Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 4

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Production Possibilities Fronier

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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

Part 1: Modeling Preferences with Utility Functions

Part 2: Some "canonical" utility functions

Preferences: Definition and Axioms

Indifference curves

The Marginal Rate of Substitution

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!

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 we model preferences mathematically?

Approach: assume consuming goods "produces" utility

Labor

Fish

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Capital

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[RESOURCES]

Utility

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[GOODS]

Fish

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Coconuts

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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

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)

MRS = {MU_1 \over MU_2} =

u(x_1,x_2) = x_1x_2

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

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

A

B

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

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

One reason to transform a utility function is to **normalize**Â 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

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

Math background: "Convex combinations"

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"

Part I: properties of **preferences**,

and how preferences can be represented by **utility functions**.

Part II: see **examples** of utility functions,

and examine how different **functional forms**

can be used to model different kinds of preferences.

**Take the time to understand this material well.**Â

It's foundational for many, many economic models.