Preferences

Rationality in Economics
Behavioral Postulate:
A decisionmaker always chooses its most preferred alternative from its set of available alternatives.
So to model choice we must model decisionmakers’ preferences.

Consumption bundle – object of consumer choice.
Consumption bundle – complete list of goods and services, the consumer can choose from.
When, where and under what circumstances matter

Consumption bundle X consists of x1, x2, …. Xn goods and services
For simplicty assume – X consists of only 2 goods: x1, x2,

Preference Relations
Comparing two different consumption bundles, x and y:
strict preference: x is more preferred than is y.
weak preference: x is as at least as preferred as is y.
indifference: x is exactly as preferred as is y.

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.

Preference Relations
denotes strict preference;
x y means that bundle x is preferred strictly to bundle y.
p
p

Preference Relations
denotes strict preference;
x y means bundle x is preferred strictly to bundle y.
~ denotes indifference; x ~ y means x and y are equally preferred.
p
p

Preference Relations
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.
denotes weak preference;
x y means x is preferred at least as much as is y.
p
p

Preference Relations
x y and y x imply x ~ y.

Preference Relations
x y and y x imply x ~ y.
x y and (not y x) imply x y.
p

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.
Can you alway tell what you choose?
Sophia’s choice

Assumptions about Preference Relations
Reflexivity: Any bundle x is always at least as preferred as itself; i.e.
x x.

Assumptions about Preference Relations
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 y and y z x z.

Transitivity –an experiment
Imagine you are hungry.
During the break you may have …….
What would you choose?
Write the first letter in the table given to you and bend the row, so that you can not see the previous choices

Transitivity –an experiment
1. Apple vs. Banana

Transitivity –an experiment
2. Banana vs. Cake

Transitivity –an experiment
3. Cake vs. Sandwich with Ham

Transitivity –an experiment
4. Sandwich with Ham vs. Sandwich with Cheese

Transitivity –an experiment
5. Apple vs. Cake

Transitivity –an experiment
6. Apple vs. Sandwich with Cheese

Transitivity –an experiment
7. Apple vs. Sandwich with Ham

Transitivity –an experiment
8. Banana vs. Sandwich with Ham

Transitivity –an experiment
9. Banana vs. Sandwich with Cheese

Transitivity –an experiment
10. Cake vs. Sandwich with Cheese

Assumptions about Preference Relations
Transitivity is a hypothesis about people’s choice behaviour – not a statement of pure logic!
We have to assume preferences are transitive to have a theory of how people make best choices

Indifference Curves
Take a reference bundle x’. The set of all bundles equally preferred to x’ is the indifference curve containing x’; the set of all bundles y ~ x’.
Since an indifference “curve” is not always a curve a better name might be an indifference “set”.

Indifference Curves
x2
x1
x”
x”’
x’ ~ x” ~ x”’
x’

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

Indifference Curves
x2
x1
x
All bundles in I1 are
strictly preferred to all in I2.
y
z
All bundles in I2 are strictly preferred to
all in I3.
I1
I2
I3

Indifference Curves
x2
x1
I(x’)
x
I(x)
WP(x), the set of
bundles weakly
preferred to x.

Indifference Curves
x2
x1
WP(x), the set of
bundles weakly
preferred to x.
WP(x)
includes I(x).
x
I(x)

Indifference Curves
x2
x1
SP(x), the set of
bundles strictly
preferred to x,
does not include
I(x).
x
I(x)

Indifference Curves Cannot Intersect
x2
x1
x
y
z
I1
I2
From I1, x ~ y. From I2, x ~ z.
Therefore y ~ z.

Indifference Curves Cannot Intersect
x2
x1
x
y
z
I1
I2
From I1, x ~ y. From I2, x ~ z.
Therefore y ~ z. But from I1 and I2 we see y z, a contradiction.
p

Slopes of Indifference Curves
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.

Slopes of Indifference Curves
Better
Worse
Good 2
Good 1
Two goods
a negatively sloped indifference curve.

Slopes of Indifference Curves
If less of a commodity is always preferred then the commodity is a bad.

Slopes of Indifference Curves
Better
Worse
Good 2
Bad 1
One good and one
bad a positively sloped indifference curve.

Slopes of Indifference Curves
If the consumer doesn’t care about a good then this good is a neutral good.

Extreme Cases of Indifference Curves; Perfect Substitutes
If a consumer always regards units of commodities 1 and 2 as equivalent, then the commodities are perfect substitutes and only the total amount of the two commodities in bundles determines their preference rank-order.

Extreme Cases of Indifference Curves; Perfect Substitutes
x2
x1
8
8
15
15
Slopes are constant at - 1.
I2
I1
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.

Extreme Cases of Indifference Curves; 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 and only the number of pairs of units of the two commodities determines the preference rank-order of bundles.

Extreme Cases of Indifference Curves; Perfect Complements
x2
x1
I1
45o
5
9
5
9
Each of (5,5), (5,9) and (9,5) contains
5 pairs so each is equally preferred.

Extreme Cases of Indifference Curves; Perfect Complements
x2
x1
I2
I1
45o
5
9
5
9
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.

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?

Indifference Curves Exhibiting Satiation
x2
x1
Satiation
(bliss)
point

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

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

Indifference Curves for 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, egs.

Indifference Curves for 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?

Indifference Curves With a Discrete Good
Gas-oline
Aircraft
0
1
2
3
4
Indifference “curves”
are collections of
discrete points.

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).

Well-Behaved Preferences
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.

Well-Behaved Preferences
Convexity
Are preferences for the sex of kids convex? I.e. will society prefer to have kids of the same sex, or different sex?

Well-Behaved Preferences -- Convexity.
x2
y2
x2+y2
2
x1
y1
x1+y1
2
x
y
z =
x+y
2
is strictly preferred to both x and y.

Well-Behaved Preferences -- Convexity.
x2
y2
x1
y1
x
y
z =(tx1+(1-t)y1, tx2+(1-t)y2)
is preferred to x and y for all 0 < t < 1.

Well-Behaved Preferences -- Convexity.
x2
y2
x1
y1
x
y
Preferences are strictly convex
when all mixtures z are strictly
preferred to their
component bundles x and y.
z

Well-Behaved Preferences -- Weak Convexity.
x’
y’
z’
Preferences are weakly convex if at least one mixture z is equally preferred to a component bundle.
x
z
y

Non-Convex Preferences
x2
y2
x1
y1
z
Better
The mixture z
is less preferred
than x or y.

More Non-Convex Preferences
x2
y2
x1
y1
z
Better
The mixture z
is less preferred
than x or y.

Slopes of Indifference Curves
The slope of an indifference curve is its marginal rate-of-substitution (MRS).
How can a MRS be calculated?

Marginal Rate of Substitution
x2
x1
x’
MRS at x’ is the slope of the
indifference curve at x’

Marginal Rate of Substitution
x2
x1
MRS at x’ is
lim {Δx2/Δx1}
Δx1 0
= dx2/dx1 at x’
Δx2
Δx1
x’

Marginal Rate of Substitution
x2
x1
dx2
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.
x’

MRS & Ind. Curve Properties
Better
Worse
Good 2
Good 1
Two goods
a negatively sloped indifference curve
MRS < 0.

MRS & Ind. Curve Properties
Better
Worse
Good 2
Bad 1
One good and one
bad a positively sloped indifference curve
MRS > 0.

MRS & Ind. Curve Properties
Good 2
Good 1
MRS = - 5
MRS = - 0.5
MRS always increases with x1 (becomes less negative) if and only if preferences are strictly
convex.

MRS & Ind. Curve Properties
x1
x2
MRS = - 0.5
MRS = - 5
MRS decreases
(becomes more negative)
as x1 increases
nonconvex preferences

MRS & Ind. Curve Properties
x2
x1
MRS
= - 0.5
MRS = - 1
MRS = - 2
MRS is not always increasing as x1 increases nonconvex
preferences.