Chapter 3. Constructing a Model of Consumer Behavior
Part A, Copyright Kwan Choi, 2009
To construct a model of consumer behavior, some basic assumptions are
needed to describe how consumers use their income to purchase bundles of
goods and services.
● First, we limit our discussion to rational consumers.
Rational consumers choose consumption bundles in a consistent way and
therefore their behavior is predictable. Irrational consumers do not make the
best possible choices or else their preferences change from moment to moment
so that their actions become unpredictable.
● Second, we want to explain how consumers choose goods and services.
Consumers also make numerous decisions that are not related to consumption,
and these choices are outside the scope.
Goods are broadly defined. Instead of explaining how consumers behave
regarding a specific good, we try to explain how consumption bundles are
chosen. A bundle may include only one good or many goods.
Consumption Possibility Set
This is the set of bundles feasible for the consumers to choose.
Remark 1: Ø  R (Null consumption is feasible).
Here the null consumption bundle Ø  (0, 0,..., 0) is an empty basket.
Remark 2: Goods are perfectly divisible.
Counterexample: goods consumed in integer quantities.
A half unit of a car is not feasible.
In Figure 1, the consumer can choose only discrete units of each consumption
good.
In this case, we cannot construct a continuous utility function. That is why we
are ruling out indivisible goods.
Convexity of Consumption Possibility Set
We also assume that consumption possibility set is convex. Let R be a
consumption set. Let x and y be feasible bundles ( x R and y  R). Then a
convex combination of x and y, z is also feasible, i.e., a line connecting two
feasible points is also feasible.
z  (1   ) x   y  R.
This implies that if a bundle is feasible, a fraction of it is also feasible, i.e.,
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x
R implies  x R, 0    1.
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The set in the top panel above is convex, because any point along the line
connecting two points in the set is also in the set. However, the crescent is not a
convex set, because some points along the line connecting a and b are not in the
set.
The interior of the Chinese word meaning convex below is not a convex set.
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Notation
Symbols



/





P
Meaning
“implies”
“is implied by”
iff (if, and only if)
Not
There exists
There does not exist
For all
Is an element of
Is not an element of
Is preferred to
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I
R
Is indifferent to
Is as good as
Ranking of Commodity Bundles
The consumer is choosing a bundle within the consumption possibility
set. The fact that the consumer chooses one implies that he is able to rank or
compare different consumption bundles.
Two distinct bundles have different quantities of various goods. When
one bundle dominates another, the choice is easy and everybody will make the
same choice. When neither of two bundles dominates the other, the choice is
more difficult.
First, we begin with comparison among bundles when dominance
relation holds.
If a  b ( a1  b1 , …, an  bn and a  b , i.e., for any good, bundle
a has more than, or as much as bundle b, and at least for one commodity bundle
a has more than bundle b), then we say that a dominates b.
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If one bundle dominates another, then the ranking of consumption bundles is
simple. All consumers prefer the dominating bundle. In Figure 3, any bundle in
the blue region dominates bundle a, whereas bundle a dominates any bundle in
the green region.
However, life is not that simple. Compare bundle a and the bundles in the white
regions. Neither dominates the other. Some consumers prefer b to a, while
others prefer a to b. This is usually the case for managers. If there is a clear
dominating choice, the manager must choose the dominating option. In all other
cases, the manager must choose between options in which neither dominates
the other.
Regarding consumer rankings of consumption bundles, we make the following
axioms. (Axioms are not proven; they are deemed self-evident)
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Axioms of Consumer Preference
1. Completeness
a P b, b P a, or a I b. (a R b, or b R a)
Given any two choices (commodity bundles), one of the following is true.
Either a is preferred to b, b is preferred to a, or a is indifferent to b.
In other words, given any two alternatives, the consumer is assumed to be able
to rank them.
2. Reflexivity
a I a.
A bundle a is always “indifferent” to itself. Thus, “I” is a reflexive relation.
“>” or “greater than” is not a reflexive relation, because “x is greater than
itself” is not true.
3. Transitivity
a P b, b P c  a P b.
a I b, b I c  a I c.
Given three alternatives, if a is preferred to b, b is preferred to c, then a is
preferred to c.
If a is indifferent to b, b is indifferent to c, then a is indifferent to c.
Many relations are not transitive. “is a friend of” is not always transitive.
Similarly, “likes” is not a transitive relation. For instance, “a likes b,” and “b
likes c” does not necessarily imply “a likes c.”
Need for a Utility Function
The preference relationship is useful to us for now. However, to facilitate our
analysis later on, it is even more useful to represent consumer preferences, not
by primitive binary relationships, but by a utility function. For instance, the
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manager consciously or unconsciously maximizes his goal, or an objective
function.
U() is a utility function, if
U(a) > U(b) iff a P b and
U(a) = U(b) iff a I b.
Translated in English, U is a utility function if it assigns a higher index to a
preferred bundle and the same index for bundles that are indifferent.
Existence
Does there exist such a utility function? Yes, if consumer preferences satisfy
completeness, reflexivity and transitivity.
How many? Infinitely many.
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Method: Draw a 45 degree line from the origin. If bundle a is not on the 45
degree line, then locate a bundle b which is indifferent to a, and assign the same
utility index, i.e., u(a) = u(b).
For any point along the 45 degree line, use the distance from the origin.
u ( x1 , x2 )  x12  x22 .
Now this distance function qualifies as a utility index. Moreover, you can
construct many others. Let
v  f (u )
be a monotone increasing function of u (That is to say, whenever u rises v also
rises, or df/du > 0). For instance,
v  u 2  x12  x22
is another utility function.
Remark: Utility is simply the level of satisfaction from consuming a bundle. If
consumer preferences satisfy completeness, reflexivity, and transitivity, then
there exists a utility function that represent the preferences.
For analytical reasons, in addition to completeness, reflexivity, and transitivity,
we also want our utility function to be continuous.
Continuity
If a P b, and b P c, then  d such that
d  (1   )a  b, and d I b.
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Figure 4. Continuity of Preference
Basically, CONTINUITY states that if consumption bundles are close together,
they will be assigned utility numbers that are close together. The utility
function is continuous if you can draw indifference curves without lifting a
pencil. That is, as you move from a to b, there is no (vertical) jump in the
utility level.
This property allows us to draw indifference curves in the consumption set.
However, not all preferences permit indifference curves.
Counterexample: Lexicographic preferences
There are some preferences that violate the assumption of continuity. One
famous counterexample is called lexicographic ordering.
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Lexicographic preferences take their name from the dictionary, where words
appear in alphabetical order.
When arranging words alphabetically, we first compare their first letters, not
paying attention to all other letters. When comparing two words, we first
compare the first letters of two words and rank them. For instance, given two
words, “car” and “apple,” apple precedes car because a precedes c. In this we
do not worry about the second or third letters in the two words.
When the two words have the same first letters, then we compare the second
letters and rank the words. For instance, “car” and “cold” have the same first
letter c. Then we proceed to compare the second letter, and rank words
accordingly. Thus, “car” precedes “cold.” If the second letters are the same,
then compare the third letters, and so on.
Similarly, if a consumer has a lexicographic preferences, he first compares the
amounts of one good these bundles contain. He chooses the bundle with the
most quantity of the primary good. If the quantity of the primary goods are the
same, he then compares the quantity of the secondary good, and so on.
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Figure 5 illustrates lexicographic ordering.
In Figure 5, honor is the primary good and gold is the secondary good. This
consumer always looks for honor first, and then compares gold. Among the
three bundles, A precedes B and B precedes C. Among the points on the line
connecting A and C, one cannot draw an indifference curve through the middle
point B. (The vertical line which starts at point B is not an indifference curve,
because all points on this line is preferred to B.) Thus, a continuous utility
function cannot be constructed for lexicographic preferences.
Question: Does there exist a continuous utility function that represents
lexicographic preferences?
If x is an integer, then
U ( x, y)  x 
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y
y 1
is a utility function that represents lexicographic preferences. This is because
y /( y  1) is always smaller than one. However, it is not a continuous utility
function. It is a utility function, but it does not have indifference curves.
If x is not an integer, it cannot represent the lexicographic preferences.
If consumer preferences satisfy completeness, reflexivity, transitivity, and
continuity, then there exists a continuous ordinal utility function.
(Why make a big deal about continuity? Continuous utility functions are
differentiable. Hence they facilitate optimization to be discussed later)
Example 1: additive utility function
u ( x, y )  x  y ,
u ( x, y )  x  by, b  0.
Example 2: Multiplicative utility function
u ( x, y )  xy.
CARDINAL AND ORDINAL UTILITY
The concepts of utility functions are somewhat more complicated than they
appear at first glance.
Utility is said to be cardinally measurable if the differences in utilities are
meaningful. For instance, if u(x) = 30 and u(y) = 10, and this implies that x is
three times as good as y. A slightly weaker cardinal preference will become
useful later when we study choices involving uncertain situations.
Utility is ordinally measurable, if the utility numbers we assign to bundles
have no meaning other than to represent the ranking of these goods. In this
case, u(x) = 30 and u(y) = 10 does not mean that x is three times as good as y,
but only that x is preferred to y.
For choices without uncertainty, we need only ordinal utility functions to
explain consumer choice. If consumer choice is made under conditions of
uncertainty, a cardinal utility function is needed.
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Additional assumptions
Selfishness: An individual is interested only in his own satisfaction. His
choices are based on his preferences only. He has no regard for the impact of
his consumption on others. His utility is written as
u ( x, y, z ),
and not as u ( x, y, z, X ), where X is what others consume. This assumption
excludes altruistic behavior or envy among consumers.
Nonsatiation: More of anything is better.
If a  b and a  b (a dominates b), then a P b.
Bundle a contains at least as much of every thing as bundle b, then a P b.
Figure 6: the better set
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Convexity of Preferences
Finally, we also assume consumer preferences are convex.
If a I b, then c   a  (1   )b R b. (R is P or I, preferred or indifferent to).
That is, a convex combination of two indifferent bundles is strictly preferred to
either. This explains why consumers generally buy a little of almost everything.
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