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Characteristics of Utility Functions

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 6

Today's Agenda

  • Review some ways to draw indifference curves
  • Analyze properties of preferences and utility
  • Look at some specific utility functions and the preferences they represent

Example: draw the indifference curve for \(u(x_1,x_2) = \frac{1}{2}x_1x_2^2\) passing through (4,6).

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!

\(u(4,6) = \frac{1}{2}\times 4 \times 6^2 = 72\)

\(\frac{1}{2}x_1x_2^2 = 72\)

\(x_2^2 = \frac{144}{x}\)

\(x_2 = \frac{12}{\sqrt x_1}\)

How to Draw an Indifference Curve through a Point: Method I

Step 3: Solve for \(x_2\).

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

\(u(4,6) =\sqrt{4} \times 6 = 12\)

\(\sqrt{x_1} \times x_2 = 12\)

\(x_2 = \frac{12}{\sqrt x_1}\)

Example: draw the indifference curve for \(u(x_1,x_2) = \frac{1}{2}x_1x_2^2\) passing through (4,6).

Step 1: Derive \(MRS(x_1,x_2)\). Determine its characteristics: is it smoothly decreasing? Constant?

Step 2: Evaluate \(MRS(x_1,x_2)\) at the point.

Step 4: Sketch the right shape of the curve, so that it's tangent to the line at the point.

How to Draw an Indifference Curve through a Point: Method II

Step 3: Draw a line passing through the point with slope \(-MRS(x_1,x_2)\) 

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

\text{In both cases }MRS(x_1,x_2) = \frac{MU_1(x_1,x_2)}{MU_2(x_1,x_2)} = \frac{x_2}{2x_1}
MRS(4,6) = \frac{6}{2 \times 4} = \frac{3}{4}

This is continuously strictly decreasing in \(x_1\) and continuously strictly increasing in \(x_2\),
so the function is smooth and strictly convex and has the "normal" shape.

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"

\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

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

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.

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"

Preferences over Soda

Suppose one liter of soda gives you 20 "utils" of utility; so each one-liter bottle (good 1) gives you 20 utils,
and each two-liter bottle (good 2) gives you 40 utils.

Which other bundles give you the same utility as
12 one-liter bottles and 6 two-liter bottles (A)?

Which other bundles give you the same utility as
12 one-liter bottles and 2 two-liter bottle (B)?

12

24

Any combination that has 24 total liters

Any combination that has 16 total liters

16

20

4

8

12

24

16

20

4

8

A

B

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Preferences over Soda

Suppose one liter of soda gives you 20 "utils" of utility; so each one-liter bottle (good 1) gives you 20 utils,
and each two-liter bottle (good 2) gives you 40 utils.

Which other bundles give you the same utility as
12 one-liter bottles and 6 two-liter bottles (A)?

Which other bundles give you the same utility as
12 one-liter bottles and 2 two-liter bottle (B)?

Any combination that has 24 total liters

Any combination that has 16 total liters

What utility function represents these preferences?

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

Preferences over Tea and Biscuits

Charlotte always has 2 biscuits with every cup of tea;
if she has a plate of biscuits and one cup of tea, she'll only eat two. (And if she has a whole pot of tea and 6 biscuits, she'll stop pouring after 3 cups of tea.)

Each (cup + 2 biscuits) gives her 10 utils of joy.

Which other bundles give her the same utility as
3 cups of tea and 4 biscuits (A)?

Which other bundles give her the same utility as
1 cup of tea and 4 biscuits (B)?

3

6

Any combination that has 4 biscuits
and 2 or more cups of tea

Any combination that has 1 cup of tea and
at 2 or more biscuits

4

5

1

2

3

6

4

5

1

2

A

B

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) = ax_1 + bx_2
{1 \over a + b} u(x_1,x_2) = {a \over a + b}x_1 + {b \over a + b}x_2
\hat u(x_1,x_2) = \alpha x_1 + (1 - \alpha)x_2

[ multiply by \({1 \over a + b}\) ]

[ let \(\alpha = {a \over a + b}\) ]

= {a \over a + b}x_1 + \left [1 - {a \over a + b}\right]x_2

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What utility function represents these preferences?

Preferences over Tea and Biscuits

Charlotte always has 2 biscuits with every cup of tea;
if she has a plate of biscuits and one cup of tea, she'll only eat two. (And if she has a whole pot of tea and 6 biscuits, she'll stop pouring after 3 cups of tea.)

Each (cup + 2 biscuits) gives her 10 utils of joy.

Which other bundles give her the same utility as
3 cups of tea and 4 biscuits (A)?

Which other bundles give her the same utility as
1 cup of tea and 4 biscuits (B)?

Any combination that has 4 biscuits
and 2 or more cups of tea

Any combination that has 1 cup of tea and
at 2 or more biscuits

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

Normalizing Cobb-Douglas 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

Normalizing Cobb-Douglas 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) = x_1^ax_2^b
u(x_1,x_2)^{1 \over a + b} = x_1^{a \over a + b}x_2^{b \over a + b}
\hat u(x_1,x_2) = x_1^\alpha x_2^{1- \alpha}

[ raise to the power of \({1 \over a + b}\) ]

[ let \(\alpha = {a \over a + b}\) ]

= x_1^{a \over a + b}x_2^{\left [1 - {a \over a + b}\right]}

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

Curve Balls

u(x_1,x_2,x_3,x_4) = \min\{{x_1+x_2 \over 2},x_3 + x_4\}

Good 1: chocolate chip cookies

Good 2: snickerdoodle cookies

Good 3: vanilla ice cream

Good 4: strawberry ice cream

Curve Balls

u(x_1,x_2,x_3) = x_1 + \min\{x_2,x_3\}

Good 1: burritos

Good 2: burgers

Good 3: fries

Curve Balls

u(x_1,x_2,x_3) = \min\{x_1 + 2x_2,4x_3\}

Good 1: fries

Good 2: onion rings

Good 3: burgers

Summary

When considering two goods, there are lots of ways you might feel about them — especially how substitutable the goods are for one another, which is captured by the MRS.

Different functional forms have different MRS's; so they're good for modeling different kinds of preferences. 

Take the time to understand this material well. 
It's foundational for many, many economic models.