Utility Maximization with Budget Constraints

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 12

pollev.com/chrismakler

How did the test go, relative to how you thought it would go?

Resources

Technology

Stuff

Happiness

🏭

⌚️

🤓

Part I:
The Real Economy

⏳

⛏

Resources

Firms

Stuff

Consumers

⏳

🏭

⌚️

🤓

Part I:
The Market Economy

Resource
Owners

👷🏽‍♀️

⛏

🏦

Resources

Firms

Stuff

Consumers

⏳

🏭

⌚️

🤓

Part I:
The Market Economy

Resource
Owners

👷🏽‍♀️

⛏

🏦

💵

Firms pay wages for labor

Firms pay rent on capital

Consumers pay prices for goods

Demand

Supply

🤩

🏪

Next Five Weeks:

How do consumers and firms respond to prices?

The Consumer's Problem

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.

L_1(x_1) + L_2(x_2) = \overline L

MRT = {L_1'(x_1) \over L_2'(x_2)}

= {{dL_1 \over dx_1} \over {dL_2 \over dx_2}}

= {{1 \over MP_{L1}} \over {1 \over MP_{L2}}}

= {MP_{L2} \over MP_{L1}}

\text{Price Ratio} = {p_1 \over p_2}

p_1x_1 + p_2x_2 = m

amount of labor required to produce another unit of good 1

amount of labor required to produce another unit of good 2

amount of money required to
buy another unit of good 1

amount of money required to
buy another unit of good 2

PPF

Budget Line

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?

Utility Maximization

Next two weeks:
Maximize utility subject to a (parameterized) budget line

🍏

🍌

Last two lectures:
Maximize utility subject to a (fixed) PPF

🐟

🥥

PPF

Next two weeks:
Maximize utility subject to a (parameterized) budget line

🍏

🍌

All the math we did last week holds,
but the slope of the constraint is the price ratio,
not the MRT. (Still opportunity cost!)

Next two weeks:
Maximize utility subject to a (parameterized) budget line

🍏

🍌

"Gravitational pull" argument:

MRS > \frac{p_1}{p_2}

MRS < \frac{p_1}{p_2}

Indifference curve is
steeper than the budget line

Indifference curve is
flatter than the budget line

Moving to the right
along the budget line
would increase utility.

Moving to the left
along the budget line
would increase utility.

Next two weeks:
Maximize utility subject to a (parameterized) budget line

🍏

🍌

Can sometimes use the tangency condition
$MRS = p_1/p_2$, sometimes you have to use logic.

Next two weeks:
Maximize utility subject to a (parameterized) budget line

🍏

🍌

BL1

Big difference:

We will be solving for the optimal bundle
as a function of income and prices:

The solutions to this problem will be called the demand functions. We have to think about how the optimal bundle will change when $p_1,p_2,m$ change.

x_1^*(p_1,p_2,m)

x_2^*(p_1,p_2,m)

BL2

“Gravitational Pull" with a Budget Line

What does it mean if the MRS is
greater than the price ratio?

\frac{MU_1}{MU_2} > \frac{p_1}{p_2}

The consumer receives more utility per additional unit of good 1 than the price reflects, relative to good 2.

\frac{MU_1}{p_1} > \frac{MU_2}{p_2}

The consumer receives more
"bang for the buck"
(utils per dollar)
from good 1 than good 2.

Regardless of how you look at it, the consumer would be
better off moving to the right along the budget line --
i.e., consuming more of good 1 and less of good 2.

MRS > \frac{p_1}{p_2}

The consumer is more willing to give up good 2
to get good 1
than the market requires.

MRS and the Price Ratio: Cobb-Douglas

Important and Difficult Distinction

The budget line and indifference curves describe different things.

Indifference curves describe the "shape of the utility hill."
They do not change when prices or income change.
They do change when preferences change, but we usually assume preferences are fixed.

The budget line describes the boundary of affordable bundles;
we can think of it as a fence over the utility hill.

IF...

THEN...

The consumer's preferences are "well behaved"
-- smooth, strictly convex, and strictly monotonic

$MRS=0$ along the horizontal axis ($x_2 = 0$)

The budget line is a simple straight line

The optimal consumption bundle will be characterized by two equations:

MRS = \frac{p_1}{p_2}

p_1x_1 + p_2x_2 = m

More generally: the optimal bundle may be found using the Lagrange method

$MRS \rightarrow \infty$ along the vertical axis ($x_1 \rightarrow 0$)

How do you tell if a preferences are "well behaved"?

Strictly monotonic

Strictly convex

Smooth

$MU_1 > 0$ and $MU_2 > 0$ for any $x_1,x_2$

$\frac{\partial MRS}{\partial x_1} \le 0$ and $\frac{\partial MRS}{ \partial x_2} \ge 0$, with at least one strict

MRS has no "jumps" (not defined piecewise)

Continuous

Utility function has no "jumps" (not defined piecewise)

(i.e., indifference curves get flatter as you move down and to the right)

u(x_1,x_2)

\max

The Lagrange Method

x_1,x_2

\text{s.t.}

p_1x_1 + p_2x_2

\le m

Cost of Bundle X

Income

Utility

\max

The Lagrange Method

x_1,x_2

\text{s.t.}

m - p_1x_1 - p_2x_2

Income left over

u(x_1,x_2)

Utility

\ge 0

The Lagrange Method

m - p_1x_1 - p_2x_2

Income left over

\mathcal{L}(x_1,x_2,\lambda)=

\lambda

u(x_1,x_2)

(

)

Utility

(utils)

(dollars)

utils/dollar

\frac{\partial \mathcal{L}}{\partial x_1} = \frac{\partial u}{\partial x_1} - \lambda p_1

First Order Conditions

\frac{\partial \mathcal{L}}{\partial x_2} = \frac{\partial u}{\partial x_2} - \lambda p_2

\frac{\partial \mathcal{L}}{\partial \lambda} = m - p_1x_1 - p_2x_2

= 0 \Rightarrow \lambda = \frac{MU_1}{p_1}

= 0 \Rightarrow \lambda = \frac{MU_2}{p_2}

"Bang for your buck" condition: marginal utility from last dollar spent on every good must be the same!

\text{Also: }\frac{\partial \mathcal{L}}{\partial m} = \lambda = \text{bang for your buck, in }\frac{\text{utils}}{\$}

\text{Set two expressions }

\text{for }\lambda \text{ equal:}

\frac{MU_1}{p_1} = \frac{MU_2}{p_2}

The Lagrange Method

m - p_1x_1 - p_2x_2

\mathcal{L}(x_1,x_2,\lambda)=

\lambda

u(x_1,x_2)

(

)

The Tangency Condition

What happens when the price of a good increases or decreases?

The Tangency Condition when Lagrange sometimes works

What happens when income decreases?

pollev.com/chrismakler

If your utility function is

$u(x_1,x_2) = 4x_1 + 2x_2$,

when will you buy only good 1?

Cobb-Douglas

Quasilinear

Perfect Substitutes

MRS = {ax_2 \over bx_1}

MRS = v'(x_1)

MRS = {a \over b}

u(x_1,x_2)=x_1^ax_2^b

u(x_1,x_2)=v(x_1) + x_2

u(x_1,x_2)=ax_1 + bx_2

TANGENCY CONDITION

x_2 = {p_1 \over p_2} \times {b \over a}x_1

v'(x_1) = {p_1 \over p_2}

{a \over b} = {p_1 \over p_2}

Ray from origin, will always intersect budget line =>

Lagrange always works

Vertical line, may or may not intersect BL in first quadrant =>

Lagrange sometimes works

Tries to equate two constants, which you just can't do =>

Lagrange never works

Kinked Budget Constraints

Nonlinear electricity rates
Gift cards
Quantity discounts
"Buying" lower prices

(more on these in the interactive lecture notes)

How to Solve a Kinked Constraint Problem

Evaluate the MRS at the kink
Compare it to the price ratio on either side of the kink
If $MRS > {p_1 \over p_2}$ to the right of the kink, the solution is to the right of the kink; use the equation for that line and maximize.
If $MRS < {p_1 \over p_2}$ to the left of the kink, the solution is to the right of the kink; use the equation for that line and maximize.
If the MRS is between the price ratios, then the solution is at the kink.
Note: if the price ratio to the left of the kink is greater than the price ratio to the left, it's more complicated...you could have two potential solutions! Maximize subject to each of the constraints and compare utility at the respective maxima.

Next class, we'll derive the demand functions: that is, the optimal choice as a function of prices and income.

Notice that your optimal choice depends on the prices of goods and your income.

Utility Maximization with Budget Constraints

Part I: The Real Economy

Part I: The Market Economy

Part I: The Market Economy

Next Five Weeks:

How do consumers and firms respond to prices?

The Consumer's Problem

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

Utility Maximization

Next two weeks: Maximize utility subject to a (parameterized) budget line

Last two lectures: Maximize utility subject to a (fixed) PPF

Next two weeks: Maximize utility subject to a (parameterized) budget line

Next two weeks: Maximize utility subject to a (parameterized) budget line

Next two weeks: Maximize utility subject to a (parameterized) budget line

Next two weeks: Maximize utility subject to a (parameterized) budget line

Big difference:

“Gravitational Pull" with a Budget Line

What does it mean if the MRS is greater than the price ratio?

Important and Difficult Distinction

The Lagrange Method

The Lagrange Method

The Lagrange Method

The Lagrange Method

The Tangency Condition

The Tangency Condition when Lagrange sometimes works

Kinked Budget Constraints

How to Solve a Kinked Constraint Problem

Part I:
The Real Economy

Part I:
The Market Economy

Part I:
The Market Economy

Next two weeks:
Maximize utility subject to a (parameterized) budget line

Last two lectures:
Maximize utility subject to a (fixed) PPF

Next two weeks:
Maximize utility subject to a (parameterized) budget line

Next two weeks:
Maximize utility subject to a (parameterized) budget line

Next two weeks:
Maximize utility subject to a (parameterized) budget line

Next two weeks:
Maximize utility subject to a (parameterized) budget line

What does it mean if the MRS is
greater than the price ratio?