Download - Lecture # 05 Consumer Preferences and the Concept of Utility (cont.) Lecturer: Martin Paredes

Lecture # 05Lecture # 05

Consumer Preferences and Consumer Preferences and the Concept of Utility (cont.)the Concept of Utility (cont.)

Lecturer: Martin ParedesLecturer: Martin Paredes

2

1. Indifference Curves (end)2. The Marginal Rate of Substitution3. The Utility Function

Marginal Utility 4. Some Special Functional Forms

3

Definition: An Indifference Curve is the set of all baskets for which the consumer is indifferent

Definition: An Indifference Map illustrates the set of indifference curves for a particular consumer

4

1. Completeness Each basket lies on only one

indifference curve

2. Monotonicity Indifference curves have negative slope Indifference curves are not “thick”

5x

y

•A

6x

y

Preferred to A

•A

7x

y

Preferred to A

•ALess preferred

8

IC1

x

y

Preferred to A

•ALess preferred

9

IC1

x

y

•A

•B

10

3. Transitivity Indifference curves do not cross

4. Averages preferred to extremes Indifference curves are bowed toward

the origin (convex to the origin).

11x

y

•A

IC1

• Suppose a consumer is indifferent between A and C

• Suppose that B preferred to A.B

•

C•

12x

y

•A•

B

•

IC1IC2

C

It cannot be the case that an IC contains both B and C

Why? because, by definition of IC the consumer is:• Indifferent between A & C• Indifferent between B & C Hence he should be indifferent

between A & B (by transitivity).

=> Contradiction.

13x

y

•A

•B IC1

14x

y

•A

•B

•(.5A, .5B)

IC1

15

IC2

x

y

•A

•B

•(.5A, .5B)

IC1

16

There are several ways to define the Marginal Rate of Substitution

Definition 1: It is the maximum rate at which the consumer would be willing to substitute a little more of good x for a little less of good y in order to leave the consumer just indifferent between consuming the old basket or the new basket

17

Definition 2: It is the negative of the slope of the indifference curve:

MRSx,y = — dy (for a constant level of

dx preference)

18

An indifference curve exhibits a diminishing marginal rate of substitution:

1. The more of good x you have, the more you are willing to give up to get a little of good y.

2. The indifference curves • Get flatter as we move out along the

horizontal axis• Get steeper as we move up along the

vertical axis.

19

Example: The Diminishing Marginal Rate of Substitution

20

Definition: The utility function measures the level of satisfaction that a consumer receives from any basket of goods.

21

The utility function assigns a number to each basket More preferred baskets get a higher

number than less preferred baskets.

Utility is an ordinal concept The precise magnitude of the number

that the function assigns has no significance.

22

Ordinal ranking gives information about the order in which a consumer ranks baskets E.g. a consumer may prefer A to B, but

we cannot know how much more she likes A to B

Cardinal ranking gives information about the intensity of a consumer’s preferences. We can measure the strength of a

consumer’s preference for A over B.

23

Example: Consider the result of an exam

• An ordinal ranking lists the students in order of their performance

E.g., Harry did best, Sean did second best, Betty did third best, and so on.

• A cardinal ranking gives the marks of the exam,

based on an absolute marking standard E.g. Harry got 90, Sean got 85, Betty got 80, and so on.

24

Implications of an ordinal utility function:

Difference in magnitudes of utility have no interpretation per se

Utility is not comparable across individuals Any transformation of a utility function that

preserves the original ranking of bundles is an equally good representation of preferences.eg. U = xy U = xy + 2 U = 2xy

all represent the same preferences.

25

10 = xy

x

y

20 5

2

5

Example: Utility and a single indifference curve

26

Example: Utility and a single indifference curve

10 = xy

20 = xy

x

y

Preference direction

20 5

2

5

27

Definition: The marginal utility of good x is the additional utility that the consumer gets from consuming a little more of x

MUx = dU dx

It is is the slope of the utility function with respect to x.

It assumes that the consumption of all other goods in consumer’s basket remain constant.

28

Definition: The principle of diminishing marginal utility states that the marginal utility of a good falls as consumption of that good increases.

Note: A positive marginal utility implies monotonicity.

29

Example: Relative Income and Life Satisfaction (within nations)

Relative Income Percent > “Satisfied”Lowest quartile 70Second quartile 78Third quartile 82Highest quartile 85

Source: Hirshleifer, Jack and D. Hirshleifer, Price Theory and Applications. Sixth Edition. Prentice Hall: Upper Saddle River, New Jersey. 1998.

30

We can express the MRS for any basket as a ratio of the marginal utilities of the goods in that basket

Suppose the consumer changes the level of consumption of x and y. Using differentials:

dU = MUx . dx + MUy . dy Along a particular indifference curve, dU =

0, so:0 = MUx . dx + MUy . dy

31

Solving for dy/dx:dy = _ MUx

dx MUy

By definition, MRSx,y is the negative of the slope of the indifference curve:

MRSx,y = MUx

MUy

32

Diminishing marginal utility implies the indifference curves are convex to the origin (implies averages preferred to extremes)

33

Example:U= (xy)0.5

MUx=y0.5/2x0.5

MUy=x0.5/2y0.5

• Marginal utility is positive for both goods:=> Monotonicity satisfied

• Diminishing marginal utility for both goods=> Averages preferred to extremes

• Marginal rate of substitution:MRSx,y = MUx = y

MUy x• Indifference curves do not intersect the axes

34

Example: Graphing Indifference Curves

IC1

x

y

35

IC1

IC2

x

y


Example: Graphing Indifference Curves

36

1. Cobb-Douglas (“Standard case”)U = Axy

where: + = 1; A, , positive constants

Properties:MUx = Ax-1y

MUy = Axy-1

MRSx,y = y x

37

Example: Cobb-Douglas

IC1

x

y

38

IC1

IC2

x

y


Example: Cobb-Douglas

39

2. Perfect Substitutes:U = Ax + By

where: A,B are positive constants

Properties:MUx = A

MUy = B

MRSx,y = A (constant MRS) B

40

Example: Perfect Substitutes (butter and margarine)

x0

y

IC1

41

x0

y

IC1IC2


42

x0

y

IC1IC2 IC3

Slope = -A/B


43

3. Perfect Complements:U = min {Ax,By}

where: A,B are positive constants

Properties:MUx = A or 0

MUy = B or 0

MRSx,y = 0 or or undefined

44

Example: Perfect Complements (nuts and bolts)

x0

y

IC1

45

Example: Perfect Complements (nuts and bolts)

x0

y

IC1

IC2

46

4. Quasi-Linear Utility Functions:U = v(x) + Ay

where: A is a positive constant, and v(0) = 0

Properties:MUx = v’(x)

MUy = A

MRSx,y = v’(x) (constant for any x)

A

47

•x

y

0

IC1

Example: Quasi-linear Preferences (consumption of beverages)

48

Example: Quasi-linear Preferences (consumption of beverages)

••

IC’s have same slopes on anyvertical line

x

y

0

IC2

IC1

49

1. Characterization of consumer preferences

without any restrictions imposed by budget

2. Minimal assumptions on preferences to get interesting conclusions on demand…seem to be satisfied for most people. (ordinal utility function)