /P y )x

6. 5 Two goods available: x and y I = $10 P x = $1 P y = $2 Budget line: 1x + 2y = 10 …BL 1 Or… y = 5 – x/2

7. 6 I/P X = 10 Y X A B I/P Y = 5 y=5-1/2x; 10=2y+x -P X /P Y = -1/2

8. 7 I/P X = 10 Y X A C B I/P Y = 5 Point A: one only consumes y Point B: one only consumes x. Point D: consumes a mixture Point C: consumes a mixture while not spending the entire budget Point E: unobtainable unless prices or income change D E

9. 8 4.2 Shifts in the Budget Constraint The budget line will change if any of its components change: –Income (shift of the budget line) –Prices of x and/or y (rotation of the budget line)

10. 9 Y X 10 5 6 12 y = 6 - x/2; 12=2y+x If Income increases, people have more money to spend on both goods The budget line will shift out Shift of a budget line – Income Increase y=5-1/2x; 10=2y+x

11. 10 Y X 5 I = $10 P X = $1 P Y = $3 y =5-1/2x; 10=2y+x y = 3.33 - x/3 3.33 10 If the price of Y rises, the budget line gets flatter and the vertical intercept shifts down (as seen here) If the price of Y falls, the budget line gets steeper and the vertical intercept shifts up Rotation of a Budget Line

12. 11 4.3 Maximizing Utility Subject to a Budget Constraint Consumers cannot have everything; they can only purchase what their budget will allow A consumer’s budget will allow for many different bundles of goods –Each bundle will give a different utility –A rational consumer will purchase the bundle that maximizes their utility

13. 12 Y X IC 2 BL 0 C B IC 1 A D Point A: affordable, doesn’t maximize utility Point B: unaffordable Point C: affordable (with income left over) but doesn’t maximize utility Point D: affordable, maximizes utility

14. 13 Maximizing Utility Subject to a Budget Constraint Maximize utility (which depends on x and y) by choosing x and y…. Subject to the constraint that the amount you spend on X and Y must not exceed income –(Generally, people spend all their income, so less than or equal to becomes equal to)

15. 14 Maximizing Utility Subject to a Budget Constraint Exogenous variables: P x, P y and Income Endogenous variables: x and y (chosen) and Utility (outcome)

16. 15 M E IC 2 BL=5-1/2E 0 C IC 1 A D: M=3, E=4 Max U(exercise, movie) given P exercise =$10 P movie =$20, Income=$100 (s.t. P e e+P m M=$100) 5 10 4 3

17. 16 Tangency

18. 17 Interior Optimum A basket is an INTERIOR OPTIMUM if positive amounts of all goods are purchased and the indifference curve is tangent to the budget line Note for more than 2 goods:

19. 18 Optimization Steps 1)Derive the BUDGET LINE 2) Calculate the point of tangency 3) Use (1) and (2) to solve for the maximizing point 4) Conclude and confirm budget is spent

20. 19 Maximization Example Vincent “fingers” McGiny enjoys two things: shooting people (s) and throwing bricks through windows (b). His MU s =2b and MU b =3s. A clip of bullets costs Fingers $2 (he’s a poor shot and needs the whole clip per hit) while a brick costs $1. If Fingers has $20, what should he to do maximize his utility?

21. 20 Optimization Steps 1)P s S+P b B=I 2S+B=20 2) MU s /MU b =P s /P b 2B/3S=2/1 2B=6S B=3S

22. 21 Optimization Steps 3)2S+B=20 2S+3S=20 5S=20 S=4 B=3S B=3(4)=12

23. 22 Optimization Steps 4) Fingers buys 12 bricks and 4 clips of bullets in order to maximize his utility. Check: 2S+B=20 2(4)+12=20 20=20

24. 23 Corner Solutions Interior Optimums occur when positive amounts of both goods are consumed to maximize utility Not everyone will maximize utility by consuming both goods: –Not everyone buys a Porsche –Not everyone values ballet shoes highly When utility is maximized and one good is not consumed, a Corner Solution exists

25. 24 Porsches Food IC 2 BL 0 B IC 1 A Point A consumes positive amounts of both goods, but does not maximize utility Point B maximizes utility while consuming only 1 good.

26. 25 Finding a corner solution 1)Solve for equilibrium as normal using the tangency condition: MU x /P x =MU y /P y 2) If either good is negative, zero of that good is consumed. (Unless negatives are valid) 3) Recalculate the basket that maximizes utility (using budget constraint).

27. 26 Two goods available: Siamese Cats and Dachund Dogs: I = $200 P c = $100 P d = $50 Utility is such that: MU c =d MU d =5+c

28. 27 Optimization Steps 1)P c c+P d d=I 100c+50d=200 2) MU c /MU d =P c /P d d/(5+c)=100/50 50d=500+100c d=10+2c

29. 28 Optimization Steps 3)100c+50d=200 100c+50(10+2c)=200 200c=-300 c=-3/2, therefore c=0 100(0)+50d=200 d=4

30. 29 Optimization Steps 4) Buying 4 dogs and no cats will maximize your utility: Budget Check: 100c+50d=200 100(0)+50(4)=200 200=200

31. 30 In the case of perfect compliments, utility is maximized when goods are consumed in a set ratio, which simplifies our calculations: Example: Let U(X,Y) = min(X,Y). Let I = $1000, Px = $50 and PY = $200. What is the optimal consumption basket? We know that to maximize utility x=y therefore: 50x+200y=1000 50x+200x=1000 4=x=y

32. 31 Example: Let U(X,Y) = min(X,Y). Let I = $1000, P x = $50 and P Y = $200. What is the optimal consumption basket? Budget line: Y = $5 - X/4

33. 32 Thus far, we have considered utility maximization: -Given one’s budget constraint, maximize utility (ie: buying the best lunch affordable) Sometimes one wishes to achieve a level of utility for the least cost possible – cost minimization -Given one’s required utility, what is the least one can spend? (ie: buying the cheapest lunch that will fill you up)

34. 33 The mirror image of the original (primal) utility optimization problem is called the expenditure minimization problem. Min P x X + P y Y (X,Y) subject to: U(X,Y) = U* where: U* is a target level of utility.

35. 34 Minimization Steps 1) Calculate the point of tangency (as per maximization) 2) Use the point of tangency and the UTILITY constraint to solve for the minimizing x and y 3) Conclude and confirm the minimum utility is reached and calculate expenditure

36. 35 Minimization Example Vincent “Hawaii” McGiny is a cheap mobster who enjoys the ham and pinapples on his Hawaiian pizzas. Every dinner he aims to achieve a utility of 18 by eating a slice of pizza which gives him utility of U=ham*pineapple. (MU h =p, MU p =h) If a slice of ham costs 10 cents and a piece of pineapple costs 20 cents, minimize Hawaii's expenditure

37. 36 Minimization Steps 1) MU h /MU p =P h /P p p/h=10/20 20p=10h 2p=h

38. 37 Minimization Steps 2) U=hp 18=2p(p) 9=p 2 3=p U=hp 18=h3 6=h

39. 38 Minimization Steps 3) I=P h h+P p P I=0.1(6)+0.2(3) I=1.20 Hawaii spends $1.20 for a satisfying slice of pizza. Check: U=ph 18=3(6) 18=18

40. 39 Optimization Comparison Utility MaximizationExpend. Minimization Given prices and utility formulas Given EXPENDITUREGiven UTILITY Solve tangency condition Substitute into budget constraint Substitute into utility formula Solve for UTILITY Solve for EXPENDITURE

41. 40 Composite Goods In reality, people consume more than one good Economists often want to study one good by graphing that good on the x axis and ALL other goods on the y axis The good on the Y axis is a COMPOSITE GOOD with default price P y =1

42. 41 Lava Lamps Composite Good 0 1.5 300 A IC 1 1 Composite goods allow an economist to study choices revolving around 1 good

43. 42 Composite Goods Application – Coupons vs. Cash Often governments consider equilibrium consumption of goods (such as essentials – food etc) to be less than optimal Governments then have 2 main options to increase consumption of these goods: vouchers/coupons or cash subsidies Cash subsidies are administratively easier but may not be optimal…..

44. 43 I/P h Food (units) Composite good, units F A F Min Example: Consider a situation where an individual consumes F a food, yet the government considers the minimum food an individual needs as F Min A I

45. 44 I/P h Food (units) Composite good, units F A F Min One option is to offer a cash subsidy to increase food consumption. However, some consumers will spend some of this cash subsidy on other goods (ie: drugs) A I (I+S)/P h I+S B

46. 45 I/P h Food (units) Composite good, units F A F Min In order to limit the increase in composite goods, the government can issue food vouchers instead, resulting in the new yellow kinked budget curve A I (I+S)/P h I+S B I+V (I+V)/P h C

47. 46 (I+V)/P h Food (units) Composite good, units F A F Min Note: A Kinked budget line due to a voucher often offers less total utility than a cash subsidy equal to the voucher C I+V D

48. 47 Composite Goods Application – Joining a Club Often consumers have the option of joining a club in order to save on goods purchased –Ie) Book or CD club –Ie) Chapters Rewards, CostCo, etc. While some consumers will benefit from joining the club, others will not

49. 48 CDs (number) Composite Good 0 15 300 A IC 1 10 Originally, a consumer buys 10 CDs at $20 per CD before joining the club

50. 49 CDs (number) Composite Good 0 15 20 300 200 A B IC 1 IC 2 Joining the club requires a membership fee of $100, which shifts in the budget line. At the same time however, CD’s now cost $10 each, shifting the BL’s x- intercept as shown. 10 This consumer benefits from joining the club

51. 50 CDs (number) Composite Good 0 15 20 300 200 A B IC 2 IC 1 Not every consumer will benefit from joining the club

52. 51 Application 3 - Borrowing and Lending Composite goods can also explain why people save or borrow money Consider 2 time periods, now and the future, each with income (I 1 and I 2 ) and an interest rate r If you spend nothing today, you can spend I 2 +I 1 (1+r) in the future If you will spend nothing in the future, you can spend I 1 +I 2 /(1+r) today

53. 52 C 1, spending this year ($) C 2, spending next year ($) I1I1 I2I2 C 1B C 2B I 1 +I 2 /(1+r) I 2 + I 1 (1+r) This budget line represents all possible saving or borrowing opportunities At point A, everything is spent as it is made At point B, money is borrowed this year At point C, money is saved for the future B A C C 2C

54. 53 C 1, spending this year ($) C 2, spending next year ($) I1I1 I2I2 C 1B C 2B I 1 +I 2 /(1+r) I 2 + I 1 (1+r) For an individual consumer, borrowing may give a higher lifetime utility What does this assume? B A IC 1 IC 2

55. 54 Application 4 - Quantity Discounts Some companies offer discounts for quantities beyond a certain point –Ie: Photocopying: 2 cents per sheet up to 100 then 1 cent after that Quantity discounts can entice consumers to purchase more, resulting in higher utility for the consumer and higher profits for the firm

56. 55 Photocopies Composite Good 0 300 6 A B 100 N M O A price of 2 cents per copy results in budget line NM while a quantity discount after 100 copies results in kinked budget line NAO

57. 56 4.5 Revealed Preferences Often a consumer’s preference can be inferred without indifference curves Mathematically, when a consumer decides between 2 baskets:

58. 57 Weak Axiom of Revealed Preference (WARP)

59. 58 Rationality Check Revealed preferences can determine whether an agent is acting rationally This can be done mathematically or graphically -Remember that graphically any point to the northeast is preferred

60. 59 Example: Consumer Choice that Fails to Maximize Utility Two goods, X and Y: I = $24 (P X,P Y ) = (4,2) (BL 1 ) (P’ x,P’ Y ) = (3,3) (BL 2 ) (X A,Y A ) = (5,2) (Basket A) (X B,Y B ) = (2,6) (Basket B) Basket A chosen when BL is BL 1 Basket B chosen when BL is BL 2

61. 60 P’ x X B + P’ y Y B = 3(2)+3(6) = 24 P’ x X A + P’ y Y A = 3(5)+3(2) = 21 B is chosen when it is more expensive; B  A There is a contradiction Weak Axiom of Revealed Preference is Violated Consumer is not rational P x X A + P y Y A = 4(5)+2(2) = 24 P x X B + P y Y B = 4(2)+2(6) = 20 A is chosen when it is more expensive; A  B

62. 61 X Composite good D A B C BL 1 BL 2 6 8 8 12 -At Budget line 1, pick basket A: A  C, C  B therefore A  B -At Budget line 2, pick basket B: B  D, D  A therefore B  A CONTRADICTION! Example: Revealed Preference Analysis

63. 62 Chapter 4 Key Concepts  The budget constraint  Shifts in the budget constraint  Maximizing Utility  Tangency  Corner solutions  Perfect Compliments  Minimizing Expenditure  Duality  Composite Goods

64. 63 Chapter 4 Key Concepts  Revealed Preferences  Weak Axiom of Revealed Preferences  Graphical application  Dogs are better than cats  If you ignore the notes you can miss things