1 Chapter 4 - Consumer ChoicePeople 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

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

3 Total expenditure on basket (x,y): Pxx + PyyThe Budget Constraint Assume only two goods available: x and y Price of x = Px Price of y = Py Income = I Total expenditure on basket (x,y): Pxx + Pyy The Basket is Affordable if total expenditure does not exceed total Income: Pxx + Pyy ≤ I

4 Definitions 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: Pxx+Pyy=I The graphable BUDGET LINE is the set of baskets that are just affordable: y=I/Py-(Px/Py)x

5 Example Two goods available: x and y I = $10 Px = $1 Py = $2 = 10 …BL1Budget line: 1x + 2y = 10 …BL1 Or… y = 5 – x/2

6 • B • The Budget Line Y A y=5-1/2x; 10=2y+x I/PY= 5 -PX/PY = -1/2 XI/PX = 10

7 • • • B • • 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 Y A • I/PY= 5 E D • • B • • C X I/PX = 10

8 4.2 Shifts in the Budget ConstraintThe 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)

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

10 Rotation of a Budget LineIf 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 Y I = $10 PX = $1 PY = $3 5 y=5-1/2x; 10=2y+x 3.33 y = x/3 10 X

11 4.3 Maximizing Utility Subject to a Budget ConstraintConsumers 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

12 • • • • Maximizing UtilityPoint 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 BL D • • B • C IC2 • A IC1 X

13 Maximizing Utility Subject to a Budget ConstraintMaximize 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)

14 Maximizing Utility Subject to a Budget ConstraintExogenous variables: Px, Py and Income Endogenous variables: x and y (chosen) and Utility (outcome)

15 • • • Maximizing Utility Utility is always maximized at theMax U(exercise, movie) given Pexercise=$10 Pmovie=$20, Income=$100 (s.t. Pee+PmM=$100) BL=5-1/2E Utility is always maximized at the tangent to the indifference curve 5 D: M=3, E=4 • 3 • C IC2 • A IC1 E 10 4

16 Tangency

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:

18 Optimization Steps Derive the BUDGET LINE2) Calculate the point of tangency 3) Use (1) and (2) to solve for the maximizing point 4) Find the Optimum (if possible) and conclude.

19 Maximization Example Vincent “fingers” McGiny enjoys two things: skittles (s) and bubble gum (b). His MUs=2b and MUb=3s. A pack of skittles costs $2 while bubble gum costs $1. If Fingers has $20, what should he to do maximize his utility?

20 Optimization Steps PsS+PbB=I 2S+B=20 2) MUs/MUb=Ps/Pb 2B/3S=2/1 2B=6S B=3S

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

22 Optional Budget Check:Optimization Steps 4) Fingers buys 12 bubble gum and 4 packs of skittles in order to maximize his utility. Optional Budget Check: 2S+B=20 2(4)+12=20 20=20

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

24 Corner Solutions Porsches Point A consumes positive amounts of both goods, but does not maximize utility Point B maximizes utility while consuming only 1 good. IC1 IC2 BL • A B • Food

25 Finding a corner solutionSolve for equilibrium as normal using the tangency condition: MUx/Px=MUy/Py 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).

26 Example Two goods available: Siamese Cats and Dachund Dogs: I = $200Pc = $100 Pd = $50 Utility is such that: MUc=d MUd=5+c

27 Optimization Steps Pcc+Pdd=I 100c+50d=200 2) MUc/MUd=Pc/Pd d/(5+c)=100/50 50d= c d=10+2c

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

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

30 Perfect Compliments 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

31 Utility is maximized at 4 when x and y are equal to 4.U= min(X,Y). U= min(4,4) =4 Utility is maximized at 4 when x and y are equal to 4. Budget line: Y = $5 - X/4

32 4.4 Minimizing ExpenditureThus 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)

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

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) Calculate expenditure and conclude.

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. (MUh=p, MUp=h) If a slice of ham costs 10 cents and a piece of pineapple costs 20 cents, minimize Hawaii's expenditure

36 Minimization Steps 1) MUh/MUp=Ph/Pp p/h=10/20 20p=10h 2p=h

37 Minimization Steps 2) U=hp 18=2p(p) 9=p2 3=p U=hp 18=h3 6=h

38 Minimization Steps 3) I=Phh+PpP I=0.1(6)+0.2(3) I=1.20 Hawaii’s minimum expenditure for a slice of pizza is $1.20 if he wants a utility of 18. Optional Check: U=ph 18=3(6) 18=18

39 Optimization ComparisonUtility Maximization Expend. Minimization Given prices and utility formulas Given prices and utility formulas Given EXPENDITURE Given UTILITY Solve tangency condition Solve tangency condition Substitute into budget constraint Substitute into utility formula Solve for UTILITY Solve for EXPENDITURE

40 Composite Goods In reality, people consume more than one goodEconomists 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 Py=1

41 Composite Good 300 Composite goods allow an economist to study choices revolving around 1 good IC1 • A 1 1.5 Lava Lamps

42 Composite Goods Application – Coupons vs. CashOften 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…..

43 Composite good, units Example: Consider a situation where an individual consumes Fa food, yet the government considers the minimum food an individual needs as FMin I A Food (units) FA FMin I/Ph

44 Composite good, units 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) I+S B • I A • Food (units) FA FMin I/Ph (I+S)/Ph

45 Composite good, units In order to limit the increase in composite goods, the government can issue food vouchers instead, resulting in the new yellow kinked budget curve I+S I+V B • • I A C • Food (units) FA FMin I/Ph (I+V)/Ph (I+S)/Ph

46 Composite good, units Note: A Kinked budget line due to a voucher often offers less total utility than a cash subsidy equal to the voucher I+V D • • C Food (units) FA FMin (I+V)/Ph

47 Composite Goods Application – Joining a ClubOften 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

48 Composite Good Originally, a consumer buys 10 CDs at $20 per CD before joining the club 300 IC1 • A 10 15 CDs (number)

49 Composite Good 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. 300 200 IC1 This consumer benefits from joining the club IC2 • A B • 10 CDs (number)

50 • • Not every consumer will benefit from joining the clubComposite Good Not every consumer will benefit from joining the club 300 A • 200 IC1 • B IC2 CDs (number)

51 Application 3 - Borrowing and LendingComposite goods can also explain why people save or borrow money Consider 2 time periods, now and the future, each with income (I1 and I2) and an interest rate r If you spend nothing today, you can spend I2+I1(1+r) in the future If you will spend nothing in the future, you can spend I1+I2/(1+r) today

52 This budget line represents all possible saving or borrowing opportunitiesAt 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 C2, spending next year ($) I2+ I1(1+r) C • C2C • I2 A C2B • B C1, spending this year ($) C2C I1 C1B I1+I2/(1+r)

53 C2, spending next year ($)For an individual consumer, borrowing may give a higher lifetime utility What does this assume? I2+ I1(1+r) • I2 A C2B • B IC2 IC1 C1, spending this year ($) I1 C1B I1+I2/(1+r)

54 Application 4 - Quantity DiscountsSome 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

55 Composite Good 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 N 6 • B • A M O 100 300 Photocopies

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

57 Weak Axiom of Revealed Preference (WARP)

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

59 Example: Consumer Choice that Fails to Maximize UtilityTwo goods, X and Y: I = $24 (PX,PY) = (4,2) (BL1) (P’x,P’Y) = (3,3) (BL2) (XA,YA) = (5,2) (Basket A) (XB,YB) = (2,6) (Basket B) Basket A chosen when BL is BL1 Basket B chosen when BL is BL2

60 A is chosen when it is more expensive; A  BPxXA + PyYA = 4(5)+2(2) = 24 PxXB + PyYB = 4(2)+2(6) = 20 A is chosen when it is more expensive; A  B P’xXB + P’yYB = 3(2)+3(6) = 24 P’xXA + P’yYA = 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

61 Example: Revealed Preference AnalysisComposite good -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! 12 BL1 • C 8 • B • D • A BL2 X

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

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