Course: Microeconomics
Text: Varian’s Intermediate
Microeconomics
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Last chapter we talk about what is
affordable or feasible to consumers.
 This time we talk about preferences:
what the consumer like more and what
they like less.
 As a rational agent, a consumer chooses
the option in the budget set that is
highest in their preference order (i.e.
one likes the most).

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Let x, y are consumption bundles.

denotes strict preference so
x y means that bundle x is preferred
strictly to bundle y.
 ~ denotes indifference; x ~ y means x
and y are equally preferred.
f denotes weak preference;

~f
x ~ y means x is preferred at least as
much as is y.

p
p
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


Strict preference, weak preference and
indifference are all preference relations.
Particularly, they are ordinal relations;
i.e. they state only the order in which
bundles are preferred.
It has no indication of how much they
like one versus the other.
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x imply x ~ y.
f y and y f
~
~
 x f y and (not y f x) imply x
~
~
x
p

y.
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Completeness: For any two bundles x
and y it is always possible to make the
statement that either
x f y
~
or
y f x.
~
 Bundles are always comparable.
 If both are true, then they are
indifferent to the individual.

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
Reflexivity: Any bundle x is always at
least as preferred as itself; i.e.
x
f x.
~
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
Transitivity: If
x is at least as preferred as y, and
y is at least as preferred as z, then
x is at least as preferred as z; i.e.
x

z
f y and y f
~
~
x
f z.
~
It avoids circular preference, and ensure
that there exists the best bundle.
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

Take a reference bundle x’. The set of
all bundles equally preferred to x’ is the
indifference curve (set) containing x’; the
set of all bundles {y: y ~ x’}.
Weakly preferred set: bundles that are
weakly preferred to x’. {y: y f
x’}.
~
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x2
x’ ~ x” ~ x”’
x’
x”
x”’
The consumer is
indifferent between
every point on the
indifference curve.
x1
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p
x
x
p
x2
z
z
y
y
If consumer prefers more
to less for each goods, all
bundles on the northeast
of the indifference curve
are strictly preferred to x,
and all bundles
southwest of the
indifference curve are
less preferred to x.
x1
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x2
I1
All bundles in I1 are
strictly preferred to
all in I2.
x
z
I2
y
I3
All bundles in I2
are strictly
preferred to all in
I3.
x1
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x2
WP(x), the set of
x bundles weakly
preferred to x.
I(x)
I(x’)
x1
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x2
WP(x), the set of
x bundles weakly
preferred to x.
WP(x)
includes
I(x)
I(x).
x1
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x2
SP(x), the set of
x bundles strictly
preferred to x,
does not
include
I(x)
I(x).
x1
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I1
I2
x
p
x2
From I1, x ~ y.
From I2, x ~ z.
Therefore y ~ z. But because
I1 and I2 represent distinct
level of preference, we see
y z, a contradiction.
y
z
x1
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


When more of a commodity is always
preferred, the commodity is a good.
If every commodity is a good then
indifference curves are negatively sloped.
It is because when one has more of one
good, one has to get less of another to make
this bundle indifferent to the original one.
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Good 2
Two goods
a negatively sloped
indifference curve.
Good 1
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


If less of a commodity is always
preferred then the commodity is a bad.
e.g. rotten fruits; tobacco smoke (if you
do not smoke)
If one good is good and the other is bad,
then the indifference curve would be
upward sloping.
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Good 2
One good and one
bad
a
positively sloped
indifference curve.
Bad 1
If you want more of the good, you also have to
get more of the bad so that you are indifferent
between them.
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


If one just do not care about whether or
how much to have a commodity, this is
called a neutral good.
E.g.: goods that you don’t use and do not
care about their existence.
If one commodity is neutral, the other is
good, the indifference curve would be
vertical / horizontal.
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


If a consumer always regards units of
commodities 1 and 2 as equivalent, then
the commodities are perfect substitutes.
Only the total amount (or a weighted sum)
of the two commodities in bundles
determines their preference rank-order.
Example: orange juice of two different
brands. Apartment in different locations.
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x2
15 I2
8
I1
Slopes are constant at - 1.
Bundles in I2 all have a total
of 15 units and are strictly
preferred to all bundles in
I1, which have a total of
only 8 units in them.
8
15
x1
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If a consumer always consumes
commodities 1 and 2 in fixed proportion
(e.g. one-to-one), then the commodities
are perfect complements.
 Only the number of pairs of units of the
two commodities determines the
preference rank-order of bundles.
 E.g.: left shoes/right shoes; computer
and monitor.

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x2
45o Each of (5,5), (5,9)
and (9,5) contains
5 pairs so each is
equally
preferred.
9
5
I
1
5
9
x1
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x2
Since each of (5,5),
(5,9) and (9,5)
contains 5 pairs,
each is less
I2 preferred than
the bundle (9,9)
I which contains 9
1
pairs.
45o
9
5
5
9
x1
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


A bundle strictly preferred to any others
is a satiation point or a bliss point.
The satiation point is the best bundle.
More of anything are not better.
What do indifference curves look like for
preferences exhibiting satiation?
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x2
Satiation
(bliss)
point
x1
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x2
Better
Satiation
(bliss)
point
x1
30
x2
Bette
r
Satiation
(bliss)
point
x1
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

A commodity is infinitely divisible if it
can be acquired in any quantity; e.g.
water or cheese.
A commodity is discrete if it comes in
unit lumps of 1, 2, 3, … and so on; e.g.
aircraft, ships and refrigerators.
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
Suppose commodity 2 is an infinitely
divisible good (gasoline) while
commodity 1 is a discrete good (aircraft).
What do indifference “curves” look like?
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Gasoline
Indifference “curves”
are collections of
discrete points.
0
1
2
3
4 Aircraft
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Typical assumptions of preferences
 A preference relation is “well-behaved” if
it is


monotonic and convex.
Monotonicity: More of any commodity is
always preferred (i.e. no satiation and
every commodity is a good).
 This implies a negatively sloped IC.

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
Convexity: Mixtures of bundles are (at
least weakly) preferred to the bundles
themselves. E.g., the 50-50 mixture of
the bundles x and y is
z = (0.5)x + (0.5)y.
z is at least as preferred as x or y.
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x2
x
x+y Is (weakly)
z=
2 preferred to both
x and y.
y
x2+y2
2
y2
x1
x1+y
1
2
y1
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x2
x
z =(tx1+(1-t)y1, tx2+(1-t)y2)
is preferred to x and y
for all 0 < t < 1.
y
y2
x1
y1
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x2
x
Preferences are strictly convex
when all mixtures z
are strictly
preferred to their
z
component
bundles x and y.
y
y2
x1
y1
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The mixture z
is less preferred
than x or y.
One likes both,
but does want
to consume
together.
x2
z
y2
x1
y1
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x2
The mixture z
is less preferred
than x or y.
z
y2
x1
y1
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


It represents a more balanced preference
rather than a preference that induces a
specialization.
It is natural to consume the goods
involved in positive amount.
It also implies a diminishing marginal
rate of substitution.
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


The negative of the slope of an indifference
curve is its marginal rate of substitution
(MRS).
Note: this is slightly different from the textbook, but my
definition is more popular.
This represents the maximum amount of x2
one is willing to give up per unit of x1 at a
specific consumption bundle.
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x2
x’
MRS at x’ is the
(negative) slope of the
indifference curve at x’
x1
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x2
Dx2
x’
MRS at x’ is
lim -{Dx2/Dx1}
Dx1
0
= -dx2/dx1 at x’
Dx1
x1
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x2
dx2 x’
dx1
-dx2 = MRS ´ dx1 so, at x’,
MRS is the rate at
which the consumer is
only just willing to
exchange commodity 2
for a small amount of
commodity 1.
x1
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Good 2
Two goods
a negatively sloped
indifference curve
MRS > 0.
Good 1
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Good 2
One good and one
bad
a
positively sloped
indifference curve
MRS < 0.
Bad 1
Because instead of giving up, you have to
obtain more good 2 for you to be willing to
accept more good 1.
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Good 2
MRS = 5
MRS always decreases
with x1 if and only if
preferences are strictly
convex.
MRS = 0.5
Good 1
We call it a diminishing marginal
rate of substitution.
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x2
MRS = 0.5
MRS increases
as x1 increases
non-convex preferences
MRS = 5
x1
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x2
MRS is not always
decreasing as x1 increases
non-convex preferences.
MRS=1.5
MRS
= 0.5
MRS = 2
x1
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


In this chapter, we talk about how we can
specify consumer’s preference towards
different consumption bundles.
We can use indifferent curve to depict
different kinds of preferences.
The marginal rate of substitution is the
slope of indifference curve. It represents
the willingness to substitute one good for
another one.
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


We have talked about preference and
indifference curve in this chapter.
To put preference in a more
mathematically convenient way, we
introduce the utility function in the coming
chapter.
Then we can put together preference/utility
and budget constraint to analyze consumer
choices.
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