Chapter 3 Preferences

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Chapter 3Preferences
2
Introduction

• The economic model of consumer behavior is very
  simple people choose the best things they can
  afford.
• The last chapter was devoted to clarifying the
  meaning of can afford.
• This chapter will be devoted to clarifying the
  economic concept of best things. A consumer
  always chooses his most preferred one from his
  set of available alternatives.
• So to model consumers choices we must model
  their preferences.

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Preference Relations

• Let x, y are consumption bundles.
• denotes strict preference x y means
  that bundle x is strictly preferred to bundle y.
• denotes indifference x y means x and y are
  equally preferred.
• denotes weak preference x y means x is
  preferred at least as much as is y.

p
p
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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.
• x y and y x imply x y.
• x y and (not y x) imply x y.

p
5
Assumptions about Preference Relations

• Completeness For any two bundles x and y it is
  always possible to make the statement that either
  x y or
  y x.
• Bundles are always comparable.
• Again, if both are true, then they are
  indifferent to the individual.

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Assumptions about Preference Relations

• Reflexivity Any bundle x is always at least as
  preferred as itself i.e.
  x x.

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Assumptions about Preference Relations

• Transitivity Ifx is at least as preferred as
  y, andy is at least as preferred as z, thenx is
  at least as preferred as z i.e. x y and
  y z x z.

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Indifference Curves

• Take a reference bundle x. The set of all
  bundles equally preferred to x is the
  indifference curve (set) containing x i.e., the
  set of all bundles y y x.
• Weakly preferred set bundles that are weakly
  preferred to x. y y x.

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Indifference Curves
x2
x
x x x
x
x
x1
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Indifference Curves
x2
z x y
p
p
x
If the consumer prefers more to less for each
good, all bundles to the northeast of the
indifference curve are strictly preferred to x,
and all bundles to the southwest of the
indifference curve are less preferred to x.
z
y
x1
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Indifference Curves
I1
All bundles in I1 are strictly preferred to all
in I2.
x2
x
z
I2
All bundles in I2 are strictly preferred to
all in I3.
y
I3
x1
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Indifference Curves
x2
WP(x), the set of bundles weakly preferred to
x.
x
I(x)
I(x)
x1
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Indifference Curves
x2
WP(x), the set of bundles weakly preferred to
x.
x
WP(x) includes I(x).
I(x)
x1
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Indifference Curves
x2
SP(x), the set of bundles strictly preferred
to x, does not include
I(x).
x
I(x)
x1
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Indifference Curves Cannot Intersect
From I1, x y. From I2, x z. Therefore y
z. However, I1 and I2 represent different levels
of preference. Contradiction!
I2
x2
I1
x
y
z
x1
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Goods

• 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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Slopes of Indifference Curves
Good 2
Two goodsa negatively sloped indifference curve.
Better
Worse
Good 1
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Bads

• 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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Slopes of Indifference Curves
Good 2
One good and onebad a positively
sloped indifference curve.
Better
Worse
Bad 1
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Neutrals

• If one just do not care about whether or how much
  to have a commodity, then the commodity is called
  a neutral good.
• e.g. goods that you dont 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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Slopes of Indifference Curves
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Perfect Substitutes

• 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.
• e.g. orange juice of two different brands.

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Perfect Substitutes
x2
Slopes are constant at - 1.
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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
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Perfect Complements

• 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.

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Perfect Complements
x2
Each of (5,5), (5,9) and (9,5) contains5 pairs
so each is equally preferred.
45o
9
5
I1
x1
5
9
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Perfect Complements
x2
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
x1
5
9
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Preferences Exhibiting Satiation

• A bundle strictly preferred to any other is a
  satiation point or a bliss point.
• What do indifference curves look like for
  preferences exhibiting satiation?

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Indifference Curves Exhibiting Satiation
x2
Satiation(bliss)point
x1
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Indifference Curves Exhibiting Satiation
x2
Better
Better
Satiation(bliss)point
Better
x1
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Indifference Curves Exhibiting Satiation
x2
Better
Better
Satiation(bliss)point
Better
x1
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Discrete Commodities

• 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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Discrete Commodities

• 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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Indifference Curves With a Discrete Good
Gasoline
Indifference curvesare collections ofdiscrete
points.
Aircraft
0
1
2
3
4
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Well-Behaved 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).
• Monotonicity implies that indifference curves are
  negatively sloped.

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Well-Behaved Preferences

• Convexity Mixtures of bundles are (at least
  weakly) preferred to the bundles themselves. For
  example, 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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Well-Behaved Preferences -- Convexity
x
x2
xy
z is preferred to both x and y.
x2y2
z
2
2
y
y2
x1y1
x1
y1
2
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Well-Behaved Preferences -- Convexity
x
x2
z (tx1(1-t)y1, tx2(1-t)y2)
is preferred to x and y for all 0 lt t lt 1.
y
y2
x1
y1
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Well-Behaved Preferences -- Convexity
Preferences are strictly convex
when all mixtures z
are strictly preferred to
their component
bundles x and y.
x
x2
z
y
y2
x1
y1
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Weak Convexity
Preferences are weakly convex if at least one
mixture z is equally preferred to a component
bundle.
x
z
x
z
y
y
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Non-Convex Preferences
x2
Better
The mixture zis less preferred than x or y.
z
y2
x1
y1
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More Non-Convex Preferences
x2
Better
The mixture zis less preferred than x or y.
z
y2
x1
y1
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Slopes of Indifference Curves

• The slope of an indifference curve is its
  marginal rate-of-substitution (MRS).
• The MRS measures the rate at which the consumer
  is just willing to substitute one good for the
  other.

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Marginal Rate of Substitution
x2
MRS at x is the slope of theindifference curve
at x
x
x1
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Marginal Rate of Substitution
x2
MRS at x is lim Dx2/Dx1 Dx1 0
dx2/dx1 at x
x
Dx2
Dx1
x1
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Marginal Rate of Substitution
x2
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.
x
dx2
dx1
x1
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MRS Ind. Curve Properties
Good 2
Two goodsa negatively sloped indifference curve
Better
MRS lt 0.
Worse
Good 1
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MRS Ind. Curve Properties
Good 2
One good and onebad a positively
sloped indifference curve
Better
MRS gt 0.
Worse
Bad 1
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MRS Ind. Curve Properties
Good 2
MRS - 5
MRS always increases (decreases in absolute
value) with x1 (becomes less negative) 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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MRS Ind. Curve Properties
x2
MRS decreases(becomes more negative)as x1
increases with nonconvex preferences.
MRS - 0.5
MRS - 5
x1
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MRS Ind. Curve Properties
x2
MRS is not always increasing as x1 increases with
nonconvex preferences.
MRS - 1
MRS - 0.5
MRS - 2
x1