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# Preferences and Utility

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 5

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Production Possibilities Fronier

Feasible

## Last Time: The PPF

## Lecture 2: Production Functions

Labor

Fish

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Capital

Coconuts

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[GOODS]

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⛏

[RESOURCES]

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🥥

🙂

😀

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## Today: Preferences

How does Chuck rank

__all__ __possible__ __combinations__

of fish and coconuts?

Goal: find the best combination *within his production possibilities set.*

Feasible

## Today's Agenda

Today: Modeling Preferences with Utility Functions

Friday: Some "canonical" utility functions

Preferences: Definition and Axioms

Indifference curves

The Marginal Rate of Substitution

Quantifying Happiness

The Mathematics of Utility Functions

Perfect Substitutes

Perfect Complements

Cobb-Douglas

Quasilinear

# Preferences

## Preferences: Ordinal Ranking of Options

Given a choice between option A and option B, an agent might have different preferences:

The agent **strictly prefers** A to B.

The agent **strictly disprefers** A to B.

The agent **weakly prefers** A to B.

The agent **weakly disprefers** A to B.

The agent is **indifferent between **A and B.

## Sidebar: “Strictly" vs. “Weakly"

The agent **strictly prefers** A to B.

The agent **weakly prefers** A to B.

## Preference Axioms

**Complete**

**Transitive**

Any two options can be compared.

If \(A\) is preferred to \(B\), and \(B\) is preferred to \(C\),

then \(A\) is preferred to \(C\).

Together, these assumptions mean that we can rank

all possible choices in a coherent way.

Special case: choosing between **bundles**

containing different **quantities of goods**.

# Preferences over Quantities

Example: “good 1” is apples, “good 2” is bananas, and “good 3” is cantaloupes:

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General framework: choosing between anything

# Special Case: Two Goods

Good 1 \((x_1)\)

Good 2 \((x_2)\)

Completeness axiom:

any two bundles can be compared.

Implication: given any bundle \(A\),

the choice space may be divided

into three regions:

preferred to A

dispreferred to A

indifferent to A

Indifference curves cannot cross!

A

The **indifference curve through A** connects all the bundles indifferent to A.

Indifference curve

through A

# Marginal Rate of Substitution

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Suppose you were indifferent between the following two bundles:

Starting at bundle X,

you would be willing

to give up 4 bananas

to get 2 apples

Let apples be good 1, and bananas be good 2.

Starting at bundle Y,

you would be willing

to give up 2 apples

to get 4 bananas

Visually: the MRS is the magnitude of the slope

of an indifference curve

# Quantifying Preferences

How do I love thee? Let me count the ways.

I love thee to the depth and breadth and height

My soul can reach, when feeling out of sight

For the ends of Being and ideal Grace.

I love thee to the level of everyday’s

Most quiet need, by sun and candlelight.

I love thee freely, as men strive for Right;

I love thee purely, as they turn from Praise.

I love thee with the passion put to use

In my old griefs, and with my childhood’s faith.

I love thee with a love I seemed to lose

With my lost saints,—I love thee with the breath,

Smiles, tears, of all my life!—and, if God choose,

I shall but love thee better after death.

Elizabeth Barrett Browning

Sonnets from the Portugese 43

# 18th/19th Centuries: Utilitarianism

thegreatest happiness of the greatest number

is the foundation of morals and legislation.

Jeremy Bentham

the utilitarian standard...

is not the agent's own greatest happiness,

but thegreatest amount of happiness, altogether.

John Stuart Mill

*Introduction to the Principles of Morals and Legislation* (1789)

*Utilitarianism* (1861)

How do we model preferences mathematically?

Approach: assume consuming goods "produces" utility

## Production Functions

Labor

Fish

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Capital

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[RESOURCES]

## Utility Functions

Utility

😀

[GOODS]

Fish

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Coconuts

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## Preferences: Ordinal Ranking of Options

Given a choice between option A and option B, an agent might have different preferences:

The agent **strictly prefers** A to B.

The agent **strictly disprefers** A to B.

The agent **weakly prefers** A to B.

The agent **weakly disprefers** A to B.

The agent is **indifferent between **A and B.

## Representing Preferences with a Utility Function

"A is strictly preferred to B"

**Words**

**Preferences**

**Utility**

"A is weakly preferred to B"

"A is indifferent to B"

"A is weakly dispreferred to B"

"A is strictly dispreferred to B"

Suppose the "utility function"

assigns a real number (in "utils")

to every possible consumption bundle

We get **completeness** because any two numbers can be compared,

and we get **transitivity** because that's a property of the operator ">"

An indifference curve is a set of all bundles between which a consumer is indifferent.

If a consumer is indifferent between two bundles (A ~ B), then \(u(a_1,a_2) = u(b_1,b_2)\)

Therefore, an indifference curve is a set of all consumption bundles which are assigned the same number of "utils" by the function \(u(x_1,x_2)\)

Likewise, set of bundles preferred to some bundle A is the a set of all consumption bundles which are assigned a *greater* number of "utils" by \(u(x_1,x_2)\)

Do we have to take the

number of "utils" seriously?

## From XKCD:

Just like the "volume" on an amp, "utils" are an arbitrary scale...saying you're "11" happy from a bundle doesn't mean anything!

### All we want to use utility functions for

is to describe **preference orderings**.

## It doesn't matter that “utils" are nonsense.

### As long as the utility function generates the correct **indifference map**,

it doesn't matter what the **level of utility** at each indifference curve is.

# The Mathematics of Utility Functions

# Marginal Utility

Given a utility function \(u(x_1,x_2)\),

we can interpret the **partial derivatives**

as the "marginal utility" from

another unit of either good:

# Indifference Curves and the MRS

Along an indifference curve, all bundles will produce the same amount of utility

In other words, each indifference curve

is a **level set** of the utility function.

The slope of an indifference curve is the MRS. By the implicit function theorem,

(Note: we'll treat this as a positive number, just like the MRTS and the MRT)

If you give up \(\Delta x_2\) units of good 2, how much utility do you lose?

If you get \(\Delta x_1\) units of good 1, how much utility do you gain?

If you end up with the same utility as you begin with:

pollev.com/chrismakler

What is the MRS of the utility function \(u(x_1,x_2) = x_1x_2\)?

MRS = 4

MRS = 1

# Transforming Utility Functions

Applying a **positive monotonic transformation** to a utility function doesn't affect

the way it ranks bundles.

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).

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).

Example: \(\hat u(x_1,x_2) = \ln(u(x_1,x_2))\)

pollev.com/chrismakler

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"

# 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

# 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"

## Next Steps

- Section: practice drawing indifference curves from utility functions
- Friday: look at specific utility functions and talk about the preferences they model
- Homework due Saturday night
- Next: maximizing utility subject to a PPF

#### Econ 50 | Lecture 05

By Chris Makler

# Econ 50 | Lecture 05

Preferences and Utility Functions

- 437