Chapter Three

1
Chapter Three

• Preferences

2
Preferences

• Recall, two things are relevant in determining
  what consumers choose to purchase.
• Budget Constraint (what they can afford)
• Preferences over goods (what they like)

3
Rationality in Economics

• Behavioral PostulateA decisionmaker always
  chooses its most preferred alternative from its
  set of available alternatives.
• So to model choice we must model the decision
  makers preferences.

4
Preferences

• We first define some notation used to describe
  preferences over two different consumption
  bundles A and B.
• For example, suppose there are only two
  consumption goods Apples and Oranges,
• A might refer to 2 apples and 3 oranges
• B might refer to 3 apples and 2 oranges.

5
Preference Relations

• Comparing two different consumption bundles, A
  and B
• strict preference A is preferred to B ( A is
  strictly better than B).
• weak preference A is at least as preferred as B
  ( at least as good but may be better).
• indifference A is exactly as good as B (consumer
  is indifferent between the two alternatives).

6
Preference Relations

• 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.

7
Preference Relations
p

• denotes strict preference so A B
  means that bundle A is preferred strictly to
  bundle B.
• denotes indifference A B means A and B are
  equally preferred.
• denotes weak preferenceA B means A is
  preferred at least as much as is B or A is weakly
  preferred to B.

p
8
Preference Relations-Implications

• A B and B A imply A B.

9
Preference Relations-Implications

• A B and B A imply A B.
• A B and (not B A) imply A B.

p
10
Preferences Assumptions

• In order to model decision making we make three
  assumptions about how consumers perceive
  different bundles of goods.
• Completeness
• Reflexivity
• Transitivity

11
Assumptions about Preference Relations

• Completeness For any two bundles A and B it is
  always possible to make the statement that either
  A B or
  B A.

12
Assumptions about Preference Relations

• Completeness requires that consumers have a
  well-defined preference ordering between any two
  alternatives.
• The consumer knows how to choose between two
  alternatives.
• Completeness also requires an explicit statement
  that the consumer is indifferent between two
  identical bundles.

13
Assumptions about Preference Relations

• Reflexivity Any bundle A is at least as good as
  itself i.e. A A.

14
Assumptions about Preference Relations

• Reflexivity is implied by completeness.
• Reflexivity requires an explicit statement that
  the consumer is indifferent between two identical
  bundles.

15
Assumptions about Preference Relations

• Transitivity IfA is at least as preferred as
  B, andB is at least as preferred as C, thenA is
  at least as preferred as C i.e. A B and B
  C A C.
• Note transitivity also requires that if

16
Assumptions about Preference Relations

• If preferences are complete, reflexive, and
  transitive, they are said to be rational.

17
Example 3.1 Preference Relations

• Consider the group of people A, B, and C and the
  preference relation strictly taller than.
• Is this preference relation complete, reflexive,
  and transitive?

18
Example 3.1 Preference Relations

• Complete No, because there is not an explicit
  statement about how to rank individuals of the
  same height.
• If people are the same height, then they cannot
  be ranked.
• Reflexive No, because there is not an explicit
  statement about how to rank individuals of the
  same height.
• A person cannot be strictly taller than his or
  herself.
• Transitive Yes. If A is strictly taller than B,
  and B is strictly taller than C, then it must be
  the case that A is strictly taller than C.

19
Example 3.2 Preference Relations

• Suppose Coach X ranks players according to
  strength, speed, and obedience.
• If a player is better than another player in two
  of these three categories, then that player is
  preferred. Otherwise, Coach X is indifferent
  between them.
• Player A very strong, very slow, and fairly
  obedient.
• Player B moderately strong, very fast, and very
  disobedient.
• Player C very weak, moderately fast, and very
  obedient.
• Are Coach Xs preferences complete, reflexive and
  transitive?

20
Example 3.2 Preference Relations

• Complete Yes, because Coach X will either
  strictly prefer or be indifferent between any two
  players.
• Reflexive Yes. Reflexivity requires that a
  player be at least as good in at least two
  categories as itself, which is true.
• Transitive No. We know that A is strictly
  preferred to B, B is strictly preferred to C, but
  C is strictly preferred to A. This violates
  transitivity.

21
Assumptions about Preference Relations

• Although one can come up with examples that
  violate these three fundamental assumptions about
  preference relations, we will from here on out
  assume that completeness, transitivity, and
  reflexivity hold in the preferences that we model.

22
Graphical Representation of Preferences

• We will focus on the two good case.
• There are many possible consumption bundles
  available to a consumer.
• We need a way of showing how consumers feel about
  these different consumption bundles relative to
  others.
• Use Indifference Curves

23
Graphical Representation of Preferences

• Given a particular consumption bundle, an
  indifference curve maps out all the consumption
  bundles that are equal in the eyes of the
  consumer.
• In other words, along an indifference curve, the
  consumer is indifferent to all consumption
  bundles.

24
Indifference Curves

• Take a reference bundle x. The set of all
  bundles equally preferred to x defines the
  indifference curve containing x
• the set of all bundles for which the consumer is
  indifferent to x.

25
Indifference Curves
If all of these points are on the same
indifference curve, then x x x
x2
x
x
x
x1
26
Indifference Curves

• Once we have an indifference curve, assuming that
  more of any one commodity is better, we can
  identify how consumers prefer points on the
  indifference curve to points not on the
  indifference curve.

27
Indifference Curves
x2
z x y
p
p
z
x
y
x1
28
Indifference Curves

• Assuming that more of both goods is preferred to
  fewer, then we know that higher indifference
  curves (to the right) will be preferred to
  consumption bundles on lower indifference curves.

29
Indifference Curves
All bundles on I1 are strictly preferred to all
on I2.
x2
z
x
I1
All bundles on I2 are strictly preferred to
all on I3.
I2
y
I3
x1
30
Indifference curve

• Define the weakly preferred set of bundle x as
  all points that are at least as good as
  consumption bundle x.
• Define the strictly preferred set as all points
  that are strictly preferred to consumption bundle
  x.

31
Indifference Curves
Weakly preferred set to x
x2
WP(x), the set of bundles weakly preferred to
x.
x
WP(x) includes I(x).
I(x)
x1
32
Indifference Curves
Strictly preferred set to x
x2
SP(x), the set of bundles strictly preferred
to x, does not include
I(x).
x
I(x)
x1
33
Indifference Curves

• Indifference curves representing distinct levels
  of preference cannot cross each other.
• If this were not true, transitivity would be
  violated.

34
Indifference Curves Cannot Intersect
Since x and y are on distinct ICs, one must be
strictly preferred. Assume that x y. From
I1, x z. From I2, z y. By transitivity, y
x. But this is violated by our original
assumption.
I2
x2
I1
x
z
y
x1
35
Drawing Indifference Curves TIP

• Many different types of preferences can be drawn
  using indifference curves.
• Tip on drawing an indifference curve
• Pick an arbitrary point to be on the curve.
• If one of the commodities increased, what would
  have to happen to the other good to keep the
  consumer equally well off?

36
Slopes of Indifference Curves Goods, Bads, and
Neutrals

• For most of the class, we will focus on
  commodities that consumers like (goods), but we
  could also consider indifference curves for
  commodities that consumers dont like (bads) or
  commodities that consumers dont care about one
  way or the other (neutrals).

37
Slopes of Indifference Curves Goods, Bads, and
Neutrals

• For a good, more is weakly preferred.
• When more of at least one commodity is always
  strictly preferred, then indifference curves are
  negatively sloped.

38
Slopes of Indifference Curves Goods, Bads, and
Neutrals
Good 2
a negatively sloped indifference curve.
Better
Worse
Good 1
39
Slopes of Indifference Curves Goods, Bads, and
Neutrals

• If less of a commodity is always preferred then
  the commodity is a bad.
• How would an indifference curve with one good and
  one bad look?

40
Slopes of Indifference Curves Goods, Bads, and
Neutrals
Good 2
One good and onebad a positively
sloped indifference curve.
Better
Worse
Bad 1
41
Slopes of Indifference Curves Goods, Bads, and
Neutrals

• Neutrals are commodities that the consumer
  doesnt care about one way or the other.
• If x1 is a good and x2 is a neutral, then what do
  indifference curves look like?

42
Slopes of Indifference Curves Goods, Bads, and
Neutrals
neutral 2
One good and oneneutral a vertical
indifference curve.
Worse
Better
good 1
43
Extreme Cases of Indifference Curves

• We now consider how to draw indifference curves
  under the following extreme cases.
• Perfect substitutes
• Perfect complements
• Satiated preferences

44
Perfect Substitutes

• If a consumer always regards units of commodities
  1 and 2 as equivalent (at least in some constant
  proportion), then the commodities are perfect
  substitutes.
• What do these indifference curves look like?

45
Example 3.3 Perfect Substitutes

• Consider the case when x1 and x2 are identical in
  the eyes of the consumer so that the consumer is
  willing to trade them 1 for 1.

46
Example 3.3 Perfect Substitutes
x2
Slopes are constant at - 1.
15
I2
Bundles in I2 all have a totalof 15 units and
are strictly preferred to all bundles in
I1, which have a total of only 8 units
in them.
8
I1
x1
8
15
47
Example 3.4 Perfect Substitutes

• Suppose that Bertha consumes two goods 8 oz
  beers and 16 oz beers.
• Bertha doesnt care what type of can she buys.
  She only cares about the total amount of beer
  that she can consume.
• Draw a couple of indifference curves for Bertha.

48
Example 3.4 Perfect Substitutes
8oz beers
Slopes are constant at - 2. For every 16 oz beer,
she is willing to give up 2 of the 8oz
beers.
16
I2
8
I1
16 oz beers
4
8
49
Perfect Complements

• If a consumer always consumes commodities 1 and 2
  in fixed proportion (for example one-to-one),
  then the commodities are perfect complements
• Here goods are consumed together in a perfect
  proportion.
• What do these indifference curves look like?

50
Example 3.5 Perfect Complements

• Consider the case where the consumer consumes 1
  unit of commodity 1 for every unit of commodity 2
  that she consumes.
• For example, we could think of commodity 1 as
  being right shoes and commodity 2 as being a left
  shoes.

51
Example 3.5 Perfect Complements
Left shoes
Each of (5,5), (5,9) and (9,5) contains5 pairs
so each is equally preferred.
45o
9
5
I1
Right shoes
5
9
52
Example 3.5 Perfect Complements
Left shoes
Since each of (5,5), (5,9) and (9,5) contains 5
pairs, each is less preferred than the bundle
(9,9) which contains 9 pairs.
45o
9
I2
5
I1
Right shoes
5
9
53
Example 3.6 Perfect Complements

• Jane consumes coffee and sugar in perfect
  proportion.
• For every cup of coffee she consumes 2 spoons of
  sugar.
• Draw a couple of indifference curves exhibiting
  these preferences.

54
Example 3.6 Perfect Complements
coffee
Slope of line connecting the kink points is .5.
For every spoon of sugar, she consumes .5 cup of
coffee.
9
I2
5
I1
sugar
18
10
55
Preferences Exhibiting Satiation

• A bundle strictly preferred to any other, if it
  exists, is a satiation point or a bliss point.
• For example, suppose the two goods consumed are
  chocolate cake and coke. If a consumer is really
  thirsty and hungry, then more of these two goods
  is better, but if too much is consumed then the
  consumer may become sick and worse off.
• After the satiation point, these goods become
  bads.
• What do indifference curves look like for
  preferences exhibiting satiation?

56
Indifference Curves Exhibiting Satiation
x2
Satiation(bliss)point
x1
57
Indifference Curves Exhibiting Satiation
x2
Better
Better
Satiation(bliss)point
Better
x1
58
Indifference Curves Exhibiting Satiation
x2
Better
Better
Satiation(bliss)point
Better
x1
59
Well-Behaved Preferences Monotonic and Convex

• A preference relation is well-behaved if it is
• monotonic and convex.

60
Well-Behaved Preferences Monotonic

• Monotonicity If it is true that for any two
  bundles of goods A and B, where A has at least as
  much of all goods as B and strictly more of at
  least one, that A is strictly preferred to B,
  then preferences are monotonic.
• If monotonicity holds, the indifference curve
  will have a negative slope.

61
Well-Behaved Preferences Convexity

• Convexity Consumers prefer a mix of goods to
  having all of one or the other

62
Well-Behaved Preferences Convexity

• Averages of bundles are (at least weakly)
  preferred to the bundles themselves.
• For example, take any two points on an
  indifference curve and connect them with a line.
• If any point on this line is at least as good as
  the points on the indifference curve, preferences
  are said to be convex.

63
Well-Behaved Preferences Convexity

• Preferences are strictly convex if all points on
  the interior of the connecting line are strictly
  above the indifference curve.
• Preferences are weakly convex if at least one
  point on the interior of the line is equally
  preferred to points on the indifference curve.

64
Well-Behaved Preferences Convexity
x is strictly preferred to both x and
x. These preferences are strictly
convex. Convexity shows a preference for
diversity / a mix.
x
x2
x
x2
x
x2
x1
x1
x1
65
Well-Behaved Preferences Convexity

• Are preferences for perfect substitutes and
  perfect complements weakly or strictly convex?
• Both are weakly convex because one can find at
  least one point on the connecting line that is
  equally as good as the endpoints.

66
Well-Behaved Preferences Convexity

• Another way of stating (weak) convexity is that
  the weakly preferred set is a convex set.
• A convex set is a set where if you connect any
  two points in the set you will not leave the set.

67
Non-Convex Preferences
The mixture x is less preferred than x or
x. Also, the weakly preferred set is not
convex These preferences show a preference for
extremes. Non-convex preferences.
x2
Better
x
x2
x1
x1
68
More Non-Convex Preferences

• Preferences that are non-convex exhibit a
  preference for extremes.
• Consumers like to consume one or the other but
  they do not like to consume the two goods
  together at all.
• Example ice cream and anchovies

69
More Non-Convex Preferences
The mixture xis less preferred than x or
x. The weakly preferred set is not a convex
set. These are not convex preferences.
x2
Better
x
x2
x1
x1
70
Well-Behaved Preferences Convexity

• We can also describe a mathematical criteria for
  (strict and weak) convexity.
• If for any constant t between 0 and 1, and any
  two points x and x on the same indifference
  curve,
• Preferences are weakly convex if
• tx(1-t)x is weakly preferred to x
• Preferences are strictly convex if
• tx(1-t)x is strictly preferred to x

71
Slopes of Indifference Curves

• The slope of an indifference curve is called the
  marginal rate-of-substitution (MRS).
• The MRS represents how much of x2 a consumer
  would be willing to give up in order to get a
  little more of the x1 good.
• This is the individual consumers tradeoff not
  the market tradeoff.
• How can a MRS be calculated?

72
Marginal Rate of Substitution
MRS at x is the slope of theindifference curve
at x. If an equation for the indifference curve
is known x2(x1). Then the MRS can be calculated
as dx2/dx1 evaluated at a particular point
.
x2
x
x1
73
EX. 3.7 Calculating MRS

• Suppose the equation for the indifference curve
  is given by x2(x1)100/x1.
• What is the MRS at the point (10, 10)?

74
EX. 3.7 Calculating MRS

• dx2/dx1 -100/ (x12)
• Evaluated at (10, 10),
• dx2/dx1 (10,10) -100/ (100)-1
• This implies that at this point, the consumer is
  willing to trade 1 unit of x2 in order to get 1
  unit of the x1 good for small trades.

75
MRS EX. 3.7 continued
x2
MRS at (10,10) is -1.
Slope is -1.
10
x1
10
76
Diminishing MRS
Good 2
MRS - 5
Notice that the slope gets flatter as good 1
increases.
MRS - 0.5
Good 1
77
Diminishing MRS

• This property is sometimes called diminishing MRS
  (the MRS gets smaller as x1 goes up).
• This comes from the interpretation of convexity
  as exhibiting a preference for diversity.
• If there is a lot of x2 and little x1, then the
  consumer is willing to trade a lot of x2 for one
  more unit of x1.
• On the other hand, if the consumer has very
  little x2 and a lot of x1, then the consumer is
  not willing to give up much x2 for an additional
  unit of x1.
• For two goods, the absolute value of the MRS
  always decreases as x1 increases, if preferences
  are strictly convex.

78
MRS Ind. Curve Properties
x2
Here for these nonconvex preferences, the MRS
increases (becomes steeper) as x1 increases
MRS - 0.5
MRS - 5
x1
79
MRS Ind. Curve Properties
Here again, we do not have diminishing MRS over
all ranges of goods.
x2
MRS - 1
MRS - 0.5
MRS - 2
x1
80
MRS Ind. Curve Properties

• Preferences exhibiting perfect substitutes have
  linear indifference curves.
• This implies that they have a constant MRS.