Chapter 1 Preference and Utility Function

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					Chapter 1: Preference and Utility Function



1.1. Scientific Approach to the Study of Human Behavior

This text starts with an inquiry into an individual’s preferences, since each individual’s
economic activities are motivated by the pursuit of his or her happiness and satisfaction.
This distinguishes economics from the natural sciences, and also poses some particularly
difficult tasks for economics. The study of interactions among individuals who try to
meet their desires has three important features. First, each individual’s behavior is driven
by his or her purpose in the economic arena. Hence, we need an objective function to
describe the purpose of a player, who is not merely a passive object as in physics or
chemistry. But the objective function of an individual may be in conflict with those of
others. The study of such conflicts, and their reconciliation, distinguish economics from
physics, chemistry, and other natural sciences. The mathematical approach to optimizing
an objective function subject to s     ome constraints is used to describe self-interested
behavior. The notion of optimization is beyond physical- or chemical-state equations.
     Second, while an individual’s desires play a central role in economic analysis, these are
not directly observable. Satisfaction of an individual’s desires is not only intangible, but also
involves that individual’s subjective judgment of value. This implies that it is difficult, if not
impossible, to measure degrees of satisfaction, and to conduct absolutely objective
experiments. Most economists follow the approach of the thought experiment to investigate an
individual’s behavior. That is, they make some assumptions about what the individual desires,
about the constraints facing her, and about her behavior in achieving her objectives subject to
those constraints. Then, through rigorous mathematical deduction, they establish the
connection between the intangible desires, the behavior, and the resulting tangible phenomena
(such as quantities and prices of goods purchased). The relationship between the tangible
quantities and prices has a certain correspondence to the assumed properties of the intangible
desire and the individual’s behavior. Economists can thus infer the intangible things from the
tangible relationship. If observable phenomena are compatible with the relationships
generated by mathematical deduction on the basis of the assumptions about intangible things,
then economists take the assumptions about desires and behavior to be acceptable as working
hypotheses. Ot herwise, the theories are falsified by observations. This procedure is referred to
as the scientific approach to positive analysis in economics. In positive analysis, economists
do not claim that their assumptions are always right. What they try to do is to establish the
hypotheses (theories) that can be tested against observations. The working hypotheses can
then be either falsified or verified.
    Finally, interactions between active players in the economy may generate something that
                            m
is much more than the si ple sum of individuals’ behaviors. An analogy may illustrate the
point. The human body comprises a heap of molecules, but it contains something that never
exists in a single molecule. We call this thing “life” or “spirit”. The spirit of an economic
system comprising many individuals is, for instance, “price,” “money,” or “equilibrium”,
which never exists in a single individual’s activity in the absence of others’ interactions with


                                               23
her. Superadditivity and network effects (see chapters 2, 5, 6, 11, and 13) are other examples
of analogues to life and spirits in molecular biology. The emergence of some phenomena from
interactions among individual elements that are much more than the simple addition of those
individual elements is a feature of the theory of complexity. This is an exciting feature of
economics as well. This, together with the first feature that we have discussed, makes
economics more complex than biology, physics, and chemistry. In chapters 4 and 6, we shall
investigate how the spirit of an economy, equilibrium prices, emerges from interactions
between individuals’ behaviors.
    In section 1.2, we make some assumptions about an individual’s preferences that ensure
the existence of a utility function, which allows for the application of mathematical
programming to the study of individual behavior. Further assumptions, relating to individuals’
desires for diverse consumption and to the second order condition for a neoclassical
consumer’s decision problem, are made in section 1.3. The economic meaning of the
assumptions is then discussed.


Questions to Ask Yourself when Reading this Chapter

    Why do we need the notion of a utility function to do a thought experiment?
    Under what conditions can an individual’s preferences be represented by a utility function?
    What assumptions about utility functions and preferences do economists make when they
use utility functions to represent preferences? Why do they need these assumptions?
    What are the relationships between the important assumptions?



1.2. Preference and Utility Function

Assume that xi is the amount of good i that is consumed and x is an m dimension column
vector representing a bundle of m goods that is consumed. Then a consumption set is defined
by

(1.1)             X ⊂ R +m = {x∈ R m: xi ≥ 0, for i = 1, 2, … , m},

where R+m denotes a set of m dimension vectors whose elements are nonnegative real numbers.
We read X ⊂ R+m as “set X is contained in set R +m” which implies that each element of set X is
an element of set R+m. X may or may not equal set R+m. If the amount of a consumption good
must be an integer, or if the maximum amount of a consumption good is limited (for instance,
leisure time for each day cannot be greater than 24 hours), then the consumption set X is a
subset of R +m and X ≠ R +m. In the rest of this volume, we assume that X = R +m, unless
indicated otherwise.
    The objectives of the decision-maker are summarized in a preference relation, which we
denote by . is a binary relation on the consumption set X, allowing the comparison of pairs
of alternative consumption bundles x, y∈X. We read x y as “x is at least as good as y.” From
  , we can derive two other important relations on X: φ and ~.



                                              24
     The strict preference relation, φ, is defined by x φ y iff (if and only if) x y but not y x.
x φ y is read “x is preferred to y.” The indifference relation, ~, is defined by x ~ y iff x y and
y x. x ~ y is read “x is indifferent to y.”
     In order to obtain some intuitive understanding of the concept of preference, which
may seem to have no connection to our daily experience, you may conduct an experiment
with your mum. Before the experiment, first prepare a piece of paper with two dim ension
coordinates as shown in Fig. 1.1. A point x1 on the horizontal axis represents the amount
of apples consumed and a point x2 on the vertical axis represents the amount of bread
consumed. Let a vector x = (x1, x2) represent a bundle of consumption of the two goods.
Suppose that x(1) = (8, 2) denotes a bundle of consumption with x1 = 8 (8 units of apples)
and x2 = 2 (2 units of bread). You may draw a vertical line that cuts point x1 = 8 and a
horizontal line that cuts point x2 = 2. The intersection point of the two lines is x(1) = (8,
2). All points that are in the first quadrant, including the horizontal and vertical axes,
constitute the consumption set. Following the same method, you can plot points x(2) = (2,
8), x(3) = (3, 9), x(4) = (9, 3), x(5) = (4, 4), and other points that represent different
bundles of consumption of the two goods, as shown in Fig. 1.1(a). Now, the experiment
is ready.




     (a) Consumption                   (b) Indifference                  (c) Non-transitive
          bundles                           curves                            preferences

                 Figure 1.1: Mum’s Preferences and Indifference Curves

     You can then ask your mum to compare each pair of consumption bundles i and j,
and in each case to choose one and only one of the three answers:
              x(i) φ x(j),      x(j) φ x(i),      x(i) ~x(j).
Suppose that her answers are:
              x(3) ∼ x(4) φ x(1) ∼ x(2) ∼ x(5).
If you connect all equivalent points with smooth curves, you will obtain the two
indifference curves shown in Fig. 1.1(b). All points on the same indifference curve
generate the same degree of satisfaction. Hence, when your mum answered your
questions, she indirectly told you her preferences and her degree of satisfaction for each
consumption bundle. The degree is what economists call utility.




                                               25
                                Figure 1.2: Utility Function

      Now if we consider the degree of satisfaction in addition to consumption bundles
themselves, there are three dimensions: x1, x2, and utility u. In order to view the third
dimension, we draw the graph in Fig. 1.2. Note that the origin in the x1-x2 coordinates in
Fig. 1.1. is now moved to the bottom of the graph. The new vertical axis represents the
degree of satisfaction, or utility. The two indifference curves in Fig. 1.1(b) are now
copied in the x1-x2 plane in Fig. 1.2 as the broken curves A and B. We now move
indifference curve A (representing a lower degree of satisfaction) straight up to the
position of curve A’ and move indifference curve B (representing a higher degree of
satisfaction) up to the position of curve B’. If we do this also for other indifference curves,
then we can obtain many similarly moved indifference curves in the three dimension
space, where the indifference curves with higher degrees of satisfaction lie above those
with lower degrees of satisfaction. All of the resulting indifference curves would
constitute the surface of a hill, as shown in Fig. 1.2. This surface is the geometric
representation of a utility function that indicates the relationship between the degree of
satisfaction and the amounts of the two goods consumed. For any point on the surface,
for instance point D, we can find corresponding amounts of apples and bread, x1(D) and
x2(D) respectively, and the corresponding degree of satisfaction u(D). That is, we have a
utility function such that

(1.2)             u = f (x1, x2).

However, the degree of satisfaction is an ordinal rather than a cardinal concept. That is, it
is independent of the unit of measurement of utility. The same set of preferences may be
represented by multiple utility functions, indicated by more or less closely spaced
indifference curves, as long as the order between any pair of indifference curves is
preserved. In other words, a steeper or a flatter hill surface may represent the same
preferences that are associated with a set of the indifference curves on x1- x2 plane.



                                              26
     Following our intuitive illustration of the notion of utility function, let us now define
the concept rigorously.
     A function u: X→ R is a utility function representing preference relations if, for all      ®æ¦¡¤Æ
x1, x2∈X,                                                                                           :R
                                                                                                 §R°£
                                                                                                 ®æ¦¡¤Æ
              x1   x2 iff u(x1) ≥ u(x2).

For instance, the previous experiment with your mum indicates that your mum’s
preference can be represented by the utility function

              u = f (x1 x2) = x1x2.

Since a utility function maps multi-dimension variables into a one-dimension real number,
it will remarkably facilitate the analysis of self-interested behavior by applying
mathematical programming which optimizes an objective function subject to constraints.
Then we need not repeat the time consuming experiment that you did with your mum.
But it is conceivable that some irrational preferences may not be representable by a utility
function. Hence, we are interested in the question: what kinds of preferences can be
represented by a utility function? The answer to the question is given by the existence
theorem of the utility function, proved by Debreu (1959).
      We first prove that only rational preferences can be represented by a utility function.
Then we examine other properties of preferences that are essential for the existence of a
utility function.
     The preference relation on X is rational if it possesses the following two properties:      ®æ¦¡¤Æ
    (i) Completeness. For any x, y∈X, we have x y, or y x, or x~y.                               ®æ¦¡¤Æ
    (ii) Transitivity. For any x, y, z∈X, if x y and y z, then x z.                              ®æ¦¡¤Æ
    If preferences are not complete, the individual cannot make judgments about what she
wants for at least some pair of consumption bundles. In that case we cannot use a utility
function to represent her preferences. If an individual’s preferences are not transitive,
they must involve inconsistency. For instance, the two indifference curves in Fig. 1.1(c)
are associated with a preference relation that violates transitivity. The intersection of the
two curves implies x(1)~C~x(4), while the fact that point x(4) is higher than point x(1)
implies x(4) φ x(1), which contradicts x(1)~x(4). Similarly, x(3)~C~x(2) contradicts
x(3) φ x(2). A person with such preferences is said to be irrational. She is inconsistent in
judging what she wants. Of course such preferences cannot be represented by a utility
function. To verify the claim, we shall prove the following proposition.

Proposition 1.1: A preference relation       can be represented by a utility function only if
it is rational.

Proof (Mas-Collel, et al, 1995) To prove this proposition, we show that if there is a utility
function that represents preferences , then must be complete and transitive.
     Completeness. Because u(.) is a real-valued function defined on X, it must be that ∀        ®æ¦¡¤Æ
(for any) x, y∈X, either u(x)≥ u(y) or u(y)≥ u(x). But because u(.) is a utility function        ®æ¦¡¤Æ
representing , this implies that either x y or y x (recall the definition of utility
function). Hence, must be complete.


                                             27
      Transitivity. Suppose that x y and y z. Because u(.) represents , we must have            §R°£
                                                                                                   :x
u(x)≥ u(y) and u( y) ≥ u(z). Therefore, u(x)≥ u(z). Because u(.) represents , this implies x
   z. Thus, we have shown that x y and y z imply x z, and so transitivity is
established. Q.E.D.
      Now we consider an example which shows that rational preferences need not be
representable by a utility function. With that motivation, we then consider further
assumptions about preferences that are essential for the existence of a utility function.
      Consider the case of the lexicographic preference relation. For simplicity, assume
that X = R+2. Define x y if either “x1>y1” or “ 1=y1 and x2 ≥y2.” This is known as the
                                                     x
lexicographic preference relation. The name derives from the way a dictionary is
organized; that is, good 1 has the highest priority in determining the preference ordering,
just as the first letter of a word does in the ordering of a dictionary. When the amount of
good 1 in two consumption bundles is the same, the amount of good 2 in the two
consumption bundles determines the consumer’s preferences. It is not difficult to verify
that the lexicographic preference is complete and transitive (see exercise 4). Nevertheless,
it can be shown that no utility function exists that represents this preference ordering.
This is intuitive. With this preference ordering, no two distinct bundles are indifferent.
Therefore, we need two-dimension real numbers to represent the preference orderings.
Hence, a utility number from the one-dimensional real line is not enough to represent the
preference ordering. A somewhat subtle argument required to establish this claim
rigorously can be found in Mas-Collel et al (1995, p. 46).
      The assumption that is needed to ensure the existence of a utility function is that the
preference relation be continuous. The preference relation on X is continuous if it is
preserved under limits. That is, for any sequence of pairs {(xn, yn)}n=1 ∝ with xn yn for all
n, x = limn→∝ xn and y = limn→∝ yn, we have x y. Continuity says that the consumer’s
preferences cannot exhibit “jumps,” with, for instance, the consumer preferring each
elements in sequence {xn} to the corresponding element in sequence {yn} but suddenly
reversing her preference at the limiting points of these sequences x and y.
      An equivalent way of stating this notion of continuity is to say that for all x, the
upper contour set {y∈X : y x} and the lower contour set {y∈X : x y} are both closed;
that is, they include their boundaries.
      We can show that lexicographic preferences are not continuous. To see this, consider
the sequence of bundles xn = (1/n, 0) and yn = (0, 1). For every n, we have xn φ yn. But
limn→∝ yn = (0,1) φ (0, 0) = limn→∝ xn. In words, as long as the first component of x is
larger than that of y, x is preferred to y even if y2 is much larger than x2. But as soon as
the first components become equal, only the second components are relevant, and so the
preference ranking is reversed at the limit points of the sequence. Lexicographic
preferences relate to a hierarchical structure of desires. For instance, a hungry man may
not be interested in feelings of accomplishment or prestige. But after his basic need for
food and clothing is met, he might desire those feelings of accomplishment and prestige.
From our daily experience we can perceive that our preferences are indeed hierarchical.
Hence, the assumption of continuous preferences that exclude hierarchical preferences
from consideration is unrealistic. However, we have to confess that we economists are
incapable of handling hierarchical preferences in terms of mathematics. Therefore, we
confine our attention within a particular layer of the hierarchical structure of desires, so
that the following theory of utility is applicable.


                                             28
     Though rationality and continuity of the preference relation are together sufficient for
the existence of a utility function, there is a further assumption that makes the proof of
the existence theorem easier and ensures some important properties of the demand
function, such as Walras’ law. This is the assumption of non-satiation. 1
     The preference relation on X is non-satiated if x, y∈X and yi > xi ∀i implies y φ x.                    ®æ¦¡¤Æ
It is strongly non-satiated if y ≥ x and y ≠ x implies y φ x.
     The assumption of non-satiation is easier to accept if x is interpreted as amounts of
goods that a consumer possesses rather than the amounts of the goods that she consumes.
This interpretation of x and the assumption of strong non-satiation, together with the
assumption of free disposal, implies that more is better.
      We are now ready to prove the existence theorem of the utility function.

Proposition 1.2: Suppose that the rational preference relations on X is continuous and                       ®æ¦¡¤Æ
non-satiated. Then there is a continuous utility function u(x) that represents .

Proof For the case of X = R+ m, we can prove the proposition with the aid of Fig. 1.3.
Denote the diagonal ray in R+m (the locus of vectors with all m components equal) by Z.
It will be convenient to let e designate the m-vector whose elements are all equal to 1.
Then α e∈Z ∀α ≥ 0.




                        Figure 1.3: Construction of a Utility Function

     Note that ∀x∈ R+m, non-satiation implies that x 0. Also note that for anyα such
that α ei > xi for i = 1, 2, … , m (as shown in Fig. 1.3), we have α e x. Non-satiation and
continuity can then be shown to imply that there is a unique value α (x) ∈[0, α ] such that
α(x) e ~ x.
     We now take α(x) as our utility function, that is, we assign a utility value u(x) = α (x)
to every x. This utility level is also depicted in Fig. 1.3. We need to check two properties
of this function. The first is that it represents the preference , that is, α(x) ≥ α(y) iff x y.
This follows from the construction of the utility function. Suppose first that α (x) ≥ α(y).
By non-satiation, this implies that α(x)e α (y)e. Since α(x)e ~ x and α(y)e ~ y, we have x
  y. Suppose, on the other hand, that x y. Then α(x)e ~ x α (y)e ~ y; and so by non-

1
 A weaker assumption of local non-satiation is made in Mas-Collel et al (1995, p. 42) and Varian (1992, p.
96).


                                                   29
satiation, we must have α(x) ≥ α (y). Hence, α(x) ≥ α(y) iff x y. The second property is
that the constructed utility function is a continuous function. Mas -Collel et al (1995, p.
48) provide a proof of this.



1.3. Convex Preference Relation, Quasi-concave Utility Function, Diminishing
Marginal Rates of Substitution, and Desire for Diverse Consumption

In addition to the conditions required for the existence of the utility function, we are also
interested in some particular properties of preferences, and the correspondence between
                                                                        ost
these properties and the features of the utility function. The m important of the
properties is convexity of the preference relation. It is important because it relates to the
second-order condition for the consumer’s decision problem and to an individual’s desire
for diverse consumption. In this section, we examine the relationships between convex
preference ordering (convex upper contour set), quasi-concave utility function,
diminishing marginal rates of substitution, and a consumer’s desire for diverse
consumption, which is essential for the interior solution in neoclassical economics and
for the trade off between economies of specialization and transaction costs in new
classical economics.




                      (a) Convex preferences           (b) Non-convex preferences

                       Figure 1.4: Convex vs. Non-convex Preferences

    The preference relation on X is convex if ∀x∈X, the upper contour set {y∈X: y x}            ®æ¦¡¤Æ
is convex; that is if y x and z x, then αy+(1-α)z x ∀α ∈[0,1]. Fig. 1.4(a) gives an             ®æ¦¡¤Æ
illustration of the definition. All points on and above the indifference curve that generates
the utility level u(x) constitute an upper contour set. For any points in the set, such as y
and z, their weighted average, which is a point on the segment between the two points, is
an element of the set. A non-convex preference relation and non-convex upper contour
set are illustrated in Fig. 1.4(b) where a weighted average of point y and z is out of the
upper contour set. It is straightforward that a convex upper contour set is equivalent to a
set of indifference curves that are convex to the origin.
    From the definition of a convex preference relation, we can see that the upper contour
set is convex iff the following statement is true.


                                             30
(1.3)         ∀ y, z ∈ X and ∀α ∈[0,1],                                                          ®æ¦¡¤Æ
              u(α y+(1-α)z) ≥ Min {u(y), u(z)}

where ∀ reads “for any.” A utility function having this property is said to be quasi-
concave. Hence, convexity of the preference relation or of the upper contour set is
equivalent to quasi-concavity of the utility function representing the preference relation.
If the close interval of α , [0,1], is replaced with the open interval (0,1), and semi-
inequality ≥ is replaced with > in (1.3), then a utility function that satisfies the condition
is said to be strictly quasi-concave.
      Now we assume that the utility function u(.) on X is twice continuously                    ®æ¦¡¤Æ
differentiable. The properties of continuity and differentiability make it easier to establish
the connection between strict quasi-concavity of the utility function and diminishing
marginal rate of substitution. But actually the utility function may not be continuous or
differentiable at any point x∈ X. Any strictly increasing transformation of a continuous         ®æ¦¡¤Æ
function representing a preference relation also represents . If the transformation is
not continuous, then the new utility function representing the continuous preference
relation is discontinuous.




                            Figure 1.5: Leontief Preferences

      Fig. 1.5 shows the graph of the Leontief preference orderings. The corresponding
                                   x
utility function is u(x) = Min { 1/a, x2/b}. It is not difficult to see that the Leontief
preference relation is convex and continuous; but it is not differentiable at the kink point.
      Assume that the continuous and differentiable utility function is u( x), where x = (x1,
x2). The equation that represents an indifference curve for a fixed utility level u1 is:

(1.4)             u(x) = u1 or f (x) ≡ u(x) - u1 = 0.

Since a convex preference relation is associated with a convex upper contour set and the
latter is associated with a set of convex indifference curves, we can establish the
connection between the first and second order derivatives of the indifference curves and
the convexity of the preference ordering. According to your high school mathematics, a
downward sloping and convex curve can be represented by function x2 = g(x1) that has a


                                              31
negative first order derivative and a positive second order derivative. Function x2 = g( x1)
is implicitly given by (1.4). Its first derivative can be derived from the implicit function
theorem.

                    dx 2    ∂u / ∂x1            dx   ∂u / ∂x 1
(1.5)                    =−           < 0 , or − 2 =           >0
                    dx 1    ∂u / ∂x 2           dx 1 ∂u / ∂x 2

where -dx2/dx1 is called marginal rate of substitution, which is the amount of good 2 that
must be given away to keep utility unchanged when the amount of good 1 is increased by
a marginal unit. A strictly convex preference relation implies strictly convex indifference
curves. Hence, the second order derivative of x2 = g(x1) should be positive. Having
differentiated (1.5), the second order derivative can be worked out as follows.
                          dx

(1.6)
                 2
               d x2
                      ≡
                        d 2
                          d x1
                               =−
                                       (    dx
                                                       )
                                  u11 + u12 dx2 u 2 − u12 + u 22 dx2 u1
                                              1
                                                               ( dx
                                                                    1
                                                                     )
                   2
                d x1     dx1                       u22

            ∂u            ∂2u            dx 2
where, ui ≡     , uij ≡            , and      is given by (1.5). Inserting (1.5) into (1.6), it can
           ∂xi          ∂ xi ∂ x j       dx1
be shown that the second order derivative in (1.6) is positive iff

(1.7)               − ( u12 u 22 − 2 u1 u 2 u 12 + u 2 2 u11 ) > 0

Condition (1.7) is equivalent to the following condition for the sign of the bordered
Hessian determinant.
                   u11 u12 u1
(1.8)              u12 u22 u2 > 0.
                   u1 u2 0
The utility function u(x) that satisfies condition (1.8) is said to have anegative definite
bordered Hessian matrix

                u11u12 u1 
                           
                u12 u22 u2  .
                           
                u1 u2 0 
This establishes the equivalence between a strictly convex preference relation and a
negative definite bordered Hessian matrix of the utility function representing the
preference relation. This deduction can be extended to utility functions with more than 2
consumption goods. For those cases, a negative definite bordered Hessian matrix requires
alternatively changing signs of all principal minors of the bordered Hessian determinant,
with a negative first order principal minor. See Chiang (1984) for more details.
     Since




                                                      32
                       dx 2                    dx 
                      d                      d - 2 
                2
              d x2     dx 1            2
                                      d x2     dx 1 
(1.9)             2
                    ≡       > 0 iff -    2
                                           ≡           < 0,
              d x1    d x1            d x1      d x1
           d x2
where −           is the marginal rate of substitution, (1.7) is equivalent to a diminishing
           d x1
marginal rate of substitution. This establishes the equivalence between a twice
continuously differentiable and strictly quasi-concave utility function and the law of
                          ate
diminishing marginal r of substitution. This law implies that as the consumption of
good 1 increases, progressively smaller amounts of good 2 must be given up in order to
maintain an unchanged utility level. Graphically, the law implies that the tangent lines of
the indifference curve become flatter as consumption of good 1 increases.
     Finally, we examine the connection between convexity of the preference relation and
the desire for diverse consumption. From Fig. 1.4(a), we can see that for a strictly convex
preference relation, a weighted average (a mixture) of the two equally valued
consumption bundles is preferred to either bundle alone. We call this feature of the
preference relation desire for diverse consumption. Hence, strict convexity of the
preference relation is equivalent to the desire for diverse consumption.
     If the signs of the expressions in (1.7), (1.8), and (1.9) are reversed, the utility
function u(x) is strictly quasi-convex. A quasi-convex utility function represents non-
convex preferences for specialized consumption. The linear utility function u = x1 + bx2 is
            -
both quasi concave and quasi-convex. It is neither strictly quasi-concave nor strictly
quasi-convex. It represents razor edge preferences between desire for specialized
consumption and desire for diverse consumption.
     So far, all the assumptions we have made about the preference relation and the utility
function relate to ex ante properties of what an individual wants. It is an ex post matter if
a consumption bundle that is actually chosen by an individual is diverse. This is because
the individual’s consumption decision depends not only on her preferences, but also on
prices, which are themselves the consequence of interactions between self-interested
behaviors, and endowments. In the text we take ex ante to mean “before individuals have
made decisions”, and ex post to mean “after individuals have made decisions and the
economy has settled down in an equilibrium which is the consequence of interactions
between all individuals’ self-interested decisions.” In the next chapter, we will see that
even if individuals prefer diverse consumption in the sense discussed above, they may
nevertheless be induced by relative prices to choose a specialized consumption pattern
when their strictly convex preferences are represented by a quasi-linear utility function.
     The relationships between convexity of the preference relation, convexity of the
upper contour set, quasi-concavity of the utility function, negative definite bordered
Hessian matrix, a diminishing marginal rate of substitution, and the desire for diverse
consumption are summarized in Fig. 1.6. Note that these relationships are based on the
existence of a twice continuously differentiable utility function.




                                             33
           Figure 1.6: Relationships Between Several Important Concepts



1.4. Ordinal vs. Cardinal Theory of Utility, and Diminishing Marginal Rate of
Substitution vs. Diminishing Marginal Utility

Since an increasing transformation of a utility function preserves the preference orderings,
it represents the same preference relation as does the original utility function (see
exercise 2). For a twice continuously differentiable utility function, its continuous and
increasing transformation preser ves not only preference orderings, but also marginal rates
of substitution. For instance, the utility function u = x1x2 can be transformed to v = u1/2 =
x11/2 x21/2. It can be shown that the transformation changes neither the marginal rates of
substitution nor the quasi-concavity of the utility function, since

                    dx2 ∂v / ∂x 1 ( dv / du)( ∂u / ∂x1 ) ∂u / ∂x1
                −      =         =                       =         .
                    dx1 ∂v / ∂x 2 ( dv / du)( ∂u / ∂x 2 ) ∂u / ∂x2

In the next chapter, we will show that demand function also will not be changed by the
increasing transformation of the utility function as long as the marginal rate of
substitution is not changed by the transformation.
    However, we can show that u(.) is not strictly concave, while v(.) is strictly concave.
A function v(x) is strictly concave if ∀y, z∈X we have

(1.10)              v(α y + (1 − α ) z ) > α v( y ) + (1 − α )v ( z) ,

where α ∈ ( 0 ,1) . It can be shown that for twice continuously differentiable v (1.10)
holds iff

(1.11)              v 11 < 0 , v 22 < 0 , v 11 v 22 − v12 2 > 0 ,

where vij ≡ ∂2v/∂xi ∂xj. (1.11) holds iff the Hessian matrix of v(.),
                 v11 v12 
                          ,
                 v12 v 22 
is negative definite. See Chiang (1984) for proof of these two propositions. ∂v/∂xi is
called marginal utility of good i. ∂2v/∂xi2 ≡ ∂(∂v/∂xi )/∂xi < 0 implies that marginal utility


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decreases as consumption of a good increases. This is referred to as diminishing marginal
utility. It is easy to show that the utility function v = x11/2 x21/2 satisfies (1.11). Hence this
utility function is strictly concave, or it displays diminishing marginal utility. But, it can
be shown that

          u11 u22 − u12 2 = −1 < 0, u11 = 0 .

Thus the utility function u = x1 x2 is not concave and it does not have diminishing
marginal utility. A property of a utility function that cannot (or can) be preserved by an
increasing transformation of that function is called a cardinal (or ordinal) property.
Hence, diminishing marginal utility is a cardinal property of a utility function, while
diminishing marginal rate of substitution and related quasi-concavity are ordinal
properties. Diminishing marginal utility is not necessary for convexity of indifference
curves, which is associated with a quasi-concave utility function; nor is it essential for
diminishing marginal rate of substitution, which relates to the second order condition for
the interior optimum consumption decision, as shown in the next chapter. It is not
difficult to show that a concave function is quasi-concave, but a quasi-concave function is
not necessarily concave.
      An increasing transformation of a utility function changes the measurement unit of
utility, preserving preference orderings as well as marginal rates of substitution. Let u =
x1x2 = 100 and v = u1/2 = x11/2 x21/2 = 10. Then the two equations generate the same
indifference curve represented by x2 = 100/x1. Hence, if we call one unit of utility in
terms of v 10 units of utility in terms of u, then the increasing transformation from u to v
does not change anything of substance in the analysis, just as the transformation of 1
kilogram to 2.2 pounds would not change real weight.
      The view that a measurement unit of utility is essential is referred to as cardinal
theory of utility, while the view that a measurement unit is not essential is referred to as
ordinal theory of utility. We shall show in the next chapter that the ordinal theory of
utility is enough for a positive analysis of demand and supply. But the cardinal theory
may be needed when some social welfare function is used for welfare analysis, though
welfare analysis may be conducted without the use of such a function.



Key Terms and Review

   Preferences and indifference curves
   Rational preferences, complete preferences, transitive preferences
   Non-satiation, strong non-satiation, and continuity of preferences
   Utility function and conditions for the existence of utility function
   Convex upper contour set, convex indifference curves, and convex preference relations
   Quasi-concave utility function
   Marginal rate of substitution
   Law of diminishing marginal rate of substitution
   Marginal utility
   Diminishing marginal utility and strictly concave utility function



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   Relationships between convex preferences, quasi-concave utility function, concave utility
function, diminishing marginal rate of substitution, negative definite bordered Hessian matrix,
and desire for diverse consumption
   Cardinal vs. ordinal theory of utility



Further Reading

Mas-Collel et al (1995, chapters 1-3), Varian (1992, chapter 7), Deaton and Muellbauer (1980),
Debreu (1959), Samuelson (1947).



Questions

1. What is meant by the scientific approach to studying an individual’s degree of satisfaction?
2. Some economists, such as Hayek, hold that economics cannot be a science because society as
    a whole always knows much more than each individual member of it. This view implies that
    what is known by even the greatest economist is only a very small part of what society knows.
                                     o
    Hence, it is an impossibility t make economics a science that is supposed to cover all
    economic knowledge in society. Comment on the view.
3. Some economists challenge the mainstream approach to studying preferences and utility
    functions by pointing out that preferences should be endogenously determined by interactions
    between self-interested behaviors rather than exogenously given. For instance, pursuit of
    relative position and peer pressure may affect the evolution of individuals’ preferences. Other
    economists, such as Becker, argue that good economic theories can explain many economic
    phenomena as the unintended consequence of interactions between self-interested behaviors
    in the absence of changes of preferences. Comment on these alternative views, and discuss
    the merits and shortcomings of the mainstream approach to studying preferences as adopted
    in this text.
4. Why do we need a utility function to represent an individual’s preferences?
5. What kind of preferences can be represented by a utility function?
6. What is the restriction that the assumption of continuous preference imposes on economic
    analysis?
7. Why do economists assume that individuals prefer diverse consumption?
8. What is the connection between convex indifference curves, quasi-concave utility function,
    diminishing marginal rate of substitution, and desire for diverse consumption?
9. What is the relationship between diminishing marginal rate of substitution and diminishing
    marginal utility?
10. What are ordinal and cardinal theories of utility?



Exercises

1. Prove the following proposition. If is rational, then (i) φ is both irreflexive (x φ x never
    holds) and transitive. (ii) ~ is reflexive (x ~ x for all x) and transitive.



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2. Show that if f: R → R is a strictly increasing function and u: X → R is a utility f
                                                                                     unction
     representing the preference relation , then the function v: X → R defined by v(x) = f(u(x)) is
     also a utility function representing the preference relation .
3.   Prove that if is strongly non-satiated, then it is non-satiated.
4.   Verify that the lexicographic ordering is complete, transitive, strongly non-satiated, and
     strictly convex.
5.   Identify the conditions under which the quasi-linear utility function u = x 1 + x 2α is strictly
     quasi-concave. Draw indifference curves represented by the utility functio n in the x1- x2 plane.
6.   Show an example of a preference relation that is not continuous but is representable by a
     utility function.
7.   Suppose that in a two -good world, the consumer’s utility function takes the form u =
     [αx 1ρ +(1-α) x 2ρ ] 1/ρ. This utility function is known as the constant elasticity of substitution (or
     CES) utility function. (a) Show that when ρ = 1, indifference curves become linear. (b) Show
     that as ρ→0, this utility function comes to represent the same preferences as the Cobb-
     Douglas utility function u = x1αx 21-α . (c ) Show that as ρ→-∝, this utility function comes to
     represent the same preferences as the Leontief utility function u = Min{x1 , x 2}.
8.   Assume that x is the amount of goods consumed, y is the amount of leisure time, L = 24-y is
     the amount of working time. The utility function is u = x αy1-α. Draw indifference curves on
     the x-L plane. Then draw indifference curves on the x-y plane.
9. Suppose the Leontief utility function is u = Min( a , b ) . Draw indifference curves on the x-y
                                                     x         y

    plane. Will utility increase if x is increased and y is unchanged? Under what condition will
    utility be raised by changes of x and y?
10. Consider the following utility functions, and in each case determine: whether they are strictly
    quasi-concave and strictly concave; whether their marginal rates of substitution diminish;
    Work out their Hessian matrix and bordered Hessian matrix; whether they are negative
    definite. (a) u = a x + ln y , a > 0; (b) u = xy / ( x + y ) ; (c) u = x ( y − a ) , a > 0; (d)
    u = ln x + b ln y , b > 0.
11. Identify from among the following those utility functions that represent the same preference
    relation. (a) u = x α y β ; (b) u = x α + y β ; (c) u = α ln x + β ln y ; (d) u = αx + β y .
12. Which of the following u      tility functions represent preferences for diverse consumption?
    (a) u = x a + y b, a , b > 0; (b)      u = a x + y 2, a > 0; (c) u = bx + y ; (d) u = Min{ x , y} .




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