Modeling Different Types of Preferences with Utility Functions

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 4

pollev.com/chrismakler

Consider the function

f(x,y) = x^2y + xy^2

What is the partial derivative of this function with respect to \(x\) evaluated at the point \((4,2)\)?

I've set up a Khan Academy course where you can practice and learn more about multivariable calculus.

 

(Link in syllabus)

Agenda for Today

Part I: Properties of Preferences

Part II: Examples of Utility Functions

  • Monotonicity
  • Convexity
  • "Well-Behaved" Preferences and the "Law of Diminishing MRS"
  • Perfect Substitutes
  • Cobb-Douglas
  • CES
  • Quasilinear

Part I: Properties of Preferences

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

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 concave.

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"

Testing for Well-Behaved Preferences

Monotonicity

Convexity

  • Monotonicity means getting more of each good makes you better off
  • Marginal utility is the change in utility as you get more of a good
  • Therefore, as long as all marginal utilities are positive, preferences are monotonic.
  • Visually: if you connect any two points on an indifference curve, does it always go through the set of preferred points?
  • Sufficient (but not necessary) conditions for convexity:
    • Preferences are monotonic, AND
    • The MRS is decreasing in \(x_1\), AND
    • The MRS is increasing in \(x_2\)
    • At least one of the last two is strict

Part II: Examples of Utility Functions

How can different functional forms can be used to model different ways people think about the relationships between goods?

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

pollev.com/chrismakler

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

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]}

pollev.com/chrismakler

The utility function

u(x_1,x_2) = x_1^2x_2^3

represents the same preferences as the utility function

u(x_1,x_2) = x_1^\alpha x_2^{1 - \alpha}

for what value of \(\alpha\)?

u(x_1,x_2) = (x_1^r+x_2^r)^{1 \over r}

Constant Elasticity of Substitution (CES) Utility

MRS = \left(x_2 \over x_1\right)^{1-r}
-\infty
1
r

COMPLEMENTS: \(r < 0\)

SUBSTITUTES: \(r > 0\)

-1
MRS = \left(x_2 \over x_1\right)^2
{1 \over 2}
MRS = \left(x_2 \over x_1\right)^{1 \over 2}
u(x_1,x_2) = (x_1^{-1}+x_2^{-1})^{-1}
u(x_1,x_2) = \left(x_1^{1 \over 2} + x_2^{1 \over 2}\right)^2

Composite Goods

You have $1,000 in your bank account.

You have a date tonight, and stop off to buy some tic tacs.
Tic tacs cost $1 per box.

  1. If we think about this as a "good 1 - good 2" problem, what is good 2?
  2. What does the MRS measure in this case?
  3. Which is changing faster: \(MU_1\) or \(MU_2\)?

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

Summary of Part I

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.