Lecture # 05

Consumer Preferences and the

Concept of Utility (cont.)


Lecturer: Martin Paredes
2
1. Indifference Curves (end)
2. The Marginal Rate of Substitution
3. 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
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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
5
x
y

A
6
x
y
Preferred to A

A
7
x
y
Preferred to A

A
Less
preferred
8
IC
1

x
y
Preferred to A

A
Less
preferred
9
IC
1

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).

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x
y

A
IC
1

Suppose a consumer is
indifferent between A and C

Suppose that B preferred to A.

B

C

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x
y

A

B

IC
1

IC
2

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.
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x
y

A

B
IC
1

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x
y

A

B

(.5A, .5B)
IC
1

15
IC
2

x
y

A

B

(.5A, .5B)
IC
1

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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

17
Definition 2: It is the negative of the slope of
the indifference curve:
MRS
x,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 consumers preferences.
We can measure the strength of a consumers
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
2
0
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
2
0
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
MU
x
= 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 consumers 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 70
Second quartile 78
Third quartile 82
Highest 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 = MU
x
. dx + MU
y
. dy
Along a particular indifference curve, dU = 0, so:
0 = MU
x
. dx + MU
y
. dy

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Solving for dy/dx:
dy = _ MU
x

dx MU
y

By definition, MRS
x,y
is the negative of the slope
of the indifference curve:
MRS
x,y
= MU
x

MU
y

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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
MU
x
=y
0.5
/2x
0.5

MU
y
=x
0.5
/2y
0.5


Marginal utility is positive for both goods:
=> Monotonicity satisfied
Diminishing marginal utility for both goods
=> Averages preferred to extremes
Marginal rate of substitution:
MRS
x,y
= MU
x
= y
MU
y
x
Indifference curves do not intersect the axes
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Example: Graphing Indifference Curves
IC
1

x
y
35
IC
1

IC
2

x
y
Preference direction
Example: Graphing Indifference Curves
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1. Cobb-Douglas (Standard case)
U = Ax

y



where: + = 1; A, , positive constants

Properties:
MU
x
= Ax
-1
y

MU
y
= Ax

y
-1
MRS
x,y
= y
x
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Example: Cobb-Douglas
IC
1

x
y
38
IC
1

IC
2

x
y
Preference direction
Example: Cobb-Douglas
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2. Perfect Substitutes:
U = Ax + By
where: A,B are positive constants

Properties:
MU
x
= A
MU
y
= B
MRS
x,y
= A (constant MRS)
B
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Example: Perfect Substitutes (butter and margarine)
x 0
y
IC
1

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x 0
y
IC
1

IC
2

Example: Perfect Substitutes (butter and margarine)
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x 0
y
IC
1

IC
2
IC
3

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:
MU
x
= A or 0
MU
y
= B or 0
MRS
x,y
= 0 or or undefined
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Example: Perfect Complements (nuts and bolts)

x 0
y
IC
1

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Example: Perfect Complements (nuts and bolts)

x 0
y
IC
1

IC
2

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4. Quasi-Linear Utility Functions:
U = v(x) + Ay
where: A is a positive constant, and v(0) = 0

Properties:
MU
x
= v(x)
MU
y
= A
MRS
x,y
= v(x) (constant for any x)
A
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x
y
0
IC
1

Example: Quasi-linear Preferences
(consumption of beverages)
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Example: Quasi-linear Preferences
(consumption of beverages)


ICs have same slopes on any
vertical line
x
y
0
IC
2

IC
1

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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 demandseem to be
satisfied for most people. (ordinal utility function)