Chapter 4 Utility

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Chapter 4Utility
2
Introduction

• Last chapter we talk about preference, describing
  the ordering of what a consumer prefers.
• For a more convenient mathematical treatment, we
  turn this ordering into a mathematical function.

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Utility Functions

• A utility function U Rn?R maps each consumption
  bundle of n goods into a real number that
  satisfies the following conditions x
  x U(x) gt U(x) x
  x U(x) lt U(x) x x
  U(x) U(x).

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Utility Functions

• Not all theoretically possible preferences have a
  utility function representation.
• Technically, a preference relation that is
  complete, transitive and continuous has a
  corresponding continuous utility function.

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Utility Functions

• Utility is an ordinal (i.e. ordering) concept.
• The number assigned only matters about ranking,
  but the sizes of numerical differences are not
  meaningful.
• For example, if U(x) 6 and U(y) 2, then
  bundle x is strictly preferred to bundle y. But
  x is not preferred three times as much as is y.

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An Example

• Consider only three bundles A, B, C.
• The following three are all valid utility
  functions of the preference.

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Utility Functions

• There is no unique utility function
  representation of a preference relation.
• Suppose U(x1,x2) x1x2 represents a preference
  relation.
• Consider the bundles (4,1), (2,3) and (2,2).
• U(2,3) 6 gt U(4,1) U(2,2) 4.That
  is, (2,3) (4,1) (2,2).

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Utility Functions

• Define V U2.
• Then V(x1,x2) x12x22 and V(2,3) 36 gt
  V(4,1) V(2,2) 16.So again, (2,3)
  (4,1) (2,2).
• V preserves the same order as U and so represents
  the same preferences.

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Utility Functions

• Define W 2U 10.
• Then W(x1,x2) 2x1x210. So, W(2,3) 22
  gt W(4,1) W(2,2) 18.
• Again, (2,3) (4,1)
  (2,2).
• W preserves the same order as U and V and so
  represents the same preferences.

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Utility Functions

• If
• U is a utility function that represents a
  preference relation and
• f is a strictly increasing function,
• then V f(U) is also a utility
  functionrepresenting .
• Clearly, V(x)gtV(y) if and only if f(V(x)) gt
  f(V(y)) by definition of increasing function.

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Ordinal vs. Cardinal

• As you will see, for our analysis of consumer
  choices, an ordinal utility is enough.
• If the numerical differences are also meaningful,
  we call it cardinal.
• For example, money, weight, height are all
  cardinal.

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Utility Functions Indiff. Curves

• An indifference curve contains equally preferred
  bundles.
• Equal preference ? same utility level.
• Therefore, all bundles on the same indifference
  curve must have the same utility level.

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Utility Functions Indiff. Curves
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Utility Functions Indiff. Curves
U6
U5
U4
U3
U2
U1
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Goods, Bads and Neutrals

• A good is a commodity which increases your
  utility (gives a more preferred bundle) when you
  have more of it.
• A bad is a commodity which decreases your utility
  (gives a less preferred bundle) when you have
  more of it.
• A neutral is a commodity which does not change
  your utility (gives an equally preferred bundle)
  when you have more of it.

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Goods, Bads and Neutrals
Utility
Utilityfunction
Units ofwater aregoods
Units ofwater arebads
Water
x
Around x units, a little extra water is a
neutral.
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Some Other Utility Functions and Their
Indifference Curves

• Consider V(x1,x2) x1 x2.
• What do the indifference curves look like?
• What relation does this function represent for
  these two goods?

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Perfect Substitutes
x2
x1 x2 5
13
x1 x2 9
9
x1 x2 13
5
V(x1,x2) x1 x2.
x1
5
9
13
These two goods are perfect substitutes for this
consumer.
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Some Other Utility Functions and Their
Indifference Curves

• Consider W(x1,x2) minx1,x2.
• What do the indifference curves look like?
• What relation does this function represent for
  these two goods?

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Perfect Complements
x2
45o
W(x1,x2) minx1,x2
minx1,x2 8
8
minx1,x2 5
5
3
minx1,x2 3
x1
3
5
8
These two goods are perfect complements for this
consumer.
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Perfect Substitutes and Perfect Complements

• In general, a utility function for perfect
  substitutes can be expressed as u (x, y)
  ax by
• And a utility function for perfect complements
  can be expressed as u (x, y) min ax ,
  by for constants a and b.

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Some Other Utility Functions and Their
Indifference Curves

• A utility function of the form
  U(x1,x2) f(x1) x2is linear in just x2 and
  is called quasi-linear.
• For example, U(x1,x2) 2x11/2 x2.

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Quasi-linear Indifference Curves
x2
Each curve is a vertically shifted copy of the
others.
x1
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Some Other Utility Functions and Their
Indifference Curves

• Any utility function of the form
  U(x1,x2) x1a x2bwith a gt 0 and b gt 0 is
  called a Cobb-Douglas utility function.
• For example, U(x1,x2) x11/2 x21/2, (a b
  1/2), and V(x1,x2) x1 x23 , (a 1, b 3).

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Cobb-Douglas Indifference Curves
x2
All curves are hyperbolic,asymptoting to, but
nevertouching any axis.
x1
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Cobb-Douglas Utility Functions

• By a monotonic transformation Vln(U)U( x, y)
  xa y b implies
• V( x, y) a ln (x) b ln(y).
• Consider another transformation WU1/(ab)W( x,
  y) xa/(ab) y b/(ab) xc y 1-c so that the
  sum of the indices becomes 1.

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Marginal Utilities

• Marginal means incremental.
• The marginal utility of commodity i is the
  rate-of-change of total utility as the quantity
  of commodity i consumed changes i.e.
  

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Marginal Utilities

• For example, if U(x1,x2) x11/2 x22, then

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Marginal Utilities and Marginal Rates of
Substitution

• The general equation for an indifference curve
  is U(x1,x2) º k, a constant.
• Totally differentiating this identity gives

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Marginal Utilities and Marginal Rates of
Substitution
It can be rearranged to
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Marginal Utilities and Marginal Rates of
Substitution
And
can be furthermore rearranged to
This is the MRS (slope of the indifference curve).
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A Note

• In some texts, economists refer to the MRS by its
  absolute value that is, as a positive number.
• However, we will still follow our convention.
• Therefore, the MRS is

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MRS and MU

• Recall that MRS measures how many units of good 2
  youre willing to sacrifice for one more unit of
  good 1 to remain the original utility level.
• One unit of good 1 is worth MU1.
• One unit of good 2 is worth MU2.
• Number of units of good 2 you are willing to
  sacrifice for one unit of good 1 is thus MU1 /
  MU2.

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An Example

• Suppose U(x1,x2) x1x2. Then

so
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An Example
U(x1,x2) x1x2
x2
8
MRS(1,8) -8/1 -8 MRS(6,6) - 6/6
-1.
6
U 36
U 8
x1
1
6
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MRS for Quasi-linear Utility Functions

• A quasi-linear utility function is of the form
  U(x1,x2) f(x1) x2.

Therefore,
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MRS for Quasi-linear Utility Functions

• MRS - f'(x1) depends only on x1 but not on x2.
  So the slopes of the indifference curves for a
  quasi-linear utility function are constant along
  any line for which x1 is constant.
• What does that make the indifference map for a
  quasi-linear utility function look like?

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MRS for Quasi-linear Utility Functions
x2
Each curve is a vertically shifted copy of the
others.
MRS -f(x1)
MRS -f(x1)
MRS is a constantalong any line for which x1
isconstant.
x1
x1
x1
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Monotonic Transformations Marginal Rates of
Substitution

• Applying a monotonic (increasing) transformation
  to a utility function representing a preference
  relation simply creates another utility function
  representing the same preference relation.
• What happens to marginal rates of substitution
  when a monotonic transformation is applied?

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Monotonic Transformations Marginal Rates of
Substitution

• For U(x1,x2) x1x2 , MRS -x2/x1.
• Create V U2 i.e. V(x1,x2) x12x22. What is
  the MRS for V?which is the same as the MRS
  for U.

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Monotonic Transformations Marginal Rates of
Substitution

• More generally, if V f(U) where f is a strictly
  increasing function, then

The MRS does not change with a monotonic
transformation. Thus, the same preference with
different utility functions still show the same
MRS.