DTINGLEYWarning.docx

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Consumer Choice
Intermediate Micro Theory
ECN 312
From Last Class
• We characterized what bundles a consumer
can afford.
• Previously, we looked at how people compare
happiness from bundles (preferences).
• But, (out of the bundles that the consumer
can afford) which one will they choose?
From Last Class
Preferences Budget (Income)

Choice (Individual Demand)
Population
Market Demand
From Last Class
Population
Market Demand
Quick Preview
y
x
A
B
C
Quick Preview
y

x
A
B
C
Which of these
bundles makes
the consumer
happiest?
Quick Preview
y
x
A
B
C
Which of these
bundles makes
the consumer
happiest?
Which ones
are

affordable?
Which one
does she
choose?
Quick Preview
y
x
A
B
C
BUT – can
she do
better than
B?
• What feasible choice will make me as happy as
possible?
• In other words – how do I get to the “best”
indifference curve without spending more money
than I have?
• In other words – how do I pick x and y to make
u(x,y) as large as possible while satisfying
Pxx + Pyy ≤ m

Decision Problem
Decision Problem - Graphically
Clothing
Food
PC = 3
PF = 2
m = 30
10
15
A B
C
D
E
Clothing
Food
PC = 3
PF = 2
m = 30

10
15
A B
C
D
E
What is wrong
with bundle A?
Clothing
Food
PC = 3
PF = 2
m = 30
10
15
A B
C
D
E

What is wrong
with bundle B?
Clothing
Food
PC = 3
PF = 2
m = 30
10
15
A B
C
D
E
What is wrong
with bundle C?
Clothing
Food
PC = 3

PF = 2
m = 30
10
15
A B
C
D
E
What is wrong
with bundle D?
Clothing
Food
PC = 3
PF = 2
m = 30
10
15
A B
C
D

E
What is wrong
with bundle E?
Clothing
Food
PC = 3
PF = 2
m = 30
10
15
E
What is true
about bundle E?
Clothing
Food
PC = 3
PF = 2
m = 30

10
15
E
1-Consumer spends all of her money
2-Slope of the IC = slope of the BL
Decision Problem Analytically
• At the optimal bundle:
– The slope of the IC will be equal to the slope of
the BL (tangency)
– All money will be spent
• The slope of the IC is given by: -MRS = -
• The slope of the BL is given by: -
• So, at the optimum: =
MUx
MUyPx
Py
MUx
MUy

Px
Py
• Let’s look at our food and clothing example:
– The prices (PC = 3, PF = 2) and income (m=30) hold
– Preferences are simple Cobb-Douglas:
u(F,C) = FC
• The slope of the IC is given by:
-MRS = - = -
• The slope of the BL is given by:
- = -
• So, at the optimum: - = - C = F
MUF
MUC
PF
PC
C
F
2
3

C
F
2
3
2
3
• We know that the consumer chooses C = F
• We know that the consumer spends all her income, i.e., we
know that the optimal bundle will satisfy the constraint:
PC C + PF F = m
3 C + 2 F = 30
• Putting these together:
3( F) + 2 F = 30
2 F + 2 F = 30
4F = 30
F* = 7.5
2
3

2
3
• We know that the consumer chooses C = F
• And we now know that F* = 7.5
• Putting these together:
C = F
C = (7.5)
C* = 5
2
3
2
3
2
3
The formal way to solve constrained
optimization problems is as a Lagrangian

optimization problems is as a Lagrangian
L = (objective function) – λ(constraint function)
optimization problems is with a Lagrangian
L = (objective function) – λ(constraint function)
L = U(x,y) – λ(Px x + Py y - m)
L = U(x,y) – λ(Px x + Py y - m)
To solve, you three first-order conditions:
1. With respect to x:
�ℒ
��
= 0
2. With respect to y:
�ℒ

��
= 0
3. With respect to L:
�ℒ
�λ
= 0
L = U(x,y) – λ(Px x + Py y - m)
1.
�ℒ
��
=
��
��
− ��
2.
�ℒ
��
=
��
��
− ��

3.
�ℒ
�λ
= �� + �� − � �� + �� = �
L = U(x,y) – λ(Px x + Py y - m)
1.
�ℒ
��
=
��
��
− ��
2.
�ℒ
��
=
��
��
− ��
3.

�ℒ
�λ
= �� + �� − � �� + �� = �
��
��
=
��
��
L = U(x,y) – λ(Px x + Py y - m)
1.
�ℒ
��
=
��
��
− ��
2.
�ℒ
��
=
��

��
− ��
3.
�ℒ
�λ
= �� + �� − � �� + �� = �
��
��
=
��
��
What about λ?
• The mathematical interpretation is how
sensitive the optimization problem is to the
constraint.
• Economists often call it the shadow price.
• In utility maximization, it tells us how much
our utility will go up by with a relaxation of
the constraint (or an increase in income). Can
you think of another way to describe λ???
Clothing

Food
PC = 3
PF = 2
m = 30
10
15
EC* = 5
F* = 7.5
Decision Problem
• In general, a consumer’s optimal bundle is
found when two conditions are satisfied:
– The MRS is equal to the price ratio (PR)
– All income is spent
• Think of this as the “usual” case. It occurs
when we have “nice” preferences and “nice”
budget lines.
What do we mean by
“nice”???
Decision Problem

• Why spend all income?
– If consumer has money left over, she could buy
more goods and be better off.
– If consumer can be better off, she’s not at the
optimum.
• Why satisfy tangency condition (MRS = PR)?
– Suppose not. Then (we’ll show this) the consumer
can be made better off by re-arranging her bundle.
– If consumer can be better off, she’s not at the
optimum.
Decision Problem
y
x
A
Suppose that you
choose a bundle A
where MRS > PR
Decision Problem
y
x

A
Now, suppose that
you increase the
amount of x by 1
unit….
Decision Problem
y
x
A
Now, suppose that
you increase the
amount of x by 1
unit….
1
Decision Problem
y
x
A
Now, suppose that

you increase the
amount of x by 1
unit….
How much y
do you have
to give up to
raise $1?
1
Decision Problem
y
x
A
Now, suppose that
you increase the
amount of x by 1
unit….
How much y
do you have
to give up to
raise $Px?
1

Decision Problem
y
x
A
Now, suppose that
you increase the
amount of x by 1
unit….
How much y
do you have
to give up to
raise $Px?
1
Px/Py
Decision Problem
y
x
A
Now, suppose that
you increase the

amount of x by 1
unit….
How much y
will you give
up to stay on
the same IC?
1
Decision Problem
y
x
1
PR
MRS
Progress so far…..
• A general rule for optimization:
- Choose the IC as far from the origin as possible, while
not exceeding the budget.
• The usual rule for optimization:
– Look for a bundle that lies at the tangency of the IC
and the budget line and exhausts all $$$.

• The usual case is when the MRS is diminishing
and the ICs do not touch the axes.
Corner
Solution
s
• Sometimes (not a “usual” case), it is better to
buy zero units of a product.
• (Almost always) happens with perfect
substitutes (can you think of one time it
wouldn’t?)
• Can happen with other preferences (as long as
the ICs have intercepts)
Perfect Substitutes
y

x
Perfect Substitutes
y
x
Perfect SubstitutesPaper Towels
10
Napkins
Paper towels and napkins with:
U = T + 2N
Price of T = 1
Price of N = 3

m = 10
3.33
A
B
C
Perfect SubstitutesPaper Towels
10
Napkins
Paper towels and napkins with:
U = T + 2N
Price of T = 1
Price of N = 3

m = 10
3.33
A
B
C
And if the
price of
napkins fell
to 10
cents???
Perfect Complements
y
x

A B
C
Perfect Complements
Jelly
Peanut Butter
U = min{PB,J}
Price of PB = 1
Price of J = 1
m = 1010
105
5

Perfect Complements
Jelly
Peanut Butter
U = min{PB,J}
Price of PB = 1
Price of J = 1
m = 1010
105
5
What if
price of J
falls to 50
cents?

Perfect Complements
Jelly
Peanut Butter
U = min{PB,J}
Price of PB = 1
Price of J = .5
m = 10
10
6.67
6.67
Intercept = 20
Example: Cobb-Douglas Utility

Consider xylophones and yo-yos with:
u(x,y) = 3x1/2y1/4
Px = 4, Py = 3, m = 12
• Cobb-
se tangency condition to find optimum
Cobb-Douglas – Solve for MRS
u(x,y) = 3x1/2y1/4
Px = 4, Py = 3, m = 12
• MRSxy= MUx/MUy
• MUx= (1/2)3 x

-1/2y1/4 = (3/2) x-1/2y1/4
• MUy= (1/4)3 x
1/2y-3/4 = (3/4) x1/2y-3/4
• MRSxy= MUx/MUy = (2y)/x
Cobb-Douglas – Set equal to price Ratio
u(x,y) = 3x1/2y1/4
Px = 4, Py = 3, m = 12
• The price ratio (negative slope of the BL)= 4/3
y = (2x)/3

Cobb-Douglas – Plug into budget
u(x,y) = 3x1/2y1/4
Px = 4, Py = 3, m = 12
• We know that at the optimum:
–
–
• Putting this together:
Example: Gas Taxes or Standards
• We have spent considerable energy learning

how to describe optimal choices between two
goods.
• What is the benefit??
– We can now explain attributes of consumer
choices that are hard with simple demand analysis
• Example: Suppose we want to reduce
gasoline consumption. Should we use mileage
standards or gas taxes?
• Gary Becker (Business Week) argues that gas
taxes are the best way. Note this this is not an
uncommon stance among economists….
• Suppose that consumers care about two things:
– The miles that they travel (T)
– All other consumption (C)

• Each product has a price and consumers have
some (limited) budget
• Questions:
– Why is the good “travel” instead of gasoline?
– What is the effect of a gas tax, holding car type
fixed?
– What is the effect of better fuel efficiency, holding
gas prices fixed?
C
T

C
T
What does a gas tax do?
C
T
C
T

What has
happened to
gas
consumption?
C
T
What does a mileage
standard (increase fuel
efficiency) do?
C

T
What does a mileage
efficiency) do?
C
T
What does a mileage
efficiency) do?
What has
happened to
gas
consumption?

• What did we learn?
– Without our analysis, it would have been hard to
understand what we know about this policy issue
and where additional evidence is needed.
– Simple models can illuminate important (and
otherwise complicated) issues
– Much of microeconomic analysis applies to the idea
that prices matter and shift behavior.
Consumer Choice: Main Ideas
• We have looked at how consumers make
choices, conditional on prices and income.
– Choices depend on preferences

– Choices depend on affordability
• Consumer goal: maximize happiness given
budget.
• “Usual” way to find optimum: set MRS = PR and
spend all money.
Consumer Budgets
Intermediate Micro Theory
ECN 312
From Last Class
• Preferences describe consumers’ opinions on
the happiness they would receive from any
bundle of goods.
• But, what will they choose? How do they

decide?
• Lines up to readings for Chapters 2 and 4
From Last Class
Population
Market Demand
From Last Class
Population

Market Demand
Budgets
• First go over how we can represent budgets
graphically
• Second go over how we can represent budgets
mathematically
Budgets
• Suppose that you live in an economy with two
goods: food (F) and clothing (C)
• These goods have prices: PF = $2 and PC = $3
• You have income of $30

Affordability
• Which bundles can we afford? How can we show
this information graphically?
– Same set of axes we used for preferences (ICs)
– What if we spent all of our money on F? What if we
spent all our money on C? (this gives intercepts)
– What if we spent some of our money on F and some
of our money on C?
Affordability
Clothing
Food
PF = 2 and PC = 3
Income = 30

Affordability
Clothing
Food
PF = 2 and PC = 3
Income = 30
What if we
spend all our
money on C?
Affordability
Clothing
Food
PF = 2 and PC = 3
Income = 30
What if we
spend all our

money on C?
10
Affordability
Clothing
Food
PF = 2 and PC = 3
Income = 30
What if we
spend all our
money on F?
10
Affordability
Clothing

Food
PF = 2 and PC = 3
Income = 30
What if we
spend all our
money on F?
10
15
Affordability
Clothing
Food
PF = 2 and PC = 3
Income = 30
What if we spend
some money of C

and some money on
F?
10
15
Affordability
Clothing
Food
PF = 2 and PC = 3
Income = 30
10
15
Affordability

Clothing
Food
PF = 2 and PC = 3
Income = 30
10
15
Affordability
Clothing
Food
PF = 2 and PC = 3
Income = 30
10
15

Slope = -
2
3
Quick Preview
Clothing
Food
C
B
A
Quick Preview
Clothing
Food
C

B
A
Which one will
make you the
happiest?
Quick Preview
Clothing
Food
C
B
A
Which one will
make you the

happiest?
Which
one will
you
choose?
• Budget constraint - Total spending must be no greater than
income:
PF*F + PC*C ≤ Income
In this case: 2*F + 3*C ≤ 30
• Budget set - The collection of affordable bundles:
All (F,C) such that PF*F + PC*C ≤ Income
• Budget line - The collection of bundles that exhaust a
consumer’s income:
All (F,C) such that PF*F + PC*C = Income
In this case: C = 10 – (2/3)F

Budgets
Budgets
In general, consider N goods in the economy (X1, X2, X3, …,
XN )
with prices (P1, P2, P3, …, PN) and consumer income (I)
• The budget constraint is:
P1 X1 + P2 X2 + P3 X3 + … + PN XN ≤ I
Σ Pi Xi ≤ I
• The budget set is all combos of X1, X2, X3, …, XN that
satisfy
the constraint
• The budget line is: Σ Pi Xi = I
i= 1
N

i= 1
N
Illustrating Budget Constraints
y
x
Finding the Intercept:
y-intercept Income divided by Py
x-intercept Income divided by Px
y
x

I
Py
I
Px
y
x
I
Py
I
Px

y
x
I
Py
I
Px
Finding the Slope:
slope -(Px divided by Py)
Px
Py
-

C
F
y-intercept 30 divided by 3
x-intercept 30 divided by 2
30
3
30
2
Finding the Slope:
slope -(2 divided by 3)
2
3
-

C
F
Income = 30
PC = 3
PF = 2
10
15
2
3
-
Income Changes
C

F
Income = 30
PC = 3
PF = 2
10
15
2
3
-
But what if
income
increases
to 45?
Income Changes
C

F
Income = 45
PC = 3
PF = 2
10
15
2
3
-
15
22.5
Income Changes
C

F
Income = 45
PC = 3
PF = 2
10
15
2
3
-
and if
income
had fallen
to 15…
15
22.5

Income Changes
y
x
Income = 15
PC = 3
PF = 2
10
15
2
3
-5
7.5
Income Changes

y
x
Income = 30
PC = 3
PF = 2
10
15
2
3
-
What if we
see a change
in the prices
of C and F?
Income Changes

C
F
Income = 30
PC = 3
PF = 2
10
15
2
3
-
What if we
see a change
in the prices
of C and F?
What if the
price of
clothing

increases to 5?
Price Changes
C
F
Income = 30
PC = 5
PF = 2
10
15
2
3
-
6

- 2
5
Price Changes
C
F
Income = 30
PC = 5
PF = 2
10
15
2
3
-
6

- 2
5
What if the
price of food
had fallen to
1.50 instead?
Price Changes
C
F
Income = 30
PC = 3
PF = 1.50
10
15
2
3

- - 1
2
20
Economic Interpretation
• Slope of the budget line:
– The rate at which the market is willing to trade y
for x. (negative slope)
– If you want another unit of x, you have to buy it
for Px. In order to get Px dollars, you need to sell
some of your y.
• To raise $1, you need to sell 1/ Py units of y.
• To raise Px dollars, you need to sell Px / Py units of y.
• Similar to MRS…

Kinked Budget Sets
• So far, prices have been constant per unit:
– The price of food was $2 for all quantities.
• In the “real world,” many prices vary with
quantity.
– Sam’s Club membership of $35
• First unit of food has price = 35 + PF
• Second unit of food has price = PF
– Cell phone service
– Quantity discounts
– Certain taxes
Example: Kinked Budget Sets
• You live in a school district with a good public school

and one good tutor, who provides an identical good of
quality education.
– The school provides 30 hours per week free.
– The tutor provides the first 5 hours per week at $10/hour
– The tutor provides time after 5 hours at $20/hour
• The only other product in the market is food (F), which
is sold at $1 per unit
• Your weekly income is $250
• You live in a school district with a good public school
and one good tutor, who provides an identical good of
quality education.
– The school provides 30 hours per week free.
– The tutor provides the first 5 hours per week at $10/hour
– The tutor provides time after 5 hours at $20/hour
• The only other product in the market is food (F), which
is sold at $1 per unit

• Your weekly income is $250 What does your
budget line look like?
Food
Education
Food
Education
Spend all
money on
food….

Food
Education
Spend all
money on
food….
250
Food
Education
Spend all
money on
education…
250

Food
Education
Spend all
money on
education…
250
45
Food
Education
250

4530 35
Food
Education
250
4530 35
What is the
slope of the BL
between 0 and
30 hours of
education?
Food

Education
250
4530 35
Slope = 0
Food
Education
250
4530 35
Slope = 0
What is the
slope of the BL
between 30 and

35 hours of
education?
Food
Education
250
4530 35
Slope = 0
Slope = -10
Food
Education

250
4530 35
Slope = 0
Slope = -10
What is the
slope of the BL
between 35 and
45 hours of
education?
Food
Education
250
4530 35

Slope = 0
Slope = -10
Slope = -20
Budgets
• Consumers’ choices are limited by what they can
afford
• We display all feasible bundles graphically with
budget lines/budget sets
• Main concepts:
– Finding the slope and intercepts of budget line
– Interpreting of slope of BC analogous to interpreting
the slope of IC
– Incorporating more complicated pricing schemes with
kinked budget lines.

Preview
• How consumers make decisions depends
fundamentally on the difference between:
1. How they are willing to trade good x for good y
2. How the market trades good x for good y
• An example from my MBA Environment
class…..
Economic Efficiency
• Bahamas on a budget -- $400
– 3 days in a tent
– Willing to pay $550
• Midrange trip -- $600
– 4 days in the airport Motel 6

• Luxury trip -- $850
– 5 days in beachside hotel
• Ultra Luxury Weeklong trip -- $1250
– 7 days in a deluxe waterfront cabana
Economic Efficiency

Net Benefits = $0
Economic Efficiency

Net Benefits = $0
Net Benefits = $250
Net Benefits = $300
Net Benefits = $150
Economic Efficiency

Net Benefits = $300
Net Benefits = $150
Economic Efficiency

Net Benefits = $300
Net Benefits = $150
You value this
trip $150 less
Economic Efficiency

Net Benefits = $300
Net Benefits = $150
You value this
trip $150 less
BUT YOU SAVE
$400
Economic Efficiency

Net Benefits = $0
Net Benefits = $150
Economic Efficiency

Net Benefits = $0
Net Benefits = $150
You value this
trip $200 less
BUT YOU SAVE
$250

Economic Efficiency
Net Benefits = $0
You value this
trip $350 less
BUT YOU SAVE
ONLY $200

Consumer Preferences
Intermediate Microeconomic Theory
ECN 312
From Last Class
• Price determines quantity demanded.
• Determinants of demand:
– Income
– Population
– Preferences
– Prices and availability of other goods
• Goal is to understand market demand. We’ll do
this piece by piece.

From Last Class
Population
Market Demand
From Last Class
Population
Market Demand

Modelling Consumer Choice
• We are going to describe the consumer’s
choice problem (and solution) using
mathematical tools
• We think that this is the best way to describe
behavior
• It also allow us to use 20,000 year of
knowledge in mathematical solutions (google
Ishango bone)
What would
happen if I asked
Ronnie about
Pythagoras’

Theorem?
Do you think that
Ronnie knows
anything about
the sides of a
triangle?
Consumers’ Choice Problem
• The solution to a consumer’s choice problem
is a bundle of goods that is preferred to all
other feasible (affordable) bundles.
• A bundle (or basket) of goods is a collection of
items that a person might consume.
– 2 Notre Dame t-shirt, 1 Nissan Leaf

– 1 slice of pizza, 1 Pepsi
– 1 slice of pizza today, 1 slice of pizza tomorrow
Assumptions on Bundles
• We make two key assumptions on bundles:
– Non-negativity: the components of a bundle
always have non-negative (≥ 0) quantity.
– Divisibility: The components of a bundle can take
on any non-negative value.
Consider two goods: x and y
Assumptions 1 & 2 imply:
y

x
Assumptions 1 & 2 imply:
y
x
Every (x,y) in this
quadrant is
possible
• We make three key assumptions on preferences:
– Completeness: Every pair of bundles can be ranked.

Either I prefer A to B, B to A, or I am indifferent.
– Transitivity: Rankings are rational. If I prefer A to B and
B to C, then I prefer A to C.
– Non-satiation: More is better than less.
Assumptions on Preferences
Assumption 3 implies:
y
x
A
3
2

?
y
x
A
3
2
Worse

y
x
A
3
2
Worse
?

y
x
A
3
2
Worse
Better
y
x

A
3
2
Worse
Better
?
?
Indifference Curves
• An indifference curve shows all combinations
of two goods that yield the same level of
satisfaction/happiness/utility to a person.
• Different people may have different looking
indifference curves for the same pair of goods.

• Indifference curves between different pairs of
goods will generally look different for the
same person.
Indifference Curves
y
x
Indifference curves are like
the lines on a topographical
map – a third dimension
projected onto a two
dimensional graph
Aside….

Indifference Curves
y
x
Indifference Curves
y
x
Indifference Curves
A
C
B
4
3

2
1 3 4
y
x
Indifference Curves
A
C
B
4
3
2

1 3 4
Which bundle is
preferred?
y
x
Indifference Curves
A
C
B
4
3
2

1 3 4
Which bundle is
preferred?
But A has the
most of y….
y
x
Indifference Curves
• Rules:
– Indifference curves cannot cross.
– Indifference curves slope downwards (for `goods’)
Indifference Curves

• Rules:
What about pollution,
disease, hunger,
homework?
Indifference Curves
• Rules:
disease, hunger,

homework?
pizza
1 hour doing homework
Indifference Curves
• Rules:
disease, hunger,
homework?
1 hour doing homework
pizza

Indifference Curves
• Rules:
disease, hunger,
homework?
1 hour free time
pizza
Marginal Rate of Substitution
• How much of good y would you give up for
another unit of good x?

• Mathematically, MRS = (-1)(slope of I.C.)
Since our indifference
curves have a negative
slope, our MRS will be
positive.
y
x
MRS
1

y
x
MRS
1
MRS = (-1)(slope)
≈ (-1)(rise/run)
• Diminishing marginal rate of substitution
• As we move down an indifference curve, MRS
typically falls.
– Averages are better than extremes.
– Indifference curves are usually convex.

1 1
MRSA
MRSB
A
B
y
x
1 1
MRSA

MRSB
A
B
A: More y than x
B: More x than y
MRSA > MRSB
y
x
• Why can’t indifference curves cross?
y
x

• Why can’t indifference curves cross?
y
x
• Lets look at two consumers with different tastes
y
x
ICAnn
ICBob

• Lets look at two consumers with different tastes
y
x
ICAnn
ICBob
Who has stronger
preferences for X
(relative to Y)?
Indifference Curves
• Extreme Case #1 – Perfect Substitutes
– No balance necessary. Extremes are as good as
averages.
– Constant MRS, ICs have constant slope

– Examples:
Indifference Curves
• Extreme Case #1 – Perfect Substitutes
– No balance necessary. Extremes are as good as
averages.
– Constant MRS, ICs have constant slope
– Examples:
• Coke and Pepsi
• Blue pens and black pens
• Margarine and butter
• Equal and Sweet’n Low

Perfect Substitutes
Blue
Pens
Black Pens
6
4
2
4 62
Perfect Substitutes
Blue
Pens
Black Pens
6
4

2
4 62
You are indifferent
between Blue and
Black Pens at all
quantities….
Indifference Curves
• Extreme Case #2 – Perfect Complements
– These goods are used in fixed proportions.
– MRS is very sensitive to proportion of goods.
– ICs are kinked.
– Examples:

Indifference Curves
• Extreme Case #2 – Perfect Complements
– These goods are used in fixed proportions.
– MRS is very sensitive to proportion of goods.
– ICs are kinked.
– Examples:
• Left shoes and Right shoes
• Peanut butter and Jelly
• Apple pie and Ice cream
• Coffee and Milk (coffee and cream, coffee and sugar)
Perfect Complements

Coffee
6
4
2
Milk1 2 3
I like 2 parts coffee
to 1 part milk
Perfect Complements
Left Shoes
Right Shoes1 2 3
3
2
1

Example – Auto Prices
• See “A Different Beat: Toyota Raises Prices
While Detroit Cuts Deeply” New York Times
• In the summer of 2005:
– US firms introduced “employee discounts” for all
– GM introduced first, other US firms followed
– Toyota (and Honda) did not
• (One) Economic question: Why does Ford
care about GM’s pricing while Toyota does
not?
GM
Ford

GM
Ford
GM and Ford may not be
perfect substitutes, but
they’re pretty substitutable
(relatively)
GM
Toyota
GM

Toyota
GM and Toyota are relatively
less substitutable
Utility
• We can do all our analysis of consumers’
choices with indifference curves (graphically)
• We can also express consumers’ tastes with a
utility function (mathematically)
• Utility function: a rule for translating bundles
into a numerical value for “happiness”
Utility
• Utility functions allow us to state consumer
choice as an optimization problem and use

calculus.
• Rules:
– A utility function U associates a total utility number
with each possible bundle. Example: U(x,y) = 20
– All bundles on an IC have the same level of utility.
– Preferred ICs have higher levels of utility.
Utility
• Each indifference curve is a level set of a
utility function
• A level set is set of input values such that the
function takes on some constant value
• When there are two inputs, a level set may
also be called a level curve, a contour line, or
an isocurve

Utility
y
x
U(x,y) = 30
U(x,y) = 20
U(x,y) = 10
Utility
• Suppose the only products in the economy are
food (F) and clothing (C). Possible utility functions
are:
– U(F,C) = 2F + C
– U(F,C) = 3F½C¼

• What is the difference between these utility
functions?
Utility
• Suppose the only products in the economy are
food (F) and clothing (C). Possible utility functions
are:
– U(F,C) = 2F + C
– U(F,C) = 3F½C¼
• What is the difference between these utility
functions?
– They imply different preferences over food and clothing.
Utility
Important:

• Numerical values of utility levels are unimportant.
All that matters is that better ICs have higher
levels of U.
• Utility functions provide ordinal information, not
cardinal.
Utility
• Bundle A is (F,C) = (2, 3). B is (F,C) = (3, 2)
– If U = 2F + C, then UA = 7 and UB = 8
– If U = 4F + 2C + 1, then UA = 15 and UB = 17
• Both versions of the utility function rank A and B
(and all other bundles!) in the same order.
• Both imply: Choose B
Utility

Cobb-Douglas utility functions
• We will see these frequently because of their
convenient properties.
Example: Food and clothing, with U = FC.
What are the utility levels for the following bundles?
(0,0)
Utility

(0,0) U = 0
Utility
.
(0,2)

Utility
• These have the form U(X,Y) = a
(0,2) U = 0
Utility
•

(1,1)
Utility
(1,1) U = 1

Utility
(1,2)
Utility

(1,2) U = 2
Cobb-Douglas Utility
U = FC
Food
Clothing
U = 30
U = 20
U = 10

U = FC
Food
Clothing
But what if U = 2FC?
U = 2FC
Food
Clothing
But what if U = 2FC?
Essentially nothing…

U = 60
U = 40
U = 20
U = FC
Food
Clothing
But what if U = F2C?
U = 20
U = 10

U = FC
Food
Clothing
But what if U = F2C?
Indifference curves
get steeper.
U = 30
U = 20
U = 10
Utility
Perfect substitutes
• These have the form U(X,Y) = aX + bY.

Example: Paper towels and napkins, with U = P + N
(0,0)
Utility
Perfect substitutes
(0,0) U = 0
Utility
Perfect substitutes

(1,3)
Utility
Perfect substitutes
(1,3) U = 4

Utility
Perfect substitutes
(3,1)
Utility
Perfect substitutes
(3,1) U = 4

Perfect SubstitutesNapkins
3
1
Paper Towels1 3
U = 4
U = 7
U = N + P
3
1

Paper Towels1 3
U = 4
U = 7
But what happens if
U = 2N + 2P?
U = N + P
3
1
Paper Towels1 3
U = 8
U = 14

U = 2N + 2P
3
1
Paper Towels1 3
U = 8
U = 14
U = 2N + 2P
But what
happens if
U = 2N + P?

3
1
Paper Towels1 3
U = 5
U = 7
U = 2N + P
Utility
Perfect complements
• These have the form U(X,Y) = c∙min(aX, bY)
Example: Peanut butter and jelly, with U = min(P,J)

(3,1)
Utility
Perfect complements
(3,1) U = 1
Utility
Perfect complements

(2,4)
Utility
Perfect complements
(2,4) U = 2
Perfect Complements
J
P1 2 3

3
2
1
U = min(P,J)
U = 3
U = 2
U = 1
Perfect Complements
J
P1 2 3
3
2

1
U = min(P,J)
But what
happens if
U = 2*min(P,J)
or
U = min(2P,2J) ?
U = 3
U = 2
U = 1
Perfect Complements
J
P1 2 3

3
2
1
U = 2*min(P,J)
U = 6
U = 4
U = 2
Perfect Complements
J
P1 2 3
3
2

1
U = 2*min(P,J)
U = 6
U = 4
U = 2
We saw
already what
happens if
U = min(P,2J)
Marginal Utility
• What is the additional benefit of one more X,
conditional on your current bundle?
• What is the utility gain from a small change in X,

while the amount of Y is held fixed?
(for well-behaved preferences)
– U always increases with X (U never decreases)
– Increase in U is greatest when quantity of X is small
Marginal Utility
U(X,YA)
U
X0 1 7 8
Total Utility Curve
Marginal Utility
U(X,YA)

U
X0 1 7 8
U(X,YB)
Marginal Utility
• Change in utility from a small change in
quantity of one good in a consumption
bundle.
• MU of X is the derivative of U with respect to
X, treating Y as a constant.
U = X + Y
Marginal Utility

bundle.
U = X + Y MUX = 1
Marginal Utility
bundle.
U = 4X + Y
Marginal Utility

bundle.
U = 4X + Y MUX = 4
Marginal Utility
bundle.
U = 4X*Y
Marginal Utility

bundle.
U = 4X*Y MUX = 4Y
Marginal Utility
• Importance of MU to microeconomics
– main goal is to understand decision-making
– When should you decide to do anything???
Marginal Utility and ICs
We can define MUY in the same way we defined MUX

Y
X
Y
X
MRSXY =
MUX
MUY

Y
X
MRSXY =
MUX
MUY
∙
∙
∙ ∙
1 1
MRSA
MRSB
A
B

Y
X
MRSXY =
MUX
MUY
∙
∙
∙ ∙
1 1
MRSA
MRSB
A

B
Why is
MRSA
>MRSB?
• MRSXY = MUX/MUY
– Cobb Douglas: U = XaYb MRS =
– Perfect Substitutes: U = aX + bY MRS =
– Perfect Complements: U = min(aX,bY) MRS = 0 or ∞
aY
bX
a
b

Problem Set 2 - due 1/26/2017
ECN 312 - Intermediate Microeconomic Theory
Spring 2017
Please make sure that your problem set is legible and stapled.
Please write your name
and section number on your problem set. On your graphs, please
label axes, curves,
intercepts, and all points relevant to the question. Explain your
work thoroughly in
order to receive full credit. In particular, “True or False?
Explain” questions must
be accompanied by an explanation to receive any credit.
Possible points are given in
parentheses.
1. (20) Consider the following three schedules, which specify
Leslie’s preferences over two goods,
earrings (E) and necklaces (N).

Utility = 100
E N
10 6
11 5
12 4
Utility = 101
E N
10 7
11 6
12 5
Utility = 103
E N
10 8
11 7
12 6
True or False? Explain.
(a) (7) The marginal utility of necklaces diminishes as N
increases, holding E constant.
(b) (7) The marginal utility of earrings increases as E increases,

holding N constant.
(c) (6) These preferences give rise to indifference curves that
satisfy the convexity property.
2. (20) Consider the following utility function, which specifies
Giulia’s preferences over two goods,
x-ray goggles (x) and yellow sweaters (y). (Please note that this
is an example of a quasi-linear
utility function.)
U(x, y) = 14
√
x + 7y
(a) (5) Plot an indifference curve map for this utility function
(please put x on the horizontal
axis and y on the vertical one). In order to do so, follow the
following instructions:
i. Consider some fixed level of utility, U0. Write and expression
for y as a function x and
U0.
ii. Consider a particular value for U0 (in this case, choose U0 =

35) and calculate the value
of y for x = 0, 1, 2, 3, 4, and 5.
iii. Plotting these values will yield the U0 = 35 indifference
curve for this utility function.
iv. Repeat this process for U1 = 42.
(b) (5) What is the marginal utility of x? What is the marginal
utility of y?
(c) (5) What is the slope of the U0 indifference curve at x = 4 ?
Interpret this number in terms
of Giulia’s willingness to trade y for x.
(d) (5) What happens to the slope of the indifference curve as
we move from U0 to U1 at x = 4?
3. (10) Consider Cruz who has preferences over hours playing
video games (V) and hours playing
board games (B). Putting V on the horizontal axis and B on the
vertical axis, draw at least three
indifference curves describing Cruz’s preferences in the

following scenarios.
(a) (5) Cruz’s level of utility today is given by the activity he
spends highest number of hours
on today.
(b) (5) Cruz’s level of utility today is given by the activity he
spend the lowest number of hours
on today.
2

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