Notation and assumptions
Graphing preferences
Preferences
Intermediate Micro
Lecture 3
Chapter 3 of Varian
Properties/Assumptions
MRS
Notation and assumptions
Graphing preferences
Properties/Assumptions
The central question of economics
Microeconomics: study of decision-making under scarcity
Scarcity: last topic
Decision-making: next 3 topics
1. Ranking of options
2. Putting numbers to the rankings
3. Modeling the decision
Applications and interpretations follow
MRS
Notation and assumptions
Graphing preferences
Properties/Assumptions
Context
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Usually use 2-good examples and exercises
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Very restrictive (”Strong assumption”)
Fine for specific contexts
Methods apply for 2 to ∞ goods
Real-world: infinitely many goods
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Context matters, not just item
MRS
Notation and assumptions
Graphing preferences
Properties/Assumptions
MRS
Ranking consumption bundles
Consumption bundles (x1 , x2 ) and (y1 , y2 )
Strict preference
(x1 , x2 ) (y1 , y2 )
(x1 , x2 ) is preferred to (y1 , y2 )
∼
Indifferent
(x1 , x2 ) ∼ (y1 , y2 )
(x1 , x2 ) is just as good as (y1 , y2 )
Weak preference
(x1 , x2 ) (y1 , y2 )
(x1 , x2 ) is at least as good
as (y1 , y2 )
Notation and assumptions
Graphing preferences
Properties/Assumptions
Properties of preferences
Axioms of an individual’s preferences
1. Completeness: Either A B or B A
2. Reflexivity: A A
3. Transitivity: If A B and B C , then A C
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If A B and B C , then A C
If A B and B C , then A C
If A B and B C , then A C
”Axioms” means we will always assume these properties
**Preferences deal with bundles, not goods!**
MRS
Notation and assumptions
Graphing preferences
Properties/Assumptions
Properties of preferences
3 options: Roosevelt (R), Taft (T ), Wilson (W )
Does each voter follow 3 axioms?
Hershel (H):
RT
RW
T W
Kent (K):
RT
R≺W
T ≺W
Josiah (J):
R≺T
R≺W
T W
MRS
Notation and assumptions
Graphing preferences
Properties/Assumptions
Properties of preferences within a group
H :R T W
J :T W R
K :W R T
Election between R&T
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2 for R, 1 for T
Election between R&W
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2 for W , 1 for R
Election between T &W
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2 for T , 1 for W
Transitive?
Election between R, T , W
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1 for R, 1 for T , 1 for W
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Who wins?
Primary between R&T , then
election against W
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Who wins?
MRS
Notation and assumptions
Graphing preferences
Properties/Assumptions
Properties of preferences within a group
H :R T W
J :T W R
K :W R T
Election between R&T
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2 for R, 1 for T
Election between R&W
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2 for W , 1 for R
Election between T &W
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2 for T , 1 for W
Transitive?
Election between R, T , W
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1 for R, 1 for T , 1 for W
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Who wins?
Primary between R&T , then
election against W
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Who wins?
MRS
Notation and assumptions
Graphing preferences
Graphing preferences
How to illustrate preference
relations
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Pick bundle X = (x1 , x2 )
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Find all Y X
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Weakly preferred set:
Graph of all such Y
bundles
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Indifference curve:
Border of weakly
preferred set
Properties/Assumptions
MRS
Notation and assumptions
Graphing preferences
Indifference Curves
Different indifference curves can’t cross!
Suppose X Y
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Draw indifference curve
through X
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Draw indifference curve
through Y
Implication if they cross
at Z ?
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Hint: axioms
Properties/Assumptions
MRS
Notation and assumptions
Graphing preferences
Perfect substitutes
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Indifference curves are
straight
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Constant slope
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At any (x1 , x2 ), same
willingness to trade
between good 1 and 2
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Gallon gas from Exxon vs
gallon gas from 7-11
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Small bag of chips vs
large bag of chips
Properties/Assumptions
MRS
Notation and assumptions
Graphing preferences
Perfect complements
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Indifference curves are
right angles
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Indifferent between kink
point and increase of a
single good
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Left shoes and right
shoes
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Bicycles and handlebar
tape
Properties/Assumptions
MRS
Notation and assumptions
In-between
Graphing preferences
Properties/Assumptions
MRS
Notation and assumptions
Graphing preferences
Bads
I
x1 good, x2 bad
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Willing to decrease 1 for
decrease in 2
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Pollution, traffic,
coconut, ...
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Neutrals neither good
nor bad
Properties/Assumptions
MRS
Notation and assumptions
Graphing preferences
Satiation
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Some best point (x̄1 , x̄2 )
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Satiation point, or bliss
point
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True of most specific
goods
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What about $?
Properties/Assumptions
MRS
Notation and assumptions
Graphing preferences
Discrete goods
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Discrete good: a good
only available in
indivisible units
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Examples: cars, housing,
pets
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Indifference sets are
collections of points
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Preferred sets are
collections of rays
Properties/Assumptions
MRS
Notation and assumptions
Graphing preferences
Properties/Assumptions
Properties/Assumptions
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Assume properties of preferences
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Not true of all situations
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Monotonicity
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Convex preferences
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Well-behaved indifference curves: Monotonicity, convexity,
and 3 axioms make for well-behaved preferences
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Handy for studying decisions
MRS
Notation and assumptions
Graphing preferences
Properties/Assumptions
MRS
Monotonicity
y1 ≥ x1 and y2 ≥ x2 ,
at least one strictly
⇓
(y1 , y2 ) (x1 , x2 )
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More is always better
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Moving up and/or right
goes to higher
indifference curve
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Indifference curves have
negative slope
Notation and assumptions
Graphing preferences
Properties/Assumptions
(Strictly) convex preferences
(x1 , x2 ) ∼ (y1 , y2 )
⇓
(αx1 + (1 − α)y1 , αx2 + (1 − α)y2 ) (x1 , x2 )
∀αs.t0 < α < 1
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Take any 2 consumption bundles from same indifference curve
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Find a (weighted) average
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The average is preferred to either extreme
MRS
Notation and assumptions
Graphing preferences
Properties/Assumptions
Weakly convex preferences
(x1 , x2 ) ∼ (y1 , y2 )
⇓
(αx1 + (1 − α)y1 , αx2 + (1 − α)y2 ) (x1 , x2 )
∀αs.t0 < α < 1
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Take any 2 consumption bundles from same indifference curve
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Find a (weighted) average
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The average is weakly preferred (at least as good) to either
extreme
MRS
Notation and assumptions
Graphing preferences
Properties/Assumptions
Convex preferences
Strictly convex preferences
Weakly convex preferences
MRS
Notation and assumptions
Graphing preferences
Marginal rate of substitution
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Preferences
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Rank bundles
Show willingness to
exchange
Given bundle (x1 , x2 ),
indifferent with
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Gain small ∆x2
Lose small ∆x1
I ∆x2
∆x1
I Value
of good 1 relative
to good 2
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(−1∗)Slope of
indifference curve
Properties/Assumptions
MRS
Notation and assumptions
Graphing preferences
Marginal rate of substitution
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If offered trade:
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From (x1 , x2 )
To anywhere on red
line
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Will choose to change
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Higher indifference curve
Properties/Assumptions
MRS
Notation and assumptions
Graphing preferences
Marginal rate of substitution
2
lim∆x1 →0 ∆x
∆x1
I dx2
dx1
I Marginal
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rate
of substitution: Slope of
indifference curve at
(x1 , x2 )
No benefit to trading on
these terms:
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(x1 , x2 )
”price” of good 2 in
terms of good 1
= −MRS
Properties/Assumptions
MRS
Notation and assumptions
Graphing preferences
Properties/Assumptions
Marginal rate of substitution
I
With monotone, convex preferences
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MRS is slope of line tangent to indifference curve at (x1 , x2 )
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Unwilling to trade from (x1 , x2 ) along MRS line
MRS
Notation and assumptions
Graphing preferences
Properties/Assumptions
Diminishing marginal rate of substitution
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Monotone, convex
preferences
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Diminishing marginal
rate of substitution:
MRS goes towards zero
as x1 ↑
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Intuition?
MRS
Notation and assumptions
Perfect substitutes
Graphing preferences
Properties/Assumptions
MRS
Notation and assumptions
Graphing preferences
Perfect complements
Properties/Assumptions
MRS