Chapter Three A Consumer Constrained Choice. © 2007 Pearson Addison-Wesley. All rights...

Chapter Three

A Consumer ConstrainedChoice

© 2007 Pearson Addison-Wesley. All rights reserved. 3–2

Consumer Choice

• In this chapter, we examine four main topics.– Preferences– Utility– Budget Constraint– Constrained Consumer Choice


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 to Bundle (like at least as much as ) and weakly prefers Bundle to Bundle , the consumer also weakly prefers Bundle to Bundle .

y

zx

zzy

x

y


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).


• 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

Properties of Consumer Preferences

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


Indifference Curves

• All indifference curve maps must have four important properties:– Bundles on indifference curves farther from

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

– There is an indifference curve through every possible bundle.

– Indifference curve cannot cross.– Indifference curves slope downward.


Figure 3.1 Bundles of Pizzas and Burritos Lisa Might Consume

(a)

302515

Z, Pizzas per semester

25

20

15

10

5

da

b

e

c

f

A

B

(b)

302515


25

20

15

10 da

b I1

e

c

f


Figure 3.1 Bundles of Pizzas and Burritos Lisa Might Consume (cont’d)

(c)

302515


25

20

15

10d

I0

I1

I2e

c

f


Figure 3.2 Impossible Indifference Curves

(a) Crossing


I1

I 0

a

be

(b) Upward Sloping


I

a

b


Figure 3.2 Impossible Indifference Curves (cont’d)

a

b

(c) Thick


I


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 Marginal Utility

• Marginal Utility– the extra utility that a consumer gets from

consuming the last unit of a good

• Thus marginal utility is the slope of the utility function

Z

UMU

Z


Willingness to Substitute Between Goods

• Marginal Rate of Substitution (MRS)– the maximum amount of one good a

consumer will sacrifice to obtain one more unit of another good


Willingness to Substitute Between Goods• The marginal rate of substitution of burritos

for pizza is

where is the number of pizzas Lisa will give up to get more burritos or vice versa and pizza (Z) is on the horizontal axis. The marginal rate of substitution is the slope of the indifference curve.

BMRS

Z

BZ


Utility and Marginal Rates of Substitution

• The marginal rate of substitution (MRS) is the slope of the indifference curve. The marginal rate of substitution can also be expressed in terms of marginal utilities.

• The marginal rate of substitution can be written as

Z

B

MUBMRS

Z MU

Utility and Marginal Rates of Substitution

• Utility function U(qz, qb) where qz is the quantity of pizza and qb is the quantity of burritos.

• Marginal utility of pizza = Uz.

• Marginal utility of burritos = Ub

• MRS = dqb /dqz = - Uz / Ub



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 approaches zero as we move down and to the right along an indifference curve.



Figure 3.3 Marginal Rate of Substitution

(a) Indifference Curve Convex to the Origin

5

3

8

1–1

1

12

0

–2

–3

3 4 5 6Z, Pizzas per semester

a

b

c

d

I

(b) Indifference Curve Concave to the Origin

5

7

1

1

2

0

–2

–3

3 4 5 6Z, Pizzas per semester

a

b

c

I



• 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 3.4 Perfect Substitutes, Perfect Complements, Imperfect Substitutes

(a) Perfect Substitutes

1 2 3 4

Pepsi, Cans per week

1

0

2

3

4

I1 I2 I3 I4

(b) Perfect Complements

1 2 3

Pie, Slices per week

1

2

3

0

I1

I2

I3

a

d

e c

b


Figure 3.4 Perfect Substitutes, Perfect Complements, Imperfect Substitutes (cont’d)

(c) Imperfect Substitutes


I


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

Pbqb + Pzqz = Y

where Pbqb is the amount she spends on burritos and Pzqz is the amount she spends on pizzas.



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


Slope of the Budget Constraint

• Marginal Rate of Transformation (MRT)– 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:

Z

B

PBMRT

Z P


Figure 3.5 Budget Constraint

Opportunity set

50 = Y/pZ

L1 (pZ = $1, Y = $50)

25 = Y/pB

20

10

100 30Z, Pizzas per semester

a

b

c

d


Effect of A Change in Price 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.


Figure 3.5-1 Changes in the Budget Constraint

(a) Price of Pizza Doubles

Loss

50

L1 (pZ = $1)

L2 (pZ = $2)

25

250Z, Pizzas per semester

(b) Income Doubles

Gain

100

L3 (Y = $100)

L1 (Y = $50)

50

25



Effect of a Change in Income on Consumption

• 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.

• 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

• The optimal bundle must be on the budget constraint. 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.


The Consumer’s Optimal Bundle• Bundles that lie on indifference curves that

cross the budget constraint (such as I1, which crosses the constraint at and ) 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.

a c


Figure 3.6 Consumer Maximization

(a) Interior Solution

Budget line

10

20

25


I 1I 2

I 3

d

fc

e

a

g

A

B

(b) Corner Solution

Budget line

25


I1

I2

I3

e


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

Z

B

MU

MU Z

B

PMRS MRT

P

Z B

Z B

MU MU

P P

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

3–43


• 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

3–44


• 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

3–45


• Lagrangian Method:

We write the equivalent Lagrangian problem as

where is the Lagrange multiplier.• The first-order conditions for an interior maximization

are

1 2

1 2 1 1 2 2, ,max ,

q qL U q q Y p q p q

1 1 11 1

0

L U

p U pq q

2 22

0

L

U pq

1 1 2 2 0

L

Y p q p q3–46


• These three first-order conditions can be solved

for the optimal values of , and .

We find that

• This optimality condition is the same as the one that we derived using the substitution method or a graphical approach.

1 2

1 2

U U

p p

1q 2q

3–47


Solved Problem 3.3: Corner Solution

SUVs per decade

1

LB

l

LN

eN

eB

1


Figure 3.7 Optimal Bundles on Convex Sections of Indifference Curves

(a) Strictly Concave Indifference Curves


e

d

I2I1 I3

e

d

I2I1

(b) Concave and Convex Indifference Curves


Budget line

Budget line


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.


Food Stamps

• Cash preferred to food stamps– Poor people who receive cash have more

choices than those who receive a comparable amount of food stamps.


Figure 3.8 Food Stamps Versus Cash

Y Y + 100

Y + 100

0 100

Food per month

Budget line withfood stamps

Budget line with cash

Originalbudget line

A

B

f

deY

C

I1

I 2

I 3

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.

U2I

3–53

Minimizing Expenditure• 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.

• We can use calculus to solve the expenditure —minimizing problem.

1 21 1 2 2

,min q q

E p q p q

1 2. . ,s tU U q q3–54

Minimizing Expenditure

• 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

3–55

Chapter Three A Consumer Constrained Choice. © 2007 Pearson Addison-Wesley. All rights...

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Transcript of Chapter Three A Consumer Constrained Choice. © 2007 Pearson Addison-Wesley. All rights...