Lecture # 05 Consumer Preferences and the Concept of Utility (cont.) Lecturer: Martin Paredes.
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Transcript of 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
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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
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
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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
5
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
35
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
38
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
41
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)