Preferences and Utility

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 3

Today's Agenda

Part 1: Modeling Preferences with Utility Functions

Part 2: Some "canonical" utility functions

Preferences: Definition and Axioms

Quantifying Happiness

Utility Functions

Indifference Curves and the MRS

Perfect Substitutes

Perfect Complements

Cobb-Douglas

Quasilinear

 

Review: 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  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.

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.

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

Quantifying Happiness

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

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 Legislation (1789)

Utilitarianism (1861)

Utility Functions

How do we model preferences mathematically?

x
f()
z = f(x,y)
y
z

Multivariable Function

u()
U = u(x_1,x_2)

Utility Function

Good 1

Good 2

Utility

Utility Functions: Cardinal Ranking of Options

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 ">" 

Indifference Curves as Level Sets

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?

Example: draw the indifference curve for \(u(x_1,x_2) = \sqrt{x_1,x_2}\) passing through (10,40).

Step 1: Evaluate \(u(x_1,x_2)\) at the point

Step 2: Set \(u(x_1,x_2)\) equal to that value.

Step 4: Plug in various values of \(x_1\) and plot!

\(u(10,40) = \sqrt{10 \times 40} = 20\)

\(\sqrt{x_1x_2} = 20\)

\(x_2 = \frac{400}{x_1}\)

How to Draw an Indifference Curve through a Point

Step 3: Solve for \(x_2\).

How would this have been different if the utility function were \(u(x_1,x_2) = 2\sqrt{x_1x_2}\)?

\(u(10, 40) = 2\sqrt{10 \times 40} = 40\)

\(2\sqrt{x_1x_2} = 40\)

\(x_2 = \frac{400}{x_1}\)

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.

MRS and Marginal Utility

Marginal Utility

Econ 1: "The additional utility you get from consuming another unit of the good"

Now that we know calculus, it's much easier:

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

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{Mathematically, all marginal utilities are positive: }MU_1 \ge 0, MU_2 \ge 0
\text{Stricly monotonic}: MU_1 > 0, MU_2 > 0

Marginal Rate of Substitution

From last time: "The amount of good 2 you would give up in order to get another unit of good 1 and be no better or worse off"

How much utility do you lose when you give up \(dx_2\) units of good 2?

How much utility do you gain when you get \(dx_1\) units of good 1?

If you gain the same amount of utility from good 1 as you lost from good 2, can you express the MRS in terms of \(dx_1\) and \(dx_2\)?

MRS = \frac{dx_2}{dx_1} = \frac{MU_1}{MU_2}
dx_2 \times MU_2
dx_1 \times MU_1

pollev.com/chrismakler

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

Transforming Utility Functions

Applying a positive monotonic transformation 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 transformation 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 transformation 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: \(\hat u(x_1,x_2) = \ln(u(x_1,x_2))\)

pollev.com/chrismakler

Normalizing Utility 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

Examples of Utility Functions

pollev.com/chrismakler

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

pollev.com/chrismakler

Perfect Complements

Goods that you like to consume
in a constant ratio.

  • Left shoes and right shoes

  • Sugar and tea

u(x_1,x_2) = \min \left\{\frac{x_1}{a},\frac{x_2}{b}\right\}

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

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

Concave

The opposite of convex: as you consume more of a good, you become more willing to give up others

  • MRS is increasing in \(x_1\) and/or decreasing in \(x_2\)

  • Indifference curves are bowed away from the origin.

e.g., u(x_1,x_2) = ax_1^2 + bx_2^2

Satiation Point

There is some ideal bundle; utility falls off as you move away from that bundle

  • Not monotonic

  • Realistic, but often the satiation point is far out of reach.

e.g., u(x_1,x_2) = 100 - (x_1 - 10)^2 - (x_2 - 10)^2

"Semi-Satiated"

One good has an ideal quantity; the other doesn't

  • Can be a combination of quasilinear and satiation point

  • Can generate the familiar linear demand curve

e.g., u(x_1,x_2) = 100 - (x_1 - 10)^2 + x_2

We previously had quantitatively defined budget sets
and qualitatively defined preferred sets.

We have now assigned a quantity of "utility" to each
consumption bundle in a way that seems rigorous.

Next step: find the consumption bundle in our budget set that maximizes our utility.

Conclusion and Next Steps

Protip: get to know these utility functions!
They're going to come back over and over again.

Econ 50 | 3 | Preferences and Utility | Fall 22

By Chris Makler

Econ 50 | 3 | Preferences and Utility | Fall 22

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