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What is an economic issue you care deeply about?
Did what we studied in Econ 50 offer a good model for this issue?
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
Checkpoint 1: October 13
Model 1: Trading
Model 3: Strategic Interactions
with Incomplete Information
Checkpoint 2: November 3
WEEK 1
WEEK 2
WEEK 3
Preferences
Exchange Economies
Production Economies
WEEK 4
WEEK 5
Analyzing a Game from a Player's POV
Static Games of Complete Information
Checkpoint 3: November 17
Final Exam: December 11 (cumulative)
WEEK 6
WEEK 7
WEEK 8
Dynamic Games of Complete Information
Static Games of Incomplete Information
Dynamic Games of Incomplete Information
WEEK 9
WEEK 10
Getting people to do what you want
Getting people to reveal information
Model 4: Interactions with Asymmetric Information
Different Types of Interactions
Model 2: Strategic Interactions with Complete Information
Grading Policy: Basically the same as Econ 50
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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 and In-Class Polls: 10% of your grade
These are challenging, and I don't expect you to be perfect; 20% bonus given -
Homework: 25% of your grade.
Max 12 points per pset, max 100 points overall -
Exams: 65% of your grade
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Lowest checkpoint: dropped
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Middle checkpoint: 10% of your grade
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Highest checkpoint: 20% of your grade
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Final exam: 35% of your grade
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Don't use AI
It will actively harm your grade.
Lecture Policy
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No electronics in class, unless you're taking notes on an iPad, in which case please sit at the front.
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Punishment: you will be my next question. :)
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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.
Review of Econ 50: Consumer Preferences
pollev.com/chrismakler
When did you take Econ 50?
How well do you remember it?
Review: Modeling Consumer 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.
Sidebar: “Strictly" vs. “Weakly"
A \succ B
A \succeq B
The agent strictly prefers A to B.
The agent weakly prefers A to B.
2
3
4
5
6
x \ge 3
x \gt 3
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.
For any choice, the choice space is the set of all options you're choosing between.
One such choice space
is the set of all bundles of commodity goods.
A=(4,3,6)
Example: “good 1” is apples, “good 2” is bananas, and
“good 3” is cherries
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B=(3,8,2)
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vectors
A=(4,3,6)
Example: “good 1” is apples, “good 2” is bananas, and
“good 3” is cherries
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B=(3,8,2)
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TWO NOTES
- Flows (e.g. apples per week) not stocks (apples)
- All quantities are infinitely divisible in this course.
Choices in general
Choices of commodity bundles
Choosing bundles of two goods
Special Case: Good 1 - Good 2 Space
x_1
x_2
Two "Goods" (e.g. apples and bananas)
A bundle is some quantity of each good
\text{Bundle }X = (x_1,x_2)
x_1 = \text{quantity of good 1 in bundle }X
x_2 = \text{quantity of good 2 in bundle }X
Can plot this in a graph with \(x_1\) on the horizontal axis and \(x_2\) on the vertical axis
A = (4,16)
B = (8,8)
A
B
4
8
12
16
20
4
8
12
16
20
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
Special Case: Good 1 - Good 2 Space
Marginal Rate of Substitution
X = (10,24)
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Y=(12,20)
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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}}
= 2\ {\text{bananas} \over \text{apple}}
Visually: the MRS is the magnitude of the slope
of an indifference curve
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
Nonmonotonic Preferences and Satiation
Some goods provide positive marginal utility only up to a point, beyond which consuming more of them actually decreases your utility.
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.
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"
How do we represent preferences mathematically?
u(A)
A=(4,3,6)
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vector
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We need a mathematical function that maps vectors onto numbers.
number of "utils"
Question: do we have to take "utils" seriously?
We can see if one bundle is preferred to another by comparing their utilities:
Marginal Utility
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}
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:
UTILS
UNITS OF GOOD 1
UTILS
UNITS OF GOOD 2
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_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 \over \Delta x_1} \approx {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}
UNITS OF GOOD 1
UNITS OF GOOD 2
Examples of Utility Functions
Perfect Substitutes
Goods that can always be exchanged at a constant rate.
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Red pencils and blue pencils, if you con't care about color
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One-dollar bills and five-dollar bills
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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.
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Used for two independent goods (neither complements nor substitutes) -- e.g., t-shirts vs hamburgers
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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."
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Marginal utility of good 2 is constant
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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.
Intertemporal Choice
Preferences over Time
u(c_1,c_2) = v(c_1)+\beta v(c_2)
\beta = \text{“between-period" discount factor}
v(c) = \ln c
v(c) = c
u(c_1,c_2) = \ln c_1 + \beta \ln c_2
Examples:
u(c_1,c_2) = c_1 + \beta c_2
A fundamental tradeoff we constantly face is
consumption today vs. consumption in the future.
c_1
c_2
Suppose our utility within any time period may be written by some function \(v(c)\).
Then our utility from some consumption stream \((c_1,c_2)\) might be written
Preferences over Time
u(c_1,c_2) = v(c_1)+\beta v(c_2)
\beta = \text{“between-period" discount factor}
v(c) = \ln c
v(c) = c
u(c_1,c_2) = \ln c_1 + \beta \ln c_2
Examples:
u(c_1,c_2) = c_1 + \beta c_2
v(c) = \text{“within-period" utility}
\displaystyle{MRS(c_1,c_2) = {v'(c_1) \over \beta v'(c_2)}}
Marginal utility of having another dollar today
Discounted MU of having another dollar tomorrow
How many future dollars would you be willing to give up to get another dollar today?
Preferences over Time
\beta = \text{“between-period" discount factor}
v(c) = \ln c
u(c_1,c_2) = \ln c_1 + \beta \ln c_2
v(c) = \text{“within-period" utility}
\displaystyle{MRS(c_1,c_2) = {v'(c_1) \over \beta v'(c_2)}}
How many future dollars would you be willing to give up to get another dollar today?
u(c_1,c_2) = v(c_1)+\beta v(c_2)
\displaystyle{= {1/c_1 \over \beta/c_2}}
\displaystyle{= {c_2 \over \beta c_1}}
What makes you more willing to give up future income for present income?
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.
Econ 51 | 01 | Welcome and Review of Econ 50
By Chris Makler
Econ 51 | 01 | Welcome and Review of Econ 50
Welcome and Review of Econ 50
