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Characteristics of
Utility Functions
Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 8
Today's Agenda
Part 1: Features of Utility Functions
Part 2: Examples of Utility Functions
Transforming Utility Functions
Monotonicity and Convexity
Perfect Substitutes
Cobb-Douglas
Quasilinear
Transforming Utility Functions
Applying a positive monotonic transformation to a utility function doesn't affect
the way it ranks bundles.
u(x_1,x_2) = x_1^{1 \over 2}x_2^{1 \over 2}
\hat u(x_1,x_2) = 2x_1^{1 \over 2}x_2^{1 \over 2}
Example: \(\hat u(x_1,x_2) = 2u(x_1,x_2)\)
u(4,16) = 8
u(9,4) = 6
u(4, 16) = 2 \times 8 = 16
u(9,4) = 2 \times 6 = 12
Transformations and the MRS
Applying a positive monotonic transformation to a utility function doesn't affect
its MRS at any bundle (and therefore generates the same indifference map).
u(x_1,x_2) = x_1^{1 \over 2}x_2^{1 \over 2}
\hat u(x_1,x_2) = 2x_1^{1 \over 2}x_2^{1 \over 2}
MU_1 = {1 \over 2}x_1^{-{1 \over 2}}x_2^{1 \over 2}
MU_2 = {1 \over 2}x_1^{{1 \over 2}}x_2^{-{1 \over 2}}
MU_1 = x_1^{-{1 \over 2}}x_2^{1 \over 2}
MU_2 = x_1^{{1 \over 2}}x_2^{-{1 \over 2}}
MRS =
\displaystyle{= {x_2 \over x_1}}
\hat{MRS} =
\displaystyle{= {x_2 \over x_1}}
Example: \(\hat u(x_1,x_2) = 2u(x_1,x_2)\)
Transformations and the MRS
Applying a positive monotonic transformation to a utility function doesn't affect
its MRS at any bundle (and therefore generates the same indifference map).
u(x_1,x_2) = x_1^{1 \over 2}x_2^{1 \over 2}
\hat u(x_1,x_2) = {1 \over 2}\ln x_1 + {1 \over 2}\ln x_2
MU_1 = {1 \over 2}x_1^{-{1 \over 2}}x_2^{1 \over 2}
MU_2 = {1 \over 2}x_1^{{1 \over 2}}x_2^{-{1 \over 2}}
MU_1 = {1 \over 2x_1}
MU_2 = {1 \over 2x_2}
MRS =
\displaystyle{= {x_2 \over x_1}}
\hat{MRS} =
\displaystyle{= {x_2 \over x_1}}
Example: \(\hat u(x_1,x_2) = \ln(u(x_1,x_2))\)
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The utility function \(u(x_1,x_2) = x_1x_2^2\) represents the same preferences as which of the following utility functions? You may select more than one answer.
Desirable Properties of Preferences
We've asserted that all (rational) preferences are complete and transitive.
There are some additional properties which are true of some preferences:
- Monotonicity
- Convexity
- Continuity
- Smoothness
Monotonic Preferences: “More is Better"
\text{For any bundles }A=(a_1,a_2)\text{ and }B=(b_1,b_2)\text{, }A \succeq B \text{ if } a_1 \ge b_1 \text{ and } a_2 \ge b_2
\text{In other words, all marginal utilities are positive: }MU_1 \ge 0, MU_2 \ge 0
Nonmonotonic Preferences and Satiation
Some goods provide positive marginal utility only up to a point, beyond which consuming more of them actually decreases your utility.
Strict vs. Weak Monotonicity
Strict monotonicity: any increase in any good strictly increases utility (\(MU > 0\) for all goods)
Weak monotonicity: no increase in any good will strictly decrease utility (\(MU \ge 0\) for all goods)
Example: Pfizer's COVID-19 vaccine has a dose of 0.3mL, Moderna's has a dose of 0.5mL
Goods vs. Bads
\text{Good: }MU > 0
\text{Bad: }MU < 0
Convex Preferences: “Variety is Better"
Take any two bundles, \(A\) and \(B\), between which you are indifferent.
Would you rather have a convex combination of those two bundles,
than have either of those bundles themselves?
If you would always answer yes, your preferences are convex.
Concave Preferences: “Variety is Worse"
Take any two bundles, \(A\) and \(B\), between which you are indifferent.
Would you rather have a convex combination of those two bundles,
than have either of those bundles themselves?
If you would always answer no, your preferences are convex.
Common Mistakes about Convexity
1. Convexity does not imply you always want equal numbers of things.
2. It's preferences which are convex, not the utility function.
Other Desirable Properties
Continuous: utility functions don't have "jumps"
Smooth: marginal utilities don't have "jumps"
Counter-example: vaccine dose example
Counter-example: Leontief/Perfect Complements utility function
Well-Behaved Preferences
If preferences are strictly monotonic, strictly convex, continuous, and smooth, then:
Indifference curves are smooth, downward-sloping, and bowed in toward the origin
The MRS is diminishing as you move down and to the right along an indifference curve
Good 1 \((x_1)\)
Good 2 \((x_2)\)
"Law of Diminishing MRS"
Examples of Utility Functions
Preferences over Soda
Suppose one liter of soda gives you 20 "utils" of utility; so each one-liter bottle (good 1) gives you 20 utils,
and each two-liter bottle (good 2) gives you 40 utils.
Which other bundles give you the same utility as
12 one-liter bottles and 6 two-liter bottles (A)?
Which other bundles give you the same utility as
12 one-liter bottles and 2 two-liter bottle (B)?
12
24
Any combination that has 24 total liters
Any combination that has 16 total liters
16
20
4
8
12
24
16
20
4
8
A
B
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Preferences over Soda
Suppose one liter of soda gives you 20 "utils" of utility; so each one-liter bottle (good 1) gives you 20 utils,
and each two-liter bottle (good 2) gives you 40 utils.
Which other bundles give you the same utility as
12 one-liter bottles and 6 two-liter bottles (A)?
Which other bundles give you the same utility as
12 one-liter bottles and 2 two-liter bottle (B)?
Any combination that has 24 total liters
Any combination that has 16 total liters
What utility function represents these preferences?
Perfect Substitutes
Goods that can always be exchanged at a constant rate.
-
Red pencils and blue pencils, if you con't care about color
-
One-dollar bills and five-dollar bills
-
One-liter bottles of soda and two-liter bottles of soda
u(x_1,x_2) = ax_1 + bx_2
Normalizing Utility Functions
One reason to transform a utility function is to normalize it.
This allows us to describe preferences using fewer parameters.
u(x_1,x_2) = ax_1 + bx_2
{1 \over a + b} u(x_1,x_2) = {a \over a + b}x_1 + {b \over a + b}x_2
\hat u(x_1,x_2) = \alpha x_1 + (1 - \alpha)x_2
[ multiply by \({1 \over a + b}\) ]
[ let \(\alpha = {a \over a + b}\) ]
= {a \over a + b}x_1 + \left [1 - {a \over a + b}\right]x_2
Cobb-Douglas
An easy mathematical form with interesting properties.
-
Used for two independent goods (neither complements nor substitutes) -- e.g., t-shirts vs hamburgers
-
Also called "constant shares" for reasons we'll see later.
u(x_1,x_2) = a \ln x_1 + b \ln x_2
u(x_1,x_2) = x_1^ax_2^b
Normalizing Cobb-Douglas Functions
One reason to transform a utility function is to normalize it.
This allows us to describe preferences using fewer parameters.
u(x_1,x_2) = a \ln x_1 + b \ln x_2
{1 \over a + b} u(x_1,x_2) = {a \over a + b}\ln x_1 + {b \over a + b}\ln x_2
\hat u(x_1,x_2) = \alpha \ln x_1 + (1 - \alpha) \ln x_2
[ multiply by \({1 \over a + b}\) ]
[ let \(\alpha = {a \over a + b}\) ]
= {a \over a + b}\ln x_1 + \left [1 - {a \over a + b}\right]\ln x_2
Normalizing Cobb-Douglas Functions
One reason to transform a utility function is to normalize it.
This allows us to describe preferences using fewer parameters.
u(x_1,x_2) = x_1^ax_2^b
u(x_1,x_2)^{1 \over a + b} = x_1^{a \over a + b}x_2^{b \over a + b}
\hat u(x_1,x_2) = x_1^\alpha x_2^{1- \alpha}
[ raise to the power of \({1 \over a + b}\) ]
[ let \(\alpha = {a \over a + b}\) ]
= x_1^{a \over a + b}x_2^{\left [1 - {a \over a + b}\right]}
Quasilinear
Generally used when Good 2 is
"dollars spent on other things."
-
Marginal utility of good 2 is constant
-
If good 2 is "dollars spent on other things," utility from good 1 is often given as if it were in dollars.
\text{e.g., }u(x_1,x_2) = \sqrt x_1 + x_2 \text{ or }u(x_1,x_2) = \ln x_1 + x_2
u(x_1,x_2) = v(x_1) + x_2
Example: draw the indifference curve for \(u(x_1,x_2) = \frac{1}{2}x_1x_2^2\) passing through (4,6).
Step 1: Evaluate \(u(x_1,x_2)\) at the point
Step 2: Set \(u(x_1,x_2)\) equal to that value.
Step 4: Plug in various values of \(x_1\) and plot!
\(u(4,6) = \frac{1}{2}\times 4 \times 6^2 = 72\)
\(\frac{1}{2}x_1x_2^2 = 72\)
\(x_2^2 = \frac{144}{x}\)
\(x_2 = \frac{12}{\sqrt x_1}\)
How to Draw an Indifference Curve through a Point: Method I
Step 3: Solve for \(x_2\).
How would this have been different if the utility function were \(u(x_1,x_2) = \sqrt{x_1} \times x_2\)?
\(u(4,6) =\sqrt{4} \times 6 = 12\)
\(\sqrt{x_1} \times x_2 = 12\)
\(x_2 = \frac{12}{\sqrt x_1}\)
Example: draw the indifference curve for \(u(x_1,x_2) = \frac{1}{2}x_1x_2^2\) passing through (4,6).
Step 1: Derive \(MRS(x_1,x_2)\). Determine its characteristics: is it smoothly decreasing? Constant?
Step 2: Evaluate \(MRS(x_1,x_2)\) at the point.
Step 4: Sketch the right shape of the curve, so that it's tangent to the line at the point.
How to Draw an Indifference Curve through a Point: Method II
Step 3: Draw a line passing through the point with slope \(-MRS(x_1,x_2)\)
How would this have been different if the utility function were \(u(x_1,x_2) = \sqrt{x_1} \times x_2\)?
\text{In both cases }MRS(x_1,x_2) = \frac{MU_1(x_1,x_2)}{MU_2(x_1,x_2)} = \frac{x_2}{2x_1}
MRS(4,6) = \frac{6}{2 \times 4} = \frac{3}{4}
This is continuously strictly decreasing in \(x_1\) and continuously strictly increasing in \(x_2\),
so the function is smooth and strictly convex and has the "normal" shape.
Summary
When considering two goods, there are lots of ways you might feel about them — especially how substitutable the goods are for one another, which is captured by the MRS.
Different functional forms have different MRS's; so they're good for modeling different kinds of preferences.
Take the time to understand this material well.
It's foundational for many, many economic models.
Econ 50 | Lecture 08
By Chris Makler
Econ 50 | Lecture 08
Preferences and Utility Functions
