### Econ 281 Chapter 3 - University of Alberta

```Section 2 - Consumer Theory
• Consumer theory attempts to explain
why consumers choose one good or
bundle of goods over another good or
bundle of goods.
1
Section 2 - Consumer Theory
• In Consumer Theory we will discuss:
–Consumer Utility (Chapter 3)
–Utility and Constraints
(Chapter 4)
–Utility and the Demand Curve
(Chapter 5)
2
Chapter 3 – Consumer Preferences
and the Concept of Utility
• Every day, people make choices about
what they prefer:
–Buy a red car with a sunroof or a green
truck with 4 wheel drive
–Buy Edo and a chocolate milk or
MacDonalds and a Diet Coke
–Buy a computer with a 24” LCD screen
and windows or a computer with a 21”
screen with an apple on it
3
Chapter 3 – Consumer Preferences
and the Concept of Utility
• In this chapter we will study:
3.1 Preferences and Ranking
3.2 Utility Functions
3.3 Indifference Curves
3.4 Marginal Rates of Substitution
3.5 Special Utility Functions
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3.1 Preference Definitions
• Basket (bundle) – any combination of
goods and services
–3 hot dogs, 2 pop and 1 ice cream
–Haircut, manicure and 20 min massage
–2 punches in the gut, 1 kick in the groin
• Consumer preferences – any ranking of
–I prefer 2 hot dogs and a coke to a hot
dog, coke and ice cream
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Preference Assumptions
1) Preferences are complete
- A consumer can always rank
preferences:
a) A is preferred to B: A  B
b) B is prefered to A: B  A
c) A consumer is indifferent between A
and B: A ≈ B
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Preference Example
1)Preferences are complete
- “I would rather go to a movie with Bobby than go
skiing with Mark.” (valid)
-“I prefer a computer with a good video card and large
screen to a computer with a good sound card and
good speakers.” (valid)
- “I hate everyone equally!” (valid)
- “I can’t decide whether Ruth or Victoria is hotter!”
(invalid)
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Preference Assumptions
2) Preferences are transitive
- Choices are consistent:
If
a) A is preferred to B: A  B
b) B is preferred to C: B  C
then
a) A is preferred to C: A  C
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Preference Example
2) Choices are transitive
- “I would rather see the movie Star Wars than Tears
and Feelings. I prefer seeing Oceans 13 to Star
Wars. Therefore, I prefer Oceans 13 to Tears and
Feelings.” (valid)
-“Ruth is hotter than Victoria and Susan is cuter than
Ruth. Victoria is more attractive than Susan,
however.” (invalid)
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Preference Assumptions
3) More is better
- A consumer always prefers having
more of a good
Examples:
-“I prefer seven hot dogs to 3.”
-“It’s better to have loved and lost than
never to have loved at all!”
-“2 heads are better than 1.”
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Ranking
Ordinal Ranking
-baskets are ordered or compared to each
other without any quantitative
information or intensity of preference
-ie: “I like dogs more than cats.”
Cardinal Ranking
-ie: “I like dogs ten times more than
cats.”
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3.2 Utility
•util: unit of pleasure.
• utility: a number that represents
the level of satisfaction that the
consumer derives from consuming
a specific quantity of a good.
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Total Utility, Marginal Utility
• TU (total utility):
–the total amount of satisfaction that you
get from consuming a product.
• MU (marginal utility):
–the increase in TU that comes about as
a result of consuming one more unit of
the product.
–The slope of the total utility function
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Marginal Utility
• If one more unit of a good is consumed, the
marginal utility is equal to the increased utility
from that extra good
• If more than one additional good is consumed:
Utility
MU 
Goods
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Law of Diminishing MU
• The MU (marginal utility) of a good or
service will decline as more units of that
good or service are consumed.
•Marginal utility is what counts for
rational consumer decisions.
• The “More is Better” assumption is
violated if MU ever becomes negative (ie:
eating 23 pieces of pizza)
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Total utility is
maximized...
34
Marginal Utility (utils per week)
Total Utility (utils per week)
Maximizing Unconstrained Utility
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22
0
2
3 4 5 6 7 8
Performances per Week
10
8
6
4
2
0
2 3
4
…where marginal
utility equals zero.
5
6
7
Performances per Week
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Marginal Utility Example
• Let Utility (U) depend on how much pizza you
eat (P), therefore
U(P) 2 P
• Therefore the first piece of pizza gives you 2
“utils” of pleasure, but 4 pieces of pizza give
you 4 “utils” of pleasure, not 10….marginal
utility is diminishing:
1
MU(P)
P
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Utility With 2 or More Goods
• Utility and 1 good can be measured on a 2dimensional graph
• Utility and 2 goods must be measured on a 3dimensional graph
• Here Marginal Utility uses the ceteris paribus
assumption: how does utility change when 1 good
changes, everything else held constant?
U
MU x 
X
y held constant
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Marginal Utility Example
• Let Utility (U) depend on how much pizza (P)
and hot dogs (H) you eat, therefore
U(P) 2 PH
• If hot dogs are held constant, each additional
pizza yields less utility:
MU(P)
H
P
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3.3 Indifference Curves
• 3 dimensional graphs are difficult to
graph and understand
• In practice, consumer preference is
graphed using 2 goods on the X and
Y axis and INDIFFERENCE CURVES
• Each INDIFFERENCE CURVE plots
all the goods combinations that yield
the same utility; that a person is
indifferent between
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y
Consider the utility function U=(xy)1/2.
Each indifference curve below shows
all the baskets of a given utility level.
Consumers are indifferent between
2
•
•
•
1
•
U=2
U=√2
0
1
2
4
x
21
y
From the indifference curves, we know
that:
A ≈ B, C ≈ D
C A &
D A &
A
2
•
1
•
C
•
B
C  B,
D B
D
•
U=2
U=√2
0
1
2
4
x
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1). Completeness => each basket lies on
only one indifference curve
2). Transitivity => indifference curves do
not cross
3). Negative Slope => when a consumer
likes both goods (MUa and MUb are positive),
the indifference curve is downward sloping
4). Thin curves => indifference curves are
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not “thick”
y
IC1
B  A. (different indifference curves)
A ≈ C (same indifference curve)
B ≈ C (same indifference curve)
Therefore:
IC2
B ≈ A by transitivity
B
•
A
•
C
•
x
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y
More of any good is more preferred and
less of a good is less preferred, so an
indifference curve cannot extend into
areas I or II; it must slope downward
I: Preferred to A
•
A
II: Less
preferred
IC1
x
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y
Since more is better, baskets B and C
should be preferred to basket A
BUT
they all lie on the same indifference
curve, implying indifference.
C
2
•
•
•
A
B
1
2
1
0
4
x
26
• Note that some specialized models
produce indifference curves that violate
one or more of our assumptions
• These models may still be useful, but their
violations must always be kept in mind
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North
Example of “more is better” violation
•B
C
•
•
A
IC3
University
Of Alberta
IC2
IC1
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East
Example: Goods people don’t care about
Are these curves Complete?
Yes
Are these curves Transitive?
Catfish in the city
Yes
0
IC1 IC2
IC3
IC4
Do these curves have Negative Slope?
No
Preference direction
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Movies watched
3.4 Marginal Rate of Substitution
(MRS)
• All along an individual’s indifference curve, an
individual consumes different baskets of goods while
remaining at the same utility
• The individual is willing to SUBSTITUTE one good for
another
• An individual must be compensated by an increase in
one good if the other good decreases
– Ie) if Bob is equally happy with 3 hot dogs and 1
soda or 2 hot dogs and 2 soda, he is willing to give
up 1 hot dog for 1 soda or 1 soda for 1 hot dog
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Marginal Rate of Substitution (MRS)
• The marginal rate of substitution (MRS) is the change
(loss) in one good needed to offset the change (gain)
in another good
– In this case, MRS is the trade-off (loss) of y for a
small increase in x
-”The Marginal Rate of Substitution of x for y”
-x is gained, so how much y must be given up
-alternately, if x is given up, how much y can we
get?
• The MRS is equal to the SLOPE of the indifference
curve (slope of the tangent to the indifference curve)
MRS x , y
 y

x
utilityconstant
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Marginal Rate of Substitution (MRS)
U  MU x x  MU y y
but since U  0 as one moves along the indifference curve,
-MUx xMU y y
MUx  y
MUy x
 y
x
utilityconstant
MU
x  MRS

utilityconstant
x, y
MUy
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MRS Example
Let Ut ility  x  y
MUx 
1
2 x
, MUy 
1
2 y
1
MRSx, y
MUx 2 x


1
MUy
2 y
MRSx, y 
y
x
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Diminishing MRS
• In general, people tend to value more what they
have less of:
– Ie) If Frank has 30 chicken wings and 1
Pepsi, he is very willing to give up wings for
another Pepsi. If Frank has 10 chicken wings
and 2 Pepsi’s, he is less willing to give up
wings for Pepsi
• Therefore MRSx,y diminishes as x increases
along the indifference curve
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Pepsi
Diminishing Marginal Utility
•
Very willing to give up Pepsi for wings
(steep slope=high MRS)
Less willing to give up Pepsi for wings
(flat slope = low MRS)
•
IC1
Wings
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Diminishing MRS
• Due to Diminishing MRS, most indifference
curves are “bowed” towards the origin (0,0)
– As seen in the above graph
• If Diminishing MRS does not hold (ie: trading
quarters for loonies), the graph is not bowed
towards the origin
Exercise: Let Utility=(Pepsi)(Wings). For a utility
level of 16, sketch the graph and see if
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Diminishing MRS applies.
3.4 Special Utility Functions
• In Economics, utility functions dealing with 2
categories of goods create unique indifferent
curves:
– Perfect Substitutes
– Perfect Compliments
• Furthermore, 2 utility functions are widely used
by Economists for their desirable properties:
– Cobb-Douglas Utility Function
– Quasi-Linear Utility Function
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Perfect Substitutes
• Goods that are perfect substitutes can always
be substituted for each other on a 1-to-1 basis
– If a restaurant doesn’t carry Pepsi, you order
– Therefore, MRSCoke, Pepsi=1 and
MUCoke/MUPepsi =1
– In general, MRS=a constant, and indifference
curves are a straight line
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Perfect Substitutes: U = Ax + By
Where: A, B positive constants
MUx = A
MUy = B
MRSx,y = A/B
 1 unit of x is equal to
B/A units of y everywhere
(constant MRS).
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Example: Perfect Substitutes (Tylenol, Extra-Strength Tylenol)
y
Slope = -A/B
IC1
0
IC2
IC3
x
40
Perfect Compliments
• Some goods are only useful in a set ratio to
each other; extra of one good is useless without
extra of the other:
– Shoes: 1 Left shoe for every Right shoe
– Cars: 4 full-size tires for every car
– Kraft Dinner: 6 cups of water for every packet
– Marriage: 1 Bride for Every Groom
• Indifference curves are right angles
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3. Perfect Complements: U = A min(x,y)
where: A is a positive constant.
MUX = 0 or A
MUY = 0 or A
MRSX,Y is 0 (horizontal)
or infinite (vertical)
or undefined (at corner)
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Example: Perfect Complements (nuts and bolts)
y
IC2
IC1
0
x
43
Cobb-Douglas Utility Function
• The Cobb-Douglas Utility function is the holy
grail of economic models for a variety of
reasons:
– It’s straightforward
– It’s easily modified to suit the model
– It has desirable mathematical properties
• The Cobb-Douglas Utility function also yields
“STANDARD” indifference curves
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1. Cobb-Douglas: U = Axy
where:  +  = 1; A, , positive constants
MUX = Ax-1y (Positive)
MUY = Axy-1 (Positive)
MRSx,y = (y)/(x)
“Standard” case:
Downward sloping IC, diminishing MRS
45
y
Example: Cobb-Douglas
(speed vs. maneuverability)
Preference direction
IC2
IC1
x
46
Quasi-Linear Utility Function
• Quasi-Linear Utility Functions often explain
consumer behavior without an overly complex
model
– It’s effective
– It’s simple
– It has a catchy name – “Quasi”
• In a Quasi-Linear Utility Function, MRS is equal
for all points above and below each other:
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U = v(x) + Ay
Where: A is a positive constant.
MUx = v’(x) = V(x)/x, where  small
MUy = A
Example:
U=4(x)1/2+2y
MUx=2/(x) ½ MUy=2
-Useful if one good’s consumption changes
little (ie:soap)
-linear in Y, non-linear in X (hence quasi48
linear)
movies
Example: Quasi-linear Preferences
(movies and toothpaste)
IC’s have same slopes on any
vertical line
•
•
0
IC2
IC1
toothpaste
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Chapter 3 Key Concepts
Preferences
Preference Assumptions
Utility
Marginal Utility
Diminishing Marginal Utility
Indifference Curves
Indifference Curve Properties
Marginal Rate of Substitution
Diminishing MRS
50
Chapter 3 Key Concepts
Special Utility Functions
Perfect Substitutes
Perfect Compliments
Cobb-Douglas
Quasi-Linear
51
```