Chapter 5: Utility Maximization Problemsd.umn.edu/~watanabe/econ401su09/doc/umpho.pdfChapter 5: Utility Maximization Problems Instructor: ... 4 Summary 2/39. ... (Chapter 3), (Chapter

Intro Solving UMP Extreme Cases Σ

Econ 401 Price Theory

Chapter 5: Utility Maximization Problems

Instructor: Hiroki Watanabe

Summer 2009

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1 Introduction

2 Solving UMPBudget Line Meets Indifference CurvesTangencyFind the Exact Solutions

3 Extreme CasesPerfect SubstitutesPerfect Complements

4 Summary

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So far we have discussed:$(Chapter 2)� (Chapter 3), (Chapter 4)

Now combine the above to predict Greg’sconsumption behavior given his budget constraintand preferences.Situational background:

Greg wants to have as many cheesecakes and teaas possible.Greg’s budget constraint does not allow him tochoose (xC,xT) =(infinite, infinite).Greg has to make choices.

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We assume that Greg picks the most preferredcombination among what he can afford.The framework to analyze Greg’s choice behavior iscalled utility maximization problem (UMP for short).

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Greg’s decision making process is summarized asfollows:

Utility Maximization Problem

Greg chooses the bundle (xC,xT) that gives him thehighest utility level , among the affordable bundles. Inother words, he

maxu(xC,xT) subject to pCxC + pTxT = m.

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1 Introduction



4 Summary

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Budget Line Meets Indifference Curves

How exactly do we find Greg’s optimal bundlex∗ = (x∗C ,x

∗T )?

Consider the case whenu(xC,xT) = xCxTm = 60,(pC,pT) = (4,3).

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0 5 10 15 200

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Cheesecakes (xC)

Tea

(x T

)

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Figure:

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Figure:

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Budget Line Meets Indifference Curves25

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Figure:

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Figure:

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Tangency

Tangency Condition

At the optimal bundle, the indifference curve is tangentto the budget constraint (for standard preferences), i.e.,they both have the same slope at x∗.

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Tangency

What does the tangency condition imply?The slope of IC denotes MRS (MWTP): the cups oftea Greg is willing to give up for one slice ofcheesecake.The slope of budgte line denotes relative price (op.cost): the cups of tea Greg has to give up for oneslice of cheesecake.Greg idea of the tea’s worth coincides withmarket’s idea of the tea’s worth when he choosesthe right amount.

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Tangency

What if the tangency condition is not met?Suppose MRS is larger than the relative price (IC issteeper than the budget line).

Greg is willing to give up 30 cups of tea for a slice ofcheesecake,while he has to sell only 4

3 cups of tea to buy a sliceof cheesecake.It’s wise to increase cheesecake consumption andreduce cups of tea.The current bundle is not optimal.

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Tangency

Suppose MRS is smaller than the relative price (IC issteeper than the budget line).

Greg is willing to give up .1 cups of tea for a slice ofcheesecake,while he has to sell 43 cups of tea to buy a slice ofcheesecake.It’s wise to reduce cheesecake consumption andincrease cups of tea.The current bundle is not optimal.

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Find the Exact Solutions

Where exactly is the bundget line tangent to theindifferent curve?We need a little bit of calculus to find Greg’soptimal bundle.

UMPmaxu(xC,xT) = xCxT subject to 4xC + 3xT = 60.

MRS at (x1,x2) is given by −x2x1.

The relative price is −43 .

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Tangency condition:

−xTxC

=−43⇒ xT =

4

3xC. (1)

There are lots of bundles (xC,xT) satisfying (1).The bundle also has to be affordable:

4xC + 3xT = 60⇒ 4xC + 34

3xC = 60.

xT = 7.5 and xC = 10.Conclude x∗ = (10,7.5).

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ExerciseGreg spends his income ($5) on coins (xC) and tea (xT).Price is given by (pC,pT) = (2,1). His preferences arerepresented by u(xC,xT) = xC +

pxT.

1 Write down the budget line.2 Write Greg’s utility maximization problem.3 What is the tangency condition in this example?

(MRS at (xC,xT) is −2pxT).4 Which bundle will Greg buy?

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Tea

(x T

)

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Figure:

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1 Introduction



4 Summary

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There are some exceptions where tangencycondition does not apply.

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Perfect Substitutes

Consider six-packs and bottles of Corona (x6,x1).Greg’s MRS is 6 everywhere.Consider three cases:

1 Market rate of exchange is higher than 6.2 Market rate of exchange is lower than 6.3 Market rate of exchange is exactly 6.

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Perfect Substitutes

Case Ê

Let’s say the market rate of exchange is 24 (i.e., 30bottles of Corona is traded for 1 six-pack).Greg is willing to give up 6 bottles for 1 six-pack.Greg will spend all his income on six-packs.This type of solution is called a corner solution (e.g.,(8,0), (35,0), (0,68) etc.)

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Perfect Substitutes

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tles

(x1)

Figure:

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Perfect Substitutes

Case Ë

Let’s say the market rate of exchange is 1 (i.e., 1bottle of Corona is traded for 1 six-pack).Greg is willing to give up 6 bottles for 1 six-pack.Greg will spend all his income on bottles.

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Perfect Substitutes

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tles

(x1)

Figure:

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Perfect Substitutes

Case Ì

The market rate of exchange is 6 (i.e., 6 bottles ofCorona is traded for 1 six-pack).Greg is willing to give up 6 bottles for 1 six-pack.Greg can choose any bundle on his budget line.

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Perfect Substitutes

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Six−Packs (x6)

Bot

tles

(x1)

Figure:

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Perfect Substitutes

DiscussionIf a bottle of Corona is sold at $2, a six-pack isusually sold at less than $12.Does that mean nobody buys Corona by the bottle?

Dharma’s MRS might be less than 6.She is willing to give up 1 six-pack for 4 bottles(MRS= 4).So, there are Corona sold by the bottle, and thereare people who buy Corona by the bottle.

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Perfect Substitutes

ExerciseGreg has $12 and spend his income on Coke & Pepsi(xC,xP). Suppose Coke is sold at $4 while Pepsi is soldat $2.

1 Find Greg’s optimal bundle.2 Confirm your answer using indifference curves and

the budget line.

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Perfect Substitutes

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Coke (xC)

Pep

si (

x P)

Figure:

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Perfect Complements

For perfect complements, MRS is not defined.That does not mean the solution to UMP does notexist.Consider left gloves (x1) and right gloves (x2).

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Perfect Complements

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ht G

love

s (x

2)

Figure:

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Perfect Complements

Optimal bundle x∗ for left gloves and right gloves is

1 on the budget line and2 on the 45 degree line.

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Perfect Complements

ExerciseConsider a bundle (cereal, milk)= (xC,xM).

Greg says he can’t have cereals without milk andthe only time he drink milk is when he eats hiscereals.Greg’s preferred cereal-milk ratio is 2 to 3.(pC,pM) = (6,4).m = 24.

1 Draw his budget line.2 Draw indifference curves at (2,3) and (4,6).3 Mark the bundle Greg will choose on your graph.

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Perfect Complements

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Milk

(x M

)

Figure:

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Perfect Complements

Greg’s preferred bundles are lined up on the rayxM = 3

2xC.The optimal bundle is

1 on the ray xM = 32xC and

2 on the budget line xM = −32 xC + 6.

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1 Introduction



4 Summary

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Set up the problem.Tangency condition for standard cases.Extreme cases.

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Documents

Chapter 5: Utility Maximization Problemsd.umn.edu/~watanabe/econ401su09/doc/umpho.pdfChapter 5: Utility Maximization Problems Instructor: ... 4 Summary 2/39. ... (Chapter 3), (Chapter