Intermediate Microeconomics
UTILITY
BEN VAN KAMMEN, PHD
PURDUE UNIVERSITY
Outline
To put this part of the class in perspective,
consumer choice is the underlying explanation for
the demand curve.
◦As utility and preferences are discussed, consider
how it relates to the demand curve.
Define and describe the problem that rational
consumers solve.
◦ Define the consumer’s objective function and describe its
properties.
◦ Describe the constraints facing a consumer’s choices.
Utility
“The satisfaction that a person receives from his or
her economic activities.”
Two obvious definitions:
◦ Good: Something that increases a person’s utility, ceteris
paribus.
◦ Examples: food, good books, Pontiac Trans Ams.
◦ Bad: Something that decreases a person’s utility, ceteris
paribus.
◦ Examples: litter, taking away of a Trans Am.
Ceteris Paribus: “other things being equal.”
Utility Functions
How can we formalize (“model”) the concept of
utility mathematically?
If you can account for all the things that affect a
consumer’s utility, a utility function could be
specified with all of those things as arguments.
◦ Utility is a function of all goods and bads for a given
consumer.
Example
𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈
= 𝑈𝑈(𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓, 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏, 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐴𝐴𝐴𝐴𝐴𝐴, 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙, 𝑛𝑛𝑛𝑛𝑛𝑛 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇
− 𝐴𝐴𝐴𝐴 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡, 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 . . . )
As the example illustrates, there are many things that
affect utility.
◦ This is something that our model will have to simplify.
Preferences
Another thing to consider when modeling utility is
how each good affects the consumer’s utility.
◦ Inevitably, some people will like a given good more than
others. e.g., I like Jelly Belly® jelly beans more than 99%
of the population, but I hate frosting—something that
many people enjoy.
◦ So in addition to representing a list of all goods and bads,
a utility function also has to account for each good’s
effect on utility.
◦ Note: this is also something that will have to be simplified to make
the model workable.
Preferences
Lastly a utility function should account for the possibility
that the marginal utility of a good changes depending on
how many we already have . . .
◦think back to the water example and diminishing marginal
utility.
To recapitulate
Modeling utility involves writing a mathematical expression
that:
1) contains an exhaustive list of goods and bads,
2) specifies the marginal utility for each combination of
those goods and bads, and
3) can be analyzed by a human being using a minimum of
calculus.
◦Clearly something in the preceding list will have to be
compromised.
◦My suggestions are #1 and #2.
Properties of Utility Functions
Completeness: there is a parable about a donkey
that found himself between a trough of oats and a
trough of hay and starved to death because he
couldn’t decide.
We will assume consumers are always able to say
which bundle of consumption they prefer.
You could say that utility is “defined”, in the
mathematical sense, for all relevant bundles.
Properties of Utility Functions
Transitivity: if consumption bundle X is preferred to bundle Y, and
bundle Y is preferred to bundle Z, transitivity implies that X is preferred
to Z.
◦ Analog in geometry: 𝑋𝑋 > 𝑌𝑌 > 𝑍𝑍 → 𝑋𝑋 > 𝑍𝑍.
More of an economic good is, well, better.
◦ Goods have nonnegative marginal utility.
◦ Consumers prefer more of a good to less, ceteris paribus.
Modeling Utility
Time to abstract away from reality to make some
analytical sense of consumers’ preferences.
Assumption: a consumer’s utility is a function of
only 2 goods: jelly beans and Pabst Blue Ribbon
beer.
◦ To make this assumption seem less extreme, tell yourself
that the consumer has all the other things that he wants
and the other goods he consumes does not change, i.e.,
ceteris paribus.
The function is: 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 = 𝑈𝑈(𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽, 𝑃𝑃𝑃𝑃𝑃𝑃).
Graphing Utility
There are 3 ways to graph this utility function in 2dimensional space.
1.Graph utility as a function only of Jbeans, holding the level
of PBR constant.
2.Graph utility as a function only of PBR, holding the level of
JBeans constant.
3.Graph quantity of one good as a function of quantity of the
other good, holding the level of utility constant.
The
st
1
way of graphing utility
Do jelly beans have diminishing marginal utility in this graph?
The
nd
2
way of graphing utility
The
rd
3
way of graphing utility
Note that this function has a negative slope that takes form,
Δ𝑃𝑃𝑃𝑃𝑃𝑃
.
Δ𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽
Indifference curves
The 3rd method, yields a graph of an indifference
curve:
◦ “a curve that shows all the combinations of goods
that provide the same level of utility.”
Each point on the curve is a combination of JBeans
and PBR, and each point gives the consumer a
utility level of 𝑈𝑈0.
The consumer is indifferent to all points on this
curve, so it is called an indifference curve.
Properties of indifference curves
When analyzing utility as a function of two goods:
◦Indifference curves slope downward.
◦Indifference curves do not intersect.
◦Indifference curves are convex (“bowed toward the
axis”).
◦An increase in utility is shown by an indifference curve
that is further from the origin.
Indifference Curves Slope Downward
Begin from some point “A” on the indifference curve, 𝑈𝑈0. Move
to a point “B” that is directly to the left of “A” and not on the
indifference curve.
◦ Conceptually take away some of the consumer’s jelly beans without
changing his PBR level (Δ𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽 < 0).
◦ The consumer will certainly not be indifferent to this change.
◦ In order to make him indifferent, you would have to give him some more
PBR (Δ𝑃𝑃𝑃𝑃𝑃𝑃 > 0) and move him up to point “C”.
Taking the two changes together we have a negative change in
jelly beans and a positive change in Pabst, i.e., a negative slope.
What the consumer has effectively done is “substitute” PBR for
jelly beans and move upward along the indifference curve.
Indifference curves do not intersect
This is something that would make an economist’s head
explode.
Look at the point where the two curves on the next slide
intersect; that would be two different levels of utility for the
same bundle.
◦ The same bundle of consumption cannot give the consumer two
different levels of utility!!!
◦ I know; it would freak me out, too.
What Kind of Messed up Preferences
are These?
Indifference curves are convex
The slope of an indifference curve is called the marginal rate
of substitution “MRS”.
Indifference curves get “flatter” as you move from left to
right along them.
◦ Getting flatter means the slope is decreasing.
So the MRS is decreasing as you go from left to right.
The quantity of JBeans the consumer is willing to substitute
for PBR is continually decreasing as you go from left to right.
◦ This is a result of diminishing marginal utility, which we will examine
more later.
◦ For now suffice it to say that indifference curves are convex.
Variation in utility (graphically)
Note: U2 > U1 > U0.
Marginal Rate of Substitution
“The rate at which a consumer is willing to reduce
consumption of one good when he or she gets one
more unit of another good. The slope of an
indifference curve.”
◦ Abbreviation: MRS
Diminishing MRS: Diminishing marginal utility
leads consumers to prefer balanced consumption
bundles to skewed bundles. This gives indifference
curves their convex shape.
Particular Preferences
Preferences affect the shape of indifference curves.
Consider the following cases.
◦A useless good.
◦An economic bad.
◦Perfect substitutes.
◦Perfect complements.
A Useless Good
The consumer cannot increase utility by consuming more of the useless good.
A Bad and a Good
MRS is positive; increasing the bad requires that you increase the good to
stay at a given level of utility.
Perfect Substitutes
Perfect substitutes have a constant marginal rate of substitution. Their
indifference curves are linear.
Perfect complements; “Hi,
Bob”
Goods are consumed in fixed proportions. Each good has zero marginal
utility. Unless you have more of the other good, you can’t increase utility.
Utility maximization
The goal of the consumer is to maximize utility. In the
absence of scarcity, he would choose to do so by consuming
an infinite quantity of all goods.
◦ But goods are scarce.
A consumer maximizes utility subject to a budget constraint.
Budget constraint
“The limit that income places on the combinations
of goods that a consumer can buy.”
◦ Each consumer has a finite capacity (income) for
purchasing goods.
◦ In the two-good model, he allocates his income between
the two goods as follows:
Income = [Expenditure on X] + [Expenditure on Y]
◦ Expenditure on each good equals the quantity of the
good multiplied by its price:
Income = I = PX*X + PY*Y
Budget constraint graphically
The vertical intercept is
𝐼𝐼
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃
, and the horizontal intercept is
𝐼𝐼
𝑃𝑃𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽
.
Price ratio
The intercepts of the budget constraint are found by setting
the quantity of one good to zero:
𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∗ 𝑄𝑄𝑃𝑃𝑃𝑃𝑃𝑃 + 𝑃𝑃𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽 ∗ 0 → 𝑄𝑄𝑃𝑃𝑃𝑃𝑃𝑃 =
𝐼𝐼
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃
𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∗ 0 + 𝑃𝑃𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽 ∗ 𝑄𝑄𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽0 → 𝑄𝑄𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽 =
𝐼𝐼
𝑃𝑃𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽
The slope of the budget constraint is the ratio of the two
prices:
◦ Rise over run implies:
𝐼𝐼
𝐼𝐼
𝑃𝑃𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽
÷
=
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃
Constrained Utility
Maximization
If the consumer has a budget constraint, he consumes a
combination of two goods that gets him on the highest
indifference curve possible.
◦ This occurs where the marginal rate of substitution equals the price
ratio:
𝑀𝑀𝑀𝑀𝑀𝑀 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅.
Graphically this occurs where the budget constraint and
indifference curve are tangent to one another.
Constrained utility maximization
Why is this Optimal?
Consider the alternatives:
◦ The consumer can get more utility than 𝑈𝑈0 by consuming more of
each good.
◦ The consumer cannot attain utility 𝑈𝑈2 because he doesn’t have
enough income.
Alternative utility levels
Only U1 is attainable and optimal.
Further examples
Constrained utility maximization for perfect
substitutes, perfect complements.
Consumers facing the same budget constraints
make different optimal choices because of
differences in preferences.
If it makes you uncomfortable to consider a utility
function of only two goods, you can think of the
good on the vertical axis as a “bundle of all other
goods” with a price equal to the index of prices of
the goods in the bundle.
Conclusion
By simplifying the number of goods and assuming a
functional form for utility, it is simple to predict what an
individual will consume, given prices.
How will their consumption change when prices change?
The next topic we will discuss is how utility maximization
produces a demand curve.