Welcome &
Review of Consumer Theory

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 1

Today's Agenda

Part 1: Course Overview

Part 2: Review of Consumer Theory

Themes of the course

Course schedule

Policies

TA Intros

Good 1 - Good 2 Space

Budget Constraints

Preferences and Utility

Optimal Choice

Demand

Three Central Themes

Efficiency and Equity
Time
Information

Weeks 1-3

Efficiency and Equity

Weeks 8-9

Game Theory II: Asymmetric Information

Quarter Rhythm

Weeks 4-7

Game Theory I: Perfect Information

Thursday 2/20

Midterm 2

Friday 3/21,
12:15-3:15pm

Final Exam

Tuesday 1/28

Midterm 1

Week 10

Externalities and Public Goods

Monday

Reading and quiz for Tuesday's lecture

Thursday

Lecture; do second half of problem set exercises

Thursday & Friday

Section; office hours

Weekly Rhythm (Suggested)

Wednesday

Reading and quiz for Thursday's lecture

Tuesday

Lecture; do first half of problem set exercises

Weekend

Review material from the week
Do practice exam problems
Finish & hand in problem set

Grading Policy: Basically the same as Econ 50

This course is not graded on a curve.
If everyone gets an A, everyone gets an A;
if everyone gets a B, everyone gets a B.
Reading quizzes: 5% of your grade
These are challenging, and I don't expect you to be perfect; 20% bonus given
In-class polls: 5% of your grade.
Graded for correctness, not just completion; but 20% added to your grade.
Homework: 25% of your grade.
Max 14 points per pset, max 100 points overall
Exams: 65% of your grade (10% lower midterm, 20% higher midterm, 35% final)
See syllabus for information on missed exams.

Most of you will get 100% (or close) on your quizzes, pollev, and homework;
your grade will largely be determined by the exams.

So, use the quizzes, lecture, and homework to prepare for the exams!

Lecture Policy

No electronics in class, unless you're taking notes on an iPad, in which case please sit at the front.
Punishment: you will be my next question. :)
There will be a 10-minute break in the middle of each class to stretch and check in with your electronic life. Music suggestions appreciated.

Course Web Sites

All content is posted/linked within Canvas.

Each lecture has its own module with everything you need to know about that lecture.

Please use Ed Discussions to ask questions (not email).

Please upload your homework to Gradescope by 8am the morning after it's due.

UCSD Videos

UCSD has video libraries for both intermediate micro and game theory...and they've let you have them for free! :) Information about how to sign up and which videos are most relevant will be posted on ED.

Review of Econ 50: Consumer Theory

pollev.com/chrismakler

When did you take Econ 50?

How well do you remember it?

Good 1 - Good 2 Space

Two "Goods" : Good 1 and Good 2

\text{Bundle }X\text{ may be written }(x_1,x_2)

\text{Bundle }X\text{ may be written }(x_1,x_2)

x_1 = \text{quantity of good 1 in bundle }X

x_1 = \text{quantity of good 1 in bundle }X

x_2 = \text{quantity of good 2 in bundle }X

x_2 = \text{quantity of good 2 in bundle }X

A = (40, 160)

A = (40, 160)

B = (80,80)

B = (80,80)

\text{Examples:}

\text{Examples:}

$A$

$B$

Prices

$A$

$B$

Let's assume all goods have a single, constant price associated with them;

so every unit of good 1 costs $p_1$
and every unit of good 2 costs $p_2$

Monetary value (cost) of bundle $X = (x_1,x_2)$ :

p_1x_1 + p_2x_2

p_1x_1 + p_2x_2

If $p_1 = 2$ and $p_2 = 1$ , what is the cost of bundle $A = (40,160)$ ?

What is the cost of bundle $B = (80,80)$ at those prices?

m = \text{money income}

m = \text{money income}

p_1 = \text{price of good 1}

p_1 = \text{price of good 1}

p_2 = \text{price of good 2}

p_2 = \text{price of good 2}

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

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

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

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

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

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

Budget Constraints

\text{Example: } p_1 = 2, p_2 = 1, m = 240

\text{Example: } p_1 = 2, p_2 = 1, m = 240

$A$

$B$

We can write down the set of all points that have the same monetary value; in Econ 50 these were "budget constraints" or sometimes "isocost lines."

Spend all $240 on good 1

Spend all $240 on good 2

Equation of line: $2x_1 + x_2 = 240$

= \frac{240}{2} = 120

= \frac{240}{2} = 120

= \frac{240}{1} = 240

= \frac{240}{1} = 240

More generally,
equation of the budget line: $p_1x_1 + p_2x_2 = m$

= -\frac{2}{1} =-2

= -\frac{2}{1} =-2

Preferences

Definition Review:

Indifference Curves

Preferred/Dispreferred Sets

Marginal Rate of Substitution

Preferences and Utility

Preferences: Ordinal Ranking of Options

Given a choice between option A and option B, an agent might have different preferences:

A \succ B

A \succ B

A \succeq B

A \succeq B

A \sim B

A \sim B

A \preceq B

A \preceq B

A \prec 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.

Special case: choosing between bundles
containing different quantities of goods.

Preferences over Quantities

A=(4,3,6)

A=(4,3,6)

Example: “good 1” is apples, “good 2” is bananas, and “good 3” is cantaloupes:

🍏🍏🍏🍏

🍌🍌🍌

🍈🍈🍈🍈🍈🍈

B=(3,8,2)

B=(3,8,2)

🍏🍏🍏

🍌🍌🍌🍌🍌🍌🍌🍌

🍈🍈

General framework: choosing between anything

Special Case: Two Goods

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 A

indifferent to A

Indifference curves cannot cross!

A

The indifference curve through A connects all the bundles indifferent to A.

Indifference curve
through A

Marginal Rate of Substitution

X = (10,24)

X = (10,24)

🍏🍏

🍌🍌🍌🍌

Y=(12,20)

Y=(12,20)

🍏🍏

🍌🍌🍌🍌

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}}

MRS = {4 \text{ bananas} \over 2 \text{ apples}}

= 2\ {\text{bananas} \over \text{apple}}

= 2\ {\text{bananas} \over \text{apple}}

Representing Preferences with a Utility Function

u(a_1,a_2) > u(b_1,b_2)

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) \ge u(b_1,b_2)

u(a_1,a_2) = 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) \le u(b_1,b_2)

u(a_1,a_2) < u(b_1,b_2)

u(a_1,a_2) < u(b_1,b_2)

u(x_1,x_2)

u(x_1,x_2)

"A is strictly preferred to B"

Words

Preferences

Utility

A \succ B

A \succ B

A \succeq B

A \succeq B

A \sim B

A \sim B

A \preceq B

A \preceq B

A \prec 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 ">"

Do we have to take the
number of "utils" seriously?

Marginal Utility

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

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}

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
as the "marginal utility" from
another unit of either good:

Indifference Curves and the MRS

Along an indifference curve, all bundles will produce the same amount of utility

In other words, each indifference curve
is a level set 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}

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)

If you give up $\Delta x_2$ units of good 2, how much utility do you lose?

If you get $\Delta x_1$ units of good 1, how much utility do you gain?

\Delta u \approx \Delta x_2 \times MU_2

\Delta u \approx \Delta x_2 \times MU_2

\Delta u \approx \Delta x_1 \times MU_1

\Delta u \approx \Delta x_1 \times MU_1

If you end up with the same utility as you begin with:

\Delta x_2 \times MU_2 \approx \Delta x_1 \times MU_1

\Delta x_2 \times MU_2 \approx \Delta x_1 \times MU_1

{\Delta x_2 \over \Delta x_1} \approx {MU_1 \over MU_2}

{\Delta x_2 \over \Delta x_1} \approx {MU_1 \over MU_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{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

\text{In other words, all marginal utilities are positive: }MU_1 \ge 0, MU_2 \ge 0

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.

Concave Preferences: “Variety is Worse"

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 no, your preferences are concave.

MRS > \frac{p_1}{p_2}

MRS > \frac{p_1}{p_2}

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

More willing to give up good 2
than the market requires

Less willing to give up good 2
than the market requires

The “Gravitational Pull" Towards Optimality

POINT A

POINT B

Gravitational Pull Argument

move to the right along the budget line

move to the left along the budget line

MRS > {p_1 \over p_2}

MRS > {p_1 \over p_2}

MRS < {p_1 \over p_2}

MRS < {p_1 \over p_2}

IF...

THEN...

The consumer's utility function is "well behaved" -- smooth, strictly convex, and strictly monotonic

The indifference curves do not cross the axes

The budget line is a simple straight line

The optimal consumption bundle will be characterized by two equations:

MRS = \frac{p_1}{p_2}

MRS = \frac{p_1}{p_2}

p_1x_1 + p_2x_2 = m

p_1x_1 + p_2x_2 = m

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

Optimal Choice

Otherwise, the optimal bundle may lie at a corner,
a kink in the indifference curve, or a kink in the budget line.
No matter what, you can use the "gravitational pull" argument!

Write an equation for the tangency condition.
Write an equation for the budget line.
Solve for $x_1^*$ or $x_2^*$ .
Plug value from (3) into either equation (1) or (2).

u(x_1,x_2) = x_1x_2

u(x_1,x_2) = x_1x_2

Solving for Optimality when Calculus Works

p_1 = 2, p_2 = 1, m = 240

p_1 = 2, p_2 = 1, m = 240

x_1^*(p_1,p_2,m)

x_1^*(p_1,p_2,m)

x_2^*(p_1,p_2,m)

x_2^*(p_1,p_2,m)

(Gross) demand functions are mathematical expressions
of endogenous choices as a function of exogenous variables (prices, income).

(Gross) Demand Functions

u(x_1,x_2) = x_1x_2

u(x_1,x_2) = x_1x_2

p_1x_1 + p_2x_2 = m

p_1x_1 + p_2x_2 = m

x_1^*(p_1,p_2,m) = \frac{a}{a+b}\times \frac{m}{p_1}

x_1^*(p_1,p_2,m) = \frac{a}{a+b}\times \frac{m}{p_1}

For a Cobb-Douglas utility function of the form

Special Case: The “Cobb-Douglas Rule"

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

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

The demand functions will be

x_2^*(p_1,p_2,m) = \frac{b}{a+b}\times \frac{m}{p_2}

x_2^*(p_1,p_2,m) = \frac{b}{a+b}\times \frac{m}{p_2}

That is, the consumer will spend fraction $a/(a+b)$ of their income on good 1, and fraction $b/(a+b)$ of their income on good 2.

This shortcut is very much worth memorizing! We'll use it a lot in the next few weeks in place of going through the whole optimization process.

pollev.com/chrismakler

Find the optimal bundle for the Cobb-Douglas utility function is

u(x_1,x_2) = \ln x_1 + \tfrac{1}{4} \ln x_2

u(x_1,x_2) = \ln x_1 + \tfrac{1}{4} \ln x_2

and the budget constraint is

1.2 x_1 + x_2 = 60

1.2 x_1 + x_2 = 60

Functional forms for utility functions:

u(x_1,x_2) = av(x_1) + bv(x_2)

u(x_1,x_2) = av(x_1) + bv(x_2)

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

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

1. Weighted average of some common
"one-good" utility function $v(x)$ :

2. "Quasilinear": one good enters linearly
(in this case $x_2$ ), another nonlinearly:

v(x) = \ln x

v(x) = \ln x

v(x) = \sqrt{x}

v(x) = \sqrt{x}

v(x) = x

v(x) = x

v(x) = x^2

v(x) = x^2

u(x_1,x_2) = a \ln x_1 + b \ln x_2

u(x_1,x_2) = a \ln x_1 + b \ln x_2

u(x_1,x_2) = a \sqrt{x_1} + b\sqrt{x_2}

u(x_1,x_2) = a \sqrt{x_1} + b\sqrt{x_2}

u(x_1,x_2) = ax_1 + bx_2

u(x_1,x_2) = ax_1 + bx_2

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

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

Cobb-Douglas (decreasing MRS)

Weak Substitutes (decreasing MRS)

Perfect Substitutes (constant MRS)

Concave (increasing MRS)

To Do Before Next Class

Be sure you've filled out the section survey.

Do the reading and the quiz -- due at 11:15am on Thursday!

Look over the summary notes for this class.