Lecture # 05Lecture # 05

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

Lecturer: Martin ParedesLecturer: Martin Paredes

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

Marginal Utility 4. Some Special Functional Forms

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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

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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

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IC1

x

y

Preferred to A

•ALess preferred

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IC1

x

y

•A

•B

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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

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IC2

x

y

•A

•B

•(.5A, .5B)

IC1

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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

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

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

dx preference)

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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.

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Example: The Diminishing Marginal Rate of Substitution

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

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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.

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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.

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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.

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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.

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10 = xy

x

y

20 5

2

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Example: Utility and a single indifference curve

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Example: Utility and a single indifference curve

10 = xy

20 = xy

x

y

Preference direction

20 5

2

5

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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.

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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.

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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.

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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

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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

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

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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

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Example: Graphing Indifference Curves

IC1

x

y

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IC1

IC2

x

y

Preference direction

Example: Graphing Indifference Curves

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

where: + = 1; A, , positive constants

Properties:MUx = Ax-1y

MUy = Axy-1

MRSx,y = y x

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Example: Cobb-Douglas

IC1

x

y

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IC1

IC2

x

y

Preference direction

Example: Cobb-Douglas

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2. Perfect Substitutes:U = Ax + By

where: A,B are positive constants

Properties:MUx = A

MUy = B

MRSx,y = A (constant MRS) B

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Example: Perfect Substitutes (butter and margarine)

x0

y

IC1

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x0

y

IC1IC2

Example: Perfect Substitutes (butter and margarine)

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x0

y

IC1IC2 IC3

Slope = -A/B

Example: Perfect Substitutes (butter and margarine)

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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

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Example: Perfect Complements (nuts and bolts)

x0

y

IC1

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Example: Perfect Complements (nuts and bolts)

x0

y

IC1

IC2

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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

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•x

y

0

IC1

Example: Quasi-linear Preferences (consumption of beverages)

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Example: Quasi-linear Preferences (consumption of beverages)

••

IC’s have same slopes on anyvertical line

x

y

0

IC2

IC1

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 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)