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1.3. Utility

1.3. Utility Text book chapter 4

Point of views:
- Historically, Utility was thought as a numeric measure of a consumer’s happiness 
Consumer try to maximize utility
- Questions: How do we measure?
How do we quantify utility from different choices
How do we compare utility between people
- Instead of above, start to think about utility as being constructive from preferences,
preferences describe choices
Utility is seen only as a way to describe preferences
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1.3.1. Utility function and ordinal utility

• Utility function: Takes consumption bundles and translates them into number
• More preferred bundles have higher utility than less preferred bundles:
(x1, x2) > (y1, y2) if and only if u (x1, x2) > u (y1, y2)
• Denote: u(.): utility function

• Ordinal utility: Absolute numbers do not matter, just the ranking of the
bundles
•  Size of utility difference between bundles does not matter

Examples
• order the bundles in the same way
• consumer prefers A to B and B to C
• All of the ways indicated are valid utility functions that describe the
same preferences
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1.3.2. Monotonic transformations

• Since only the ranking matter, no unique way to assign utility to
consumption bundles  infinite number of way
• Ex: if u(x1, x2) present a way to assign utility numbers to the bundles (x1,
x2) then multiplying u(x1, x2) by 2 (or any positive number), that is an
other way to assign utility.
• (positive) Monotonic function: Transform a set of number into another set
that maintains the order of the numbers. If u1 > u2  f (u1) > f (u2)
• present a monotonic transformation by a function f(u)

• Examples: Multiply by a positive number (f(u) = 3u), adding any number (f(u) = u
+ 7), raising u to an odd power (f(u) = 𝑢 ), ….
• The rate of change of f(u) as u changes
• Always has a positive rate of change

Not monotonic function, sometimes

Monotonic function, always increasing
increases, sometimes decreases.
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Which of the following are monotonic transformations?

(1) u = 2v − 13;
(2) u = −1/𝑣 ;
The text said that raising a number to an odd
(3) u = 1/𝑣 ; power was a monotonic transformation. What
(4) u = ln v; about raising a number to an even power? Is
this a monotonic transformation?
(5) u = −𝑒 ; (Hint: consider the case f(u) = u2.)
(6) u = 𝑣 ;
(7) u = 𝑣 for v > 0;
(8) u = 𝑣 for v < 0

1.3.3. Utility Functions & Indifferent curves

• Consider bundles (4,1), (2,3) and (2,2).
• Suppose (2,3) > (4,1) ~ (2,2).
• Assign to these bundles any numbers that preserve the preference ordering;
e.g. U(2,3) = 6 > U(4,1) = U(2,2) = 4.
• Call these numbers utility levels.
• An indifference curve contains equally preferred bundles.
• Equal preference  same utility level.
• Therefore, all bundles in an indifference curve have the same utility level.
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Utility Functions & ICs

• So the bundles (4,1)
and (2,2) are in the IC x2
with utility level U=4.
• But the bundle (2,3) is
in the IC with utility
level U=6.
• On an IC diagram, this
preference
information looks like: U6
U4

Utility Functions & ICs

• Comparing more bundles will
create a larger collection
x2 of all ICs
and a better description of the
consumer’s preferences.

• The collection of all ICs for a given

preference relation is an
indifference map.
• An indifference map is equivalent U6
to a utility function. U4
U2
x1
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Some examples of utility functions

• Function: u(x1, x2)
• Draw IC: Plot all point (x1,x2) that have u(x1,x2) = k
• Each difference value of k, get a different IC
• Level set: Set of all (x1,x2) such that u(x1, x2) = a
constant (k)
• Typical IC: k = x1x2; x2 = k/x1
 Graphic: Shown ICs

Some examples of utility functions

• Utility function:
• How do its difference curves look like?

Since u(x1, x2) > 0  v(x1,x2) is monotonic

transformation of u(x1,x2)
v(x1,x2) has exactly the same shaped indifferent
curves of u(x1, x2)  Graphic

 The labeling of IC will be different

The label 1, 2, 3, … now will be 1, 4, 9, …
The set bundles that v(x1, x2) = 9, exactly the same as the set of bundles that has u(x1, x2) =3
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Perfect substitution
• The red pencil and blue pencil example: All the matter to the consumer
was the total number of pencils.
• Measure utility by the total number of pencils.
• Utility function u(x1, x2) = x1+x2 satisfy 2 conditions:
• This utility function constant along the indifference curves
• A higher label to more-preferred bundles?
• Monotonic transformation of u(x1, x2), ex:
v(x1, x2) = 𝑥 + 𝑥 = 𝑥 + 2𝑥 𝑥 + 𝑥
• Utility function if good 1 is twice as valuable to consumer as good2:
u(x1,x2) = 2𝑥 + 𝑥
• Utility function present perfect substitution preferences:
u(x1,x2) = a𝑥 + 𝑏𝑥 , slope = -a/b

Perfect complement
x2

45o
• Remind example of left shoe and right shoe
• Consumer care about the number of pairs of
shoe  Choose the number of pairs of
shoes as the utility function min{x1,x2} = 8
8
• The number of complete pairs: minimum of
the number right shoes and the number of 5 min{x1,x2} = 5
left shoes, thus the utility function for
perfect complement: u(x1,x2) = min{x1,x2} 3 min{x1,x2} = 3

• Ex: take bundle min{10,10} = min{11,10} = x1

10 3 5 8
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• Second example: consumer always uses 2 teaspoons of sugar with

each cup of tea
• If x1 is the number of cups of tea available and x2 is the number of
teaspoons of sugar, utility function: u(x1,x2) = min{𝑥 𝑥 }.
• In general, a utility function that describes perfect-complement
preferences is given by:
u(x1, x2) = min{a𝑥 + 𝑏𝑥 }
• a and b are positive numbers that indicate the proportions in which the
goods are consumed

Quasi-linear utility function

• Quasi-linear utility function : the preferences are x2
vertically shifted version: Each curve is a vertically
x2= k – v(x1) shifted copy of the others.
• ie: The height of each difference curve is some
function of x1 plus a constant. There’s a general
form: U(x1,x2) = k = v(x1) + x2
• Quasi-linear (partly linear): The utility function is
linear in good 2, but (maybe) nonlinear in good 1
• E.g.: u(x1,x2) = ln x1 + x2; u(x1,x2) = 𝑥 +
𝑥 ; u(x1,x2) = 𝑥 + 𝑥
x1
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Cobb-Douglas utility function

• Cobb-Douglas utility function:
u(x1,x2) = x1c x2d
with a > 0 and b > 0.
• E.g. U(x1,x2) = x11/2 x21/2 (a = b = 1/2)
V(x1,x2) = x1 x23 (a = 1, b = 3)
• Cobb-Douglas preferences are the
standard example of IC that look well-
behaved
• All curves are hyperbolic,
asymptoting to, but never
touching any axis.

1.3.4. Marginal Utilities (MU)

• The marginal utility: measures the rate of change in utility (U) associated
with a small change in the amount of good 1(x1) or good 2 (x2)

• MU if x1 change:

• MU if x2 change:
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1.3.5. Marginal Utility and Marginal Rate of

Substitution (MU and MRS)
• Consider a change in the consumption of each good, (Δx1, Δx2), that keeps
utility constant—U(x1,x2)  k (k is a constant), that is, a change in consumption
that moves us along the indifference curve. Then we must have:
• Totally differentiating this identity gives: 𝑀𝑈 𝑥 + 𝑀𝑈 𝑥 = 𝑈 = 0

 MRS = =-

• Negative sign: If get more of good 1, have to get less of good 2 in order to keep
the same level of utility.
• However, MRS is normally refer by its absolute value – as a positive number

Monotonic Transformations & MRS

• For U(x1,x2) = x1x2, the MRS = -x2/x1.
• Create V = U2; i.e. V(x1,x2) = x12x22.
• What is the MRS for V?
 V /  x1 2 x1 x22 x
MRS     2
 V /  x2 2 x12 x2 x1
• which is the same as the MRS for U.
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Summary
1. A utility function is simply a way to represent or summarize
a preference 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.
3. The marginal rate of substitution, MRS, can be calculated
from the utility function via the formula MRS = Δx2/Δx1 =
−MU1/MU2.

Review questions
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