Lecture 4 Notes: Utility functions

From preferences to numbers
Cardinal v ordinal
Utility
Intermediate Micro
Lecture 4
Chapter 4 of Varian
Examples
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Preferences and decision-making
1. Last lecture: Ranking consumption bundles by
preference/indifference
2. Today: Assigning values (numbers) to bundles
3. Next lecture: Using values to model decisions
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Utility function
I
Assume we know preferences, indifference curves
I
Surveys, market data...
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Utility function u(x1 , x2 ) is a function that describes
preferences
I
Original idea: measure of happiness with consumption bundle
Choice of u(·) must correctly give
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1. Indifference curves
2. Ranking of bundles
MRS
From preferences to numbers
Cardinal v ordinal
Utility function
I
u(x1 , x2 ) > u(y1 , y2 ) ⇔ (x1 , x2 ) (y1 , y2 )
I
Level sets of u(·) are indifference curves
Level set is all pairs (x1 , x2 )
I
I
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so that u(x1 , x2 ) = k
for a given k
Examples
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Utility function for perfect substitutes
I
xa = minutes with
Albuquerque radio
station
I
xb = minutes with Boise
radio station (same
playlist)
I
Indifference curves: lines
with slope = −1
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Utility function for perfect substitutes
I
I
Indifference curve for 2
hours of Albuquerque
radio
This indifference curve
I
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General indifference
curve
I
I
120 = xa + xb
k = xa + xb
A utility function
I
u(xa , xb ) = xa + xb
MRS
From preferences to numbers
Cardinal v ordinal
Examples
MRS
Choosing u(x) given indifference curves
I
Previous slide: u(x) for
indifference curve =
distance along xa -axis
from origin
I
I
u(xa , xb ) = xa + xb
Book’s suggestion: u(x)
= distance along
45-degree line from
origin
I
u(xa , xb ) =
√
xa + xb
From preferences to numbers
Cardinal v ordinal
Examples
Ordinal utility
u(xa , xb ) = xa + xb
√
u(xa , xb ) = xa + xb
I
Both are correct
I
I
Ordinal utility: The
utility function ranks
bundles correctly
Based solely on what
we can observe about
preferences
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Cardinal utility
u(xa , xb ) = xa + xb
√
u(xa , xb ) = xa + xb
I
Only one, or neither, is
correct
I
I
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Cardinal utility: The
utility function returns
the right value
(number) for each
bundle
Assumes we can
accurately measure
utility
Can use to compare
two individuals’ utility
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Monotonic transformation
I
Monotonic transformation: A
function whose outputs keep the
same order as its inputs
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If u1 > u2 , then f (u1 ) > f (u2 )
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Any function f with f 0 (u) > 0
for all (applicable) u
Monotonic transformation of a
utility function has same ordinal
properties.
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I
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Same indifference curves
Same ranking of bundles
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Lexicographic preferences
Not all preferences allow for a utility function
Example: Lexicographic preferences
I (x1 , x2 ) (y1 , y2 ) iff
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I
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[x1 > y1 ], or
[x1 = y1 and x2 > y2 ]
Cookie Monster preferences
I
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2 cookies and 1 apple better than 1 cookie and 50 apples
2 cookies and 1 apple better than 2 cookies and 0 apples
MRS
From preferences to numbers
Cardinal v ordinal
Examples
When is there a utility function?
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Utility function exists only if
preferences are continuous
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Preferences are ”smooth”
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Continuous preferences give
at-least-as-good sets that are
closed
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A closed set includes its
boundaries
I
Monotone, convexity not
necessary for utility function
Lexicographic preferences
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Choosing a utility function
How we assigned utility function for perfect substitutes:
1. Draw indifference curves
2. Find function with level sets that look like indifference curves
3. Check fit/set constants
Next few slides: utility functions we will frequently work with
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Perfect substitutes
u(x1 , x2 ) = ax1 + bx2
I
Solve for x2 , get
indifference curves
straight lines
I
Slope of − ba
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Perfect compliments
u(x1 , x2 ) = min{ax1 , bx2 }
I
Indifference curves are 90
degree angle around
points where ax1 = bx2
I
What are a and b when
x1 = left shoes, and x2 =
right shoes?
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Quasilinear utility function
u(x1 , x2 ) = v (x1 ) + x2
I
v (·) is some function of
x1
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Indifference curves are
parallel, vertical shifts of
one another
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Cobb-Douglas utility function
u(x1 , x2 ) = x1c x2d
c, d > 0
I
Gives monotone, convex
preferences
I
Easy to work with
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Cobb-Douglas example
u(x1 , x2 ) = x13 x25
Use monotonic transformations
1
1st step: f (u) = u 8
3
5
u(x1 , x2 ) = x18 x28
2nd step: g (u) = ln(u)
3
5
u(x1 , x2 ) = ln(x1 ) + ln(x2 )
8
8
MRS
From preferences to numbers
Cardinal v ordinal
Marginal utility
I
Effect of small increase
in x1 on utility
I
Marginal utility wrt x1
I
MUx1 =
I
Can do same analysis
with x2
∂u(x1 ,x2 )
∂x1
Examples
MRS
From preferences to numbers
Cardinal v ordinal
Marginal utility - Example
u(x1 , x2 ) = x1 + 2x2
(x1 , x2 ) = (9, 8)
I
Draw indifference curve
through (9, 8)
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What is
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What is
∂u
∂x1 ?
∂u
∂x2 ?
Examples
MRS
From preferences to numbers
Cardinal v ordinal
Marginal utility - Example
u(x1 , x2 ) =
√
x1 + 2x2
(x1 , x2 ) = (9, 8)
I
Draw indifference curve
through (9, 8)
I
What is
I
What is
∂u
∂x1 ?
∂u
∂x2 ?
Examples
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Marginal utility and ordinal utility
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MU changes with choice of utility function
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Even for monotonic transformations
∂u(x1 ,x2 )/∂x1
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∂u(x1 ,x2 )/∂x2
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does not change with monotonic transformation
This ratio is MRS, or (−1∗) slope of indifference curve
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Recall MRS is |slope|
MRS
From preferences to numbers
Cardinal v ordinal
Examples
MU and MRS
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Make small change dx1 to x1
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Choose small change dx2 to x2 so u is unchanged
∂u
∂u
dx1 +
dx2
∂x1
∂x2
...(full proof on board and in book)
∂u/∂x1
slope = −
∂u/∂x2
du = 0 =
I
MRS =
∂u/∂x1
∂u/∂x2
MRS
From preferences to numbers
Cardinal v ordinal
Examples
MU and MRS
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Can also show monotonic transformations do not affect
slope/MRS
For 2 utility functions
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Same indifference curves ⇒ Same MRS (at all bundles)
Same MRS (at all bundles) ⇒ Same indifference curves
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Can observe MRS in real world (next chapter)
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Can find utility function that fits ordinal properties
MRS
From preferences to numbers
Cardinal v ordinal
Examples
Calculating MRS
Example: For the following, calculate MRS as a function of x1 and
x2 , and find the value of the MRS at (1, 2):
1. u(x) = x13 x22
√
2. u(x) = x1 + 21 x2
MRS

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