Specific Utility Functions; Budget Constraints

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 5

Agenda for Today

Part I: Examples of Utility Functions

Part II: Budget Constraints

Perfect Substitutes
Cobb-Douglas
CES
Quasilinear

Choice Space vs Feasible Sets
Constructing a Budget Set
Changes in the Budget Set

(save kinked constraints for Monday)

Part I: Examples of Utility Functions

How can different functional forms can be used to model different ways people think about the relationships between goods?

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

pollev.com/chrismakler

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

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]}

u(x_1,x_2) = (x_1^r+x_2^r)^{1 \over r}

Constant Elasticity of Substitution (CES) Utility

MRS = \left(x_2 \over x_1\right)^{1-r}

-\infty

COMPLEMENTS: $r < 0$

SUBSTITUTES: $r > 0$

-1

MRS = \left(x_2 \over x_1\right)^2

{1 \over 2}

MRS = \left(x_2 \over x_1\right)^{1 \over 2}

u(x_1,x_2) = (x_1^{-1}+x_2^{-1})^{-1}

u(x_1,x_2) = \left(x_1^{1 \over 2} + x_2^{1 \over 2}\right)^2

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

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.

Summary of Part I

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.

Part II: Budget Constraints

Choice space:
all possible options

Feasible set:
all options available to you

Optimal choice:
Your best choice(s) of the ones available to you

Constrained Optimization

Choice Space
(all colleges plus alternatives)

Feasible Set
(colleges you got into)

Your optimal choice!

Preferences

Preferences describe how the agent ranks all options in the choice space.

For example, we'll assume that you could rank all possible colleges
(and other options for what to do after high school) based upon your preferences.

Preference Ranking

Found a startup

Harvard

Stanford

Play Xbox in parents' basement

Cal

Choice space

Feasible set

Optimal
choice!

Found a startup

Stanford

Cal

Harvard

Play XBox in parents' basement

Optimal choice is the highest-ranking option in the feasible set.

One type of feasible set: the budget set

Prices and Expenditure

Suppose each good has a constant price
(so every unit of the good costs the same)

p_1 = \text{price of good 1}

p_2 = \text{price of good 2}

x_1 = \text{quantity of good 1}

x_2 = \text{quantity of good 2}

p_1x_1 = \text{amount spent on good 1}

p_2x_2 = \text{amount spent on good 2}

p_1x_1 + p_2x_2 = \text{cost of buying bundle }X

Affordability

Suppose you have a given income $m$
to spend on goods 1 and 2.

Then bundle $X = (x_1,x_2)$ is affordable if

p_1x_1 + p_2x_2 \le m

The feasible set, or budget set, is the set of all affordable bundles.

Example: suppose you have $m = \$24$ to spend on two goods.

Good 1 costs $p_1 = \$4$ per unit.

Good 2 costs $p_2 = \$2$ per unit.

Is the bundle (2,4) affordable (in your budget set)? What about the bundle (4,6)?

Draw your budget set.

How would it change if the price of good 2 rose to $p_2' = \$6$ per unit?

How would it change if your income rose to $m' = \$32$?

m = \text{money income}

p_1 = \text{price of good 1}

p_2 = \text{price of good 2}

\text{horizontal intercept} = \frac{m}{p_1}

\text{vertical intercept} = \frac{m}{p_2}

\text{slope of budget line} = -\frac{p_1}{p_2}

Budget Line

pollev.com/chrismakler

Holding income and the price of good 2 constant, an increase in the price of good 1 will cause the budget line to become:

steeper‎

flatter

‎it depends on the‎level of income

‎it depends on the‎ price of good 2‎

Interpreting the Slope of the Budget Line

Example:

Apples cost 50 cents each

Bananas cost 25 cents each

Slope of the budget line represents the opportunity cost of consuming good 1, as dictated by market prices.

-\frac{p_1}{p_2} = -2 \text{ bananas per apple}

In other words: it is the amount of good 2 the market requires you to give up in order to get another unit of good 1.

pollev.com/chrismakler

If apples (good 1) cost $0.80 each,
and bananas (good 2) cost $0.20 each, what is the magnitude (absolute value) of the slope of the budget line?‎

Composite Goods

You have $100 in your pocket.

You see a cart selling apples (good 1) for $2 per pound.

Plot your budget line.
What is "good 2"?
What does the bundle (10,80) signify?
What is the slope of the budget line, and what are its units?

Econ 50 | Spring 25 | Lecture 5

By Chris Makler

Econ 50 | Spring 25 | Lecture 5

The Consumer's Problem: Utility Maximization subject to a Budget Constraint

Specific Utility Functions; Budget Constraints

Agenda for Today

Part I: Examples of Utility Functions

Preferences over Soda

Preferences over Soda

Perfect Substitutes

Goods that can always be exchanged at a constant rate.

Cobb-Douglas

An easy mathematical form with interesting properties.

Normalizing Cobb-Douglas Functions

Normalizing Cobb-Douglas Functions

Constant Elasticity of Substitution (CES) Utility

Quasilinear

Generally used when Good 2 is "dollars spent on other things."

Summary of Part I

Part II: Budget Constraints

Constrained Optimization

Preferences

One type of feasible set: the budget set

Prices and Expenditure

Affordability

The feasible set, or budget set, is the set of all affordable bundles.

Budget Line

Interpreting the Slope of the Budget Line

Composite Goods

Econ 50 | Spring 25 | Lecture 5

More from Chris Makler

Generally used when Good 2 is
"dollars spent on other things."