© 2008 Pearson Addison Wesley. All rights reserved Chapter Three A Consumer’s Constrained Choice

© 2008 Pearson Addison Wesley. All rights reserved

Chapter Three

A Consumer’s Constrained Choice

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© 2008 Pearson Addison Wesley. All rights reserved. 3-2

消費支出結構(%) 86年 96年增減百分點

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家庭收支調查 _消費支出


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86 年消費支出結構 96 年消費支出結構

家庭收支調查 _消費性支出 _食品類項目表


主食品米、米製品，麥麵，雜糧，農牧戶自種自食設算。

副食品肉類，魚貝及水產品，蔬菜類，蛋類，油脂類，調味品類。農牧戶自種自食設算。

乳酪類奶粉，訂購鮮牛、羊奶，其他奶製品，如奶油、煉乳、養樂多.....等。

水果類水果，自種自食設算，其他加工水果與乾果，檳榔。其他茶，咖啡、可可，砂糖(方糖)，糖果、冰淇淋等其他糖類。

婚生壽慶喪祭宴費凡上列各類食物因婚生壽慶喪祭及地區性拜拜而增加購買食品費金額，例如滿月油飯、蛋糕、喜餅、建醮宴客等 (初1、15拜拜之食品。

在外伙食費1.向現場有調理(煎煮炒炸、沖泡)之餐飲服務者，所購買可即時依消費者需求調整口味且供立即食用之餐飲費。2.平均每月搭伙：如學生餐費；就業；托兒所、幼稚園點心費。

Chapter Outline

• In this chapter, we examine four main topics.

– Preferences

– Utility

– Budget Constraint

– Constrained Consumer Choice


© 2007 Pearson Addison-Wesley. All rights reserved. 3–6

Preferences

• To explain consumer behavior, economists assume that consumers have a set of tastes or preferences that they use to guide them in choosing between goods. These tastes differ substantially among individuals.


Properties of Consumer Preferences

• Economists make five critical assumptions about the properties of consumers’ preferences.

• For brevity, these properties are referred to as completeness, transitivity, more is better, continuity, and strict convexity.


Completeness

• The completeness property holds that, when facing a choice between any two bundles of goods, a consumer can rank them so that one and only one of the following relationships is true: The consumer prefers the first bundle to the second, prefers the second to the first, or is indifferent between them.


Transitivity

• The transitivity (or what some people refer to

as rationality) property is that a consumer’s

preferences over bundles is consistent in the

sense that, if the consumer weakly prefers Bundle z to Bundle y (like z at least as much as

y ) and weakly prefers Bundle y to Bundle x ,

the consumer also weakly prefers Bundle z to

Bundle x .


• good– a commodity for which more is preferred to

less, at least at some levels of consumption

• bad– something for which less is preferred to

more, such as pollution

• We now concentrate on goods.

More is Better


More is Better

• The more-is-better property holds that, all else the same, more of a commodity is better than less of it (always wanting more is known as nonsatiation).

Continuity

• The continuity property holds that if a consumer prefer Bundle a to Bundle b, then the consumer prefers Bundle c to b if c is very close to a.

• The purpose of this assumption is to rule out sudden preference reversals in response to small change in the characteristics of a bundle.


Strict Convexity

• Strict convexity of preferences means that consumers prefer averages to extremes.

• If Bundle a and Bundle b are distinct bundles and the consumer prefers both of these bundles to Bundle c, then the consumer prefers a weighted average of a and b, a + (1-)b (where 0 < < 1), to Bundle c.



Preference Maps

• One of the simplest ways to summarize information about a consumer’s preferences is to create a graphical interpretation—a map—of them.


• indifference curve– the set of all bundles of goods that a

consumer views as being equally desirable

• indifference map (or preference map)– a complete set of indifference curves that

summarize a consumer’s tastes or preferences

Preference Maps


Figure 3.1Bundles of Pizzas and Burritos Lisa Might Consume


Indifference Curves

• All indifference curve maps must have five important properties:1. Bundles on indifference curves farther from

the origin are preferred to those on indifference curves closer to the origin.

2. There is an indifference curve through every possible bundle.

3. Indifference curves cannot cross.

4. Indifference curves slope downward.

5. Indifference curves cannot be thick


Figure 3.2Impossible Indifference Curves


Utility

• Utility– a set of numerical values that reflect the

relative rankings of various bundles of goods

• Utility Function– the relationship between utility values and

every possible bundle of goods


Ordinal Preferences

• If we know only consumers’ relative rankings of bundles, our measure of pleasure is ordinal rather than cardinal. An ordinal measure is one that tells us the relative ranking of two things but not how much more one rank is than another.

• Because utility is an ordinal measure, we should not put any weight on the absolute differences between the utility associated with one bundle and another. We care only about the relative utility or ranking of the two bundles.

Utility and Indifference Curves

• In short, an indifference curve consists of all those bundles that correspond to a particular utility measure.

• A utility function is . This expression determines all those bundles of and that give her utility of pleasure.


1q 2q U


Willingness to Substitute Between Goods

• Marginal Utility

– the slope of the utility function– the extra utility that a consumer gets from

consuming the last unit of a good–

11

1 Uq

Uq

ofutilitymarginal

艾爾弗雷德 ·馬歇爾 (Alfred Marshall)

Born26 July 1842(1842-07-26)

Died13 July 1924 (aged 81)

Nationality British

Occupation Economist

Known for Principles of Economics (1890)


• 英國經濟學家 (1842年 7月 26日－1924年 7月 13日 )。

• 出生在英國倫敦南岸的柏孟塞（ Bermondsey）。

• 他的著作《經濟學原理》 (1890 的 ) 集中了供求理論、邊際效用理論和生產成本，形成了一個邏輯的整體。

• 這部著作使他成為當時的最顯赫的經濟學家之一，並在很長時間裡是英國大學科堂的經濟學課本。

• 馬歇爾曾任劍橋大學教授，凱恩斯是他的學生之一。



• Marginal Rate of Substitution (MRS)– the slope at a point of an indifference curve– the maximum amount of one good that a

consumer will sacrifice (trade) to obtain one more unit of another good

– The marginal rate of substitution can also be expressed in terms of marginal utilities.

2

1

1

2

d

d

U

U

q

qMRS


• If • Let be the implicit function that shows how

much it takes to keep utility at given .• Then • with respect to :

• Because is a constnat, .


),( 21 qqUU )( 12 qq

2q U 1q

))(,( 121 qqqUU

1q

1

221

1

2

2

121

1

121

1 d

d

d

d))(,())(,(

d

d

q

qUU

q

q

q

qqqU

q

qqqU

q

U

U 0/ 1 dqUd

2

1

2

1

1

2

/

/

qU

qU

U

U

dq

dqMRS


Figure 3.3Marginal Rate of Substitution

1

2

d

d

q

qMRS

1

2

d

dslope

q

q

Solved Problem 3.2Cobb-Douglas Utility Function

• Suppose that Jackie has what is known as a Cobb-Douglas utility function:

Where is a positive constant, is the number of music CDs she buys a year, and is the number of movie DVDs she buys.

• What is her marginal rate of substitution?


aaqqU 121

a 1q2q

Solved Problem 3.2Cobb-Douglas Utility Function

• Marginal utilities of CDs is :

• Marginal utilities of DVDs is :

• Express her marginal rate of substitution in terms of her marginal utilities is :


1

2112

111

),(

q

qqUaqaqU aa

2

21212

),()1()1(

q

qqUaqqaU aa


Curvature of Indifference Curves

• An indifference curve doesn’t have to be

convex, but casual observation suggests that

most people’s indifference curves are convex.

• When people have a lot of one good, they are

willing to give up a relatively large amount of it

to get a good of which they have relatively little.

However, after that first trade, they are willing to

give up less of the first good to get the same

amount more of the second good.


• This willingness to trade fewer burritos for one

more pizza as we move down and to the right

along the indifference curve reflects a

diminishing marginal rate of substitution:

The marginal rate of substitution (MRS)

approaches zero─becomes flatter or sloped─as

we move down and to the right along an

indifference curve.




• Perfect Substitutes– goods that a consumer is completely

indifferent as to which to consume– U(C, G) = iC + jG

• Perfect Complements– goods that a consumer is interested in

consuming only in fixed proportions– U(A, V) = min(iA, jV)


Figure 3.4Perfect Substitutes, Perfect Complements, Imperfect Substitutes


Budget Constraint

• Consumers maximize their well-being subject to constraints. The most important constraint most of us face in deciding what to consume is our personal budget constraint.

• For simplicity, we assume that each consumer has a fixed amount of money to spend now, so we can use the terms budget and income interchangeably.

Budget Constraint

• If Lisa spends all her budget, Y, on pizza and burritos, then

where p1q1 is the amount she spends on

pizzas and p2q2 is the amount she spends

on burritos.



Budget Constraint

• Budget Line (or Budget Constraint )

– the bundles of goods that can be bought if

the entire budget is spent on those goods at

given prices

• Opportunity Set

– all the bundles a consumer can buy, including

all the bundles inside the budget constraint

and on the budget constraint


Figure 3.5Budget Constraint


Slope of the Budget Constraint

• Marginal Rate of Transformation (MRT)– the slope of the budget line– the trade-off the market imposes on the

consumer in terms of the amount of one good the consumer must give up to obtain more of the other good

• The marginal rate of transformation is the rate at which Lisa can trade burritos for pizza in the marketplace:


Effect of A Change on Consumption

• If the price of pizza doubles but the price of

burritos is unchanged, the budget constraint

swings in toward the origin in panel a of Figure

3.5-1.

• If the consumer’s income increases, the

consumer can buy more of all goods. The

budget constraint shifts outward—away from

the origin—and is parallel to the origin

constraint in panel b of Figure 3.5-1.


Figure 3.5-1 Changes in the Budget Constraint

(a) Price of Pizza Doubles

Loss

50

L1 (p1 = $1)

L 2 (p1 = $2)

25

250

(b) Income Doubles

Gain

100

L3 (Y = $100)

L1 (Y = $50)

50

25

500


Effect of a Change on Consumption

• A change in income affects only the position and not the slope of the budget line.

• The slope is determined solely by the relative prices of pizza and burritos.


The Consumer’s Optimal Bundle

• Bundles that lie on indifference curves above the constraint, such as those on I3, are not in the opportunity set.

• For any bundle inside the constraint (such as d on I1), there is another bundle on the constraint with more of at least one of the two goods, and hence she prefers that bundle.

• Therefore, the optimal bundle must lie on the budget constraint.



• Bundles that lie on indifference curves that cross

the budget constraint (such as I1, which crosses the constraint at a and c ) are less desirable

than certain other bundles on the constraint.

• Thus the optimal bundle must lie on the budget

constraint and be on an indifference curve that

does not cross it. Such a bundle is the

consumer’s optimum.


• The optimal bundle must lie on an indifference

curve that touches the budget constraint

but does not cross it.



Figure 3.6Consumer Maximization


Interior Solution

• For the indifference curve I2 to touch the budget

constraint but not cross it, it must be tangent to

the budget constraint: The budget constraint

and the indifference curve have the same slope

at the point e where they touch.

Rearranging terms, this condition is equivalent to

Corner Solution

• There are two ways for an optimal bundle to lie

on an indifference curve that touches the budget

constraint but does not cross it. The first is an

interior solution. The other possibility is called a

corner solution.

– Indifference curve does not cross the constraint into

the opportunity set

– Indifference curve is not tangent to the budget line.



Figure 3.7Optimal Bundles on Convex Sections of Indifference Curves


Buying Where More is Better

• If both goods are consumed in positive quantities and their prices are positive, more of either good must be preferred to less.

• In summary, we do not observe consumer optima at bundles where indifference curves are concave or consumers are satiated. Thus we can safely assume that indifference curves are convex and that consumers prefer more to less in the ranges of goods that we actually observe.

Maximizing Utility Subject to a Constraint Using Calculus

• The utility maximization problem can be written as the following:

• Two approaches to solve this problem:

Substitution and Lagrangian Method.

1 21 2

,max ( , )q qU q q

1 1 2 2. . s t Y p q p q

3–49


• Substitution:

We can substitute the budget constraint into the utility function.

• So we can use standard maximization techniques to solve it.

2

2 22

1

max , q

Y p qU q

p

1 2 21 2

2 1 2 2 1 1 2 1

0

q p pdU U U U UU U

dq q q q p q q p

3–50


• By rearranging these terms. We have the

marginal rate of substitution equals the

marginal rate of transformation:

1 1

2 2

U p

MRS MRTU p

3–51


• Lagrangian Method:

We write the equivalent Lagrangian problem as

where is the Lagrange multiplier.



• The first-order conditions for an interior maximization are :

• These three first-order conditions can be solved

for the optimal values of , and .

We find that 1 2

1 2

U U

p p

1q 2q

3–53


• This optimality condition is the same as the

one that we derived using the substitution

method or a graphical approach.

3–54

Minimizing Expenditure

• Consider the alternative problem where we ask how Lisa can make the lowest possible expenditure to maintain her utility at a particular level, ,which corresponds to indifference curve .

• The rule for minimizing expenditure while achieving a given level of utility is to choose the lowest expenditure such that the budget line touches — is tangent to — the relevant indifference curve.

U

3–55

2I


• Thus solving either of the two problems –

maximizing utility subject to a budget

constraint or minimizing expenditure subject

to maintaining a given level of utility – yields

the same optimal values.

3–56


• We can use calculus to solve the expenditure —

minimizing problem.

• The solution of this problem is an expression of the

minimum expenditure as a function of the price and

the specified utility level:

• We call this expression the expenditure function.

1 2, ,E E p p U

3–57

1 21 1 2 2

,min q q

E p q p q

1 2. . ,s tU U q q


Figure 3.8Minimizing Expenditure

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© 2008 Pearson Addison Wesley. All rights reserved Chapter Three A Consumer’s Constrained Choice