Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Econ 401 Price Theory
Chapter 3: Preferences
Instructor: Hiroki Watanabe
Summer 2009
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Intro
Pref.
1
2
3
4
5
6
7
Assumptions
Indifference Curves
Example Trinity
Σ
Introduction
Preference Relations
Assumptions
Rational Preferences
Well-Behaved Preferences
Indifference Curves
Indifference Curves
Preferred Sets
Convexity and Indifference Curves
Example Preferences
Perfect Substitutes
Perfect Complements
Trinity
Marginal Willingness to Pay
Comparison to Relative Prices
Summary
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Chapter 2 covered $ part of consumer theory.
Now we move on to , part.
How do we describe ,?
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Fact
1
Numbers are easy to compare (Chapter 4).
2
Bundles are hard to compare (Chapter 3).
1
xC = 1 vs yC = 10.
2
(xC , xT ) = (10, 3) vs (yC , yT ) = (3, 12).
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
We assume that a decision-maker always chooses
his most preferred alternative from his set of
available alternatives.
We need to some tools to describe decision-maker’s
preferences.
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Intro
Pref.
1
2
3
4
5
6
7
Assumptions
Indifference Curves
Example Trinity
Σ
Introduction
Preference Relations
Assumptions
Rational Preferences
Well-Behaved Preferences
Indifference Curves
Indifference Curves
Preferred Sets
Convexity and Indifference Curves
Example Preferences
Perfect Substitutes
Perfect Complements
Trinity
Marginal Willingness to Pay
Comparison to Relative Prices
Summary
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Comparing two different bundles x = (x1 , x2 ) and
y = (y1 , y2 ).
Preference (x y): You like x at least as much as y.
Say x is preferred to y.
Indifference (x ∼ y): Consuming x or y doesn’t
make any difference to you.
If you prefer x = (xC , xT ) = (10, 3) to
y = (yC , yT ) = (3, 6), we write (10, 3) (3, 6).
If you prefer (yC , yT ) = (3, 6) to x = (xC , xT ), we write
(3, 6) (10, 3).
If you are indifferent between (xC , xT ) = (10, 3) and
(yC , yT ) = (3, 6), we write (10, 3) ∼ (3, 6).
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Note x ∼ y is equivalent to
x y and y x,
i.e., if you like x at least as much as y and y at least
as much as x at the same time, then you’re
indifferent between x and y.
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Introduction
Preference Relations
Assumptions
Rational Preferences
Well-Behaved Preferences
Indifference Curves
Indifference Curves
Preferred Sets
Convexity and Indifference Curves
Example Preferences
Perfect Substitutes
Perfect Complements
Trinity
Marginal Willingness to Pay
Comparison to Relative Prices
Summary
1
2
3
4
5
6
7
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Rational Preferences
Rational Preferences
Your preference is rational if your preference relation is
complete and transitive.
1
Complete: For any bundle x = (x1 , x2 ), y = (y1 , y2 ),
you can say whether x y, y x or both.
2
Transitive: For x = (x1 , x2 ), y = (y1 , y2 ), and
z = (z1 , z2 ), if x y and y z, then x z.
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Rational Preferences
Example
At Kayak’s, the following combinations of cheesecakes
and tea are available:
x = (xC , xT ) = (1, 2).
y = (yC , yT ) = (2, 1).
z = (zC , zT ) = (3, 4).
Greg’s preferences are given by
y x,
x z,
y z.
Dharma’s preferences are given by
x y,
z x.
Which one has rational preferences?
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Rational Preferences
Greg’s preferences are complete because he
compares all the choices.
Greg’s preferences are transitive because y x,
x z should imply y z and he does have that
relation.
Dharma’s preferences are not complete because
she doesn’t compare y and z.
Dharma’s preferences are not transitive because
z x, x y should imply z y but she doesn’t have
that.
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Rational Preferences
We exclude irrational agents like Dharma (she
cannot make a choice between y and z).
While Greg’s preferences ARE rational, his
preference y = (2, 1) z = (3, 4) doesn’t sound
convincing.
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Well-Behaved Preferences
Well-Behaved Preferences
Your preferences are well-behaved if your preferences
are
1
Monotonic
2
Convex.
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Well-Behaved Preferences
Monotonicity
More of any commodity is always preferred.
If x1 ≥ y1 and x2 ≥ y2 , then x y.
If is monotonic, for x = (xC , xT ) = (1, 2)
z = (zC , zT ) = (3, 4) x because 3 ≥ 1 and 4 ≥ 2.
The bundles lying to the northeast of x is preferred
to x itself.
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Well-Behaved Preferences
T
•z
4
2
•
C
1
3
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Well-Behaved Preferences
Convex Preferences
If you prefer mixtures of bundles are preferred to the
bundles themselves, your preferences are convex.
If your
preferences
are convex and
1
9
x=
∼y=
, then
9
1
9
5
1
z = .5
+ .5
=
x.
9
1
5
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Well-Behaved Preferences
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Well-Behaved Preferences
The mixture doesn’t have to be 50-50.
1
9
7
z0 = .25
+ .75
=
x
9
1
2
1
9
2
z00 = .75
+ .25
=
x
9
1
7
1
9
1.8
z000 = .9
+ .1
=
x.
9
1
8.2
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Well-Behaved Preferences
10
9
x=(1, 9)
90−10
75−25
50−50
25−75
y=(9, 1)
8
Tea (x2)
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
Cheesecakes (x1)
8
9
10
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1
Intro
Pref.
1
2
3
4
5
6
7
Assumptions
Indifference Curves
Example Trinity
Σ
Introduction
Preference Relations
Assumptions
Rational Preferences
Well-Behaved Preferences
Indifference Curves
Indifference Curves
Preferred Sets
Convexity and Indifference Curves
Example Preferences
Perfect Substitutes
Perfect Complements
Trinity
Marginal Willingness to Pay
Comparison to Relative Prices
Summary
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Indifference Curves
Now we have tool to describe Greg’s preferences: & ∼.
For example, we can describe Greg’s preferences
as:
(2, 2) (1, 1)
(1, 3) ∼ (2, 2)
(3, 2) (2, 2)
(3, 2) (1, 1)
..
.
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Indifference Curves
While does represent what Greg prefers, it is hard
to visualize it.
We have a device called the indifference curve to
picture preferences.
Indifference Curves
The indifference curve at a bundle x0 = (x10 , x20 ) is a
collection of bundles that is equally preferred to x0 .
If a bundle x = (x1 , x2 ) and y = (y1 , y2 ) is on the
same indifference curve, then Greg is indifferent
between them: x ∼ y.
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Indifference Curves
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Indifference Curves
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Intro
Pref.
Assumptions
Indifference Curves
Figure:
Example Trinity
Σ
Indifference Curves
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Indifference Curves
Greg’s indifference curve does not go across
another indifference curve if Greg is rational.
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Indifference Curves
Figure:
x and z is on the same indifference curve so that
x ∼ z.
likewise z ∼ y.
By transitivity, x ∼ y.
x and y are on the different curves so that x is not
∼ y.
contradiction.
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Preferred Sets
We have another visual aid:
Preferred Set
A preferred set to a bundle x0 = (x10 , x20 ) is a collection of
all the bundles that is preferred to x0 .
If y ∼ x, then y is in the preferred set of a bundle x.
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Preferred Sets
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Preferred Sets
If Greg’s rational, his indifference curve is
downward sloping.
2
•
3
2
•
1
3
O
6
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Preferred Sets
Exercise
Greg’s preferences for cheesecakes and tea (xC , xT ) are
given by:
(2, 2) (1, 1)
(1, 4) ∼ (2, 2)
(3, 1) ∼ (1, 4)
(2, 3) (2, 2).
1
Sketch an indifference curve at (xC , xT ) = (2, 2).
2
Is a bundle (2, 3) contained in the preferred set to
(2, 2)?
1
How about (1, 1)?
3
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Preferred Sets
5
3
T
Tea (x )
4
2
1
0
0
1
2
Cheesecakes (xC)
3
4
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1
Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Convexity and Indifference Curves
What does Greg’s preferred set look like when his
preferences exhibit convexity?
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Convexity and Indifference Curves
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Intro
Pref.
Assumptions
Indifference Curves
Figure:
Example Trinity
Σ
Convexity and Indifference Curves
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Convexity and Indifference Curves
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Intro
Pref.
1
2
3
4
5
6
7
Assumptions
Indifference Curves
Figure:
Example Trinity
Σ
Introduction
Preference Relations
Assumptions
Rational Preferences
Well-Behaved Preferences
Indifference Curves
Indifference Curves
Preferred Sets
Convexity and Indifference Curves
Example Preferences
Perfect Substitutes
Perfect Complements
Trinity
Marginal Willingness to Pay
Comparison to Relative Prices
Summary
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Two extreme examples of preferences and one
example of linear preferences.
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Perfect Substitutes
Perfect Substitutes
If Greg always regards units of commodities 1 and 2 as
equivalent, then the commodities are perfect
substitutes.
E.g., Coke & Pepsi.
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Perfect Substitutes
The indifference curve at (xC , xP ) = (2, 2) includes
0
1
2
3
4
,
,
,
,
.
4
3
2
1
0
(What do the bundles above have in common?)
The indifference curve at (xC , xP ) = (2, 0) includes
2
1
0
1.5
,
,
,
.
0
1
2
.5
(They all sum up to 2).
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Perfect Substitutes
4
6
six−packs (x )
3
2
1
0
0
1
2
bottles (x1)
3
4
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1
Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Perfect Substitutes
The commodities do not have to be
interchangeable on the one-to-one basis.
Consider a bundle of six-packs and bottles of
Corona (x6 , x1 ):
(0, 12) ∼ (1, 6)
(0, 12) ∼ (2, 0).
They all satisfy
6x6 + x1 = 12,
i.e., any bundle on the indifference curve at (12, 0)
contains 12 bottles of Corona in total.
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Perfect Substitutes
Likewise,
(4, 0) ∼ (3, 6)
(4, 0) ∼ (2, 12)
(4, 0) ∼ (1, 18)
(4, 0) ∼ (0, 24).
They all satisfy
x1 + 6x6 = 24.
It is likely that
(4, 0) (2, 0).
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Perfect Substitutes
24
bottles (x1)
18
12
6
0
0
1
2
six−packs (x6)
3
4
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1
Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Perfect Substitutes
Exercise
Draw the indifference curve at the bundle
(x5 , x1 ) = (2, 10), where x5 denotes # of nickels and x1
denotes # pennies.
Note (2, 10) amounts to 20 cents. What other
combinations of pennies and nickels brings you 20
cents?
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Perfect Substitutes
20
Pennies (x1)
15
10
5
0
0
1
2
Nickels (x5)
3
4
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1
Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Perfect Complements
Perfect substitutes are completely interchangeable
and one commodity is replaced by the other
commodity.
Perfect complements are the other extreme case:
Perfect Complements
If Greg always consumes commodities 1 and 2 in fixed
proportion, then the commodities are perfect
complements. Only the # of pairs of the two
commodities (as opposed to # of each commodities)
determines his preferences.
E.g., left glove and right glove.
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Perfect Complements
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Intro
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Assumptions
Indifference Curves
Figure:
Example Trinity
Σ
Perfect Complements
Exercise
Draw Greg’s indifference curve at (cereal, milk)= (2, 5).
Greg says he can’t have cereals without milk and
the only time he has milk is when he eats his
cereals.
Greg’s preferred cereal-milk ratio is 2 to 3.
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Perfect Complements
6
5
M
Milk (x )
4
3
2
1
0
0
1
2
3
Cereal (xC)
4
5
6
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1
Intro
Pref.
1
2
3
4
5
6
7
Assumptions
Indifference Curves
Example Trinity
Σ
Introduction
Preference Relations
Assumptions
Rational Preferences
Well-Behaved Preferences
Indifference Curves
Indifference Curves
Preferred Sets
Convexity and Indifference Curves
Example Preferences
Perfect Substitutes
Perfect Complements
Trinity
Marginal Willingness to Pay
Comparison to Relative Prices
Summary
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Marginal Willingness to Pay
Just like the slope of the budget line has the
meaning (recall trinity),
the slope of the indifference curve has an important
meaning.
Marginal Willingness to Pay
If Greg is just willing to give up one slice of cheesecake
for a cups of tea, then a is called his marginal
willingness to pay.
What does he mean by "just willing"?
E.g., if (4, 5) ∼ (3, 7), then Greg is just willing to give
up 1 slice of cheesecake for 2 cups of tea. He is
indifferent between those two bundles. MWTP is 2.
If (8, 2) ∼ (7, 6), then Greg is just willing to give up 1
slice of cheesecake for 4 cups of tea. His MWTP at
(8, 2) is 4.
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Marginal Willingness to Pay
In general
xC
xT
∼
xC − 1
.
xT + MRS
"Marginal" = additional one unit.
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Intro
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Assumptions
Indifference Curves
Example Trinity
Σ
Marginal Willingness to Pay
24
bottles (x1)
18
12
6
0
0
1
2
six−packs (x6)
3
4
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1
Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Marginal Willingness to Pay
The slope of indifference curves represents the
marginal willingness to pay.
MWTP is also referred to as marginal rate of
substitution.
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Marginal Willingness to Pay
T
1
C
O
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Marginal Willingness to Pay
MRS usually varies depending on the bundle x.
Consider two bundles:
x = (xC , xT ) = (1, 29384720396)
y = (yC , yT ) = (3240894603, 1).
MRS at x is pretty large: Greg is willing to give up
lots of cups of tea for a slice of cheesecake.
MRS at y is pretty small: Greg is willing to give up
only few cups of tea for a slice of cheesecake (he
already has many cheesecakes).
1
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Marginal Willingness to Pay
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Intro
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Assumptions
Indifference Curves
Figure:
Example Trinity
Σ
Comparison to Relative Prices
Notice the difference:
Relative price is the cups of tea Greg has to sell
(give up) to get one slice of cheesecakes to keep to
his budget.
Marginal willingness to pay is the cups of tea Greg is
willing to give up to get one slice of cheesecakes to
remain as happy as before.
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Intro
Pref.
1
2
3
4
5
6
7
Assumptions
Indifference Curves
Example Trinity
Σ
Introduction
Preference Relations
Assumptions
Rational Preferences
Well-Behaved Preferences
Indifference Curves
Indifference Curves
Preferred Sets
Convexity and Indifference Curves
Example Preferences
Perfect Substitutes
Perfect Complements
Trinity
Marginal Willingness to Pay
Comparison to Relative Prices
Summary
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Intro
Pref.
Assumptions
Indifference Curves
Example Trinity
Σ
Describe preferences.
Rational, well-behaved, perfect substitutes, perfect
complements.
Indifference curves, preferred sets.
Trinity.
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