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 constraint4.2 Shifts in the budget constraint4.3 Maximizing Utility4.4 Minimizing Expenditure4.5 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

Price of x = Px Price of y = Py Income = I

Total expenditure on basket (x,y): Pxx + Pyy

Assume only two goods available: x and y

The Basket is Affordable if total expenditure does not exceed totalIncome: Pxx + Pyy ≤ I

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: Pxx+Pyy=I

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

5

Two goods available: x and y

I = $10Px = $1Py = $2

Budget line: 1x + 2y

= 10 …BL1

Or… y = 5 – x/2

6

I/PX = 10

Y

X

•A

B

•

I/PY= 5 y=5-1/2x; 10=2y+x

-PX/PY = -1/2

7

I/PX = 10

Y

X

•

•

A

C B

•

I/PY= 5

Point A: one only consumes yPoint B: one only consumes x. Point D: consumes a mixturePoint C: consumes a mixture while not spending the entire budgetPoint E: unobtainable unless prices or income change

D

• •E

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)

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

10

Y

X

5

I = $10PX = $1PY = $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

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

12

Y

X

•IC2

BL

0

C •• B

• IC1

A

D

Point A: affordable, doesn’t maximize utilityPoint B: unaffordablePoint C: affordable (with income left over) but doesn’t maximize utilityPoint D: affordable, maximizes utility

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)

IyPxPts

yxU

yx

yx

..

),(max),(

14

Maximizing Utility Subject to a Budget Constraint

• Exogenous variables: Px, Py and Income

• Endogenous variables: x and y (chosen) and Utility (outcome)

IyPxPts

yxU

yx

yx

..

),(max),(

15

M

E

•IC2

BL=5-1/2E

0

C •• IC1

A

D: M=3, E=4

Max U(exercise, movie) given Pexercise=$10 Pmovie=$20, Income=$100(s.t. Pee+PmM=$100)5

104

3

16

Tangency

yx,

,

PPor

BL, theo tangent tis IC theSince

/ IC theof Slope

/ linebudget theof Slope

: thatRecall

yx

y

x

y

xyx

yxyx

yx

MUMU

P

P

MU

MUMRS

MUMUMRS

PP

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:

zyx PPPzyx MUMUMU

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

19

Maximization ExampleVincent “fingers” McGiny enjoys two things: shooting people (s) and throwing bricks through windows (b). His MUs=2b and MUb=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?

20

Optimization Steps

1) PsS+PbB=I

2S+B=20

2) MUs/MUb=Ps/Pb

2B/3S=2/1

2B=6S

B=3S

21

Optimization Steps

3) 2S+B=20

2S+3S=20

5S=20

S=4

B=3S

B=3(4)=12

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

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

Porsches

Food

IC2

BL

0

• B

•

IC1

A

Point A consumes positive amounts of both goods, but does not maximize utilityPoint B maximizes utility while consuming only 1 good.

25

Finding a corner solution

1) Solve 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

Two goods available: Siamese Cats and Dachund Dogs:

I = $200Pc = $100Pd = $50

Utility is such that:

MUc=d

MUd=5+c

27

Optimization Steps

1) Pcc+Pdd=I

100c+50d=200

2) MUc/MUd=Pc/Pd

d/(5+c)=100/50

50d=500+100c

d=10+2c

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

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

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=100050x+200x=10004=x=y

31

Example: Let U(X,Y) = min(X,Y). Let I = $1000, Px = $50 and PY = $200. What is the optimal consumption basket?

Budget line: Y = $5 - X/4

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)

33

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. If U* is the maximum utility from the utility

problem, then an interior optimum, if it exists, of the expenditure problem also solves the utility problem.

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

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=hp18=2p(p)9=p2

3=p

U=hp18=h36=h

38

Minimization Steps

3) I=Phh+PpPI=0.1(6)+0.2(3)I=1.20

Hawaii spends $1.20 for a satisfying slice of pizza.

Check: U=ph18=3(6)18=18

39

Optimization Comparison

Utility Maximization Expend. MinimizationGiven prices and

utility formulasGiven 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 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 Py=1

41Lava Lamps

Composite Good

0 1.5

300

• A

IC1

1

Composite goods allow an economist to study choices revolving around 1 good

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…..

43

I/PhFood (units)

Composite good, units

FA FMin

Example: Consider a situation where an individual consumes Fa food, yet the government considers the minimum food an individual needs as FMin

AI

44

I/PhFood (units)

Composite good, units

FA FMin

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)

AI

(I+S)/Ph

I+S

B

•

•

45

I/PhFood (units)

Composite good, units

FA FMin

In order to limit the increase in composite goods, the government can issue food vouchers instead, resulting in the new yellow kinked budget curveAI

(I+S)/Ph

I+S

BI+V

(I+V)/Ph

C•

••

46

(I+V)/PhFood (units)

Composite good, units

FA FMin

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

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

48CDs (number)

Composite Good

0 15

300

• A

IC1

10

Originally, a consumer buys 10 CDs at $20 per CD before joining the club

49CDs (number)

Composite Good

0 15 20

300

200

•• A

B

IC1

IC2

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

50CDs (number)

Composite Good

0 15 20

300

200

•

•A

BIC2

IC1

Not every consumer will benefit from joining the club

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 (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

C1, spendingthis year ($)

C2, spending next year ($)

I1

I2

C1B

C2B

I1+I2/(1+r)

I2+I1(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•

C2C

C2C

53

C1, spendingthis year ($)

C2, spending next year ($)

I1

I2

C1B

C2B

I1+I2/(1+r)

I2+I1(1+r)

For an individual consumer, borrowing may give a higher lifetime utility

What does this assume?•

•B

A

IC1

IC2

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

55Photocopies

Composite Good

0 300

6

••

AB

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

56

4.5 Revealed Preferences

• Often a consumer’s preference can be inferred without indifference curves

• Mathematically, when a consumer decides between 2 baskets:

B A

:chosen isA and B than more costsA If-

B A or BA means BA :Note

B) topreferred weakly is(A BA

:chosen isA and same cost the B andA basketsboth If-

57

Weak Axiom of Revealed Preference(WARP)

rational be cheaper to isit it when chose

must you expensive, more isit n basket whe abuy you If :ie

)"Preference Revealed of AxiomWeak ("

chosen is B and YPXPYPXP

thatbecannot It

chosen, isA and YPXPYPXP

:

By'

Bx'

Ay'

Ax'

ByBxAyAx

If

Therefore

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 Utility

Two 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

P’xXB + P’yYB = 3(2)+3(6) = 24P’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

PxXA + PyYA = 4(5)+2(2) = 24PxXB + PyYB = 4(2)+2(6) = 20

A is chosen when it is more expensive; A B

61

X

Composite good

•D

A•

• B

•C

BL1

BL2

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

62

Chapter 4 Key ConceptsThe budget constraintShifts in the budget constraintMaximizing Utility

TangencyCorner solutionsPerfect Compliments

Minimizing ExpenditureDualityComposite Goods

63

Chapter 4 Key ConceptsRevealed Preferences

Weak Axiom of Revealed PreferencesGraphical application

Dogs are better than catsIf you ignore the notes you can miss

things