The Consumer’s Optimization Problem

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The Consumer’s Optimization Problem

• Individual consumption decisions are made with the goal of maximizing total satisfaction from consuming various goods and services

5-2

Consumer Theory

• Assumes buyers are completely informed about:• Range of products available

• Prices of all products

• Capacity of products to satisfy

• Their income

5-3

Indifference Curves

• Locus of points representing different bundles of goods, each of which yields the same level of total utility

• Negatively sloped & convex

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Properties of Consumer Preferences

• Completeness• For every pair of consumption bundles, A and B, the

consumer can say one of the following:• A is preferred to B• B is preferred to A• The consumer is

indifferent between A and B

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• Transitivity• If X is preferred to Y, and Y is preferred to Z,

then X must be preferred to Z

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• Nonsatiation• More of a good is always preferred to less

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Utility

• The benefits consumers obtain from the goods and services they consume is called utility.

• A utility function shows an individual’s perception of the utility level attained from consuming each conceivable bundle of goods

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Marginal Utility

• Addition to total utility attributable to the addition of one unit of a good to the current rate of consumption, holding constant the amounts of all other goods consumed

• MU= Changes in Total Utility /

Change in No of Units Consumed MU U X

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Constrained Utility Maximization (Figure 5.8)

A•

I

C•

•B

II

R

T

Quantity of burgers

Qu

anti

ty o

f piz

zas

0 8020 10040 60

10

20

30

40

50

7010 9030 50

•E

III

•DIV

45

15

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Marginal Rate of Substitution

• MRS shows the rate at which one good can be substituted for another while keeping utility constant• Negative of the slope of the indifference curve

• Diminishes along the indifference curve as X increases & Y decreases

• Ratio of the marginal utilities of the goods

X

Y

MUYMRS

X MU

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How to get MRS Formula

• U = ƒ(XY) (Here, U= Utility; X,Y= 2 goods)

by differentiation,

dU= ƒ1dX + ƒ2dY Here, ƒ1= ΔU/ ΔX = MUx

ƒ2= ΔU/ ΔY = MUy

as per indifference curve, U remains constant. So, U = 0

→ ƒ1dX + ƒ2dY = 0

→ ƒ1dX = - ƒ2dY

→ ƒ1 / ƒ2 = - dX/dY

as indifference curve is neutral, so….

IdX/dYI = I ƒ1 / ƒ2 I = MUx/ MUy

So, MRSxy = MUx/ MUy

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Slope of an Indifference Curve & the MRS (Figure 5.3)

Quantity of good X

Qu

an

tity

of

go

od

Y

0

I

C (360,320)

600

800

A

B

T

T’

360

320

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MRS = slope of indifference curve = slope of tangent line

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The slope is 35/35 = 1

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MRS = − ΔY /ΔX = 5 /10 = 1 2

5

10

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5

10

Before, − ΔY /ΔX = 5/10 or 1/ 2, After, − ΔX/ ΔY = 10/5 or 2

10:5 or 2:1

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Consumer’s Budget Line

• Shows all possible commodity bundles that can be purchased at given prices with a fixed money income

X YM P X P Y

X

Y Y

PMY X

P P

or

5-19

Consumer’s Budget Constraint (Figure 5.5)

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Typical Budget Line (Figure 5.6)

Quanti

ty o

f Y

Quantity of X

X

Y Y

PMY X

P P

•

•

A

B

Y

MP

X

MP

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Shifting Budget Lines (Figure 5.7)

Panel B – Changes in price of X

200

100A

B

250

D

R

N

120

240

Qu

anti

ty o

f Y

Quantity of X

Panel A – Changes in money income

Qu

anti

ty o

f Y

Quantity of X

A

B

100F

Z

80

160

200

125

C

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The following figure shows a portion of a consumer’s indifference map. Theconsumer faces the budget line ZL, and the price of Y is $20.

The consumer's income = $__________. The price of X is $_____________.

600

20

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The equation for the budget line ZL is Y = ______________________.

30 - 1x

30/ 30

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What combination of X and Y would the consumer choose? Why?

15X and 15Y

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The marginal rate of substitution at the combination in part c is __________.

MRS=PMRS=Pxx / P / PYY

= 20 / 20= 20 / 20 = 1= 1

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If the budget line pivots to ZM, the consumer chooses _______ units of good X and _________ units of good Y.

10

15

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Along budget line ZM, the price of X is $_________ and the price of Y is$________.30

20

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The new MRS is equal to __________.MRS= 30/ 20 =1MRS= 30/ 20 =1

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The figure below shows a portion of a consumer’s indifference map, and a budget line. The consumer’s income is $1,200 and the price of Y is $6.

Using the given budget line, what is one point on the consumer’s demand for X?(Both Price & Quantity)

Px = $1,200/200 = $6 and X = 100

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The figure below shows a portion of a consumer’s indifference map, and a budget line. The consumer’s income is $1,200 and the price of Y is $6.

Pivot the budget line and derive two other points on the consumer’s demand for X.

At A, Px = $1,200/100 = $12 and X = 50

At B, Px = $1,200/200 = $6 and X = 100

At C, Px = $1,200/300 = $4 and X = 150

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Market Demand

• Market demand is a list of prices and the quantities consumers are willing and able

• to purchase at each price in the list, other things being held constant.

• Marketdemand is derived by horizontally summing the demand curves for all the individuals in the market.

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Derivation of Market Demand

Quantity demanded

Price Consumer 1 Consumer 2 Consumer 3Market

demand

$6

2

1

5

4

3

3

1213

5

8

10

0

7

10

1

3

5

0

6

8

0

1

4

3

2531

6

1219

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Derivation of Market Demand Figure (5.10)

The Consumer’s Optimization Problem

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