ECON 205
Professor Joel Watson & Professor Joel Sobel
September 11, 2008
ii
Preface
These notes are based on the lecture course given by Professor Joel Watson at
the University of California San Diego during summer 2005. They have been
revised and expande
Econ 205 - Slides from Lecture 14
Joel Sobel
September 10, 2010
Econ 205
Sobel
Theorem (Lagrange Multipliers)
Theorem
If x solves max f (x ) subject to G (x ) = 0 then there exists such
that Df (x ) = DG (x ).
It is standard to write the equation
m
i gi (
Econ 205 - Slides from Lecture 13
Joel Sobel
September 10, 2010
Econ 205
Sobel
Second Order Conditions
For a critical point x we have by Taylors Theorem that
f (x + h) f (x )
h
2
=
1t 2
2 h D f (z)h
2
h
We could write h = t v, where v = 1. So
12t 2
t v D
Econ 205 - Slides from Lecture 12
Joel Sobel
September 8, 2010
Econ 205
Sobel
Monotone Comparative Statics
Loyalty to Stanford
Comparative Statics Without Calculus
Optimizer Set Valued
No concavity
No dierentiability
Motivating Example
x () arg max f (x ,
Econ 205 - Slides from Lecture 11
Joel Sobel
September 7, 2010
Econ 205
Sobel
Inverse and Implicit Functions: Introduction
Theme:
Whatever you know about linear functions is true locally about
dierentiable functions.
Econ 205
Sobel
Inverse Functions: Revi
Econ 205 - Slides from Lecture 10
Joel Sobel
September 2, 2010
Econ 205
Sobel
Example
2
Find the tangent plane to cfw_x | x1 x2 x3 = 6 R3 at
x = (2, 5, 2).
2
If you let f (x) = x1 x2 x3 , then this is a level set of f for value 6.
f f f
,
,
)
x1 x2 x3
=
Econ 205 - Slides from Lecture 9
Joel Sobel
September 2, 2010
Econ 205
Sobel
Representing Functions
Graph: Gr (f ) = cfw_(x , y ) : y = f (x ).
If f : Rn R, then the graph is a subset of Rn+1 .
If n = 2, then the graph is a subset of R3 , so someone with
Econ 205 - Slides from Lecture 8
Joel Sobel
September 1, 2010
Econ 205
Sobel
Computational Facts
1. det AB = det BA = det A det B
2. If D is a diagonal matrix, then det D is equal to the product
of its diagonal elements.
3. det A is equal to the product o
Econ 205 - Slides from Lecture 7
Joel Sobel
August 31, 2010
Econ 205
Sobel
Linear Algebra: Main Theory
A linear combination of a collection of vectors cfw_x1 , . . . , xk is a
vector of the form k=1 i xi for some scalars 1 , . . . , k .
i
Rn has the prop
Econ 205 - Slides from Lecture 6
Joel Sobel
August 30, 2010
Econ 205
Sobel
Linear Algebra Basics
Denition
n-dimensional Euclidean Space:
Rn = R R R R
If X and Y are sets, then
X Y
cfw_(x , y ) | x X , y Y
so
Rn = cfw_(x1 , x2 , . . . , xn ) | xi R, i =
Econ 205 - Slides from Lecture 5
Joel Sobel
August 27, 2010
Econ 205
Sobel
Univariate Optimization
max f (x ) subject to x S
1. Objective function what you are trying to optimize (f ).
2. Constraint Set possible values (S ).
3. Solution: x such that (a) x
Econ 205 - Slides from Lecture 4
Joel Sobel
August 26, 2010
Econ 205
Sobel
Mean-Value Theorem
Theorem (Mean Value Theorem)
If f is real valued and continuous on [a, b ], and dierentiable on
(a, b ), then a point c (a, b ) such that
f (b ) f (a) = (b a)f (
Econ 205 - Slides from Lecture 3
Joel Sobel
August 25, 2010
Econ 205
Sobel
Dierentiation
Calculus ideas:
1. Linear functions are easy to deal with.
2. Not all functions are linear.
3. Lots of functions are locally linear.
4. So: sacrice global analysis of
Econ 205 - Slides from Lecture 2
Joel Sobel
August 31, 2010
Econ 205
Sobel
Limits of Functions
Take any function f ,
f : X R
for X = (c , d ) (so note that it is an open set) and take a X .
Denition
y is called the limit of f from the right at a (right ha
Econ 205 - Slides from Lecture 1
Joel Sobel
August 23, 2010
Econ 205
Sobel
Warning
I cant start without assuming that something is common
knowledge. You can nd basic denitions of Sets and Set
Operations (Union, Intersection, . . . ) in the lecture notes.
Economics 205 Final Examination Fall 2010
Instructions.
1. You have three hours to complete this examination.
2. You may use scratch paper, but please write your nal answers (including your complete
arguments) on these sheets.
3. You may use one page of n
Labor Supply
Chapter 9
1
Question
A doctor earns $150 an hour
He wants to take 2 hours off from work to see a movie
The ticket costs $8
How many dollars worth of other consumption must he
give up to see the movie?
(A)
(B)
(C)
(D)
0
Cant tell
$8
$308
2
Sup
Econ 205 - Slides from Lecture 15
Joel Sobel
September 10, 2010
Econ 205
Sobel
Simple Allocation
k
max x1 xn subject to
xi = K .
i =1
This problem has an equality constraint. If v = x1 xn , then we
can write the rst-order conditions as v = xi for all i .
Economics 205, Fall 2000
Final Examination
September 18, 2000
Instructions. Try to answer all nine problems. (Read all of the questions now
and start on the ones that seem easiest.) Think before you write. Make your
answers as complete and rigorous as pos
Intertemporal Choice
Chapter 10
1
Intertemporal Choice
m1 & m2 are incomes in periods 1 & 2.
c1 & c2 are consumptions in periods 1 & 2.
p1 & p2 are prices in periods 1 & 2.
r is the interest rate.
Given (m1,m2) and (p1,p2), what is the most
preferred inte
Intro
DU Anomalies
Present-Biased Preferences
Beta-Delta Model
Intro
DU Anomalies
Beta-Delta Model
Intertemporal Choice: Background
(Exponential) Discounted Utility model (Samuelson, 1937):
T t
U t (ct , . . . , cT ) =
k u(ct+k ),
k=0
1
where = ( 1+ ).
Giving According to GARP: An Experimental Test of the Consistency of Preferences for
Altruism
Author(s): James Andreoni and John Miller
Source: Econometrica, Vol. 70, No. 2 (Mar., 2002), pp. 737-753
Published by: The Econometric Society
Stable URL: http:/
American Economic Review: Papers & Proceedings 2013, 103(3): 484488
http:/dx.doi.org/10.1257/aer.103.3.484
Income and Substitution Effects of Estate Taxation
By James R. Hines, Jr.*
Despite the long US history of taxing estates,
the economic consequences
Intro
Preview
Econ 205A: Microeconomic Theory
Intro
Preview
Outline
Introduction to 205A
Course Information
What is this course about?
Preview of Econ 205A
Intro
Preview
Class Information
Read the syllabus
Web: https:/gauchospace.ucsb.edu
Instructor: Za
Lagrangians
Kuhn-Tucker
Interpretation of the multiplier
Lagrangians and Kuhn-Tucker:
Constrained Optimization with multiple and/or
non-binding constraints
Lagrangians
Kuhn-Tucker
Interpretation of the multiplier
Consider a basic utility maximization prob
Not-so-basic Constrained Optimization
Outline
Q: What is the role of mathematics in economics?
A: While math is a really great tool, it still is just a tool.
Consider these questions relating to intertemporal choice:
Econ class: Mr. K will live for two p
Econ 205 - Supplementary Slides on Optimization
Joel Sobel
September 10, 2010
Econ 205
Sobel
I didnt lecture on this
I wont test you on this.
Econ 205
Sobel
Saddle Points
To begin the study, we need a notion of a direction that is not
ruled out by constra
Economics 205, Fall 2001: Quiz 3
Instructions. Try to answer all three problems. (Read all of the questions
now and start on the ones that seem easiest.) Make your answers as complete
and rigorous as possible. In particular, give reasons for your computat
Economics 205, Fall 2001: Suggested Answers to Quiz 2
Comments. There were 36 points total (9 points for each question). Class
performance: Min: 13; Max: 36; Median: 30. Several people either misunderstood Part b of Question 1 or do not understand the rel
Economics 205, Fall 2001: Quiz 2
Instructions. Try to answer all four problems. (Read all of the questions
now and start on the ones that seem easiest.) Make your answers as complete
and rigorous as possible. In particular, give reasons for your computati