Constructing A Model Of Consumer Behavior

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+ l ) x l y R.

This implies that if a bundle is feasible, a fraction of it is also feasible, i.e.,

2 x R implies lx R, 0

3 4 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.

5 Notation

Symbols 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 P Is preferred to

6 I Is indifferent to R 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 b 1, …, an b n 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.

7 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)

8 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

9 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.

10 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.

2 2 u( x1 , x 2 )= x 1 + x 2 .

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,

2 2 2 v= u = x1 + x 2 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 -l ) a + l b , and d I b.

11 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.

12 13 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.

14 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

y U( x , y ) = x + y +1

15 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.

16 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

17 Convexity of Preferences

Finally, we also assume consumer preferences are convex.

If a I b, then c=l a +(1 - l ) 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.