Consumers’ preferences
ECO61
Udayan Roy
Fall 2008
Goods bundles
Origin
Preferences
• Consumers have preferences that they can use
to compare different goods bundles
• The preferences may be over goods bundles
consumed by oneself or over goods bundles
consumed by someone else
– For example, a parent may have preferences over
various bundles of food and clothing bought by
the parent but consumed by a child
Assumptions about Preference
Orderings
• Completeness: the consumer is able to rank all
possible bundles of goods and services.
– For any two bundles A and B, the consumer knows
whether A is better, or B is better, or they are equally
good
• Transitivity: for any three bundles A, B, and C, if A
is at least as good as B and B is at least as good as
C, then A is at least as good as C.
• These two assumptions imply the ranking
principle
The Ranking Principle
• A consumer can rank, in order of preference,
all potentially available alternatives
Assumption: More-Is-Better
• Other things equal, more of a good is
preferred to less.
– We ignore goods that are harmful or poisonous,
for which more is not better than less. Such goods
are jokingly referred to as ‘bads’
Indifference
Z2
D
W2
W is worse than A. Z is
better than A. So, on the
line joining W and Z,
there must exist a goods
bundle such as B that the
consumer considers
equally good as A. By
using this logic
repeatedly, we can find
many other bundles—
such as B, C, and D—that
are equally good as A.
Indeed, for any
consumption bundle, it
is possible to find other
bundles that are equally
good
Origin
An Indifference Curve
An indifference curve is a
set of consumption
bundles that the
consumer prefers equally
K is inferior and L is
superior to the bundles on
the indifference curve
Origin
Part of an Indifference Map
Origin
Properties of Indifference Maps
1. Bundles on indifference curves farther from
the origin are preferred to those on
indifference curves closer to the origin.
2. There is an indifference curve through every
possible bundle.
3. Indifference curves cannot cross.
4. Indifference curves slope downward.
Impossible Indifference Curves
B, Burritos per semester
• Lisa is indifferent
between e and a, and
also between e and b…
e
b
a
I1
I0
Z, Pizzas per semester
– so by transitivity she
should also be indifferent
between a and b…
– but this is impossible,
since b must be preferred
to a given it has more of
both goods.
• Lisa is indifferent
between b and a
since both points are
in the same
indifference curve…
– But this contradicts
the “more is better”
assumption. Can you
tell why?
– Yes, b has more of
both and hence it
should be preferred
over a.
B, Burritos per semester
Impossible Indifference Curves
b
a
I
Z, Pizzas per semester
Impossible Indifference Curves
Substitution Between Goods
• Economic decisions involve trade-offs
• Indifference curves provide information on
the amount of one good that the consumer is
willing to give up to gain a unit of another
good
4-14
Rates of Substitution
• Consider moving along an indifference curve, from
one bundle to another
• This is the same as taking away units of one good
and compensating the consumer for the loss by
adding units of another good
• Slope of the indifference curve shows how much of
the second good is needed to make up for a loss of
the first good
4-15
Figure 4.8: Rates of Substitution
• Look at the move from
bundle A to C
• Consumer loses 1 soup (S
= -1); gains 2 bread (B =
+2)
• A and C are equally
desirable
• Slope of indifference curve
= B/S = -2
• Consumer is willing to
substitute for soup with
bread at 2 ounces per pint
4-16
Marginal Rate of Substitution
• The marginal rate of substitution for X with Y, MRSXY, is the
rate at which a consumer must adjust Y to maintain the
same level of well-being when X changes by a tiny amount,
from a given starting point
MRS XY   Y X
MRS XY  slope of indifferen ce curve
• Tells us how much Y a consumer needs to compensate for
losing a little bit of X, per unit of X
• Tells us the maximum amount of Y a consumer would be
willing to pay per additional unit of X
• That is, MRSXY is the consumer’s willingness to pay Y for a
unit of X
4-17
Figure 4.9: Marginal Rate of Substitution
• Slope = B/S = 3/(-2) =
-3/2
• MRSSB= -B/S=-3/(-2) = 3/2
• The slope—and its negative, the
MRS—at bundle A can be
approximated by the slope of
the line AD, or the line AE, or
the line AF, etc.
• But the precise value is obtained
from the slope of the line that is
tangent to the indifference
curve at bundle A.
4-18
What Determines Rates of Substitution?
• Tastes
– Preferences for one good over another affect the slope of
an indifference curve and MRS
• Starting point on the indifference curve; the initial
goods bundle
– People like variety. So most indifference curves get flatter
as we move from top left to bottom right
– Link between slope and MRS implies that MRS declines;
the amount of Y required to compensate for a given
change in X decreases as X increases
• One gets bored with X as consumption of X increases. Therefore,
one needs less Y to compensate for a unit loss of X
4-19
Figure 4.10: Indifference Curves and
Consumer Tastes
4-20
Preferences and time
• To a non-economist, food is food is food.
• To an economist, “food delivered this year” and
“food delivered next year” are different goods
Preferences and chance
• To an economist, “food delivered tomorrow if
it is sunny” and “food delivered tomorrow if
there is a hurricane” are different goods
Figure 4.11: MRS along an Indifference
Curve
4-23
Perfect Substitutes and Complements
• Two products are perfect substitutes if their
functions are identical; in such a case, a consumer is
willing to swap one for the other at a fixed rate
• Two products are perfect complements if they are
valuable only when used together in fixed
proportions
4-24
Figure 4.12: Perfect Substitutes
MRSRE = ½
4-25
Figure 4.13: Perfect Complements
4-26
Utility
• Recall that under the completeness and
transitivity assumptions, the ranking principle is
true:
– the consumer can rank all bundles according to her
preference
• Therefore, the consumer can assign a number to
each bundle such that the numbers assigned to
the bundles represent the consumer’s
preferences
• The number assigned to a bundle is called its
utility
Utility functions
• If the utility numbers assigned by a consumer to
the various consumption bundles can be
represented by a mathematical formula, that
formula is called a utility function
• Example:
– Consider two goods, food and clothing and let the
quantities consumed be F and C.
– Then, the formula U(F,C) = F  C can be used to assign
a number to any bundle. (For example, if F = 11 and C
= 3, then U = 33.)
– And if the assigned numbers agree with the
consumer’s preference ranking, then the formula is a
utility function.
CONSUMER PREFERENCES
•
Utility and Utility Functions
● utility
Numerical score representing the satisfaction that a
consumer gets from a given market basket.
● utility function
Formula that assigns a level of utility to individual
market baskets.
Utility Functions and Indifference Curves
A utility function can be
represented by a set of
indifference curves, each
with a numerical
indicator.
This figure shows three
indifference curves (with
utility levels of 25, 50,
and 100, respectively)
associated with the utility
function:
u(F,C) = FC
Indifference Curves for the Utility Function
U=FS
Marginal Utility
• Marginal utility is the increase in a consumer’s
utility resulting from the addition of a very small
amount of some good, per unit of the good
MU X  U X
4-31
MU and MRS
• Consider changes in
consumption, X and
Y, that leave utility
unchanged
• A small change in X, X,
causes utility to change
by MUXX
• Small change in Y, Y,
causes utility to change
by MUYY
• If we stay on same
indifference curve, then
MUXX + MUYY = 0.
Therefore,
MU X X  MU Y Y  0
MU X X   MU Y Y
MU X
X   Y
MU Y
MU X
Y

 MRS XY
MU Y
X
4-32
Utility and
Marginal Utility
U, Utils
(a) Utility
0
U = 20
MU Z 
Z = 1
1
2
3
4
5
6
7
8
9
U
Z
10
Z, Pizzas per semester
(b) Marginal Utility
MU Z, Marginal utility of pizza
• Marginal utility is the
slope of the utility
function as we hold the
quantity of the other
good constant.
Utility function, U (10, Z )
250
230
• As Lisa consumes more
pizza, holding her
consumption of burritos
constant at 10, her total
utility, U, increases…
– and her marginal
utility of pizza, MUZ,
decreases (though it
remains positive).
350
130
20
0
MU Z
1
2
3
4
5
6
7
8
9
10
Z, Pizzas per semester
Ordinal utility
• The indifference map of the utility function U =
XY will look identical to the indifference map of
the utility function V = (XY)2 = U2 or of the utility
function W = (XY)2 + 12 = U2 + 12
• That is, the way a utility function ranks various
goods bundles is unchanged if the utility numbers
given to every bundle are transformed in an
order-preserving manner
• The utility numbers themselves are unimportant
• Only the implied rankings are important
Ordinal utility
• As was just claimed, the indifference map of
the utility function U = XY will look identical to
the indifference map of the utility function V =
(XY)2 = U2 or of the utility function W = (XY)2 +
12 = U2 + 12
• In particular, MRSXY at any goods bundle will
be unaffected if the utility numbers given to
every bundle are transformed in an orderpreserving manner
Figure 4.12: Perfect Substitutes
Utility function: U = 2E + R
MRSRE = ½
4-36
Figure 4.13: Perfect Complements
Utility function: U = min{R, L}
4-37
Quasi-linear utility
• U = f(X) + Y
– Example: U = X0.5 + Y
Y
X
• MRSXY depends on X but
not on Y
• That is, at any value of X, all
indifference curves have
the same slope
• As all indifference curves
are parallel to each other,
the vertical distance
between any two
indifference curves is
always the same
• We will see later why this
utility function is significant