Transcript App 6

Indifference Curves and Utility
Maximization
CHAPTER
6 Appendix
© 2003 South-Western/Thomson Learning
1
Indifference Curves and Utility
Maximization
Marginal utility analysis requires some
numerical measure of utility in order to
determine the optimal consumption
combinations
Economists have developed another,
more general, approach to utility and
consumer behavior
This approach does not require that
numbers be attached to specific levels
of utility
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Indifference Curves and Utility
Maximization
All this new approach requires is that
consumers be able to rank their
preferences for various combinations of
goods
Specifically, the consumer should be able
to say whether
Combination A is preferred to combination B
Combination B is preferred to combination A.
or
Both combinations are equally preferred
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Consumer Preferences
Indifference curve shows all
combinations of goods that provide the
consumer with the same satisfaction, or
the same utility
Thus, the consumer finds all
combinations on a curve equally
preferred
Since each of the alternative bundles of
goods yields the same level of utility,
the consumer is indifferent about which
combination is actually consumed
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Exhibit 9: An Indifference Curve
•Suppose there are only two goods
available: pizzas and movie videos
•Point a shows the consumption
bundle consisting of 1 pizza
and 8 video rentals
•Suppose a person is given a
choice at point a or some other
combination
10
a
8
•Holding utility constant, how
many video rentals would a
person be willing to give up to get
a second pizza?
5
4
3
•Moving from point a to point b, we are
willing to give up 4 videos to get a second
pizza (total utility is the same at points a and
b); the marginal utility of another pizza per
week is just sufficient to compensate for the
utility lost from decreasing video purchases
by 4 movies per week.
2
b
c
d
I
0
1 2 3 4 5
10
Pizzas per week
5
Exhibit 9: An Indifference Curve
In moving from point b to c, again
total utility is constant; the person is
willing to give up only 1 video for
another pizza. Once at point c, the
person is willing to give up another
video only if they get two more pizzas
in return  combination d consists
of 5 pizzas and 2 videos
Points a, b, c, and d can be connected
to form the indifference curve, I,
which represents possible
combinations of pizza and videos that
would keep the person at the same
level of total utility.
Combinations of goods along an
indifference curve reflect some
constant, though unspecified, level of
total utility
10
a
8
5
4
3
b
c
d
2
I
0
1 2 3 4 5
10
Pizzas per week
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Indifference Curves
For a person to remain indifferent
among consumption combinations, the
increase in utility from eating more
pizza must just offset the decrease in
utility from watching fewer videos
Thus, along an indifference curve, there
is an inverse relationship between the
quantity of one good consumed and the
quantity of another consumed 
indifference curves slope down
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Indifference Curves
Indifference curves are also convex to
the origin  they are bowed inward
toward the origin
The curve gets flatter as you move
down it
The marginal rate of substitution, or
MRS, between pizza and videos
indicates the number of videos that the
consumer is willing to give up to get
one more pizza, while maintaining the
same level of total utility
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Marginal Rate of Substitution
The MRS measures the consumers
willingness to trade videos for pizza 
depends on the amount of each good
the consumer is consuming at the time
Mathematically, the MRS is equal to the
absolute value of the slope of the
indifference curve
For example, in moving from combination a
to combination b, the consumer is willing to
give up 4 videos to get 1 more pizza  slope
between these two points equals –4  MRS
= 4; from b to c, MRS = 1
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Marginal Rate of Substitution
The law of diminishing marginal rate of
substitution says that as a persons
consumption of pizza increases, the
number of videos that they are willing
to give up to get another pizza declines
This implies that the indifference curve
has a diminishing slope  as we move
down the indifference curve, the
consumption of pizza increases and the
marginal utility of additional pizza
declines
10
Indifference Map
We can use the same approach to
generate a series of indifference curves,
called an indifference map  graphical
representation of a consumer’s tastes
Each curve in the map reflects a
different level of utility
Exhibit 10 illustrates an indifference
map for a particular consumer
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Each indifference curve represents
a different level of utility and each
consumer has a unique indifference
map based on their preferences
Because both goods yield marginal
utility, the consumer prefers more
of each, rather than less  curves
farther from the origin represent
greater consumption levels 
higher levels of utility  total
utility along I2 higher than along I1,
I3 higher than I2, etc
This can be seen by drawing a
ray from the origin and
following it to higher
indifference curves
Video rentals per week
Exhibit 10: An Indifference Map
10
5
I2
curve reflects greater amounts of both goods
I3
I1
0
The combination on each successive indifference
I4
5
10
Pizzas per week
12
If indifference curves
crossed, such as point i,
then every point on
indifference curve I and
every point on curve I',
would have to reflect the
same level of utility as at
point i
But point k is a combination
with more pizza and more
videos than point j  must
represent a higher level of
utility
Video rentals per week
Exhibit 11: Indifference Curves Do Not Intersect
k
j
i
I'
I
0
Pizzas per week
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Properties of Indifference Curves
A particular indifference curve reflects a
constant level of utility  the consumer
is indifferent among all consumption
combinations along a given curve
If total utility is to remain constant, an
increase in the consumption of one
good must be offset by a decrease in
the consumption of the other good 
indifference curves slope downward
Higher indifference curves represent
higher levels of utility
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Properties of Indifference Curves
Because of the law of diminishing
marginal rate of substitution,
indifference curves are bowed in toward
the origin
Indifference curves do not intersect
Indifference curves are a graphical
representation of a consumer’s tastes
for the two goods
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Properties of Indifference Curves
Once we have the consumer’s
indifference may, we turn to the issue of
how much of each good will be
consumed?
To answer this question, we must
consider the relative prices of the two
goods and the consumer’s income
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Budget Line
Budget line depicts all possible
combinations of movies and pizzas,
given prices and your budget
Suppose movies rent for $4, pizza sells
for $8, and the budget is $40 per week
 if you spend the entire $40 on videos,
consumer can purchase 10 videos, and
if on pizzas person can afford 5 per
week
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Exhibit 12: A Budget Line
These two intercepts are then
connected to form the budget
line.
The budget line defines all
possible combinations of the
two goods, pizza and videos,
that can be purchased, given
prices and income  can be
thought of as a consumption
possibilities frontier
Video rentals per week
Given the prices and budget, the
budget line meets the vertical axis at
10 videos and meets the horizontal
axis at 5 pizzas
10
5
At the point where the budget line meets
0
the vertical axis, the maximum number
of videos you can rent equals income
divided by the video rental price = I / pv
and for the horizontal axis is I / pp
Slope of the budget line
indicates what it costs the
consumer in terms of
foregone video rentals to get
another pizza
–pp –$8
Slope =
=
= –2
pv
$4
5
10
Pizzas per week
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Summary
The indifference curve indicates what
the consumer is willing to buy
The budget line shows what the
consumer is able to buy
When the indifference curve and the
budget line are combined, we find the
quantities of each good the consumer is
both willing and able to buy
Exhibit 13 illustrates this
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The utility-maximizing consumer
will select a combination along the
budget line that lies on the highest
attainable indifference curve
Given prices and income, this
occurs at point e, where I2 just
touches, or is tangent to, the
budget line  buy 3 pizzas at $8
each and rent 4 videos at $4 each
Other attainable combinations
along the budget line reflect
lower levels of utility. For
example, point a is on the budget
line  a combination that can be
purchased. However, point a lies
on a lower indifference curve.
Video rentals per week
Exhibit 13: Utility Maximization
10
a
5
4
e
I1
0
3
5
I2
I3
10
Pizzas per week
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Consumer Equilibrium
Consumer equilibrium occurs where the
slope of the indifference curve is equal
to the slope of the budget line
Recall that the absolute value of the
slope of the indifference curve is the
marginal rate of substitution, and the
absolute value of the slope of the
budget line equals the price ratio
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Consumer Equilibrium
Thus,
MRS = Pp / Pv
Further, the marginal rate of
substitution of pizzas for video rentals
can be found from the marginal utilities
of pizza and video  MRS = MUp / MUv
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Consumer Equilibrium
In fact, the absolute value of the slope
of the indifference curve equals
MUp/MUv and the slope of the budget
line equals pp / pv  the equilibrium
condition for the indifference curve
approach can be written as
MUP  MUV
PP
PV
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Effects of a Change in Price
What happens to the consumer’s
equilibrium consumption when there is
a change in price?
The answer can be found by using
indifference curve approach to derive
the demand curve
Exhibit 14 illustrates this process
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Exhibit 14: Effect of a Drop in Price of Pizza
The initial equilibrium here is at point e with 3
pizzas and 4 videos.
If the price of pizza falls from $8 to $6 per unit,
other things constant, the price drop means that
if the entire budget were devoted to pizza, the
consumer could purchase 6.67 pizzas (40 / 6)
Thus, even though money income has remained
the same, real income has increased because of
the lower price of pizza.
Since the rental price of videos has not changed,
the maximum number of videos that can be rented
remains the same at 10.
10
5
4
e
e"
As a result of the price change, the lower end of the
budget line rotates out from 5 to 6.67
After the price change, the new equilibrium occurs
at e", where pizza purchases increase from 3 to 4,
and, as it happens, video rentals remains at 4.
I
0
3 4 5
7
Pizzas per week
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Exhibit 14: Effect of a Drop in Price of Pizza
The demand curve is shown in the lower panel
and depicts how price and quantity demanded
are related
Specifically, when the price of pizza falls from
$8 per unit to $6 per unit, other things assumed
constant, the quantity demanded increases
from 3 to 4
Videos per week
(a) 10
5
4
I
0
Price per pizza
(b)
Since the consumer is on a higher indifference
curve at e", the consumer is clearly better off
after the price reduction  the consumer
surplus has increased
e"
e
$8
6
3 4 5
7
Pizzas per week
e
e"
D
0
3 4
Pizzas per week
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Income and Substitution Effects
The law of demand was initially
explained in terms of an income effect
and a substitution effect
With indifference curve analysis we
have the analytical tools to examine
these two effects more precisely
Exhibit 15 illustrates this process
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Suppose the price of pizza falls from $8 to
$4, other things constant  consumer
can now purchase a maximum of 10
pizzas with a budget of $40.
The budget intercept line rotates out
from 5 to 10 pizzas.
The increase in the quantity of pizzas
demanded can be broken down into the
substitution and the income effect of a
price change. When the price of pizza
falls, the change in the ratio of the price
of pizza to the price of video rentals
shows up through the change in the slope
of the budget line.
Video rentals per week
Exhibit 15: Substitution and Income Effects
10
e*
5
4
e
I*
I
0
3
5
To derive the substitution effect, let’s assume that you must
maintain the same level of utility after the price change as before
 consumer’s utility level has not changed but the relative prices
you face have changed.
10
Pizzas per week
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A new budget line reflecting the change in
relative prices, not a change in utility is
shown by the dashed line CF.
Given the new set of relative prices,
the consumer would increase the
quantity of pizza demanded to the
point on the indifference curve I is
just tangent to the dashed budget line
 utility is at the initial level with the
new relative prices, but adjust income
so that utility remains the same
Video rentals per week
Exhibit 15: Substitution and Income Effects
10
C
5
4
e*
e
I*
e'
The consumer moves down along the
3 4 5
0
indifference curve I to point e', renting fewer
videos but buying more pizzas. These
Substitution
changes reflect the substitution effect of lower effect
prices of pizza
I
F
10
Pizzas per week
Since consumption bundle e' represents the same level of utility as consumption
bundle e, the consumer is neither better or worse off at point e'.
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But at point e', the consumer has not
spent the full budget. The consumer’s
real income has increased because of the
lower price of pizza  he is able to attain
point e* on indifference curve I*.
At this point, the consumer buys 5
pizzas and rent 5 videos. Because prices
are held constant during the move from
e’ to e*, the change in consumption is
due solely to a change in real income 
the change in the quantity demanded
from 4 to 5 reflects the income effect of
the lower pizza price
The substitution effect is shown by the
move from point e to point e' in response
to a change in the relative price of pizza,
with utility held constant along I. The
income effect is shown by the move from
e' to e* in response to a change in real
income with relative prices constant
Video rentals per week
Exhibit 15: Substitution and Income Effects
10
C
5
4
e*
e
I*
e'
I
0
3 4 5
Substitution
effect
F
10
Pizzas per week
Income
effect
30