Tastes/Preferences
Indifference Curves
Rationality in Economics
Rationality Behavioral Postulate:
“Rational Economic Man”
The decision-maker chooses the
most preferred bundle from the set of
available bundles.
We must model:
Set of available bundles; and
The decision-maker’s preferences.
PREFERENCES
X is the bundle (x1,x2) and Y is the bundle (y1,y2)
f
~ Weakly preferred
Bundle X is as least as good as bundle Y
(X f Y)
~
~ Indifferent
Bundle X is equivalent to bundle Y (X ~ Y)
f Strictly preferred
Bundle X is preferred to bundle Y (X f Y)
PREFERENCES: Axioms
1. Completeness
{A f B or Bf A or A ~ B}
Any two bundles can be compared.
2. Reflexive
{A f A }
~
Any bundle is at least as good as itself.
3. Transitivity
{If Af B and Bf C then A f C}
Non-satiation assumption (I.e. goods, not bads)
Axioms
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 f y and y f z x f z.
~ ~ ~
PREFERENCES
Intransitivity?
AfB Bf C C fA
Starting at C
Willing to pay to get to B
Willing to pay to get to A
Willing to pay to get to C
Willing to pay to get to B …
“Money Pump” Argument
(I.e. proof by contradiction)
INDIFFERENCE CURVES
The indifference curve
x2 through any particular
x1 consumption bundle
consists of all bundles
of products that leave
the consumer
x2 indifferent to the given
bundle.
I(x’)
x3
x1 ~ x2 ~ x3
x1
INDIFFERENCE CURVES
x2 z
p x
p y
x
z
y
x1
INDIFFERENCE CURVES
I1 All bundles in I1 are
x2
x strictly preferred to
all in I2.
z
I2
All bundles in I2 are
y strictly preferred to
I3
all in I3.
x1
INDIFFERENCE CURVES
x2
WP(x), the set of
x bundles weakly
preferred to x.
I(x’)
x1
INTERSECTING
INDIFFERENCE CURVES?
x2 I2 From I1, x ~ y
I1 From I2, x ~ z
Therefore y ~ z?
x
y
z
x1
INTERSECTING
INDIFFERENCE CURVES?
x2 I2 But from I1 and
I1 I2 we see y f z.
There is a
contradiction.
x
y
z
x1
SLOPES OF INDIFFERENCE CURVES?
When more of a product is always
preferred, the product is a good.
If every product is a good then
indifference curves are negatively
sloped.
SLOPES OF INDIFFERENCE CURVES?
Good 2 Two “goods” therefore
a negatively sloped
indifference curve.
Good 1
SLOPES OF INDIFFERENCE CURVES?
Ifless of a product is always
preferred then the product is a “bad”.
SLOPES OF INDIFFERENCE CURVES?
Good 2 One “good” and one
“bad” therefore a
positively sloped
indifference curve.
Bad 1
PERFECT SUBSITIUTES
Ifa consumer always regards units
of products 1 and 2 as equivalent,
then the products are perfect
substitutes and only the total amount
of the two products matters.
PERFECT SUBSITIUTES
x2
Slopes are constant at - 1.
I2
Examples?
I1
x1
PERFECT COMPLEMENTS
Ifa consumer always consumes
products 1 and 2 in fixed proportion
(e.g. one-to-one), then the products
are perfect complements and only
the number of pairs of units of the
two products matters.
PERFECT COMPLEMENTS
x2
45o Example: Each of
(5,5), (5,9) and (9,5)
is equally preferred
9
5 I1
5 9 x1
PERFECT COMPLEMENTS
x2
45o Each of (5,5), (5,9)
and (9,5) is less
preferred than the
bundle (9,9).
9 I2
5 I1
5 9 x1
WELL BEHAVED PREFERENCES
A preference relation is “well-behaved”
if it is monotonic and convex.
Monotonicity: More of any product is
always preferred (i.e. every product is a
good, no satiation).
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.
WELL BEHAVED PREFERENCES
Monotonicity
more of either product is better
indifference curves have negative
slopes
Convexity
averages are preferred to extremes
slopes get flatter as you move further
to the right (not obvious yet)
WELL BEHAVED PREFERENCES
Convexity
x2 x
z is strictly
x2+y2 x+y preferred to both
z=
2 x and y
2
y
y2
x1 x1+y y1
1
2
WELL BEHAVED PREFERENCES
Convexity
x2 x
z =(tx1+(1-t)y1, tx2+(1-t)y2)
is preferred to x and y
for all 0 0
Bad 1
MRS
Good 2 MRS decreases (in
absolute terms) as
MRS = (-) 5 x1 increases if and
only if preferences
are strictly convex.
Intuition?
MRS = (-) 0.5 Good 1
MRS
x2
MRS = (-) 0.5 If MRS increases (in
absolute terms) as x1
increases non-convex
preferences
MRS = (-) 5
x1
MRS
x2 MRS is not
always
decreasing as
MRS = - 1 x1 increases
MRS - non-
= - 0.5 MRS = - 2 convex
preferences.
x1