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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
Preferences Budget (Income)
Choice (Individual Demand)
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
Decision Problem - Graphically
Clothing
Food
PC = 3
PF = 2
m = 30
10
15
A B
C
D
E
What is wrong
with bundle A?
Decision Problem - Graphically
Clothing
Food
PC = 3
PF = 2
m = 30
10
15
A B
C
D
E
What is wrong
with bundle B?
Decision Problem - Graphically
Clothing
Food
PC = 3
PF = 2
m = 30
10
15
A B
C
D
E
What is wrong
with bundle C?
Decision Problem - Graphically
Clothing
Food
PC = 3
PF = 2
m = 30
10
15
A B
C
D
E
What is wrong
with bundle D?
Decision Problem - Graphically
Clothing
Food
PC = 3
PF = 2
m = 30
10
15
A B
C
D
E
What is wrong
with bundle E?
Decision Problem - Graphically
Clothing
Food
PC = 3
PF = 2
m = 30
10
15
E
What is true
about bundle E?
Decision Problem - Graphically
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
Decision Problem Analytically
• 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
Decision Problem Analytically
• 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
Decision Problem Analytically
• 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
Decision Problem Analytically
The formal way to solve constrained
optimization problems is as a Lagrangian
Decision Problem Analytically
The formal way to solve constrained
optimization problems is as a Lagrangian
L = (objective function) – λ(constraint function)
Decision Problem Analytically
The formal way to solve constrained
optimization problems is with a Lagrangian
L = (objective function) – λ(constraint function)
L = U(x,y) – λ(Px x + Py y - m)
Decision Problem Analytically
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
Decision Problem Analytically
L = U(x,y) – λ(Px x + Py y - m)
To solve, you three first-order conditions:
1.
�ℒ
��
=
��
��
− ��� ���
2.
�ℒ
��
=
��
��
− ��� ���
3.
�ℒ
�λ
= ��� + ��� − � ��� + ��� = �
Decision Problem Analytically
L = U(x,y) – λ(Px x + Py y - m)
To solve, you three first-order conditions:
1.
�ℒ
��
=
��
��
− ��� ���
2.
�ℒ
��
=
��
��
− ��� ���
3.
�ℒ
�λ
= ��� + ��� − � ��� + ��� = �
���
���
=
��
��
Decision Problem Analytically
L = U(x,y) – λ(Px x + Py y - m)
To solve, you three first-order conditions:
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???
Decision Problem - Graphically
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
Consider xylophones and yo-yos with:
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
Consider xylophones and yo-yos with:
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
Consider xylophones and yo-yos with:
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?
Example: Gas Taxes or Standards
• 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
Example: Gas Taxes or Standards
• 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?
Example: Gas Taxes or Standards
C
T
Example: Gas Taxes or Standards
C
T
What does a gas tax do?
Example: Gas Taxes or Standards
C
T
What does a gas tax do?
Example: Gas Taxes or Standards
C
T
What does a gas tax do?
What has
happened to
gas
consumption?
Example: Gas Taxes or Standards
C
T
What does a mileage
standard (increase fuel
efficiency) do?
Example: Gas Taxes or Standards
C
T
What does a mileage
standard (increase fuel
efficiency) do?
Example: Gas Taxes or Standards
C
T
What does a mileage
standard (increase fuel
efficiency) do?
What has
happened to
gas
consumption?
Example: Gas Taxes or Standards
• 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
Preferences Budget (Income)
Choice (Individual Demand)
Population
Market Demand
From Last Class
Preferences Budget (Income)
Choice (Individual Demand)
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
Illustrating Budget Constraints
y
x
Finding the Intercept:
y-intercept Income divided by Py
x-intercept Income divided by Px
I
Py
I
Px
Illustrating Budget Constraints
y
x
Finding the Intercept:
y-intercept Income divided by Py
x-intercept Income divided by Px
I
Py
I
Px
Illustrating Budget Constraints
y
x
Finding the Intercept:
y-intercept Income divided by Py
x-intercept Income divided by Px
I
Py
I
Px
Finding the Slope:
slope -(Px divided by Py)
Px
Py
-
Illustrating Budget Constraints
C
F
Finding the Intercept:
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
-
Illustrating Budget Constraints
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
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 What does your
budget line look like?
Example: Kinked Budget Sets
Food
Education
Example: Kinked Budget Sets
Food
Education
Spend all
money on
food….
Example: Kinked Budget Sets
Food
Education
Spend all
money on
food….
250
Example: Kinked Budget Sets
Food
Education
Spend all
money on
education…
250
Example: Kinked Budget Sets
Food
Education
Spend all
money on
education…
250
45
Example: Kinked Budget Sets
Food
Education
250
4530 35
Example: Kinked Budget Sets
Food
Education
250
4530 35
What is the
slope of the BL
between 0 and
30 hours of
education?
Example: Kinked Budget Sets
Food
Education
250
4530 35
Slope = 0
Example: Kinked Budget Sets
Food
Education
250
4530 35
Slope = 0
What is the
slope of the BL
between 30 and
35 hours of
education?
Example: Kinked Budget Sets
Food
Education
250
4530 35
Slope = 0
Slope = -10
Example: Kinked Budget Sets
Food
Education
250
4530 35
Slope = 0
Slope = -10
What is the
slope of the BL
between 35 and
45 hours of
education?
Example: Kinked Budget Sets
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
– Willing to pay $900
• Luxury trip -- $850
– 5 days in beachside hotel
– Willing to pay $1100
• Ultra Luxury Weeklong trip -- $1250
– 7 days in a deluxe waterfront cabana
– Willing to pay $1250
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
– Willing to pay $900
• Luxury trip -- $850
– 5 days in beachside hotel
– Willing to pay $1100
• Ultra Luxury Weeklong trip -- $1250
– 7 days in a deluxe waterfront cabana
– Willing to pay $1250
Net Benefits = $0
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
– Willing to pay $900
• Luxury trip -- $850
– 5 days in beachside hotel
– Willing to pay $1100
• Ultra Luxury Weeklong trip -- $1250
– 7 days in a deluxe waterfront cabana
– Willing to pay $1250
Net Benefits = $0
Net Benefits = $250
Net Benefits = $300
Net Benefits = $150
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
– Willing to pay $900
• Luxury trip -- $850
– 5 days in beachside hotel
– Willing to pay $1100
• Ultra Luxury Weeklong trip -- $1250
– 7 days in a deluxe waterfront cabana
– Willing to pay $1250
Net Benefits = $0
Net Benefits = $250
Net Benefits = $300
Net Benefits = $150
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
– Willing to pay $900
• Luxury trip -- $850
– 5 days in beachside hotel
– Willing to pay $1100
• Ultra Luxury Weeklong trip -- $1250
– 7 days in a deluxe waterfront cabana
– Willing to pay $1250
Net Benefits = $300
Net Benefits = $150
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
– Willing to pay $900
• Luxury trip -- $850
– 5 days in beachside hotel
– Willing to pay $1100
• Ultra Luxury Weeklong trip -- $1250
– 7 days in a deluxe waterfront cabana
– Willing to pay $1250
Net Benefits = $300
Net Benefits = $150
You value this
trip $150 less
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
– Willing to pay $900
• Luxury trip -- $850
– 5 days in beachside hotel
– Willing to pay $1100
• Ultra Luxury Weeklong trip -- $1250
– 7 days in a deluxe waterfront cabana
– Willing to pay $1250
Net Benefits = $300
Net Benefits = $150
You value this
trip $150 less
BUT YOU SAVE
$400
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
– Willing to pay $900
• Luxury trip -- $850
– 5 days in beachside hotel
– Willing to pay $1100
• Ultra Luxury Weeklong trip -- $1250
– 7 days in a deluxe waterfront cabana
– Willing to pay $1250
Net Benefits = $0
Net Benefits = $150
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
– Willing to pay $900
• Luxury trip -- $850
– 5 days in beachside hotel
– Willing to pay $1100
• Ultra Luxury Weeklong trip -- $1250
– 7 days in a deluxe waterfront cabana
– Willing to pay $1250
Net Benefits = $0
Net Benefits = $150
You value this
trip $200 less
BUT YOU SAVE
$250
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
– Willing to pay $900
• Luxury trip -- $850
– 5 days in beachside hotel
– Willing to pay $1100
• Ultra Luxury Weeklong trip -- $1250
– 7 days in a deluxe waterfront cabana
– Willing to pay $1250
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
Preferences Budget (Income)
Choice (Individual Demand)
Population
Market Demand
From Last Class
Preferences Budget (Income)
Choice (Individual Demand)
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)
Modelling Consumer Choice
Modelling Consumer Choice
What would
happen if I asked
Ronnie about
Pythagoras’
Theorem?
Modelling Consumer Choice
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.
Assumptions on Bundles
Consider two goods: x and y
Assumptions 1 & 2 imply:
y
x
Assumptions on Bundles
Consider two goods: x and y
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
Assumptions on Preferences
Consider two goods: x and y
Assumption 3 implies:
y
x
A
3
2
?
Assumptions on Preferences
Consider two goods: x and y
Assumption 3 implies:
y
x
A
3
2
Worse
Assumptions on Preferences
Consider two goods: x and y
Assumption 3 implies:
y
x
A
3
2
Worse
?
Assumptions on Preferences
Consider two goods: x and y
Assumption 3 implies:
y
x
A
3
2
Worse
Better
Assumptions on Preferences
Consider two goods: x and y
Assumption 3 implies:
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:
– Indifference curves cannot cross.
– Indifference curves slope downwards (for `goods’)
What about pollution,
disease, hunger,
homework?
Indifference Curves
• Rules:
– Indifference curves cannot cross.
– Indifference curves slope downwards (for `goods’)
What about pollution,
disease, hunger,
homework?
pizza
1 hour doing homework
Indifference Curves
• Rules:
– Indifference curves cannot cross.
– Indifference curves slope downwards (for `goods’)
What about pollution,
disease, hunger,
homework?
1 hour doing homework
pizza
Indifference Curves
• Rules:
– Indifference curves cannot cross.
– Indifference curves slope downwards (for `goods’)
What about pollution,
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.
Marginal Rate of Substitution
y
x
MRS
1
Marginal Rate of Substitution
y
x
MRS
1
MRS = (-1)(slope)
≈ (-1)(rise/run)
Marginal Rate of Substitution
• 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.
Marginal Rate of Substitution
1 1
MRSA
MRSB
A
B
y
x
Marginal Rate of Substitution
1 1
MRSA
MRSB
A
B
A: More y than x
B: More x than y
MRSA > MRSB
y
x
Marginal Rate of Substitution
• Why can’t indifference curves cross?
y
x
Marginal Rate of Substitution
• Why can’t indifference curves cross?
y
x
Marginal Rate of Substitution
• Lets look at two consumers with different tastes
y
x
ICAnn
ICBob
Marginal Rate of Substitution
• 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?
Example – Auto Prices
GM
Ford
Example – Auto Prices
GM
Ford
GM and Ford may not be
perfect substitutes, but
they’re pretty substitutable
(relatively)
Example – Auto Prices
GM
Toyota
Example – Auto Prices
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
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) U = 0
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?
(0,2)
Utility
Cobb-Douglas utility functions
• These have the form U(X,Y) = a
• 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?
(0,2) U = 0
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?
(1,1)
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?
(1,1) U = 1
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?
(1,2)
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?
(1,2) U = 2
Cobb-Douglas Utility
U = FC
Food
Clothing
U = 30
U = 20
U = 10
Cobb-Douglas Utility
U = FC
Food
Clothing
But what if U = 2FC?
Cobb-Douglas Utility
U = 2FC
Food
Clothing
But what if U = 2FC?
Essentially nothing…
U = 60
U = 40
U = 20
Cobb-Douglas Utility
U = FC
Food
Clothing
But what if U = F2C?
U = 20
U = 10
Cobb-Douglas Utility
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
What are the utility levels for the following bundles?
(0,0)
Utility
Perfect substitutes
• These have the form U(X,Y) = aX + bY.
Example: Paper towels and napkins, with U = P + N
What are the utility levels for the following bundles?
(0,0) U = 0
Utility
Perfect substitutes
• These have the form U(X,Y) = aX + bY.
Example: Paper towels and napkins, with U = P + N
What are the utility levels for the following bundles?
(1,3)
Utility
Perfect substitutes
• These have the form U(X,Y) = aX + bY.
Example: Paper towels and napkins, with U = P + N
What are the utility levels for the following bundles?
(1,3) U = 4
Utility
Perfect substitutes
• These have the form U(X,Y) = aX + bY.
Example: Paper towels and napkins, with U = P + N
What are the utility levels for the following bundles?
(3,1)
Utility
Perfect substitutes
• These have the form U(X,Y) = aX + bY.
Example: Paper towels and napkins, with U = P + N
What are the utility levels for the following bundles?
(3,1) U = 4
Perfect SubstitutesNapkins
3
1
Paper Towels1 3
U = 4
U = 7
U = N + P
Perfect SubstitutesNapkins
3
1
Paper Towels1 3
U = 4
U = 7
But what happens if
U = 2N + 2P?
U = N + P
Perfect SubstitutesNapkins
3
1
Paper Towels1 3
U = 8
U = 14
U = 2N + 2P
Perfect SubstitutesNapkins
3
1
Paper Towels1 3
U = 8
U = 14
U = 2N + 2P
But what
happens if
U = 2N + P?
Perfect SubstitutesNapkins
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)
What are the utility levels for the following bundles?
(3,1)
Utility
Perfect complements
• These have the form U(X,Y) = c∙min(aX, bY)
Example: Peanut butter and jelly, with U = min(P,J)
What are the utility levels for the following bundles?
(3,1) U = 1
Utility
Perfect complements
• These have the form U(X,Y) = c∙min(aX, bY)
Example: Peanut butter and jelly, with U = min(P,J)
What are the utility levels for the following bundles?
(2,4)
Utility
Perfect complements
• These have the form U(X,Y) = c∙min(aX, bY)
Example: Peanut butter and jelly, with U = min(P,J)
What are the utility levels for the following bundles?
(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
• 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 MUX = 1
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 = 4X + Y
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 = 4X + Y MUX = 4
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 = 4X*Y
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 = 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
Marginal Utility and ICs
We can define MUY in the same way we defined MUX
Y
X
Marginal Utility and ICs
We can define MUY in the same way we defined MUX
Y
X
MRSXY =
MUX
MUY
Marginal Utility and ICs
We can define MUY in the same way we defined MUX
Y
X
MRSXY =
MUX
MUY
∙
∙
∙ ∙
1 1
MRSA
MRSB
A
B
Marginal Utility and ICs
We can define MUY in the same way we defined MUX
Y
X
MRSXY =
MUX
MUY
∙
∙
∙ ∙
1 1
MRSA
MRSB
A
B
Why is
MRSA
>MRSB?
Marginal Rate of Substitution
• 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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DTINGLEYWarning.docx

  • 2. Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 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)
  • 3. Choice (Individual Demand) Population Market Demand From Last Class Preferences Budget (Income) Choice (Individual Demand) Population Market Demand Quick Preview y x A B C Quick Preview y
  • 4. 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
  • 5. 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
  • 6. Decision Problem Decision Problem - Graphically Clothing Food PC = 3 PF = 2 m = 30 10 15 A B C D E Decision Problem - Graphically Clothing Food PC = 3 PF = 2 m = 30
  • 7. 10 15 A B C D E What is wrong with bundle A? Decision Problem - Graphically Clothing Food PC = 3 PF = 2 m = 30 10 15 A B C D E
  • 8. What is wrong with bundle B? Decision Problem - Graphically Clothing Food PC = 3 PF = 2 m = 30 10 15 A B C D E What is wrong with bundle C? Decision Problem - Graphically Clothing Food PC = 3
  • 9. PF = 2 m = 30 10 15 A B C D E What is wrong with bundle D? Decision Problem - Graphically Clothing Food PC = 3 PF = 2 m = 30 10 15 A B C D
  • 10. E What is wrong with bundle E? Decision Problem - Graphically Clothing Food PC = 3 PF = 2 m = 30 10 15 E What is true about bundle E? Decision Problem - Graphically Clothing Food PC = 3 PF = 2 m = 30
  • 11. 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
  • 12. Px Py Decision Problem Analytically • 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
  • 13. C F 2 3 2 3 Decision Problem Analytically • 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
  • 14. 2 3 Decision Problem Analytically • 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 Decision Problem Analytically The formal way to solve constrained optimization problems is as a Lagrangian
  • 15. Decision Problem Analytically The formal way to solve constrained optimization problems is as a Lagrangian L = (objective function) – λ(constraint function) Decision Problem Analytically The formal way to solve constrained optimization problems is with a Lagrangian L = (objective function) – λ(constraint function) L = U(x,y) – λ(Px x + Py y - m) Decision Problem Analytically 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
  • 16. �� = 0 3. With respect to L: �ℒ �λ = 0 Decision Problem Analytically L = U(x,y) – λ(Px x + Py y - m) To solve, you three first-order conditions: 1. �ℒ �� = �� �� − ��� ��� 2. �ℒ �� = �� �� − ��� ���
  • 17. 3. �ℒ �λ = ��� + ��� − � ��� + ��� = � Decision Problem Analytically L = U(x,y) – λ(Px x + Py y - m) To solve, you three first-order conditions: 1. �ℒ �� = �� �� − ��� ��� 2. �ℒ �� = �� �� − ��� ��� 3.
  • 18. �ℒ �λ = ��� + ��� − � ��� + ��� = � ��� ��� = �� �� Decision Problem Analytically L = U(x,y) – λ(Px x + Py y - m) To solve, you three first-order conditions: 1. �ℒ �� = �� �� − ��� ��� 2. �ℒ �� = ��
  • 19. �� − ��� ��� 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??? Decision Problem - Graphically Clothing
  • 20. 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
  • 21. • 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
  • 22. 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
  • 23. 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
  • 24. 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
  • 25. 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 $$$.
  • 26. • 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
  • 27. 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
  • 28. 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
  • 29. m = 10 3.33 A B C And if the price of napkins fell to 10 cents??? Perfect Complements y x
  • 30. A B C Perfect Complements Jelly Peanut Butter U = min{PB,J} Price of PB = 1 Price of J = 1 m = 1010 105 5
  • 31. 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?
  • 32. 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
  • 33. 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 Consider xylophones and yo-yos with: u(x,y) = 3x1/2y1/4 Px = 4, Py = 3, m = 12 • MRSxy= MUx/MUy • MUx= (1/2)3 x
  • 34. -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 Consider xylophones and yo-yos with: 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
  • 35. Cobb-Douglas – Plug into budget Consider xylophones and yo-yos with: 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
  • 36. 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? Example: Gas Taxes or Standards • 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)
  • 37. • Each product has a price and consumers have some (limited) budget Example: Gas Taxes or Standards • 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? Example: Gas Taxes or Standards C T
  • 38. Example: Gas Taxes or Standards C T What does a gas tax do? Example: Gas Taxes or Standards C T What does a gas tax do? Example: Gas Taxes or Standards C T
  • 39. What does a gas tax do? What has happened to gas consumption? Example: Gas Taxes or Standards C T What does a mileage standard (increase fuel efficiency) do? Example: Gas Taxes or Standards C
  • 40. T What does a mileage standard (increase fuel efficiency) do? Example: Gas Taxes or Standards C T What does a mileage standard (increase fuel efficiency) do? What has happened to gas consumption?
  • 41. Example: Gas Taxes or Standards • 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
  • 42. – 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
  • 43. decide? • Lines up to readings for Chapters 2 and 4 From Last Class Preferences Budget (Income) Choice (Individual Demand) Population Market Demand From Last Class Preferences Budget (Income) Choice (Individual Demand) Population
  • 44. 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
  • 45. 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
  • 46. 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
  • 47. 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
  • 48. 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
  • 49. and some money on F? 10 15 Affordability Clothing Food PF = 2 and PC = 3 Income = 30 10 15 Affordability
  • 50. Clothing Food PF = 2 and PC = 3 Income = 30 10 15 Affordability Clothing Food PF = 2 and PC = 3 Income = 30 10 15
  • 51. Slope = - 2 3 Quick Preview Clothing Food C B A Quick Preview Clothing Food C
  • 52. B A Which one will make you the happiest? Quick Preview Clothing Food C B A Which one will make you the
  • 53. 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
  • 54. 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
  • 55. i= 1 N Illustrating Budget Constraints y x Finding the Intercept: y-intercept Income divided by Py x-intercept Income divided by Px Illustrating Budget Constraints y x Finding the Intercept: y-intercept Income divided by Py
  • 56. x-intercept Income divided by Px I Py I Px Illustrating Budget Constraints y x Finding the Intercept: y-intercept Income divided by Py x-intercept Income divided by Px I Py I Px
  • 57. Illustrating Budget Constraints y x Finding the Intercept: y-intercept Income divided by Py x-intercept Income divided by Px I Py I Px Finding the Slope: slope -(Px divided by Py) Px Py -
  • 58. Illustrating Budget Constraints C F Finding the Intercept: 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 -
  • 59. Illustrating Budget Constraints C F Income = 30 PC = 3 PF = 2 10 15 2 3 - Income Changes C
  • 60. F Income = 30 PC = 3 PF = 2 10 15 2 3 - But what if income increases to 45? Income Changes C
  • 61. F Income = 45 PC = 3 PF = 2 10 15 2 3 - 15 22.5 Income Changes C
  • 62. F Income = 45 PC = 3 PF = 2 10 15 2 3 - and if income had fallen to 15… 15 22.5
  • 63. Income Changes y x Income = 15 PC = 3 PF = 2 10 15 2 3 -5 7.5 Income Changes
  • 64. 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
  • 65. 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
  • 66. increases to 5? Price Changes C F Income = 30 PC = 5 PF = 2 10 15 2 3 - 6
  • 67. - 2 5 Price Changes C F Income = 30 PC = 5 PF = 2 10 15 2 3 - 6
  • 68. - 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
  • 69. - - 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…
  • 70. 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
  • 71. 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 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
  • 72. • Your weekly income is $250 What does your budget line look like? Example: Kinked Budget Sets Food Education Example: Kinked Budget Sets Food Education Spend all money on food…. Example: Kinked Budget Sets
  • 73. Food Education Spend all money on food…. 250 Example: Kinked Budget Sets Food Education Spend all money on education… 250
  • 74. Example: Kinked Budget Sets Food Education Spend all money on education… 250 45 Example: Kinked Budget Sets Food Education 250
  • 75. 4530 35 Example: Kinked Budget Sets Food Education 250 4530 35 What is the slope of the BL between 0 and 30 hours of education? Example: Kinked Budget Sets Food
  • 76. Education 250 4530 35 Slope = 0 Example: Kinked Budget Sets Food Education 250 4530 35 Slope = 0 What is the slope of the BL between 30 and
  • 77. 35 hours of education? Example: Kinked Budget Sets Food Education 250 4530 35 Slope = 0 Slope = -10 Example: Kinked Budget Sets Food Education
  • 78. 250 4530 35 Slope = 0 Slope = -10 What is the slope of the BL between 35 and 45 hours of education? Example: Kinked Budget Sets Food Education 250 4530 35
  • 79. 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.
  • 80. 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
  • 81. – Willing to pay $900 • Luxury trip -- $850 – 5 days in beachside hotel – Willing to pay $1100 • Ultra Luxury Weeklong trip -- $1250 – 7 days in a deluxe waterfront cabana – Willing to pay $1250 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 – Willing to pay $900 • Luxury trip -- $850 – 5 days in beachside hotel – Willing to pay $1100
  • 82. • Ultra Luxury Weeklong trip -- $1250 – 7 days in a deluxe waterfront cabana – Willing to pay $1250 Net Benefits = $0 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 – Willing to pay $900 • Luxury trip -- $850 – 5 days in beachside hotel – Willing to pay $1100 • Ultra Luxury Weeklong trip -- $1250 – 7 days in a deluxe waterfront cabana
  • 83. – Willing to pay $1250 Net Benefits = $0 Net Benefits = $250 Net Benefits = $300 Net Benefits = $150 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 – Willing to pay $900 • Luxury trip -- $850 – 5 days in beachside hotel – Willing to pay $1100
  • 84. • Ultra Luxury Weeklong trip -- $1250 – 7 days in a deluxe waterfront cabana – Willing to pay $1250 Net Benefits = $0 Net Benefits = $250 Net Benefits = $300 Net Benefits = $150 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 – Willing to pay $900
  • 85. • Luxury trip -- $850 – 5 days in beachside hotel – Willing to pay $1100 • Ultra Luxury Weeklong trip -- $1250 – 7 days in a deluxe waterfront cabana – Willing to pay $1250 Net Benefits = $300 Net Benefits = $150 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 – Willing to pay $900 • Luxury trip -- $850
  • 86. – 5 days in beachside hotel – Willing to pay $1100 • Ultra Luxury Weeklong trip -- $1250 – 7 days in a deluxe waterfront cabana – Willing to pay $1250 Net Benefits = $300 Net Benefits = $150 You value this trip $150 less 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 – Willing to pay $900
  • 87. • Luxury trip -- $850 – 5 days in beachside hotel – Willing to pay $1100 • Ultra Luxury Weeklong trip -- $1250 – 7 days in a deluxe waterfront cabana – Willing to pay $1250 Net Benefits = $300 Net Benefits = $150 You value this trip $150 less BUT YOU SAVE $400 Economic Efficiency • Bahamas on a budget -- $400 – 3 days in a tent
  • 88. – Willing to pay $550 • Midrange trip -- $600 – 4 days in the airport Motel 6 – Willing to pay $900 • Luxury trip -- $850 – 5 days in beachside hotel – Willing to pay $1100 • Ultra Luxury Weeklong trip -- $1250 – 7 days in a deluxe waterfront cabana – Willing to pay $1250 Net Benefits = $0 Net Benefits = $150 Economic Efficiency • Bahamas on a budget -- $400 – 3 days in a tent – Willing to pay $550
  • 89. • Midrange trip -- $600 – 4 days in the airport Motel 6 – Willing to pay $900 • Luxury trip -- $850 – 5 days in beachside hotel – Willing to pay $1100 • Ultra Luxury Weeklong trip -- $1250 – 7 days in a deluxe waterfront cabana – Willing to pay $1250 Net Benefits = $0 Net Benefits = $150 You value this trip $200 less BUT YOU SAVE $250
  • 90. 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 – Willing to pay $900 • Luxury trip -- $850 – 5 days in beachside hotel – Willing to pay $1100 • Ultra Luxury Weeklong trip -- $1250 – 7 days in a deluxe waterfront cabana – Willing to pay $1250 Net Benefits = $0 You value this trip $350 less BUT YOU SAVE ONLY $200
  • 91. 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.
  • 92. From Last Class Preferences Budget (Income) Choice (Individual Demand) Population Market Demand From Last Class Preferences Budget (Income) Choice (Individual Demand) Population Market Demand
  • 93. 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) Modelling Consumer Choice Modelling Consumer Choice What would happen if I asked Ronnie about Pythagoras’
  • 94. Theorem? Modelling Consumer Choice 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
  • 95. – 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. Assumptions on Bundles Consider two goods: x and y Assumptions 1 & 2 imply: y
  • 96. x Assumptions on Bundles Consider two goods: x and y 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.
  • 97. 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 Assumptions on Preferences Consider two goods: x and y Assumption 3 implies: y x A 3 2
  • 98. ? Assumptions on Preferences Consider two goods: x and y Assumption 3 implies: y x A 3 2 Worse Assumptions on Preferences
  • 99. Consider two goods: x and y Assumption 3 implies: y x A 3 2 Worse ? Assumptions on Preferences Consider two goods: x and y Assumption 3 implies:
  • 100. y x A 3 2 Worse Better Assumptions on Preferences Consider two goods: x and y Assumption 3 implies: y x
  • 101. 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.
  • 102. • 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….
  • 104. 2 1 3 4 y x Indifference Curves A C B 4 3 2
  • 105. 1 3 4 Which bundle is preferred? y x Indifference Curves A C B 4 3 2
  • 106. 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
  • 107. • Rules: – Indifference curves cannot cross. – Indifference curves slope downwards (for `goods’) What about pollution, disease, hunger, homework? Indifference Curves • Rules: – Indifference curves cannot cross. – Indifference curves slope downwards (for `goods’) What about pollution, disease, hunger,
  • 108. homework? pizza 1 hour doing homework Indifference Curves • Rules: – Indifference curves cannot cross. – Indifference curves slope downwards (for `goods’) What about pollution, disease, hunger, homework? 1 hour doing homework pizza
  • 109. Indifference Curves • Rules: – Indifference curves cannot cross. – Indifference curves slope downwards (for `goods’) What about pollution, 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?
  • 110. • Mathematically, MRS = (-1)(slope of I.C.) Since our indifference curves have a negative slope, our MRS will be positive. Marginal Rate of Substitution y x MRS 1 Marginal Rate of Substitution
  • 111. y x MRS 1 MRS = (-1)(slope) ≈ (-1)(rise/run) Marginal Rate of Substitution • 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.
  • 112. Marginal Rate of Substitution 1 1 MRSA MRSB A B y x Marginal Rate of Substitution 1 1 MRSA
  • 113. MRSB A B A: More y than x B: More x than y MRSA > MRSB y x Marginal Rate of Substitution • Why can’t indifference curves cross? y x
  • 114. Marginal Rate of Substitution • Why can’t indifference curves cross? y x Marginal Rate of Substitution • Lets look at two consumers with different tastes y x ICAnn ICBob Marginal Rate of Substitution
  • 115. • 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
  • 116. – 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
  • 117. Perfect Substitutes Blue Pens Black Pens 6 4 2 4 62 Perfect Substitutes Blue Pens Black Pens 6 4
  • 118. 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:
  • 119. 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
  • 120. 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
  • 121. 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? Example – Auto Prices GM Ford
  • 122. Example – Auto Prices GM Ford GM and Ford may not be perfect substitutes, but they’re pretty substitutable (relatively) Example – Auto Prices GM Toyota Example – Auto Prices GM
  • 123. 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
  • 124. 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
  • 125. 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
  • 126. • 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:
  • 127. • 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
  • 128. 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 Cobb-Douglas utility functions • We will see these frequently because of their convenient properties.
  • 129. Example: Food and clothing, with U = FC. What are the utility levels for the following bundles? (0,0) U = 0 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? (0,2)
  • 130. Utility Cobb-Douglas utility functions • These have the form U(X,Y) = a • 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? (0,2) U = 0 Utility Cobb-Douglas utility functions • • We will see these frequently because of their convenient properties.
  • 131. Example: Food and clothing, with U = FC. What are the utility levels for the following bundles? (1,1) 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? (1,1) U = 1
  • 132. 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? (1,2) Utility Cobb-Douglas utility functions • We will see these frequently because of their
  • 133. convenient properties. Example: Food and clothing, with U = FC. What are the utility levels for the following bundles? (1,2) U = 2 Cobb-Douglas Utility U = FC Food Clothing U = 30 U = 20 U = 10
  • 134. Cobb-Douglas Utility U = FC Food Clothing But what if U = 2FC? Cobb-Douglas Utility U = 2FC Food Clothing But what if U = 2FC? Essentially nothing…
  • 135. U = 60 U = 40 U = 20 Cobb-Douglas Utility U = FC Food Clothing But what if U = F2C? U = 20 U = 10 Cobb-Douglas Utility
  • 136. 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.
  • 137. Example: Paper towels and napkins, with U = P + N What are the utility levels for the following bundles? (0,0) Utility Perfect substitutes • These have the form U(X,Y) = aX + bY. Example: Paper towels and napkins, with U = P + N What are the utility levels for the following bundles? (0,0) U = 0 Utility Perfect substitutes
  • 138. • These have the form U(X,Y) = aX + bY. Example: Paper towels and napkins, with U = P + N What are the utility levels for the following bundles? (1,3) Utility Perfect substitutes • These have the form U(X,Y) = aX + bY. Example: Paper towels and napkins, with U = P + N What are the utility levels for the following bundles? (1,3) U = 4
  • 139. Utility Perfect substitutes • These have the form U(X,Y) = aX + bY. Example: Paper towels and napkins, with U = P + N What are the utility levels for the following bundles? (3,1) Utility Perfect substitutes • These have the form U(X,Y) = aX + bY. Example: Paper towels and napkins, with U = P + N What are the utility levels for the following bundles? (3,1) U = 4
  • 140. Perfect SubstitutesNapkins 3 1 Paper Towels1 3 U = 4 U = 7 U = N + P Perfect SubstitutesNapkins 3 1
  • 141. Paper Towels1 3 U = 4 U = 7 But what happens if U = 2N + 2P? U = N + P Perfect SubstitutesNapkins 3 1 Paper Towels1 3 U = 8 U = 14
  • 142. U = 2N + 2P Perfect SubstitutesNapkins 3 1 Paper Towels1 3 U = 8 U = 14 U = 2N + 2P But what happens if U = 2N + P? Perfect SubstitutesNapkins
  • 143. 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) What are the utility levels for the following bundles?
  • 144. (3,1) Utility Perfect complements • These have the form U(X,Y) = c∙min(aX, bY) Example: Peanut butter and jelly, with U = min(P,J) What are the utility levels for the following bundles? (3,1) U = 1 Utility Perfect complements • These have the form U(X,Y) = c∙min(aX, bY) Example: Peanut butter and jelly, with U = min(P,J)
  • 145. What are the utility levels for the following bundles? (2,4) Utility Perfect complements • These have the form U(X,Y) = c∙min(aX, bY) Example: Peanut butter and jelly, with U = min(P,J) What are the utility levels for the following bundles? (2,4) U = 2 Perfect Complements J P1 2 3
  • 146. 3 2 1 U = min(P,J) U = 3 U = 2 U = 1 Perfect Complements J P1 2 3 3 2
  • 147. 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
  • 148. 3 2 1 U = 2*min(P,J) U = 6 U = 4 U = 2 Perfect Complements J P1 2 3 3 2
  • 149. 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,
  • 150. 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)
  • 151. 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 • Change in utility from a small change in
  • 152. 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 MUX = 1 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 = 4X + Y Marginal Utility
  • 153. • 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 = 4X + Y MUX = 4 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 = 4X*Y Marginal Utility
  • 154. • 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 = 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
  • 155. Marginal Utility and ICs We can define MUY in the same way we defined MUX Y X Marginal Utility and ICs We can define MUY in the same way we defined MUX Y X MRSXY = MUX MUY Marginal Utility and ICs
  • 156. We can define MUY in the same way we defined MUX Y X MRSXY = MUX MUY ∙ ∙ ∙ ∙ 1 1 MRSA MRSB A B
  • 157. Marginal Utility and ICs We can define MUY in the same way we defined MUX Y X MRSXY = MUX MUY ∙ ∙ ∙ ∙ 1 1 MRSA MRSB A
  • 158. B Why is MRSA >MRSB? Marginal Rate of Substitution • 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
  • 159. 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).
  • 160. 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,
  • 161. 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 =
  • 162. 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
  • 163. 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