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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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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

Β 

Preferences

Preferences: Ordinal Ranking of Options

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

A \succ B
A \succeq B
A \sim B
A \preceq B
A \prec B

The agent strictly prefers A to B.

The agent strictly disprefers Β A to B.

The agent weakly prefers Β A to B.

The agent weakly disprefers Β A to B.

The agent is indifferent between A and B.

Sidebar: β€œStrictly" vs. β€œWeakly"

A \succ B
A \succeq B

The agent strictly prefers A to B.

The agent weakly prefers Β A to B.

2
3
4
5
6
x \ge 3
x \gt 3

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 \(C\).

\text{For any options }A\text{ and }B\text{, either }A \succeq B \text{ or } B \succeq A
\text{If }A \succeq B \text{ and } B \succeq C\text{, then } A \succeq 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

A=(4,3,6)

Example: β€œgood 1” is apples, β€œgood 2” is bananas, and β€œgood 3” is cantaloupes:

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B=(3,8,2)

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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 AΒ 

indifferent to AΒ 

Indifference curves cannot cross!

A

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

Indifference curve
through A

Marginal Rate of Substitution

X = (10,24)

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Y=(12,20)

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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

MRS = {4 \text{ bananas} \over 2 \text{ apples}}
= 2\ {\text{bananas} \over \text{apple}}

Visually: the MRS is the magnitude of the slope
of an indifference curve

Quantifying Preferences

How do I love thee?Β  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,β€”I love thee with the breath,
Smiles, tears, of all my life!β€”and, if God choose,
I shall but love thee better after death.

Elizabeth Barrett Browning
Sonnets from the Portugese 43

18th/19th Centuries: Utilitarianism

the greatest 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 the greatest amount of happiness, altogether.

John Stuart Mill

Introduction to the Principles of Morals and LegislationΒ (1789)

UtilitarianismΒ (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

πŸ˜€

[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:

A \succ B
A \succeq B
A \sim B
A \preceq B
A \prec B

The agent strictly prefers A to B.

The agent strictly disprefers Β A to B.

The agent weakly prefers Β A to B.

The agent weakly disprefers Β A to B.

The agent is indifferent between A and B.

Representing Preferences with a Utility Function

u(a_1,a_2) > u(b_1,b_2)
u(a_1,a_2) \ge u(b_1,b_2)
u(a_1,a_2) = u(b_1,b_2)
u(a_1,a_2) \le u(b_1,b_2)
u(a_1,a_2) < u(b_1,b_2)
u(x_1,x_2)

"A is strictly preferred to B"

Words

Preferences

Utility

A \succ B
A \succeq B
A \sim B
A \preceq B
A \prec B

"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 ">"Β 

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 β€œutils" are nonsense.

As long as the utility function generates the correct indifference map,
it doesn't matter what the level of utilityΒ at each indifference curve is.

The Mathematics of Utility Functions

Marginal Utility

MU_1(x_1,x_2) = {\partial u(x_1,x_2) \over \partial x_1}
MU_2(x_1,x_2) = {\partial u(x_1,x_2) \over \partial x_2}

Given a utility function \(u(x_1,x_2)\),
we can interpret the partial derivativesΒ 
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 setΒ of the utility function.

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

MRS(x_1,x_2) = \frac{\partial u(x_1,x_2) \over \partial x_1}{\partial u(x_1,x_2) \over \partial x_2} = {MU_1 \over MU_2}

(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?

\Delta u \approx \Delta x_2 \times MU_2
\Delta u \approx \Delta x_1 \times MU_1

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

\Delta x_2 \times MU_2 \approx \Delta x_1 \times MU_1
{\Delta x_2 \over \Delta x_1} \approx {MU_1 \over MU_2}

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

\displaystyle{MRS = {MU_1 \over MU_2} =}
u(x_1,x_2) = x_1x_2
\displaystyle{MU_1 = {\partial u(x_1,x_2) \over \partial x_1} =}
\displaystyle{MU_2 = {\partial u(x_1,x_2) \over \partial x_1} =}
4 \times 16
A = (4,16)
B = (8,8)
8 \times 8
x_2
x_1
\displaystyle{x_2 \over x_1}
= 64
= 64
16
8
4
8
\displaystyle{16 \over 4}
=4
\displaystyle{8 \over 8}
=1

MRS = 4

MRS = 1

Transforming Utility Functions

Applying a positive monotonic transformationΒ 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 transformationΒ 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 transformationΒ 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 completeΒ and transitive.

There are some additional properties which are true of someΒ preferences:

  • Monotonicity
  • Convexity
  • Continuity
  • Smoothness

Monotonic Preferences: β€œMore 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 decreasesΒ 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: β€œVariety 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: β€œVariety 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 preferencesΒ 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

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