Consumer behavior (introduction)
Managerial Economics
Moon Oulatta, PhD
Department of Economics
Consumer behavior (basic introduction)
Learning objectives
Understanding consumer preferences.
Be familiar with the consumer optimization problem: the
characteristics of a utility function, consumer preferences,
and the budget constraint.
Use these slides as additional learning material to your
recorded videos, book readings, and in-class solutions.
Consumer behavior (basic introduction)
The consumer optimization problem
The household chooses a combination of goods to maximize its
well-being given a limited amount of income. Therefore, consumer
theory incorporates three key factors:
budget constraint: consumers have a fixed amount of income that
they must allocate between goods in order to maximize their
well-being.
consumer preference: reflects a consistent systematic pattern of
consumption and describes the manner in which the consumer
prefers to consume a combination of goods.
utility function: a mathematical function that reflects the consumer
preference and assigns a level utility to different baskets of goods.
For theoretical simplicity, introductory courses in economics usually
focus on two different types of goods to model the utility function.
Consumer behavior (basic introduction)
Utility function
Utility can be defined as the pleasure or satisfaction that a
consumer derives from consuming various market baskets. Consider
two types of utility functions:
ordinal utility: focuses exclusively on ranking market baskets from
least to most preferred. Thus, any numerical values generated from
an ordinal utility function is strictly used for ranking purposes and
cannot be used to draw any conclusions other than ranking.
cardinal utility: describes the magnitude of consumer preferences,
which is difficult to measure, because assigning an exact value to
satisfaction is categorically impossible.
Consumer behavior (basic introduction)
Axioms of consumer preference
The axioms of consumer preferences are important to develop a
sound theory of consumer behavior. For simplicity, we summarize
the key axioms of consumer preferences as follows:
completeness: preferences for goods are assumed to be complete,
meaning that a consumer should always be able to rank market
baskets.
transitivity: consumers preferences should be consistent. If a
consumer prefers basket 1 to basket 2 and basket 2 to basket 3.
Then the consumer should also prefer basket 1 to basket 3.
non-satiation: the goods that enter the utility function are assumed
to be desirable, which means that more of one good is better than
less goods. Consumers are never satiated (satisfied).
Consumer behavior (basic introduction)
What is an indifference curve?
The indifference curve connects the market baskets that provide
the same level of utility to the consumer.
The shape of the indifference curve gives us information about
consumer preferences.
For two goods: the higher the indifference curve, the more of both
goods you consume, which means that every consumer wants to be
on the highest indifference curve.
Consumer behavior (basic introduction)
Indifference curve (slope)
The slope of an indifference curve is referred to as the marginal
rate of substitution, which describes the maximum amount of one
good a consumer is willing to give up in order to acquire an extra
unit of another good.
Here, income is fixed and the consumer is indifferent across all
market baskets that lie on the indifference curve. To consume an
extra unit of one good, the consumer must give up units of the
other good; this ensures that the level of utility remains constant
along the indifference curve.
The change in utility that arises from consuming an extra unit of
one good while giving up units of the other good balances out: this
is why every bundle on an indifference curve provides the same level
of utility as you move up and down on the indifference curve.
Consumer behavior (basic introduction)
Indifference curve (slope)
Let’s assume that a consumer consumes two goods: shoes (X
1
) and
food (X
2
). Moreover, for consumption preferences, she prefers
averages rather than extremes. For simplicity, let’s assume an
implicit utility (U
∗
) defined as follows
U
∗
= U(X
1
, X
2
) (1)
It follows that the total change in utility (dU
∗
) that arises from
consuming an extra unit of food (X
2
), while giving up units of shoes
(X
1
) can be explained by totally differentiating Equation (1) as
follows
dU
∗
99K U
X
1
(X
1
, X
2
)dX
1
+ U
X
2
(X
1
, X
2
)dX
2
= 0
Consumer behavior (basic introduction)
Indifference curve (slope)
From which we can solve for the slope of the indifference curve by
rearranging the previous equation as follows
−
dX
1
dX
2
=
U
X
2
(X
1
, X
2
)
U
X
1
(X
1
, X
2
)
(2)
where Equation (2) states that the slope of the indifference curve
(here it is the marginal rate of substitution of food for shoes) is
negative and equal to ratio of the marginal utilities of both goods.
Consumer behavior (basic introduction)
Budget constraint
Consumer theory assumes that consumers face an income
constraint: there is a fixed amount of income that must be spent
on all goods in order to maximize utility from consumption (that’s a
key principle).
A budget constraint can be expressed as a simple linear equation
that defines the total amount of goods that a consumer can acquire
with a limited amount of income (I).
With two goods (X
1
and X
2
), the budget line can be expressed as
follows
I = P
1
X
1
+ P
2
X
2
(3)
where Equation (3) represents the budget constraint. P
1
is the price
of X
1
and P
2
is the price X
2
. The total amount of goods that can
be purchased cannot exceed total income (I ).
Consumer behavior (basic introduction)
Optimal consumer choice
Given the implicit utility function, consumer preference, and the
budget constraint, we can determine the optimal consumption
bundle.
The optimal consumer choice occurs at the point of tangency,
where the slope of of the indifference curve is equal to the slope of
the budget constraint as follows
MRS ≡
U
X
2
(X
1
, X
2
)
U
X
1
(X
1
, X
2
)
=
P
2
P
1
(4)
The equal marginal principle implies that all income must be
spent and the marginal utilities per dollar are equal as follows
U
X
1
(X
1
, X
2
)
P
1
=
U
X
2
(X
1
, X
2
)
P
2
Consumer behavior (basic introduction)
Consumer preferences
There are multiple types of consumer preferences in economics. For
this lecture, I will consider the following:
strictly convex preferences: the goods are imperfect substitutes in
consumption, and the consumers prefers averages in contrast to
extremes.
perfect complements: the consumer prefers to consume both goods
in fixed proportions.
perfect substitutes: when a consumer perceives both goods as
perfect substitutes.
Consumer behavior (basic introduction)
Strictly convex preferences
The consumer maximization problem consists of maximizing the
following the following Cobb-Douglas utility function
MAX
{X
1
,X
2
}
n
X
α
1
X
(1−α)
2
o
subject to a simple budget constraint
I = P
1
X
1
+ P
2
X
2
Consumer behavior (basic introduction)
Strictly convex preferences
Secondly write down the Lagrangian as follows
Φ =
n
X
α
1
X
(1−α)
2
o
+ λ(I − P
1
X
1
+ P
2
X
2
)
The the first-order conditions are given by
∂Φ
∂X
1
αX
α−1
1
X
(1−α)
2
− λP
1
= 0
∂Φ
X
2
(1 − α)X
α
1
X
(−α)
2
− λP
2
= 0
∂Φ
∂λ
↠ I − P
1
X
1
+ P
2
X
2
= 0
Consumer behavior (basic introduction)
Strictly convex preferences
Solve for λ from the first two (FOCs) to get the marginal utility
per dollar for both goods as follows
λ =
αX
(α−1)
1
X
(1−α)
2
P
1
(5)
λ =
(1 − α)X
α
1
X
−α
2
P
2
(6)
set equations (5) and (6) equal to each as follows
α
(1 − α)
X
(α−1)
1
X
(1−α)
2
X
α
1
X
−α
2
=
P
1
P
2
simplifying the above as follows
X
(1−α+α)
2
X
(α−α+1)
1
=
(1 − α)
α
P
1
P
2
Consumer behavior (basic introduction)
Strictly convex preferences
Simplifying the above and solving for optimal X
2
yields the following
condition
X
2
=
(1 − α)
α
P
1
P
2
X
1
(7)
plug back equation (7) into your budget constraint to arrive at the
following
I = P
1
X
1
+ P
2
(1 − α)
α
P
1
P
2
X
1
simplifying the above as follows
I = P
1
X
1
+
(1 − α)
α
P
1
X
1
factoring out P
1
X
1
and simplifying the above yields the following
solution
I = P
1
X
1
α + 1 − α
α
Consumer behavior (basic introduction)
Strictly convex preferences
Then from the above we can solve for the optimal demand function
for X
1
as follows
X
∗
1
= α
I
P
1
(8)
to solve for the optimal demand for good X
2
, you can plug equation
(8) into equation (7) to derive the optimal demand for good (2) as
follows
X
2
=
(1 − α)
α
P
1
P
2
α
I
P
1
simplifying further yields
X
∗
2
= (1 − α)
I
P
2
Here the optimal demand functions are referred to as the
Marshallian demands or the uncompensated demand curves.
Consumer behavior (basic introduction)
Strictly convex preferences
For strictly convex preferences, the shape of our indifference curve is
bowed in and strictly convex to the origin: this emphasizes the
principle of diminishing marginal utility or a declining marginal
rate of substitution (slope becomes less negative as we move down
the indifference curve).
For strictly convex preferences, it follows that the Marshallian
demands are positively related to income and negatively related to
prices.
Consumer behavior (basic introduction)
Perfect substitute preferences
It is possible for two goods to be perfect substitutes, which means
that the consumer is equally satisfied in consuming each individual
good or a combination of both goods. This choice depends on the
marginal utilities per dollar for both goods.
Under these types of preferences, both goods exhibit a high degree
of substitution when it comes to price changes (they are perfect
substitutes).
Consumer behavior (basic introduction)
Perfect substitute preferences
The consumer problem can be defined as follows
Φ = αX
1
+ βX
2
+ λ(I − P
1
X
1
+ P
2
X
2
)
The the first-order conditions are given by
∂Φ
X
1
↠ α − λP
1
= 0
∂Φ
X
2
↠ β − λP
2
= 0
∂Φ
∂λ
↠ I − P
1
X
1
+ P
2
X
2
= 0
Consumer behavior (basic introduction)
Perfect substitute preferences
We have three specific solutions for the case of perfect substitutes.
The first two cases produce a corner solution:
If the marginal utility per dollar for X
1
is greater than the marginal
utility per dollar for X
2
, then the consumer does not consume both
goods. She only consumes X
1
:
α
P
1
>
β
P
2
X
1
=
I
P
1
; X
2
= 0
Alternatively, the consumer only chooses X
2
when the marginal utility
per dollar for X
2
is greater than the marginal utility per dollar for X
1
,
β
P
2
>
α
P
1
X
2
=
I
P
2
; X
1
= 0
Consumer behavior (basic introduction)
Perfect substitute preferences
When the marginal utilities per dollar are equal
α
P
1
=
β
P
2
it implies that the indifference curve coincides with the budget
constraint, which means that the consumer can consume any
combination of both goods along the budget constraint.
For goods that are perfect substitutes, the marginal rate of
substitution (slope) for both goods is always constant (think of a
downward-sloping straight line): consuming an extra unit of one
good requires giving up a fixed quantity of the other good. Here,
the utility function remains convex but not strictly convex.
Consumer behavior (basic introduction)
Perfect complements preferences
Some consumers prefer to consume goods in fixed proportions: this
is the case, where both goods are complements in consumption.
Two goods are perfect complements, when consuming one good only
increases utility if the other good is consumed in a specific manner.
For complementary goods, the utility function is defined by the
minimum of X
1
and X
2
. Moreover, the utility function is not easily
differentiable because of the L-shaped indifference curve. However,
the consumer problem can be solved by finding the point at which
the indifference curve kinks.
Consumer behavior (basic introduction)
Perfect complements preferences
The consumer maximization problem consists of maximizing the
following Leontief utility function
U = min(X
1
, X
2
) (9)
subject to the constraint that all income is spent on both goods
I = P
1
X
1
+ P
2
X
2
Consumer behavior (basic introduction)
Perfect complements preferences
We know that utility is a minimum of X
1
and X
2
. Hence the optimal
condition requires that the consumer consumes both goods in fixed
proportions as follows
X
1
= X
2
(10)
We can plug Equation (10) into the budget constraint and derive
X
∗
2
, then plug X
∗
2
back into equation (10) to obtain the
uncompensated demand function for X
∗
1
. Following the latter
guidelines, the Marshallian demands are derived as follows:
X
∗
1
=
I
(P
1
+ P
2
)
; X
∗
2
=
I
(P
1
+ P
2
)
Note that because both goods are perfect complements, the optimal
demand functions depend negatively on the prices of both goods.
Consumer behavior (basic introduction)
Conclusion
The objective of this lecture is to introduce MBA students to some
basic principles in consumer theory in order to facilitate the in-class
lectures, which rely on more advanced topics.
At the minimum, this lecture introduces some key principles that are
useful to understand consumer theory. In class, I am going to discuss
other utility functions (for example, we will go over the CES utility
function and the quasi-linear utility function).
Understanding consumer preferences is critical for economic
modeling: for example, we will examine the properties of different
utility functions to determine which one is more appropriate for
modeling different types of goods.
