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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
Three Central Themes
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Efficiency and Equity
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Time
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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
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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: 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
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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.
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)
x_1 = \text{quantity of good 1 in bundle }X
x_2 = \text{quantity of good 2 in bundle }X
A = (40, 160)
B = (80,80)
\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
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}
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 Constraints
\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}{1} = 240
More generally,
equation of the budget line: \(p_1x_1 + p_2x_2 = m\)
= -\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 \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.
Special case: choosing between bundles
containing different quantities of goods.
Preferences over Quantities
A=(4,3,6)
Example: “good 1” is apples, “good 2” is bananas, and “good 3” is cantaloupes:
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B=(3,8,2)
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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)
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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}}
Representing Preferences with a Utility Function
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) = 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(x_1,x_2)
"A is strictly preferred to B"
Words
Preferences
Utility
A \succ B
A \succeq B
A \sim B
A \preceq 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_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}
(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_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}
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
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}
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}
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}
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
Solving for Optimality when Calculus Works
p_1 = 2, p_2 = 1, m = 240
x_1^*(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
p_1x_1 + p_2x_2 = m
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
The demand functions will be
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
and the budget constraint is
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) = 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) = \sqrt{x}
v(x) = x
v(x) = 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) = ax_1 + bx_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.
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
