Lecture # 05
Consumer Preferences and the
Concept of Utility (cont.)
Lecturer: Martin Paredes
1. Indifference Curves (end)
2. The Marginal Rate of Substitution
3. 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”
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y
•
A
x
5
y
Preferred to A
•
A
x
6
y
Preferred to A
•
A
Less
preferred
x
7
y
Preferred to A
•
A
Less
preferred
IC1
x
8
y
A
•
•
B
IC1
x
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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).
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y
• Suppose a consumer is
indifferent between A and C
• Suppose that B preferred to A.
IC1
•
A
•
B
C
•
x
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y
IC1
IC2
•
A
•
B
 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.
C
•
x
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y
A
•
•
B
IC1
x
13
y
A
•
(.5A, .5B)
•
•
B
IC1
x
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y
A
•
(.5A, .5B)
•
IC2
•
B
IC1
x
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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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y
Example: Utility and a single indifference curve
5
2
0
10 = xy
2
5
x
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y
Example: Utility and a single indifference curve
Preference direction
5
20 = xy
2
0
10 = xy
2
5
x
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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
Lowest quartile
Second quartile
Third quartile
Highest quartile
Percent > “Satisfied”
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78
82
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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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y
Example: Graphing Indifference Curves
IC1
x
34
y
Example: Graphing Indifference Curves
Preference direction
IC2
IC1
x
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1. Cobb-Douglas (“Standard case”)
U = Axy
where:  +  = 1; A, , positive constants
Properties:
MUx = Ax-1y
MUy = Axy-1
MRSx,y = y
x
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y
Example: Cobb-Douglas
IC1
x
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y
Example: Cobb-Douglas
Preference direction
IC2
IC1
x
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2. Perfect Substitutes:
U = Ax + By
where: A,B are positive constants
Properties:
MUx = A
MUy = B
MRSx,y = A
B
(constant MRS)
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y
Example: Perfect Substitutes (butter and margarine)
IC1
0
x
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y
Example: Perfect Substitutes (butter and margarine)
IC1
0
IC2
x
41
y
Example: Perfect Substitutes (butter and margarine)
Slope = -A/B
IC1
0
IC2
IC3
x
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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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y
Example: Perfect Complements (nuts and bolts)
IC1
0
x
44
y
Example: Perfect Complements (nuts and bolts)
IC2
IC1
0
x
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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)
A
(constant for any x)
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Example: Quasi-linear Preferences
(consumption of beverages)
y
IC1
•
0
x
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Example: Quasi-linear Preferences
(consumption of beverages)
y
IC2
IC1
•
•
0
IC’s have same slopes on any
vertical line
x
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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)
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