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Utility Maximization
Consumption
Why do we consume so much? Because we agreed that more is better because it can
make us happier (“Lead a Richer
Life!”). In fact, we can argue that:
The goal of consumers is to spend their wealth and income on goods and services
in a way that allows them to obtain the greatest or total utility (utility).
Given this assumption, our goal as economists is to think rationally and use the
information to make predictions about consumption (economic) activities.
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There are two basic laws (that we have already discussed) can explain the rational or
utility maximizing consumer behavior.
1. Law of Demand. If P declines then Q demanded rise. It is driven by the
basic notion that “more is better” up to certain saturation point.
Income effect – When Px declines then the “Purchasing Power” of
income rise. We can consume “more” of Qx.
Substitution effect - When Px declines then “more” of its Qx will be
substituted for consumption.
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2. Law of Diminishing Returns. In this case, more KISSes you consume, they
provide less of marginal utility.
Marginal Utility (MU)
MU
Demand
1 2 3 4 5 6 7 8 9 10 # KISS
Give the referenced two laws that govern our consumption (economic activities), how
would you maximize your total utility (TU happiness)?
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Study the relationship between TU and MU graphical analyses. They suggest that:
TU is at its maximum when MU is at zero. This means that more is better up to the
saturation point. Beyond this point, more becomes worse.
Note that time (t) affects the location of saturation point on the X axis. In other words,
time affects the rate of diminishing return when consuming.
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Can we model this?
Yes, we can model this simple consumption model. More importantly, it can help us
predict (rational) consumer behaviors.
Now, how do we spend our income to help obtain the maximum total utility? Use your
income to consume much as you can on any particular good (e.g. wine) until you had it
enough or before you get sick! (MU of W = 0)
However, the real world is more complicated. Man does not live by wine alone. We
consume varied goods (services).
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Therefore, we consume a combination of goods (services) until their consumption leads
to individual marginal utility equals to zero at a particular time.
Optimal Consumption (Max. Total Utility)
MU of Wine = 0
MU of Beer = 0
MU of Cigar = 0
Etc…
Interestingly, this outcome assumes that we have no income (or wealth) constraint or
have unlimited resources. Therefore, we can consume all we want to achieve MU = 0
all goods.
However, in the real world, our income (or wealth) is limited. Therefore, with the income
or budget constraint, we are unable to get to MU = 0. Why? Cannot get all we want.
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Case Study
Given the budget of $12, how much Wine, Beer and Cigar (units) should you consume as
a utility maximize?
Note that with $36 budget, you can consume enough Wine, Beer and Cigar to
achieve MU of W, B & C = 0.
Price: $1 for a unit of Wine, Beer, Cigar
# of MU of MU of MU of
Consumption Wine Beer Cigar
1 5 5 5
2 4 4 4
3 3 3 3
More is Diminishing
Better +
4 2 2 2
5 1 1 1 Returns -
6 0 0 0
Case of Irrational One:
2 Wines $2
4 Beers $4
6 Cigars $6
Total $12
Marginal Utility (Given Consumption Choices of 2W, 4B & 6C):
MU of Wine = 4
MU of Beer = 2
MU of Cigar = 0
Total Utility (Given Consumption Choices of 2B, 4B & 6C):
1 Wine 9 Utilities (5 + 4)
4 Beers 14 Utilities
6 Cigars 15 Utilities
Total 38 Utilities
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Can we do better than TU = 38?
Remember, we want to utilize our limited income to maximize total utility. What
combination of goods should be consumed to maximize the utility?
(Try it…)
# of MU of MU of MU of
Consumption Wine Beer Cigar
1 5 5 5
2 4 4 4
3 3 3 3
4 2 2 2
5 1 1 1
6 0 0 0
Case of Rational One:
4 Wines $4
4 Beers $4
4 Cigars $4
Total $12
Marginal Utility (Given Consumption Choices of 4B, 4B, & 4C):
MU of Wine = 2
MU of Beer = 2
MU of Cigar = 2
Total Utility:
4 Wines 14 Utilities
4 Beers 14 Utilities
4 Cigars 14 Utilities
Total 42 Utilities
By achieving MUw = MUb = MUc, the TU is maximized for a given budge or income
scarcity!
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What happens if the unit prices are different?
1 Unit of Wine = $1
1 Unit of Beer = $1
1 Unit of Cigar = $2
Do we now need to change our consumption behavior? Yes. We are always looking for
best deals.
P =$1 P=$1 P =$2
# of MU of MU of MU of
Consumption Wine Beer Cigar
1 5 5 5
2 4 4 4
3 3 3 3
4 2 2 2
5 1 1 1
6 0 0 0
(Class exercise)
You have to think in terms of how should I spend my next or marginal dollar (e.g.
marginal cost and benefit analysis).
Marginal utility of dollar is used to assess the extra cost of utility for consuming different
goods. Get the best deal (utility) or the most bang for your last dollar.
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MU per Dollar or Bang for Dollar
# of MU of MU of MU of Cigar
Consumption Wine Beer
1 5 5 5/2 or 2.5
2 4 4 4/2 or 2.0
3 3 3 3/2 or 1.5
4 2 2 2/2 or 1.0
5 1 1 ½ or 0.5
6 0 0 0/2 or 0.0
Note than, TU is maximized when MUb / Pb = MUb / Pb = Muc / Pc for a given budget.
Case of Rational One:
4 Wines $4
4 Beers $4
2 Cigars $4
Total $12
Marginal Utility (Given Consumption Choices 4B, 4B & 2 C):
MU of Wine per Dollar = 2
MU of Beer per Dollar = 2
MU of Cigar per Dollar = 2
Total Utility:
4 Wines 14 Utilities
4 Beers 14 Utilities
2 Cigars 4.5 Utilities
Total 32.5 (Less than 42 because lower income)
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Utility-Maximization Rule
The Utility-Maximization rule states: To maximize satisfaction (utility), the consumer
allocates his or her money income so that the last dollar spent on each product yields the
same amount of extra (marginal) utility.
Personally, happiness is achieved when you have an appropriate and efficient balance in
your life.
The utility-maximization rule can be used to solve a long-standing economic puzzle such
as the diamond and water paradox.
Why water (essential good) price is lower than diamond (non-essential good) price?
Hint: MU of Water / P of Water = MU of Diamond / P of Diamond
1 utility / $1 = 1000 utilities / $1000
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How come? The water supply is high. Therefore, its price is low. We consume a lot of
water until its MU is low. Whereas, the diamond supply is low.
mu
Consumer Surplus
s
mu = p
q
Note that a diamond cartel controls the global supply of diamonds or keep the diamond
supply curve far to the left. Thus, its price remains high. Therefore, we consume a few
diamonds thus its MU if high.
The utility maximization rule can also help to explain the current puzzles related to our
market (economic) activities.
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Meet a great entertainer who understands applies the utility maximization rule to make a
long lasting career.
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Group Question 5.1
Why movie theaters charge so much for their popcorn and sell less? Hint: Profit
maximization (via Total Revenue Test).
Why movie theaters price their popcorns so that the price difference between large
and small varies by just $.50? Hint: Profit maximization (via Utility Maximization
Rule)
Why do rock bands charge $50 for their concerts when they know that they could
sell out at $100? Hint: Profit & Utility Maximization
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De we have a model (map)?
Yes, we do have an elegant model based on indifference curves. This model incorporates
the two laws of consumer consumption behavior (which discussed initially) to help us
predict our consumption behavior given limited income.
Indifference curve is an individual and subjective line or curve that shows alternative
combinations of goods that yields the equal satisfaction or utility.
Therefore, the individual is indifferent between consuming A or B along the given
indifference curve. Hence, the model can not be used to predict consumption behavior of
multiple individuals.
A
6 C
3 D
I or 10 Utilities
2 4 6 B
I = 10 utilities
C = 6A & 2 B = I
D = 3A & 4 B = I
The consumer is indifferent between C & D. Why? The level of utilities does not change
(e.g. I = 10 utilities).
Therefore, the consumer is indifferent to alternative combinations of A & B as long as the
overall utility does not change.
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Why is the indifference curve down sloping? To maintain its overall level of given
utility, if more of A then less of B… This shows a negative relationship or trade off to
maintain the constant level of utilities.
Why is the indifference curve convex in shape? If more of A then a lot less of B… It
shows the effect of the diminishing returns
Note that the indifference curve model elegantly incorporates the Law of Demand (more
is better) and Diminishing Returns (more is worse).
The Indifference Map
Saturation Point on the
A Hills of Happiness…
mu=0
E Higher Utility
6 C
F
I2
3 D
I1
origin
2 4 6 mu=0 B
The indifference map depicts all possible levels of utilities attainable from various set
combination of goods. F > D since more of A & B is better!
Moving away from the origin, each successive indifference curve represents a higher
level of utilities until the saturation point is reached.
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What about the limited income or budget?
We need to incorporate the budget Line. In most cases, we never may get to our
saturation point since we have limited resources.
#A
10 Budget / Price of A = #A P of A = $1
D ($10 / $1 = 10)
C
utilities
5
Budget / Price of B = #B P of B = $2
($10 / $2 = 5)
To maximize the total utility, which combination of A or B is preferred? D or C?
Answer: C
Why? It is on the higher utility or indifference curve! Note that at D, it is rational to
trade or exchange A for B at 2 to 1 ratio (relative prices) and move down along the
budget line and get to C (or higher indifference curve).
Observe that to maintain the constant utility level, the relative trade off between 2A and
1B at the point C implies MUA = 1 and MUB = 2. Therefore given:
MUa / Pa = MUb / Pb 1U / $1 = 2 U / $ 2
We comply with the utility maximization rule.
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18
Can indifference curves intersect?
As long as you are rational, they do not intersect. What are we saying about the given
preference set?
A
C D
F
E
B
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Rationality assumes a transitivity relation. As you can see in this case,
D > C since more B is consumed
F > E since more A is consumed
D = E on the same indifference curve
C = F on the same indifference curve
D > C E > F and E < F Irrational preference!
Acting like a two year old!
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Can we derive a demand curve?
We should be able to since it incorporates the key determinants of demand.
A
Price of B falls from $2 to $1
Budget / Pb (down) = B (up)
B
P of B
$2
$1
Demand Curve
B
Finally, can the model also explain why an increase in income is good?
Note that the increased income shifts the budget line to the right and enable the individual
to achieve a higher utility curve.
More Stuff is Better (up to the saturation point)!
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Note that the model is only good if it can help us to predict our economic behavior. We
will try to make predictions.
How would winning a $100K lottery affect consumption of steaks (normal good) and
hotdogs (inferior good)?
(Why don’t you try to draw this indifference curve map…)
What about the cave person?
The model can predict the change in his consumption pattern between fruits and buffalo
if a herd of buffalo passes by his cave and reduce the required hunting time by 50
percent?
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Group Question 5.2
Review the indifference curve information (three next slides) at
http://ingrimayne.saintjoe.edu/econ/MaximizingBeha/Indifference.html
Draw the indifference curve map model to rationally or systematically predict the
consumption behavior of Professor John.
Professor John spends all of his income on food, beer and clothing. He gets a pay raise
from $3,000 a month to $4,000 a month (wow!!! unbelievable). At the same time, the
food prices increase by 50 percent.
How do you think Professor John will change his spending pattern as a result of
the changes in his income?
Can you say whether Professor John is better off or worse off in his new
situation?
After the changes in his income, food, beer and clothing prices rise by 50 percent.
How do you think Professor John change his spending pattern?
Can you say whether Professor John is better off or worse off in his new
situation?
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