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Which Econ-related song is playing

right now?

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 5

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Production Possibilities Fronier

Feasible

Labor

Fish

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Capital

Coconuts

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[GOODS]

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â

[RESOURCES]

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How does Chuck rank

__all__ __possible__ __combinations__

of fish and coconuts?

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

Feasible

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

Â

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** Â A to B.

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

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

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

A \succ B

A \succeq B

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

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

2

3

4

5

6

x \ge 3

x \gt 3

**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**.

A=(4,3,6)

Example: âgood 1â is apples, âgood 2â is bananas, and âgood 3â is cantaloupes:

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B=(3,8,2)

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General framework: choosing between anything

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 AÂ

indifferent to AÂ

Indifference curves cannot cross!

A

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

Indifference curve

through A

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

How do I love thee?Â 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 everydayâs

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 childhoodâs faith.

I love thee with a love I seemed to lose

With my lost saints,âI love thee with the breath,

Smiles, tears, of all my life!âand, if God choose,

I shall but love thee better after death.

Elizabeth Barrett Browning

Sonnets from the Portugese 43

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*Â (1789)

*Utilitarianism*Â (1861)

How do we model preferences mathematically?

Approach: assume consuming goods "produces" utility

Labor

Fish

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Capital

â³

â

[RESOURCES]

Utility

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[GOODS]

Fish

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Coconuts

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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** Â A to B.

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

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

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

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 ">"Â

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?

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

is to describe

it doesn't matter what the

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 derivatives**Â

as the "marginal utility" from

another unit of either good:

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

In other words, each indifference curve

is a **level set**Â 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

Applying a **positive monotonic transformation**Â 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

Applying a **positive monotonic transformation**Â 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: \(\hat u(x_1,x_2) = 2u(x_1,x_2)\)

**positive monotonic transformation**Â 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 \(u(x_1,x_2) = x_1x_2^2\)â represents the same preferences as which of the following utility functions? You may select more than one answer.

We've asserted that all (rational) preferences are *complete*Â and *transitive*.

There are some additional properties which are true of *some*Â preferences:

- Monotonicity
- Convexity
- Continuity
- Smoothness

\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

Some goods provide positive marginal utility only up to a point, beyond which consuming more of them actually *decreases*Â your utility.

**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

\text{Good: }MU > 0

\text{Bad: }MU < 0

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.**

Take any two bundles, \(A\) and \(B\), between which you are **indifferent**.

than have either of those bundles themselves?

**If you would always answer no, your preferences are convex.**

1. Convexity does not imply you always want equal numbers of things.

2. It's *preferences*Â which are convex, not the *utility function*.

**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*

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"

- 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