ECON2316ConsumerBehavior

8/12/2019 ECON2316ConsumerBehavior

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Intro Lagrangian Utility Indifference Curves Budget Constraint Optimality

ECON2316: Microeconomic TheoryCh. 04: Consumer Behavior

Mr. David Hom

Northeastern University Department of Economics

January 23, 2014

Mr. David Hom Ch. 04: Consumer Behavior 1 / 69


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Table of Contents (1/2)

1 IntroductionMotivation

2 Constrained MaximizationFundamentals

3 Preferences and UtilityFundamentals



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Table of Contents (2/2)

4 Indifference CurvesFundamentals

Marginal Rate of SubstitutionCurvature and the Relationship Between Goods

5 Income and Budget Constraint

Standard ConstraintNon-Standard Constraint



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Motivation

Solid Foundation

Use seemingly self-evident facts for a solid foundation

Confidence in predictionAvoid accusations of ad-hoc analysis

Life is essentially a series of constrained optimization decisions


I L i U ili I diff C B d C i O i li


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Fundamentals

Lagrange Multiplier Method (1/2)

Lagrange Multiplier Method

Let f(x1, . . . , xn) be the objective function to be optimized and

g(x1, . . . , xn

) = cbe the condition to be satisfied where c is anarbitrary constant. Then the Lagrangian is:

L = f () + (c g(x1, . . . , xn)) (1)

Optimization is either maximization or minimizationFocus only on strict (equality) single-constraint situations


I t L i Utilit I diff C B d t C t i t O ti lit


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Fundamentals

Lagrange Multiplier Method (2/2)

The solution is the n-tuple (x1 , . . . , x

n ) that satisfies:

L

x1

= . . . = 0

...

L

xn= . . . = 0

L

= c g(x1, . . . , xn) = 0



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Fundamentals

Worked Example (2/4)

The first order conditions (F.O.C.s) are:

L

L = W 2 = 0

L

W = L 2 = 0

L

= X 2 L 2 W= 0

Use the first two F.O.C.s to eliminate :

W 2 = L 2

W= L




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Fundamentals


Substitute into the last F.O.C.:

X 2 L 2 L = 0

L = X

4

Solve for the other, unsolved variable:

W= L

W= X

4




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Fundamentals


Use either of the first two F.O.C.s to solve for :

W 2 = 0

= W

2Substitute for W:

=X

4

2

= X8

is the change in the objective function given a unit relaxation ofthe constraint




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g g y g p y

Fundamentals

Interpreting

Solving for is not required unless requested, but helpful

Here, answers the question, By how much would areaincrease if fencing increased by one unit?

can answer more interesting questions in other contextsIf the objective is health and the constraint is budget:

How much healthier would we be if we allocated one moreunit of currency to health?

If the objective is taste and the constraint is calories:How much tastier would this be if we increased caloric countby one unit?




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g g y g p y

Fundamentals

Four Properties of Preferences

Completeness (rankability): given any two (or more)consumption bundles, consumers can determine whether oneis preferred to another or both are equal

Allows for comparisons of even disparate goods

Non-satiation: more is never less-preferred to less

Free disposal: can always dispose of goods that may exhibitsatiation

Transitivity: for any three bundles A, B, and C, if A ispreferred to B and B is preferred to C, then A is preferred to C

Diminishing marginal benefit: each additional unit of a goodis less beneficial than the prior ones

The more of a good that a consumer already has, the lesswilling he is to give up any other good in exchange for more ofthat goodTendency to prefer a variety



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Fundamentals

Properties of Utility (1/2)

Non-satiation implies:

U

xi>0 i= 1, . . . , n

Marginal utility is positive; diminishing marginal benefit implies:

2U

x2i


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Fundamentals

Properties of Utility (2/2)

Utility is ordinal (rankable), not cardinal (countable)

Can say which bundles are preferred to others

Cannot say how much more one bundle is preferred toanother

Can use numerical comparison for a single individual

Cannot make interpersonal comparisons of utility

All monotonic (order-preserving) transformations of utility areidentical



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Fundamentals

Definition of an Indifference Curve

Indifference Curve

An indifference curveis a graphical representation of alltwo-good consumption bundles among which a consumer derives

the same level of utility.

Compares two goods only

Ease of construction and analysis

Downward-sloping

Loss of one good must be compensated by gain in another tomaintain equal utility




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Fundamentals

Graphical Representation

Figure 1: Building an Indifference Curve; Source: Goolsbee, Levitt and Syverson,Microeconomics, 1st ed. (New York: Worth Publishers, 2013), 117, Print.

(a): loss in nearby friends must be compensated by increase insquare footage (A B, B C)

(b): connect all points of equal utility indifference curve



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Fundamentals


Figure 2: A Consumers Indifference Curves; Source: Goolsbee, Levitt and Syverson,Microeconomics, 1st ed. (New York: Worth Publishers, 2013), 117, Print.

Bundles along U2 are preferred to those on U1 because for anyamount of friends, more square footage is better than less

Equivalently, for any amount of square footage, more friendsis better than less




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Fundamentals


Figure 3: Indifference Curves Cannot

Cross; Source: Goolsbee, Levitt andSyverson, Microeconomics, 1st ed. (NewYork: Worth Publishers, 2013), 118, Print.

Bundles D and F must beequal (lie on same I.C.)

Bundles E and F must be

equal (lie on same I.C.)

If D = F and E = F, then D= E

But bundle E must be better

than D (more of both)D = E and D < E is acontradiction




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Marginal Rate of Substitution

Definition of the Marginal Rate of Substitution


Themarginal rate of substitution of X for Y (MRSXY) is therate at which a consumer is willing to trade off the good on thehorizontal axis for the good on the vertical axis holding utility

constant. It is equal to the negative slope of the indifference curve:

MRSXY= Y

X (3)

Answers the question, How many units of good Y would theconsumer be willing to give up to get one more unit of goodX?

Negative so that the result is a positive number



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Marginal Rate of Substitution and Marginal Utility (2/3)

Rearrange the equation to solve for slope:

dY

dX =

UXUY

= MUXMUY

From before:

MRSXY=

Y

X

Therefore, MRS also equals negative the ratio of the marginalutility of X to that of Y




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Marginal Rate of Substitution and Marginal Utility (3/3)

Consider two bundles: one where X is small and Y is large, andvice-versa

MUX is larger when X is smaller (same for Y)

The I.C. will be steeper when X is small and Y is large

Follows from diminishing marginal utility




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Application: Value of an American Football Team

Crooker, John R. and Aju J. Fenn, Estimating Local WelfareGenerated by an NFL Team under Credible Threat of Relocation,Southern Economic Journal76, 1 (2009): 198223.

Minnesota Vikings franchise threatened to leave town for anew stadium

Two economists surveyed Minnesotans, How much wouldyour household be willing to pay in extra taxes for a newstadium?

Average value = $571.60; across 1.3M households

$750M

$750M = MRS of Vikings for consumption

Drawback: hypothetical vs. actual choice




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Application: Phone Service Buyers

Economides, Nicholas, Katja Seim, and V. Brian Viard,Quantifying the Benefits of Entry into Local Phone Service,RAND Journal of Economics39, 3 (2008): 699730.

Measured utility and indifference curves of local and regionalcalls in NY (19992003)

Internet-enabled households had higher MRS of local toregional (1 vs. .5)

Willing to give up more regional calls for a given amount of

localKey: local calls worth more to Internet-enabled households

Dial-up internet was billed as a local call



M i l R f S b i i


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Figure 4: New Yorkers Preferences for Local and Regional Phone Calls, 19992003;Source: Goolsbee, Levitt and Syverson, Microeconomics, 1st ed. (New York: Worth

Publishers, 2013), 126, Print.

Internet-enabled households are more willing to trade regionalcalls for local calls

Steeper slopeLocal calls worth more (higher utility: Internet)



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Curvature and the Relationship Between Goods


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Figure 5: Indifference Curves for PerfectSubstitutes; Source: Goolsbee, Levitt andSyverson, Microeconomics, 1st ed. (NewYork: Worth Publishers, 2013), 128, Print.

U(X,Y) = X+ 4Y

MRS = slope = 14Consumer is always willing

to give up 1 12-oz bag inexchange for 4 3-oz bags ofpotato chips

Note that any a and b suchthat b = 4a is valid

Note also that the rate neednot be 1:1





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Perfect Complements

Usefulness of a good depends on its proportion with respect toanother

Ignore selling of goods

Have the general form:

U(X,Y) = min {aX, bY}

Having more of either good does not increase utility withoutmore of the other





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Figure 6: Indifference Curves for PerfectComplements; Source: Goolsbee, Levittand Syverson, Microeconomics, 1st ed.(New York: Worth Publishers, 2013), 129,Print.

U(X,Y) = min {L,R}

3 pairs of shoes (D) is betterthan 2 (A)

3 right shoes and 2 left

shoes (B) is no better than2 pairs (A)

2 right shoes and 3 leftshoes (C) is also no better

than 2 pairs (B)





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Same Consumer, Same Goods, Different IndifferenceCurves

Curvature can change depending on utility levelStill cannot cross

Usually reflects different preferences

Different utility functions





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p


Figure 7: The Same Consumer Can HaveIndifference Curves with Different Shapes;Source: Goolsbee, Levitt and Syverson,Microeconomics, 1st ed. (New York:Worth Publishers, 2013), 130, Print.

UA: almost perfectsubstitutes

Any fruit is better thanno fruit

UB: almost perfectcomplements

Variety is important





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p

Bads

Some things (e.g. pollution) are worse as quantity increases

Can still use indifference curves to graph themInterpretations differ slightly

Can also define absence of a bad as a new good

Reverts back to usual analysis



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Graphical Representation (2/2)

Figure 9: Indifference Curves for theAbsence of a Bad; Source: Goolsbee,Levitt and Syverson, Microeconomics, 1st

ed. (New York: Worth Publishers, 2013),133, Print.

Saved commute time is agood

To maintain utility, fewerfriends can be

compensated with a fastercommute



Standard Constraint


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Assumptions

Finite income

Two goods

Positive prices

Enough supply to satisfy any consumer even if s/he devotesall of his/her income to purchasing it

No saving or borrowing allowed



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Standard Constraint


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Consider two goods: socks (S) and T-shirts (T). The price ofsocks and T-shirts is $5 and $10 respectively. If a consumer hasincome of $50, then the budget constraint is:

5 S+ 10 T= 50

To find the intercepts, set each variable to zero and solve for theother in turn:

5 (0) + 10 T= 50

TInt= 55 (S) + 10 (0) = 50

SInt= 10



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Standard Constraint


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Price or Income Changes and the Budget Constraint

Important parameters:

m = PX

PY

XInt=

I

PX

YInt= I

PY

Change in income only parallel shift (I is not in m)

Increase outward; decrease inward

Change in price only pivot (change relevant intercept)

Increase inward; decrease outward



Standard Constraint


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Figure 11: The Effects of Price or Income Changes on the Budget Constraint;Source: Goolsbee, Levitt and Syverson, Microeconomics, 1st ed. (New York: WorthPublishers, 2013), 136, Print.

(a): Increase in the price of socks sock intercept (X)shifts inward

(b): Increase in the price of T-shirts T-shirt intercept(Y) shifts inward (downward)

(c): Decrease in income parallel shift inward



Non-Standard Constraint

Q D


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Quantity Discounts

Sometimes, firms offer lower prices for bulk (volume)

purchasesBudget constraint will have a kink (sharp bend) where the newpricing starts




G hi l R i


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Figure 12: Quantity Discounts and theBudget Constraint; Source: Goolsbee,Levitt and Syverson, Microeconomics, 1st


After 600 phone minutes,they are cheaper

Cheaper good on y-axis higher intercept/slope

Dashed line area: newlyfeasible set

Could influence theconsumer to buy more

phone minutes




Q tit Li it


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Quantity Limits

Sometimes, governments (or parents) limit the maximumamount of a good consumers can buy

Budget constraint will have a kink (sharp bend) where thelimit occurs

Examples: gasoline during the 1970s oil price spike, warrationing




G hi l R t ti


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Figure 13: Quantity Limits and theBudget Constraint; Source: Goolsbee,Levitt and Syverson, Microeconomics, 1st


No more than 600 phoneminutes are purchaseable

Constraint turnshorizontal at 600 phone

minutesRed area: newly infeasibleset

Could influence theconsumer to buy less phoneminutes



General Solution

G hi l R s t ti


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Figure 14: The Consumers OptimalChoice; Source: Goolsbee, Levitt andSyverson, Microeconomics, 1st ed. (NewYork: Worth Publishers, 2013), 141, Print.

E is unaffordable

B through D cannot beoptimal

A is feasible (on the

budget constraint) andhas higher utility (U2farther from the originthan U1)

A is optimal

Is the bundle at thetangency of the highestI.C. and the B.C.



General Solution

Mathematical Representation (1/4)


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Let X and Y be the quantities of two goods X and Y, PX and PY

their respective prices, I be income, and U(X, Y) be theconsumers utility function . Then the consumers problem is:

maxX,Y

L = U(X, Y) + (I PX X PY Y)

The F.O.C.s are:

L

X =

U

X PX= 0

L

Y =

U

Y

PY= 0

L

= I PX X PY Y= 0



General Solution



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Rewrite the first two F.O.C.s so that the term is on one side:

U

X = PX

UY = P

Y

Divide one by the other to eliminate :

UX

UY

=

PX

PY



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General Solution



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Rewrite the optimality condition:

UX

PX

=UY

PY

Each side is the marginal utility per dollar spent on that good

Bang for the buck

If the equality does not hold, the consumer can increae utility

by reallocating (not increasing) expenditure



General Solution

Application Revisited: Phone Service Buyers


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Application Revisited: Phone Service Buyers

Phone companies usually offer a variety of plans

High fixed fee, low piece (per call) rateLow fixed fee, high piece (per call) rate

Different plans can increase consumer utility



General Solution



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Figure 15: Paying a Fixed Fee to Reduce

the Price of Local Calls; Source: Goolsbee,Levitt and Syverson, Microeconomics, 1st


BC2: high fixed fee, lowerrate per local call

High fixed fee reduces themaximum number of

regional callsLower rate increases themaximum number of localcalls

Internet-enabled households

increase utiilty with the highfixed fee plan

I2 unaffordable otherwise



Corner Solution



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Figure 16: A Corner Solution; Source:Goolsbee, Levitt and Syverson,Microeconomics, 1st ed. (New York:Worth Publishers, 2013), 147, Print.

U3: unattainable

Outside (above) the B.C.

U1: suboptimal

Some feasible bundlesconfer more utility

U2: optimal

Optimal bundle consistsonly of one good



Corner Solution

Definition of a Corner Solution


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Definition of a Corner Solution

Corner Solution

A corner solution is a utility-maximizing bundle where the

consumer purchases only one good

Tangency is violated

Consumer would like even more of the good he is buying andeven less of the one he is not



Expenditure Minimization

Dual Approach


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Dual Approach

Maximizing utility subject to a budget: fix a budget, find thehighest utility possible

Minimizing expenditure subject to a level of utility: fixhappiness, find the cheapest bundle possible

Twin approaches to the same problem






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p p

Figure 17: Utility Maximization Versus Expenditure Minimization; Source: Goolsbee,Levitt and Syverson, Microeconomics, 1st ed. (New York: Worth Publishers, 2013),150, Print.

(a): given BC, what is the highest utility?

Not U1 (suboptimal) or U2 (unaffordable) but U*

(b): given U*, what is the cheapest bundle possible?

Not BCA (too cheap) or BCB (excessive) but BC*






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p ( / )

Let there be two goods: movies (M) and video games (V). Theprices of movies and video games are $20 and $60 respectively.Suppose a consumer has income of $600 and utility over moviesand video games given by:

U(M, V) = M2V3

What combination of M and V will maximize utility?






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p ( / )

Set up the Lagrangian:

maxM,V

L = M2V3 + (600 20M 60V)

Set up the F.O.C.s:

L

M= 2MV3 20 = 0

L

V = 3M2V2 60 = 0

L

= 600 20M 60V



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Let there be two goods: movies (M) and video games (V). The

prices of movies and video games are $20 and $60 respectively.Suppose a consumer has desired utility level of 31,104; what is hismost cost-effective bundle?






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Set up the Lagrangian:

minM,V

L = 20M+ 60V+

U M2V3

Set up the F.O.C.s:

L

M= 20 2MV3 = 0

L

V = 60 3M2V2 = 0

L

= U M2V3 = 0






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Isolate in the first two F.O.C.s:

20 = 2MV3

60 = 3M2V2

Divide the second equation by the first to eliminate and obtainan intermediate equation relating the two choice variables:

60

20=

3M2V2

2MV3

3 = 3M2V

2V= M






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Substitute into the utility constraint:

31, 104 (2V)2 V3 = 0

7, 776 = V5

V= 6

Now use the intermediate equation to solve for M:

M= 12

Verify that utility is at target:

122 63 = 31, 104

Solve for expenditure:

20 (12) + 60 (6) = 600



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