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Unformatted text preview: CHAPTER 4 UTILITY In Victorian days, philosophers and economists talked blithely of “utility”
as an indicator of a person’s overall wellbeing. Utility was thought of as
a numeric measure of a person’s happiness. Given this idea, it was natural
to think of consumers making choices so as to maximize their utility, that
is, to make themselves as happy as possible. The trouble is that these classical economists never really described how
we were to measure utility. How are we supposed to quantify the “amount”
of utility associated with different choices? Is one person’s utility the same
as another’s? What would it mean to say that an extra candy bar would
give me twice as much utility as an extra carrot? Does the concept of utility
have any independent meaning other than its being what people maximize? Because of these conceptual problems, economists have abandoned the
old—fashioned View of utility as being a measure of happiness. Instead,
the theory of consumer behavior has been reformulated entirely in terms
of consumer preferences, and utility is seen only as a way to describe
preferences. Economists gradually came to recognize that all that mattered about
utility as far as choice behavior was concerned was whether one bundle
had a higher utility than anotherihow much higher didn’t really matter. UTILITY 55 Originally, preferences were deﬁned in terms of utility: to say a bundle
(3:1, 2:2) was preferred to a bundle (yhyg) meant that the Xbundle had a
higher utility than the ybundle. But now we tend to think of things the
other way around. The preferences of the consumer are the fundamen—
tal description useful for analyzing choice, and utility is simply a way of
describing preferences. A utility function is a way of assigning a number to every possible
consumption bundle such that more—preferred bundles get assigned larger
numbers than lesspreferred bundles. That is, a bundle (171,312) is preferred
to a bundle (311,312) if and only if the utility of ($1,902) is larger than the
utility of (yhyg): in symbols, (331,332) > (yhyg) if and only if u(:r1,x2) >
”(311,92) The only property of a utility assignment that is important is how it
orders the bundles of goods. The magnitude of the utility function is only
important insofar as it ranks the different consumption bundles; the size of
the utility difference between any two consumption bundles doesn’t matter.
Because of this emphasis on ordering bundles of goods, this kind of utility
is referred to as ordinal utility. Consider for example Table 4.1, where we have illustrated several dif
ferent ways of assigning utilities to three bundles of goods, all of which
order the bundles in the same way. In this example, the consumer prefers
A to B and B to C. All of the ways indicated are valid utility functions
that describe the same preferences because they all have the property that
A is assigned a higher number than B, which in turn is assigned a higher
number than C. Different ways to assign utilities. Bundle U1 U2 U3
A 3 17 —1
B 2 10 —2
C 1 .002 —3 Since only the ranking of the bundles matters, there can be no unique
way to assign utilities to bundles of goods. If we can ﬁnd one way to assign
utility numbers to bundles of goods, we can find an inﬁnite number of
ways to do it. If u(:r1,a:2) represents a way to assign utility numbers to
the bundles (271,232), then multiplying u(;1:1,$2) by 2 (or any other positive
number) is just as good a way to assign utilities. Multiplication by 2 is an example of a monotonic transformation. A 56 UTILITY (Ch. 4) monotonic transformation is a way of transforming one set of numbers into
another set of numbers in a way that preserves the order of the numbers. We typically represent a monotonic transformation by a function ﬂu)
that transforms each number u into some other number ﬂu), in a way
that preserves the order of the numbers in the sense that ul > U2 implies
ﬂul) > f (U2) A monotonic transformation and a monotonic function are
essentially the same thing. Examples of monotonic transformations are multiplication by a positive
number (e.g., ﬂu) : 3a), adding any number (e.g., ﬂu) : u+ 17), raising
u to an odd power (e.g., ﬂu) = u3), and so on.1 The rate of change of ﬂu) as u changes can be measured by looking at
the change in f between two values of u, divided by the change in u: g z f(U2)f(u1)_ Au ug — U1 For a monotonic transformation, f (ug) — f (ul) always has the same sign as
U2 — ul. Thus a monotonic function always has a positive rate of change.
This means that the graph of a monotonic function will always have a
positive slope, as depicted in Figure 4.1A. _ _ __"..A' ' .'  ' "'B _
II 'I 3. .A posrtwe monotonic transformation Panel A illustrates
' “a monotonic function—ﬁne that is. aIWays inereasing. Panel B ' illustrates a function that is not menotonic, since it sometimes
increases and sometimes decreaSes; 1 What we are calling a “monotonic transformation” is, strictly speaking, called a “posi
tive monotonic transformation,” in order to distinguish it from a “negative monotonic
transformation,” which is one that reverses the order of the numbers. Monotonic
transformations are sometimes called “monotonous transformations,” which seems
unfair, since they can actually be quite interesting. CARDlNAL UTILITY 57 If f (a) is any monotonic transformation of a utility function that repre
sents some particular preferences, then f (u(3:1, 332)) is also a utility function
that represents those same preferences. Why? The argument is given in the following three statements: 1. To say that u(3:1, x2) represents some particular preferences means that
“(9311552) > ”@1792) if and only if (931,302) > (91,112) 2. But if f(u) is a monotonic transformation, then ’u.(.1‘1,$2) > u(y1,y2) if
and only if f(u($1,$2)) > f(U(’y1,y2)) 3. Therefore, f(u(a:1,a‘2)) > f(a(yl,y2)) if and only if ($1,392) > (311,142),
so the function f (u) represents the preferences in the same way as the
original utility function u(:c1, $2). We summarize this discussion by stating the following principle: a mono
tonic transformation of a utility function is a utility function that represents
the same preferences as the original utility function. Geometrically, a utility function is a way to label indifference curves.
Since every bundle on an indifference curve must have the same utility, a
utility function is a way of assigning numbers to the different indifference
curves in a way that higher indifference curves get assigned larger num
bers. Seen from this point of View a monotonic transformation is just a
relabeling of indifference curves. As long as indifference curves containing
more—preferred bundles get a larger label than indifference curves contain
ing less—preferred bundles7 the labeling will represent the same preferences. 4.1 Cardinal Utility There are some theories of utility that attach a signiﬁcance to the magni—
tude of utility. These are known as cardinal utility theories. In a theory
of cardinal utility, the size of the utility difference between two bundles of
goods is supposed to have some sort of signiﬁcance. We know how to tell whether a given person prefers one bundle of goods
to another: we simply offer him or her a choice between the two bundles
and see which one is chosen. Thus we know how to assign an ordinal utility
to the two bundles of goods: we just assign a higher utility to the chosen
bundle than to the rejected bundle. Any assignment that does this will be
a utility function. Thus we have an operational criterion for determining
whether one bundle has a higher utility than another bundle for some
individual. But how do we tell if a person likes one bundle twice as much as another?
How could you even tell if you like one bundle twice as much as another? One could propose various deﬁnitions for this kind of assignment: I like
one bundle twice as much as another if I am willing to pay twice as much
for it. Or, I like one bundle twice as much as another if I am willing to run 58 UTILITY (Ch. 4) twice as far to get it, or to wait twice as long, or to gamble for it at twice
the odds. There is nothing wrong with any of these deﬁnitions; each one would
give rise to a way of assigning utility levels in which the magnitude of the
numbers assigned had some operational signiﬁcance. But there isn’t much
right about them either. Although each of them is a possible interpretation
of what it means to want one thing twice as much as another, none of them
appears to be an especially compelling interpretation of that statement. Even if we did ﬁnd a way of assigning utility magnitudes that seemed
to be especially compelling, what good would it do us in describing choice
behavior? To tell whether one bundle or another will be chosen, we only
have to know which is preferredkwhich has the larger utility. Knowing
how much larger doesn’t add anything to our description of choice. Since
cardinal utility isn’t needed to describe choice behavior and there is no
compelling way to assign cardinal utilities anyway, we will stick with a
purely ordinal utility framework. 4.2 Constructing a Utility Function But are we assured that there is any way to assign ordinal utilities? Given
a preference ordering can we always ﬁnd a utility function that will order
bundles of goods in the same way as those preferences? Is there a utility
function that describes any reasonable preference ordering? Not all kinds of preferences can be represented by a utility function.
For example, suppose that someone had intransitive preferences so that
A >— B > C > A. Then a utility function for these preferences would have
to consist of numbers u(A), u(B), and u(C) such that u(A) > u(B) >
u(C) > u(A). But this is impossible. However, if we rule out perverse cases like intransitive preferences, it
turns out that we will typically be able to find a utility function to represent
preferences. We will illustrate one construction here, and another one in
Chapter 14. Suppose that we are given an indifference map as in Figure 4.2. We know
that a utility function is a way to label the indifference curves such that
higher indifference curves get larger numbers. How can we do this? One easy way is to draw the diagonal line illustrated and label each
indifference curve with its distance from the origin measured along the
line. How do we know that this is a utility function? It is not hard to see that
if preferences are monotonic then the line through the origin must intersect
every indifference curve exactly once. Thus every bundle is getting a label,
and those bundles on higher indifference curves are getting larger labels?
and that’s all it takes to be a utility function. SOME EXAMPLES OF UTILITY FUNCTIONS 59 __Méa$uresdistance* _
'. .from Origin  .‘ ' . . ' 'i'1iidiszerén¢e3"3"i..... . u . . ... 53223313"?
Constructing a attrity'ﬁin‘cftionlrmmmdzﬂ’emnce nurses; . .
Draw aldiagﬁﬂal line andlabeleachgiridiﬁereneeeurre With how _
far it is from the originmeasnred alengithe_.line.‘: ' '   ' ‘ ' This gives us one way to ﬁnd a labeling of indifference curves, at least as
long as preferences are monotonic. This won’t always be the most natural
way in any given case, but at least it shows that the idea of an ordinal utility
function is pretty general: nearly any kind of “reasonable” preferences can
be represented by a utility function. 4.3 Some Examples of Utility Functions In Chapter 3 we described some examples of preferences and the indiffer
ence curves that represented them. We can also represent these preferences
by utility functions. If you are given a utility function, 1431mm), it is rel
atively easy to draw the indifference curves: you just plot all the points
(1:1,m2) such that u(:1:1,$2) equals a constant. In mathematics, the set of all (3:1, (1:2) such that u(x1,:r2) equals a constant IS called a level set. For
each different value of the constant, you get a different indifference curve. EXAMPLE: Indifference Curves from UtiIity Suppose that the utility function is given by: u(:r1, m2) = $1332. What do
the indifference curves look like? 60 UTILITY (Ch. 4) We know that a typical indifference curve is just the set of all 331 and 332
such that k = 331172 for some constant k. Solving for 932 as a function of 3:1,
we see that a typical indifference curve has the formula: $2 = —.
$1 This curve is depicted in Figure 4.3 for k = 1, 2, 3   . Indifference
curves X1 Indifference curves. The indifference curves 1: == $1502 for
different values of k. “Maul—M‘— Let’s consider another example. Suppose that we were given a utility
function v(:r1,:z:2) 2 using. What do its indifference curves look like? By
the standard rules of algebra we know that: v(a:1,:c2) 2 £333 : (x1x2)2 : u(:c1,$2)2. Thus the utility function v(m1,$2) is just the square of the utility func—
tion u(:c1,a:2). Since £431,332) cannot be negative, it follows that v(a:1, 3:2)
is a monotonic transformation of the previous utility function, u(:r1,:r2).
This means that the utility function Man, :52) 2 ﬂag has to have exactly
the same shaped indifference curves as those depicted in Figure 4.3. The
labeling of the indifference curves will be differentﬂthe labels that were
1, 2, 3, ~   Will now be 1,4, 9,   —~but the set of bundles that has 11(1'1, 322) = SOME EXAMPLES OF UTILITY FUNCTIONS 61 9 is exactly the same as the set of bundles that has u(cr1,:r2) = 3. Thus
v(a:1,:1:2) describes exactly the same preferences as U($1,.T2) since it orders
all of the bundles in the same way. Going the other direction—finding a utility function that represents some
indifference curves~is somewhat more difficult. There are two ways to
proceed. The first way is mathematical. Given the indifference curves, we
want to ﬁnd a function that is constant along each indifference curve and
that assigns higher values to higher indifference curves. The second way is a bit more intuitive. Given a description of the pref—
erences, we try to think about what the consumer is trying to maximize,
what combination of the goods describes the choice behavior of the con—
sumer. This may seem a little vague at the moment, but it will be more
meaningful after we discuss a few examples. Perfect Substitutes Remember the red pencil and blue pencil example? All that mattered to
the consumer was the total number of pencils. Thus it is natural to measure
utility by the total number of pencils. Therefore we provisionally pick the
utility function u(:r1, m2) : 331 +132. Does this work? Just ask two things: is
this utility functiou constant along the indifference curves? Does it assign
a higher label to morepreferred bundles? The answer to both questions is
yes, so we have a utility function. Of course, this isn’t the only utility function that we could use. We could
also use the square of the number of pencils. Thus the utility function
v(a:1,w2) = (:31 + 3:2)2 : m? + 2:121:32 + 33% will also represent the perfect«
substitutes preferences, as would any other monotonic transformation of
u($1, 932). What if the consumer is willing to substitute good 1 for good 2 at a rate
that is different from oneto—one? Suppose, for example, that the consumer
would require two units of good 2 to compensate him for giving up one unit
of good 1. This means that good 1 is twice as valuable to the consumer as
good 2. The utility function therefore takes the form Man, 332) : 2.11 + 372.
Note that this utility yields indifference curves with a SIOpe of #2. In general, preferences for perfect substitutes can be represented by a
utility function of the form u(331,:r2) : (1:171 + bxg. Here a and b are some positive numbers that measure the “value” of goods
1 and 2 to the consumer. Note that the slope of a typical indifference curve
is given by —a/b. 62 UTILITY (Ch. 4) Perfect Complements This is the left shoe—right shoe case. In these preferences the consumer only
cares about the number of pairs of shoes he has, so it is natural to choose
the number of pairs of shoes as the utility function. The number of complete
pairs of shoes that you have is the minimum of the number of right shoes
you have, :31, and the number of left shoes you have, 3:2. Thus the utility
function for perfect complements takes the form u(a:1, :52) : min{;r1,.rg}. To verify that this utility function actually works, pick a bundle of goods
such as (10,10). If we add one more unit of good 1 we get (11,10),
which should leave us on the same indifference curve. Does it? Yes, since
min{10, 10} = min{11, 10} = 10. So u(3:1, 9:2) = min{zt1,:c2} is a possible utility function to describe per—
fect complements. As usual, any monotonic transformation would be suit
able as well. What about the case where the consumer wants to consume the goods
in some proportion other than oneto—one? For example, what about the
consumer who always uses 2 teaspoons of sugar with each cup of tea? If 3:1
is the number of cups of tea available and 1172 is the number of teaspoons
of sugar available, then the number of correctly sweetened cups of tea will
be min{:r1, éﬂig}. This is a little tricky so we Should stop to think about it. If the number
of cups of tea is greater than half the number of teaspoons of sugar, then
we know that we won’t be able to put 2 teaspoons of sugar in each cup.
In this case, we will only end up with %LL‘2 correctly sweetened cups of tea.
(Substitute some numbers in for 331 and $2 to convince yourself.) Of course, any monotonic transformation of this utility function will
describe the same preferences. For example, we might want to multiply by
2 to get rid of the fraction. This gives us the utility function u(:c1,a:2) =
min{2;r1,:cg}. In general, a utility function that describes perfect—complement prefer—
ences is given by 15(351, (£2) ‘—‘ min{ax1, b.1132}, where a and b are positive numbers that indicate the proportions in which
the goods are consumed. Quasilinear Preferences Here’s a shape of indifference curves that we haven’t seen before. Suppose
that a consumer has indifference curves that are vertical translates of one
another, as in Figure 4.4. This means that all of the indifference curves are
just vertically “shifted” versions of one indifference curve. It follows that SOME EXAMPLES OF UTILITY FUNCTIONS 63 the equation for an indifference curve takes the form $2 = k — v(:131), where k is a different constant for each indifference curve. This equation says that the height of each indiﬁerence curve is some functiOn of £161, —'U(:c1), plus a
constant k. Higher values of k give higher indifference curves. (The minus sign is only a convention; we’ll see why it is convenient below.) Indifference
curves X1 Quasilinear preferences. Each indifference Curve'isa verti—'_ 
cally shifted version of a singlﬂ indifference curve. ' ' The natural way to label indifference curves here is with kmroughly
speaking, the height of the indifference curve along the vertical axis. Solv ing for k and setting it equal to utility, we have
u($1,$2) = k = 1,7(131) I .2132. In this case the utility function is linear in good 2, but (possibly) non—
linear in good 1', hence the name quasilinear utility, meaning “partly
linear” utility. Speciﬁc examples of quasilinear utility would be u($1, :32) =
ﬂ? + $2, or u(:c1,:::2) : In :51 + 332. Quasilinear utility functions are not
particularly realistic, but they are very easy to work with, as we’ll see in
several examples later on in the book. Cobb—Douglas Preferences Another commonly used utility function is the CobbDouglas utility func— tion
u(a:1,:c2) 2' 3511733, 64 UTILITY (Ch. 4) where c and d are positive numbers that describe the preferences of the
consumer.2 The Cobb—Douglas utility function will be useful in several examples.
The preferences represented by the CobbDouglas utility function have the
general shape depicted in Figure 4.5. In Figure 4.5A, we have illustrated the
indifference curves for c = 1/2, d = 1/2. In Figure 4.5B, we have illustrated
the indifference curves for c : 1/5, d = 4/5. Note how different values of
the parameters c and (1 lead to different shapes of the indifference curves. . X1 X1
A C=ll2 d=1l2 B C=ll5d=4l5 // { Cobb—Douglas indifference curves. Panel A shows the case where c = 1/2, of = 1/2 and panel B shows the case where
c = 1/5, (1 = 4/5. _______________________.._.______.—————————— Cobb—Douglas indifference curves look just like the nice convex mono—
tonic indifference curves that we referred to as “well—behaved indifference
curves” in Chapter 3. CobbDouglas preferences are the standard exam
ple of indifference curves that look well—behaved, and in fact the formula
describing them is about the simplest algebraic expression that generates
well—behaved preferences. We‘ll ﬁnd Cobb—Douglas preferences quite useful
to present algebraic examples of the economic ideas we’ll study later. Of course a monotonic transformation of the CobbDouglas utility func~
tion will represent exactly the same preferences, and it is useful to see a
couple of examples of these transformations. 2 Paul Douglas was a twentieth—century economist at the University of Chicago who
later became a US. senator. Charles Cobb was a mathematician at Amherst College.
The CobbDouglas functional form was originally used to study production behavior. MARGINAL UTILITY 6'5 First, if we take the natural log of utility, the product of the terms will
become a sum so that we have v(w1, 3:2) : immiscg) = cln 331 + dln mg. The indifference curves for this utility function will look just like the ones
for the ﬁrst Cobb—Douglas function, since the logarithm is a monotonic
transformation. (For a brief review of natural logarithms, see the Mathe
matical Appendix at the end of the book.) For the second example, suppose that we start with the CobbaDouglas
form v(m1,m2) : mfxg. Then raising utility to the 1/(c + d) power, we have c d. 33f” 11:; d. Now deﬁne a new number c c+d' We can now write our utility function as a: v(:r1,:32) = 3:?fo
This means that we can always take a monotonic transformation of the
CobbDouglas utility function that make the exponents sum to 1. This
will turn out to have a useful interpretation later on.
The CobbDouglas utility function can be expressed in a variety of ways;
you should learn to recognize them, as this family of preferences is very
useful for examples. 4.4 Marginal Utility Consider a consumer who is consuming some bundle of goods, (231,172).
How does this consumer’s utility change as we give him or her a little more
of good 1? This rate of change is called the marginal utility with respect
to good 1. We write it as M U1 and think of it as being a ratio, AU u(2:1 + A351, 332) — u(a:1,:r2)
U : —— : ——“—‘—‘’—‘—"
M 1 A221 A151 ’ that measures the rate of change in utility (AU) associated with a small
change in the amount of good 1 (A531). Note that the amount of good 2 is
held ﬁxed in this calculation.3 3 See the appendix to this chapter for a calculus treatment of marginal utility. 66 UTILITY (Ch. 4) This deﬁnition implies that to calculate the change in utility associated
with a small change in consumption of good 1, we can just multiply the
change in consumption by the marginal utility of the good: AU 2 MU1 A561 .
The marginal utility with respect to good 2 is deﬁned in a similar manner: MU2 : LU : W22
13552 A372 Note that when we compute the marginal utility with respect to good 2 we
keep the amount of good 1 constant. We can calculate the change in utility
associated with a change in the consumption of good 2 by the formula AU 2 MUgACBQ. It is important to realize that the magnitude of marginal utility depends
on the magnitude of utility. Thus it depends on the particular way that we
choose to measure utility. If we multiplied utility by 2, then marginal utility
would also be multiplied by 2. We would still have a perfectly valid utility
function in that it would represent the same preferences, but it would just
be scaled differently. This means that marginal utility itself has no behavioral content. How
can we calculate marginal utility from a consumer’s choice behavior? We
can’t. Choice behavior only reveals information about the way a consumer
Tanks different bundles of goods. Marginal utility depends on the partic—
ular utility function that we use to reflect the preference ordering and its
magnitude has no particular signiﬁcance. However, it turns out that mar—
ginal utility can be used to calculate something that does have behavioral
content, as we will see in the next section. 4.5 Marginal Utility and MRS A utility function u(:c1, :32) can be used to measure the marginal rate of
substitution (MRS) defined in Chapter 3. Recall that the MRS measures
the slope of the indifference curve at a given bundle of goods; it can be
interpreted as the rate at which a consumer is just willing to substitute a
small amount of good 2 for good 1. This interpretation gives us a simple way to calculate the MRS. Con—
sider a change in the consumption of each good, (A331, A552), that keeps
utility constant—that is, a change in consumption that moves us along the
indifference curve. Then we must have MUlACL'l ‘l— MU2AIL‘2 = AU 2 0 UTILITY FOR COMMUTING ()7 Solving for the slope of the indifference curve we have A332 MU1
M = —_ : 1 . . (Note that we have 2 over 1 on the left—hand side of the equation and 1
over 2 011 the right—hand side. Don’t get confused!) The algebraic sign of the MRS is negative: if you get more of good 1 you
have to get less of good 2 in order to keep the same level of utility. However,
it gets very tedious to keep track of that pesky minus sign, so economists
often refer to the MRS by its absolute valuekthat is, as a positive number.
We’ll follow this convention as long as no confusion will result. Now here is the interesting thing about the MRS calculation: the MRS
can be measured by observing a person’s actual behavior~rwe ﬁnd that
rate of exchange where he or she is just willing to stay put, as described in
Chapter 3. The utility function, and therefore the marginal utility function, is not
uniquely determined. Any monotonic transformation of a utility function
leaves you with another equally valid utility function. Thus, if we multiply
utility by 2, for example, the marginal utility is multiplied by 2. Thus the
magnitude of the marginal utility function depends on the choice of utility
function, which is arbitrary. It doesn’t depend 011 behavior alone; instead
it depends on the utility function that we use to describe behavior. But the ratio of marginal utilities gives us an observable magnitude‘
namely the marginal rate of substitution. The ratio of marginal utilities
is independent of the particular transformation of the utility function you choose to use. Look at what happens if you multiply utility by 2. The MRS becomes
21% U 1 2MU2' The 2s just cancel out, so the MRS remains the same. The same sort of thing occurs when we take any monotonic transforma
tion of a utility function. Taking a monotonic transformation is just rela—
beling the indifference curves, and the calculation for the MRS described
above is concerned with moving along a given indifference curve. Even
though the marginal utilities are changed by monotonic transformations,
the ratio of marginal utilities is independent of the particular way chosen
to represent the preferences. MRS = — 4.6 Utility for Commuting Utility functions are basically ways of describing choice behavior: if a bun—
dle of goods X is chosen when a bundle of goods Y is available, then X
must have a higher utility than Y. By examining choices consumers make
we can estimate a utility function to describe their behavior. 68 UTILITY (Ch. 4) This idea has been widely applied in the ﬁeld of transportation economics
to study consumers’ commuting behavior. In most large cities commuters
have a choice between taking public transit or driving to work. Each of
these alternatives can be thought of as representing a bundle of different
characteristics: travel time, waiting time, out—of—pocket costs, comfort, con—
venience, and so on. We could let 331 be the amount of travel time involved
in each kind of transportation, $2 the amount of waiting time for each kind,
and so on. If (3:1, 2:2, . . . ,arn) represents the values of 71 different characteristics of
driving, say, and (y1, y2, . . . , y”) represents the values of taking the bus, we
can consider a model where the consumer decides to drive or take the bus
depending on whether he prefers one bundle of characteristics to the other. More speciﬁcally, let us suppose that the average consumer’s preferences
for characteristics can be represented by a utility function of the form U(331»$2,  «  7331:) = 51$1+ ﬁ2$2 + ' ' ' + [37155717 where the coefficients )31,ﬁg, and so on are unknown parameters. Any
monotonic transformation of this utility function would describe the choice
behavior equally well, of course, but the linear form is especially easy to
work with from a statistical point of view. Suppose now that we observe a number of similar consumers making
choices between driving and taking the bus based on the particular pattern
of commute times, costs, and so on that they face. There are statistical
techniques that can be used to ﬁnd the values of the coefﬁcients at for 2' :
1, . . . ,n that best ﬁt the observed pattern of choices by a set of consumers.
These statistical techniques give a way to estimate the utility function for
different transportation modes. One study reports a utility function that had the form4 U(Tl/V, TT, C) : —0.147TW  0.0411TT — 2.24C, (4.2) where TW 2 total walking time to and from bus or car
TT 2 total time of trip in minutes
0 = total cost of trip in dollars The estimated utility function in the DomenichMcFadden book correctly
described the choice between auto and bus transport for 93 percent of the
households in their sample. 4 See Thomas Domenich and Daniel McFadden7 Urban Travel Demand (NorthHolland
Publishing Company, 1975). The estimation procedure in this book also incorporated
various demographic characteristics of the households in addition to the purely eco~
nomic variables described here. Daniel McFadden was awarded the Nobel Prize in
economics in 2000 for his work in developing techniques to estimate models of this
sort. SUMMARY 69 The coefﬁcients on the variables in Equation (4.2) describe the weight
that an average household places on the various characteristics of their
commuting trips; that is, the marginal utility of each characteristic. The
ratio of one coefﬁcient to another measures the marginal rate of substitu
tion between one characteristic and another. For example, the ratio of the
marginal utility of walking time to the marginal utility of total time indi—
cates that walking time is viewed as being roughly 3 times as onerous as
travel time by the average consumer. In other words, the consumer would
be willing to substitute 3 minutes of additional travel time to save 1 minute
of walking time. Similarly, the ratio of cost to travel time indicates the average consumer’s
tradeoff between these two variables. In this study, the average commuter
valued a minute of commute time at 0.0411/2.24 = 0.0183 dollars per
minute, which is $1.10 per hour. For comparison, the hourly wage for the
average commuter in 1967, the year of the study, was about $2 .85 an hour. Such estimated utility functions can be very valuable for determining
whether or not it is worthwhile to make some change in the public trans~
portation system. For example, in the above utility function one of the
signiﬁcant factors explaining mode choice is the time involved in taking
the trip. The city transit authority can, at some cost, add more buses to
reduce this travel time. But will the number of extra riders warrant the
increased expense? Given a utility function and a sample of consumers we can forecast which
consumers will drive and which consumers will choose to take the bus. This
will give us some idea as to whether the revenue will be sufﬁcient to cover
the extra cost. Furthermore, we can use the marginal rate of substitution to estimate
the value that each consumer places on the reduced travel time. We saw
above that in the DomenichMcFadden study the average commuter in
1967 valued commute time at a rate of $1.10 per hour. Thus the commuter
should be willing to pay about $0.37 to cut 20 minutes from his or her
trip. This number gives us a measure of the dollar beneﬁt of providing
more timely bus service. This beneﬁt must be compared to the cost of
providing more timely bus service in order to determine if such provision
is worthwhile. Having a quantitative measure of beneﬁt will certainly be
helpful in making a rational decision about tranSport policy. Summary 1. A utility function is simply a way to represent or summarize a prefer
ence ordering. The numerical magnitudes of utility levels have no intrinsic
meaning. 2. Thus, given any one utility function, any monotonic transformation of
it will represent the same preferences. 70 UTILITY (Ch‘ 4) 3. The marginal rate of substitution, MRS7 can be calculated from the
utility function via the formula MRS 2 Asa/A351 2 —MU1/MU2. REVIEW QUESTIONS 1. The text said that raising a number to an odd power was a monotonic
transformation. What about raising a number to an even power? Is this a
monotonic transformation? (Hint: consider the case f (u) 2 112.) 2. Which of the following are monotonic transformations? (1) u 2 2'0 — 13;
(2) u 2 ~1/n2; (3) u 2 1/122; (4) u 21m); (5) u 2 ~e””; (6) u 2 v2;
(7)1;21)2 forv>0; (8)”:U2 forv<0. 3. We claimed in the text that if preferences were monotonic, then a diag—
onal line through the origin would intersect each indifference curve exactly
once. Can you prove this rigorously? (Hint: what would happen if it
intersected some indifference curve twice?) 4. What kind of preferences are represented by a utility function of the
form u(:1:1,:r2) 2 V501 + 1102? What about the utility function v(:c1,:v2) 2
13331 ‘I‘ 13352? 5. What kind of preferences are represented by a utility function of the form
u(a:1, x2) 2 x1 + 4:32? Is the utility function v(x1,:c2) 2 23% + 2$1,/:1:2 +562
a monotonic transformation of u(:z:1,a:2)? 6. Consider the utility function u(m1,:c2) 2 MCI/'11:} What kind of pref— erences does it represent? Is the function v(:c1,$2) 2 aﬁmg a monotonic transformation of U(II}1,£L‘2)? Is the function w(a:1,a:2) 2 933.12% a monotonic transformation of u(a:1, x2 )7 7. Can you explain Why taking a monotonic transformation of a utility
function doesn’t change the marginal rate of substitution? APPENDIX 3’ First, let us clarify what is meant by “marginal utility. As elsewhere in eco—
nomics, “marginal” just means a derivative. So the marginal utility of good 1 is
just 14351 + A$1,£E2) — ’LL($1,:E2) __ 811(171, $2)
A$10 A231 6331 . Note that we have used the partial derivative here, since the marginal utility
of good 1 is computed holding good 2 fixed. APPENDIX 71 Now We can rephrase the derivation of the MRS given in the text using calculus.
We'll do it two ways: ﬁrst by using differentials, and second by using implicit
functions. For the ﬁrst method, we consider making a change (d$1,dﬂ$2) that keeps utility
constant. So we want 8u(:c1,:c2) _ 6u(a:1,ct2)
du — 85131 d$1 + 8352 The first term measures the increase in utility from the small change dim, and
the second term measures the increase in utility from the small change de‘2. We
want to pick these changes so that the total change in utility, do, is zero. Solving
for d1€2/d$1 gives us (1.132 = 0. drm _ 8u(:c1,:c2)/6:131
E _ _5u(m1,332)/3:c2’
which is just the calculus analog of equation (4.1) in the text,
As for the second method, we now think of the indifference curve as being
described by a function $2 (11). That is, for each value of 321, the function 532(1‘1) tells us how much :32 we need to get on that speciﬁc indifference curve. Thus the
function 332(5151) has to satisfy the identity u($1,x2($1)) E k, where k is the utility label of the indifference curve in question.
We can differentiate both sides of this identity with respect to 9:1 to get
8u(.’c1,m2) + 81.4931, 1132) 61132031)
811:1 812 BIL'l
Notice that $1 occurs in two places in this identity, so changing 221 will change
the function in two ways, and we have to take the derivative at each place that as; appears.
We then solve this equation for film ($1)/3m1 to ﬁnd 6m2(:c1)_ 8u(:r1,a:2)/8:1:1
33m _ (rilii($1,a"2)/8:c27 =0. just as we had before. The implicit function method is a little more rigorous, but the differential
method is more direct, as long as you don’t do something silly. Suppose that we take a monotonic transformation of a utility function, say,
v(:r1,a:2) : f(u(a:1, 222)). Let’s calculate the MRS for this utility function. Using
the chain rule
811/8301 df/du (Chi/83:1
80/8362 ‘af/eu alt/3:62 Bit/83:1
” air/am since the 6 f /8u term cancels out from both the numerator and denominator.
This shows that the MRS is independent of the utility representation This gives a useful way to recognize preferences that are represented by dif—
ferent utility functions: given two utility functions, just compute the marginal
rates of substitution and see if they are the same. If they are, then the two
utility functions have the same indifference curves. If the direction of increasing
preference is the same for each utility function, then the underlying preferences
must be the same. ll MRS:— 72 UTHJTY(Ch.4) EXAMPLE: Cobb—Douglas Preferences The MRS for CobbDouglas preferences is easy to calculate by using the formula
derived above. If we choose the log representation where
u(m1,m2) : clnml + dlnmg, then we have 814:0], m2)/8a:1
—8u($1,$2)/3m2
6/31
_d/$2
C$2 d$1l Note that the MRS only depends on the ratio of the two parameters and the
quantity of the two goods in this case.
What if we choose the exponent representation where MRS u(a:1,:c2) 2 £353? Then we have _8u($1,$2)/8m1
314271, $2)/89:2 c—l d
C$1 $2
1 MRS Fl C _
dalmg
C$2
(151217 which is the same as we had before. Of course you knew all along that a monotonic
transformation couldn’t change the marginal rate of substitution! ...
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 Fall '09
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